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DNA Separation and sequencing by Electric Field-Flow Fractionation (EFFF) in a microchannel

University of Florida Institutional Repository

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DNA SEPARATION AND SEQUENCING BY ELECTRIC FIELD-FLOW FRACTIONATION (EFFF) IN A MICROCHANNEL By ZHI CHEN A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE UNIVERSITY OF FLORIDA 2003

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Copyright 2003 by Zhi Chen

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ACKNOWLEDGMENTS This work was performed under careful and elaborate instructions of Dr. Anuj Chauhan. During this work, Dr. Chauhan gave me invaluable help and direction, which guided me when I struggled with difficulties and questions. In addition, the lessons he taught me about how to approach a project will help me in my future research and work. Also, my laboratory colleagues gave me many helpful suggestions. I deeply appreciate their help. We also acknowledge the financial support of NASA (NAG 10-316), the Engineering Research Center (ERC) for Particle Science and Technology at the University of Florida, The National Science Foundation (NSF Grant EEC-94-02989), and the Industrial Partners of the ERC for support of this research. iii

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TABLE OF CONTENTS Page ACKNOWLEDGMENTS .................................................................................................iii LIST OF TABLES .............................................................................................................vi LIST OF FIGURES ..........................................................................................................vii ABSTRACT .......................................................................................................................ix CHAPTER 1 BACKGROUND ........................................................................................................1 Deoxyribonucleic Acid..............................................................................................1 What Forms DNA and RNA Molecules ...........................................................1 Structure of DNA Molecules ............................................................................5 Primary structure .....................................................................................5 Code and gene .........................................................................................6 The 3D structure ......................................................................................9 Stability ........................................................................................................... 12 Melting ..................................................................................................12 Degradation of DNA molecules ............................................................13 Separation of DNA molecules ........................................................................13 Fluidic Properties of DNA Molecules ............................................................19 Mobility of DNA molecules in free solution ........................................19 Molecular diffusivity in free solution ...................................................20 2 THEORY ..................................................................................................................22 Derivation of Equations about EFFF Using Perturbation Analysis .........................22 Comparison with Brenners Theory .........................................................................29 3 RESULTS AND DISCUSSION ..............................................................................32 Limiting Cases ..........................................................................................................32 Dependence of the Mean Velocity on eyU and Pe ...................................................34 Dependence of D* on eyU and Pe .............................................................................35 Separation of DNA Fragments of Different Lengths ...............................................36 Separation Efficiency ...............................................................................................37 iv

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Comparison of Lateral and Axial Electric Field ......................................................43 Comparison with Other Promising Methods for DNA Separation ..........................47 4 DNA SEQUENCING ATTEMPT ...........................................................................49 5 CONCLUSIONS ......................................................................................................54 APPENDIX NOMENCLATURE......................................................................................57 LIST OF REFERENCES ...................................................................................................59 BIOGRAPHICAL SKETCH .............................................................................................63 v

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LIST OF TABLES Table page 1-1 Genomes of prominent organisms ..............................................................................6 1-2 Genetic code (mRNA) ................................................................................................7 3-1 Comparison of EFFF with other techniques ............................................................48 vi

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LIST OF FIGURES Figure page 1-1 Basic structure of DNA molecules .............................................................................2 1-2 Structure of pentose with numbering .........................................................................3 1-3 Structure of bases in DNA and RNA .........................................................................3 1-4 Examples of deamination which involves the removal of an amino group. Accidental deamination may change the cytosine to uracil, or the methylated cytosine to thymine ....................................................................................................5 1-5 Scheme of single strand DNA ....................................................................................6 1-6 Gene structure of DNA strand ....................................................................................8 1-7 Example of dA-dT and dG-dC base pair as found within DNA double helix ...........9 1-8 Normal right-handed "double helix" structure of DNA, also known as the B form .......................................................................................................................10 1-9 Comparison between B form and Z form .................................................................11 2-1 Schematic of the 2D channel ....................................................................................22 3-1 Dependency of (D*-R)/Pe2 on the product of Pe and eyU The dotted line is the small approximation (Equation 3-2), and the dashed line is the large approximation (Equation 3-3) ......................................................................33 3-2 Dependency of mean velocity *U on the product of P eyU The dotted line is the small approximation (Equation 3-4), and the dashed line is the large approximation (Equation 3-5) ......................................................................34 3-3 Separation of a 1:1 mixture of DNA strands of different sizes into two separate Gaussian peaks. =1mm/s, D1 = 1x10-9, D2 = 2x10-9, eyU =1, Pe1=10, h = 10 m ..................................................................................................37 vii

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and Pe for separation of molecules of the same size but different charges. e 1yU 3-4 Dependency of L/h on and Pe for separation of DNA strands of different sizes. eyUey = e2yU = U Pe1 = Pe, and Pe1/Pe2 = D2/D1 = 10 .............................38 3-5 Dependency of L/h on eyUey and Pe for separation of DNA strands of different sizes. e1yU = e2yU = U Pe1 = Pe, and Pe1/Pe2 = D2/D1 = 2 ................................39 3-6 Dependency of L/h on eyU and Pe for separation of molecules of the same size but different charges. eyU = e1yU e2yU / e1yU = 10, and Pe = Pe1 = Pe2,i.e., D2 = D1 ..............................................................................................................41 3-7 Dependency of L/h on eyU eyU = e1yU e2yU / e1yU = 2, and Pe = Pe1 = Pe2, i.e., D2 = D1 .............................................................................................................42 3-8 Dependency of L/h on exU and Pe for an axial field for separation of molecules of the same size but different charges. Pe = Pe1 = Pe2, exU = e1xU and e2xU / e1xU = 2 ..................................................................................................... 44 3-9 Dependency of the difference of mean velocities of pulses of two kinds of molecules (Z2=2 Z1, Pe2/Pe1 = 2) on the product of Pe = Pe1=2 and eyU ( eyU = e1yU ) ................................................................................................................ 46 3-10 Dependency of the difference of mean velocities of pulses of two kinds of molecules on an electric field. The dash-dot line is for the axial electric field and solid lines are of the lateral electric field for different Pe. In these plots Pe2/Pe1 =2 .......................................................................................................47 4-1 L/h required to separate DNA fragments that differ in length by a single base pair vs. the number of base pairs in the smaller fragment. h =1m, Pe = 1, eyU = 0.4. The largest value of L/h represents the size of the channel required for separation. The dashed line is calculated from the large approximations. ........................................................................................................50 4-2 Length and Time required to separate DNA fragments that differ in length by a single base pair vs. the number of base pairs in the smaller fragment. h = 10m, Pe = 200, = 0.01, =0.02m/s .......................................................52 eyU viii

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Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Master of Science DNA SEPARATION AND SEQUENCING BY ELECTRIC FIELD-FLOW FRACTIONATION (EFFF) IN A MICROCHANNEL By Zhi Chen August, 2003 Chair: Anuj Chauhan Major Department: Chemical Engineering DNA separation is a core activity in biology, especially in mapping and sequencing DNA molecules. Separation of DNA molecules of different chain lengths by electrophoresis is difficult because the ratio of the total charge to viscous resistance is independent of the chain length. We modeled using a lateral electric field in a microchannel to separate DNA fragments in different sizes based on Taylor dispersion theory. With this method, we can perform the separation in free solution, which avoids the difficulty of loading gel in commonly used capillary gel electrophoresis (GE) or complicated fabrications in an artificial sieving matrix. During the research, we found that the lateral electric field will build a concentration profile in the lateral direction, which has a different distribution for different DNA molecules. Theoretically, the product of Pe and determines the shape of the profile. The variant profiles combined with the parabolic velocity profile along the eyU ix

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lateral direction engender different mean velocities of pulse. Based on this fact, we acquired length-dependent separation of DNA molecules. Furthermore, EFFF can also effectively separate DNA strands differing in size by a single base pair, and thus can be used to sequence DNA. The separation step in sequencing 500 base pair long DNA molecules can be done in 1.7 hours by a 0.72-meter long, 10-micron thick channel. However, since this result is from theoretical calculations, we need further experiments to test it. x

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CHAPTER 1 BACKGROUND Deoxyribonucleic Acid (DNA) Nucleic acids are important kind of biological molecules that exist in every living being. These molecules were discovered in 1868 by a young Swiss scientist when he refined an organic substance with a very large phosphorous content from a bandage. But the biological functions of nucleic acids were not recognized until 1944, when scientist Avery did the famous pneumobacillus transform experiment. This experiment successfully proved that nucleic acids but not proteins are the genetic substances in a living organic body. The next milestone in the field of molecular biology occurred in 1953 when Watson and Crick discovered the double helix structure of DNA molecules. Then in the 1970s, recombination technology was invented, leading to the foundation of the new field of Gene technology. Since then, researchers have tried to modify genes in organisms for a variety of applications. Most living cells contain two kinds of nucleic acidsdeoxyribonucleic acid (DNA) and ribonucleic acid (RNA). Most cells have both of them with DNA existing in nucleolus and RNA existing in the cytoplasm. However viruses usually contain either only DNA or RNA. What Forms DNA and RNA Molecules DNA and RNA are polymers known as polynucleotide, in which the monomer units are nucleotides. Each nucleotide comprises three partspentose, base and phosphate group [1]. In DNA or RNA, each nucleotide contains only one phosphate 1

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2 group, but some cellular free nucleotides (such as ADP and ATP, which are important species in the energy transfer and storage cycle) may contain more than one. Figure 1-1. Basic structure of DNA molecules. (Reprinted with permission from The Web Book Publications. 2003. Molecular biology web book. Available from URL: http://www.web-books.com/MoBio Site last visited July 2003) The carbon atoms on the bases (purine and pyrmidine rings) are numbered as 1,2,3,4,5 and thus to avoid confusion the carbon atoms on the pentose are numbered 1', 2', 3', 4', and 5' (Figure 1-2). Two kinds of pentose exist in nucleic acid'-deoxyribose and ribose. The difference between the two is that the deoxyribose lacks a hydroxyl group at the 2'-position [2]. The deoxyribose pentose is present in DNA and the ribose pentose is present in RNA. In both the 2'-deoxyribose and ribose, the hydroxyl groups on the 5'and 3'carbons link to the phosphate groups. Five different kinds of bases are present in nucleic acids. They are adenine, guanine, cytosine, thymine and uracil. Figure 1-3 shows the molecular structure of these bases.

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3 Figure 1-2. Structure of pentose with numbering Figure 1-3. Structure of bases in DNA and RNA (Reprinted with permission from The Web Book Publications. 2003. Molecular biology web book. Available from URL: http://www.web-books.com/MoBio Site last visited July 2003)

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4 Four of these five bases are present in DNA. They are given one letter abbreviations as shorthand (A is for adenine; G is for guanine; C is for cytosine; T is for thymine). In the RNA molecules, there are also four types of bases; A, G, C also exist in RNA, but T is replaced by U(uracil). If the phosphate groups in a nucleotide are removed, it becomes a nucleoside, which consists of one of the bases covalently attached to the 1' position of a pentose. The five different nucleosides of DNA and RNA are deoxyadenosine (dA), deoxyguanosine (dG), deoxycytosine (dC), deoxythymidine (dT) and deoxyuracil (dU), which is present only in RNA. The bases in RNA and DNA form pairsA-T and G-C in DNA, and A-U and G-C in RNA. The chemical structure of uracil is simpler than thymine and uracil can pair perfectly with adenine. Thus, it puzzled researchers that A pairs with T rather than U in a DNA strand. This issue was finally resolved after we understood the details of the repairing mechanisms in DNA. As an evolution source, mutations may occur under the influence of external factors (UV radiation, exposure to chemical agents, etc.) or cellular processes (accidental deamination, replication errors, etc.). Figure 1-4 shows one of these mutation factors: the deamination of cytosine. Cytosine is one of four bases in DNA molecules. As shown above, Cytosine can be mutated to uracil by deamination process. Since DNA does not contain uracil, this mutation can be easily detected and repaired by base excision. If DNA were made up of uracil, the cytosine to uracil mutation could hardly be corrected. This explains why DNA

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5 chooses thymine, instead of uracil, even though the chemical structure of uracil is simpler than thymine. Figure 1-4. Examples of deamination which involves the removal of an amino group. Accidental deamination may change the cytosine to uracil, or the methylated cytosine to thymine (Reprinted with permission from The Web Book Publications. 2003. Molecular biology web book. Available from URL: http://www.web-books.com/MoBio Site last visited July 2003) Structure of DNA Molecules Primary structure As mentioned above, DNA molecules consists of A,T,C,G [3]. They are joined by the 3-5 phosphate bonds into a strand (Figure 1-5). The sequence of these four nucleotides determines the genetic information contained in the DNA molecules, and different creatures have different sequence and

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6 length of DNA molecules. Table 1-1 shows the typical length of DNA molecules in various organisms. Figure 1-5. Scheme of single strand DNA Table 1-1. Genomes of prominent organisms Organism Genome size (Mb*) Gene number Hepatitis D virus 0.0017 1 Hepatitis B virus 0.0032 4 HIV-1 0.0092 9 Bacteriophage l 0.0485 80 Escherichia coli 4.6392 4400 S. cerevisiae (yeast) 12.155 6300 C. elegans (nematode) 97. 19000 D. melanogaster (fruit fly) 137. 13600 Mus musculus (mouse) 3000. ? Homo sapiens (human) 3000 30000** 1 Mb = 1 million base pairs (for double-stranded DNA or RNA) or 1 million bases (for single-stranded DNA or RNA). ** The total number of human genes is still quite controversial. It could be as high as 75,000 [see a paper published on July 4, 2001 ]. Note: Reprinted with permission from The Web Book Publications. 2003. Molecular biology web book. Available from URL: http://www.web-books.com/MoBio Site last visited July 2003.

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7 Code and gene Scientists have found that each three continuous nucleotides within the DNA encode a protein and have drawn Table 1-2, which shows the correspondence between the codes and proteins. Table 1-2. Genetic code (mRNA) 2nd position (middle) 1st position (5' end) U C A G 3rd position (3' end) U Phe F Phe F Leu L Leu L Ser S Ser S Ser S Ser S Tyr Y Tyr Y STOP STOP Cys C Cys C STOP Trp W U C A G C Leu L Leu L Leu L Leu L Pro P Pro P Pro P Pro P His H His H Gln Q Gln Q Arg R Arg R Arg R Arg R U C A G A Ile I Ile I Ile I Met M Thr T Thr T Thr T Thr T Asn N Asn N Lys K Lys K Ser S Ser S Arg R Arg R U C A G G Val V Val V Val V Val V Ala A Ala A Ala A Ala A Asp D Asp D Glu E Glu E Gly G Gly G Gly G Gly G U C A G Synthesis of a peptide always starts from methionine (Met), coded by AUG. The stop codon (UAA, UAG or UGA) signals the end of a peptide. Note: Reprinted with permission from The Web Book Publications. 2003. Molecular biology web book. Available from URL: http://www.web-books.com/MoBio Site last visited July 2003. By definition, a gene includes the entire nucleic acid sequence used for the expression of one product (peptide or RNA). Such sequence may be separated into two categoriesregulatory region and transcriptional region. The regulatory region controls the activation of the gene, which could be near or far from the transcriptional region. And the transcriptional region consists of exons and introns. After the transcription introns will be removed, whereas exons remains encoding a peptide or functional RNA

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8 Figure 1-6 shows the secondary structure of DNA molecule, which is made up of genes, pseudogenes and extragenic region. Pseudogenes are nonfunctional genes, which often comes from mutation of genes that happens in the duplication process. However, because duplicated genes commonly have many copies, the organism can still survive even if some of them become nonfunctional. Figure 1-6. Gene structure of DNA strand (Reprinted with permission from The Web Book Publications. 2003. Molecular biology web book. Available from URL: http://www.web-books.com/MoBio Site last visited July 2003) A certain piece of DNA sequence often repeats several times in the total DNA of a cell. Experimentally, the number of repeated copies is classified on the basis of DNA reassociation kinetics [1]. The entire DNA is first randomly cleaved into fragments with an average size of about 1000 bp. Then, they are heated to separate the strands of each fragment. Subsequently, temperature is reduced to allow strand reassociation. If a fragment contains a sequence which is repeated many times in the total DNA, it will have greater chance to find a complementary strand and reassociate more quickly than other fragments with less repetitive sequences.

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9 Based on the reassociation rate, DNA sequences are divided into three classes: highly repetitive, moderately repetitive and single copy. The 3D structure In a DNA molecule, the two strands intertwine with each other, forming a double helix structure. This structure was first discovered by James D. Watson and Francis Crick in 1953. In this structure, the sugar-phosphate backbones of the two DNA strands wind around the helix axis and the bases of the individual nucleotides are on the inside of the helix. Within the DNA double helix, base A forms 2 hydrogen bonds with T on the opposite strand and G forms 3 hydrogen bonds with C. Figure 1-7. Example of dA-dT and dG-dC base pair as found within DNA double helix (Reprinted with permission from The Web Book Publications. 2003. Molecular biology web book. Available from URL: http://www.webbooks.com/MoBio Site last visited July 2003)

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10 The helix turns a round every 10 base pairs (Figure 1-8). Since the distance between two base pairs is 0.34 nm, the length is about 3.4 nm per turn for DNA molecule. The intertwined strands make two grooves of different widths the major groove and the minor groove, which may bind with specific proteins. The human DNA molecule in a diploid cell, if fully extended, would have a total length of 1.7 meters. If one unfolds all of the DNA molecules in the body, one could reach the moon for 6000 times! [1] Figure 1-8. Normal right-handed "double helix" structure of DNA, also known as the B form (Reprinted with permission from The Web Book Publications. 2003. Molecular biology web book. Available from URL: http://www.webbooks.com/MoBio Site last visited July 2003)

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11 In a solution with higher salt concentrations or with alcohol added, the DNA structure may change from normally B form to an A form, which is still right-handed. But every 2.3 nm makes a turn with 11 base pairs in it. Another DNA structure is called the Z form because it seems to zigzag, which is left-handed on rotation. One turn includes 4.6 nm, comprising 12 base pairs. The DNA molecule with alternating G-C sequences in alcohol or high salt solution tends to have such structure. Figure 1-9. Comparison between B form and Z form (Reprinted with permission from The Web Book Publications. 2003. Molecular biology web book. Available from URL: http://www.web-books.com/MoBio Site last visited July 2003)

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12 Stability [3,4] In this part, we will introduce two aspects of DNA molecules. One is the melting of helix double strand DNA, the other is the degradation. Melting Melting is the term given to the separation of the two strands of a DNA molecule, which is also called denaturation. Several factors make the DNA molecules relatively stable. The DNA strands in a double helix are held together by the H-bonds between the bases. These bonds (also called Watson-Crick) attach two strands together. Moreover, base pairs sit on the top of each other at a rotation of 36 and there is strong interaction between all adjacent base pairs. This interaction, called stacking interaction, stabilizes the DNA double helix. Additionally, the phosphate groups must be neutralized (by Na+ or Mg2+ ions) to allow the negatively charged phosphates to be in close proximity. As introduced above, two hydrogen bonds exist between A and T and three exist between G and C. If a solution of DNA is heated, the hydrogen bonds will break at high temperatures, and the stacking interactions will become weak. As some researchers concluded, the most important contribution to DNA helix stability is the stacking of the bases on top of one another. Thus, in order to denature DNA, we must overcome the stacking energies that provide cohesion between adjacent base pairs. Since AT pairs have only two hydrogen bonds, they are easier to undergo severance. Together with the fact that the stacking energies are less for AT-rich regions, AT rich area tends to separate compared to GC rich area. As a result, the base composition of the DNA influences the melting temperature (Tm), at which two DNA strands separate. The greater the proportion of G-C base pairs in the DNA is, the higher the Tm is. However,

