kinetic modeling of virus growth in cells · frontiers of cell-level modeling of virus infections....

Kinetic Modeling of Virus Growth in Cells

John Yin,a Jacob Redovicha

aDepartment of Chemical and Biological Engineering, Wisconsin Institute for Discovery, University ofWisconsin-Madison, Madison, Wisconsin, USA

SUMMARY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2EARLY MODELS OF VIRUS GROWTH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

Bacteriophage T4, a Double-Stranded DNA Virus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6A Single-Stranded Positive-Sense RNA Virus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

BACTERIOPHAGE Q� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8Kinetics of Q� RNA Replication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8Positive Feedback in RNA Replication during Q� Growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9Robustness of Simulated Phage Growth with Respect to Parameter Values . . . . . . . . . . . . . 9Efficient Use of Host Energy Resources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10Refinement of Phage Q� Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10Testing Antiviral Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

BACTERIOPHAGE T7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11Antiviral Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12From Transcriptome to Proteome . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13Genome Organization Affects Virus Fitness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14Synthetic Biology Test of Model Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14How the Host Cell Environment Can Affect Virus Fitness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15Epistasis: Quantitative Assessment of Genetic Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15Robustness versus Fragility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

BACTERIOPHAGE M13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17HIV-1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

An Anti-HIV Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19INFLUENZA A VIRUS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

Control of Viral RNA Synthesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20Kinetics of Defective Interfering Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

POLIOVIRUS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21Optimal Resource Use within Cells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

VSV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22Effects of Genome Organization on Virus Growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

BACULOVIRUS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23HBV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24HCV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25HSV-1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25FRONTIERS FOR MODELING VIRUS GROWTH IN CELLS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

Other Viruses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26Host Cell Physiology and Innate Immune Responses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27Single-Cell and Single-Virus Tracking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27Multiscale Modeling of Virus Infection Spread . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29AUTHOR BIOS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

SUMMARY When a virus infects a host cell, it hijacks the biosynthetic capacity ofthe cell to produce virus progeny, a process that may take less than an hour ormore than a week. The overall time required for a virus to reproduce depends col-lectively on the rates of multiple steps in the infection process, including initial bind-ing of the virus particle to the surface of the cell, virus internalization and release ofthe viral genome within the cell, decoding of the genome to make viral proteins,replication of the genome, assembly of progeny virus particles, and release of theseparticles into the extracellular environment. For a large number of virus types, much

Published 28 March 2018

Citation Yin J, Redovich J. 2018. Kineticmodeling of virus growth in cells. MicrobiolMol Biol Rev 82:e00066-17. https://doi.org/10.1128/MMBR.00066-17.

Copyright © 2018 American Society forMicrobiology. All Rights Reserved.

Address correspondence to John Yin,[email protected].

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has been learned about the molecular mechanisms and rates of the various steps.However, in only relatively few cases during the last 50 years has an attempt beenmade— using mathematical modeling—to account for how the different steps con-tribute to the overall timing and productivity of the infection cycle in a cell. Here wereview the initial case studies, which include studies of the one-step growth behav-ior of viruses that infect bacteria (Q�, T7, and M13), human immunodeficiency virus,influenza A virus, poliovirus, vesicular stomatitis virus, baculovirus, hepatitis B and Cviruses, and herpes simplex virus. Further, we consider how such models enable oneto explore how cellular resources are utilized and how antiviral strategies might bedesigned to resist escape. Finally, we highlight challenges and opportunities at thefrontiers of cell-level modeling of virus infections.

KEYWORDS DNA virus, RNA virus, bacteriophages, biophysics, computationalbiology, computer modeling, growth modeling, kinetics, mathematical modeling,molecular biology

INTRODUCTION

Amajor challenge in biology is to predict how organisms will behave based on howthey interact with their environments. This is hard because essential behaviors,

such as how organisms reproduce or develop, depend on sensing and responding todiverse environmental factors, often involving the activation and expression of multiplegenes as well as coordinated interaction among multiple gene products. To addressthis challenge, it may help to start small by targeting the simplest organisms, whosegrowth and development are encoded by the shortest genomes, involving a manage-able number of essential genes and interactions.

