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Hou et al. BMC Genetics 2012, 13:91 http://www.biomedcentral.com/14712156/13/91 GenetiBMC Genetics Systems mapping of HIV1 infection Wei Hou1'2, Yihan Suil, Zhong Wang3, Yaqun Wang3, Ningtao Wang3, Jingyuan Liu3, Yao Li4, Maureen Goodenow5, Li Yin5, Zuoheng Wang6 and Rongling Wul'3* Abstract Mathematical models of viral dynamics in vivo provide incredible insights into the mechanisms for the nonlinear interaction between virus and host cell populations, the dynamics of viral drug resistance, and the way to eliminate virus infection from individual patients by drug treatment. The integration of these mathematical models with highthroughput genetic and genomic data within a statistical framework will raise a hope for effective treatment of infections with HIV virus through developing potent antiviral drugs based on individual patients' genetic makeup. In this opinion article, we will show a conceptual model for mapping and dictating a comprehensive picture of genetic control mechanisms for viral dynamics through incorporating a group of differential equations that quantify the emergent properties of a system. Introduction To control HIV1 virus, antiviral drugs have been devel oped to prevent the infection of new viral cells or stop alreadyinfected cells from producing infectious virus particles by inhibiting specific viral enzymes [1,2]. Because of the multifactorial complexity of viralhost association, however, the development and delivery of clinically more beneficial novel antiviral drugs have proved a difficult goal [3]. In this essay, we argue that this bottleneck may be overcome by merging two recent advances in mathematical biology and genotyping tech niques toward precision medicine. First, viraldrug inter actions constitute a complex dynamic system, in which different types of viral cells, including uninfected cells, infected cells, and free virus particles, cooperate with each other and together fight with host immune cells to determine the pattern of viral change in response to drugs [46]. A number of sophisticated mathematical models have been developed to describe viral dynamics in vivo, providing incredible insights into the mechan isms for the nonlinear interaction between virus and host cell populations, the dynamics of viral drug resistance, and the way to eliminate virus infection from patients by drug treatment [715]. Second, the combination between *Correspondence rwu@phs psuedu Center for Computational Biology, Beijing Forestry University, Beijing 100081, China Center for Statistical Genetics, Pennsylvania State University, Hershey, PA 17033, USA Full list of author information is available at the end of the article novel instruments and an increasing understanding of mo lecular genetics has led to the birth of highthroughput genotyping assays such as single nucleotide polymorphisms (SNPs). Through mapping or associating concrete nucleo tides or their combinations with the dynamic process of HIV infection [16,17], we can precisely taxonomize this disease by its underlying genomic and molecular causes, thereby enabling the application of precision medicine to diagnose and treat it. Systems mapping: a novel tool to dissect complex traits Beyond a traditional mapping strategy focusing on the static performance of a trait, systems mapping dissolves the phenotype of the trait into its structural, functional or metabolic components through design principles of biological systems, maps the interrelationships and coordination of these components and identifies genes involved in the key pathways that cause the endpoint phenotype [1823]. Systems mapping not only preserves the capacity of functional mapping [2426] to study the dynamic pattern of genetic control on a time and space scale, but also shows a unique advantage in revealing the dynamic behavior of the genetic correlations among different but developmentally related traits. Its methodo logical innovation is to integrate mathematical aspects of phenotype formation and progression into a genetic mapping framework to test the interplay between genes and development. Various differential equations which have been instrumental for studying nonlinear and 2012 Hou et al., licensee BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Biole led Central Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Hou et al. BMC Genetics 2012, 13:91 http://www.biomedcentral.com/14712156/13/91 complex dynamics in engineering [27] have shown increasing value and power to quantify the emergent properties of a biological system and interpret experi mental results [912,28,29]. The past two decades have witnessed an excellent suc cess in modeling HIV dynamics with differential equa tions [912]. Treating viralhost interactions as a system, Appendix 1 gives a basic model composed of three ordinary differential equations (ODE) for describing the shortterm overall dynamics of uninfected cells (x), infected cells (y), and free virus particles (v). These three components together determine the extent and process of pathogenesis according to six ODE parameters, i.e., the rates of production and death of uninfected cells, the rate of production of infected cells from free viruses, the rate of death of infected cells, and the rates of produc tion and death of new viruses from infected cells. Thus, by changing the values of these parameters singly or in combination, the dynamic properties of viral infection, such as viral half life, the limiting ratio of infected to uninfected cells, and the basic reproductive ratio of the virus, can be quantified and predicted [10]. By embed ding a system of ODEs within a mixture model frame work (Appendix 1), we can use systems mapping to identify specific host genes and their interactions for the pattern of viral dynamics and infection inside a host body. Figure 1 illustrates the characterization of a hypothesized gene that contributes to variation in viral dynamic behavior. Per these genotypespecific changes, an optimal strategy for HIV treatment in terms of the dose and time at which an antiviral drug is admini strated can be determined, thus providing a first step toward personalized medicine [23]. In practice, a drug may be resisted if HIV1 viruses mutate to create new strains [30]. The emergence of drug resistance is a consequence of evolution and pre sents a response to pressures imposed on the viruses. Different viruses vary in their sensitivity to the drug used and some with greater fitness may be capable of sur viving drug treatment [31,32]. In order to understand how viruses are resistant to drugs through mutation, the basic model of Appendix 1 should be expanded to include three additional variables, cells infected by mutant virus, mutant virus particles, and the probability of mutation from wildtype to resistant mutant during reverse transcription of viral RNA into proviral DNA [9]. Systems mapping shows tremendous power to detect genes for virus drug resistance [21] and predict the dy namics of drug resistance (Figure 2). Systems mapping can not only better interpret the genetic mechanisms of drug resistance from experimental data, but also pro vide scientific guidance on the administration of new antiviral drugs. Mapping triple genome interactions It has been widely accepted that the symptoms and severity of infectious diseases are determined by pathogen host specificity through cellular, biochemical and signal exchanges [4,3335]. This specificity, established by undermining a host's immunological ability to mount an immune response against a particular pathogen, is found to be under genetic determination. Current genetic stud ies of pathogenhost systems focus on either the host or the pathogen genome, but there is increasing recognition that the complete genetic architecture of pathogenhost specificity, described by the number, position, effect, plei otropy, and epistasis among genes, involves interactive components from both host and viral genomes [3538]. In other words, the infection phenotype does not merely result from additive effects of host and pathogen geno types, but also from specific interactions between the two genomes [35,37]. While many molecular studies define pathogenhost interactions, regardless of the type of hosts, epidemiological AA V.