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13 experimentally, at temperatures higher than 80C the GC pairs will also melt, and the DNA will become single stranded which will be present in coiled and unstructured forms. Several methods can be used to obtain the melting of double strand DNA: Reduction of Salt Concentrationas the salt concentration is reduced, the phosphate groups are no longer neutralized by Na+ or Mg2+ ions and the negative charges of phosphate group tend to force the strands apart. Extreme of pHit alters the ionization states of the groups on the bases which provide and accept the H-bonds. Commonly, linear DNA molecules will denature and precipitate when ph is above 12. Increase in Temperaturewhen temperature of a DNA solution increases to a certain value, which is called the melting point (Tm), the strands separate. Degradation of DNA molecules Degradation of DNA molecules is related to the breakup of phosphoric bonds and consequently the cleavage of DNA chain. Various factors contribute to the degradation of DNA: chemical, physical and enzymatic, etc. Commonly, the effects that result degradation are much stronger than that for melting. Prolonged heat treatment may result in DNA hydrolysis which degrades the DNA. Low pH may increase chemical modifications and hydrolysis of DNA. For example, at low pH (pH 4), maize DNA and plasmid DNA were rapidly degraded [5]. Under low pH conditions, what will happen first in the degradation of DNA is the depurination of the nucleic acid backbone. After that, hydrolysis of adjacent 3'-5'-phosphodiester linkages occurs, resulting in measurable shortening of DNA strands [6]. Enzymatic degradation of DNA by nucleases may also occur on prolonged storage. Adding EDTA can inhibit the activity of DNA enzyme by chelating metal ions with valence of 2, so that the storage time can be above 5 years under -70C. Separation of DNA molecules Microfluidics is a sub area of the microelectromechanical systems(MEMS) and is mainly concerned with moving fluids and then performing various unit operations in micron-sized channels. Microfluidics is rapidly becoming a very important area of

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14 research due to numerous potential applications in separation and analysis. The current trend in this field is towards development of chips that can accomplish reactions, separations, and detection at a very rapid rate, such as a chip that can separate DNA fragments of different lengths and detect them. DNA electrophoresis has become a very important separation technique in molecular biology and, in particular, in the genome project. DNA fragments are first separated by chain length and are later processed to read the sequence of the bases that form the genetic code of all living organisms. This technique is also indispensable in forensic applications for identifying a person from a tissue sample [7]. However, separation of DNA fragments of different chain lengths by electrophoresis is difficult because the velocity of the charged DNA molecules due to an axial electric field is independent of the chain length. The reason of this independency is that the mobility of a DNA molecule is approximately inversely proportional to the length while the total charge is directly proportional to the length. This difficulty is traditionally overcome by performing the electrophoresis in columns filled with gels. In these processes, because negatively charged DNA surrounded by positive counterions moves through a matrix such as an agarose gel, the mobility is no longer inversely proportion to the length. Thus, DNA chains of different lengths traverse the gel at different speeds and separate in a series of bands. In gel electrophoresis, the electric field can either be continuous or pulsed. Continuous field electrophoresis is useful for separating DNA molecules of sizes below approximately 20000 base pairs. The migration rates of DNA strands above this size is almost independent of the length of the strand except at very low voltages with which it takes an excessively long time to accomplish separation. Pulsed field gel

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15 electrophoresis (PFGE) was developed to separate longer DNA fragments, which can not be separated by the conventional gel electrophoresis. PFGE utilizes a pulsed electric field, which changes directions continually, resulting in changes in migration directions. These changes lead to a stronger dependency of the net migration rates on the DNA chain length, even if the chains are longer than about 20000 base pairs. Although gel electrophoresis can separate DNA fragments, there are some problems associated with its use in DNA separation. Bubbles can form in the gel during an operation, resulting in variations in DNA electrophoretic mobility [8]. Moreover, the separation by electrophoresis of DNA fragments larger than 40000 base pairs using gel is slow; which is still one of the slowest steps in the genome project. This kind of separation typically takes more than 20 hours because a low-intensity and pulsating field is used to separate DNA fragments to prevent the long fragments from being damaged by high temperatures that may result under large fields [9]. To eliminate the temperature increase during separation, researchers developed capillary electrophoresis, which has a high surface-area-to-volume ratio, providing rapid elimination of heat and allowing application of high electric fields without a substantial temperature increase [8,10,11]. The use of capillary sequencers in the genome project resulted in an eight-fold increase in the sequencing capacity and output [12]. However, preparing uniform, homogeneous, bubble free and stable gel-filled capillaries is difficult, especially for separation of DNA fragments, which commonly involves many parallel lanes running simultaneously. Recent advances in microfabrication techniques have led to production of microfluidic devices frequently referred to as a lab-on-a-chip that can perform a

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16 number of unit-operations such as reactions, separations, detection, etc., at a much higher throughput. Gel-based DNA separations are not convenient in such devices because of the difficulty in loading the gel [9]. Thus, gels have been replaced with polymeric solutions as the sieving mediums. Electrophoresis in a free medium can also separate DNA fragments but it requires precise modifications to the DNA molecules [13]. Microfabricated obstacles such as posts [14], self-assembling colloids [15], entropic barriers [16], and Brownian ratchets [17,18] have also shown to be effective at separating DNA strands. Craighead et. al. used an entropic trapping system, which consists of alternating thick (0.65-1.6 m) and thin (90nm) regions in a channel flow. Since larger molecules need to reach higher entropic states to enter the thinner regions, they spend less time in the channel and exit the channel earlier than smaller ones. In this method, the number of traps is one of the most important factors that controls the separation effect. In their experiment, they did not achieve good separation for DNA molecules (24.5, 48.5, 73.0, 97.0kbp) until the number of traps reached 3700 and the total separation time reached 40 mins [19]. Turner et. al. fabricated artificial arrays of posts in a microchannel by lithography. The diameter of the posts and the interval between them were both 100nm small enough to provide a strong sieving effect. They tested separation of 7.2 kbp and 43 kpb DNA strands and obtained a ratio of 2 between the mean velocities of these two kinds of DNA strands [20]. Baron et al. used un-crosslinked polymer solution which provides sparse sieving and thus has low resistance to DNA molecules. By this method, they reduced the operation time to about 20mins. But the separation for DNA with large molecule weight is still not satisfactory [21]. Viovy et al. used magnetic fields to drive

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17 superparamagnetic particles to form a post matrix with the interparticle distance to be about 5.7m. They successfully separated large DNA molecules (15, 33.5, and 48.5kbp) in only 10-15mins. Furthermore, when the magnetic fields are released, the viscosity of the fluid in the pipe becomes low. Consequently, this method avoids the difficulty of loading gel that exists in gel-capillary electrophoresis. Bader et al. [17,18] created a spatially periodic anisotropic potential energy field to trap the molecules at the potential energy minima. As a result of pulsating application of an electric field, the molecules that diffuse outside a trap when the field is released are attracted to the next trap. In this method, the smaller molecules with large diffusivity have larger migration speeds, and the larger molecules have lower speeds. This difference in speed leads to separation. However, the optimal DNA separation technique should accomplish separation without any sieving medium and should be amenable to online modification to accomplish separation for a wide range of DNA sizes. Our proposed strategy utilizes lateral electric fields and no sieving medium, and the amplitude of the field can be adjusted to separate different DNA compositions. Essentially, this method is called electric field-flow fractionation (EFFF) [22-24], which is a derivation of field-flow fractionation (FFF). Giddings first proposed FFF in 1966 [25]. The basic idea is to use a field in the direction perpendicular to flow and form a concentration profile on the cross section [26]. When charged DNA molecules flow through channels in the presence of lateral fields, i.e., fields perpendicular to the flow direction, they experience an attractive force towards the wall of the opposite polarity. In the absence of any field, each DNA molecule has an equal probability of accessing different streamlines in a time scale larger than h 2 /D, where h is the height of the channel, and D is the molecular diffusivity.

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18 However, due to the electric field, the molecules on average access streamlines closer to the wall, which results in a mean axial velocity smaller than the mean fluid velocity. We shall show later that the enhancement in concentration near the wall is greater for the more slowly diffusing molecules, and thus their mean velocity is reduced more than the mean velocity of the faster diffusing molecules. If a slug of DNA molecules is introduced into a channel with lateral electric fields, the difference in mean velocities leads to separation of the molecules into bands, and the bands of smaller molecules travel faster. FFF has also been used in size based particle separation using gravity or centrifugal acceleration as the lateral force [27-29]. New trends in FFF are thermal field-flow fractionation (TFFF) [30-32] and its application in bioseparation [33,34]. In this paper, we analyze the Taylor dispersion of charged molecules such as DNA in a microchannel with pressure driven flow under lateral fields by using regular perturbation techniques. Based on our investigation, we propose a new scheme for separating DNA molecules in channels by application of lateral electric fields without using any gel or polymeric solution as sieving mediums. Brenner used the method of moments to obtain the Taylor dispersion coefficient for shear flow in a channel accompanied by a lateral flow [35]. In our proposed technique we have Poiseuille flow in a channel along with a lateral flow driven by an electric field. We obtain the dispersion coefficient by using a regular perturbation scheme. In this paper we restrict our analysis to a 2D channel because the qualitative behavior of the DNA separation is expected to be the same in 3D even though quantitatively the results may differ.

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19 Fluidic Properties of DNA Molecules Since we are studying the separation of DNA in free solutions, it is important to understand the behavior of DNA molecules in free solutions, particularly the mobility and diffusivity. Mobility of DNA molecules in free solution Most researchers define the velocity of DNA molecules under unit electric field intensity as its mobility. Consequently, the unit of mobility is m 2 /(s*volt). After the invention of capillary electrophoresis (CE), it is possible to measure the mobility of DNA molecules in free solution accurately. However, one needs to ensure that the capillary walls are coated to eliminate the electroosmosis flow (EOF) of the solvent so that the data obtained is accurate. CE experiments have shown that the mobility of small DNA strands increases with size but levels off beyond a critical size. However, there are discrepancies on the critical size beyond which the DNA mobility is independent of size. In two separate studies Stellwagen et al. determined the critical size to be 400bp and 170bp [36,37]. From the trend of changing of mobility with DNA size, researchers concluded that small, relatively rigid DNA molecules experience greater friction with the solvent. Most researchers believe that electrolyte friction contributes to this phenomenon. This friction is an additional source of friction induced in the bulk solvent by the migrating polyionsthe DNA molecules. As we know, the counterions in the solvent will build a double layer around the DNA molecules. When the DNA molecules are in static state, the double layer will reach an equilibrium state. However, when these DNA molecules migrate, the counterions will change the distribution around the DNA molecules. Before a new equilibrium state is established, which needs some time to achieve, the counterion

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20 cloud will create a fluctuating force on the DNA molecules. Researchers believe that this is the origin of the dependence of the mobility of small DNA molecules on the size. Molecular diffusivity in free solution The molecular diffusivity is another important parameter that affects the separation efficiency. Several methods can be used to measure the molecular diffusivity of DNA molecules in free solutions: capillary electrophoresis(CE), NMR, Dynamic light scattering. Based on Stellwagens research [38], these three methods give comparable results. Here, we introduce the most commonly used methodthe stop-flow method in capillary electrophoresis. In this method, the voltage is turned off after the analytes have migrated nearly half way along the capillary channel. Then, these analytes are left there for a certain period time, within which band broadening occurs without the intervention of electric field. After that, the electric field is turned on again until all analytes go through the whole channel and are detected by the detector located at the end of the capillary. By using Equation 1-1 ( is the band variance) t)M(D2)t(02e2 (1-1) where t is the time during which the field is turned off, and by repeating the experiment for different times, we can obtain the molecular diffusivity D 0 (the slope of the curve). By further analysis, several researchers got an accordant result about the relationship between the molecular diffusivity and the size of DNA molecules. Stellwagen, Sorlie and Pecora, et al. got the scaling law For long flexible polyers, a classic theory, called Florys theory, is successfully used to describe the asymptotic behavior )03.068.0(M/1~D 5/3M/1~D.

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21 To find a model to calculate the molecular diffusivity of DNA molecules in free solution, Axel E.Nkodo, et al, tried several models [39]. And they found that their diffusion data agree well with the Zimm theory, which is used for a nonfree draining polymer. Therefore, they concluded that one can use Zimm equation to predict the diffustion coefficient of DNA molecules in free solutions fairly accurately with a good model for the hydrodynamic radius R H (M). And they recommend the equations for R H (M). For short rod-like fragments, one can use Equation 1-2 ))d/L/(ln(L3kT~D (1-2) Here, is the viscosity of fluid, L is the length of DNA molecule; d is the diameter of the DNA molecule. When the molecular size is medium compared to their persistence length, the Kratky-Porod equation provides an excellent model for R H (M). As for very long molecules, Florys scaling law applies. Additionally, some researchers did some experiment to test the effect of the intensity of electric field on the molecular diffusivity of DNA molecules in free solutions. The result is that the electric field does not change the diffusivity much within applicable conditions. This result is good since we can use high voltage to acquire high velocity of DNA molecules in the free solutions without considering much about the changing of diffusion.

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CHAPTER 2 THEORY Derivation of Equations About EFFF Using Perturbation Analysis Figure 2-1 shows the geometry of the 2D channel along with the electrodes for applying the lateral electric field. L and h are the channels length and height respectively, and the channel is infinitely wide in the third direction. The approximate values of L and h are about 2 cm and 5 microns respectively. Thus, continuum is still valid for flow in the channel. As we know, large Knudsen number(the ratio of mean free path to the dimension of the channel) invalidates Navier-Stokes equations. Commonly, when knudson number is larger than 0.01, the traditional continuum based equation becomes inaccurate. The mean free path of particles in gas of about 1 atm is around 70nm. Consequently, the height should be larger than 7 microns so that the Navier-Stokes equation applies. But for particles in liquid, the mean free path is much smaller than 70nm. As a result, even when the channel height is 1 m, our analysis still stands. Figure 2-1. Schematic of the 2D channel 22

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23 Consider diffusion of a solute in a 2D pressure driven flow in the channel. The convection-diffusion equation for the solute is 2222||eyycDxcDycuxcutc (2-1) where c is the solute concentration, u is the fluid velocity in the axial (x) direction, and are the diffusion coefficients in the direction parallel and perpendicular to flow, respectively. u is the velocity of the molecules in the lateral direction due to the electric field and can be estimated by the Smoluchowski equation, D ,D|| ey Er0ey u, where r and are the fluids dielectric constant and viscosity, respectively, is the permittivity of vacuum, and is the zeta potential. In the limit of large double layer thickness which occurs when the salt concentration is low, the electrophoretic velocity can equivalently be expressed as, ykTeZDtD u,where Z is the effective charge on the polyion, is the electric potential, k is the Boltzman constant and T ey 2 t is the temperature. Since DNA is a stable molecule, we do not propose to use any salts in our separations and thus we use the above expression to estimate the electrophoretic velocity. The flow and the lateral electric field are expected to stretch the DNA molecule in the axial direction. Therefore, the strands will have different diffusivities in the x and y directions. Generally, the diffusion coefficient of a cylindrical molecule with a large aspect ratio like a DNA strand in the direction parallel to flow is about twice the diffusion coefficient perpendicular to flow, i.e., [40]. Although the extent of stretching and consequently the D||

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24 diffusivity varies across the cross-section due to the difference in shear rate, for simplicity we treat the DNA strands as stretched cylinders at every lateral position. The mobility of the negatively charged DNA molecules will be reduced by the positive counter-ions surrounding the DNA. This electroviscous effect is small for spherical molecules, which are similar to cylindrical molecules in this respect [41], so, we will neglect it in our analysis. We also neglect the shear-induced diffusion and the presence of other charged species such as salts in the solution in the model developed below. Outside the nm-thin double layer the fluid is electroneutral, and the velocity of charged molecules due to the electric fields in the y direction is constant. Thus, Equation 2-1 becomes )ycxcR(Dycuxcutc2222ey (2-2) where and we denote as D. 2DDR||/ D The boundary conditions for solving Equation 2-2 are 0cuycDey at y = 0,h. (2-3) The boundary conditions (Equation 2-3) are strictly valid only at the wall and not at the outer edge of the double layer, which is the boundary of the domain in which our differential equation is valid. Still, since the double layer is very thin (~ nm), and the time scale for attaining steady state inside the double layer is very short, we neglect the total flux of the DNA molecules from the bulk to the double layer.

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25 From the momentum equation, we get x)ecZ(yuxp0ii22 (2-4) Due to electroneutrality in the bulk, the velocity profile remains unaffected by the lateral electric field. Thus, the fluid velocity profile in the axial direction is parabolic, i.e., ))h/y(h/y(u6u2 (2-5) where the mean velocity in the channel is 2hxp31u (2-6) Here again we neglect the change in the axial velocity across the double layer because it is very thin. Our model shows that the electric field will affect the mean velocity of the molecules only if the electric field driven velocity in the lateral direction is comparable to the mean velocity, i.e., u~yTkDZeuttey (2-7) The approximate values of D, h and u are 10 -9 m 2 /s, 10 m and 1mm/s, respectively. Using these values in Equation 2-7 and assuming Z ~ 1, which is a very conservative assumption, gives V1.0DZeku~t hTt In addition, there will be a potential drop of about a volt in each of the double layers at the wall. We note that we are neglecting adsorption of molecules at the channel walls and the streaming potential that may result because of the charge variation in the double layer. However, streaming

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26 potential alters the mean velocity of all the molecules by the same amount and does not affect the dispersivity. Our aim is to determine the Taylor dispersion of a pulse of solute introduced into the channel at t = 0. In a reference frame moving with a velocity u, the mean velocity of the pulse(comprising pure kind of DNA molecules), Equation 2-2 becomes )ycxcR(Dycuxc)uu(tc2222ey* (2-8) Since we are interested in long-term dispersion, the appropriate time scale is L/ where L is the total channel length, and is the mean fluid velocity. In this time, a pulse will spread to a width of about uDL/~ l, which is the appropriate length scale in the x direction. These scales ensure that the convective time scale is comparable to the diffusive time scale in the axial direction. The scaling gives 2221DhuhDhuhLuLDll (2-9) where 1Lh~hl since 1~Dhu Pe We use the following de-dimensionalization: u / LtT U = u/, u/uU** U = u/, C = c/c ey ey 0 X = x/l, Y = y/h (2-10) where L is the length of the channel; l is the width of the pulse as it exits the channel; h is the height; is the average velocity of the flow; and Pe is the Peclet number based on

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27 D =. In dimensionless form, Equation 2-8 and the boundary conditions (Equation 2-3) become D 22222ey2*YC1XCRYCUPeXC)UU(PeTC (2-11) 0CPeUYCey at Y = 0,1. (2-12) We assume a regular expansion for C in ................CCCC2210 (2-13) Substituting the regular expansion for C into Equation 2-11 gives the following sets of equations and boundary conditions to different orders in (1/ 2 ): 2020eyYCYCPeU ; 0CPeUYC0ey0 at Y = 0, 1 (2-14) )YPeUexp()T,X(ACey0 (1/): 2121ey0*YCYCPeUXC)UU(Pe ; 0CPeUYC1ey1 (2-15) Substituting C 0 from Equation 2-14 gives 2121eyey*YCYCPeU)YPeUexp(XA)UU(Pe (2-16) Integrating Equation 2-16 in Y from 0 to 1 gives 10ey*10eydY)YPeUexp(UdY)YPeUexp(U (2-17) Equation 2-17 gives the average velocity of pulse.

