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His indepth knowledge in computational geometry and mathematics helped me a lot in completing my research. His working style and philosophy were also been beneficial to me. TABLE OF CONTENTS page ACKNOWLEDGMENT S ............ ...... ._._ ...............3.... LIST OF FIGURES .............. ...............6..... AB S TRAC T ......_ ................. ............_........7 1 INTRODUCTION TO THE PROBLEM OF HELIX PACKING ................ ............... .....8 M otiv ati on ................. ...............8............ .... Literature Review .................. ..... ... ..... ........ Problem Description, Definitions and Assumptions .............. ...............10.... 2 REVIEW OF BIINCIDENCE FEASIBILITY REGIONS SOLVING IN HELIX PACKING ................. ...............13................. Review of Helix Packing via Constraint Solving ................. ............. .. ............. .....1 Review of Biincidence Feasibility Regions Solving in Angle Approach .............................14 Parametrization of a Dumbbell Sliding on a Fixed Dumbbell (Solving for # and ly)............... ...............15.. Moving a Helix to a Position of Its Dumbbell ................. .......... ... .....................17 Helix Hinge Motion and Intersection (Solving for a and P) ................... ...............17 3 IMPLEMENTATION OF BIINCIDENCE FEASIBLITY REGIONS SOLVING ALGORITHEM ................. ...............20................. Implementation Environment .............. ... ........... ...............20....... Overview on the Structure of Implementation ................ ........ ......... ................20 Implementation: Phase 1............... ...............20... Solving # ............. ...............21..... Solving ly ............... .... .. ........ ........2 Translate the Whole Helix to Make Sure the Chosen Dumbbell Lies in the Configuration ................ ...............22................. Solving a and P Pair .............. ...............23.... User Specified Dumbbell Solving ................. ......... ...............24. .... Output the Result ................. ...............24................ Implementation: Phase 2............... ...............25... 4 CONCLU SION............... ...............2 Effectiveness of BiIncidence Feasibility Regions Solving ................. ........................28 Implementation of the Algorithm ................. ...............28....___ ..... Challenge and Future Development .............. ...............28.... APPENDIX INPUT HELIX DATA ................. ...............30................ LIST OF REFERENCE S ................. ...............3.. 1......... ... BIOGRAPHICAL SKETCH .............. ...............32.... LIST OF FIGURES FiMr page 11 The a helix ........... ....... ...............8.... 12 A helix is a rigid 3D obj ect consisting of a set of m spheres (atoms) in fixed positions. ................ ...............11................. 13 General case and a special case of incidences between 2 helices ................. .................1 1 22 Biincidence of a pair of dumbbells and the parametrization ................. ........._.._. ......15 23 Sphere intersection when rotating about the hinge ........._..._. ....._... ........_..._.....19 31 Structure of output file ........_................. .........._. .......2 32 Line segment to represent the a region ................._..._.._ ...............25... 33 Sample output of Phase 2 with a regions............... ...............26 34 Sample output of Phase 2 without a regions ................. ....._._. .................2 Abstract of Thesis Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Master of Science IMPLEMENTATION OF BIINCIDENCE FEASIBILITY REGIONS IN HELIX PACKING VIA CONSTRAINT SOLVING By Zhongjie Li August 2008 Chair: Jorg Peters Major: Computer Engineering By introducing computational geometry and computer science methods to solving the helix packing problem in biochemistry, we can assist the research. An algorithm of helix packing via constraint solving is being explored by the researchers at the University of Florida, which is based on a parametrization of the configuration space. As the first step of the algorithm, an angle based approach has been brought out to parametrize the first two incidences between two helices and therefore locate the third incidence. The algorithm successfully reduces our search space to fourdimensional space. Later on, the algorithm has been implemented in C++ in two phases. Phase 1 processes the input helix data with the algorithm then outputs the result as a text file. Phase 2 takes the output of Phase 1, and displays it with OpenGL. CHAPTER 1 INTRODUCTION TO THE PROBLEM OF HELIX PACKING Motivation Since different structure aspects of proteins have been discovered by biologists in the mid 20th Century, scientists started to focus on revealing the essential mechanism of the transitions between these structure forms, such as folding that enables proteins to be able to perform their biological function. Fig. 11 shows an alpha helix which acts a critical role in understanding helix packing, a type of protein folding. However, because experimentally determining the three dimensional structure of a protein is often very difficult and expensive, scientists started to research on the possible ways of making predictions on the protein structure. Figure 11. The a helix. (Source: http://www.uic. edu/classes/bios/biosl100/lecturesf04am/lect02 .htm Last accessed July, 2008). A number of factors exists that make protein structure prediction a very difficult task. The two main problems are that the number of possible protein structures is extremely large, and that the physical basis of protein structural stability is not fully understood. As a result, any protein structure prediction method needs a way to explore the space of possible structures efficiently (a search strategy), and a way to identify the most plausible structure (an energy function). Thus predicting the protein structure accurately remains a big challenge today. In atomistic level, helices are often modeled as rigid 3D obj ects. Hence due to their geometric nature, Computational Geometry methods can be introduced into this Hield. However, because of the complex structure of the helix, brutal force collision detection is often used in exploring the packing space. As a tradeoff, this method consumes high computational power. In contrast to improving the computation capacity of computers, scientists are also seeking more efficient algorithms to solve the packing space with constraints. Literature Review The packing of transmembrane helices was modeled at the residual and atomistic levels, respectively. For predictions at the residual level, the helixhelix and helixlipid interactions were described by a set of knowledgebased energy functions. For predictions at the atomistic level, the CHARMM19 force Hield (Chen and Xu, 2005) was employed. In 2002, Fleishman and BenTal developed a scoring function and a computational methodology for predicting the tertiary fold of a pair of alpha helices such that its chances of being tightly packed are maximized. The scoring function was constructed based on the qualitative insights gained in the past two decades from the solved structures of transmembrane (TM) and soluble proteins (Fleishman and BenTal, 2002). Basically, they reward the formation of contacts between small amino acid residues such as Gly, Cys, and Ser, that are known to promote dimerization of helices, and penalize the burial of large amino acid residues such as Arg and Trp. Thus in computation, they can maximize the number of contacts between residues. This kind of solution is often called knowledgebased approach because of using known contact potentials, and the packing is modeled at the residual level instead of atoms level. A similar solution has been introduced in 2007 which uses coarsegrained knowledgebased potentials to score the mutual configuration TM helices (Wendel and Gohlke, 2008). Direct simulation of protein folding in atomic detail, via methods such as molecular dynamics with a suitable energy function, is also a promising direction of the research. Because of the high computational cost of it, distributed computing technology has been introduced into this field. In 2000, a distributed computing proj ect designed to perform computationally intensive simulations of protein folding and other molecular dynamics has been launched by Stanford University. The proj ect is named as Folding@home and with the computational power volunteered by thousands or even millions private