Systems mapping of HIV-1 infection

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Systems mapping of HIV-1 infection
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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 high-throughput 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.
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doi:10.1186/1471-2156-13-91 Cite this article as: Hou et al.: Systems mapping of HIV-1 infection. BMC Genetics 2012 13:91.

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GenetiBMC
Genetics


Systems mapping of HIV-1 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
high-throughput 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 HIV-1 virus, antiviral drugs have been devel-
oped to prevent the infection of new viral cells or stop
already-infected cells from producing infectious virus
particles by inhibiting specific viral enzymes [1,2].
Because of the multifactorial complexity of viral-host
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, viral-drug 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 [4-6]. 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 [7-15]. 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 high-throughput
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 end-point
phenotype [18-23]. Systems mapping not only preserves
the capacity of functional mapping [24-26] 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 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.







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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 [9-12,28,29].
The past two decades have witnessed an excellent suc-
cess in modeling HIV dynamics with differential equa-
tions [9-12]. Treating viral-host interactions as a system,
Appendix 1 gives a basic model composed of three
ordinary differential equations (ODE) for describing the
short-term 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 genotype-specific 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 HIV-1 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 wild-type 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 [4,33-35]. 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 stud-
ies of pathogen-host systems focus on either the host or
the pathogen genome, but there is increasing recognition
that the complete genetic architecture of pathogen-host
specificity, described by the number, position, effect, plei-
otropy, and epistasis among genes, involves interactive
components from both host and viral genomes [35-38].
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 pathogen-host
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 HIV-1 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 life-times 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.


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0


A








0 10 20 30 40 50 60


0 10 20 30 40 50 60
Time (Days)
Figure 2 Simulated genotype-specific 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 wild-type virus (solid line) and mutant virus (dash lines) (B), and
free virus, v, for wild-type 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 [39-41]. 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 triple-genome
interactions from viruses, transmitters and recipients.
Systems mapping described in Appendix 2 should
be embedded within Li et al.'s [42] unifying model to


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0 10 20 30 40 50 60






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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 high-order epistasis
from the viral, recipient and transmitter genomes. Given
a growing body of evidence that high-order epistasis is
an important determinant of the genetic architecture of
complex traits [43-45], 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 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 HIV-1 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 well-established
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 spatio-temporal 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 non-linear 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. Large-scale 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 system-level
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 life-times
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.


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Resistance model
When a treatment is used to control HIV-1, 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 wild-type
virus, cells infected by mutant virus, free wild-type virus,
and free mutant virus, respectively [10]. The mutation
rate between wild-type and mutant is given by e (in both
directions). For a small e, the basic reproductive ratios
of wild-type 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 wild-type and resistant
mutant, the mutation rate of virus replication, and the
relative replication rates of wild-type 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 wild-type 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 HIV-infected 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 model-based likelihood to estimate genotype-specific
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 Y---vi
yY y_ )


with Xx,, XY, and Xv, being (Ti x Ti) covariance matrices
of time-dependent x, y and v values, respectively, and


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elements off-diagonal 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 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. [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 Expectation-Maximization (EM) algorithm.
To demonstrate the usefulness of systems mapping,
we assume a sample of n HIV-infected 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 viral-host 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 life-times 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 gene-gene interactions and
genome-genome 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