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28 1)exp()()exp(1212)exp(66U2* (2-18) where = (2-19) eyPeU In Equation 2-16 we assume )Y(GXAC1 (2-20) This gives 22eyey*YGYGPeU)YPeUexp()UU(Pe (2-21) Solving Equation 2-21 with boundary conditions gives Y32223Ye)constY2Y6Y3)1e()Ye(12(PeG (2-22) and the constraint determines the const in Equation 2-22. However, this const does not affect the mean velocity and the dispersion coefficient. 0GdY10 0 : 2222022ey1*0YCXCRYCPeUXC)UU(PeTC ; 0CPeUYC2ey2 at Y = 0,1 (2-23) Integrating Equation 2-23, using the boundary conditions and using Equation 2-24 ]1)PeU[exp(PeUAdYCCeyey1000 (2-24)

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29 gives ]dY)Y(G)UU(1eUPeR[XCTC10*PeUey22020ey (2-25) Thus, the dimensionless dispersion coefficient D is ]dY)Y(G)UU(1eUPeR[D10*PeUey2*ey (2-26) Substituting G from Equation 2-22 into Equation 2-26 and integrating gives ))1e/(()727202016e2016e6048e720e72e144e24e720e504e6048e144e24e504e720(PeRD6323323324222223422* (2-27) Comparison with Brenners Theory Howard Brenner used the method of moments to calculate *U and D in FFF with a shear flow between two infinite plates. To test our method, we solve the same problem with the perturbation method. According to Brenner, the equation for mean velocity of pulse is ),1(),2(hGu* and ),1(),2(U* (2-28) where, G' is the parameter for v=G'h (v(h) is the velocity profile for shear flow), =PeU y e and a0nd)exp(),1n( (2-29) To obtain the effective diffusivity, one needs to solve Equation 2-30, )UU)(Yexp(]dYBd)Y[exp(dYd* (B.C., 1,0Y@0dYBd ) (2-30)

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30 The result for D is: )(kPeRDv2* (2-31) 10*vdY)UU)(Yexp(Be1)(k (2-32) ttancons)1e()Ye(2YB2Y2 (2-33) After considerable simplification ]),2(),3(3),1(),2(2[]),1(),2([a4)(k2v (2-34) Comparing Equation 2-32 with our result Equation 2-26, we get )Yexp(BPe)Y(G )Yexp(Pe)Y(GB (2-35) substituting it into Equation 2-30 gives )UU(e)]ePeGPeeYG(e[dYd*YYYY (2-36) )UU(eePeGPeeYG2PeeYG(e)ePeGPeeYG(e*YY2YY22YYYY (2-37) Simplifying it gives Y*22ePe)UU(YGYG (2-38) Comparing Equation 2-38 with Equation 2-21, we find that they are actually the same (since the direction of lateral velocity in Brenners model is opposite to that in ours,

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31 should be replaced by when comparing these two results). Thus as expected, the two techniques yield the same results.

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CHAPTER 3 RESULTS AND DISCUSSION Limiting Cases The dispersion coefficient depends on the Peclet number and In the limit that approaches zero, we expect eyU eyU *U and D to approach the respective value for a 2D pressure driven flow in a channel without electric field, which are 2**Pe2101RD ; 1U (3-1) eyU = 0 implies = Pe = 0. To check whether our results match Equation 3-1, we expand our results for D eyU in the limit of This gives 0 ))(O166320089180012101(PeRD5422* (3-2) To leading order, Equation 3-2 reduces to 2Pe2101R which is the same as Equation 3-1. Also, we expand Equation 2-27 as goes to infinity. The result is )201672072(PeRD6542* (3-3) Figure 3-1 shows the curves from Equation 3-3, 3-2, and 2-27. The asymptotic solutions match the numerical solution if < 2 or >8. 32

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33 0 5 10 15 20 25 30 0 0.002 0.004 0.006 0.008 0.01 (D*-R)/Pe2 UyePe* 0 5 10 15 20 25 30 0 0.002 0.004 0.006 0.008 0.01 (D*-R)/Pe2 Uye UyePe* Figure 3-1. Dependency of (D -R)/Pe 2 on the product of Pe and. The dotted line is the small approximation (Equation 3-2), and the dashed line is the large approximation (Equation 3-3) eyU Similarly the asymptotic behavior of U in the limits of small and large is 039916801100800125201601-1U8642* (3-4) 126U2* (3-5) The result for *U in the limit of also reduces to 1 to leading order in Figure 3-2 shows the comparison of these asymptotic results and the numerical results from Equation 2-18. The small and the large results match in the limit of <2 and >40, respectively. These asymptotic results help us in understanding the physics of the dispersion and the DNA separation, as discussed below. 0

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34 0 10 20 30 40 50 60 70 80 0 0.5 1 1.5 Mean velocity of pulse UyePe* 0 10 20 30 40 50 60 70 80 0 0.5 1 1.5 Mean velocity of pulse Uye UyePe* Figure 3-2. Dependency of mean velocity *U on the product of Pe and The dotted line is the small approximation (Equation 3-4), and the dashed line is the large approximation (Equation 3-5) eyU Dependence of the Mean Velocity on and Pe eyU Figure 3-2 shows the dependence of the mean velocity on and Pe. The mean velocity of pulse depends only on the product of U and Pe. U changes the mean velocity of the pulse because the presence of an electric field leads to a higher concentration of the charged particles near the wall of the opposite polarity. The lateral concentration profile is a balance of the dimensionless electric flux, which is equal to and the dimensionless lateral diffusive flux, which is equal to eyUey ey cPeUey YC Thus, an increase in either Pe or leads to an increase in the electric flux that has to be balanced by a larger diffusive flux, which leads to particle buildup in a thinner region near the wall. Since the particles near the wall access streamlines with a smaller velocity than that eyU

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35 in the middle, an increase in Pe reduces the mean velocity of the pulse. As discussed above, in the limit of Pe approaching zero, the mean velocity approaches the mean fluid velocity, i.e., eyU eyU 1 U* 2Pe/ Dependence of D on and Pe eyU The effective dispersion coefficient D depends separately on U and Pe. However, ey *RD depends only on the product of U and Pe (Figure 3-1). At small with an increase of the particle concentration near the wall of opposite polarity (Y = 1 in our case) begins to increase, and at the same time the particle concentration near Y = 0 begins to decrease. However, a significant number of particles still exist near the center. The increase of results in an average deceleration of the particles (Figure 3-1), but a significant number of particles still travel at the maximum fluid velocity. This results in an increase in D ey and a consequent spread of the pulse. At larger only very few particles exist near the center as most of the particles are concentrated in a thin layer near the wall, and any further increase in leads to thinning of this layer. Thus, the maximum velocity of the majority of the particles goes down, resulting in a smaller spread of the pulse. Finally, as approaches infinity, the mean particle velocity approaches zero, and the dispersion coefficient approaches the molecular diffusivity. Figure 3-1 shows that the maximum value of 2 *Pe/RD is about .007. This implies that the convective contribution to dispersion is at most .007 Pe 2 Thus, even at Pe = 10, the convective contribution is only about 35% of the diffusive contribution R, which is approximately equal to 2.

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36 Separation of DNA Fragments of Different Lengths DNA fragments of different lengths have the same because the total charge on the molecule is directly proportional to the length (the charge is from the phosphoric structure), and the diffusion coefficient is inversely proportional to the length. Thus, a pure axial field cannot separate DNA molecules in free solution. However, as shown above, in the presence of lateral fields, the mean velocity of the molecules depends on the product of and Pe, where eyU eyU yZTkheuyTkDZeDhuUPettttey (3-6) In Equation 3-6, ,T t y and h are fixed for all the DNA molecules. Thus, the product in Equation 3-6 only depends on Z, which is directly proportional to the length of the DNA fragments. Since charge Z, and consequently *U and D are different for molecules of different sizes, a mixture of DNA fragments of different sizes separates into bands that contain same size DNA molecules, and these bands travel with their mean velocity and disperse as a Gaussian with the dispersion coefficient corresponding to their chain length. Thus, we can separate DNA strands according to their sizes by applying a lateral field instead of an axial field. Figure 3-3 shows the separation of a pulse containing two types of DNA molecules into two individual peaks as the molecules traverse the channel. At t = 0, a 1:1 mixture of two types of DNA molecules is introduced as a pulse at the channel entrance. For this simulation, the ratio of the diffusion coefficients of the two types of molecules is 2, and all the other physical

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37 constants are given in the caption. Figure 3-3 shows that, as time progresses, the DNA molecules separate into two separate Gaussian distributions. 0 0.2 0.4 0.6 0.8 1 LengthRelative concentration Pulse introduced at t=0 21tut=t1t=t2 01.252.53.755x 10-3m 0 0.2 0.4 0.6 0.8 1 LengthRelative concentration Pulse introduced at t=0 21tut=t1t=t2 01.252.53.755x 10-3m Figure 3-3. Separation of a 1:1 mixture of DNA strands of different sizes into two separate Gaussian peaks. =1mm/s, D 1 = 1x10 -9 D 2 = 2x10 -9 =1, Pe eyU 1 =10, h = 10 m Separation Efficiency Consider separation of two types of DNA molecules in a channel. We assume that when the distance between two pulse centers is larger than 3 times of the sum of their half widths, they are separated, i.e., )tDD4tDD4(3t)uu(*22*11*1*2 (3-7) where the subscripts indicate the two different DNA fragments. If the channel is of length L, the time available for separation is the time taken by the faster moving species through the channel, i.e., )u,umax(/*2*1 L. Substituting for t, and expressing all the variables in dimensionless form gives

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38 2*1*212*2*1*2*11]UUDDDD)[U,Umax(Pe136h/L (3-8) In the discussion below, we used L/h to indicate the efficiency of separation, i.e., smaller L/h implies more efficient separation. 0.5 1 1.5 2 2.5 3 0 500 1000 1500 L/hPe=5Pe=7Pe=9Pe=10 UyeUyeUye 0.5 1 1.5 2 2.5 3 0 500 1000 1500 L/hPe=5Pe=7Pe=9Pe=10 UyeUyeUye Figure 3-4. Dependency of L/h on and Pe for separation of DNA strands of different sizes. U = = Pe eyU e1y e2yU eyU 1 = Pe, and Pe 1 /Pe 2 = D 2 /D 1 = 10 In Figures 3-4 and 3-5, we show the dependence of L/h on Pe and in the case of =, which corresponds to DNA fragments of different lengths. Figure 3-5 is similar to Figure 3-4; the only difference is the ratio D eyU e1yU e2yUU 2 /D 1 Figure 3-5 shows that increasing which is physically equivalent to increasing the electric field, leads to a reduction in L/h required for separation. As increases, the mean velocities of both kinds of molecules decrease (Figure 3-1). But the dispersion coefficients do not change ey eyU Pe

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39 significantly because they are very close to the diffusive value of R for small Pe(Pe<10). Thus, L/h is primarily determined by the ratio 2*1*212UU1PeU 0.511.522.53050010001500L/hUyeUyeUyePe=5Pe=7Pe=9Pe=10 0.511.522.53050010001500L/hUyeUyeUyePe=5Pe=7Pe=9Pe=10 Figure 3-5. Dependency of L/h on and Pe for separation of DNA strands of different sizes. U = = Pe eyU e1y e2yU eyU 1 = Pe, and Pe 1 /Pe 2 = D 2 /D 1 = 2 As shown earlier, in the small regime 2*6011~U thus, 221224ey12*1*21*2PePeUPe1~UU1PeU2 Since the ratio Pe 2 /Pe 1 is fixed, 4ey51221224ey1UPe~PePeUPe14eyUeyU Thus, an increase in either Pe or leads to a reduction in L/h in the regime of small The constant Pe plots in Figure 3-4 and 3-5 show the dependency when is small. Also, the constant Pe curves shift down eyU

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40 with increasing Pe, due to the Pe -5 dependency shown in the above scaling. On the other side, as shown earlier, in the limit of large /6~U* thus, ey2122121ey2*1*21*2U~PePePePePePeU~UU1PeUeyU4eyUU This implies that in the large regime and at O(1) Pe, L/h becomes independent of Pe and begins to increase with an increase in as shown in Figure 3-4. Since L/h scales as in small regime, and as in the large regime, it must have a minimum. The minimum in L/h is clearly visible in Figure 3-5. In Figure 3-4, the minimum occurs for slightly larger than shown in the Figure 3-5. Physically, the minimum arises because at small field strength, the molecules accumulate near the wall, but a finite thickness of the region of accumulation still remains. Since the thickness of the region is different for the two types of molecules, the mean velocities of the two types of molecules differ. However, as the field strength becomes very large, both the mean velocities approach zero, and thus their difference also approaches zero. Subsequently, the difference in the mean velocities is zero for zero field because both the mean velocities are equal to the fluid velocity, and is also zero at very large fields because both the mean velocities approach zero; this implies that a maximum in the difference between the mean velocities of the two types of molecules must exist at some intermediate field. This maximum combined with other secondary effects results in a minimum in L/h required for separation. eyU ey

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41 0.511.522.5302004006008001000 UyeL/hPe=1Pe=3Pe=5Pe=10 0.511.522.5302004006008001000 Uye UyeL/hPe=1Pe=3Pe=5Pe=10 Figure 3-6. Dependency of L/h on and Pe for separation of molecules of the same size but different charges. = / = 10, and Pe = Pe eyU eyU e1yU e2yU e1yU 1 = Pe 2 i.e., D 2 = D 1 The effect of changing Pe while keeping fixed is more difficult to understand physically. Due to the dedimensionalization of the only way to change Pe while keeping fixed is to increase the fluid velocity and the field by the same factor. As a result, if we want to verify the effect of only an increase in the mean velocity , we need to increase Pe and concurrently reduce Thus, we first move to the smaller value and then follow the larger Pe curve. This keeps Pe and consequently D eyUeyUey eyU U eyU eyU and *U unchanged, and thus, L/h ~ 1/Pe. Physically, this inverse dependency on the mean fluid velocity arises because the dimensional mean velocity of the molecules depends linearly on . Thus, an increase in results in a linear increase in the difference

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42 between the mean velocities of the two types of molecules, i.e., *2*1uu The distance between the peaks at the channel exit is independent of because although *2*1uu increases linearly with , the time spent by the molecules in the channel varies inversely with . However, because D s do not change with changes in only , the spread of each of the Gaussians decreases with an increase in due to the reduction of time spent in the channel. Consequently, the spread of the peaks becomes smaller making it easier to separate the two types of DNA. 0.511.522.5302004006008001000 UyeL/hPe=10Pe=7Pe=5Pe=3 Pe=9 0.511.522.5302004006008001000 Uye UyeL/hPe=10Pe=7Pe=5Pe=3 Pe=9 Figure 3-7. Dependency of L/h on and Pe for separation of molecules of the same size but different charges. = / = 2, and Pe=Pe eyU eyU e1yU e2yU e1yU 1 = Pe 2 i.e., D 2 = D 1 In Figure 3-6 and 3-7, we explore the separation for particles having same diffusivity but different charges. By comparing them, we find that under the same Pe and

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43 e1yU larger results in better separation. This result is similar to the effect of an increase in D e1ye2yU/U 2 /D 1 shown in Figures 3-4 and 3-5. Furthermore, all the trends discussed above for the effect of Pe and U on L/h for separation shown in Figures 3-4 and 3-5 persist in Figures 3-6 and 3-7 because the arguments presented above are valid even when the Peclet numbers are the same for the two types of molecules and their are different. Thus, an increase in Pe for fixed reduces L/h, and an increase in for a fixed Pe first reduces L/h at small (=Pe), and then increases L/h at larger resulting in a minimum. ey eyUeyU eyUeyU Comparison of Lateral and Axial Electric Field As shown above, a lateral field can be used to separate particles in instances in which axial fields are ineffective because the ratio of the charge to the viscous resistance is the same for all the molecules. In this section, we wish to compare the effectiveness of lateral fields with axial fields in cases in which pure axial fields can result separation, i.e., in cases where the ratio of charge to viscous resistance is different for the molecules that need to be separated. As a special case, we consider two kinds of particles with equal and isotropic diffusivities, and Z 2 = 2 Z 1 In this case, the two types of molecules have the same Peclet number but different lateral electric velocities: and Furthermore, axial electric fields simply alter the mean velocity of the molecules without affecting the dispersion coefficient of molecules, i.e., e1ye2yU2U e1xe2xU2U 2Pe21011 **DU ex and U1 We adopt a same definition for the length of the channel needed for separation for the axial fields. Thus, we get

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44 2e1xe2x2*2*12*1*22*2*1]UU210/Pe12)[U,Umax(Pe136]UU210/Pe12[u)u,umax(Pe136h/L (3-9) Figure 3-8 plots the L/h required for separation in the above case as a function of Pe and The L/h required for separation decreases with an increase in e1xU exUex because an increase in electric field strength increases the difference between the mean velocities of the two kinds of molecules without affecting their dispersivities. At small Equation 3-9 gives L/h ~ and, at large it gives L/h ~ U, 2exU exU 1exU Also, at small Pe, L/h ~ 1/Pe, and, at large Pe, L/h~Pe. 0.5 1 1.5 2 2.5 3 0 200 400 600 800 1000 UxeL/hPe=10Pe=7Pe=5Pe=3Pe=1 0.5 1 1.5 2 2.5 3 0 200 400 600 800 1000 Uxe UxeL/hPe=10Pe=7Pe=5Pe=3Pe=1 Figure 3-8. Dependency of L/h on and Pe for an axial field for separation of molecules of the same size but different charges. Pe = Pe exU 1 = Pe 2 U and = 2 exU = e1x e2xU/ e1xU

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45 To compare the axial and the lateral fields, we compute the ratio of the lateral and the axial electric fields that result in the same difference in mean velocity of two molecules with For a pure axial electric field 1Z22Z )ZZ(DTk)uu()ZZ(DTk)uu(x12tt*2*112tte2xe1x (3-10) For a pure lateral field 1tte1yDZTkuy (3-11) where is the electric velocity required to obtain the mean velocity difference of e1yu *2 *1uu This relationship cannot be expressed analytically but is shown graphically in Figure 3-9. Dividing Equation 3-11 by 3-10 gives ])UU(PeU[PeZ)ZZ(])uu(Peu[PeZ)ZZ()uu(Zu)ZZ(xy*2*1e1y112*2*1e1y112*2*11e1y12 (3-12) From Figure 3-9, the minimum value of )UU(PeU*2*1e1y is about 14. This means that if Z 2 = 2 Z 1 in order to achieve the same velocity difference, the ratio of lateral electric field to axial field must be more than 14 Pe1 Thus, lateral fields could be more effective than axial fields above Pe = 14. Furthermore, at large Pe, D in lateral fields is smaller than the D in axial fields, which further adds to the effectiveness of the lateral fields. However, a lateral electric field always reduces the mean velocity of charged particles making it less than the mean velocity of fluid flow. Therefore, the maximum velocity

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46 difference between different kinds of charged particles is less than the mean velocity of the flow. An axial electric field, on the other hand, does not have this limit. Figure 3-10 shows the dependency of *1*2UU on for the axial fields (dash-dot line) and on for the lateral fields (solid lines; each line corresponds to a different Pe). In the region to the right of the dashed line, the axial fields are more effective. This region corresponds to Pe < 14 but only for exU eyU *1*2UU < 0.25. Using axial fields is the only way to achieve mean velocity differences larger than 0.25. Further research shows this critical velocity difference will increase with an increase in Z 2 /Z 1 and approach 1 as the ratio approaches infinity. As a result, in certain cases lateral fields could be more effective even in instances where axial fields can also accomplish separation. ye ye ye 0 5 10 15 20 25 30 0 0.05 0.1 0.15 0.2 0.25 0.3 U1-U2 U 0 5 10 15 20 25 30 0 0.05 0.1 0.15 0.2 0.25 0.3 U1-U2 U UPe Figure 3-9. Dependency of the difference of mean velocities of pulses of two kinds of molecules(Z 2 =2 Z 1 Pe 2 /Pe 1 = 2) on the product of Pe = Pe 1 =2 and (=) eyU eyU e1yU

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47 0 1 2 3 0 0.1 0.2 0.3 U1-U2 Uye Uxeor Pe=30 Pe=14Pe=6 Pe=2 0 1 2 3 0 0.1 0.2 0.3 U1-U2 Uye Uxeor Uye Uye Uxe Uxeor Pe=30 Pe=14Pe=6 Pe=2 Figure 3-10. Dependency of the difference of mean velocities of pulses of two kinds of molecules on an electric field. The dash-dot line is for the axial electric field and solid lines are of the lateral electric field for different Pe. In these plots Pe 2 /Pe 1 = 2 Comparison with Other Promising Methods for DNA Separation Entropic trapping has been successfully employed to separate a mixture of 24.5 kbp, 48.5 kbp, 73 kbp and 97 kbp DNA fragments in a 1.5 cm long channel in about 40 minutes[42,43]. Our simulations show that the same separation can be accomplished in a 80 m channel by our proposed method. The operating time is about 1.5 minutes. The parameters for the two methods are listed in Table 3-1. Table 3-1 also shows the comparison of the proposed methods with the technique proposed by Doyle et al. that uses a self assembled matrix of magnetic particles as a sieving media. The authors do not report the channel length in the paper, but the time needed for separation is longer than