computer users across the globe (Shirts and Pande, 2000). It is one of the most powerful distributed computing clusters in the world. The goal of Folding@home is aimed on understanding protein folding, misfolding, and related diseases. With the assistance of the proj ect, a number of achievements have been reached till today, such as the simulation of folding of a small a helical protein in atomistic detail (Zagrovic et al., 2002) and atomistic protein folding simulations (Pande et al., 2003). Problem Description, Definitions and Assumptions In order to explore the space of possible structures in helix packing more efficiently, we are seeking for an algorithm with formal guarantees for sampling the configuration space of 2 packed helices with collision avoidance constraints by introducing computational geometry and other computer science technologies into this field. In our problem, according to the assumption, helix is a rigid 3D obj ect consisting of a set of m spheres (atoms) in fixed positions (Fig. 12), and the spheres can be intersected with others in the same helix. But in the process of packing, the spheres of different helices must not intersect. This collision avoidance is called packing constraint. It increased the difficulty of our problem due to the nonlinear feature it brought. A configuration is defined by the relative position and orientation of 2 helices for a fixed position of the first helix. Hence a configuration has 6 independent degrees of freedom, thus can be viewed as a point in 916 Space. If a configuration satisfied packing constraint, then we call it a packing configuration. A configuration of 2 helices is an extreme configuration if its incidences rigidly Eix one helix to the other one. Generically, 6 incidences fix two helices respect to each other, but in some specific cases, 6 incidences may not be enough to form an extreme configuration (Fig. 13). Here we will mainly focus on the general case. Figure 12. A helix is a rigid 3D object consisting of a set of m spheres (atoms) in fixed positions. (a) (b) Figure 13. General case and a special case of incidences between 2 helices. Doted lines represent the connections within a helix. Solid lines represent the connections between helix. (a) is the general case where 6 incidences can fix the configuration of 2 helices. (b) is a special case where although 6 incidences happened, the configuration is still not fixed because right helix can rotate about the hinge of the connection between 2 atoms in left helix. (Source: Meera Sitharam, Heping Gao, Jorg Peters. 2007. Helix packing via Constraint Solving). Our algorithm should contain every extreme packing configuration in the sample result, or a configuration arbitrarily close to it (within an input desired precision). For other configurations, we should be able to sample them with any desired (user input) precision. On the other hand, one of the noticeable drawbacks of existent algorithm on exploring the feasible regions of helix packing is that it will sample one extreme configuration multiple times. As a main factor in increasing the efficiency, our algorithm should outputs each sample configuration exactly once or at least same number of times (constant times). In the following chapters, we will first introduce the algorithm developed by a group of researchers at the University of Florida. Then we will focus on the first step of the algorithm and an implementation of it. CHAPTER 2 REVIEW OF BIINCIDENCE FEASIBILITY REGIONS SOLVING IN HELIX PACKING Review of Helix Packing via Constraint Solving In early 2007, the idea of exploring the feasible regions of helix packing via constraint solving was explored by Dr. Meera Sitharam, Heping Gao, and Dr. Jorg Peters at the University of Florida. In their algorithm, the incidences between 2 helices will be parametrized, and based on this parametrization, the next incidence can be located in a range described by one or a set of equations achieved by applying collision detection between spheres. The algorithm will start from the parametrization on 2 incidences, then by recursively applying the principle, the number of contacts will be increased and hence the extreme configurations can be found. As the first step of this algorithm, 2 incidences between 2 helices will be parametrized. To be more specific on the 2 incidences, the term of"Dumbbell" has been defined as a pair of intersecting spheres (a,, a2 ) Within a helix a. A biincidence between a pair of helices a, b is hence defined as dumbbells (a,, a2 )and( (b, b2) touch each other: a, is incident on b, and a2 is incident on b2 (Fig. 21). Note that an incidence between two spheres does not pin down the point of incidence. Therefore each such incidence removes only one degree of freedom, not three. Within the biincidence, the packing constraint should also be enforced. Thus we can derive equations according to this constraint to parametrize the biincidence configuration space. The parametrization can be done in both the distances and the angles between the dumbbells. In the distance based approach, the constraint can be reflected by the distance between a, and b2 > and a2 and b But here we will focus on the angle based approach, which will be reviewed later. After the biincidence has been fixed by the parameters, we can further rotate the helices about the hinge of the dumbbells respectively, and use certain parameters to locate the next incidence. Therefore, in the next step, we can seek for a method to parametrize the 3incidences then locate the 4th .incidence accordingly, and so on. Figure 21. Dumbbell (a,, a2 ) fTOm helix a and dumbbell (b,, b2 ) fTOm helix b form a bi incidence. (Source: Meera Sitharam, Heping Gao, Jorg Peters. 2007. Helix packing via constraint solving). Review of Biincidence Feasibility Regions Solving in Angle Approach As we stated before, the biincidence space consists of one dumbbell from each of the two helices. Dr. Jorg Peters at the University of Florida derived the parametrization of the bi incidence with angle approach in his draft "Biindicence feasibility regions in helix packing via constraint solving". For simplicity, we position one dumbbell (Da ) on the x axis and one of its spheres centered at the origin. The rest of Helix a will be translated accordingly. Then we move the dumbbell (D, ) of the other helix b to a position that form a biincidence contact with D . By specifying the two angles (# and ly) of the biincidence, we fix the configuration between these two dumbbells. For each sampled configuration, we will also translate the rest of Helix b accordingly. Then by sampling on P which denotes the angle Helix b rotates about the hinge of Db, We can determine the feasible region of a, the angle Helix a rotates about the hinge of D, Therefore, the output of solving biincidence feasibility regions will be sets of angles (, ,y, a, P ). Parametrization of a Dumbbell Sliding on a Fixed Dumbbell (Solving for and ly) We fix one of the dumbbells to have its axis aligned with the x axis and one endpoint at the origin (Fig. 22). Figure 22. Biincidence of a pair of dumbbells and the parametrization (Source: Meera Sitharam, Jorg Peters, and James Pence. 2007. Biincidence feasibility regions in helix packing via constraint solving). x ,, x1" the endpoints of dumbbell i, (21) r, r,', the corresponding radii, (22) r ,the sum of radii of x) and x (23) d', the distance between x', and x," within dumbbell i, (24) d the axis formed by connecting x,O, and xf (25) For a clear view of the figure, although in practice a dumbbell should consist of two intersecting spheres, here we do not require that constraint. That is, d' > > + r4' need not hold. And since our algorithm here does not rely on that constraint, the result is still valid. The parametrization places *x: on a circle in the xy plane with parameter # of radius r, about xr' on a circle with parameter ly of a cone around d. dx := x)Y x',' I= (\d") + 2d" cos # + + I) (26) To keep the biincidence property, the feasible region for the parameters # and ly is constrained by ~: dx > (To + r1)2 (27) ~: d, <: (l + d")2 (28) ~: d, < (r0 + d1) (29) w: Ix, ~ x > (r" + ;> ) (210) Since (28) is trivial which always holds, by combining (27) and (29), we have the region of # : (T + ')2 (r )2 (d") (To + d' ) (r )2 (d") < cos# < (211) 2rnd" 2rnd" Then for each feasible #, we only need to find out ly that will not cause intersection between dumbbells. We note that xs = (7) Define a as a:= "2z , Then the first coordinate of x), is (d'" + r, cos #)a r, sin sin l ( ~)" a" Therefore the left hand side of (210) becomes 10Ix: ~ x = (r ) 