Why focus on viruses? Many viruses can readily be cultured, facilitating their detailedstudy, and the relatively simple life cycles of many viruses have for many decades beenamenable to molecular-level dissection and characterization. More broadly, viruses canaffect the behavior of ecosystems and play significant roles in human health anddisease. Studies on viruses that infect bacteria, the bacteriophages, played a key role inseminal discoveries of molecular biology. For example, early studies on phage T4 byHershey and Chase provided compelling evidence of the role of nucleic acids, notproteins, as the material that encodes genetic information. Later work on phage T4 byBrenner, Jacob, and Meselson led to our mechanistic understanding of protein trans-lation, while Jacob and Monod’s studies on phage lambda provided the earliestunderstanding of how the expression of genes could be regulated (1, 2). The one-stepgrowth method for phages developed by Delbrück and Ellis set a foundation for thefirst quantitative study of the virus growth cycle (1), and still further work on animalviruses has contributed to the understanding of how eukaryotic cells regulate theirgrowth and how loss of such regulation can lead to cancer (3).

Every genome in nature encodes multiple processes, and genomes of viruses are noexception. In an appropriate environment of a living cell, the release of a genome froman invading virus can take command, directing material and energy resources awayfrom cellular processes and toward the synthesis of components that are essential forvirus growth, i.e., viral mRNA, viral proteins, viral genomes, and lipids of viral mem-branes. Assembly of these and other parts produces progeny virus particles that, uponrelease by the cell, may then infect other susceptible cells. For many viruses, includingviruses that infect microbes, plants, animals, and humans, essential molecular processesof intracellular development have been elucidated. However, despite the relativelyshort lengths of virus genomes, the networks of reactions that define virus growth andtheir interactions with their host cells remain complex. These networks can containmultiple positive and negative feedbacks that make it challenging to predict howperturbations to viral or host cellular functions, either by genetic engineering or by thepresence of drugs that specifically target viral or cellular functions, will quantitativelyinfluence virus production or cell survival. To begin to systematically address thischallenge, one may build mathematical models to account for the essential processes

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(Fig. 1). Here we review efforts aimed at modeling how viruses reproduce within cells,and we describe how such quantitative models have begun to provide insights into theintegrated behavior of virus growth. Further, we show how such models can serve asa basis for the development of new antiviral strategies and shed light on the evolutionof viruses.

To set the stage, we use the remainder of the introduction to provide somebackground on virus genomes and the process of virus growth within cells. Further-more, we suggest how the integration of knowledge on viruses at the molecular andcellular levels can be enriched by perspectives of physical scientists, mathematicians,and engineers.

Virus genome sizes span several orders of magnitude. The largest known viralgenome, which belongs to Pandoravirus (2,770 kb) (4), is larger than the smallestgenome of a free-standing bacterium, which belongs to Mycoplasma genitalium (580kb) (5). The smallest viral genomes are less than 10 kb long, and they include genomesfor hepatitis B virus (HBV) (3 kb) and phage Q� (4 kb). Most known virus genomes fallin the size range of 5 to 100 kb and encode up to several hundred proteins (Fig. 2). The

FIG 1 Modeling integrates diverse data to predict virus growth in cells. Descriptions of the molecular functions encoded by the viral genome are used to writeequations that describe how levels of viral mRNA, protein, and genomes change over the course of infection. The equations integrate kinetic and otherbiochemical and biophysical data as parameter values. They are typically solved computationally by numerical integration to yield predicted concentrations ofintermediates and final product (virus) levels over the course of infection.

FIG 2 Most virus genomes encode fewer than 100 proteins. Virus genomes are relatively small, with mostbeing fewer than 100 kb long, encoding about 100 proteins; 1 kb of sequence encodes about oneprotein. The first genome of any organism to be sequenced was that of phage phiX174 (5.39 kb),completed in 1977.

Kinetic Modeling of Virus Growth in Cells Microbiology and Molecular Biology Reviews