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48 that for our proposed techniques. These comparisons show that our technique is promising. In our model we have not taken into account the entrance and exit effects, and thus the length of the channel required for separation will actually be longer than the results shown in Table 3-1. In this situation, we propose that the electric field is only applied in the fully developed region and thus the mean velocity of the molecules at the entrance and the exit will be the same as the fluid velocity, and the region in between where the fields are applied will contributes to separation. There are other factors that may reduce the separation efficiency of our technique such as the extra dispersion caused by the effect of the walls in the third direction. Also, in our model the thickness of the region in which the molecules reside near the wall of the opposite polarity scales as D/. For the longest DNA molecules, the thickness of this layer is about 10 nm. Clearly, accumulation of molecules in such a thin region will be affected by the double layers surrounding the DNA molecules, and this may impact the separation efficiency. Thus we believe that our current model only serves as a guide in designing the best separation strategy. However, the results of our simulations show that EFFF in microchannel is certainly a promising technique for separating DNA fragments. eyU Table 3-1 Comparison of EFFF with other techniques sample (kbp) method Total time (min) Length of channel Continuous method a) 5 126.2 m 15,33.5,48.5 magnetic 15 Not mentioned Continuous method b) 1.5 80 m 24.5,48.5,73,97 Entropic trapping 40 1.5 cm a) h=1 m, =0.3, =0.1mm/s eyU b) h=1 m, =0.2, =0.1mm/s eyU

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CHAPTER 4 DNA SEQUENCING ATTEMPT To read the genome of an organism, chromosomes, which range in size from 50 million to 250 million bases, are broken into much shorter pieces, about 500 base pairs in length. In the Sanger process, each of the smaller pieces are primed for replication and then added to four beakers, each containing all the four bases A, T, C and G needed for replication. However, in each beaker, a fraction of one type of nucleotides is defective; the replication stops at these nucleotides. Each replication reaction then proceeds until a reaction-terminating nucleotide is incorporated into the growing strand, whereupon replication stops. Thus, in a beaker containing a defective A, the length of the replicated fragments corresponds to location of T. The last step of the sequencing is then separation of these fragments, which differ in length by only one base pair, and this is accomplished by gel electrophoresis. We therefore seek to determine the efficiency of our system at sequencing DNA. To sequence a fragment N base pair long, we need to separate a mixture of bases of lengths varying from 1 to N. In Figure 4-1 we plot the L/h of a channel required to separate a DNA fragment n bases long from a fragment n+1 bases long. To calculate L/h (n) we use Equation 3-8 with a slight modification; the factor 36 is replaced by 16 because a spacing of two times the sum of the half widths between two Gaussian peaks is enough to identify them as separate peaks, but we need a spacing of about three times to separate them. In Equation 3-8, we identify species 1 as the nth fragment and species 2 as 49

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50 the (n+1)th fragment. Thus, D )1n/(nD/12 and nPePe1 where Pe is the Peclet number for a single nucleotide. 0 100 200 300 400 500 0 0.5 1 1.5 2 2.5x 106 Number of base pairsL/h 0 100 200 300 400 500 0 0.5 1 1.5 2 2.5x 106 Number of base pairsL/h Figure 4-1. L/h required to separate DNA fragments that differ in length by a single base pair vs. the number of base pairs in the smaller fragment. h =1m, Pe = 1, = 0.4. The largest value of L/h represents the size of the channel required for separation. The dashed line is calculated from the large approximations. eyU Figure 4-1 plots L/h required for separation as a function of the maximum number of the base pairs n for fixed Pe and For a given Pe and (h = 1m, Pe = 1, Uye=0.4), L/h first decreases because of an increase in Pe eyU eyU 1 and then begins to increase because the effect of an increase in D 2 /D 1 dominates over the effect of an increase in Pe. Since the Peclet number will proportionally increase with the increase in the number of bases, it is easy to reach large values of the product of the Peclet number and Usually separation of the two largest fragments, i.e., 500 and 499 base pair long eyU

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51 fragments, is the hardest. Therefore, to optimize the sequencing, we will first focus in the large regime. Using the large approximations gives L/h ~ n 2 (In this limit D* is equal to D). The dashed curve in Figure 4-1 is the large Pe asymptotic result. Plotting the L/h vs U eyU y e on log-log axes gives a slope of 2, which agrees with above analysis. The time required for separation of fragments in DNA sequencing will be )n(U* /L where )n(U* is the mean velocity of the slowest moving, i.e., the longest, DNA fragment. Large approximation gives T ~ n 3 Pe ( ) 2eyUey2Un2 Thus, one could accomplish faster separation by reducing Pe and However, these expressions are based on large approximation, so Pe and cannot be made too small. In fact, if Pe is too small, the diffusivity of the DNA molecules begins to increase (Figure 3-2), resulting in long operating time and large required channel length. Specifically when both Pe and are reduced, the length needed to separate the first several base pairs increases dramatically because the difference in velocities between the first few fragments becomes small. Therefore, to get the best performance, we need a large Pe to limit the diffusion of large molecules (according to Equation 3-3). However, we cannot make Pe arbitrarily large because doing so will increase the pressure required to pump the fluid. Note that making Pe large by increasing h will also increase L. Therefore, we essentially have an optimization problem in which we need to minimize L, t and P by manipulating , h and To guide us in this optimization we use the large results, i.e., eyU/L eyU eyU eyU eyU ~h (3-13) 2ey3hUn~t (3-14)

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52 Furthermore, for 2D Poiseuille flow 2hLu~P (3-15) According to Equation 3-15, we choose a relatively large h to provide large and Pe. At the same time, we reduce U to balance the effect on L and t of increasing h. After h and are fixed, we choose a Pe that is large enough to control the diffusivity. Note that we use the large approximation only as a guide, and use the appropriate equations to finally determine L/h and t. We finally conclude that 1-500 base pairs can be separated in a 0.72-meter long, 10-micron thick channel in 1.7 hrs at Pe = 200 and Figure 4-2 shows this result. ey eyU 01.0Uey 0 100 200 300 400 500 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 Number of base pairsTime(hr)Length(m) 00.20.40.60.8 0 100 200 300 400 500 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 Number of base pairsTime(hr)Length(m) 00.20.40.60.8 Figure 4-2. Length and Time required to separate DNA fragments that differ in length by a single base pair vs. the number of base pairs in the smaller fragment. h = 10m, Pe = 200, = 0.01, =0.02m/s eyU

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53 The length of the channel is about 0.72m, which is too long for fabrication on a chip. However, one could potentially fabricate about 36 parallel channels, each about 2 cm long, and join them at the ends to fabricate a channel with straight segments joined by curved ends. In such a channel, we need to evaluate the extra dispersion caused by the curved ends.

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CHAPTER 5 CONCLUSIONS Application of lateral fields affects the mean velocity and the dispersion coefficient of colloidal particles undergoing Poiseuille flow in a 2D channel. The dimensionless mean velocity *U depends on the product of the lateral velocity due to electric field yTkuDzeUttey and the Peclet number. The convective contribution to the dispersion coefficient is of the form The mean velocity of the particles decreases monotonically with an increase in but )PeU(fPeey2PeUey 2*Pe/RD has a maximum at a value of ~ 4. This maximum arises when the thickness of the region near the wall where a majority of the particles accumulate is about h/2. PeUey Since the mean velocity of the particles under a lateral field depends on the charge Z but not on the product of the diffusion coefficient and the charge, colloidal particles such as DNA molecules that have the same ratio of charge to viscous resistance can be separated on the basis of their lengths on a chip by applying lateral electric fields. Axial fields cannot accomplish this separation unless the channel is packed with a gel. Thus, our proposed strategy of accomplishing separation by lateral fields may offer a solution to separating DNA on a chip. The length of the channel required for separation depends on the ratio of the diffusion coefficients of the two types of molecules that need to be separated and on the Pe and Lateral fields can also be used to separate molecules that can already be effectively separated by purely axial fields, and, in certain instances eyU 54

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55 such as large Pe, the lateral fields require smaller field strengths than the axial fields. However, lateral fields are limited by the fact that the maximum difference in the mean velocity of two types of molecules that need to be separated is less than the mean velocity of the fluid. Axial fields do not suffer from this limitation. Lateral fields can also be used for sequencing DNA on a chip. The separation step in DNA sequencing requires separating a mixture of DNA fragments that range in size from 1 to 500 base pairs and differ in length by a single base pair. Lateral fields can accomplish such a high-resolution separation in channels that are about 0.72m long. With current micro lithographic techniques, such a channel can be incorporated into a chip by fabricating about 36 parallel channels, each 2 cm long, and eventually joining them to acquire a single channel. In such a channel we will need to consider the extra dispersion caused by the curves that link two successive straight channels. While analyzing the dispersion of molecules under lateral electric field, we assumed that the fluid away from the double layer is electroneutral, that the electric field is constant in the bulk of the fluid, and that DNA molecules are fully charged and aligned parallel to the flow. Also, we neglected the double layer at the electrodes, the interaction of the molecules with the wall, and the reduction in mobility due to the counterion cloud surrounding the charged colloidal particles. We also presented our analysis for a 2D channel, where the presence of walls has demonstrated a first order effect on the dispersion of molecules in Poiseuille flow in a channel in the absence of electric fields. In addition, the flow of current in the lateral direction will result in generation of oxygen and carbon dioxide that may affect the hydrodynamics of the flow. The DNA molecules may also be affected and possibly damaged by the high lateral fields. Thus, while our

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56 proposed separation technique is promising, more experimental and theoretical work needs to be done to determine the effectiveness of lateral fields in accomplishing separation of DNA on a chip.

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APPENDIX NOMENCLATURES A Variable defined in Equation 2-14 c Concentration of a kind of particle in fluid D Molecular diffusion coefficient D* Dimensionless effective diffusion coefficient G Variable defined in Equation 2-20 h Height of the channel k t Thermal constant k v Function defined in Equation 2-31 l Width of the pulse as it exits the channel L Length of the channel P Pressure Pe Peclet number h/D R Ratio of D to || D t Time T Dimensionless time T t Temperature u Velocity of flow at a certain position and time *u Mean velocity of a pulse consisting of one kind of particle Mean velocity of flow 57

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58 U Dimensionless velocity of flow *U Dimensionless u exu Velocity of a charged particle in the x-direction due to electric field exU Dimensionless u ex eyu Velocity of a charged particle in the y-direction due to a lateral electric field eyU Dimensionless yu e x Position on the x-axis X Dimensionless position in the x-axis y Position on the y-axis Y Dimensionless position in the y-axis Z Number of charges of a particle Variable defined in Equation 2-19 Function defined in Equation 2-29 Intensity of an electric field Perturbation, the ratio of h to l, or the square root of h/L Viscosity of the fluid

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LIST OF REFERENCES 1. The Web Book Publications. 2003. Molecular biology web book. Available from URL: http://www.web-books.com/MoBio Site last visited July 2003. 2. Hallick RB. 1995. A Molecular Graphics companion to an Introductory Course in Biology or Biochemistry. Available from URL: http://www.blc.arizona.edu/Molecular_Graphics/DNA_Structure/DNA_Tutorial.H TML Site last visited July 2003. 3. Boxall N. 2003. DNA components. Available from URL: http://www.massey.ac.nz/~wwbioch/DNA/tutehome/tutetext.htm Site last visited July 2003. 4. Sarma RH. 1996. DNA stability. Available from URL: http://www.albany.edu/~achm110/dnastability.html Site last visited July 2003. 5. Kuiper HA. 2000. Progress report 2: Results and Milestones 2nd year. Available from URL: http://www.entransfood.com/RTDprojects/gmobility/GMOBILITY.htm Site last visited July 2003. 6. S. Karger AG, Basel CH. 2001. Safety considerations of DNA in foods. Available from URL: http://www.ilsi.org/file/RPDNAinfoods.pdf Site last visited July 2003. 7. Watson A. A new breed of high-tech detectives. Science 2000; 289: 850-854. 8. Barron AE, Blanch HW, Soane DS. A transient entanglement coupling mechanism for DNA separation by capillary electrophoresis in ultradilute polymer solutions. Electrophoresis 1994; 15: 597-615. 9. Sassi AP, Barron A, Alonso-Amigo MG, Hion DY, Yu JS, Soane DS, Hooper HH. Electrophoresis of DNA in noval thermoreversible matrices. Electrophoresis 1996; 17: 1460-1469. 10. Hu L, Harrison JD, Masliyah JH. Numerical model of electrokinetic flow for capillary electrophoresis. Journal of Colloid and Interface Science 1999; 215: 300-312. 11. Long D, Stone HA, Ajdari A. Electroosmotic flows created by surface defects in capillary electrophoresis. Journal of Colloid and Interface Science 1999; 212: 338-349. 59

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60 12. Meldrum DR. Sequencing genomes and beyond. Science 2001; 292: 515-517. 13. Ren H, Karger AE, Oaks F, Menchen S, Slater GW, Drouin G. Separating DNA sequencing fragments without a sieving matrix. Electrophoresis 1999; 20: 2501-2509. 14. Volkmuth WD, Duke T, Austin RH, Cox EC. Trapping of branched DNA in microfabricated structures. Proceedings of the National Academy of Sciences of the United States of America 2002; 92: 6887-6891. 15. Doyle PS, Bibette J, Bancaud A, Viovy JL. Self-assembled magnetic matrices for DNA separation chips. Science 2002; 295: 2237-2237. 16. Dorfman KD, Brenner H. Generalized Taylor-Aris dispersion in discrete spatially periodic networks: Microfluidic applications. Physical Review E 2002; 65: 021103~1-18. 17. Bader JS, Hammond RW, Henck SA, Deem MW, Mcdermott GA, Bustillo JM, Simpson JW, Mulhern GT, Rothberg JM. DNA transport by a micromachined Brownian ratchet device. PNAS 1999; 96: 13165-13169. 18. Hammond RW, Bader JS, Henck SA, Deem MW, Mcdermott GA, Bustillo JM, Rothberg JM. Differential transport of DNA by a rectified Brownian motion device. Electrophoresis 2000; 21: 74-80. 19. Han J, Craighead HG. Characterization and optimization of an entropic trap for DNA separation. Analytical Chemistry 2002; 74: 394-401. 20. Turner SW, Perez AM, Lopez A, Craighead HG. Monolithic nanofluid sieving structures for DNA manipulation. American Vacuum Society 1998; 16: 3835-3840. 21. Barron AE, Sunada WM, Blanch HW. The effects of polymer properties on DNA separations by capillary electrophoresis in uncross-linked polymer solutions. Electrophoresis 1996[0]; 17: 744-757. 22. Tri N, Caldwell K, Beckett R. Development of Electrical Field-Flow Fractionation. Analytical Chemistry 2000; 72:1823-1829. 23. Tsukahara S, Yamanaka K, Watarai H. Flow Fractionation of Microparticles under a Dielectrophoretic Field in a Quadrupole Electrode Capillary. Analytical Chemistry 2001; 73: 5661-5668. 24. Gale BK, Caldwell KD, Frazier AB. Geometric Scaling Effects in Electrical Field Flow Fractionation. 1. Theoretical Analysis. Analytical Chemistry 2001; 73: 2345-2351. 25. Myers MN. Overview of Field-Flow Fractionation. Journal of Microcolumn Separations 1997; 9: 151-162.

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61 26. Shiue MP, Pearlstein AJ. Free-Solution Electrophoresis with Amplification of Small Mobility Differences by Helical Flow. Journal of Chromatograph A 1995; 707: 87-103. 27. Contado C, Reschiglian P, Faccini S, Zattoni A, Dondi F. Continuous split-flow thin cell and gravitational field-flow fractionation of wheat starch particles. Journal of Chromatograph A 2000; 871: 449-460. 28. Hecker R, Fawell PD, Jefferson A, Farrow JB. Flow Field-Flow Fractionation of high-molecular-weight polyacrylamide. Journal of Chromatograph A 1999; 837: 139-151. 29. Reschiglian P, Torsi G. Determination of Particle Size Distribution by Gravitational Field-Flow Fractionation: Dimensional Characterization of Silica Particles. Chromatographia 1995; 40: 467-473. 30. Mes EPC, Kok WT, Tijssen R. Sub-micron particle analysis by thermal field-flow fractionation and multi-angle light scattering detection. Chromatographia 2001; 53: 697-703. 31. Vastamki P, Jussila M, Riekkola ML. Development of Continuously Operating Two-Dimensional Thermal Field-Flow Fractionation Equipment. Separation Science and Technology 2001; 36: 2535-2545. 32. Semenov SN. Combined thermal field-flow fractionation-capillary electrophoresis. Analytical Communication 1998; 35: 229-233. 33. Sanz R, Torsello B, Reschiglian P, Puignou L, Galceran MT. Improved performance of gravitational field-flow fractionation for screening wine-making yeast varieties. Journal of Chromatography A 2002; 966: 135-143. 34. Saenton S, Lee HK, Gao Y, Ranville J, Williams SKR. Evaluation of Different Field-Flow Fractionation Techniques for Separating Bacteria. Separation Science and Technology, 35, 1761-1775, 2000. 35. Howard B, Edwards DA. macrotransport processes. Stoneham(MA): Butterworth-Heinemann; 1993. 36. Stellwagen NC, Gelfi C, Righetti PG. The free solution mobility of DNA. Biopolymers 1997; 42: 687-703. 37. Stellwagen E, Stellwagen NC. Determining the electrophoretic mobility and translational diffusion coefficients of DNA molecules in free solution. Electrophoresis 2002; 23: 2794-2803. 38. Stellwagen NC, Gelfi C, Righetti PG. The use of gel and capillary electrophoresis to investigate some of the fundamental physical properties of DNA. Electrophoresis 2002; 23: 167-175.

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62 39. Nkodo AE, Garnier JM, Tinland B, Ren H. Diffusion coefficient of DNA molecules during free solution electrophoresis. Electrophoresis 2001; 22: 2424-2432. 40. Brenner H. Rheology of a dilute suspension of axisymmetric Brownian particles. International Journal of Multiphase Flow 1974; 1: 195-341. 41. Warszynski P. Coupling of hydrodynamic and electric interactions in adsorption of colloidal particles. Advances in Colloid and Interface Science 2000; 84: 47-142. 42. Han J, Craighead HG. Characterization and optimization of an entropic trap for DNA separation. Analytical Chemistry 2002; 74: 394-401. 43. Turner SW, Perez AM, Lopez A, Craighead HG. Monolithic nanofluid sieving structures for DNA manipulation. American Vacuum Society 1998; 16: 3835-3840.

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BIOGRAPHICAL SKETCH I was born in Apr 23, 1976 in He Zhang, a small town in Gui Zhou province, China. My father is a teacher of physics in a high school and my mother is a doctor. I established strong interest in science since childhood due to the intellectual surrounding provided by my family. My wide region of reading earned me honors in various competitions of high school. With competitive scores in the National Entrance Examination, I was admitted by the most prestigious university of ChinaTsinghua University. I urged myself in my undergraduate study in Tsinghua University, took five-year courses in four years and got high scores in most courses. As a result, I graduated one year earlier than my peers, ranking top 5% in my department of 120 students and entered the graduate program of Biochemical Engineering in 1998, waived of the entrance examination. In graduate stage, I ranked 10% in my class. During seven years in Tsinghua University, I participated in several projects. In my undergraduate diploma project, I studied the measurement of solubility of sodium sulfate in supercritical fluid, which is a part of the research of SuperCritical Water Oxidation (SCWO), a promising method for dealing with wastewater. Deeply absorbed in this wonderful supercritical world, I searched the literature extensively, discussed with professors, and did experiments carefully. Finally, I got satisfactory results, and my diploma got a high score/100. In 1999, I took part in a project to undertake middle-scaled amplification of the production of PHB (poly--hydroxybutyrate, a kind of biodegradable plastic) with 63

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64 E.Coli., which was a part of a Ninth Five-year National Key Project of China. Under my active and successful participation, we found and eliminated the scattering of nitrogen during sterilization and improved the distribution of air input. The density of bacteria reached 120g/l and the production of PHB extended to 80g/l, far beyond the original goal. The amplification succeeded and won me the honor of the first prize of outstanding performance in field practice of my department. After the practice, I began my thesis work under the guidance of Prof. Zhongyao Shen, the Vice-Dean of School of Life Sciences and Engineering in Tsinghua University. My work focused on the coupling of fermentation and separation. In the first year, I applied the coupling of fermentation and ion exchange on the production of 2-Keto Gulonic Acid, the direct precursor of Vitamin C. However, this research was abandoned because an impossibility coming from the fermentation system. After that, my main interest was on the coupling of fermentation and membrane separation in the production of acrylamide from acrylotrile. During the process, I acquire insights on membrane, fermentation, ion exchange, and operation of analytical equipment. Finally, I got a high enzyme activity from the fermentation, which is the highest value on documents. After I graduated from Tsinghua University in 2001, I came to the Department of Chemical Engineering, University of Florida to pursue advanced education. My research focuses on separation processes with microchannel and electric fields. And we have already got some promising results. This thesis is some of my research results. After I get master degree, I will further my work on this project, such as doing experiments to testify our theoretical results in this thesis. Below is a list of my published papers.