\ 64.7 *' X 30.0 "Y 19.4 0 10 20 30 40 50 60 Time (Days) Aa y 0 10 20 30 40 50 60 Time (Days) aa 0 10 20 30 40 50 60 Time (Days) Figure 1 Numerical simulation showing how a gene affects the dynamics of HIV1 infection, composed of uninfected cells (x), infected cells (y), and virus particles (v), as described by a basic model (1) in Appendix 1. The simulated gene has three genotypes M, Aa and ao, each displaying a different time trajectory for each of these three cell types. Based on these differences, one can test and determine how the gene affects the emerging properties of viral dynamic system, such as average lifetimes of different cell types and the points of three variables (indicated by triangles) when the system converges to an equilibrium state. The parameter values are (A, d, /3, a, k, u) (10, 0.01, 0.005, 0.5, 10, 3), (12, 0.01, 0.005, 0.6, 8, 3), and (12, 0.008, 0.005, 0.55, 8, 4) for genotypes AA, A and aoo, respectively. Page 2 of 7 Hou et al. BMC Genetics 2012, 13:91 http://www.biomedcentral.com/14712156/13/91 0 A 0 10 20 30 40 50 60 0 10 20 30 40 50 60 Time (Days) Figure 2 Simulated genotypespecific differences in the dynamics of drug resistance as described by a model (2) in Appendix 1. The system simulation focuses on uninfected cell, x (A), infected cells, y, for wildtype virus (solid line) and mutant virus (dash lines) (B), and free virus, v, for wildtype virus (solid line) and mutant virus (dash line) (C), and relative frequency of mutant virus in free virus (solid line) and infected cell population (dash line) (D). models distinguish the difference of hosts as a recipient and transmitter to better characterize the epidemic struc ture of disease infection, given that infectious diseases like HIV/AIDS are transmitted from an infected person to an other [3941]. From this point of view, the infection outcome should be determined differently but simultan eously by genes from transmitters and recipients. To chart a comprehensive picture of genetic control mechan isms for viral dynamics, we need to address the questions of how genes from viral and host genomes interact to in fluence viral dynamics and how genetic interactions between recipients and transmitters of virus play a part in the dynamic behavior of viruses. Li et al. [42] pioneered the unification of quantitative genetic theory and epi demiological dynamics for characterizing triplegenome interactions from viruses, transmitters and recipients. Systems mapping described in Appendix 2 should be embedded within Li et al.'s [42] unifying model to Page 3 of 7 0 10 20 30 40 50 60 Hou et al. BMC Genetics 2012, 13:91 http://www.biomedcentral.com/14712156/13/91 include the interactions of genes derived from the three genomes. This integration allows main genetic effects and epistatic interactions expressed at the genome level to be tested and characterized, including additive effects from the haploidd) viral genome, additive and dominant effects from the transmitter genome, additive and dom inant effect from the recipient genome as well as all possible interactions among these main effects. It is interesting to note that the integrated system mapping is capable of estimating and testing highorder epistasis from the viral, recipient and transmitter genomes. Given a growing body of evidence that highorder epistasis is an important determinant of the genetic architecture of complex traits [4345], systems mapping should be equipped with triple genome interaction modeling. It should be pointed out that virus evolves through gene recombination and mutations. The genetic machin eries that cause viral evolution can be incorporated into systems mapping without technical difficulty. Through such incorporation, systems mapping will provide a use ful and timely incentive to detect the genetic control mechanisms of viral dynamics and antivirus drug resist ance dynamics and ultimately to design personalized medicine to treat HIV1 infection from increasingly available genome and HIV data worldwide. Toward precision medicine A major challenge that faces drug development and delivery for controlling viral diseases is to develop com putational models for analyzing and predicting the dynamics of decline in virus load during drug therapy and further providing estimates of the rate of emergence of resistant virus. The integration of wellestablished mathematical models for viral dynamics with high throughput genetic and genomic data within a statistical framework will raise a hope for effective diagnosis and treatment of infections with HIV virus through develop ing potent antiviral drugs based on individual patients' genetic makeup. In this opinion article, we have provided a synthetic framework for systems mapping of viral dynamics dur ing its progression to AIDS. This framework is equipped with unified mathematical and statistical power to extract genetic information from messy data and possess the analytical and modeling efficiency which does not exist for traditional approaches. By fitting the rate of change of virus infection with clinically meaningful mathematical models, the spatiotemporal pattern of genetic control can be illustrated and predicted over a range of time and space scales. Statistical modeling allows the estimation of mathematical parameters that specify genetic effects on viral dynamics. By genotyping both host and viral genomes, systems mapping is able to identify which viral genes and which human genes from recipients and transmitters determine viral dynamics additively or through nonlinear interactions. In this sense, it paves a new way to chart a comprehensive picture of the genetic architecture of viral infection. An increasing trend in drug development is to inte grate it with systems biology aimed to gain deep insights into biological responses. Largescale gene, protein and metabolite (omics) data that found the building blocks of complex systems have become essential parts of the drug industry to design and deliver new drug [46,47]. However, the true wealth of systems biology will critic ally rely upon the way of how to incorporate it into human cell and tissue function that affects pathogen esis. By integrating knowledge of organ and systemlevel responses and omics data, systems mapping will help to prioritize targets and design clinical trials, promising to improve decision making in pharmaceutical development. Appendix 1. Mathematical models of viral dynamics Basic model Bonhoeffer et al. [10] developed a basic model for short term virus dynamics. The model includes three variables: uninfected cells, x, infected cells, y, and free virus parti cles, v. These three types of cells interact with each other to determine the dynamic changes of virus in a host's body, which can be described by a system of differential equations: A dx fxv fxv ay ky uv where uninfected cells are yielded at a constant rate, 1, and die at the rate dx; free virus infects uninfected cells to yield infected cells at rate fixv; infected cells die at rate ay; and new virus is yielded from infected cells at rate ky and dies at rate uv. The system (1) is defined by six parameters (A,d,p,a,k,u) and some initial conditions about x, y, and v. The dynamic pattern of this system can be determined and predicted by the change of these parameters and the initial conditions of x, y, and v. The basic reproductive ratio of the virus is defined as Ro = fAk/(adu). If Ro is larger than one, then system converges in damped oscillations to the equilibrium x* = au/(pfk), y* = /a  du/(pk), and v* = Ak/(au) d/p. The average lifetimes of uninfected cells, infected cells, and free virus are given by 1/d, 1/a, and 1/u, respectively. The average number of virus particles produced over the lifetime of a single infected cell (the burst size) is given by k/a. Page 4 of 7 Hou et al. BMC Genetics 2012, 13:91 http://www.biomedcentral.com/14712156/13/91 Resistance model When a treatment is used to control HIV1, the viruses will produce the resistance to the drug through mutation. The dynamics of drug resistance can be modeled by A dx fxv imxvm i(1 E)xv ay PExv + imXVm aym ky uv kmym UVm where y, ym, v, and Vm denote cells infected by wildtype virus, cells infected by mutant virus, free wildtype virus, and free mutant virus, respectively [10]. The mutation rate between wildtype and mutant is given by e (in both directions). For a small e, the basic reproductive ratios of wildtype and mutant virus are Ro = f/Ak/(adu) and Rom = imAkm(adu). Model (2) shows that the expected pretreatment frequency of resistant mutant depends on the number of point mutations between wildtype and resistant mutant, the mutation rate of virus replication, and the relative replication rates of wildtype virus, resistant mutant, and all intermediate mutants. Whether or not resistant virus is present in a patient before ther apy will crucially depend on the population size of infected cells. Cell diversity model The infected cells may harbor actively replicating virus (yl), latent virus (y2) and defective virus (ys). The basic model (1) can be expanded to include these three types, expressed as A dx fxv qwpxv awyw, w kyi + cy2 uv 1,2,3 (3) where q1, q2, and q3 (q, + q2 + q3 = 1) are the propor tions that the cell will immediately enter active viral rep lication at a rate of virus production k, become latently infected with the virus at a (much slower) rate of virus production c, and produce a defective provirus that will not produce any offspring virus, respectively; and a,, a2, and a3 are the decay rates of actively produ cing cells, latently infected cells, and defectively infected cells, respectively. The basic reproductive ratio of the wildtype is Ro = PAA/(du). If Ro is larger than one, then system con verges to the equilibrium x = ul(pA), y = (A ), y2 =2f y y1=3 i y* and v*= IA where A= kq + Ial al2 A full model of viral dynamics can be obtained by uni fying the resistance model and cell diversity model to form a system of nine ODEs, expressed as yw Ywm (2) Vm A dx fxv mxvm qwf(1 e)xv awyw, w = 1,2, 3 qpExv + qw3mixvm awywm, w = 1, 2, 3 ky5 + cy2 MV kmylm + Cmy2m UVm This group of ODEs provides a comprehensive descrip tion of how viral loads change their rate in a time course, how infected cells are generated in response to the emergence of viral particles, and how viral mutation impacts on viral dynamics and drug resistance dynamics. The emerging properties of system (4) were discussed in ref. [10], which can be integrated with systems mapping described in Appendix 2. Appendix 2. Systems mapping of viral dynamics Systems mapping allows the