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larger virus genomes, such as those of Epstein-Barr virus, smallpox, and Pandoravirus,are composed of DNA. In contrast, many of the smaller virus genomes, such as thoseof human immunodeficiency virus type 1 (HIV-1), influenza virus, and hepatitis C virus(HCV), are made of RNA. These genomes may be composed of one or more segmentsof either single-stranded or double-stranded DNA or RNA. Despite these biochemicaldifferences in the genomes of DNA and RNA viruses, all viruses must make mRNA, andall viruses use the host cell translation machinery to synthesize viral proteins. Virusesthat carry single-stranded RNA genomes may possess positive- or negative-sense RNA,and a positive-sense RNA genome may immediately serve as an mRNA template for thetranslation of viral proteins. Figure 3 shows how the genomes of different virusesemploy different viral or host polymerase activities to synthesize RNA and, ultimately,protein. Viruses that carry double-stranded DNA (dsDNA) genomes, such as Epstein-Barr virus, use the host cellular RNA polymerase (RNA Pol) for transcription of viralmRNA, while other dsDNA viruses, such as smallpox virus, use their own viral DNA-dependent RNA polymerase. Viruses that carry positive-sense RNA genomes, such ashepatitis C virus and poliovirus, may immediately use their genomes as mRNA tem-plates to direct the synthesis of virus proteins, while viruses that carry negative-senseRNA genomes, such as influenza or Ebola virus, employ a viral RNA-dependent RNApolymerase to make positive-sense RNA, which may then serve as a template forprotein synthesis. Double-stranded RNA viruses, such as rotavirus, which can causediarrhea in children, also employ a viral RNA polymerase to make viral mRNA. Retro-viruses, most notably HIV-1, carry two copies of a positive-sense RNA genome, butthis RNA is not translated at the outset of infection. Instead, the retroviral genomeis reverse transcribed by a viral RNA-dependent DNA polymerase to make viral DNAthat is integrated into the host cellular DNA, which later serves as the template tomake viral mRNA. A more extensive description of the diverse strategies that virusesuse to express their genome-encoded functions and to replicate their genomes isavailable elsewhere (6).

Infection cycles follow a series of basic steps that are shared by many viruses(Fig. 4). These steps generally include (i) binding of the virus particle to the cell

FIG 3 To replicate, all viruses must make mRNA and protein. Single-stranded positive-sense RNA genomes, designated (�)ssRNA, are of the same sense asmRNA, so they can immediately serve as templates for protein synthesis.




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FIG 4 Each step of virus reproduction can be described by a differential equation. (1) Binding. A free virus particle initially adsorbs to the surface of a livinghost cell, a process that is usually mediated by proteins on the surface of the virus particle and specific receptor proteins on the surface of the host cell. Theequation expresses the binding event as a mass action process that depends on levels of free (unbound) virus and free (unoccupied) receptors. Further, theequation describes how levels of bound virus fall as they enter the cell. Here, for simplicity, we neglect the possible dynamics of the receptor, which can recycleto the surface or be internalized. (2) Entry. The genome of the virus, often accompanied by copackaged viral proteins, is delivered into the host cell, where itgains access to the protein synthesis machinery and other resources of the cell. The equation accounts for the appearance of genomes that are supplied byviral entry as well as the increase in genome level owing to replication and their depletion owing to their decay. (3) Transcription. The virus genome is usedas a template to produce different viral mRNA transcripts. Different mRNAs (denoted by the subscripted variable i in the equation) are accounted for bypotentially different rates of transcription (promoter strengths), and they depend on genome levels. For positive-sense RNA viruses, the transcription step isnot included because the viral genome serves as the template for translation. (4) Translation. The viral mRNAs recruit the protein synthesis machinery of thehost cell to produce different viral proteins. (5) Assembly. Virus proteins self-assemble to form aggregates (procapsids). (6) Encapsidation. The packaging ofprogeny viral genomes into viral procapsids yields intact viral progeny. The equation accounts for the dependence of ith protein synthesis on the level of ithmRNA and also for depletion of free proteins by processes that assemble them into capsids and processes that cause them to decay. (7) Release. Viral progenyare liberated from the host cell, and encounters between the viral progeny and other susceptible host cells initiate further rounds of growth. Concurrentprocesses (not shown, for simplicity) include a diverse range of cellular responses to infection, such as activation of cellular defenses (restriction or clusteredregularly interspaced short palindromic repeat [CRISPR]-Cas responses in bacteria or interferon-mediated innate immune signaling in mammalian host cells),induction of cell suicide (apoptosis) responses, and shutdown of host biosynthetic functions.




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surface, (ii) entry of the viral genome into the cytoplasm or nucleus, (iii) transcrip-tion of viral mRNA, (iv) translation of viral proteins, (v) replication of viral genomes,(vi) assembly and packaging of virus particles, and (vii) release of progeny virusparticles into the extracellular environment. Some viruses, notably HIV-1 and variousherpesviruses, can establish a state of latent infection in which their host cells do notactively produce virus progeny but instead turn off most genes while retaining theability to switch to productive infection in response to certain changes that result inconditions that have been selected to enable viral persistence over a range of resource-rich or -poor host environments. Quantitative and mechanistic models have beendeveloped to study various partial facets of the virus infection cycle, including virusparticle adsorption to cells (7–9), virus entry (10–13), transcription (14–18), RNA splicing(19, 20), protein translation (21–23), genome replication (24–26), assembly of virusparticles (27–29), packaging of virus particles (30–32), and release or budding ofprogeny virus particles from cells (33). We do not review this extensive literature here;instead, our focus is on models that have sought to be more comprehensive, integrat-ing quantitative descriptions of most of the essential steps in the virus infection cycle.Many books provide a useful introduction and serve as references on viruses and theirgrowth. Among the most accessible to the nonspecialist is Sompayrac’s introduction topathogenic viruses (34). A more-detailed reference in which strategies of specificviruses are used to illustrate broader concepts of virology is that by Flint and colleagues(35), and a more comprehensive reference, which provides detailed reviews of specificvirus families, is Fields Virology (36).