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65 1. Zhi Chen, Xudong Sun,Yue Shi,et al. Study on Production of Acrylamide by Microbial Method ()Culture of bacterium cells and expression of high activity of nitrile hydratase. CHINESE JOURNAL OF BIOTECHNOLOGY, 2002 Vol.18 No.1,p55 2. Zhi Chen, Xudong Sun, Yue Shi, et al. Study on Production of Acrylamide by Microbial Method ( )Enzyme catalytic kinetics and de-active dynamics of nitrile hydratase. CHINESE JOURNAL OF BIOTECHNOLOGY, 2002 Vol.18 No.2, p225 3. Xiang Botao,Wang Tao,Chen Zhi,et al. The Solubility of Sodium Sulfate in Supercritical Water. CHEMICAL ENGINEERING, 2001 Vol.29 No.1,p72 4. Sun Xudong, Chen Zhi, Shi yue, et al. Studies on Bioprocess and Bioreactors Used in Bioconversion for Acrylamide. CHEMICAL INDUSTRY AND ENGINEERING PROGRESS, 2002 Vol.21 No.5, p319


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Title: DNA Separation and sequencing by Electric Field-Flow Fractionation (EFFF) in a microchannel
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Creator: Chen, Zhi ( Author, Primary )
Publication Date: 2003
Copyright Date: 2003

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Material Information

Title: DNA Separation and sequencing by Electric Field-Flow Fractionation (EFFF) in a microchannel
Physical Description: Mixed Material
Creator: Chen, Zhi ( Author, Primary )
Publication Date: 2003
Copyright Date: 2003

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Source Institution: University of Florida
Holding Location: University of Florida
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DNA SEPARATION AND SEQUENCING BY ELECTRIC FIELD-FLOW
FRACTIONATION (EFFF) IN A MICROCHANNEL
















By

ZHI CHEN


A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
MASTER OF SCIENCE

UNIVERSITY OF FLORIDA


2003


































Copyright 2003

by

Zhi Chen















ACKNOWLEDGMENTS

This work was performed under careful and elaborate instructions of Dr. Anuj

Chauhan. During this work, Dr. Chauhan gave me invaluable help and direction, which

guided me when I struggled with difficulties and questions. In addition, the lessons he

taught me about how to approach a project will help me in my future research and work.

Also, my laboratory colleagues gave me many helpful suggestions. I deeply appreciate

their help.

We also acknowledge the financial support of NASA (NAG 10-316), the

Engineering Research Center (ERC) for Particle Science and Technology at the

University of Florida, The National Science Foundation (NSF Grant EEC-94-02989), and

the Industrial Partners of the ERC for support of this research.
















TABLE OF CONTENTS
Page

A C K N O W L E D G M E N T S ................................................................................................. iii

LIST OF TABLES ....................................................... ............ ....... ....... vi

L IST O F F IG U R E S .... ...... ................................................ .. .. ..... .............. vii

ABSTRACT .............. .......................................... ix

CHAPTER

1 BA CK GR OU N D .................. ............................. .. ........... .............. .

D eoxyribonucleic A cid .................................................. ................................ 1
What Forms DNA and RNA Molecules......................................................
Structure of D N A M olecules ..................................... .................................... 5
Prim ary structure ......................................... ............... ...... .. ...... .. 5
Code and gene .................................... ........................... ...... 6
The 3D structure ....................................................... .... .. ........ ..
Stab ility .................................................................................................. 12
M e ltin g ............................................................................................ 1 2
Degradation of DNA molecules........................................................... 13
Separation of D N A m olecules ................................... ........... .................. 13
Fluidic Properties of DNA M olecules .......................................................... 19
Mobility of DNA molecules in free solution ................................... 19
Molecular diffusivity in free solution ....................................... 20

2 T H E O R Y ................................................................................................ ..... 22

Derivation of Equations about EFFF Using Perturbation Analysis.........................22
Com prison w ith Brenner's Theory ................................... ................................... 29

3 RESULTS AND DISCU SSION ........................................ ......................... 32

L im itin g C ases................................... ............................32
Dependence of the Mean Velocity on Uy and Pe..................................... 34
Dependence of D* on U and Pe............................................................... 35
Separation of DNA Fragments of Different Lengths.............................................36
Separation E efficiency ........................ ................ ................... .. ...... 37










Comparison of Lateral and Axial Electric Field ....................................................43
Comparison with Other Promising Methods for DNA Separation .........................47

4 DNA SEQUENCING ATTEMPT ........................................ ........................ 49

5 C O N C L U SIO N S ................................................................. .............. ...... 54

APPENDIX N OM EN CLATURE........................................................... ............... 57

L IST O F R EFE R E N C E S ............................................................................ ...............59

B IO G R A PH IC A L SK E T C H ...................................................................... ..................63












































v
















LIST OF TABLES

Table page

1-1 Genomes of prominent organism s.............................................................................6

1-2 G enetic code (m R N A )................................................................ ....................... 7

3-1 Comparison of EFFF with other techniques .........................................................48















LIST OF FIGURES


Figure p

1-1 Basic structure of DN A m olecules.................................... .......................... ......... 2

1-2 Structure of pentose with numbering ......................... .................................... 3

1-3 Structure of bases in DN A and RN A ........................................ ...... ............... 3

1-4 Examples of deamination which involves the removal of an amino group.
Accidental deamination may change the cytosine to uracil, or the methylated
cytosine to thym ine ...................... .................. ............................. .5

1-5 Schem e of single strand D N A ............................................................. .................

1-6 G ene structure of D N A strand .............................. ........................... .....................8

1-7 Example of dA-dT and dG-dC base pair as found within DNA double helix ..........9

1-8 Normal right-handed "double helix" structure of DNA, also known as the B
fo rm ...................................................................................... 1 0

1-9 Comparison between B form and Z form............................................ 11

2-1 Schem atic of the 2D channel............... ................. ....................... ............... 22

3-1 Dependency of (D*-R)/Pe2 on the product of Pe and U The dotted line
is the small a approximation (Equation 3-2), and the dashed line is the
large a approxim ation (Equation 3-3) .................................... ......... ............... 33

3-2 Dependency of mean velocity U on the product of P U The dotted
line is the small a approximation (Equation 3-4), and the dashed line is the
large a approximation (Equation 3-5) ........... .............................. ............... 34

3-3 Separation of a 1:1 mixture of DNA strands of different sizes into two
separate Gaussian peaks. =lmm/s, D1 = x109, D2 2x109, U =1,
P ei= 10 h = 10 tm ................................................ ................ 3 7









3-4 Dependency of L/h on Uy and Pe for separation of DNA strands of different
sizes. Uy, = U2 = U, Pei = Pe, and Per/Pe2 =D2/D = 10 ..........................38

3-5 Dependency of L/h on Uy and Pe for separation of DNA strands of different
sizes. U = U = U, Pei = Pe, and Per/Pe2 = D2/D1 = 2 .............................39

3-6 Dependency of L/h on U6 and Pe for separation of molecules of the same
size but different charges. U = UPe U, /U = 10, and Pe = Pei = Pe2,
3-7 Dependency of L/h on Uy and Pe for separation of molecules of the same
size but different charges. U = Uyl, U2 /U1 = 2, and Pe = Pei = Pe2,
i.e ., D 2 = D .............................................................................................................. 4 1



3-8 Dependency of L/h on Ue and Pe for an axial field for separation of molecules of the same
size but different charges. Uy = Ue, Ue 2/Uej 2, and Pe = Pei = Pe2,

i.e ., D 2 = D 1 ............................................................................................................. 4 2

3-8 Dependency of L/h on Ue and Pe for an axial field for separation of
molecules of the same size but different charges. Pe = Pei = Pe2, U = U,1
and U e /U e = 2 .......................................................... .. .......... 44

3-9 Dependency of the difference of mean velocities of pulses of two kinds of
molecules (Z2=2 Z1, Pe2/Pei = 2) on the product of Pe = Pej=2 and Ue
y
(U y = U ) ........ .............. .................... ............... .....................46

3-10 Dependency of the difference of mean velocities of pulses of two kinds of
molecules on an electric field. The dash-dot line is for the axial electric
field and solid lines are of the lateral electric field for different Pe. In these
p lots P e2/P ei = 2 .................................................... ................ 4 7

4-1 L/h required to separate DNA fragments that differ in length by a single
base pair vs. the number of base pairs in the smaller fragment. h =1 tm,
Pe = 1, Ue = 0.4. The largest value of L/h represents the size of the channel
required for separation. The dashed line is calculated from the large ac
approxim nations. ...............................................................50

4-2 Length and Time required to separate DNA fragments that differ in length
by a single base pair vs. the number of base pairs in the smaller fragment.
h = 10[tm, Pe = 200, Ue = 0.01, =0.02m/s ....................................... .......... 52
Y















Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Master of Science

DNA SEPARATION AND SEQUENCING BY ELECTRIC FIELD-FLOW
FRACTIONATION (EFFF) IN A MICROCHANNEL

By

Zhi Chen

August, 2003

Chair: Anuj Chauhan
Major Department: Chemical Engineering

DNA separation is a core activity in biology, especially in mapping and sequencing

DNA molecules. Separation of DNA molecules of different chain lengths by

electrophoresis is difficult because the ratio of the total charge to viscous resistance is

independent of the chain length. We modeled using a lateral electric field in a

microchannel to separate DNA fragments in different sizes based on Taylor dispersion

theory. With this method, we can perform the separation in free solution, which avoids

the difficulty of loading gel in commonly used capillary gel electrophoresis (GE) or

complicated fabrications in an artificial sieving matrix.

During the research, we found that the lateral electric field will build a

concentration profile in the lateral direction, which has a different distribution for

different DNA molecules. Theoretically, the product of Pe and Uy determines the shape

of the profile. The variant profiles combined with the parabolic velocity profile along the









lateral direction engender different mean velocities of pulse. Based on this fact, we

acquired length-dependent separation of DNA molecules.

Furthermore, EFFF can also effectively separate DNA strands differing in size by a

single base pair, and thus can be used to sequence DNA. The separation step in

sequencing 500 base pair long DNA molecules can be done in 1.7 hours by a 0.72-meter

long, 10-micron thick channel. However, since this result is from theoretical calculations,

we need further experiments to test it.














CHAPTER 1
BACKGROUND

Deoxyribonucleic Acid (DNA)

Nucleic acids are important kind of biological molecules that exist in every living

being. These molecules were discovered in 1868 by a young Swiss scientist when he

refined an organic substance with a very large phosphorous content from a bandage. But

the biological functions of nucleic acids were not recognized until 1944, when scientist

Avery did the famous pneumobacillus transform experiment. This experiment

successfully proved that nucleic acids but not proteins are the genetic substances in a

living organic body. The next milestone in the field of molecular biology occurred in

1953 when Watson and Crick discovered the double helix structure of DNA molecules.

Then in the 1970s, recombination technology was invented, leading to the foundation of

the new field of Gene technology. Since then, researchers have tried to modify genes in

organisms for a variety of applications.

Most living cells contain two kinds of nucleic acids-deoxyribonucleic acid (DNA)

and ribonucleic acid (RNA). Most cells have both of them with DNA existing in

nucleolus and RNA existing in the cytoplasm. However viruses usually contain either

only DNA or RNA.

What Forms DNA and RNA Molecules

DNA and RNA are polymers known as polynucleotide, in which the monomer

units are nucleotides. Each nucleotide comprises three parts-pentose, base and

phosphate group [1]. In DNA or RNA, each nucleotide contains only one phosphate









group, but some cellular free nucleotides (such as ADP and ATP, which are important

species in the energy transfer and storage cycle) may contain more than one.












Pentose





Figure 1-1. Basic structure of DNA molecules. (Reprinted with permission from The
Web Book Publications. 2003. Molecular biology web book. Available from
URL: http://www.web-books.com/MoBio. Site last visited July 2003)


The carbon atoms on the bases purinee and pyrmidine rings) are numbered as

1,2,3,4,5 and thus to avoid confusion the carbon atoms on the pentose are numbered 1', 2',

3', 4', and 5' (Figure 1-2). Two kinds of pentose exist in nucleic acid-2'-deoxyribose

and ribose. The difference between the two is that the deoxyribose lacks a hydroxyl

group at the 2'-position [2]. The deoxyribose pentose is present in DNA and the ribose

pentose is present in RNA. In both the 2'-deoxyribose and ribose, the hydroxyl groups on

the 5'- and 3'- carbons link to the phosphate groups.

Five different kinds of bases are present in nucleic acids. They are adenine,

guanine, cytosine, thymine and uracil. Figure 1-3 shows the molecular structure of these

bases.




























Figure 1-2. Structure of pentose with numbering


Figure 1-3. Structure of bases in DNA and RNA (Reprinted with permission from The
Web Book Publications. 2003. Molecular biology web book. Available from
URL: http://www.web-books.com/MoBio. Site last visited July 2003)










Four of these five bases are present in DNA. They are given one letter

abbreviations as shorthand (A is for adenine; G is for guanine; C is for cytosine; T is for

thymine). In the RNA molecules, there are also four types of bases; A, G, C also exist in

RNA, but T is replaced by U(uracil).

If the phosphate groups in a nucleotide are removed, it becomes a nucleoside,

which consists of one of the bases covalently attached to the 1' position of a pentose. The

five different nucleosides of DNA and RNA are deoxyadenosine (dA), deoxyguanosine

(dG), deoxycytosine (dC), deoxythymidine (dT) and deoxyuracil (dU), which is present

only in RNA.

The bases in RNA and DNA form pairs-A-T and G-C in DNA, and A-U and G-C

in RNA. The chemical structure of uracil is simpler than thymine and uracil can pair

perfectly with adenine. Thus, it puzzled researchers that A pairs with T rather than U in a

DNA strand. This issue was finally resolved after we understood the details of the

repairing mechanisms in DNA.

As an evolution source, mutations may occur under the influence of external factors

(UV radiation, exposure to chemical agents, etc.) or cellular processes (accidental

deamination, replication errors, etc.). Figure 1-4 shows one of these mutation factors: the

deamination of cytosine.

Cytosine is one of four bases in DNA molecules. As shown above, Cytosine can be

mutated to uracil by deamination process. Since DNA does not contain uracil, this

mutation can be easily detected and repaired by base excision. If DNA were made up of

uracil, the cytosine to uracil mutation could hardly be corrected. This explains why DNA









chooses thymine, instead of uracil, even though the chemical structure of uracil is simpler

than thymine.


Cytosine


Uracil


NH2

N CH


0 N
H


NH2

N C---C H 3


H

5-methylcytosine


Deamination








Deamination
______>


0








0
HN CH
H


O NI


HN CCcH


H


Thymine


Figure 1-4. Examples of deamination which involves the removal of an amino group.
Accidental deamination may change the cytosine to uracil, or the methylated
cytosine to thymine (Reprinted with permission from The Web Book
Publications. 2003. Molecular biology web book. Available from URL:
http://www.web-books.com/MoBio. Site last visited July 2003)


Structure of DNA Molecules

Primary structure

As mentioned above, DNA molecules consists of A,T,C,G [3]. They are joined by

the 3'-5' phosphate bonds into a strand (Figure 1-5).

The sequence of these four nucleotides determines the genetic information

contained in the DNA molecules, and different creatures have different sequence and









length of DNA molecules. Table 1-1 shows the typical length of DNA molecules in

various organisms.


5'-Phosphate


3'-OH


Figure 1-5. Scheme of single strand DNA


Table 1-1. Genomes of prominent organisms


Organism Genome size (Mb*) Gene number
Hepatitis D virus 0.0017 1
Hepatitis B virus 0.0032 4
HIV-1 0.0092 9
Bacteriophage 1 0.0485 80
Escherichia coli 4.6392 4400
S. cerevisiae (yeast) 12.155 6300
C. elegans (nematode) 97. 19000
D. melanogaster (fruit fly) 137. 13600
Mus musculus (mouse) 3000. ?
Homo sapiens (human) 3000 30000**
* 1 Mb = 1 million base pairs (for double-stranded DNA or RNA) or 1 million bases (for single-
stranded DNA or RNA).
** The total number of human genes is still quite controversial. It could be as high as 75,000 [see
a paper published on July 4, 2001].
Note: Reprinted with permission from The Web Book Publications. 2003. Molecular biology web
book. Available from URL: http://www.web-books.com/MoBio. Site last visited July 2003.









Code and gene

Scientists have found that each three continuous nucleotides within the DNA

encode a protein and have drawn Table 1-2, which shows the correspondence between

the codes and proteins.


Table 1-2. Genetic code (mRNA)
1st position 2nd position (middle) 3rd position
(5' end) (3' end)
U C A G
Phe F Ser S Tyr Y Cys C U
Phe F Ser S Tyr Y Cys C C
Leu L Ser S STOP STOP A
Leu L Ser S STOP Trp W G
Leu L Pro P His H Arg R U
Leu L Pro P His H Arg R C
LeuL Pro P Gn Q Arg R A
Leu L Pro P Gln Q Arg R G
Ile I Thr T Asn N Ser S U
Ile I Thr T Asn N Ser S C
Ile I Thr T Lys K Arg R A
Met M Thr T Lys K Arg R G
Val V Ala A Asp D Gly G U
GVal V Ala A Asp D Gly G C
Val V Ala A Glu E Gly G A
Val V Ala A Glu E Gly G G
Synthesis of a peptide always starts from methionine (Met), coded by AUG. The stop codon
(UAA, UAG or UGA) signals the end of a peptide.
Note: Reprinted with permission from The Web Book Publications. 2003. Molecular biology
web book. Available from URL: http://www.web-books.com/MoBio. Site last visited July 2003.

By definition, a gene includes the entire nucleic acid sequence used for the

expression of one product (peptide or RNA). Such sequence may be separated into two

categories-regulatory region and transcriptional region. The regulatory region controls

the activation of the gene, which could be near or far from the transcriptional region.

And the transcriptional region consists of exons and introns. After the transcription

introns will be removed, whereas exons remains encoding a peptide or functional RNA









Figure 1-6 shows the secondary structure of DNA molecule, which is made up of

genes, pseudogenes and extragenic region. Pseudogenes are nonfunctional genes, which

often comes from mutation of genes that happens in the duplication process. However,

because duplicated genes commonly have many copies, the organism can still survive

even if some of them become nonfunctional.



Exon


Intron


DNA


SGene Pseudogene Extragenic region



Figure 1-6. Gene structure of DNA strand (Reprinted with permission from The Web
Book Publications. 2003. Molecular biology web book. Available from URL:
http://www.web-books.com/MoBio. Site last visited July 2003)


A certain piece of DNA sequence often repeats several times in the total DNA of a

cell. Experimentally, the number of repeated copies is classified on the basis of DNA

reassociation kinetics [1]. The entire DNA is first randomly cleaved into fragments with

an average size of about 1000 bp. Then, they are heated to separate the strands of each

fragment. Subsequently, temperature is reduced to allow strand reassociation. If a

fragment contains a sequence which is repeated many times in the total DNA, it will have

greater chance to find a complementary strand and reassociate more quickly than other

fragments with less repetitive sequences.









Based on the reassociation rate, DNA sequences are divided into three classes:

highly repetitive, moderately repetitive and single copy.