genes and genetic interactions for viral dynamics to be identified by incorporating ODEs into a mapping framework. Consider a segregating popula tion composed of n HIVinfected patients genotyped for a set of molecular markers. These patients were repeated sampled to measure uninfected cells (x), infected cells (y) and viral load (v) in their plasma at a series of time points. If specific genes exist to affect the system (1) in Appendix 1, the parameters that specify the system should be different among genotypes. Genetic mapping uses a mix ture modelbased likelihood to estimate genotypespecific parameters. This likelihood is expressed as n L(x;y;v) =7 [&)(ifl(xi, y,vi) +... + G)(Jli(xi,yivi)l i=i (1) where xi = (xi(t), ..., x(tTi)) y = (y(ti), ..., y(tTi)) and vi = (Vi(ti), ..., vi(tTi)) are the phenotypic values of x, y, and v for subject i measured at Ti time points, o(il is the conditional probability of QTL genotype (j = 1, ..., ]) given the marker genotype of patient i, andf (xi,yi,vi) is a multivariate normal distribution with expected mean vector for patient i that belongs to genotype j, (m I m I m vjli)(mxjli(t), ,il I (tTi);;i, (tl ),l , in, I (tTi); mvli(ti),..., mVji (tTi)) (2) and covariance matrix for subject i, ( Yi = yx, Y\V', Ly, Lyev Zvy, Eve yY y,,, vYi Yvi yY y_ ) with Xx,, XY, and Xv, being (Ti x Ti) covariance matrices of timedependent x, y and v values, respectively, and Page 5 of 7 x, Hou et al. BMC Genetics 2012, 13:91 http://www.biomedcentral.com/14712156/13/91 elements offdiagonal being a (T, x Ti) systematical covariance matrix between the two variables. For a natural population, the conditional probability of functional genotype given a marker genotype (jl;;) is expressed in terms of the linkage disequilibria between different loci [48]. In systems mapping, we incorporate ODEs (1) of Appendix 1 into mixture model (1) to esti mate genotypic means (2) specified by ODE param eters for different genotypes, expressed as (Aj,dj,p1,aj,kj,uj) for j = 1, ..., J. Since x, y and v variables obey dynamic system (1) of Appendix 1, the derivatives of genotypic means can be expressed in a similar way. Let glii(t,jyli) denote the genotypic derivative for variable k (k = x, y, or z), i.e., dt We use flji to denote the genotypic mean of variable j for individual i belonging to genotype j at an arbitrary point in a time course. The RungeKutta fourth order algorithm can be used to solve the ODEs. Next, we need to model the covariance structure by using a parsimonious and flexible approach such as an autoregressive, antedependence, autoregressive moving average, or nonparametric and semiparametric approaches. Yap et al. [49] provided a discussion of how to choose a general approach for covariance structure modeling. In likelihood (1), the conditional probabilities of functional genotypes given marker genotypes can be expressed as a function of recombination fractions for an experimental cross population or linkage disequilibria for a natural population [48,50]. The estimation of the recombination fractions or linkage disequilibria can be implemented with the ExpectationMaximization (EM) algorithm. To demonstrate the usefulness of systems mapping, we assume a sample of n HIVinfected patients drawn from a natural human population at random. The sam ple is analyzed by systems mapping, leading to the detec tion of a molecular marker which is associated with a QTL that determines the dynamics of drug resistance in a way described by (2) in Appendix 1. At the QTL detected, there are three genotypes AA, Aa and aa, each with a different set of curve parameters (A, d, f!, P3m, a, k, k,,, u, e) estimated by systems mapping. We assume that these parameters are estimated as (10, 0.01, 0.005, 0.02, 0.5, 10, 10, 3, 0.0001) for genotype AA, (12, 0.01, 0.005, 0.02, 0.6, 8, 8, 3, 0.0001) for genotype Aa, and (12, 0.008, 0.005, 0.02, 0.55, 8, 12, 4, 0.0001) for genotype aa. Using these estimated values, we draw the curves of drug resistance dynamics for each genotype (Figure 2). Pro nounced differences in the form of these curves indicate that the QTL plays an important part in determining the resistance dynamics of drugs used to treat HIV/AIDS. The model for systems mapping described above can be expanded in two aspects, mathematical and genetic, to better characterize the genetic architecture of viral dynamics. The mathematical expansions are to incorpor ate the drug resistance model (2), the cell diversity model (3) and the unifying resistance and cell diversity model (4). These expansions allow the functional genes operating at different pathways of viralhost reactions to be identified and mapped, making system mapping more clinically feasible and meaningful. The genetic expan sions aim to not only model individual genes from the host or pathogen genome but also characterize epistatic interactions between genes from different genomes. This can be done by expanding the conditional probability of functional genes given marker genotypes 6)jl using a framework derived by Li et al. [42]. By formulating and testing novel hypotheses, system mapping can address many basic questions. For example, they are 1) How do DNA variants regulate viral dynamics? 2) How do these genes affect the average lifetimes of uninfected cells, infected cells, and free virus, respectively? 3) How do genes determine the emergence and progression of drug resistance? 4) Are there specific genes that control the possibility of virus eradication by antiviral drug? 5) How important are genegene interactions and genomegenome interactions to the dynamic behavior of viral load with or without treatment? Acknowledgements This work is supported by Florida Center for AIDS Research Incentive Award, NIH/NIDA R01 DA031017, and NIH/UL1RRO330184 Author details Center for Computational Biology, Beijing Forestry University, Beijing 100081, China 2Department of Biostatistics, University of Florida, Gainesville, FL 32611, USA Center for Statistical Genetics, Pennsylvania State University, Hershey, PA 17033, USA 4Division of Public Health Sciences, Fred Hutchinson Cancer Research Center, Seattle, WA 98109, USA SDepartment of Pathology, Immunology and Laboratory Medicine, University of Florida, Gainesville, FL 32610, USA 6Division of Biostatistics, Yale University, New Haven, CT 06510, USA Received: 9 May 2012 Accepted: 27 September 2012 Published: 23 October 2012 References 1 Smith K, Powers KA, Kashuba AD, Cohen MS HIV1 treatment as prevention: the good, the bad, and the challenges. 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REPORT xmlns http:www.fcla.edudlsmddaitss xmlns:xsi http:www.w3.org2001XMLSchemainstance xsi:schemaLocation http:www.fcla.edudlsmddaitssdaitssReport.xsd INGEST IEID E8BF4ZRUO_UQRSFB INGEST_TIME 20130305T20:12:46Z PACKAGE AA00013471_00001 AGREEMENT_INFO ACCOUNT UF PROJECT UFDC FILES PAGE 1 CORRESPONDENCEOpenAccessSystemsmappingofHIV1infectionWeiHou1,2,YihanSui1,ZhongWang3,YaqunWang3,NingtaoWang3,JingyuanLiu3,YaoLi4,MaureenGoodenow5, LiYin5,ZuohengWang6andRonglingWu1,3*AbstractMathematicalmodelsofviraldynamics invivo provideincredibleinsightsintothemechanismsforthenonlinear interactionbetweenvirusandhostcellpopulations,thedynamicsofviraldrugresistance,andthewaytoeliminate virusinfectionfromindividualpatientsbydrugtreatment.Theintegrationofthesemathematicalmodelswith highthroughputgeneticandgenomicdatawithinastatisticalframeworkwillraiseahopeforeffectivetreatment ofinfectionswithHIVvirusthroughdevelopingpotentantiviraldrugsbasedonindividualpatients genetic makeup.Inthisopinionarticle,wewillshowaconceptualmodelformappinganddictatingacomprehensive pictureofgeneticcontrolmechanismsforviraldynamicsthroughincorporatingagroupofdifferentialequations thatquantifytheemergentpropertiesofasystem.IntroductionTocontrolHIV1virus,antiviraldrugshavebeendevelopedtopreventtheinfectionofnewviralcellsorstop alreadyinfectedcellsfromproducinginfectiousvirus particlesbyinhibitingspecificviralenzymes[1,2]. Becauseofthemultifactorialcomplexityofviralhost association,however,thedevelopmentanddeliveryof clinicallymorebeneficialnovelantiviraldrugshave provedadifficultgoal[3].Inthisessay,wearguethat thisbottleneckmaybeovercomebymergingtworecent advancesinmathematicalbiologyandgenotypingtechniquestowardprecisionmedicine.First,viraldruginteractionsconstituteacomplexdynamicsystem,inwhich differenttypesofviralcells,includinguninfectedcells, infectedcells,andfreevirusparticles,cooperatewith eachotherandtogetherfightwithhostimmunecellsto determinethepatternofviralchangeinresponseto drugs[46].Anumberofsophisticatedmathematical modelshavebeendevelopedtodescribeviraldynamics invivo ,providingincredibleinsightsintothemechanismsforthenonlinearinteractionbetweenvirusand hostcellpopulations,thedynamicsofviraldrugresistance, andthewaytoeliminatevirusinfectionfrompatientsby drugtreatment[715].Second,thecombinationbetween novelinstrumentsandanincreasingunderstandingofmoleculargeneticshasledtothebirthofhighthroughput genotypingassayssuchassing lenucleotidepolymorphisms (SNPs).Throughmappingorassociatingconcretenucleotidesortheircombinations withthedynamicprocessof HIVinfection[16,17],wecanpreciselytaxonomizethis diseasebyitsunderlyinggenomicandmolecularcauses, therebyenablingtheapplicationofprecisionmedicineto diagnoseandtreatit.Systemsmapping:anoveltooltodissect complextraitsBeyondatraditionalmappingstrategyfocusingonthe staticperformanceofatrait,systemsmappingdissolves