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

Abstract

Mathematical models of viral dynamics

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

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

The past two decades have witnessed an excellent success in modeling HIV dynamics with differential equations

Numerical simulation showing how a gene affects the dynamics of HIV-1 infection, composed of uninfected cells (

**Numerical simulation showing how a gene affects the dynamics of HIV-1 infection, composed of uninfected cells (****), infected cells (****), and virus particles (****), as described by a basic model (1) in Appendix 1.** The simulated gene has three genotypes

In practice, a drug may be resisted if HIV-1 viruses mutate to create new strains

Simulated 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, **A**), infected cells, **B**), and free virus, **C**), and relative frequency of mutant virus in free virus (solid line) and infected cell population (dash line) (**D**).

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

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

Systems mapping described in Appendix 2 should be embedded within Li et al.’s

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

Appendix 1. Mathematical models of viral dynamics

Basic model

Bonhoeffer et al.

where uninfected cells are yielded at a constant rate,

The dynamic pattern of this system can be determined and predicted by the change of these parameters and the initial conditions of _{0} = _{0} is larger than one, then system converges in damped oscillations to the equilibrium ^{
*
} = ^{
*
} = ^{
*
} =

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

where _{
m
}, _{
m
} denote cells infected by wild-type virus, cells infected by mutant virus, free wild-type virus, and free mutant virus, respectively _{0} = _{0m
} = _{
m
}
_{
m
}/(

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 (_{1}), latent virus (_{2}) and defective virus (_{3}). The basic model (1) can be expanded to include these three types, expressed as

where _{1}, _{2}, and _{3} (_{1} + _{2} + _{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 _{1}, _{2}, and _{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 _{0} = _{0} is larger than one, then system converges to the equilibrium ^{
*
} =

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

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.

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

where x_{
i
} = (_{
i
}(_{1}), …, ^{
t
}
_{
i
})) , y_{
i
} = (_{
i
}(_{1}), …, ^{
t
}
_{
i
})) and v_{
i
} = (_{
i
}(_{1}), …, _{
i
}(^{
t
}
_{
i
})) are the phenotypic values of _{
i
} time points, _{
j|i
} is the conditional probability of QTL genotype _{
j
}(x_{
i
},y_{
i
},v_{
i
}) is a multivariate normal distribution with expected mean vector for patient

and covariance matrix for subject

with _{
i
} × _{
i
}) covariance matrices of time-dependent _{
i
} × _{
i
}) systematical covariance matrix between the two variables.

For a natural population, the conditional probability of functional genotype given a marker genotype (_{
j|i
}) is expressed in terms of the linkage disequilibria between different loci _{
j
}
_{
j
}
_{
j
}
_{
j
}
_{
j
}
_{
j
}) for _{
kj|i
}(_{
kj|i
}) denote the genotypic derivative for variable

We use _{
kj|i
} to denote the genotypic mean of variable

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.

To demonstrate the usefulness of systems mapping, we assume a sample of _{
m
}
_{
m
},

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 _{
j|i
} using a framework derived by Li et al.

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/UL1RR0330184.