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doi:10.1186/1471-2156-13-91
Cite this article as: Hou et al Systems mapping of HIV-1 infection. BMC
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CORRESPONDENCEOpenAccessSystemsmappingofHIV-1infectionWeiHou1,2,YihanSui1,ZhongWang3,YaqunWang3,NingtaoWang3,JingyuanLiu3,YaoLi4,MaureenGoodenow5, LiYin5,ZuohengWang6andRonglingWu1,3*AbstractMathematicalmodelsofviraldynamics invivo provideincredibleinsightsintothemechanismsforthenonlinear interactionbetweenvirusandhostcellpopulations,thedynamicsofviraldrugresistance,andthewaytoeliminate virusinfectionfromindividualpatientsbydrugtreatment.Theintegrationofthesemathematicalmodelswith high-throughputgeneticandgenomicdatawithinastatisticalframeworkwillraiseahopeforeffectivetreatment ofinfectionswithHIVvirusthroughdevelopingpotentantiviraldrugsbasedonindividualpatients ’ genetic makeup.Inthisopinionarticle,wewillshowaconceptualmodelformappinganddictatingacomprehensive pictureofgeneticcontrolmechanismsforviraldynamicsthroughincorporatingagroupofdifferentialequations thatquantifytheemergentpropertiesofasystem.IntroductionTocontrolHIV-1virus,antiviraldrugshavebeendevelopedtopreventtheinfectionofnewviralcellsorstop already-infectedcellsfromproducinginfectiousvirus particlesbyinhibitingspecificviralenzymes[1,2]. Becauseofthemultifactorialcomplexityofviral-host association,however,thedevelopmentanddeliveryof clinicallymorebeneficialnovelantiviraldrugshave provedadifficultgoal[3].Inthisessay,wearguethat thisbottleneckmaybeovercomebymergingtworecent advancesinmathematicalbiologyandgenotypingtechniquestowardprecisionmedicine.First,viral-druginteractionsconstituteacomplexdynamicsystem,inwhich differenttypesofviralcells,includinguninfectedcells, infectedcells,andfreevirusparticles,cooperatewith eachotherandtogetherfightwithhostimmunecellsto determinethepatternofviralchangeinresponseto drugs[4-6].Anumberofsophisticatedmathematical modelshavebeendevelopedtodescribeviraldynamics invivo ,providingincredibleinsightsintothemechanismsforthenonlinearinteractionbetweenvirusand hostcellpopulations,thedynamicsofviraldrugresistance, andthewaytoeliminatevirusinfectionfrompatientsby drugtreatment[7-15].Second,thecombinationbetween novelinstrumentsandanincreasingunderstandingofmoleculargeneticshasledtothebirthofhigh-throughput genotypingassayssuchassing lenucleotidepolymorphisms (SNPs).Throughmappingorassociatingconcretenucleotidesortheircombinations withthedynamicprocessof HIVinfection[16,17],wecanpreciselytaxonomizethis diseasebyitsunderlyinggenomicandmolecularcauses, therebyenablingtheapplicationofprecisionmedicineto diagnoseandtreatit.Systemsmapping:anoveltooltodissect complextraitsBeyondatraditionalmappingstrategyfocusingonthe staticperformanceofatrait,systemsmappingdissolves thephenotypeofthetraitintoitsstructural,functional ormetaboliccomponentsthroughdesignprinciples ofbiologicalsystems,mapstheinterrelationshipsand coordinationofthesecomponentsandidentifiesgenes involvedinthekeypathwaysthatcausetheend-point phenotype[18-23].Systemsmappingnotonlypreserves thecapacityoffunctionalmapping[24-26]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/1471-2156/13/91