thephenotypeofthetraitintoitsstructural,functional ormetaboliccomponentsthroughdesignprinciples ofbiologicalsystems,mapstheinterrelationshipsand coordinationofthesecomponentsandidentifiesgenes involvedinthekeypathwaysthatcausetheendpoint phenotype[1823].Systemsmappingnotonlypreserves thecapacityoffunctionalmapping[2426]tostudythe dynamicpatternofgeneticcontrolonatimeandspace scale,butalsoshowsauniqueadvantageinrevealingthe dynamicbehaviorofthegeneticcorrelationsamong differentbutdevelopmentallyrelatedtraits.Itsmethodologicalinnovationistointegratemathematicalaspects ofphenotypeformationandprogressionintoagenetic mappingframeworktotesttheinterplaybetweengenes anddevelopment.Variousdifferentialequationswhich havebeeninstrumentalforstudyingnonlinearand *Correspondence: rwu@phs.psu.edu1CenterforComputationalBiology,BeijingForestryUniversity,Beijing 100081,China3CenterforStatisticalGenetics,PennsylvaniaStateUniversity,Hershey,PA 17033,USA Fulllistofauthorinformationisavailableattheendofthearticle 2012Houetal.;licenseeBioMedCentralLtd.ThisisanOpenAccessarticledistributedunderthetermsoftheCreative CommonsAttributionLicense(http://creativecommons.org/licenses/by/2.0),whichpermitsunrestricteduse,distribution,and reproductioninanymedium,providedtheoriginalworkisproperlycited.Hou etal.BMCGenetics 2012, 13 :91 http://www.biomedcentral.com/14712156/13/91 PAGE 2 complexdynamicsinengineering[27]haveshown increasingvalueandpowertoquantifytheemergent propertiesofabiologicalsystemandinterpretexperimentalresults[912,28,29]. ThepasttwodecadeshavewitnessedanexcellentsuccessinmodelingHIVdynamicswithdifferentialequations[912].Treatingviralhostinteractionsasasystem, Appendix1givesabasicmodelcomposedofthree ordinarydifferentialequations(ODE)fordescribingthe shorttermoveralldynamicsofuninfectedcells( x ), infectedcells( y ),andfreevirusparticles( v ).Thesethree componentstogetherdeterminetheextentandprocess ofpathogenesisaccordingtosixODEparameters,i.e., theratesofproductionanddeathofuninfectedcells,the rateofproductionofinfectedcellsfromfreeviruses,the rateofdeathofinfectedcells,andtheratesofproductionanddeathofnewvirusesfrominfectedcells.Thus, bychangingthevaluesoftheseparameterssinglyorin combination,thedynamicpropertiesofviralinfection, suchasviralhalflife,thelimitingratioofinfectedto uninfectedcells,andthebasicreproductiveratioofthe virus,canbequantifiedandpredicted[10].ByembeddingasystemofODEswithinamixturemodelframework(Appendix1),wecanusesystemsmappingto identifyspecifichostgenesandtheirinteractionsfor thepatternofviraldynamicsandinfectioninsidea hostbody.Figure1illustratesthecharacterizationofa hypothesizedgenethatcontributestovariationinviral dynamicbehavior.Perthesegenotypespecificchanges, anoptimalstrategyforHIVtreatmentintermsofthe doseandtimeatwhichanantiviraldrugisadministratedcanbedetermined,thusprovidingafirststep towardpersonalizedmedicine[23]. Inpractice,adrugmayberesistedifHIV1viruses mutatetocreatenewstrains[30].Theemergenceof drugresistanceisaconsequenceofevolutionandpresentsaresponsetopressuresimposedontheviruses. Differentvirusesvaryintheirsensitivitytothedrugused andsomewithgreaterfitnessmaybecapableofsurvivingdrugtreatment[31,32].Inordertounderstand howvirusesareresistanttodrugsthroughmutation, thebasicmodelofAppendix1shouldbeexpandedto includethreeadditionalvariables,cellsinfectedby mutantvirus,mutantvirusparticles,andtheprobability ofmutationfromwildtypetoresistantmutantduring reversetranscriptionofviralRNAintoproviralDNA [9].Systemsmappingshowstremendouspowertodetect genesforvirusdrugresistance[21]andpredictthedynamicsofdrugresistance(Figure2).Systemsmapping cannotonlybetterinterpretthegeneticmechanisms ofdrugresistancefromexperimentaldata,butalsoprovidescientificguidanceontheadministrationofnew antiviraldrugs.MappingtriplegenomeinteractionsIthasbeenwidelyacceptedthatthesymptomsand severityofinfectiousdiseasesaredeterminedbypathogenhostspecificitythroughcellular,biochemicalandsignal exchanges[4,3335].Thisspecificity,establishedby underminingahost simmunologicalabilitytomountan immuneresponseagainstaparticularpathogen,isfound tobeundergeneticdetermination.Currentgeneticstudiesofpathogenhostsystemsfocusoneitherthehostor thepathogengenome,butthereisincreasingrecognition thatthecompletegeneticarchitectureofpathogenhost specificity,describedbythenumber,position,effect,pleiotropy,andepistasisamonggenes,involvesinteractive componentsfrombothhostandviralgenomes[3538]. Inotherwords,theinfectionphenotypedoesnotmerely resultfromadditiveeffectsofhostandpathogengenotypes,butalsofromspecificinteractionsbetweenthe twogenomes[35,37]. Whilemanymolecularstudiesdefinepathogenhost interactions,regardlessofthe typeofhosts,epidemiological Figure1 NumericalsimulationshowinghowageneaffectsthedynamicsofHIV1infection,composedofuninfectedcells( x ),infected cells( y ),andvirusparticles( v ),asdescribedbyabasicmodel(1)inAppendix1. Thesimulatedgenehasthreegenotypes AA Aa and aa eachdisplayingadifferenttimetrajectoryforeachofthesethreecelltypes.Basedonthesedifferences,onecantestanddeterminehowthe geneaffectstheemergingpropertiesofviraldynamicsystem,suchasaveragelifetimesofdifferentcelltypesandthepointsofthreevariables (indicatedbytriangles)whenthesystemconvergestoanequilibriumstate.Theparametervaluesare( d a k u )=(10,0.01,0.005,0.5,10,3), (12,0.01,0.005,0.6,8,3),and(12,0.008,0.005,0.55,8,4)forgenotypes AA Aa and aa ,respectively. Hou etal.BMCGenetics 2012, 13 :91 Page2of7 http://www.biomedcentral.com/14712156/13/91 PAGE 3 modelsdistinguishthedifferenceofhostsasarecipient andtransmittertobettercharacterizetheepidemicstructureofdiseaseinfection,giventhatinfectiousdiseaseslike HIV/AIDSaretransmittedfromaninfectedpersontoanother[3941].Fromthispointofview,theinfection outcomeshouldbedetermineddifferentlybutsimultaneouslybygenesfromtransmittersandrecipients.To chartacomprehensivepictureofgeneticcontrolmechanismsforviraldynamics,weneedtoaddressthequestions ofhowgenesfromviralandhostgenomesinteracttoinfluenceviraldynamicsandhowgeneticinteractions betweenrecipientsandtransmittersofvirusplayapartin thedynamicbehaviorofviruses.Lietal.[42]pioneered theunificationofquantitativegenetictheoryandepidemiologicaldynamicsforcharacterizingtriplegenome interactionsfromviruses,transmittersandrecipients. SystemsmappingdescribedinAppendix2should beembeddedwithinLietal. s[42]unifyingmodelto Figure2 Simulatedgenotypespecificdifferencesinthedynamicsofdrugresistanceasdescribedbyamodel(2)inAppendix1. Thesystemsimulationfocusesonuninfectedcell, x ( A ),infectedcells, y ,forwildtypevirus(solidline)andmutantvirus(dashlines)( B ),and freevirus, v ,forwildtypevirus(solidline)andmutantvirus(dashline)( C ),andrelativefrequencyofmutantvirusinfreevirus(solidline)and infectedcellpopulation(dashline)( D ). Hou etal.BMCGenetics 2012, 13 :91 Page3of7 http://www.biomedcentral.com/14712156/13/91 PAGE 4 includetheinteractionsofgenesderivedfromthethree genomes.Thisintegrationallowsmaingeneticeffects andepistaticinteractionsexpressedatthegenomelevel tobetestedandcharacterized,includingadditiveeffects fromthe(haploid)viralgenome,additiveanddominant effectsfromthetransmittergenome,additiveanddominanteffectfromtherecipientgenomeaswellasall possibleinteractionsamongthesemaineffects.Itis interestingtonotethattheintegratedsystemmapping iscapableofestimatingandtestinghighorderepistasis fromtheviral,recipientandtransmittergenomes.Given agrowingbodyofevidencethathighorderepistasisis animportantdeterminantofthegeneticarchitectureof complextraits[4345],systemsmappingshouldbe equippedwithtriplegenomeinteractionmodeling. Itshouldbepointedoutthatvirusevolvesthrough generecombinationandmutations.Thegeneticmachineriesthatcauseviralevolutioncanbeincorporatedinto systemsmappingwithouttechnicaldifficulty.Through suchincorporation,systemsmappingwillprovideausefulandtimelyincentivetodetectthegeneticcontrol mechanismsofviraldynamicsandantivirusdrugresistancedynamicsandultimatelytodesignpersonalized medicinetotreatHIV1infectionfromincreasingly availablegenomeandHIVdataworldwide.TowardprecisionmedicineAmajorchallengethatfacesdrugdevelopmentand deliveryforcontrollingviraldiseasesistodevelopcomputationalmodelsforanalyzingandpredictingthe dynamicsofdeclineinvirusloadduringdrugtherapy andfurtherprovidingestimatesoftherateofemergence ofresistantvirus.Theintegrationofwellestablished mathematicalmodelsforviraldynamicswithhighthroughputgeneticandgenomicdatawithinastatistical frameworkwillraiseahopeforeffectivediagnosisand treatmentofinfectionswithHIVvirusthroughdevelopingpotentantiviraldrugsbasedonindividualpatients geneticmakeup. Inthisopinionarticle,wehaveprovidedasynthetic frameworkforsystemsmappingofviraldynamicsduringitsprogressiontoAIDS.Thisframeworkisequipped withunifiedmathematicalandstatisticalpowerto extractgeneticinformationfrommessydataandpossess theanalyticalandmodelingefficiencywhichdoesnot existfortraditionalapproaches.Byfittingtherateof changeofvirusinfectionwithclinicallymeaningful mathematicalmodels,thespatiotemporalpatternof geneticcontrolcanbeillustratedandpredictedovera rangeoftimeandspacescales.Statisticalmodeling allowstheestimationofmathematicalparametersthat specifygeneticeffectsonviraldynamics.Bygenotyping bothhostandviralgenomes,systemsmappingisableto identifywhichviralgenesandwhichhumangenesfrom recipientsandtransmittersdetermineviraldynamics additivelyorthroughnonlinearinteractions.Inthis sense,itpavesanewwaytochartacomprehensive pictureofthegeneticarchitectureofviralinfection. Anincreasingtrendindrugdevelopmentistointegrateitwithsystemsbiologyaimedtogaindeepinsights intobiologicalresponses.Largescalegene,proteinand metabolite(omics)datathatfoundthebuildingblocks ofcomplexsystemshavebecomeessentialpartsofthe drugindustrytodesignanddelivernewdrug[46,47]. However,thetruewealthofsystemsbiologywillcriticallyrelyuponthewayofhowtoincorporateitinto humancellandtissuefunctionthataffectspathogenesis.Byintegratingknowledgeoforganandsystemlevel responsesandomicsdata,systemsmappingwillhelpto prioritizetargetsanddesignclinicaltrials,promisingto improvedecisionmakinginpharmaceuticaldevelopment.Appendix1.Mathematicalmodelsof viraldynamicsBasicmodelBonhoefferetal.