2d''((d' + r, cos )ar sin Qsin7 yl( )~> a )+(d')> Here we define k, as: rl sin ),It> () d' 2d2 k, :=1 (212) (rO)2 + (d')2 (r0 + rC )2 (dO +r, cosf)a 2d0 The constraint on ly will be: <: sin r (213) Moving a Helix to a Position of Its Dumbbell Since in our current stage, the dumbbell lies on x axis, Da, will be a test dumbbell which is preset, we need only translate Db .The main idea is to first move Helix b from its original position Ob to where the sphere x: overlapped with current xf Then we calculate the rotation mlatrlix to rotate~ vectrI (x1 ,x) to L mlake xg also overlapped with current one. At last we apply this matrix to the whole helix. The detailed translation process will be provided in the next chapter. Helix Hinge Motion and Intersection (Solving for a and P) The dumbbell axes serve as hinges around which the remainder of the helix can rotate by angles (a, P). Here we only give each dumbbell a and b one additional sphere az, respectively b3, rather than checking all m spheres. To extend the parametrization to a full helix, we can test all the spheres start from the nearest ones to the dumbbell, then find the intersection of all the regions. For a fixed feasible (#,ly), we seek, for each choice of a3 and b3, the intersection of the feasible (a P) regions (hinge angles so that the two additional spheres do not intersect one another) (Fig. 23). P"(7): the position for sphere attached to dumbbell i, where y indicates the rotation angle. y = 0 represents the initial position of P" (214) 6, : the origin of the rotation of P" (215) p" : corresponding radius of the additional sphere (216) To avoid intersect between a3 and b3, the following equation should be satisfied: IPo(a)PL(Bf) 2 0 1 2) (217) Set P := P' (0) x~ (218) x, x h := (219) cp := cos sp := smn (220) 2 2 v(): (c s))PI + 2c y,P x h + 2s(h :h (221) k(P):= vf )(2 230 As a constraint between a and p, the feasible region is characterized by cos a v2 (P> + Sin a v3 (P) < k(P) (223) Figure 23. Sphere intersection when rotating about the hinge (Source: Meera Sitharam, Jorg Peters, and James Pence. 2007. Biincidence feasibility regions in Helix packing via Constraint Solving). Therefore we has constrained the feasible region for each of 4, ly, a, S By combining them together, the feasible region will be in W4 Space. CHAPTER 3 IMPLEMENTATION OF BIINCIDENCE FEASIBLITY REGIONS SOLVING ALGORITHEM Implementation Environment Here we use C++ to implement the algorithm. The develop environment is Microsoft Visual C++. To display the result of 4, y, a and P we included the library of OpenGL into our program to draw them on the screen. Overview on the Structure of Implementation The program is divided into two phases. The first phase reads the data of the helix, and with a specified preset test helix we apply the algorithm of biincidence feasibility regions solving on them. It will have a formatted text file of the sampled 4, l, a and P as the output. It can be used to solve all the valid dumbbells in the input helix as well as any user specified dumbbell only. The second phase takes the output of the first one, then with the data on both helices (the input one and the preset test helix), it displays the result in a 3D scene with the shapes of helices. Implementation: Phase 1 The input data of a helix is organized as a text file, specifying the position and the radius of each sphere in the helix. For a sample input, please refer to the section of Appendix. First of all, we open the file in C++ and read all the data into memory. Also, we specify our test helix. In the form of Sphere(center~x, center4y, center~z, radius), the three spheres of test helix are: Ir)\t.9>herre_1(0, 0, 0, 0.5), testr.9>herre2(1, 0, 0, 0.5), and testSphere_3(0, 1, 0, ). As for the algorithm, rlev.9>helre_1 serves as the sphere located on xO, testSphere_2 will be located on x , and testSphere_3 will be as Then we do a search on the intersection between each pair of spheres in the input helix. If we found an intersection pair, they will form the dumbbell that we will work on. Solving # For each pair of dumbbells, the first read in sphere will serve as the one located on x and the second one will be at x' Also we get the radius~ ofA+ both and the distance between their centers. Since solving # and ly doesn't require the dumbbells to be actually touched, we do not need to translate the chosen dumbbell of the helix to its configuration for now. Then by applying equation (21 1), we can have the range of cos # In C++, due to the precision of calculation, the result can be slightly out of the allowed range. For example, in some of the cases cos # can be equal to 1.0000001. Therefore, after each calculation requires meaningful boundary such as sin and cos, we do a manual round up, such as: if (lowerBoundCosPhi < 1) { lowverBoundCosPhi = 1; if(upperBoundCosPhi > 1) { upper~oundCosPhi = 1; We noticed that in C++, the acos function will give out the result in the region of [0, 7t]. However, because the constraint only apply to cos value, there should be two feasible regions. So here we use the result of acos to get the other region also. Solving ly Based on the result regions, we make an even sampling on the feasible values. Then for each feasible we apply equation (213) to get the feasible region of sin ly . An issue here is that the divider of (213) can possibly be 0, which will cause unexpectable r d,2 (d1)2 r2 ) nete ae eg result of l The 0 factor can be from sin # or I(~ 0 2_x 2 n ihrcae eg d 2dz back to check the original constraint d,2 (d1)2 r2 +d) 31 (ro' + rO 2 (0 2 2dad x 2 0+d 31 2dz If it holds, then we assert that all of lu values are valid, thus ri/e [0,21i]. Otherwise, there is no valid ly. Also we will need to check if the equation asks sin l > 1 If so, we also assure that there is no valid region for l . After all boundary checks have been done, we use asin function in C++ to solve l Then we project the result to the region of [0,21i]. It may also cut the region into two. Then we sample in the region of feasible ly and work for each one. Translate the Whole Helix to Make Sure the Chosen Dumbbell Lies in the Configuration Before we proceed with solving a and P, we need to translate the whole helix to the position specified as the chosen and l . Based on previous process, we now can get the position of x ,and xf Now we refer the original position of the dumbbell spheres as x~ o and x: o. By subtracting x~ o from x: _o, we are going to have a vector denoted as V, Correspondingly, subtracting x frmxwilgv out another vector denoted as Va. Because the length of V, and Vc are same, the process of translating V, to Vc consists of only two parts, translation which refers to moving the vector, and rotation. In the rotation process, we use x~ and x~ o as the origins respectively. Our first step is to compute the rotation matrix to put V, into the plane x y Then we rotate V, about z axis to align it to x axis. At last we rotate it about y axis and z axis to make it align with Va. After rotation, the process of translation is trivial since we only need to compute the difference between x o and xf Then by applying the 4 rotation matrices and 1 translation matrix, we can successfully move the whole helix. In order to ensure the correctness of the transformation, we deployed a piece of code to check the result. Solving a and P Pair Recall equation (223) cos a v2 ( ) + Sin a v3 (P) < k(P) To translate this equation into a form more suitable for solving, set R := (v2 2 3 2;c~iZ (32) If Re 0 we calculate an offset angle ii as v2(P ii:= acos( ) (33) Then (223) is equivalent to R cos(a /) < k(P) (3 4) Our approach to get the (a, P) pair is to sample on P in the region of [0,27t]. Therefore, for each p sample value, we have a range of cost a ii), and we are able to get the feasible region of a If there is no valid a the sample value of P is consequently not valid. Since a is constrained by a cos function, the result can still have two regions. We use a similar technique as in solving # to handle the situation. User Specified Dumbbell Solving As an additional feature of the implementation, besides solving all the dumbbells of a helix, our program allows the user to specify one dumbbell and get its result accordingly. User will need to indicate the index number of the two spheres of the dumbbell. If the two cannot form a dumbbell, an error message will be out. Output the Result As we running the program, after each valid a region is given out, we will write a line to the output text file. The structure of the output content is organized as a tree (Fig. 31). *** Figure 31. Structure of output file Implementation: Phase 2 Since the result ( ly, a, p) forms a space in 914, a shape in 3D is not enough to represent the whole feasible region. Therefore we first map the selected #, ly, and P to x, y , and z axis respectively. Then we use a colored line segment to represent the valid a region for each 3D space selected point (Fig. 32). Red represents not valid regions, while blue represents valid regions. not valid Valid valid 0 27r Figure 32. Line segment to represent the a region Besides the feasible region, the program also displays the input helix with solid red spheres as selected dumbbell and wired blue spheres as other spheres. And the test helix is displayed in its own coordinate system as well. For a better view, the program can be switched between showing (f ly, P) regions, which will only consists of points, and showing the whole space. In Fig 33, the input helix is on the left side, and the test helix lies on the right side. Centered is a wired yellow cube represents the boundary of the whole space. The strong red line is x axis representing #, the strong yellow line is y axis representing y, and the strong blue line is z axis representing P Except y y/ others are mapping [0,2xi] region to [5, +5], while y y is mapping [0,2x2] to [0, + 10]. In Fig. 34, we only display ( y/, P) regions as points in the same coordinate system. Figure 33. Sample output of Phase 2 with a regions. 26 Figure 34. Sample output of Phase 2 without a regions. 27 CHAPTER 4 CONCLUSION Effectiveness of BiIncidence Feasibility Regions Solving In the process of sampling the configuration space of 2 helices so that they do not intersect, we are particularly interested in including those extreme configurations in our samples. By bi incidence parameterization, we can effectively locate the feasibility regions and hence the first 3 incidences between the 2 helices avoiding sampling of collision detection. In general, this process will leave the configuration space with 3 degrees of freedom. Implementation of the Algorithm In the implementation process, we noticed that in the process of solving ly, the divider of equation (213), k, can possibly be 0. This will cause unexpected result in the program. However, due to equation (210), we can interpret this case correctly by checking equation (31). Besides, our implementation handles the precision and return values of acos and asin in C++ properly . The result of the implementation program has been tested to be correct. And there are also codes in the program for testing purpose after some important steps such as the translation of the whole helix. In phase two, the output of phase one can be handled correctly and user can also interact with the display to get a better view on the feasible regions. Challenge and Future Development After 3 incidences have been specified by our approach, we can do collision detection based on these constraints. This is also more efficient than doing collision detection in the earliest stage. A better way, to be explored, is to parameterize a triple incidence, then 4 incidences, etc. This would also prevent "holes" due to insufficient sampling. At present, with sampling, a P value in between of 2 adjacent sampled valid P s may not be valid so we cannot guarantee that we have found all the extreme points in the result region. Possibly, the first step to parametrize a triple incidences is abstracting the configurations to the normal of the triangles formed by the 3 spheres of each helix respectively. Also in the implementation, a potential improvement is that during our computation on a and p what we did is apply the equation for each pair of the spheres. However, in practice some of the spheres, such as two far away spheres, can be trivially tested with a boundary on all of their possible locations. This method can bring improvement on the complexity of the algorithm although how much it may improve is depending on the exact configuration of the two helices. < APPENDIX INPUT HELIX DATA The file hlx~full.dat has 54 rows containing the data of 12 distinct amino acid sequences. Each row represents an atom. The 2nd, 3rd, and 4th COlumns are the xyz position of the spheres respectively. The 5th COlumn is for the radius of each sphere. Contents of the file are below. 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10 10 11 11 12 12 13 13 14 14 15 15 16 16 17 17 18 18 19 19 4.731 3 5.689 3.046 0 4.302 1 1.7062 1.7062 0.325 C 1.975 ; 1.381 0.800 2.934 2.208 4.214 3.782 5.742 5.025 7.457 [ 6.792 ( 8.860 8.084 10.183 8.981 11.846 11.172 13.472 13.100 14.770 14.770 16.219 15.282 17.943 17.292 19.421 18.808 20.710 19.395 1.827 0.561 3.261 1.447 ).089 3.383 ..511 5.169 !.640 0.910 !.640 0.910 0.141 1.278 2.373 2.468 1.717 1.725 2.084 2.754 1.622 2.652 2.802 2.913 2.061 0.897 2.091 1.501 1.415 0.822 2.603 0.574 0.639 2.475 0.239 3.652 2.583 1.038 3.627 0.494 0.708 1.988 0.457 3.325 1.852 0.270 2.637 1.228 0.922 2.291 1.448 2.662 2.539 0.890 2.539 0.890 0.778 2.037 2.270 2.261 1.235 1.315 1.256 2.547 2.254 1.080 3.220 0.963 1.520 2.416 0.876 4.189 2.9 3.5 2.9 3.9 2.9 2.4 2.9 2.7 2.9 3.1 2.9 3.2 2.9 2.7 2.9 3.2 2.9 3.2 2.9 3.2 2.9 3.2 2.9 3.2 2.9 2.7 2.9 2.4 2.9 3.3 2.9 3.0 2.9 2.7 2.9 3.7 2.9 2.4 20 20 21 21 22 22 23 23 24 24 25 25 26 26 27 27 23.992 23.059 25.334 24.538 26.711 25.950 28.416 28.063 29.973 29.589 31.254 30.138 32.777 32.392 0.160 1.195 2.753 3.541 0.012 1.008 1.620 1.877 1.700 2.311 2.200 2.233 1.279 1.908 1.652 3.001 0.775 1.631 3.012 3.579 0.038 0.563 0.948 1.124 2.592 4.177 2.579 2.488 0.731 1.901 2.9 3.5 2.9 3.2 2.9 3.0 2.9 2.7 2.9 2.7 2.9 3.5 2.9 2.7 2.9 3.0 34.494 0.563 33.837 0.188 22.309 1.701 1.199 22.309 1.701 1.199 LIST OF REFERENCES Chen, Z., and Y. Xu. 2005. Multiscale hierarchical structure prediction of helical transmembrane proteins. Computational Systems Bioinformatics Conference, 2005. 203 207. Fleishman, S. J., and N. BenTal. 2002. A novel scoring function for predicting the conformations of tightly packed pairs of transmembrane a helices. J Mol. Biol. 321:363378. Javadpour, M. M., M. Oilers, M. Groesbeek, and S. O. Smith. 1999. Helix packing in polytopic membrane proteins: role of glycine in transmembrane helix association. Biophysical Journal. 77:16091618. Pande, V. S., I. Baker, J. Chapman, S. P. Elmer, S. Khaliq, S. M. Larson, Y. M. Rhee, M. R. Shirts, C. D. Snow, E. J. Sorin, and B. Zagrovic. 2003. Atomistic protein folding simulations on the submillisecond time scale using worldwide distributed computing. Biopolymers. 68:91109. Shirts, M. R., and V. Pande. 2000. Screen savers of the world, unite! Science. 290:19031904. Wendel, C., and H. Gohlke. 2008. Predicting transmembrane helix pair configurations with knowledgebased distancedependent pair potentials. Proteins. 70:984999. Zagrovic, B., C. D. Snow, M. R. Shirts, and V. S. Pande. 2002. Simulation of folding of a small alphahelical protein in atomistic detail using worldwidedistributed computing. J. Mol. Biol. 323:927937. BIOGRAPHICAL SKETCH I received my Bachelor of Engineering degree in computer science and technology from Beihang University, one of the top schools in China. During my undergraduate study there, I have been selected as a visiting student to study at the University of Alberta in Canada for one year, where I learned computer graphics and developed an interest in related fields. After my graduate, I continued my study in computer engineering at the University of Florida as a master' s candidate. Because of my excellent academic records there, I have been selected for Outstanding Achievement Awards twice. PAGE 1 IMPLEMENTATION OF BIINCIDENCE FEAS IBILITY REGIONS IN HELIX PACKING VIA CONSTRAINT SOLVING By ZHONGJIE LI A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLOR IDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE UNIVERSITY OF FLORIDA 2008 1 PAGE 2 2008 Zhongjie Li 2 PAGE 3 ACKNOWLEDGMENTS I thank Dr. Jorg Peters for his advisory on my work. His indepth knowledge in computational geometry and mathematics helped me a lot in completing my research. His working style and philosophy were al so been beneficial to me. 3 PAGE 4 TABLE OF CONTENTS page ACKNOWLEDGMENTS ...............................................................................................................3 LIST OF FIGURES .........................................................................................................................6 ABSTRACT .....................................................................................................................................7 1 INTRODUCTION TO THE PROB LEM OF HELIX PACKING...........................................8 Motivation .................................................................................................................................8 Literature Review .....................................................................................................................9 Problem Description, Definitions and Assumptions ..............................................................10 2 REVIEW OF BIINCIDENCE FEASIB ILITY REGIONS SOLVING IN HELIX PACKING........................................................................................................................ .......13 Review of Helix Packing via Constraint Solving...................................................................13 Review of Biincidence Feasibilit y Regions Solving in Angle Approach .............................14 Parametrization of a Dumbbell Sliding on a Fixed Dumbbell (Solving for and ).................................................................................................15 Moving a Helix to a Position of Its Dumbbell ................................................................17 Helix Hinge Motion and Intersection (Solving for and ).......................................17 3 IMPLEMENTATION OF BIINCIDENC E FEASIBILITY REGIONS SOLVING ALGORITHEM......................................................................................................................20 Implementation Environment .................................................................................................20 Overview on the Structure of Implementation .......................................................................20 Implementation: Phase 1 .........................................................................................................20 Solving ........................................................................................................................21 Solving ........................................................................................................................21 Translate the Whole Helix to Make Sure the Chosen Dumbbell Lies in the Configuration...............................................................................................................22 Solving and Pair ....................................................................................................23 User Specified Dumbbell Solving ...................................................................................24 Output the Result .............................................................................................................24 Implementation: Phase 2 .........................................................................................................25 4 CONCLUSION................................................................................................................... ....28 Effectiveness of BiIncidence Feasibility Regions Solving ...................................................28 Implementation of the Algorithm ...........................................................................................28 Challenge and Future Development .......................................................................................28 4 PAGE 5 APPENDIX INPUT HELIX DATA...........................................................................................30 LIST OF REFERENCES ...............................................................................................................31 BIOGRAPHICAL SKETCH .........................................................................................................32 5 PAGE 6 LIST OF FIGURES Figure page 11 The helix .........................................................................................................................8 12 A helix is a rigid 3D obj ect consisting of a set of spheres (atoms) in fixed positions. ............................................................................................................................11 m 13 General case and a special case of incidences between 2 helices. .....................................11 22 Biincidence of a pair of dumbbells and the parametrization ............................................15 23 Sphere intersection when rotating about the hinge............................................................19 31 Structure of output file .......................................................................................................24 32 Line segment to represent the region ............................................................................25 33 Sample output of Phase 2 with regions. ........................................................................26 34 Sample output of Phase 2 without regions. ...................................................................27 6 PAGE 7 Abstract of Thesis Presen ted to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Master of Science IMPLEMENTATION OF BIINCIDENCE FEAS IBILITY REGIONS IN HELIX PACKING VIA CONSTRAINT SOLVING By Zhongjie Li August 2008 Chair: Jorg Peters Major: Computer Engineering By introducing computational geometry and com puter science methods to solving the helix packing problem in biochemistry, we can assist the research. An algorithm of helix packing via constraint solving is being explor ed by the researchers at the Un iversity of Florida, which is based on a parametrization of the configuration space. As the first step of the algorithm, an angle based approach has been brought out to parametri ze the first two incidences between two helices and therefore locate the third incidence. The al gorithm successfully reduces our search space to fourdimensional space. Later on, the algorithm ha s been implemented in C++ in two phases. Phase 1 processes the input helix data with the algorithm then outputs the result as a text file. Phase 2 takes the output of Phase 1, and displays it with OpenGL. 7 PAGE 8 CHAPTER 1 INTRODUCTION TO THE PROB LEM OF HELIX PACKING Motivation Since different structure aspect s of proteins have been disc overed by biologists in the mid 20 th Century, scientists started to focus on revealing the essential mechanism of the transitions between these structure forms, such as folding that enables proteins to be able to perform their biological function. Fig. 11 show s an alpha helix which acts a critical role in understanding helix packing, a type of protein folding. However, because experimentally determining the three dimensional structure of a protei n is often very difficult and expe nsive, scientists started to research on the possible ways of making predictions on the protein structure. Figure 11. The helix. (Source: http://www.uic.edu/classes/bios/b ios100/lecturesf04am/lect02.htm Last accessed July, 2008). A number of factors exists that make protein structure prediction a very difficult task. The two main problems are that the number of possible pr otein structures is extr emely large, and that the physical basis of protein stru ctural stability is not fully un derstood. As a result, any protein structure prediction method needs a way to explore the space of possible stru ctures efficiently (a search strategy), and a way to identify the most plausible structure (a n energy function). Thus predicting the protein structure accurate ly remains a big challenge today. 8 PAGE 9 In atomistic level, helices are often modele d as rigid 3D objects. Hence due to their geometric nature, Computational Geometry methods can be introduced into this field. However, because of the complex structure of the helix, brut al force collision detection is often used in exploring the packing space. As a tradeoff, th is method consumes high computational power. In contrast to improving the computa tion capacity of computers, scientists are also seeking more efficient algorithms to solve the packing space with constraints. Literature Review The packing of transmembrane helices was mo deled at the residual and atomistic levels, respectively. For predictions at th e residual level, the helixhelix a nd helixlipid interactions were described by a set of knowledgebased energy func tions. For predictions at the atomistic level, the CHARMM19 force field (Chen and Xu, 2005) was employed. In 2002, Fleishman and BenTal developed a scoring function and a computational methodology for predicting the tertiary fold of a pair of alpha helic es such that its chances of being tightly packed are maximized. The sc oring function was constructed based on the qualitative insights gained in the past two decades from the solved structures of transmembrane (TM) and soluble proteins (Fleishman and BenTal, 2002). Basically, they reward the formation of contacts between small amino acid residues such as Gly, Cys, and Ser, that are known to promote dimerization of helices, and penalize the burial of large amino acid residues such as Arg and Trp. Thus in computation, they can maximize the number of contacts between residues. This kind of solution is often called knowledgebased approach because of using known contact potentials, and the packing is m odeled at the residual level instead of atoms level. A similar solution has been introduced in 2007 which uses coarsegrained knowledgebased potentials to score the mutual configuration TM helices (Wendel and Gohlke, 2008). 