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complexdynamicsinengineering[27]haveshown increasingvalueandpowertoquantifytheemergent propertiesofabiologicalsystemandinterpretexperimentalresults[9-12,28,29]. ThepasttwodecadeshavewitnessedanexcellentsuccessinmodelingHIVdynamicswithdifferentialequations[9-12].Treatingviral-hostinteractionsasasystem, Appendix1givesabasicmodelcomposedofthree ordinarydifferentialequations(ODE)fordescribingthe short-termoveralldynamicsofuninfectedcells( 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.Perthesegenotype-specificchanges, anoptimalstrategyforHIVtreatmentintermsofthe doseandtimeatwhichanantiviraldrugisadministratedcanbedetermined,thusprovidingafirststep towardpersonalizedmedicine[23]. Inpractice,adrugmayberesistedifHIV-1viruses mutatetocreatenewstrains[30].Theemergenceof drugresistanceisaconsequenceofevolutionandpresentsaresponsetopressuresimposedontheviruses. Differentvirusesvaryintheirsensitivitytothedrugused andsomewithgreaterfitnessmaybecapableofsurvivingdrugtreatment[31,32].Inordertounderstand howvirusesareresistanttodrugsthroughmutation, thebasicmodelofAppendix1shouldbeexpandedto includethreeadditionalvariables,cellsinfectedby mutantvirus,mutantvirusparticles,andtheprobability ofmutationfromwild-typetoresistantmutantduring reversetranscriptionofviralRNAintoproviralDNA [9].Systemsmappingshowstremendouspowertodetect genesforvirusdrugresistance[21]andpredictthedynamicsofdrugresistance(Figure2).Systemsmapping cannotonlybetterinterpretthegeneticmechanisms ofdrugresistancefromexperimentaldata,butalsoprovidescientificguidanceontheadministrationofnew antiviraldrugs.MappingtriplegenomeinteractionsIthasbeenwidelyacceptedthatthesymptomsand severityofinfectiousdiseasesaredeterminedbypathogenhostspecificitythroughcellular,biochemicalandsignal exchanges[4,33-35].Thisspecificity,establishedby underminingahost ’ simmunologicalabilitytomountan immuneresponseagainstaparticularpathogen,isfound tobeundergeneticdetermination.Currentgeneticstudiesofpathogen-hostsystemsfocusoneitherthehostor thepathogengenome,butthereisincreasingrecognition thatthecompletegeneticarchitectureofpathogen-host specificity,describedbythenumber,position,effect,pleiotropy,andepistasisamonggenes,involvesinteractive componentsfrombothhostandviralgenomes[35-38]. Inotherwords,theinfectionphenotypedoesnotmerely resultfromadditiveeffectsofhostandpathogengenotypes,butalsofromspecificinteractionsbetweenthe twogenomes[35,37]. Whilemanymolecularstudiesdefinepathogen-host interactions,regardlessofthe typeofhosts,epidemiological Figure1 NumericalsimulationshowinghowageneaffectsthedynamicsofHIV-1infection,composedofuninfectedcells( x ),infected cells( y ),andvirusparticles( v ),asdescribedbyabasicmodel(1)inAppendix1. Thesimulatedgenehasthreegenotypes AA Aa and aa eachdisplayingadifferenttimetrajectoryforeachofthesethreecelltypes.Basedonthesedifferences,onecantestanddeterminehowthe geneaffectstheemergingpropertiesofviraldynamicsystem,suchasaveragelife-timesofdifferentcelltypesandthepointsofthreevariables (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/1471-2156/13/91

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modelsdistinguishthedifferenceofhostsasarecipient andtransmittertobettercharacterizetheepidemicstructureofdiseaseinfection,giventhatinfectiousdiseaseslike HIV/AIDSaretransmittedfromaninfectedpersontoanother[39-41].Fromthispointofview,theinfection outcomeshouldbedetermineddifferentlybutsimultaneouslybygenesfromtransmittersandrecipients.To chartacomprehensivepictureofgeneticcontrolmechanismsforviraldynamics,weneedtoaddressthequestions ofhowgenesfromviralandhostgenomesinteracttoinfluenceviraldynamicsandhowgeneticinteractions betweenrecipientsandtransmittersofvirusplayapartin thedynamicbehaviorofviruses.Lietal.[42]pioneered theunificationofquantitativegenetictheoryandepidemiologicaldynamicsforcharacterizingtriple-genome interactionsfromviruses,transmittersandrecipients. SystemsmappingdescribedinAppendix2should beembeddedwithinLietal. ’ s[42]unifyingmodelto Figure2 Simulatedgenotype-specificdifferencesinthedynamicsofdrugresistanceasdescribedbyamodel(2)inAppendix1. Thesystemsimulationfocusesonuninfectedcell, x ( A ),infectedcells, y ,forwild-typevirus(solidline)andmutantvirus(dashlines)( B ),and freevirus, v ,forwild-typevirus(solidline)andmutantvirus(dashline)( C ),andrelativefrequencyofmutantvirusinfreevirus(solidline)and infectedcellpopulation(dashline)( D ). Hou etal.BMCGenetics 2012, 13 :91 Page3of7 http://www.biomedcentral.com/1471-2156/13/91