[10]developedabasicmodelforshorttermvirusdynamics.Themodelincludesthreevariables: uninfectedcells, x ,infectedcells, y ,andfreevirusparticles, v .Thesethreetypesofcellsinteractwitheach othertodeterminethedynamicchangesofvirusina host sbody,whichcanbedescribedbyasystemof differentialequations: x dx xv y xv ay v ky uv 1 whereuninfectedcellsareyieldedataconstantrate, anddieattherate dx ;freevirusinfectsuninfectedcells toyieldinfectedcellsatrate xv ;infectedcellsdieat rate ay ;andnewvirusisyieldedfrominfectedcellsat rate ky anddiesatrate uv .Thesystem(1)isdefinedby sixparameters( d a k u )andsomeinitialconditions about x y ,and v Thedynamicpatternofthissystemcanbedetermined andpredictedbythechangeoftheseparametersandthe initialconditionsof x y ,and v .Thebasicreproductive ratioofthevirusisdefinedas R0= k /( adu ).If R0islargerthanone,thensystemconvergesindamped oscillationstotheequilibrium x*= au /( k ), y*= / a du /( k ),and v*= k /( au ) d / .Theaveragelifetimes ofuninfectedcells,infectedcells,andfreevirusaregiven by1/ d ,1/ a ,and1/ u ,respectively.Theaveragenumber ofvirusparticlesproducedoverthelifetimeofasingle infectedcell(theburstsize)isgivenby k / a .Hou etal.BMCGenetics 2012, 13 :91 Page4of7 http://www.biomedcentral.com/14712156/13/91 PAGE 5 ResistancemodelWhenatreatmentisusedtocontrolHIV1,theviruses willproducetheresistancetothedrugthrough mutation.Thedynamicsofdrugresistancecanbe modeledby x dx xv mxvmy 1 xv ay ym_ xv mxvm aymv ky uv vm_ kmym uvm 2 where y ym, v ,and vmdenotecellsinfectedbywildtype virus,cellsinfectedbymutantvirus,freewildtypevirus, andfreemutantvirus,respectively[10].Themutation ratebetweenwildtypeandmutantisgivenby (inboth directions).Forasmall ,thebasicreproductiveratios ofwildtypeandmutantvirusare R0= k /( adu )and R0 m= m km/( adu ). Model(2)showsthattheexpectedpretreatment frequencyofresistantmutantdependsonthenumber ofpointmutationsbetweenwildtypeandresistant mutant,themutationrateofvirusreplication,andthe relativereplicationratesofwildtypevirus,resistant mutant,andallintermediatemutants.Whetheror notresistantvirusispresentinapatientbeforetherapywillcruciallydependonthepopulationsizeof infectedcells.CelldiversitymodelTheinfectedcellsmayharboractivelyreplicatingvirus ( y1),latentvirus( y2)anddefectivevirus( y3).Thebasic model(1)canbeexpandedtoincludethesethreetypes, expressedas x dx xv yw_ qw xv awyw; w 1 ; 2 ; 3 v ky1 cy2 uv 3 where q1, q2,and q3( q1+ q2+ q3=1)aretheproportionsthatthecellwillimmediatelyenteractiveviralreplicationatarateofvirusproductionk,becomelatently infectedwiththevirusata(muchslower)rateof virusproductionc,andproduceadefectiveprovirus thatwillnotproduceanyoffspringvirus,respectively; and a1, a2,and a3arethedecayratesofactivelyproducingcells,latentlyinfectedcells,anddefectivelyinfected cells,respectively. Thebasicreproductiveratioofthewildtypeis R0= A /( du ).If R0islargerthanone,thensystemconvergestotheequilibrium x*= u /( A ), y 1 q1a1 du A ; y 2 a1a2 q2q1y 1; y 3 a1a3 q3q1y 1,and v uA d ,where A kq1a1 cq2a2. Afullmodelofviraldynamicscanbeobtainedbyunifyingtheresistancemodelandcelldiversitymodelto formasystemofnineODEs,expressedasx dx xv mxvmyw_ qw 1 xv awyw; w 1 ; 2 ; 3 ywm_ qw xv qwmxvm awywm; w 1 ; 2 ; 3 v ky1 cy2 uv vm_ kmy1 m cmy2 m uvm 4 ThisgroupofODEsprovidesacomprehensivedescriptionofhowviralloadschangetheirrateinatimecourse, howinfectedcellsaregeneratedinresponsetothe emergenceofviralparticles,andhowviralmutation impactsonviraldynamicsanddrugresistancedynamics. Theemergingpropertiesofsystem(4)werediscussedin ref.[10],whichcanbeintegratedwithsystemsmapping describedinAppendix2.Appendix2.SystemsmappingofviraldynamicsSystemsmappingallowsthegenesandgeneticinteractions forviraldynamicstobeidentifiedbyincorporatingODEs intoamappingframework.Considerasegregatingpopulationcomposedof n HIVinfectedpatientsgenotypedfora setofmolecularmarkers.Thesepatientswererepeated sampledtomeasureuninfectedcells( x ),infectedcells ( y )andviralload( v )intheirplasmaataseriesoftime points.Ifspecificgenesexisttoaffectthesystem(1)in Appendix1,theparametersthatspecifythesystemshould bedifferentamonggenotypes.Geneticmappingusesamixturemodelbasedlikelihoodtoestimategenotypespecific parameters.ThislikelihoodisexpressedasL x ; y ; v Yn i 11 j i f1xi; yi; vi ... J j i fJxi; yi; vi 1 wherexi=( xi( t1), ... x (tTi)),yi=( yi( t1), ... y (tTi))and vi=( vi( t1), ... vi(tTi))arethephenotypicvaluesof x y and v forsubject i measuredat Titimepoints, jiisthe conditionalprobabilityofQTLgenotype j ( j =1, ... J ) giventhemarkergenotypeofpatient i ,and fj(xi,yi,vi)isa multivariatenormaldistributionwithexpectedmean vectorforpatient i thatbelongstogenotype j mxj j i; myj j i; mvj j imxj j it1 ; ... ; mxj j i tTi ; myj j it1 ; ... ; myj j i tTi ; mvj j it1 ; ... ; mvj j i tTi 2 andcovariancematrixforsubject i, i xixiyixiviyixiyiyivivixiviyivi0 @ 1 A 3 with xi, yiand vibeing( Ti Ti)covariancematrices oftimedependent x y and v values,respectively,andHou etal.BMCGenetics 2012, 13 :91 Page5of7 http://www.biomedcentral.com/14712156/13/91 PAGE 6 elementsoffdiagonalbeinga( Ti Ti)systematical covariancematrixbetweenthetwovariables. Foranaturalpopulation,theconditionalprobabilityof functionalgenotypegivenamarkergenotype( ji)is expressedintermsofthelinkagedisequilibriabetween differentloci[48].Insystemsmapping,weincorporate ODEs(1)ofAppendix1intomixturemodel(1)toestimategenotypicmeans(2)specifiedbyODEparametersfordifferentgenotypes,expressedas( j, dj, j, aj, kj, uj) for j =1, ... J .Since x y and v variablesobeydynamic system(1)ofAppendix1,thederivativesofgenotypic meanscanbeexpressedinasimilarway.Let gkji( t kji) denotethegenotypicderivativeforvariable k ( k = x y or z ),i.e., g kj j i t ; kj j i d kj j i dt : Weuse kjitodenotethegenotypicmeanofvariable j forindividual i belongingtogenotype j atanarbitrary pointinatimecourse.TheRunge Kuttafourthorder algorithmcanbeusedtosolvetheODEs. Next,weneedtomodelthecovariancestructureby usingaparsimoniousandflexibleapproachsuchasan autoregressive,antedependence,autoregressivemoving average,ornonparametricandsemiparametricapproaches. Yapetal.[49]providedadiscussionofhowtochoosea generalapproachforcovariancestructuremodeling.In likelihood(1),theconditionalprobabilitiesoffunctional genotypesgivenmarkergenotypescanbeexpressedas afunctionofrecombinationfractionsforanexperimental crosspopulationorlinkagedisequilibriaforanatural population[48,50].Theestimationoftherecombination fractionsorlinkagedisequilibriacanbeimplementedwith theExpectationMaximization(EM)algorithm. Todemonstratetheusefulnessofsystemsmapping, weassumeasampleof n HIVinfectedpatientsdrawn fromanaturalhumanpopulationatrandom.Thesampleisanalyzedbysystemsmapping,leadingtothedetectionofamolecularmarkerwhichisassociatedwitha QTLthatdeterminesthedynamicsofdrugresistancein awaydescribedby(2)inAppendix1.AttheQTL detected,therearethreegenotypes AA Aa and aa ,each withadifferentsetofcurveparameters( d m,a k km, u, )estimatedbysystemsmapping.Weassumethat theseparametersareestimatedas(10,0.01,0.005,0.02, 0.5,10,10,3,0.0001)forgenotype AA ,(12,0.01,0.005, 0.02,0.6,8,8,3,0.0001)forgenotype Aa, and(12,0.008, 0.005,0.02,0.55,8,12,4,0.0001)forgenotype aa .Using theseestimatedvalues,wedrawthecurvesofdrug resistancedynamicsforeachgenotype(Figure2).Pronounceddifferencesintheformofthesecurvesindicate thattheQTLplaysanimportantpartindeterminingthe resistancedynamicsofdrugsusedtotreatHIV/AIDS. Themodelforsystemsmappingdescribedabovecan beexpandedintwoaspects,mathematicalandgenetic, tobettercharacterizethegeneticarchitectureofviral dynamics.Themathematicalexpansionsaretoincorporatethedrugresistancemodel(2),thecelldiversity model(3)andtheunifyingresistanceandcelldiversity model(4).Theseexpansionsallowthefunctionalgenes operatingatdifferentpathwaysofviralhostreactionsto beidentifiedandmapped,makingsystemmappingmore clinicallyfeasibleandmeaningful.Thegeneticexpansionsaimtonotonlymodelindividualgenesfromthe hostorpathogengenomebutalsocharacterizeepistatic interactionsbetweengenesfromdifferentgenomes.This canbedonebyexpandingtheconditionalprobabilityof functionalgenesgivenmarkergenotypes jiusinga frameworkderivedbyLietal.[42]. Byformulatingandtestingnovelhypotheses,system mappingcanaddressmanybasicquestions.Forexample, theyare 1)HowdoDNAvariantsregulateviraldynamics? 2)Howdothesegenesaffecttheaveragelifetimesof uninfectedcells,infectedcells,andfreevirus, respectively? 3)Howdogenesdeterminetheemergenceand progressionofdrugresistance? 4)Aretherespecificgenesthatcontrolthepossibility ofviruseradicationbyantiviraldrug? 5)Howimportantaregenegeneinteractionsand genomegenomeinteractionstothedynamic behaviorofviralloadwithorwithouttreatment?Acknowledgements ThisworkissupportedbyFloridaCenterforAIDSResearchIncentiveAward, NIH/NIDAR01DA031017,andNIH/UL1RR0330184. 