For molecular biologists and biophysicists, we show how molecular mechanismsdeduced from diverse experiments may be integrated into models that enable anintegrated perspective of their behavior. In short, we show how models can be used toexplore how individual molecules and their interactions contribute to the overalldevelopment of a minimal organism. For the physical scientist, engineer, mathemati-cian, or computer scientist, we highlight a central role for creating mathematical andcomputable descriptions of the relevant molecular processes as a systematic way toaccount for their contributions. For the experimental biologist, we emphasize a key rolefor the development of quantitative experiments (with frequent sampling and repeatmeasures) that permit estimations of parameters and their variability, an essentialprocess for validation and refinement of the models. For biomedical scientists, we showthat such models can enable exploration and identification of potentially robustantiviral strategies or means to maximize yields of virus particles for vaccine or genetherapeutic applications. Finally, for ecological and evolutionary biologists, we makethe case that the ability to calculate the growth rate or fitness of an organism canprovide a means to probe questions on how genes interact and their effects on thedesign, robustness, and adaptability of organisms. A summary of the kinetic models ofvirus growth established over the last 25 years is provided in Table 1.

EARLY MODELS OF VIRUS GROWTH

The earliest models of virus growth focused on phages, the viruses that infectbacteria. Here we present these models in chronological order, starting with phage T4and moving on to RNA phages, with an emphasis on phage Q�. We then take on phageT7, which served as a foundation for diverse modeling studies that examined antiviralstrategies, mechanistic inference from mRNA-protein expression patterns, effects ofgenome organization and host physiology on phage fitness, and computational mod-eling of epistatic interactions. We conclude the section with the model for phage M13,which was recently developed by Fisk and coworkers (37, 38) but is included in thissection to maintain continuity with the focus on phage models.

Bacteriophage T4, a Double-Stranded DNA Virus

The first models of intracellular virus replication, which were published by Barricelliand coworkers in 1971, focused on bacteriophage T4, a double-stranded DNA virus(39–41). These earliest computer simulations were used to test mechanisms of phage




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recombination when two or more parent phage strains coinfect the same host cell andcreate a progeny phage that is a cross of the parent strains. The models used numberstrings to represent locations and characteristics of experimentally determined markerson the genomes of the different strains (41), and they showed that one could feasiblystore such strings in the available memory of an early computer, enable stored stringsto interact (recombine), store the resulting progeny strings, and provide progenygenomes as the readout. In short, these works showed that it was possible to simulaterecombination between virus genomes by using an early scientific computer (IBMmodel 7094).

The models were mechanistic to the extent that they accounted for how differentgenomes might physically interact to create crossover events. They were also mecha-nistic in the implementation of rules that allowed genomes to amplify during growthand permitted such amplification processes to continue until only a viable number orexperimentally observed burst size of phage progeny was obtained (41). However, themodels were nonmechanistic in the sense that arbitrary rules were applied for caseswhere mechanisms were unknown, such as the regulation of DNA synthesis. Experi-mental data incorporated into these models included genomic locations of markersfrom classical genetic crosses and mutagenesis experiments compiled from the phageT4 literature.

A Single-Stranded Positive-Sense RNA Virus

In 1975, Knijnenburg and Kreisher initiated development of a computational modelto describe the intracellular development of an RNA phage. Their work was motivatedby an emerging mechanistic understanding from experiments on the highly coupledroles that the synthesis of viral RNA and viral proteins play in the life cycle ofsingle-stranded RNA bacteriophages MS2 and Q� (42, 43). Both MS2 and Q� carrypositive-sense RNA genomes which can serve as mRNA templates to synthesize phageproteins or to produce negative-sense RNA templates, which are essential intermedi-ates in phage genome replication. First, Knijnenburg and Kreisher explored, throughsimulation, how the probability of ribosome binding to sites for translation of thephage coat and RNA polymerase could influence the kinetics of coat and polymeraseproduction during infection. They accounted for the role of coat protein synthesis andaccumulation in the subsequent suppression of polymerase synthesis in perhaps the