The 3D structure

In a DNA molecule, the two strands intertwine with each other, forming a double

helix structure. This structure was first discovered by James D. Watson and Francis

Crick in 1953. In this structure, the sugar-phosphate backbones of the two DNA strands

wind around the helix axis and the bases of the individual nucleotides are on the inside of

the helix.

Within the DNA double helix, base A forms 2 hydrogen bonds with T on the

opposite strand and G forms 3 hydrogen bonds with C.





















Figure 1-7. Example of dA-dT and dG-dC base pair as found within DNA double helix
(Reprinted with permission from The Web Book Publications. 2003.
Molecular biology web book. Available from URL: http://www.web-
books.com/MoBio. Site last visited July 2003)









The helix turns a round every 10 base pairs (Figure 1-8). Since the distance

between two base pairs is 0.34 nm, the length is about 3.4 nm per turn for DNA

molecule. The intertwined strands make two grooves of different widths- the major

groove and the minor groove, which may bind with specific proteins.

The human DNA molecule in a diploid cell, if fully extended, would have a total

length of 1.7 meters. If one unfolds all of the DNA molecules in the body, one could

reach the moon for 6000 times! [1]




5' 3'
0.34 nm



Minor
groove



3.4 nm



Major
groove











Figure 1-8. Normal right-handed "double helix" structure of DNA, also known as the B
form (Reprinted with permission from The Web Book Publications. 2003.
Molecular biology web book. Available from URL: http://www.web-
books.com/MoBio. Site last visited July 2003)










In a solution with higher salt concentrations or with alcohol added, the DNA

structure may change from normally B form to an A form, which is still right-handed.

But every 2.3 nm makes a turn with 11 base pairs in it.

Another DNA structure is called the Z form because it seems to zigzag, which is

left-handed on rotation. One turn includes 4.6 nm, comprising 12 base pairs. The DNA

molecule with alternating G-C sequences in alcohol or high salt solution tends to have

such structure.























B Form Z Form


Figure 1-9. Comparison between B form and Z form (Reprinted with permission from
The Web Book Publications. 2003. Molecular biology web book. Available
from URL: http://www.web-books.com/MoBio. Site last visited July 2003)









Stability [3,4]

In this part, we will introduce two aspects of DNA molecules. One is the melting

of helix double strand DNA, the other is the degradation.

Melting

Melting is the term given to the separation of the two strands of a DNA molecule,

which is also called denaturation.

Several factors make the DNA molecules relatively stable. The DNA strands in a

double helix are held together by the H-bonds between the bases. These bonds (also

called Watson-Crick) attach two strands together. Moreover, base pairs sit on the top of

each other at a rotation of 360 and there is strong interaction between all adjacent base

pairs. This interaction, called stacking interaction, stabilizes the DNA double helix.

Additionally, the phosphate groups must be neutralized (by Na+ or Mg2+ ions) to allow

the negatively charged phosphates to be in close proximity.

As introduced above, two hydrogen bonds exist between A and T and three exist

between G and C. If a solution of DNA is heated, the hydrogen bonds will break at high

temperatures, and the stacking interactions will become weak. As some researchers

concluded, the most important contribution to DNA helix stability is the stacking of the

bases on top of one another. Thus, in order to denature DNA, we must overcome the

stacking energies that provide cohesion between adjacent base pairs. Since AT pairs

have only two hydrogen bonds, they are easier to undergo severance. Together with the

fact that the stacking energies are less for AT-rich regions, AT rich area tends to separate

compared to GC rich area. As a result, the base composition of the DNA influences the

melting temperature (Tm), at which two DNA strands separate. The greater the

proportion of G-C base pairs in the DNA is, the higher the Tm is. However,









experimentally, at temperatures higher than 800C the GC pairs will also melt, and the

DNA will become single stranded which will be present in coiled and unstructured forms.

Several methods can be used to obtain the melting of double strand DNA:

* Reduction of Salt Concentration-as the salt concentration is reduced, the
phosphate groups are no longer neutralized by Na+ or Mg2+ ions and the negative
charges of phosphate group tend to force the strands apart.

* Extreme of pH-it alters the ionization states of the groups on the bases which
provide and accept the H-bonds. Commonly, linear DNA molecules will denature
and precipitate when ph is above 12.

* Increase in Temperature-when temperature of a DNA solution increases to a
certain value, which is called the melting point (Tm), the strands separate.

Degradation of DNA molecules

Degradation of DNA molecules is related to the breakup of phosphoric bonds and

consequently the cleavage of DNA chain. Various factors contribute to the degradation of

DNA: chemical, physical and enzymatic, etc. Commonly, the effects that result

degradation are much stronger than that for melting.

* Prolonged heat treatment may result in DNA hydrolysis which degrades the DNA.

* Low pH may increase chemical modifications and hydrolysis of DNA. For
example, at low pH (pH 4), maize DNA and plasmid DNA were rapidly degraded
[5]. Under low pH conditions, what will happen first in the degradation of DNA is
the depurination of the nucleic acid backbone. After that, hydrolysis of adjacent 3'-
5'-phosphodiester linkages occurs, resulting in measurable shortening of DNA
strands [6].

* Enzymatic degradation of DNA by nucleases may also occur on prolonged storage.
Adding EDTA can inhibit the activity of DNA enzyme by chelating metal ions with
valence of 2, so that the storage time can be above 5 years under -700C.

Separation of DNA molecules

Microfluidics is a sub area of the microelectromechanical systems(MEMS) and is

mainly concerned with moving fluids and then performing various unit operations in

micron-sized channels. Microfluidics is rapidly becoming a very important area of









research due to numerous potential applications in separation and analysis. The current

trend in this field is towards development of chips that can accomplish reactions,

separations, and detection at a very rapid rate, such as a chip that can separate DNA

fragments of different lengths and detect them.

DNA electrophoresis has become a very important separation technique in

molecular biology and, in particular, in the genome project. DNA fragments are first

separated by chain length and are later processed to read the sequence of the bases that

form the genetic code of all living organisms. This technique is also indispensable in

forensic applications for identifying a person from a tissue sample [7]. However,

separation of DNA fragments of different chain lengths by electrophoresis is difficult

because the velocity of the charged DNA molecules due to an axial electric field is

independent of the chain length. The reason of this independency is that the mobility of a

DNA molecule is approximately inversely proportional to the length while the total

charge is directly proportional to the length. This difficulty is traditionally overcome by

performing the electrophoresis in columns filled with gels. In these processes, because

negatively charged DNA surrounded by positive counterions moves through a matrix

such as an agarose gel, the mobility is no longer inversely proportion to the length. Thus,

DNA chains of different lengths traverse the gel at different speeds and separate in a

series of bands. In gel electrophoresis, the electric field can either be continuous or

pulsed. Continuous field electrophoresis is useful for separating DNA molecules of sizes

below approximately 20000 base pairs. The migration rates of DNA strands above this

size is almost independent of the length of the strand except at very low voltages with

which it takes an excessively long time to accomplish separation. Pulsed field gel









electrophoresis (PFGE) was developed to separate longer DNA fragments, which can not

be separated by the conventional gel electrophoresis. PFGE utilizes a pulsed electric

field, which changes directions continually, resulting in changes in migration directions.

These changes lead to a stronger dependency of the net migration rates on the DNA chain

length, even if the chains are longer than about 20000 base pairs.

Although gel electrophoresis can separate DNA fragments, there are some

problems associated with its use in DNA separation. Bubbles can form in the gel during

an operation, resulting in variations in DNA electrophoretic mobility [8]. Moreover, the

separation by electrophoresis of DNA fragments larger than 40000 base pairs using gel is

slow; which is still one of the slowest steps in the genome project. This kind of separation

typically takes more than 20 hours because a low-intensity and pulsating field is used to

separate DNA fragments to prevent the long fragments from being damaged by high

temperatures that may result under large fields [9].

To eliminate the temperature increase during separation, researchers developed

capillary electrophoresis, which has a high surface-area-to-volume ratio, providing rapid

elimination of heat and allowing application of high electric fields without a substantial

temperature increase [8,10,11]. The use of capillary sequencers in the genome project

resulted in an eight-fold increase in the sequencing capacity and output [12]. However,

preparing uniform, homogeneous, bubble free and stable gel-filled capillaries is difficult,

especially for separation of DNA fragments, which commonly involves many parallel

lanes running simultaneously.

Recent advances in microfabrication techniques have led to production of

microfluidic devices frequently referred to as a "lab-on-a-chip" that can perform a









number of unit-operations such as reactions, separations, detection, etc., at a much higher

throughput. Gel-based DNA separations are not convenient in such devices because of

the difficulty in loading the gel [9]. Thus, gels have been replaced with polymeric

solutions as the sieving mediums. Electrophoresis in a free medium can also separate

DNA fragments but it requires precise modifications to the DNA molecules [13].

Microfabricated obstacles such as posts [14], self-assembling colloids [15], entropic

barriers [16], and Brownian ratchets [17,18] have also shown to be effective at separating

DNA strands.

Craighead et. al. used an entropic trapping system, which consists of alternating

thick (0.65-1.6 [tm) and thin (90nm) regions in a channel flow. Since larger molecules

need to reach higher entropic states to enter the thinner regions, they spend less time in

the channel and exit the channel earlier than smaller ones. In this method, the number of

traps is one of the most important factors that controls the separation effect. In their

experiment, they did not achieve good separation for DNA molecules (24.5, 48.5, 73.0,

97.0kbp) until the number of traps reached 3700 and the total separation time reached 40

mins [19]. Turner et. al. fabricated artificial arrays of posts in a microchannel by

lithography. The diameter of the posts and the interval between them were both 100nm-

small enough to provide a strong sieving effect. They tested separation of 7.2 kbp and 43

kpb DNA strands and obtained a ratio of 2 between the mean velocities of these two

kinds of DNA strands [20]. Baron et al. used un-crosslinked polymer solution which

provides sparse sieving and thus has low resistance to DNA molecules. By this method,

they reduced the operation time to about 20mins. But the separation for DNA with large

molecule weight is still not satisfactory [21]. Viovy et al. used magnetic fields to drive









superparamagnetic particles to form a post matrix with the interparticle distance to be

about 5.7 jtm. They successfully separated large DNA molecules (15, 33.5, and 48.5kbp)

in only 10-15mins. Furthermore, when the magnetic fields are released, the viscosity of

the fluid in the pipe becomes low. Consequently, this method avoids the difficulty of

loading gel that exists in gel-capillary electrophoresis. Bader et al. [17,18] created a

spatially periodic anisotropic potential energy field to trap the molecules at the potential

energy minima. As a result of pulsating application of an electric field, the molecules that

diffuse outside a trap when the field is released are attracted to the next trap. In this

method, the smaller molecules with large diffusivity have larger migration speeds, and

the larger molecules have lower speeds. This difference in speed leads to separation.

However, the optimal DNA separation technique should accomplish separation

without any sieving medium and should be amenable to online modification to

accomplish separation for a wide range of DNA sizes. Our proposed strategy utilizes

lateral electric fields and no sieving medium, and the amplitude of the field can be

adjusted to separate different DNA compositions. Essentially, this method is called

electric field-flow fractionation (EFFF) [22-24], which is a derivation of field-flow

fractionation (FFF). Giddings first proposed FFF in 1966 [25]. The basic idea is to use a

field in the direction perpendicular to flow and form a concentration profile on the cross

section [26]. When charged DNA molecules flow through channels in the presence of

lateral fields, i.e., fields perpendicular to the flow direction, they experience an attractive

force towards the wall of the opposite polarity. In the absence of any field, each DNA

molecule has an equal probability of accessing different streamlines in a time scale larger

than h2/D, where h is the height of the channel, and D is the molecular diffusivity.









However, due to the electric field, the molecules on average access streamlines closer to

the wall, which results in a mean axial velocity smaller than the mean fluid velocity. We

shall show later that the enhancement in concentration near the wall is greater for the

more slowly diffusing molecules, and thus their mean velocity is reduced more than the

mean velocity of the faster diffusing molecules. If a slug of DNA molecules is introduced

into a channel with lateral electric fields, the difference in mean velocities leads to

separation of the molecules into bands, and the bands of smaller molecules travel faster.

FFF has also been used in size based particle separation using gravity or centrifugal

acceleration as the lateral force [27-29]. New trends in FFF are thermal field-flow

fractionation (TFFF) [30-32] and its application in bioseparation [33,34].

In this paper, we analyze the Taylor dispersion of charged molecules such as DNA

in a microchannel with pressure driven flow under lateral fields by using regular

perturbation techniques. Based on our investigation, we propose a new scheme for

separating DNA molecules in channels by application of lateral electric fields without

using any gel or polymeric solution as sieving mediums. Brenner used the method of

moments to obtain the Taylor dispersion coefficient for shear flow in a channel

accompanied by a lateral flow [35]. In our proposed technique we have Poiseuille flow in

a channel along with a lateral flow driven by an electric field. We obtain the dispersion

coefficient by using a regular perturbation scheme. In this paper we restrict our analysis

to a 2D channel because the qualitative behavior of the DNA separation is expected to be

the same in 3D even though quantitatively the results may differ.









Fluidic Properties of DNA Molecules

Since we are studying the separation of DNA in free solutions, it is important to

understand the behavior of DNA molecules in free solutions, particularly the mobility

and diffusivity.

Mobility of DNA molecules in free solution

Most researchers define the velocity of DNA molecules under unit electric field

intensity as its mobility. Consequently, the unit of mobility is m2/(s*volt).

After the invention of capillary electrophoresis (CE), it is possible to measure the

mobility of DNA molecules in free solution accurately. However, one needs to ensure

that the capillary walls are coated to eliminate the electroosmosis flow (EOF) of the

solvent so that the data obtained is accurate.

CE experiments have shown that the mobility of small DNA strands increases with

size but levels off beyond a critical size. However, there are discrepancies on the critical

size beyond which the DNA mobility is independent of size. In two separate studies

Stellwagen et al. determined the critical size to be 400bp and 170bp [36,37].

From the trend of changing of mobility with DNA size, researchers concluded that

small, relatively rigid DNA molecules experience greater friction with the solvent. Most

researchers believe that "electrolyte friction" contributes to this phenomenon. This

friction is an additional source of friction induced in the bulk solvent by the migrating

polyions-the DNA molecules. As we know, the counterions in the solvent will build a

double layer around the DNA molecules. When the DNA molecules are in static state, the

double layer will reach an equilibrium state. However, when these DNA molecules

migrate, the counterions will change the distribution around the DNA molecules. Before

a new equilibrium state is established, which needs some time to achieve, the counterion









cloud will create a fluctuating force on the DNA molecules. Researchers believe that this

is the origin of the dependence of the mobility of small DNA molecules on the size.

Molecular diffusivity in free solution

The molecular diffusivity is another important parameter that affects the separation

efficiency.

Several methods can be used to measure the molecular diffusivity of DNA

molecules in free solutions: capillary electrophoresis(CE), NMR, Dynamic light

scattering. Based on Stellwagen's research [38], these three methods give comparable

results. Here, we introduce the most commonly used method-the stop-flow method in

capillary electrophoresis. In this method, the voltage is turned off after the analytes have

migrated nearly halfway along the capillary channel. Then, these analytes are left there

for a certain period time, within which band broadening occurs without the intervention

of electric field. After that, the electric field is turned on again until all analytes go

through the whole channel and are detected by the detector located at the end of the

capillary. By using Equation 1-1 (c is the band variance)


C2(t)= C +2D,(M)t (1-1)
where t is the time during which the field is turned off, and by repeating the experiment

for different times, we can obtain the molecular diffusivity Do (the slope of the curve).

By further analysis, several researchers got an accordant result about the

relationship between the molecular diffusivity and the size of DNA molecules.

Stellwagen, Sorlie and Pecora, et al. got the scaling law D ~ 1/M(O 68 03) For long

flexible polyers, a classic theory, called Flory's theory, is successfully used to describe

the asymptotic behavior D ~ 1/M3 .









To find a model to calculate the molecular diffusivity of DNA molecules in free

solution, Axel E.Nkodo, et al, tried several models [39]. And they found that their

diffusion data agree well with the Zimm theory, which is used for a nonfree draining

polymer. Therefore, they concluded that one can use Zimm equation to predict the

diffustion coefficient of DNA molecules in free solutions fairly accurately with a good

model for the hydrodynamic radius RH(M). And they recommend the equations for

RH(M). For short rod-like fragments, one can use Equation 1-2

kT
D (1-2)
37TrL /(ln(L / d) + y)

Here, r is the viscosity of fluid, L is the length of DNA molecule; d is the diameter of the

DNA molecule. When the molecular size is medium compared to their persistence length,

the Kratky-Porod equation provides an excellent model for RH(M). As for very long

molecules, Flory's scaling law applies.

Additionally, some researchers did some experiment to test the effect of the

intensity of electric field on the molecular diffusivity of DNA molecules in free solutions.

The result is that the electric field does not change the diffusivity much within applicable

conditions. This result is good since we can use high voltage to acquire high velocity of

DNA molecules in the free solutions without considering much about the changing of

diffusion.















CHAPTER 2
THEORY

Derivation of Equations About EFFF Using Perturbation Analysis

Figure 2-1 shows the geometry of the 2D channel along with the electrodes for

applying the lateral electric field. L and h are the channel's length and height

respectively, and the channel is infinitely wide in the third direction. The approximate

values of L and h are about 2 cm and 5 microns respectively. Thus, continuum is still

valid for flow in the channel. As we know, large Knudsen number(the ratio of mean free

path to the dimension of the channel) invalidates Navier-Stokes equations. Commonly,

when knudson number is larger than 0.01, the traditional continuum based equation

becomes inaccurate. The mean free path of particles in gas of about 1 atm is around

70nm. Consequently, the height should be larger than 7 microns so that the Navier-

Stokes equation applies. But for particles in liquid, the mean free path is much smaller

than 70nm. As a result, even when the channel height is 1 |tm, our analysis still stands.









flow in


FL 2x




Figure 2-1. Schematic of the 2D channel










Consider diffusion of a solute in a 2D pressure driven flow in the channel. The

convection-diffusion equation for the solute is


9c 9c e 9c 29c 2 c
--+U--+U- = D +--D2 (2-1)
at ax Oy ax2 ay2


where c is the solute concentration, u is the fluid velocity in the axial (x) direction, and

D D are the diffusion coefficients in the direction parallel and perpendicular to flow,


respectively. Uy is the velocity of the molecules in the lateral direction due to the


electric field and can be estimated by the Smoluchowski equation, u = E, where Sr


and [t are the fluid's dielectric constant and viscosity, respectively, So is the permittivity

of vacuum, and is the zeta potential. In the limit of large double layer thickness which

occurs when the salt concentration is low, the electrophoretic velocity can equivalently be

expressed as, u _-D eZ c~) ,where Z is the effective charge on the polyion, O is the
Y kTt cy

electric potential, k is the Boltzman constant and Tt is the temperature. Since DNA is a

stable molecule, we do not propose to use any salts in our separations and thus we use the

above expression to estimate the electrophoretic velocity. The flow and the lateral

electric field are expected to stretch the DNA molecule in the axial direction. Therefore,

the strands will have different diffusivities in the x and y directions. Generally, the

diffusion coefficient of a cylindrical molecule with a large aspect ratio like a DNA strand

in the direction parallel to flow is about twice the diffusion coefficient perpendicular to

flow, i.e., D, = 2 D [40]. Although the extent of stretching and consequently the









diffusivity varies across the cross-section due to the difference in shear rate, for

simplicity we treat the DNA strands as stretched cylinders at every lateral position. The

mobility of the negatively charged DNA molecules will be reduced by the positive

counter-ions surrounding the DNA. This electroviscous effect is small for spherical

molecules, which are similar to cylindrical molecules in this respect [41], so, we will

neglect it in our analysis. We also neglect the shear-induced diffusion and the presence

of other charged species such as salts in the solution in the model developed below.



Outside the nm-thin double layer the fluid is electroneutral, and the velocity of

charged molecules due to the electric fields in the y direction is constant. Thus, Equation

2-1 becomes


ac ac ac a2c 8 2c)
-+u-+u --= D(R- + ) (2-2)
at ax y x2 2y

where R = D /D 2, and we denote D as D.