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Submit your next manuscript to BioMed Central and take full advantage of: Convenient online submission Thorough peer review No space constraints or color gure charges Immediate publication on acceptance Inclusion in PubMed, CAS, Scopus and Google Scholar Research which is freely available for redistribution Submit your manuscript at www.biomedcentral.com/submit Hou etal.BMCGenetics 2012, 13 :91 Page7of7 http://www.biomedcentral.com/14712156/13/91 !DOCTYPE art SYSTEM 'http:www.biomedcentral.comxmlarticle.dtd' ui 147121561391 ji 14712156 fm dochead Correspondence bibl title p Systems mapping of HIV1 infection aug au id A1 snm Houfnm Weiinsr iid I1 I2 email whou@ufl.edu A2 SuiYihansuiyihansunshine@yahoo.cn A3 WangZhongI3 zwang@phs.psu.edu A4 WangYaqunyxw179@psu.edu A5 WangNingtaonxw5034@psu.edu A6 LiuJingyuanjul221@psu.edu A7 LiYaoI4 yli23@fhcrc.org A8 GoodenowMaureenI5 goodenow@pathology.ufl.edu A9 YinLiyin@pathology.ufl.edu A10 WangZuohengI6 zuoheng.wang@yale.edu A11 ca yes WuRonglingrwu@phs.psu.edu insg ins Center for Computational Biology, Beijing Forestry University, Beijing, 100081, China Department of Biostatistics, University of Florida, Gainesville, FL, 32611, USA Center for Statistical Genetics, Pennsylvania State University, Hershey, PA, 17033, USA Division of Public Health Sciences, Fred Hutchinson Cancer Research Center, Seattle, WA, 98109, USA Department of Pathology, Immunology and Laboratory Medicine, University of Florida, Gainesville, FL, 32610, USA Division of Biostatistics, Yale University, New Haven, CT, 06510, USA source BMC Genetics section Statistical and computational geneticsissn 14712156 pubdate 2012 volume 13 issue 1 fpage 91 url http://www.biomedcentral.com/14712156/13/91 xrefbib pubidlist pubid idtype doi 10.1186/147121561391pmpid 23092371 history rec date day 9month 5year 2012acc 2792012pub 23102012 cpyrt 2012collab Hou et al.; licensee BioMed Central Ltd.note This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. abs sec st Abstract Mathematical models of viral dynamics it in vivo provide incredible insights into the mechanisms for the nonlinear interaction between virus and host cell populations, the dynamics of viral drug resistance, and the way to eliminate virus infection from individual patients by drug treatment. The integration of these mathematical models with highthroughput genetic and genomic data within a statistical framework will raise a hope for effective treatment of infections with HIV virus through developing potent antiviral drugs based on individual patients’ genetic makeup. In this opinion article, we will show a conceptual model for mapping and dictating a comprehensive picture of genetic control mechanisms for viral dynamics through incorporating a group of differential equations that quantify the emergent properties of a system. bdy Introduction To control HIV1 virus, antiviral drugs have been developed to prevent the infection of new viral cells or stop alreadyinfected cells from producing infectious virus particles by inhibiting specific viral enzymes abbrgrp abbr bid B1 1 B2 2 . Because of the multifactorial complexity of viralhost association, however, the development and delivery of clinically more beneficial novel antiviral drugs have proved a difficult goal B3 3 . In this essay, we argue that this bottleneck may be overcome by merging two recent advances in mathematical biology and genotyping techniques toward precision medicine. First, viraldrug interactions constitute a complex dynamic system, in which different types of viral cells, including uninfected cells, infected cells, and free virus particles, cooperate with each other and together fight with host immune cells to determine the pattern of viral change in response to drugs B4 4 B5 5 B6 6 . A number of sophisticated mathematical models have been developed to describe viral dynamics in vivo, providing incredible insights into the mechanisms for the nonlinear interaction between virus and host cell populations, the dynamics of viral drug resistance, and the way to eliminate virus infection from patients by drug treatment B7 7 B8 8 B9 9 B10 10 B11 11 B12 12 B13 13 B14 14 B15 15 . Second, the combination between novel instruments and an increasing understanding of molecular genetics has led to the birth of highthroughput genotyping assays such as single nucleotide polymorphisms (SNPs). Through mapping or associating concrete nucleotides or their combinations with the dynamic process of HIV infection B16 16 B17 17 , we can precisely taxonomize this disease by its underlying genomic and molecular causes, thereby enabling the application of precision medicine to diagnose and treat it. Systems mapping: a novel tool to dissect complex traits Beyond a traditional mapping strategy focusing on the static performance of a trait, systems mapping dissolves the phenotype of the trait into its structural, functional or metabolic components through design principles of biological systems, maps the interrelationships and coordination of these components and identifies genes involved in the key pathways that cause the endpoint phenotype B18 18 B19 19 B20 20 B21 21 B22 22 B23 23 . Systems mapping not only preserves the capacity of functional mapping B24 24 B25 25 B26 26 to study the dynamic pattern of genetic control on a time and space scale, but also shows a unique advantage in revealing the dynamic behavior of the genetic correlations among different but developmentally related traits. Its methodological innovation is to integrate mathematical aspects of phenotype formation and progression into a genetic mapping framework to test the interplay between genes and development. Various differential equations which have been instrumental for studying nonlinear and complex dynamics in engineering B27 27 have shown increasing value and power to quantify the emergent properties of a biological system and interpret experimental results 9 10 11 12 B28 28 B29 29 .The past two decades have witnessed an excellent success in modeling HIV dynamics with differential equations 9 10 11 12 . Treating viralhost interactions as a system, Appendix 1 gives a basic model composed of three ordinary differential equations (ODE) for describing the shortterm overall dynamics of uninfected cells (x), infected cells (y), and free virus particles (v). These three components together determine the extent and process of pathogenesis according to six ODE parameters, i.e., the rates of production and death of uninfected cells, the rate of production of infected cells from free viruses, the rate of death of infected cells, and the rates of production and death of new viruses from infected cells. Thus, by changing the values of these parameters singly or in combination, the dynamic properties of viral infection, such as viral half life, the limiting ratio of infected to uninfected cells, and the basic reproductive ratio of the virus, can be quantified and predicted 10 . By embedding a system of ODEs within a mixture model framework (Appendix 1), we can use systems mapping to identify specific host genes and their interactions for the pattern of viral dynamics and infection inside a host body. Figure figr fid F1 1 illustrates the characterization of a hypothesized gene that contributes to variation in viral dynamic behavior. Per these genotypespecific changes, an optimal strategy for HIV treatment in terms of the dose and time at which an antiviral drug is administrated can be determined, thus providing a first step toward personalized medicine 23 . fig Figure 1caption Numerical simulation showing how a gene affects the dynamics of HIV1 infection, composed of uninfected cells (x), infected cells (y), and virus particles (v), as described by a basic model (1) in Appendix 1text b Numerical simulation showing how a gene affects the dynamics of HIV1 infection, composed of uninfected cells (x), infected cells (y), and virus particles (v), as described by a basic model (1) in Appendix 1. The simulated gene has three genotypes AA, Aa and aa, each displaying a different time trajectory for each of these three cell types. Based on these differences, one can test and determine how the gene affects the emerging properties of viral dynamic system, such as average lifetimes of different cell types and the points of three variables (indicated by triangles) when the system converges to an equilibrium state. The parameter values are (λ, d, β, a, k, u) = (10, 0.01, 0.005, 0.5, 10, 3), (12, 0.01, 0.005, 0.6, 8, 3), and (12, 0.008, 0.005, 0.55, 8, 4) for genotypes AA, Aa and aa, respectively. graphic file 1471215613911 In practice, a drug may be resisted if HIV1 viruses mutate to create new strains B30 30 . The emergence of drug resistance is a consequence of evolution and presents a response to pressures imposed on the viruses. Different viruses vary in their sensitivity to the drug used and some with greater fitness may be capable of surviving drug treatment B31 31 B32 32 . In order to understand how viruses are resistant to drugs through mutation, the basic model of Appendix 1 should be expanded to include three additional variables, cells infected by mutant virus, mutant virus particles, and the probability of mutation from wildtype to resistant mutant during reverse transcription of viral RNA into proviral DNA 9 . Systems mapping shows tremendous power to detect genes for virus drug resistance 21 and predict the dynamics of drug resistance (Figure F2 2). Systems mapping can not only better interpret the genetic mechanisms of drug resistance from experimental data, but also provide scientific guidance on the administration of new antiviral drugs. Figure 2Simulated genotypespecific differences in the dynamics of drug resistance as described by a model (2) in Appendix 1 Simulated genotypespecific differences in the dynamics of drug resistance as described by a model (2) in Appendix 1. The system simulation focuses on uninfected cell, x (A), infected cells, y, for wildtype virus (solid line) and mutant virus (dash lines) (B), and free virus, v, for wildtype virus (solid line) and mutant virus (dash line) (C), and relative frequency of mutant virus in free virus (solid line) and infected cell population (dash line) (D). 