TABLE 1 Intracellular kinetic models of virus growth

Class Virus

Model(s) available

Reference(s)Binding/entry Transcription

Genomereplication Translation

Particleassemblyor release

Geneinteraction Antiviral

Wet-labexperiments

dsDNA Phage T7 ● ● ● ● ● ● 67–69, 77, 80, 83Herpesvirus ● ● ● ● ● 18Baculovirus ● ● 119Baculovirus ● ● ● ● ● ● 23

ssDNA Phage M13 ● ● ● ● ● 37, 38

(�)ssRNA Phage Q� ● ● ● ● ● 47, 52Semliki Forest virus ● ● ● ● 10Poliovirus ● ● ● ● 102, 54Hepatitis C virus ● ● ● 128General ● ● ● ● 63

(�)ssRNA Vesicular stomatitis virus ● ● ● ● ● ● ● 116, 155Influenza virus ● ● ● ● ● ● 97Influenza virus ● ● ● ● ● 94

ssRNA-RT HIV-1 ● ● ● ● ● 91HIV-1 ● ● ● ● ● ● 19, 20, 88, 89HIV-1 ● ● ● ● 16

dsDNA-RT Hepatitis B virus ● ● ● ● ● 125




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earliest computational study of stochastic gene expression (44). Unfortunately, insuffi-cient details were provided to fully understand how the simulation was implemented.In subsequent work, they expanded their initial model to account for synthesis ofessential RNA and proteins of the phage (45). Their efforts were notable because theaim of their model—to better understand how the information content of the viralgenome and its interactions with viral proteins contribute to the manufacture of viralprogeny— defined a central long-term goal for the field. This work marked the earliestmechanistic modeling of virus replication within cells. Two assumptions were key. First,it was assumed that the kinetics of infection could be examined in a manner indepen-dent of the processes of the host cell. This independence or uncoupling was achievedby assuming that all essential cellular resources, including ribosomal subunits, replica-tion factors, tRNAs, amino acids, and nucleoside triphosphates (NTPs), were present inexcess throughout the infection cycle. Today, this assumption remains a useful simpli-fying step to enable one to define an initial set of equations focused on the essentialfunctions encoded by the viral genome. A second key assumption was based on theseparation of time scales between fast molecular binding release events and slowprocesses of template-directed polymerization. In short, the binding release processeswere assumed to be sufficiently fast that the participating molecules could be treatedas equilibrated with their complexes and characterized by a single equilibrium bindingconstant. This assumption had the effect of making the dynamics of virus intracellulardevelopment be governed primarily by the rates of viral transcription, genome repli-cation, and protein synthesis, which also continues to be a useful starting point for thedevelopment of virus simulations.

BACTERIOPHAGE Q�

Bacteriophage Q� possesses a single-stranded RNA genome of 4,160 nucleotidesthat carries four genes. The genome is a positive-sense RNA which can serve directly asa template for translation of phage proteins. The virus-encoded proteins include amaturation/lysis protein (A2), the coat protein, a readthrough protein (A1), and asubunit of the phage RNA-dependent RNA polymerase (replicase). It is notable that twodistinct proteins, the phage coat protein and A1 protein, are synthesized from a singlecistron. The Q� replicase normally uses the phage genome and negative-sense anti-genomes during infection. It is also able to use a large number of phage-like shorterRNAs (typically 25 to 50 nucleotides long) as templates because they retain secondarystructures essential for binding and engagement of the replicase. The Q� infectioncycle within Escherichia coli can produce about 1,000 infectious phage progeny per cellin about 45 min at 37°C, and the intracellular steps have been well characterized (43,46–48).

Kinetics of Q� RNA Replication

Detailed in vitro kinetic studies of RNA replication by the Q� replicase, along withanalysis and computer simulations, revealed features of the dynamics that establisheda foundation for a model of phage Q� growth. With an excess of high-energymonomers (NTPs), the level of RNA increases either exponentially or linearly, dependingon the relative level of replicase (49, 50). When replicase concentrations exceedtemplate concentrations, RNA growth depends only on RNA levels, so RNA levelsincrease exponentially. Alternatively, in the presence of excess templates (or limitingreplicase), RNA levels increase linearly, a condition where one may assume a quasi-steady state at the level of the elongating RNA complex (50). These in vitro modelsystems have been useful for highlighting similarities between replication of short RNAtemplates and that of full-length phage RNA genomes. For example, formation ofdouble-stranded RNA through the annealing of cRNA templates can deplete both RNAsserving as templates for replication, reducing overall rates of RNA synthesis (25).Further, rate-limiting processes for replication of short-chain RNA species tend to be atthe stage of replicase release, requiring hundreds of seconds in vitro, as opposed to invitro elongation, which occurs in tens of seconds (51). In contrast, for genome-length




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RNA, the rate-limiting process is elongation (50). For simplicity, these models omitconsideration of RNA templates that simultaneously engage more than one replicasemolecule, an assumption that may need to be relaxed for genomic templates.