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includetheinteractionsofgenesderivedfromthethree genomes.Thisintegrationallowsmaingeneticeffects andepistaticinteractionsexpressedatthegenomelevel tobetestedandcharacterized,includingadditiveeffects fromthe(haploid)viralgenome,additiveanddominant effectsfromthetransmittergenome,additiveanddominanteffectfromtherecipientgenomeaswellasall possibleinteractionsamongthesemaineffects.Itis interestingtonotethattheintegratedsystemmapping iscapableofestimatingandtestinghigh-orderepistasis fromtheviral,recipientandtransmittergenomes.Given agrowingbodyofevidencethathigh-orderepistasisis animportantdeterminantofthegeneticarchitectureof complextraits[43-45],systemsmappingshouldbe equippedwithtriplegenomeinteractionmodeling. Itshouldbepointedoutthatvirusevolvesthrough generecombinationandmutations.Thegeneticmachineriesthatcauseviralevolutioncanbeincorporatedinto systemsmappingwithouttechnicaldifficulty.Through suchincorporation,systemsmappingwillprovideausefulandtimelyincentivetodetectthegeneticcontrol mechanismsofviraldynamicsandantivirusdrugresistancedynamicsandultimatelytodesignpersonalized medicinetotreatHIV-1infectionfromincreasingly availablegenomeandHIVdataworldwide.TowardprecisionmedicineAmajorchallengethatfacesdrugdevelopmentand deliveryforcontrollingviraldiseasesistodevelopcomputationalmodelsforanalyzingandpredictingthe dynamicsofdeclineinvirusloadduringdrugtherapy andfurtherprovidingestimatesoftherateofemergence ofresistantvirus.Theintegrationofwell-established mathematicalmodelsforviraldynamicswithhighthroughputgeneticandgenomicdatawithinastatistical frameworkwillraiseahopeforeffectivediagnosisand treatmentofinfectionswithHIVvirusthroughdevelopingpotentantiviraldrugsbasedonindividualpatients ’ geneticmakeup. Inthisopinionarticle,wehaveprovidedasynthetic frameworkforsystemsmappingofviraldynamicsduringitsprogressiontoAIDS.Thisframeworkisequipped withunifiedmathematicalandstatisticalpowerto extractgeneticinformationfrommessydataandpossess theanalyticalandmodelingefficiencywhichdoesnot existfortraditionalapproaches.Byfittingtherateof changeofvirusinfectionwithclinicallymeaningful mathematicalmodels,thespatio-temporalpatternof geneticcontrolcanbeillustratedandpredictedovera rangeoftimeandspacescales.Statisticalmodeling allowstheestimationofmathematicalparametersthat specifygeneticeffectsonviraldynamics.Bygenotyping bothhostandviralgenomes,systemsmappingisableto identifywhichviralgenesandwhichhumangenesfrom recipientsandtransmittersdetermineviraldynamics additivelyorthroughnon-linearinteractions.Inthis sense,itpavesanewwaytochartacomprehensive pictureofthegeneticarchitectureofviralinfection. Anincreasingtrendindrugdevelopmentistointegrateitwithsystemsbiologyaimedtogaindeepinsights intobiologicalresponses.Large-scalegene,proteinand metabolite(omics)datathatfoundthebuildingblocks ofcomplexsystemshavebecomeessentialpartsofthe drugindustrytodesignanddelivernewdrug[46,47]. However,thetruewealthofsystemsbiologywillcriticallyrelyuponthewayofhowtoincorporateitinto humancellandtissuefunctionthataffectspathogenesis.Byintegratingknowledgeoforganandsystem-level 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 / .Theaveragelife-times 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/1471-2156/13/91

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ResistancemodelWhenatreatmentisusedtocontrolHIV-1,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 vmdenotecellsinfectedbywild-type virus,cellsinfectedbymutantvirus,freewild-typevirus, andfreemutantvirus,respectively[10].Themutation ratebetweenwild-typeandmutantisgivenby (inboth directions).Forasmall ,thebasicreproductiveratios ofwild-typeandmutantvirusare R0= k /( adu )and R0 m= m km/( adu ). Model(2)showsthattheexpectedpretreatment frequencyofresistantmutantdependsonthenumber ofpointmutationsbetweenwild-typeandresistant mutant,themutationrateofvirusreplication,andthe relativereplicationratesofwild-typevirus,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. Thebasicreproductiveratioofthewild-typeis 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 HIV-infectedpatientsgenotypedfora setofmolecularmarkers.Thesepatientswererepeated sampledtomeasureuninfectedcells( x ),infectedcells ( y )andviralload( v )intheirplasmaataseriesoftime points.Ifspecificgenesexisttoaffectthesystem(1)in Appendix1,theparametersthatspecifythesystemshould bedifferentamonggenotypes.Geneticmappingusesamixturemodel-basedlikelihoodtoestimategenotype-specific 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, j|iisthe 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 oftime-dependent x y and v values,respectively,andHou etal.BMCGenetics 2012, 13 :91 Page5of7 http://www.biomedcentral.com/1471-2156/13/91