9 PAGE 10 Direct simulation of protein folding in atom ic detail, via methods such as molecular dynamics with a suitable energy function, is also a promising direction of the research. Because of the high computational cost of it, distributed computing te chnology has been introduced into this field. In 2000, a distributed computing pr oject designed to perform computationally intensive simulations of protein folding and other molecular dynamics has been launched by Stanford University. The project is named as Folding@home and with the computational power volunteered by thousands or even millions privat e computer users across the globe (Shirts and Pande, 2000). It is one of the most powerful di stributed computing clusters in the world. The goal of Folding@home is aimed on understandin g protein folding, misfolding, and related diseases. With the assistance of the project, a number of achievements ha ve been reached till today, such as the simulation of folding of a small helical protein in atomistic detail (Zagrovic et al., 2002) and atomistic protein fold ing simulations (Pande et al., 2003). Problem Description, Definitions and Assumptions In order to explore the space of possible struct ures in helix packing more efficiently, we are seeking for an algorithm with formal guarantees for sampling the c onfiguration space of 2 packed helices with collision avoidance constraints by introducing computational geometry and other computer science technologies into this field. In our problem, according to the assumption, helix is a rigid 3D object consisting of a set of spheres (atoms) in fixed positions (Fig. 12), and the spheres can be intersected with others in the same helix. But in the process of packing, the spheres of different helices must not intersect. This collision avoidance is called packing constraint It increased the difficulty of our probl em due to the nonlinear feature it brought. m A configuration is defined by the relative position a nd orientation of 2 helices for a fixed position of the first helix. Hence a configuration has 6 independent degrees of freedom, thus can 10 PAGE 11 be viewed as a point in space. If a configuration satisfied packing constraint, then we call it a packing configuration A configuration of 2 helices is an extreme configuration if its incidences rigidly fix one helix to the other one. Generica lly, 6 incidences fix two helices respect to each other, but in some specific cases, 6 incidenc es may not be enough to form an extreme configuration (Fig. 13). Here we will mainly focus on the general case. 6 Figure 12. A helix is a rigid 3D object consisting of a set of spheres (atoms) in fixed positions. m (a) (b) Figure 13. General case and a special case of incidences between 2 helices. Doted lines represent the connections w ithin a helix. Solid lines represent the connections between helix. (a) is the general case where 6 incidences can fix the configuration of 2 helices. (b) is a speci al case where although 6 in cidences happened, the configuratio n is still not fixed becaus e righ t helix can rotate ab out the h i ng e of the connection between 2 atom s in left helix. (Source: Meera Sitharam Heping Gao, Jorg Peters. 2007. Helix packing via Constraint Solving). 11 PAGE 12 Our algorithm should contain every extreme pack ing configuration in the sample result, or a configuration arbitrarily close to it (within an input desired precision). For other configurations, we should be able to sample them with any de sired (user input) precisio n. On the other hand, one of the noticeable drawbacks of existent algorithm on exploring the feasible regions of helix packing is that it will sample one extreme conf iguration multiple times. As a main factor in increasing the efficiency, our al gorithm should outputs each sample configuration exactly once or at least same number of times (constant times). In the following chapters, we will first introduce the algorithm developed by a group of researchers at the University of Florida. Then we will focus on the first step of the algorithm and an implementation of it. 12 PAGE 13 CHAPTER 2 REVIEW OF BIINCIDENCE FEASIBILITY REGIONS SOLVING IN HELIX PACKING Review of Helix Packing via Constraint Solving In early 2007, the idea of exploring the feasib le regions of helix p acking via constraint solving was explored by Dr. Meera Sitharam, Hepi ng Gao, and Dr. Jorg Peters at the University of Florida. In their algorithm, the incidences between 2 helices will be parametrized, and based on this parametrization, the next incidence can be located in a ra nge described by one or a set of equations achieved by applying collision detection between spheres. The algorithm will start from the parametrization on 2 incidences, then by recursively applying the principle, the number of contacts will be increased and hence th e extreme configurations can be found. As the first step of this algorithm, 2 incidenc es between 2 helices will be parametrized. To be more specific on the 2 incidences, the term of Dumbbell has been defined as a pair of intersecting spheres ( ) within a helix a. A biincidence between a pair of helices a, is hence defined as dumbbells ( ) and ( ) touch each other: is incident on and is incident on (Fig. 21). Note that an incidence be tween two spheres does not pin down the point of incidence. Therefore each such incidenc e removes only one degree of freedom, not three. 1a 2a b 1a 2a 1b 2b 1a 1b 2a 2b Within the biincidence, the packing constrai nt should also be en forced. Thus we can derive equations according to this constraint to parametrize the biincidence configuration space. The parametrization can be done in both the di stances and the angles between the dumbbells. In the distance based approach, the constraint can be reflected by the distance between and and and But here we will focus on the angle based approach, which will be reviewed later. 1a 2b 2a 1b After the biincidence has been fixed by the parameters, we can further rotate the helices about the hinge of the dumbbells respectively, an d use certain parameters to locate the next 13 PAGE 14 incidence. Therefore, in the next step, we can seek for a method to parametrize the 3incidences then locate the 4 th incidence accordingly, and so on. Figure 21. Dumbbell ( ) from helix and dumbbell ( ) from helix b form a biincidence. 1a 2a a 1b 2b (Source: Meera Sitharam, Heping Gao, Jorg Peters. 2007. Helix packing via constraint solving). Review of Biincidence Feasibility Regions Solving in Angle Approach As we stated before, the biincidence space c onsists of one dumbbell from each of the two helices. Dr. Jorg Peters at the University of Florida derived the parametrization of the biincidence with angle approach in his draft Biindicen ce feasibility regions in helix packing via constraint solving. For simp licity, we position one dumbbell ( ) on the aD x axis and one of its spheres centered at the origin. The rest of Helix a will be translated accordingly. Then we move the dumbbell ( ) of the other helix b to a position that form a biincidence contact with bD aD 14 PAGE 15 By specifying the two angles ( and ) of the biincidence, we fi x the configuration between these two dumbbells. For each sampled configurati on, we will also translate the rest of Helix b accordingly. Then by sampling on which denotes the angle Helix b rotates about the hinge of we can determine the feasible region of bD the angle Helix rotates about the hinge of Therefore, the output of solving biincidence f easibility regions will be sets of angles ( a aD ,,, ). Parametrization of a Dumbbell Slidin g on a Fixed Dumbbell (Solving for and ) x We fix one of the dumbbells to have its axis aligned with the axis and one endpoint at the origin (Fig. 22). Figure 22. Biincidence of a pair of dumbbells and the parametr ization (Source: Meera Sitharam, Jorg Peters, and James Pence. 