1471215613912 Mapping triple genome interactions It has been widely accepted that the symptoms and severity of infectious diseases are determined by pathogenhost specificity through cellular, biochemical and signal exchanges 4 B33 33 B34 34 B35 35 . This specificity, established by undermining a host’s immunological ability to mount an immune response against a particular pathogen, is found to be under genetic determination. Current genetic studies of pathogenhost systems focus on either the host or the pathogen genome, but there is increasing recognition that the complete genetic architecture of pathogenhost specificity, described by the number, position, effect, pleiotropy, and epistasis among genes, involves interactive components from both host and viral genomes 35 B36 36 B37 37 B38 38 . In other words, the infection phenotype does not merely result from additive effects of host and pathogen genotypes, but also from specific interactions between the two genomes 35 37 .While many molecular studies define pathogenhost interactions, regardless of the type of hosts, epidemiological models distinguish the difference of hosts as a recipient and transmitter to better characterize the epidemic structure of disease infection, given that infectious diseases like HIV/AIDS are transmitted from an infected person to another B39 39 B40 40 B41 41 . From this point of view, the infection outcome should be determined differently but simultaneously by genes from transmitters and recipients. To chart a comprehensive picture of genetic control mechanisms for viral dynamics, we need to address the questions of how genes from viral and host genomes interact to influence viral dynamics and how genetic interactions between recipients and transmitters of virus play a part in the dynamic behavior of viruses. Li et al. B42 42 pioneered the unification of quantitative genetic theory and epidemiological dynamics for characterizing triplegenome interactions from viruses, transmitters and recipients.Systems mapping described in Appendix 2 should be embedded within Li et al.’s 42 unifying model to include the interactions of genes derived from the three genomes. This integration allows main genetic effects and epistatic interactions expressed at the genome level to be tested and characterized, including additive effects from the (haploid) viral genome, additive and dominant effects from the transmitter genome, additive and dominant effect from the recipient genome as well as all possible interactions among these main effects. It is interesting to note that the integrated system mapping is capable of estimating and testing highorder epistasis from the viral, recipient and transmitter genomes. Given a growing body of evidence that highorder epistasis is an important determinant of the genetic architecture of complex traits B43 43 B44 44 B45 45 , systems mapping should be equipped with triple genome interaction modeling.It should be pointed out that virus evolves through gene recombination and mutations. The genetic machineries that cause viral evolution can be incorporated into systems mapping without technical difficulty. Through such incorporation, systems mapping will provide a useful and timely incentive to detect the genetic control mechanisms of viral dynamics and antivirus drug resistance dynamics and ultimately to design personalized medicine to treat HIV1 infection from increasingly available genome and HIV data worldwide. Toward precision medicine A major challenge that faces drug development and delivery for controlling viral diseases is to develop computational models for analyzing and predicting the dynamics of decline in virus load during drug therapy and further providing estimates of the rate of emergence of resistant virus. The integration of wellestablished mathematical models for viral dynamics with highthroughput genetic and genomic data within a statistical framework will raise a hope for effective diagnosis and treatment of infections with HIV virus through developing potent antiviral drugs based on individual patients’ genetic makeup.In this opinion article, we have provided a synthetic framework for systems mapping of viral dynamics during its progression to AIDS. This framework is equipped with unified mathematical and statistical power to extract genetic information from messy data and possess the analytical and modeling efficiency which does not exist for traditional approaches. By fitting the rate of change of virus infection with clinically meaningful mathematical models, the spatiotemporal pattern of genetic control can be illustrated and predicted over a range of time and space scales. Statistical modeling allows the estimation of mathematical parameters that specify genetic effects on viral dynamics. By genotyping both host and viral genomes, systems mapping is able to identify which viral genes and which human genes from recipients and transmitters determine viral dynamics additively or through nonlinear interactions. In this sense, it paves a new way to chart a comprehensive picture of the genetic architecture of viral infection.An increasing trend in drug development is to integrate it with systems biology aimed to gain deep insights into biological responses. Largescale gene, protein and metabolite (omics) data that found the building blocks of complex systems have become essential parts of the drug industry to design and deliver new drug B46 46 B47 47 . However, the true wealth of systems biology will critically rely upon the way of how to incorporate it into human cell and tissue function that affects pathogenesis. By integrating knowledge of organ and systemlevel responses and omics data, systems mapping will help to prioritize targets and design clinical trials, promising to improve decision making in pharmaceutical development. Appendix 1. Mathematical models of viral dynamics Basic model Bonhoeffer et al. 10 developed a basic model for shortterm virus dynamics. The model includes three variables: uninfected cells, x, infected cells, y, and free virus particles, v. These three types of cells interact with each other to determine the dynamic changes of virus in a host’s body, which can be described by a system of differential equations: displayformula M1 m:math name 147121561391i1 xmlns:m http:www.w3.org1998MathMathML m:mtable columnalign left m:mtr m:mtd m:mover accent true m:mi x m:mo · = m:mrow λ − d x − β x v y · = β x v − a y v · = k y − u v where uninfected cells are yielded at a constant rate, λ, and die at the rate dx; free virus infects uninfected cells to yield infected cells at rate βxv; infected cells die at rate ay; and new virus is yielded from infected cells at rate ky and dies at rate uv. The system (1) is defined by six parameters (λ d β a k u) and some initial conditions about x, y, and v.The dynamic pattern of this system can be determined and predicted by the change of these parameters and the initial conditions of x, y, and v. The basic reproductive ratio of the virus is defined as R sub 0 = βλk/(adu). If R 0 is larger than one, then system converges in damped oscillations to the equilibrium x sup * = au/(βk), y * = λ/a – du/(βk), and v * = λk/(au) – d/β. The average lifetimes of uninfected cells, infected cells, and free virus are given by 1/d, 1/a, and 1/u, respectively. The average number of virus particles produced over the lifetime of a single infected cell (the burst size) is given by k/a. Resistance model When a treatment is used to control HIV1, the viruses will produce the resistance to the drug through mutation. The dynamics of drug resistance can be modeled by M2 147121561391i2 x · = λ − d x − β x v − m:msub β m x v m y · = β m:mfenced open ( close ) m:mn 1 − ε x v − a y y · m = β ε x v + β m x v m − a y m v · = k y − u v v · m = k m y m − u v m where y, y m , v, and v m denote cells infected by wildtype virus, cells infected by mutant virus, free wildtype virus, and free mutant virus, respectively 10 . The mutation rate between wildtype and mutant is given by ε (in both directions). For a small ε, the basic reproductive ratios of wildtype and mutant virus are R 0 = βλk/(adu) and R 0m = β m λk m /(adu).Model (2) shows that the expected pretreatment frequency of resistant mutant depends on the number of point mutations between wildtype and resistant mutant, the mutation rate of virus replication, and the relative replication rates of wildtype virus, resistant mutant, and all intermediate mutants. Whether or not resistant virus is present in a patient before therapy will crucially depend on the population size of infected cells. Cell diversity model The infected cells may harbor actively