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elementsoff-diagonalbeinga( Ti Ti)systematical covariancematrixbetweenthetwovariables. Foranaturalpopulation,theconditionalprobabilityof functionalgenotypegivenamarkergenotype( j|i)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 gkj|i( t kj|i) denotethegenotypicderivativeforvariable k ( k = x y or z ),i.e., g kj j i t ; kj j i d kj j i dt : Weuse kj|itodenotethegenotypicmeanofvariable 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 theExpectation-Maximization(EM)algorithm. Todemonstratetheusefulnessofsystemsmapping, weassumeasampleof n HIV-infectedpatientsdrawn 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 operatingatdifferentpathwaysofviral-hostreactionsto beidentifiedandmapped,makingsystemmappingmore clinicallyfeasibleandmeaningful.Thegeneticexpansionsaimtonotonlymodelindividualgenesfromthe hostorpathogengenomebutalsocharacterizeepistatic interactionsbetweengenesfromdifferentgenomes.This canbedonebyexpandingtheconditionalprobabilityof functionalgenesgivenmarkergenotypes j|iusinga frameworkderivedbyLietal.[42]. Byformulatingandtestingnovelhypotheses,system mappingcanaddressmanybasicquestions.Forexample, theyare 1)HowdoDNAvariantsregulateviraldynamics? 2)Howdothesegenesaffecttheaveragelife-timesof uninfectedcells,infectedcells,andfreevirus, respectively? 3)Howdogenesdeterminetheemergenceand progressionofdrugresistance? 4)Aretherespecificgenesthatcontrolthepossibility ofviruseradicationbyantiviraldrug? 5)Howimportantaregene-geneinteractionsand genome-genomeinteractionstothedynamic behaviorofviralloadwithorwithouttreatment?Acknowledgements ThisworkissupportedbyFloridaCenterforAIDSResearchIncentiveAward, NIH/NIDAR01DA031017,andNIH/UL1RR0330184. Authordetails1CenterforComputationalBiology,BeijingForestryUniversity,Beijing 100081,China.2DepartmentofBiostatistics,UniversityofFlorida,Gainesville, FL32611,USA.3CenterforStatisticalGenetics,PennsylvaniaStateUniversity, Hershey,PA17033,USA.4DivisionofPublicHealthSciences,FredHutchinson CancerResearchCenter,Seattle,WA98109,USA.5DepartmentofPathology, ImmunologyandLaboratoryMedicine,UniversityofFlorida,Gainesville,FL 32610,USA.6DivisionofBiostatistics,YaleUniversity,NewHaven,CT06510, USA. Received:9May2012Accepted:27September2012 Published:23October2012 References1.SmithK,PowersKA,KashubaAD,CohenMS: HIV-1treatmentas prevention:thegood,thebad,andthechallenges. CurrOpinHIVAIDS 2011, 6 (4):315 – 325. 2.PadianNS,McCoySI,KarimSSA,HasenN,KimJ, etal : HIVprevention transformed:thenewpreventionresearchagenda. Lancet 2011, 378: 269 – 278. 3.PadianNS,McCoySI,BalkusJE,WasserheitJN: Weighingthegoldinthe goldstandard:challengesinHIVpreventionresearch. AIDS 2010, 24: 621 – 635. 4.FellayJ,ShiannaKV,TelentiA,GoldsteinDB: HostgeneticsandHIV-1: Thefinalphase? PLoSPathog 2010, 6 (10):e1001033.Hou etal.BMCGenetics 2012, 13 :91 Page6of7 http://www.biomedcentral.com/1471-2156/13/91

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NatChemBiol 2008, 4: 682 – 690. 48.WuRL,ZengZB: Jointlinkageandlinkagedisequilibriummappingin naturalpopulations. Genetics 2001, 157: 899 –909. 49.YapJ,FanJWRL: Nonparametricmodelingofcovariancestructure infunctionalmappingofquantitativetraitloci. Biometrics 2009, 65: 1068 – 1077. 50.WuRL,MaCX,CasellaG: StatisticalGeneticsofQuantitativeTraits:Linkage, Maps,andQTL .NewYork:Springer;2007.doi:10.1186/1471-2156-13-91 Citethisarticleas: Hou etal. : SystemsmappingofHIV-1infection. BMC Genetics 2012 13 :91. 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/1471-2156/13/91