Positive Feedback in RNA Replication during Q� Growth

A computational model for phage Q� intracellular growth was published in 1991(47). In contrast with the earlier in vitro models and experiments on Q� RNA replication,in which the replicase enzyme was readily available as an initial condition, replicaselevels for the phage model were initially zero, as in actual phage Q� infections. In thephage model, an initial phage genomic template is translated by host ribosomes toproduce phage proteins. Further, the phage model accounts for the diverse competingprocesses that can engage the phage genomic RNA. It may be bound by ribosomes toserve as mRNA to make protein, or it may be bound by the replicase enzyme to serveas a template to make antigenomic RNA. Further, the genomic RNA may be bound byphage coat proteins during the process of phage particle assembly. However, thetiming and sequence of these processes are governed by the initial conditions. In theinitial absence of phage replicase and coat protein, only translation of the genomictemplate can take place. The synthesis of replicase coupled with the amplification ofthe RNA genome that encodes replicase defines an explosive positive-feedback loopthat amplifies both replicase and RNA with an initially hyperbolic dependence, asfollows: dx/dt � xn, where 1.5 � n � 2 and x represents the intracellular concentrationof the replicase or the phage genome. This amplification is more sensitive to proteinand RNA levels and more rapid than exponential amplification (n � 1). Because thisamplification depends on the availability of ribosomes to make replicase, and becauseribosome levels are finite, rapid amplification of RNA eventually reaches a steady statewhere the RNA level is balanced by the combined levels of ribosomes and replicases.Beyond the equivalence point, RNA levels increase with linear dependence on theselevels. This hyperbolic-to-linear growth transition in the phage infection model mirrorsthe exponential-to-linear growth transition observed during the in vitro amplification ofsmall RNA templates by replicase. Finally, accumulation of coat protein enables the coatto successfully compete against the ribosomes and replicases for binding to thegenomic RNA, allowing packaging of the genomes and production of phage progeny.Over the course of the simulation, NTP resources are described as “buffered,” meaningthat these building blocks for transcription and RNA replication are assumed to beunchanging. They are assumed to be supplied at the same rate as that at which theyare consumed. Ribosome levels, however, are set at a finite level.

Robustness of Simulated Phage Growth with Respect to Parameter Values

The 1991 model of phage Q� growth by Eigen and coworkers (47) focuses on theinformation flows associated with transcription, translation, and genome replication.The initial condition for the model is a single-phage genomic template in a cell-sizedvolume with a finite number of ribosomes and a fixed supply rate of substrate resourcesfor RNA synthesis. Substrate resources for translation, such as pools of amino acid-charged tRNAs, were not specified but appear to be unbounded. The simulationsterminate following packaging of genomic templates with coat proteins at a ratio of1:180. It was noted that the overall sequence of events and behavior “proved inexhaustive trials to be rather insensitive to moderate changes in the rate constantvalues,” supporting the idea that the processes and interactions do not need to befinely tuned to a narrow set of parameter ranges in order to exhibit phage-likebehavior. Eigen et al. found that the rate constants for RNA binding to replicase,ribosomes, and coat protein should be within a 100-fold range of each other and thatcoat protein binding to RNA should be balanced, i.e., favorable enough to represstranslation by competing with ribosomes or replicase for RNA but not so high as to shutdown replicase production and RNA synthesis. In this manner, the coat protein plays apivotal role in mediating early functions of genomic RNA as the template for transcrip-tion and replication, with a later purpose as a packaged product of infection. The




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coupling of reactions during phage growth creates other compensatory effects thatcontribute to robustness. For example, reducing the rate of ribosome binding to phageRNA can have a detrimental effect on phage protein synthesis, but at the same time itmay permit greater access of the replicase to the genomic template, an essential stepfor synthesis of the antigenome template (52).