The boundary conditions for solving Equation 2-2 are

ac
-Dc+uec= 0 aty= 0,h. (2-3)


The boundary conditions (Equation 2-3) are strictly valid only at the wall and not at

the outer edge of the double layer, which is the boundary of the domain in which our

differential equation is valid. Still, since the double layer is very thin (- nm), and the

time scale for attaining steady state inside the double layer is very short, we neglect the

total flux of the DNA molecules from the bulk to the double layer.









From the momentum equation, we get

ap 8 u 8B
0 = +- (- Zec,) (2-4)
ax ay- 8x

Due to electroneutrality in the bulk, the velocity profile remains unaffected by the lateral

electric field. Thus, the fluid velocity profile in the axial direction is parabolic, i.e.,


u= 6(y/h-(y/h)2) (2-5)


where the mean velocity in the channel is


< u >= (- h2 (2-6)


Here again we neglect the change in the axial velocity across the double layer because it

is very thin.

Our model shows that the electric field will affect the mean velocity of the

molecules only if the electric field driven velocity in the lateral direction is comparable to

the mean velocity, i.e.,

DZe dO
u --- (2-7)
S ktTt y

The approximate values of D, h and are 10-9 m2/s, 10 |tm and Imm/s,

respectively. Using these values in Equation 2-7 and assuming Z ~ 1, which is a very

< u > ktTth
conservative assumption, gives AO = 0.1V. In addition, there will be a
DZe

potential drop of about a volt in each of the double layers at the wall. We note that we

are neglecting adsorption of molecules at the channel walls and the streaming potential

that may result because of the charge variation in the double layer. However, streaming









potential alters the mean velocity of all the molecules by the same amount and does not

affect the dispersivity.

Our aim is to determine the Taylor dispersion of a pulse of solute introduced into

the channel at t = 0. In a reference frame moving with a velocity U*, the mean velocity

of the pulse(comprising pure kind of DNA molecules), Equation 2-2 becomes


ac _a c ae c d 2C a2C
+ (u -u )-+u = D(R- + ) (2-8)
at Ox 'y x2 2y


Since we are interested in long-term dispersion, the appropriate time scale is L/

where L is the total channel length, and is the mean fluid velocity. In this time, a

pulse will spread to a width of about / DL/< u > which is the appropriate length

scale in the x direction. These scales ensure that the convective time scale is comparable

to the diffusive time scale in the axial direction. The scaling gives


12 L L < u > h 12 h 1
-- -> -J (2-9)
D h D h D 22


h h h
where --- <1, since Pe 1.
1 L D

We use the following de-dimensionalization:

t -
T = U = u/, U = / < u >,
L/

U = u /, C = c/co, X = x/1, Y = y/h (2-10)

where L is the length of the channel; I is the width of the pulse as it exits the channel; h is

the height; is the average velocity of the flow; and Pe is the Peclet number based on









D = D. In dimensionless form, Equation 2-8 and the boundary conditions (Equation 2-3)

become


OC Pe U) OC Pe aeC
+ (U U)-+ Ue
T ae X 2 Y Y


02C 1 a2C
R- +----2
R +
aX2 F2 ay2


aC
-- +PeUC= 0 atY= 0,1. (2-12)
Y Y
We assume a regular expansion for C in ;,

C=Co +C1 +C2 2 + ................ (2-13)
Substituting the regular expansion for C into Equation 2-11 gives the following

sets of equations and boundary conditions to different orders in s.

(1/82):


PeUe C0
Y BY


02C ac0 PeUeC
-Pe'aC
Oy 2 Oy Y


0 atY = 0, 1


CO = A(X, T)exp(PeU Y)


-* C e C I 2C 1 Pe
Pe(U U) + PeUe -, 1 -Pec
aX y Y Y2 Y
Substituting Co from Equation 2-14 gives


-aA aC
Pe(U- U*) Aexp(PeU Y) + PeU 1
ItX Y 2

Integrating Equation 2-16 in Y from 0 to 1 gives


eC
yC1


02C,
aY2


fU exp(PeU;Y)dY


U' exp(PeU Y)dY


Equation 2-17 gives the average velocity of pulse.


(2-11)


(1/8):


(2-14)


(2-15)


(2-16)


(2-17)











U


where





In Equation 2-16 we assume


6+6exp(a) 12
+
cL


exp(a) -1


a= PeUe
y


8A
C1, =-G(Y)
8X


Pe(U- U)exp(PeU Y) +PeU G
SY


2G
aY2


Solving Equation 2-21 with boundary conditions gives

12(e +aY) 3Y2 6Y2 2Y3
G = Pe( +-- + const)e"
a (e -1) a aO a


and the constraint jGdY = 0 determines the const in Equation 2-22. However, this const
0

does not affect the mean velocity and the dispersion coefficient.

80


cCo
0 + Pe(U
aT


/C
ax


+PeUe a2
SBY


02C,
R +
aX2


a2C,
Y2 '
ay2


-2 PeUeC,
Yy 2


OatY=0,1


(2-23)


Integrating Equation 2-23, using the boundary conditions and using Equation 2-24


< CO >= C0dY = A [exp(PeU )-1]
PeU e
Y


-12 exp(a)
(~)


(2-18)


(2-19)


This gives


(2-20)


(2-21)


(2-22)


(2-24)









gives


a < CO > a2 eCO > [1
S[R -Pe2 j ( (U )G(Y)dY] (2-25)
aT X 2 e 1


Thus, the dimensionless dispersion coefficient D* is

Ue
D* =[R-Pe2 (U-* )G(Y)dY] (2-26)
e -1

Substituting G from Equation 2-22 into Equation 2-26 and integrating gives

D* = R- Pe2(720e a+504e"a2 -24e a4 -144e a3 -6048e2a 504e2a 2 +720e2a + 24e2a 4
-144e2aa + 72e3a 2 -720e3ua+6048e +2016e3u -2016-720a-72a2)/((e 1)3 a6)

(2-27)
Comparison with Brenner's Theory

Howard Brenner used the method of moments to calculate U* and D in FFF with

a shear flow between two infinite plates. To test our method, we solve the same problem

with the perturbation method.

According to Brenner, the equation for mean velocity of pulse is

SG'h(2, a) y7(2, a)
u = and U (2-28)
ay(l, a) ay(l, a)

where, G' is the parameter for v=G'h (v(h) is the velocity profile for shear flow),

a=PeUye, and


y(n + 1, a) = n exp(-)d (2-29)
0

To obtain the effective diffusivity, one needs to solve Equation 2-30,

d dB dB
[exp(-aY)d] = exp(-aY)(U U) (B.C., d= 0@ Y = 0,1) (2-30)
dY dY dY









The result for D* is:

D* R+Pe2 k (a)


kv(a) = 1 Jexp(-aY)(U -U*)dY
1-e 0

Y2 2(eY -aY)
B = ay) + cons tan t
a a(e 1)

After considerable simplification

4 y (2, a) y7(2, a) y 7(3, a)
k ()= [ ][2 +3 ]
a y(1, a) y(1, a) y(2, a)

Comparing Equation 2-32 with our result Equation 2-26, we get


G(Y) = -Pe B. exp(-aY) G(Y) exp(aY)
Pe

substituting it into Equation 2-30 gives


d
-[e
dY


ayG eP" G
". (-.-+--.a.e )]
-Y Pe Pe


e "' (-U*)


_aY BG eY G 2Y G e"
ae (---+--- a- )+e (-
-Y Pe Pe NY Pe


BG e" G 2 ,
+2--Y a--+--a
BY Pe Pe


e -Y(U- U)


(2-37)


Simplifying it gives

02G OG
+ a- = (U U*)Pe.e -" (2-38)
BY2 BY

Comparing Equation 2-38 with Equation 2-21, we find that they are actually the same

(since the direction of lateral velocity in Brenner's model is opposite to that in ours, a


(2-31)


(2-32)



(2-33)


(2-34)


(2-35)


(2-36)






31


should be replaced by -a when comparing these two results). Thus as expected, the two

techniques yield the same results.














CHAPTER 3
RESULTS AND DISCUSSION

Limiting Cases

The dispersion coefficient depends on the Peclet number and U In the limit that

Uy approaches zero, we expect U* and D* to approach the respective value for a 2D

pressure driven flow in a channel without electric field, which are


U*= ; D*=R+ Pe2 (3-1)
210


Uy = 0 implies a = PeUy =0. To check whether our results match Equation 3-1,

we expand our results for D* in the limit of a -> 0. This gives


D = R+Pe2( 1 + 2 89 a4 +O(a)) (3-2)
210 1800 1663200

To leading order, Equation 3-2 reduces to R + pe2, which is the same as Equation 3-1.
210

Also, we expand Equation 2-27 as a goes to infinity. The result is


D =R+pe272 720 2016
D* = R + Pe2(4 + -- ) (3-3)
a a a6

Figure 3-1 shows the curves from Equation 3-3, 3-2, and 2-27. The asymptotic solutions

match the numerical solution if a< 2 or a >8.














0.008
I

0.006


O' 0.004


0.002


0
0 5 10 15 20 25 30
Pe*U:



Figure 3-1. Dependency of (D*-R)/Pe2 on the product of Pe and U The dotted line is
the small a approximation (Equation 3-2), and the dashed line is the large a
approximation (Equation 3-3)


Similarly the asymptotic behavior of U in the limits of small and large a is


21 1 1 1 8
U* = 1 2 -6 a O0 (3-4)
60 2520 100800 3991680
U* 6 12
U 2 a o (3-5)
a a2

The result for U in the limit of a -> 0 also reduces to 1 to leading order in a. Figure 3-2

shows the comparison of these asymptotic results and the numerical results from

Equation 2-18. The small a and the large a results match in the limit of a<2 and a>40,

respectively. These asymptotic results help us in understanding the physics of the

dispersion and the DNA separation, as discussed below.











1.5




S1
C-
C


S0.5




0
0 10 20 30 40 50 60 70 80
Pe* U



Figure 3-2. Dependency of mean velocity U* on the product of Pe and U The dotted
line is the small a approximation (Equation 3-4), and the dashed line is the
large a approximation (Equation 3-5)



Dependence of the Mean Velocity on Uy and Pe


Figure 3-2 shows the dependence of the mean velocity on Uy and Pe. The mean


velocity of pulse depends only on the product of U, and Pe. Uy changes the mean


velocity of the pulse because the presence of an electric field leads to a higher

concentration of the charged particles near the wall of the opposite polarity. The lateral

concentration profile is a balance of the dimensionless electric flux, which is equal to

8C
PeU c, and the dimensionless lateral diffusive flux, which is equal to -- Thus, an
Y OY

increase in either Pe or Ue leads to an increase in the electric flux that has to be balanced


by a larger diffusive flux, which leads to particle buildup in a thinner region near the

wall. Since the particles near the wall access streamlines with a smaller velocity than that









in the middle, an increase in Pe U reduces the mean velocity of the pulse. As discussed

above, in the limit of Pe U approaching zero, the mean velocity approaches the mean

fluid velocity, i.e., U -> 1.

Dependence of D* on Uy and Pe

The effective dispersion coefficient D* depends separately on Uy and Pe.

However, (D* -R)/Pe2 depends only on a, the product of UV and Pe (Figure 3-1). At

small a, with an increase of a, the particle concentration near the wall of opposite polarity

(Y = 1 in our case) begins to increase, and at the same time the particle concentration

near Y = 0 begins to decrease. However, a significant number of particles still exist near

the center. The increase of a results in an average deceleration of the particles (Figure 3-

1), but a significant number of particles still travel at the maximum fluid velocity. This

results in an increase in D*, and a consequent spread of the pulse. At larger a, only very

few particles exist near the center as most of the particles are concentrated in a thin layer

near the wall, and any further increase in a leads to thinning of this layer. Thus, the

maximum velocity of the majority of the particles goes down, resulting in a smaller

spread of the pulse. Finally, as a approaches infinity, the mean particle velocity

approaches zero, and the dispersion coefficient approaches the molecular diffusivity.

Figure 3-1 shows that the maximum value of (D* -R)/Pe2 is about .007. This implies

that the convective contribution to dispersion is at most .007 Pe2. Thus, even at Pe = 10,

the convective contribution is only about 35% of the diffusive contribution R, which is

approximately equal to 2.









Separation of DNA Fragments of Different Lengths

DNA fragments of different lengths have the same Uy because the total charge on

the molecule is directly proportional to the length (the charge is from the phosphoric

structure), and the diffusion coefficient is inversely proportional to the length. Thus, a

pure axial field cannot separate DNA molecules in free solution. However, as shown

above, in the presence of lateral fields, the mean velocity of the molecules depends on the

product of Uy and Pe, where


hDZe OP he 8(
Pe Ue =__ > Z (3-6)
Y D ktTt By ktTt ay


In Equation 3-6, ,Tt, and h are fixed for all the DNA molecules. Thus, the
By

product in Equation 3-6 only depends on Z, which is directly proportional to the length of

the DNA fragments. Since charge Z, and consequently U* and D* are different for

molecules of different sizes, a mixture of DNA fragments of different sizes separates into

bands that contain same size DNA molecules, and these bands travel with their mean

velocity and disperse as a Gaussian with the dispersion coefficient corresponding to their

chain length. Thus, we can separate DNA strands according to their sizes by applying a

lateral field instead of an axial field. Figure 3-3 shows the separation of a pulse

containing two types of DNA molecules into two individual peaks as the molecules

traverse the channel. At t = 0, a 1:1 mixture of two types of DNA molecules is

introduced as a pulse at the channel entrance. For this simulation, the ratio of the

diffusion coefficients of the two types of molecules is 2, and all the other physical









constants are given in the caption. Figure 3-3 shows that, as time progresses, the DNA

molecules separate into two separate Gaussian distributions.


Pulse introduced at t=0

S1 e
0
,-- ]^-------

0.8

0.6
CU l
0
a tAt=t --

0.2 -=t2


0 1.25 2.5 3.75 5
Length x103m


Figure 3-3. Separation of a 1:1 mixture of DNA strands of different sizes into two
separate Gaussian peaks. =lmm/s, D1 = x109, D2 2x109, U =1,
Pei=10, h= 10 im


Separation Efficiency

Consider separation of two types of DNA molecules in a channel. We assume that

when the distance between two pulse centers is larger than 3 times of the sum of their

half widths, they are separated, i.e.,


(-Ui* )t> 3( 4DD t+ 4D D*t) (3-7)

where the subscripts indicate the two different DNA fragments. If the channel is of

length L, the time available for separation is the time taken by the faster moving species

through the channel, i.e., L /max(Ui, u). Substituting for t, and expressing all the

variables in dimensionless form gives











L> 1 --, D1 2 (3-8)
[1D
L/h 36 --max(U,1,U2)[ -. 1 2 (3-8)
PeC U2 -Uj


In the discussion below, we used L/h to indicate the efficiency of separation, i.e., smaller

L/h implies more efficient separation.



1500



Pe=5
1000



Pe=9
500

Pe=10

0 1
0.5 1 1.5 2 2.5 3
UY



Figure 3-4. Dependency of L/h on Uy and Pe for separation of DNA strands of different
sizes. Uy, = U2 = U Pei = Pe, and Pe/Pe2 = D2/D1 = 10


In Figures 3-4 and 3-5, we show the dependence of L/h on Pe and Ue in the case


of U, = U2 which corresponds to DNA fragments of different lengths. Figure 3-5 is


similar to Figure 3-4; the only difference is the ratio D2/D1. Figure 3-5 shows that

increasing U which is physically equivalent to increasing the electric field, leads to a


reduction in L/h required for separation. As Pe Uy increases, the mean velocities of both


kinds of molecules decrease (Figure 3-1). But the dispersion coefficients do not change






39


significantly because they are very close to the diffusive value of R for small Pe(Pe<10).


Thus, L/h is primarily determined by the ratio U2 -
Pe 1 U2 -U


0.5 1 1.5 2 2.5 3


Figure 3-5. Dependency of L/h on U' and Pe for separation of DNA strands of different
sizes. Ui = U2 = Uy, Pe = Pe, and Pe/Pe2 = D2/D = 2


1
As shown earlier, in the small a regime U* 1 a2, thus,
60


P,-U ; U P U 12 P )
Pe U2 1 U1 Peu) (Pe22 Pe2)2


. Since the ratio Pe2/Pei is fixed,


e 1 ~ Pe, (ue Thus, an increase in either Pe or Ue leads to a
Pe U Y Pe2 -Pel2

reduction in L/h in the regime of small a. The constant Pe plots in Figure 3-4 and 3-5

show the (U ) dependency when Ue is small. Also, the constant Pe curves shift down
Y Y









with increasing Pe, due to the Pe5 dependency shown in the above scaling. On the other

side, as shown earlier, in the limit of large a U* 6/a, thus,

U* 1 U Pe Pe .
2 2
~ Pe Pe2 Ue. This implies that in the large a regime and
PeI U ;-U PelPe2 PeY- Pej

at 0(1) Pe, L/h becomes independent of Pe and begins to increase with an increase in

U as shown in Figure 3-4.


Since L/h scales as (U) 4 in small a regime, and as Ue in the large a regime, it

must have a minimum. The minimum in L/h is clearly visible in Figure 3-5. In Figure 3-

4, the minimum occurs for slightly larger Ue than shown in the Figure 3-5. Physically,

the minimum arises because at small field strength, the molecules accumulate near the

wall, but a finite thickness of the region of accumulation still remains. Since the thickness

of the region is different for the two types of molecules, the mean velocities of the two

types of molecules differ. However, as the field strength becomes very large, both the

mean velocities approach zero, and thus their difference also approaches zero.

Subsequently, the difference in the mean velocities is zero for zero field because both the

mean velocities are equal to the fluid velocity, and is also zero at very large fields

because both the mean velocities approach zero; this implies that a maximum in the

difference between the mean velocities of the two types of molecules must exist at some

intermediate field. This maximum combined with other secondary effects results in a

minimum in L/h required for separation.











1000


800


600 ,e=l


400


200 PQ=3 Pe=5 Pe=10


0
0.5 1 1.5 2 2.5 3

U,



Figure 3-6. Dependency of L/h on Uy and Pe for separation of molecules of the same
size but different charges. Ue = U y, U e /U = 10, and Pe= Pel = Pe2,
i.e., D2 = D1


The effect of changing Pe while keeping Uy fixed is more difficult to understand


physically. Due to the dedimensionalization of U the only way to change Pe while


keeping Uy fixed is to increase the fluid velocity and the field by the same factor. As a

result, if we want to verify the effect of only an increase in the mean velocity , we

need to increase Pe and concurrently reduce U Thus, we first move to the smallerUe


value and then follow the larger Pe curve. This keeps Pe U and consequently D* and


U* unchanged, and thus, L/h 1/Pe. Physically, this inverse dependency on the mean

fluid velocity arises because the dimensional mean velocity of the molecules depends

linearly on . Thus, an increase in results in a linear increase in the difference










between the mean velocities of the two types of molecules, i.e., u, u*. The distance

between the peaks at the channel exit is independent of because although

u, u increases linearly with , the time spent by the molecules in the channel varies

inversely with . However, because D*'s do not change with changes in only , the

spread of each of the Gaussians decreases with an increase in due to the reduction of

time spent in the channel. Consequently, the spread of the peaks becomes smaller making

it easier to separate the two types of DNA.


1000


800


600


400


200


1 1.5 2 2.5 3


Figure 3-7. Dependency of L/h on Uy and Pe for separation of molecules of the same
size but different charges. Ue = U Ue /U = 2, and Pe=Pel = Pe2, i.e.,
D2 = D1


In Figure 3-6 and 3-7, we explore the separation for particles having same

diffusivity but different charges. By comparing them, we find that under the same Pe and









Uy,, larger Uy2 /Uy results in better separation. This result is similar to the effect of an

increase in D2/D1 shown in Figures 3-4 and 3-5. Furthermore, all the trends discussed

above for the effect of Pe and Uy on L/h for separation shown in Figures 3-4 and 3-5

persist in Figures 3-6 and 3-7 because the arguments presented above are valid even

when the Peclet numbers are the same for the two types of molecules and their Uy are

different. Thus, an increase in Pe for fixed Uy reduces L/h, and an increase in Ue for a

fixed Pe first reduces L/h at small a (a=Pe U ), and then increases L/h at larger a

resulting in a minimum.