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title
p Systems mapping of HIV-1 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 1471-2156
pubdate 2012
volume 13
issue 1
fpage 91
url http://www.biomedcentral.com/1471-2156/13/91
xrefbib pubidlist pubid idtype doi 10.1186/1471-2156-13-91pmpid 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 high-throughput 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 HIV-1 virus, antiviral drugs have been developed to prevent the infection of new viral cells or stop already-infected cells from producing infectious virus particles by inhibiting specific viral enzymes abbrgrp
abbr bid B1 1
B2 2
. Because of the multifactorial complexity of viral-host 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, viral-drug 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 high-throughput 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 end-point 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 viral-host interactions as a system, Appendix 1 gives a basic model composed of three ordinary differential equations (ODE) for describing the short-term 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 genotype-specific 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 HIV-1 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 HIV-1 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 life-times 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 1471-2156-13-91-1 In practice, a drug may be resisted if HIV-1 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 wild-type 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 genotype-specific differences in the dynamics of drug resistance as described by a model (2) in Appendix 1
Simulated genotype-specific 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 wild-type virus (solid line) and mutant virus (dash lines) (B), and free virus, v, for wild-type 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).
1471-2156-13-91-2
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
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 pathogen-host systems focus on either the host or the pathogen genome, but there is increasing recognition that the complete genetic architecture of pathogen-host 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 pathogen-host 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 triple-genome 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 high-order epistasis from the viral, recipient and transmitter genomes. Given a growing body of evidence that high-order 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 HIV-1 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 well-established 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 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 spatio-temporal 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 non-linear 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. Large-scale 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 system-level 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 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:
display-formula M1
m:math name 1471-2156-13-91-i1 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 life-times 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 HIV-1, the viruses will produce the resistance to the drug through mutation. The dynamics of drug resistance can be modeled by
M2
1471-2156-13-91-i2
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 wild-type virus, cells infected by mutant virus, free wild-type virus, and free mutant virus, respectively
10
. The mutation rate between wild-type and mutant is given by ε (in both directions). For a small ε, the basic reproductive ratios of wild-type 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 wild-type and resistant mutant, the mutation rate of virus replication, and the relative replication rates of wild-type 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
1471-2156-13-91-i3
x
·
=
λ

d
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where q
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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 wild-type is R
0 = βλA/(du). If R
0 is larger than one, then system converges to the equilibrium x
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1471-2156-13-91-i4
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1471-2156-13-91-i6
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=
k
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+
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.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
1471-2156-13-91-i7
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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 HIV-infected 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 model-based likelihood to estimate genotype-specific parameters. This likelihood is expressed as
M5
1471-2156-13-91-i8
L
bold x
;
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t
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i
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)) are the phenotypic values of x, y, and v for subject i measured at T
i
time points, ω
j|i
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
1471-2156-13-91-i9
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M7
1471-2156-13-91-i10
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with
1471-2156-13-91-i11
Σ
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1471-2156-13-91-i12
Σ
y
i
and
1471-2156-13-91-i13
Σ
v
i
being (T
i
× T
i
) covariance matrices of time-dependent x, y and v values, respectively, and elements off-diagonal 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
j|i
) 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
kj|i
(t
μ
kj|i
) denote the genotypic derivative for variable k (k = x, y, or z), i.e.,
M8
1471-2156-13-91-i14
g
(
k
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)
(
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)
=
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(
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)
d
t
m:mtext .
We use μ
kj|i
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 Expectation-Maximization (EM) algorithm.To demonstrate the usefulness of systems mapping, we assume a sample of n HIV-infected 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 viral-host 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
j|i
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 areindent 1 1) How do DNA variants regulate viral dynamics?2) How do these genes affect the average life-times 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 gene-gene interactions and genome-genome interactions to the dynamic behavior of viral load with or without treatment?
bm
ack
Acknowledgements
This work is supported by Florida Center for AIDS Research Incentive Award, NIH/NIDA R01 DA031017, and NIH/UL1RR0330184.
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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 high-throughput 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.
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Hou, Wei
Sui, Yihan
Wang, Zhong
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Liu, Jingyuan
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Goodenow, Maureen
Yin, Li
Wang, Zuoheng
Wu, Rongling
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BMC Genetics. 2012 Oct 23;13(1):91
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