2007. Biinciden ce feasibility regions in helix packing via constraint solving). ix0 the endpoints of dumbbell i, (21) ix1 15 PAGE 16 ir0 the corresponding radii, (22) ir1 jr the sum of radii of and (23) 0 jx 1 jx id the distance between and within dumbbell i, (24) ix0 ix1 d the axis formed by connecting and (25) 0 0x 1 1x For a clear view of the figure, although in practice a dumbbell should consist of two intersecting spheres, here we do not require that constraint. That is, need not hold. And since our algorithm here does not rely on th at constraint, the result is still valid. iiirrd10 The parametrization places on a circle in the 1 1x x y plane with parameter of radius about 1r 0 1x on a circle with parameter 1 0x of a cone around d 2 11 020 0 0 1 1)(cos2)( : rrddxxdx (26) To keep the biincidence property, the feasible region for the parameters and is constrained by : (27) 21 1 0 0 2)(rrd : (28) 20 1 2)(drdx : (29) 21 0 2)(drd : 21 0 0 1 2 0 1 1 0)( rrxx (210) Since (28) is trivial which always holds, by combining (27) and (29), we have the region of : 0 1 202 1 21 0 0 1 202 1 21 1 0 02 )()()( cos 2 )()()( dr drdr dr drrr (211) Then for each feasible we only need to find out that will not cause intersection between dumbbells. We note that 2 0 2 1 0)( rx Define a as 2 2 0 2122 )( : d rdd a 16 PAGE 17 Then the first coordinate of is 1 0x 22 0 1 1 0)(sinsin)cos( a d r rard Therefore the left hand side of (210) becomes 20 22 0 1 1 002 0 2 0 1 1 0)())(sinsin)cos((2)( da d r rarddrxx Here we define as: k 0 21 0 0 1 202 0 1 0 2 2 2 0 212 2 0 12 )()()( )cos( ) 2 )( ()(sin : d rrdr ard d rdd d r r k (212) The constraint on will be: sin 1 k (213) Moving a Helix to a Po sition of Its Dumbbell Since in our current stage, the dumbbell lies on x axis, will be a test dumbbell which is preset, we need only translate The main idea is to first move Helix b from its original position to where the sphere overlapped with current Then we calculate the rotation matrix to rotate vector ( ) to make also overlapped with current one. At last we apply this matrix to the whole helix. The detailed translation process will be provided in the next chapter. aD bD bO 1 1x 1 1x 1 1x 1 0x 1 0x Helix Hinge Motion and Intersection (Solving for and ) The dumbbell axes serve as hinges around which the remainder of the helix can rotate by angles ( ). Here we only give each dumbbell and b one additional sphere a 3a 17 PAGE 18 respectively rather than checking all spheres. To extend the parametrization to a full helix, we can test all the spheres start from the neares t ones to the dumbbell, then find the intersection of all the regions. 3b m For a fixed feasible ( ), we seek, for each choice of and the intersection of the feasible ( 3a 3b ) regions (hinge angles so that the tw o additional spheres do not intersect one another) (Fig. 23). )(iP : the position for sphere attached to dumbbell i, where indicates the rotation angle. 0 represents the initial position of i P (214) i : the origin of the rotation of i P (215) i : corresponding radius of the additional sphere (216) To avoid intersect between and the following equation s hould be satisfied: 3a 3b 210 2 1 0)()()( PP (217) Set 1 0 1)0(:xPP (218) 1 0 1 1 1 0 1 1: xx xx h (219) 2 cos: c 2 sin: s (220) 0 0)(2 2)(:)(1 0 2 221 0x hPhshPscPscxv (221) ))()()(( 2 1 :)(2102 0 2 0 v k (222) As a constraint between and the feasible region is characterized by )()(sin)(cos3 2 kv v (223) 18 PAGE 19 Figure 23. Sphere intersection when rotating ab out the hinge (Source: Meera Sitharam, Jorg Peters, and James Pence. 2007. Biincidence f easibility regions in Helix packing via Constraint Solving). Therefore we has constrained th e feasible region for each of ,,, By combining them together, the feasible region will be in space. 4 19 PAGE 20 CHAPTER 3 IMPLEMENTATION OF BIINCIDENC E FEASIBILITY REGIONS SOLVING ALGORITHEM Implementation Environment Here we use C++ to implement the algorithm. The develop environment is Microsoft Visual C++. To display the result of ,, and we included the library of OpenGL into our program to draw them on the screen. Overview on the Structure of Implementation The program is divided into two phases. The first phase reads the data of the helix, and with a specified preset test helix we apply the algorithm of biincidence feasibility regions so lving on them. It will have a formatted text file of the sampled ,, and as the output. It can be used to solve all the valid dumbbells in the input helix as well as any user specified dumbbell only. The second phase takes the output of the first one, then with the data on both helices (the input one and the preset test helix) it displays the result in a 3D scene with the shapes of helices. Implementation: Phase 1 The input data of a helix is or ganized as a text file, specifyi ng the position and the radius of each sphere in the helix. For a samp le input, please refer to the s ection of Appendix. First of all, we open the file in C++ and read all the data into memory. Also, we specify our test helix. In the form of Sphere(center_x, center_y, center_z, radius) the three spheres of test helix are : testSphere_1(0, 0, 0, 0.5), testSphere_2( 1, 0, 0, 0.5), and testSphere_3(0, 1, 0, 2 2 ) As for the algorithm, testSphere_1 serves as the sphere located on testSphere_2 will be located on and testSphere_3 will be 0 0x 0 1x 3 Then we do a search on the in tersection between each pair of 20 PAGE 21 spheres in the input helix. If we found an inte rsection pair, they will form the dumbbell that we will work on. Solving For each pair of dumbbells, the first read in sphere will serve as the one located on and the second one will be at Also we get the radius of bot h and the distance between their centers. Since solving 1 0x 1 1x and doesnt require the dumbbells to be actually touched, we do not need to translate the chosen dumbbell of the helix to its configuration for now. Then by applying equation (211), we can have the range of cos In C++, due to the precision of calculation, the result can be slightly out of the allowed range. For example, in some of the cases cos can be equal to 1.0000001. Therefore, after each calcula tion requires meaningful boundary such as sin and cos, we do a manual round up, such as: if (lowerBoundCosPhi < 1){ lowerBoundCosPhi = 1; } if (upperBoundCosPhi > 1){ upperBoundCosPhi = 1; } We noticed that in C++, the acos function will give out th e result in the region of ,0 However, because the constraint only apply to co s value, there should be two feasible regions. So here we use the result of acos to get the other region also. Solving Based on the result regions, we make an even sampling on the feasible values. Then for each feasible we apply equation (213) to get the feasib le region of sin 21 PAGE 22 An issue here is that the divi der of (213) can possibly be 0, which will cause unexpectable result of The 0 factor can be from sin or 2 2 2 0 212 2 0) 2 )( ()( d rdd d r In either case, we go back to check the original constraint 20 2 2 0 212 02 0 20 1 1 0)( 2 )( 2)()( d d rdd ddrrr (31) If it holds, then we assert that all of values are valid, thus 2,0 Otherwise, there is no valid Also we will need to check if the equation asks 1sin If so, we also assure that there is no valid region for After all boundary checks have been done, we use asin function in C++ to solve Then we project the result to the region of 2,0 It may also cut the region into two. Then we sample in the region of feasible and work for each one. Translate the Whole Helix to Make Sure the Chosen Dumbbell Lies in the Configuration Before we proceed with solving and we need to translate the whole helix to the position specified as the chosen and Based on previous process, we now can get the position of and Now we refer the original position of the dumbbell spheres as and By subtracting from we are going to have a vector denoted as Correspondingly, subtracting from will give out another vector denoted as Because the length of and are same, the process of translating to consists of only two parts, translati on which refers to moving the vector, and rotation. In the rotation process, we use and as the origins respectively. Our first step is to compute the rotation matrix to put into the plane 1 0x 1 1x ox_1 0 ox _1 1 ox_1 0 ox _1 1 oV 1 0x 1 1x cV oV cV oV cV 1 0x ox_1 0 oV y x Then we rotate about axis oV z 22 PAGE 23 to align it to x axis. At last we rotate it about y axis and axis to make it align with After rotation, the process of transla tion is trivial since we only need to compute the difference between and z cV ox_1 0 1 0x Then by applying the 4 rotation matrices and 1 translation matrix, we can successfully move the whole helix. In order to ensure the correctness of the transformation, we deployed a piece of code to check the result. Solving and Pair Recall equation (223) )()(sin)(cos3 2 kv v To translate this equation into a form more suitable for solving, set 2 3 2 2))(())((:vvR (32) If we calculate an offset angle 0 R as ) )( cos(:2 R v a (33) Then (223) is equivalent to )()cos( k R (34) Our approach to get the ( ) pair is to sample on in the region of 2,0 Therefore, for each sample value, we have a range of cos( ), and we are able to get the feasible region of If there is no valid the sample value of is consequently not valid. Since is constrained by a cos function, the result can still have two regions. We use a similar technique as in solving to handle the situation. 