Comparison of Lateral and Axial Electric Field

As shown above, a lateral field can be used to separate particles in instances in

which axial fields are ineffective because the ratio of the charge to the viscous resistance

is the same for all the molecules. In this section, we wish to compare the effectiveness of

lateral fields with axial fields in cases in which pure axial fields can result separation, i.e.,

in cases where the ratio of charge to viscous resistance is different for the molecules that

need to be separated. As a special case, we consider two kinds of particles with equal

and isotropic diffusivities, and Z2 = 2 Z1. In this case, the two types of molecules have the

same Peclet number but different lateral electric velocities: U2 = 2Uy1 and U2 = 2U .

Furthermore, axial electric fields simply alter the mean velocity of the molecules without


affecting the dispersion coefficient of molecules, i.e., U* = 1+U and D = 1+ Pe2.
210

We adopt a same definition for the length of the channel needed for separation for

the axial fields. Thus, we get










1 max(u, ) 21+Pe2/210 1 2- 1+Pe2/2102
L/h >36 [ = 36-max(U,,U2)[ ]
Pe U -U1 Pe U2 -U


(3-9)

Figure 3-8 plots the L/h required for separation in the above case as a function of

Pe and UI The L/h required for separation decreases with an increase in Ue because

an increase in electric field strength increases the difference between the mean velocities

of the two kinds of molecules without affecting their dispersivities. At small Ue,


Equation 3-9 gives L/h (U ) and, at large Ue, it gives L/h (Ue) I. Also, at small

Pe, L/h ~ 1/Pe, and, at large Pe, L/h~Pe.


1000


800


600


400


200


0


1 1.5 e 2 2.5
Ux


Figure 3-8. Dependency of L/h on Ue and Pe for an axial field for separation of
molecules of the same size but different charges. Pe = Pel = Pe2, Ue = e
and Ue2 /U~ = 2









To compare the axial and the lateral fields, we compute the ratio of the lateral and

the axial electric fields that result in the same difference in mean velocity of two

molecules with Z2 = 2Z1. For a pure axial electric field


S(ui, -u)ktTt (U,, -u )ktTt
(3-10)
9x ~ D(Z2 -Z,) D(Z2 -Z) (3-10)

For a pure lateral field


.0C u kT
= uykt (3-11)
Oy DZ1


where u is the electric velocity required to obtain the mean velocity difference of

u1 u2. This relationship cannot be expressed analytically but is shown graphically

in Figure 3-9. Dividing Equation 3-11 by 3-10 gives


aV / mI (Z2 Z)uy (Z2 -Z,) Peuye (Z2 -Z,) PeUyi
[ ] [ ] (3-12)
y /x Z,(-ut -u ZPe (- u ) ZPe (-U -U-)


PeUe
From Figure 3-9, the minimum value of 1 is about 14. This means that if
(U U2)

Z2 = 2 Z1, in order to achieve the same velocity difference, the ratio of lateral electric

field to axial field must be more than 14-. Thus, lateral fields could be more effective
Pe

than axial fields above Pe = 14. Furthermore, at large Pe, D* in lateral fields is smaller

than the D* in axial fields, which further adds to the effectiveness of the lateral fields.

However, a lateral electric field always reduces the mean velocity of charged particles

making it less than the mean velocity of fluid flow. Therefore, the maximum velocity










difference between different kinds of charged particles is less than the mean velocity of

the flow. An axial electric field, on the other hand, does not have this limit. Figure 3-10

shows the dependency of (u u1) on Ue for the axial fields (dash-dot line) and on Uy


for the lateral fields (solid lines; each line corresponds to a different Pe). In the region to

the right of the dashed line, the axial fields are more effective. This region corresponds

to Pe < 14 but only for (u U) < 0.25. Using axial fields is the only way to achieve

mean velocity differences larger than 0.25. Further research shows this critical velocity

difference will increase with an increase in Z2/Z1 and approach 1 as the ratio approaches

infinity. As a result, in certain cases lateral fields could be more effective even in

instances where axial fields can also accomplish separation.





0.3

0.25

0.2

S0.15

0.1

0.05

0
0 --------------
0 5 10 15 20 25 30
PeU



Figure 3-9. Dependency of the difference of mean velocities of pulses of two kinds of
molecules(Z2=2 Z1, Pe2/Pe1 = 2) on the product of Pe = Pei=2 and Ue
(Uy =














0.3 ,Pe=2
i/ ,, Pe=14 Pe=6



0.2




0.1




0
0 1 2 3
U; orUx



Figure 3-10. Dependency of the difference of mean velocities of pulses of two kinds of
molecules on an electric field. The dash-dot line is for the axial electric
field and solid lines are of the lateral electric field for different Pe. In these
plots Pe2/Per = 2


Comparison with Other Promising Methods for DNA Separation

Entropic trapping has been successfully employed to separate a mixture of 24.5

kbp, 48.5 kbp, 73 kbp and 97 kbp DNA fragments in a 1.5 cm long channel in about 40

minutes[42,43]. Our simulations show that the same separation can be accomplished in a

80 |tm channel by our proposed method. The operating time is about 1.5 minutes. The

parameters for the two methods are listed in Table 3-1. Table 3-1 also shows the

comparison of the proposed methods with the technique proposed by Doyle et al. that

uses a self assembled matrix of magnetic particles as a sieving media. The authors do not

report the channel length in the paper, but the time needed for separation is longer than









that for our proposed techniques. These comparisons show that our technique is

promising.

In our model we have not taken into account the entrance and exit effects, and thus

the length of the channel required for separation will actually be longer than the results

shown in Table 3-1. In this situation, we propose that the electric field is only applied in

the fully developed region and thus the mean velocity of the molecules at the entrance

and the exit will be the same as the fluid velocity, and the region in between where the

fields are applied will contributes to separation. There are other factors that may reduce

the separation efficiency of our technique such as the extra dispersion caused by the

effect of the walls in the third direction. Also, in our model the thickness of the region in

which the molecules reside near the wall of the opposite polarity scales as D/ U For the

longest DNA molecules, the thickness of this layer is about 10 nm. Clearly,

accumulation of molecules in such a thin region will be affected by the double layers

surrounding the DNA molecules, and this may impact the separation efficiency. Thus we

believe that our current model only serves as a guide in designing the best separation

strategy. However, the results of our simulations show that EFFF in microchannel is

certainly a promising technique for separating DNA fragments.


Table 3-1 Comparison of EFFF with other techniques
sample (kbp) method Total time (min) Length of channel
15,33.5,48.5 Continuous method a) 5 126.2 pm
magnetic 15 Not mentioned
24.5,48.5,73,97 Continuous method b) 1.5 80 pm
Entropic trapping 40 1.5 cm
a) h=l Ifm, Uy =0.3, =0.1mm/s
b) h=l Ifm, Ue =0.2, =0.1mm/s
y














CHAPTER 4
DNA SEQUENCING ATTEMPT

To read the genome of an organism, chromosomes, which range in size from 50

million to 250 million bases, are broken into much shorter pieces, about 500 base pairs in

length. In the Sanger process, each of the smaller pieces are primed for replication and

then added to four beakers, each containing all the four bases A, T, C and G needed for

replication. However, in each beaker, a fraction of one type of nucleotides is 'defective';

the replication stops at these nucleotides. Each replication reaction then proceeds until a

reaction-terminating nucleotide is incorporated into the growing strand, whereupon

replication stops. Thus, in a beaker containing a 'defective' A, the length of the replicated

fragments corresponds to location of T. The last step of the sequencing is then separation

of these fragments, which differ in length by only one base pair, and this is accomplished

by gel electrophoresis.

We therefore seek to determine the efficiency of our system at sequencing DNA.

To sequence a fragment N base pair long, we need to separate a mixture of bases of

lengths varying from 1 to N. In Figure 4-1 we plot the L/h of a channel required to

separate a DNA fragment n bases long from a fragment n+1 bases long. To calculate L/h

(n) we use Equation 3-8 with a slight modification; the factor 36 is replaced by 16

because a spacing of two times the sum of the half widths between two Gaussian peaks is

enough to identify them as separate peaks, but we need a spacing of about three times to

separate them. In Equation 3-8, we identify species 1 as the nth fragment and species 2 as










the (n+l)th fragment. Thus, D /D = n /(n + 1) and Pe = nPe, where Pe is the Peclet

number for a single nucleotide.


6
2 x 10
2.5






1.5
1-
S/




0.5 -



0 100 200 300 400 500
Number of base pairs


Figure 4-1.


L/h required to separate DNA fragments that differ in length by a single base
pair vs. the number of base pairs in the smaller fragment. h =1[tm, Pe = 1,
Uy = 0.4. The largest value of L/h represents the size of the channel required

for separation. The dashed line is calculated from the large ac
approximations.


Figure 4-1 plots L/h required for separation as a function of the maximum number

of the base pairs n for fixed Pe and U For a given Pe and Uy (h = l[tm, Pe = 1,


Uye=0.4), L/h first decreases because of an increase in Pei and then begins to increase

because the effect of an increase in D2/D1 dominates over the effect of an increase in Pe.

Since the Peclet number will proportionally increase with the increase in the number of

bases, it is easy to reach large values of the product of the Peclet number and U .


Usually separation of the two largest fragments, i.e., 500 and 499 base pair long









fragments, is the hardest. Therefore, to optimize the sequencing, we will first focus in the

large a regime. Using the large a approximations gives L/h n2 U (In this limit D* is

equal to D). The dashed curve in Figure 4-1 is the large Pe asymptotic result. Plotting the

L/h vs Uye on log-log axes gives a slope of 2, which agrees with above analysis. The time

required for separation of fragments in DNA sequencing will be L/ U (n), where

U (n) is the mean velocity of the slowest moving, i.e., the longest, DNA fragment.

Large a approximation gives T n3 Pe (u)2 Thus, one could accomplish faster

separation by reducing Pe and U However, these expressions are based on large a

approximation, so Pe and Ue cannot be made too small. In fact, ifPe U is too small,

the diffusivity of the DNA molecules begins to increase (Figure 3-2), resulting in long

operating time and large required channel length. Specifically when both Pe and Uy are

reduced, the length needed to separate the first several base pairs increases dramatically

because the difference in velocities between the first few fragments becomes small.

Therefore, to get the best performance, we need a large Pe to limit the diffusion of large

molecules (according to Equation 3-3). However, we cannot make Pe arbitrarily large

because doing so will increase the pressure required to pump the fluid. Note that making

Pe large by increasing h will also increase L. Therefore, we essentially have an

optimization problem in which we need to minimize L, t and P by manipulating , h

and U To guide us in this optimization we use the large a results, i.e.,


L/h n2Ue (3-13)

t n3U2h2 (3-14)
t Uy










Furthermore, for 2D Poiseuille flow


(3-15)


According to Equation 3-15, we choose a relatively large h to provide large

and Pe. At the same time, we reduce Uy to balance the effect on L and t of increasing h.


After h and Uy are fixed, we choose a Pe that is large enough to control the diffusivity.


Note that we use the large a approximation only as a guide, and use the appropriate

equations to finally determine L/h and t. We finally conclude that 1-500 base pairs can

be separated in a 0.72-meter long, 10-micron thick channel in 1.7 hrs at Pe = 200 and

Ue = 0.01. Figure 4-2 shows this result.
Y


0.8



0.6



1)-4
0.4


0.2
0.2


100 200 300 400 500
Number of base pairs


Figure 4-2. Length and Time required to separate DNA fragments that differ in length by
a single base pair vs. the number of base pairs in the smaller fragment. h =
10tm, Pe = 200, Uy = 0.01, =0.02m/s
y


P [t
h2






53


The length of the channel is about 0.72m, which is too long for fabrication on a

chip. However, one could potentially fabricate about 36 parallel channels, each about 2

cm long, and join them at the ends to fabricate a channel with straight segments joined by

curved ends. In such a channel, we need to evaluate the extra dispersion caused by the

curved ends.














CHAPTER 5
CONCLUSIONS

Application of lateral fields affects the mean velocity and the dispersion coefficient

of colloidal particles undergoing Poiseuille flow in a 2D channel. The dimensionless

mean velocity U* depends on the product of the lateral velocity due to electric field

Dze 8D
U = and the Peclet number. The convective contribution to the
ktTt ay

dispersion coefficient is of the form Pe2f(PeUe). The mean velocity of the particles

decreases monotonically with an increase in U Pe, but (D* R)/Pe2 has a maximum at

a value of U Pe 4. This maximum arises when the thickness of the region near the

wall where a majority of the particles accumulate is about h/2.

Since the mean velocity of the particles under a lateral field depends on the charge

Z but not on the product of the diffusion coefficient and the charge, colloidal particles

such as DNA molecules that have the same ratio of charge to viscous resistance can be

separated on the basis of their lengths on a chip by applying lateral electric fields. Axial

fields cannot accomplish this separation unless the channel is packed with a gel. Thus,

our proposed strategy of accomplishing separation by lateral fields may offer a solution

to separating DNA on a chip. The length of the channel required for separation depends

on the ratio of the diffusion coefficients of the two types of molecules that need to be

separated and on the Pe and U Lateral fields can also be used to separate molecules

that can already be effectively separated by purely axial fields, and, in certain instances









such as large Pe, the lateral fields require smaller field strengths than the axial fields.

However, lateral fields are limited by the fact that the maximum difference in the mean

velocity of two types of molecules that need to be separated is less than the mean velocity

of the fluid. Axial fields do not suffer from this limitation.

Lateral fields can also be used for sequencing DNA on a chip. The separation step

in DNA sequencing requires separating a mixture of DNA fragments that range in size

from 1 to 500 base pairs and differ in length by a single base pair. Lateral fields can

accomplish such a high-resolution separation in channels that are about 0.72m long.

With current micro lithographic techniques, such a channel can be incorporated into a

chip by fabricating about 36 parallel channels, each 2 cm long, and eventually joining

them to acquire a single channel. In such a channel we will need to consider the extra

dispersion caused by the curves that link two successive straight channels.

While analyzing the dispersion of molecules under lateral electric field, we

assumed that the fluid away from the double layer is electroneutral, that the electric field

is constant in the bulk of the fluid, and that DNA molecules are fully charged and aligned

parallel to the flow. Also, we neglected the double layer at the electrodes, the interaction

of the molecules with the wall, and the reduction in mobility due to the counterion cloud

surrounding the charged colloidal particles. We also presented our analysis for a 2D

channel, where the presence of walls has demonstrated a first order effect on the

dispersion of molecules in Poiseuille flow in a channel in the absence of electric fields.

In addition, the flow of current in the lateral direction will result in generation of oxygen

and carbon dioxide that may affect the hydrodynamics of the flow. The DNA molecules

may also be affected and possibly damaged by the high lateral fields. Thus, while our






56


proposed separation technique is promising, more experimental and theoretical work

needs to be done to determine the effectiveness of lateral fields in accomplishing

separation of DNA on a chip.














APPENDIX
NOMENCLATURES

A Variable defined in Equation 2-14

c Concentration of a kind of particle in fluid

D Molecular diffusion coefficient

D* Dimensionless effective diffusion coefficient

G Variable defined in Equation 2-20

h Height of the channel

kt Thermal constant

kv Function defined in Equation 2-31

1 Width of the pulse as it exits the channel

L Length of the channel

P Pressure

Pe Peclet number, h/D

R Ratio of D to Di

t Time

T Dimensionless time

Tt Temperature

u Velocity of flow at a certain position and time

u* Mean velocity of a pulse consisting of one kind of particle

Mean velocity of flow






58


U Dimensionless velocity of flow

U* Dimensionless u*


u Velocity of a charged particle in the x-direction due to electric field

Ue Dimensionless uX

uy Velocity of a charged particle in the y-direction due to a lateral electric field

U Dimensionless uy
Y Y

x Position on the x-axis

X Dimensionless position in the x-axis

y Position on the y-axis

Y Dimensionless position in the y-axis

Z Number of charges of a particle

a Variable defined in Equation 2-19

y Function defined in Equation 2-29

() Intensity of an electric field

E Perturbation, the ratio of h to 1, or the square root of h/L

(t Viscosity of the fluid















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BIOGRAPHICAL SKETCH

I was born in Apr 23, 1976 in He Zhang, a small town in Gui Zhou province,

China. My father is a teacher of physics in a high school and my mother is a doctor. I

established strong interest in science since childhood due to the intellectual surrounding

provided by my family. My wide region of reading earned me honors in various

competitions of high school. With competitive scores in the National Entrance

Examination, I was admitted by the most prestigious university of China-Tsinghua

University. I urged myself in my undergraduate study in Tsinghua University, took five-

year courses in four years and got high scores in most courses. As a result, I graduated

one year earlier than my peers, ranking top 5% in my department of 120 students and

entered the graduate program of Biochemical Engineering in 1998, waived of the

entrance examination. In graduate stage, I ranked 10% in my class.

During seven years in Tsinghua University, I participated in several projects. In

my undergraduate diploma project, I studied the measurement of solubility of sodium

sulfate in supercritical fluid, which is a part of the research of SuperCritical Water

Oxidation (SCWO), a promising method for dealing with wastewater. Deeply absorbed

in this wonderful supercritical world, I searched the literature extensively, discussed with

professors, and did experiments carefully. Finally, I got satisfactory results, and my

diploma got a high score-92/100.

In 1999, I took part in a project to undertake middle-scaled amplification of the

production of PHB (poly- 1 -hydroxybutyrate, a kind of biodegradable plastic) with









E.Coli., which was a part of a Ninth Five-year National Key Project of China. Under my

active and successful participation, we found and eliminated the scattering of nitrogen

during sterilization and improved the distribution of air input. The density of bacteria

reached 120g/l and the production of PHB extended to 80g/l, far beyond the original goal.

The amplification succeeded and won me the honor of the first prize of outstanding

performance in field practice of my department.

After the practice, I began my thesis work under the guidance of Prof Zhongyao

Shen, the Vice-Dean of School of Life Sciences and Engineering in Tsinghua University.

My work focused on the coupling of fermentation and separation. In the first year, I

applied the coupling of fermentation and ion exchange on the production of 2-Keto

Gulonic Acid, the direct precursor of Vitamin C. However, this research was abandoned

because an impossibility coming from the fermentation system. After that, my main

interest was on the coupling of fermentation and membrane separation in the production

of acrylamide from acrylotrile. During the process, I acquire insights on membrane,

fermentation, ion exchange, and operation of analytical equipment. Finally, I got a high

enzyme activity from the fermentation, which is the highest value on documents.

After I graduated from Tsinghua University in 2001, I came to the Department of

Chemical Engineering, University of Florida to pursue advanced education. My research

focuses on separation processes with microchannel and electric fields. And we have

already got some promising results. This thesis is some of my research results. After I

get master degree, I will further my work on this project, such as doing experiments to

testify our theoretical results in this thesis.

Below is a list of my published papers.






65


1. Zhi Chen, Xudong Sun,Yue Shi,et al. Study on Production of Acrylamide by
Microbial Method ( I )-Culture of bacterium cells and expression of high
activity of nitrile hydratase. CHINESE JOURNAL OF BIOTECHNOLOGY,
2002 Vol.18 No. ,p55
2. Zhi Chen, Xudong Sun, Yue Shi, et al. Study on Production of Acrylamide by
Microbial Method ( II )-Enzyme catalytic kinetics and de-active dynamics of
nitrile hydratase. CHINESE JOURNAL OF BIOTECHNOLOGY, 2002 Vol.18
No.2, p225
3. Xiang Botao,Wang Tao,Chen Zhi,et al. The Solubility of Sodium Sulfate in
Supercritical Water. CHEMICAL ENGINEERING, 2001 Vol.29 No. l,p72
4. Sun Xudong, Chen Zhi, Shi yue, et al. Studies on Bioprocess and Bioreactors
Used in Bioconversion for Acrylamide. CHEMICAL INDUSTRY AND
ENGINEERING PROGRESS, 2002 Vol.21 No.5, p319