Efficient Use of Host Energy Resources

Virus infections consume energy at every step of their intracellular growth. Kim andYin sought to estimate how biochemical energy was utilized by accounting for itsconsumption over the course of a simulated phage infection cycle (52). An energyequivalent was defined as the energy released by hydrolysis of a phosphate group fromATP, the common currency in bioenergetics. In this work, Eigen’s Q� model wasextended to account for the energy consumed in the synthesis of all phage RNA species(genome, antigenome, and mRNA) as well as the phage proteins. Synthesis of 25,000phage particles, of which 1 to 10% are typically infectious, requires 3 � 109 energyequivalents, which is about 20% of the level needed to synthesize an E. coli host cell.Of the total energy expended, an estimated 90% is directly invested in the biosynthesisof phage progeny particles. Tenfold changes in parameter values of the model in mostcases had relatively little impact on the efficiency of energy expenditures, which weredirectly related to 2-fold (at most) changes in phage yields. Together these resultssuggest that the phage has evolved to efficiently utilize available energy resourcesduring growth, and this approach has been extended to other systems, includingphage T4, influenza virus (53), and poliovirus (54).

The quantitative analysis of metabolic networks has advanced experimental andcomputational approaches to link cellular resource utilization with intracellular meta-bolic fluxes in E. coli (55). These approaches have been extended to include themetabolic demand by infections of a Q�-like positive-sense RNA phage, MS2 (56),where biosynthetic demands by the phage growth were predicted to substantiallyincrease metabolic activity in the pentose phosphate pathway.

Refinement of Phage Q� Models

Models of phage Q� growth, to date, have neglected processes of initial binding ofthe virus to its host cell, virus entry and release of the phage genome into thecytoplasm of the host cell, and lysis-mediated release of progeny phage from the cell.In particular, the timing of lysis directly affects phage yields and likely reflects anoptimization owing to ecological factors (57, 58). While it is known that the phage lysisprotein prevents cell wall biosynthesis by inhibiting an essential enzyme in the process(59), how inhibition of cell wall biosynthesis ultimately affects the timing of phageprogeny release remains to be elucidated mechanistically. For other aspects of phagegrowth, refinements in the mechanisms of other processes, such as Q� RNA replication(26) and particle assembly (60), may also enable refinements or improvements tocurrent models.

Testing Antiviral Strategies

Many current antiviral drugs are directed to inhibit specific viral functions, butviruses can develop mutations that enable them to resist such treatments. Onealternative strategy may be to develop defective viruses that parasitize normal virusinfections by diverting essential resources for virus growth to the viral parasite (61).Such defective viruses have historically been called defective interfering particles (DIPs).In the context of Q� infection, we employed our model to computationally explore thepotential effectiveness of such a parasitic antiviral strategy (62). We simulated thebehavior of minivariant 11 (MNV-11), an 87-nucleotide RNA derived from the Q�

genomic RNA that can compete with the wild-type template for the Q� replicase. Theability of this molecular parasite to very rapidly amplify itself and compete for Q�

replicase allowed it to prevent amplification of the wild-type phage genome, based onour simulations. Other strategies against positive-strand viruses, based on RNA silenc-




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ing, have also been explored by simulation and found to yield potentially diverse re-sponses, including inhibition of virus growth, oscillatory behavior, or complete clearance ofinfection (63).

BACTERIOPHAGE T7

The genome of bacteriophage T7 is a double-stranded DNA of 39,937 nucleotidesthat carries 56 genes and infects E. coli in a lytic manner, typically producing 100 phageprogeny in 40 min at 30°C. The genome enters the host cell in a gradual manner,coupled with transcription of the phage genes, initially by the E. coli RNA polymeraseand later by the phage RNA polymerase, which encounters stronger promoters as moreof the genome becomes accessible. Owing to the gradual nature of genome entry andtranscription, T7 genes are expressed at different times depending on their relativepositions in the genome.

An initial motivation of Endy et al. for developing an intracellular model forbacteriophage T7 was to better understand the behavior of phage mutants that wereselected during evolution experiments. In phage cultures, wild-type phage T7 wascontinuously passaged on recombinant hosts that constitutively expressed an essentialphage enzyme, the T7 RNA polymerase. During 30 generations, at most, phage mutantsarose that carried a deletion in the gene for this essential enzyme, making themcompletely dependent on the host-supplied enzyme for their growth. Moreover, thesemutants grew faster on the recombinant hosts than the initial wild-type phage did (64,65). Endy et al. speculated that the faster production of mutant progeny could beattributed to the faster synthesis of shorter phage genomes and the ability to dispensewith the time required to synthesize the polymerase in wild-type infections. Back-of-the-envelope calculations of the rate of T7 DNA polymerase and the transcription andtranslation rates needed to produce the enzyme appeared to account for the 10% faster(3 to 4 min) production of mutants than that of wild-type phage (66). Using thesecalculations as a basis, we assembled a more comprehensive model that accountedfor the rate of phage genome entry into the host cell, synthesis of T7 mRNAs,translation of these mRNAs to produce T7 proteins, known protein-protein interactionsand feedbacks on host and phage transcription, synthesis of T7 DNA genomes, andassembly of phage progeny (67). Over the course of developing the model, otherapplications for models of virus intracellular growth emerged. For example, simulationsbased on published mechanisms and data can be used to examine the consistency ofthe data and reveal when new results challenge the existing literature. Second, modelscan readily be extended to predict how different antiviral strategies will influence thesimulated growth dynamics. Since the simulations can readily be modified, manypotential strategies can be explored efficiently, often suggesting experiments thatmight not otherwise be performed or helping to prioritize planned experiments.