23 PAGE 24 User Specified Dumbbell Solving As an additional feature of the implementation, besides solving all the dumbbells of a helix, our program allows the user to specify one dum bbell and get its result accordingly. User will need to indicate the index number of the two sp heres of the dumbbell. If the two cannot form a dumbbell, an error message will be out. Output the Result As we running the program, after each valid region is given out, we will write a line to the output text file. The structure of the output content is organized as a tree (Fig. 31). Whole Result Sphere(0,1) Sphere(0,2) Sphere(n1,n) 1 2 SAMPLE MAX 1 2 SAMPLE MAX 1 2 SAMPLE MAX Valid Region Valid Region Valid Region Figure 31. Structure of output file 24 PAGE 25 Implementation: Phase 2 Since the result ( ) forms a space in a shape in 3D is not enough to represent the whole feasible region. Therefore we first map the selected 4 and to x y and axis respectively. Then we use a colored line segment to represent the valid z region for each 3D space selected point (Fig. 32). Red represents not valid regions, while blue represents valid regions. Figure 32. Line segment to represent the region Besides the feasible region, the program also displays the inpu t helix with solid red spheres as selected dumbbell and wired blue spheres as other spheres. And the test helix is displayed in its own coordinate system as well. For a better view, the program can be switched between showing ( ) regions, which will only consists of points, and showing the whole space. In Fig 33, the input helix is on the left si de, and the test helix lies on the right side. Centered is a wired yellow cube represents th e boundary of the whole space. The strong red line is x axis representing the strong yellow line is y axis representing and the strong blue line is axis representing z Except y others are mapping 2,0 region to [5, +5], while y is mapping 2,0 to [0, +10]. In Fig. 34, we only display ( ) regions as points in the same coordinate system. 25 PAGE 26 Figure 33. Sample output of Phase 2 with regions. 26 PAGE 27 Figure 34. Sample output of Phase 2 without regions. 27 PAGE 28 CHAPTER 4 CONCLUSION Effectiveness of BiIncidence Feasibility Regions Solving In the process of sampling the configuration spa ce of 2 helices so that they do not intersect, we are particularly interested in including those extreme configurations in our samples. By biincidence parameterization, we can effectively locate the feasibility regions a nd hence the first 3 incidences between the 2 helices avoiding sampli ng of collision detection. In general, this process will leave the configurati on space with 3 degrees of freedom. Implementation of the Algorithm In the implementation process, we noti ced that in the process of solving the divider of equation (213), can possibly be 0. This will cause unexpected result in the program. However, due to equation (210), we can interpre t this case correctly by checking equation (31). Besides, our implementation handles th e precision and re turn values of acos and asin in C++ properly. k The result of the implementation program has b een tested to be corr ect. And there are also codes in the program for testing pur pose after some important steps such as the translation of the whole helix. In phase two, the output of phase one can be handled correctly and user can also interact with the display to get a better view on the feasible regions. Challenge and Future Development After 3 incidences have been specified by our approach, we can do collision detection based on these constraints. This is also more efficient than doing collision detection in the earliest stage. A better way, to be explored, is to parameteri ze a triple incidence, then 4 incidences, etc. This would also prevent holes due to insufficient sampling. At present, with sampling, a value in between of 2 adjacent sampled valid s may not be valid so we cannot 28 PAGE 29 guarantee that we have found all the extreme points in the result re gion. Possibly, the first step to parametrize a triple incidences is abstracting the configurations to the normal of the triangles formed by the 3 spheres of each helix respectively. Also in the implementation, a potential improve ment is that during our computation on and what we did is apply the equation for each pair of the spheres. However, in practice some of the spheres, such as two far away spheres, can be trivially tested with a boundary on all of their possible locations. This method can bring improvement on the complexity of the algorithm although how much it may improve is depending on the exact configuratio n of the two helices. 29 PAGE 30 APPENDIX INPUT HELIX DATA The file hlx_full.dat has 54 rows containing th e data of 12 distinct amino acid sequences. Each row represents an atom. The 2 nd 3 rd and 4 th columns are the x yz position of the spheres respectively. The 5 th column is for the radius of each sphere. Contents of the file are below. 1 1 4.731 1.827 0.561 2.9 1 1 5.689 3.261 1.447 3.5 2 2 3.046 0.089 3.383 2.9 2 2 4.302 1.511 5.169 3.9 3 3 1.706 2.640 0.910 2.9 3 3 1.706 2.640 0.910 2.4 4 4 0.325 0.141 1.278 2.9 4 4 1.975 2.373 2.468 2.7 5 5 1.381 1.717 1.725 2.9 5 5 0.800 2.084 2.754 3.1 6 6 2.934 1.622 2.652 2.9 6 6 2.208 2.802 2.913 3.2 7 7 4.214 2.061 0.897 2.9 7 7 3.782 2.091 1.501 2.7 6 8 5.742 1.415 0.822 2.9 6 8 5.025 2.603 0.574 3.2 6 9 7.457 0.639 2.475 2.9 6 9 6.792 0.239 3.652 3.2 6 10 8.860 2.583 1.038 2.9 6 10 8.084 3.627 0.494 3.2 8 11 10.183 0.708 1.988 2.9 8 11 8.981 0.457 3.325 3.2 6 12 11.846 1.852 0.270 2.9 6 12 11.172 2.637 1.228 3.2 7 13 13.472 0.922 2.291 2.9 7 13 13.100 1.448 2.662 2.7 3 14 14.770 2.539 0.890 2.9 3 14 14.770 2.539 0.890 2.4 9 15 16.219 0.778 2.037 2.9 9 15 15.282 2.270 2.261 3.3 10 16 17.943 1.235 1.315 2.9 10 16 17.292 1.256 2.547 3.0 11 17 19.421 2.254 1.080 2.9 11 17 18.808 3.220 0.963 2.7 12 18 20.710 1.520 2.416 2.9 12 18 19.395 0.876 4.189 3.7 3 19 22.309 1.701 1.199 2.9 3 19 22.309 1.701 1.199 2.4 < PAGE 31 LIST OF REFERENCES Chen, Z., and Y. Xu. 2005. Multiscale hierarchical structure prediction of helical transmembrane proteins. Computational Systems Bioinformatics Conference, 2005 203207. Fleishman, S. J., and N. BenTal. 2002. A novel scoring function for predicting the conformations of tightly packed pairs of transmembrane helices. J. Mol. Biol 321:363. Javadpour, M. M., M. Eilers, M. Groesbeek, and S. O. Smith. 1999. Helix packing in polytopic membrane proteins: role of glycine in transmembrane helix association. Biophysical Journal 77:16091618. Pande, V. S., I. Baker, J. Chapman, S. P. Elmer, S. Khaliq, S. M. Larson, Y. M. Rhee, M. R. Shirts, C. D. Snow, E. J. Sorin, and B. Zagrovic. 2003. Atomistic protein folding simulations on the submillisecond time scal e using worldwide distributed computing. Biopolymers 68:91. Shirts, M. R., and V. Pande. 2000. Screen savers of the world, unite! Science. 290:19031904. Wendel, C., and H. Gohlke. 2008. Predicting tran smembrane helix pair configurations with knowledgebased distancedepe ndent pair potentials. Proteins 70:984999. Zagrovic, B., C. D. Snow, M. R. Shirts, and V. S. Pande. 2002. Simulation of folding of a small alphahelical protein in at omistic detail using worldwidedistributed computing. J. Mol. Biol 323:927. 31 PAGE 32 BIOGRAPHICAL SKETCH I received my Bachelor of Engineering de gree in computer science and technology from Beihang University, one of the top schools in China. During my undergraduate study there, I have been selected as a visiting student to study at the University of Alberta in Canada for one year, where I learned computer gr aphics and developed an interest in related fields. After my graduate, I continued my study in computer engineer ing at the University of Florida as a masters candidate. Because of my excellent academic record s there, I have been selected for Outstanding Achievement Awards twice. 32 