replicating virus (y 1), latent virus (y 2) and defective virus (y 3). The basic model (1) can be expanded to include these three types, expressed as M3 147121561391i3 x · = λ − d x − β x v y · w = q w β x v − a w y w , m:mspace width 0.5em w = 1 , 2 , 3 v · = k y 1 + c y 2 − u v where q 1, q 2, and q 3 (q 1 + q 2 + q 3 = 1) are the proportions that the cell will immediately enter active viral replication at a rate of virus production k, become latently infected with the virus at a (much slower) rate of virus production c, and produce a defective provirus that will not produce any offspring virus, respectively; and a 1, a 2, and a 3 are the decay rates of actively producing cells, latently infected cells, and defectively infected cells, respectively.The basic reproductive ratio of the wildtype is R 0 = βλA/(du). If R 0 is larger than one, then system converges to the equilibrium x * = u/(βA), inlineformula 147121561391i4 m:msubsup y 1 * = m:mstyle scriptlevel +1 m:mfrac q 1 a 1 λ − d u β A , y 2 * = a 1 a 2 q 2 q 1 y 1 * , y 3 * = a 1 a 3 q 3 q 1 y 1 * , and 147121561391i5 m:msup v * = λ u A − d β , where 147121561391i6 A = k q 1 a 1 + c q 2 a 2 .A full model of viral dynamics can be obtained by unifying the resistance model and cell diversity model to form a system of nine ODEs, expressed as M4 147121561391i7 x · = λ − d x − β x v − β m x v m y · w = q w β 1 − ε x v − a w y w , 1.25em w = 1 , 2 , 3 y · mathvariant italic wm = q w β ε x v + q w β m x v m − a w y wm , w = 1 , 2 , 3 v · = k y 1 + c y 2 − u v v · m = k m y 1 m + c m y 2 m − u v m This group of ODEs provides a comprehensive description of how viral loads change their rate in a time course, how infected cells are generated in response to the emergence of viral particles, and how viral mutation impacts on viral dynamics and drug resistance dynamics. The emerging properties of system (4) were discussed in ref. 10 , which can be integrated with systems mapping described in Appendix 2. Appendix 2. Systems mapping of viral dynamics Systems mapping allows the genes and genetic interactions for viral dynamics to be identified by incorporating ODEs into a mapping framework. Consider a segregating population composed of n HIVinfected patients genotyped for a set of molecular markers. These patients were repeated sampled to measure uninfected cells (x), infected cells (y) and viral load (v) in their plasma at a series of time points. If specific genes exist to affect the system (1) in Appendix 1, the parameters that specify the system should be different among genotypes. Genetic mapping uses a mixture modelbased likelihood to estimate genotypespecific parameters. This likelihood is expressed as M5 147121561391i8 L bold x ; y ; v = displaystyle m:munderover ∏ i = 1 n [ ] ω 1  i f 1 x i , y i , v i + … + ω J  i f J x i , y i , v i where x i = (x i (t 1), …, x( t T i )) y i = (y i (t 1), …, y( t T i )) and v i = (v i (t 1), …, v i ( t T i )) are the phenotypic values of x, y, and v for subject i measured at T i time points, ω ji is the conditional probability of QTL genotype j (j = 1, …, J) given the marker genotype of patient i, and f j (x i ,y i ,v i ) is a multivariate normal distribution with expected mean vector for patient i that belongs to genotype j, M6 147121561391i9 m x j  i ; m y j  i ; m v j  i ° m x j  i t 1 , … , m x j  i t T i ; m y j  i t 1 , … , m y j  i t T i ; m v j  i t 1 , … , m v j  i t T i and covariance matrix for subject i, M7 147121561391i10 Σ i = center Σ x i Σ x i y i Σ x i v i Σ y i x i Σ y i Σ y i v i Σ v i x i Σ v i y i Σ v i with 147121561391i11 Σ x i , 147121561391i12 Σ y i and 147121561391i13 Σ v i being (T i × T i ) covariance matrices of timedependent x, y and v values, respectively, and elements offdiagonal being a (T i × T i ) systematical covariance matrix between the two variables.For a natural population, the conditional probability of functional genotype given a marker genotype (ω ji ) is expressed in terms of the linkage disequilibria between different loci B48 48 . In systems mapping, we incorporate ODEs (1) of Appendix 1 into mixture model (1) to estimate genotypic means (2) specified by ODE parameters for different genotypes, expressed as (λ j d j β j a j k j u j ) for j = 1, …, J. Since x, y and v variables obey dynamic system (1) of Appendix 1, the derivatives of genotypic means can be expressed in a similar way. Let g kji (t μ kji ) denote the genotypic derivative for variable k (k = x, y, or z), i.e., M8 147121561391i14 g ( k j  i ) ( t , μ k j  i ) = d μ ( k j  i ) d t m:mtext . We use μ kji to denote the genotypic mean of variable j for individual i belonging to genotype j at an arbitrary point in a time course. The Runge–Kutta fourth order algorithm can be used to solve the ODEs.Next, we need to model the covariance structure by using a parsimonious and flexible approach such as an autoregressive, antedependence, autoregressive moving average, or nonparametric and semiparametric approaches. Yap et al. B49 49 provided a discussion of how to choose a general approach for covariance structure modeling. In likelihood (1), the conditional probabilities of functional genotypes given marker genotypes can be expressed as a function of recombination fractions for an experimental cross population or linkage disequilibria for a natural population 48 B50 50 . The estimation of the recombination fractions or linkage disequilibria can be implemented with the ExpectationMaximization (EM) algorithm.To demonstrate the usefulness of systems mapping, we assume a sample of n HIVinfected patients drawn from a natural human population at random. The sample is analyzed by systems mapping, leading to the detection of a molecular marker which is associated with a QTL that determines the dynamics of drug resistance in a way described by (2) in Appendix 1. At the QTL detected, there are three genotypes AA, Aa and aa, each with a different set of curve parameters (λ, d, β, β m , a, k, k m , u, ε) estimated by systems mapping. We assume that these parameters are estimated as (10, 0.01, 0.005, 0.02, 0.5, 10, 10, 3, 0.0001) for genotype AA, (12, 0.01, 0.005, 0.02, 0.6, 8, 8, 3, 0.0001) for genotype Aa, and (12, 0.008, 0.005, 0.02, 0.55, 8, 12, 4, 0.0001) for genotype aa. Using these estimated values, we draw the curves of drug resistance dynamics for each genotype (Figure 2). Pronounced differences in the form of these curves indicate that the QTL plays an important part in determining the resistance dynamics of drugs used to treat HIV/AIDS.The model for systems mapping described above can be expanded in two aspects, mathematical and genetic, to better characterize the genetic architecture of viral dynamics. The mathematical expansions are to incorporate the drug resistance model (2), the cell diversity model (3) and the unifying resistance and cell diversity model (4). These expansions allow the functional genes operating at different pathways of viralhost reactions to be identified and mapped, making system mapping more clinically feasible and meaningful. The genetic expansions aim to not only model individual genes from the host or pathogen genome but also characterize epistatic interactions between genes from different genomes. This can be done by expanding the conditional probability of functional genes given marker genotypes ω ji using a framework derived by Li et al. 42 .By formulating and testing novel hypotheses, system mapping can address many basic questions. 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The integration of these mathematical models with highthroughput genetic and genomic data within a statistical framework will raise a hope for effective treatment of infections with HIV virus through developing potent antiviral drugs based on individual patients’ genetic makeup. In this opinion article, we will show a conceptual model for mapping and dictating a comprehensive picture of genetic control mechanisms for viral dynamics through incorporating a group of differential equations that quantify the emergent properties of a system. http:purl.orgdcelements1.1creator Hou, Wei Sui, Yihan Wang, Zhong Wang, Yaqun Wang, Ningtao Liu, Jingyuan Li, Yao Goodenow, Maureen Yin, Li Wang, Zuoheng Wu, Rongling http:purl.orgeprinttermsisExpressedAs epdcx:valueRef swordmetsexpr1 http:purl.orgeprintentityTypeExpression http:purl.orgdcelements1.1language epdcx:vesURI http:purl.orgdctermsRFC3066 en http:purl.orgeprinttermsType http:purl.orgeprinttypeJournalArticle http:purl.orgdctermsavailable epdcx:sesURI http:purl.orgdctermsW3CDTF 20121023 http:purl.orgdcelements1.1publisher BioMed Central Ltd http:purl.orgeprinttermsstatus http:purl.orgeprinttermsStatus http:purl.orgeprintstatusPeerReviewed http:purl.orgeprinttermscopyrightHolder Wei Hou et al.; licensee BioMed Central Ltd. http:purl.orgdctermslicense http://creativecommons.org/licenses/by/2.0 http:purl.orgdctermsaccessRights http:purl.orgeprinttermsAccessRights http:purl.orgeprintaccessRightsOpenAccess http:purl.orgeprinttermsbibliographicCitation BMC Genetics. 2012 Oct 23;13(1):91 http:purl.orgdcelements1.1identifier http:purl.orgdctermsURI http://dx.doi.org/10.1186/147121561391 fileSec fileGrp swordmetsfgrp1 USE CONTENT file swordmetsfgid0 swordmetsfile1 FLocat LOCTYPE URL xlink:href 147121561391.xml swordmetsfgid1 swordmetsfile2 applicationpdf 147121561391.pdf structMap swordmetsstruct1 structure LOGICAL div swordmetsdiv1 DMDID Object swordmetsdiv2 File fptr FILEID swordmetsdiv3 