Several key assumptions defined the framework for the T7 model. Although Endy etal. only later became aware of Knijnenburg and Kreisher’s 1983 phage model, theynevertheless employed several of the same assumptions. For example, for the T7 modelit was assumed that the protein synthesis machinery of the cell (ribosomes, activatedtRNAs, and proteins) was present in excess at all stages of infection, that protein-proteinbinding interactions were sufficiently fast to be treated as equilibrated, and that theintracellular processes were “well mixed,” neglecting any spatial heterogeneities orroles of component transport in the dynamics of growth. In contrast to Knijnenburgand Kreisher’s stochastic kinetics model, however, the phage T7 model was determin-istic, as it was based on a set of coupled ordinary differential and algebraic equations.Further assumptions specifically related to details of phage T7. For example, to simulatethe assembly of the T7 virion particles, which had not yet been studied in detail, resultsfor phage P22, a dsDNA phage that is morphologically similar to T7, were employed.The final model incorporated 46 parameters spanning 27 years of the phage and E. coliliterature. Twenty of the parameters described rates of elongation by RNA polymerases(transcription), DNA polymerase (genome replication), and ribosomes (protein synthe-sis), spacing requirements for polymerases and ribosomes on their respective tem-




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plates, decay rates for mRNA, proteins, and DNA, binding constants for protein-proteininteractions, and other processes. Moreover, 16 parameters quantified the relativestrengths and initiation efficiencies of the T7 transcriptional promoters, and 10 param-eters described the stoichiometry requirements of the progeny phage particles. Noneof the parameters were adjusted to fit the final results. In addition, owing to a lack ofmechanistic understanding of the lysis step that allows progeny T7 phage to bereleased from the cell, we simulated only the production of phage progeny within thecell.

The initial T7 model was able to capture essential features of the T7 growthbehavior, based on a comparison with data that were independent of the modelformulation. The observed timing of simultaneous shutdown of host RNA polymer-ase activity and expression of phage RNA polymerase activity early in the infectioncycle was well captured by the model. The simulated order of appearance of phageproteins that are characteristically expressed during early, middle, and late stagesof the infection cycle captured trends observed from pulse-labeling experiments,though discrepancies were notable, particularly for later structural proteins, whichtended to appear earlier in the simulations than in the experiments. Finally,experimental validation of phage intracellular production was performed by lysingcells at different time points following the initiation of infection and quantifying thecorresponding intracellular level of phage particles. Simulated production of wild-type T7 phage rose at a rate similar to that for wild-type phage in our experiments,but one-step growth curves appeared about 3 to 5 min earlier than those observedexperimentally for a 30-min infection cycle. Simulations of a phage deletion mutanton a recombinant host that expresses the T7 RNA polymerase produced mutantphage earlier than that found in the wild-type phage simulations, consistent withexperimental observations.

Antiviral Strategies

We initially employed our model of phage T7 intracellular growth to simulate howdifferent antiviral strategies might influence T7 growth (67, 68). We chose to initiallytest antisense strategies because such approaches offered a convenient way to thinkmechanistically and quantitatively about the effects of targeting specific phage func-tions. For each antiviral strategy, the following three features were defined: the specificT7 mRNA target, the initial level of target-specific antisense RNA in the cell, and theequilibrium binding constant for the formation of the complex between the targetmRNA and the antisense RNA. Here the binding constant for complex formation can beviewed as a parameter that describes the intrinsic potency of the antisense drug againstits mRNA target. We expanded our model of T7 intracellular development to includeantisense drugs that targeted mRNA encoding either the major coat (capsid) protein orthe T7 RNA polymerase (RNA Pol). Key results suggest that drugs that target differentessential components can produce qualitatively different effects on growth (Fig. 5). RNAPol transcribes phage proteins that inhibit the function of the host transcriptionmachinery, and this machinery is essential for expression of the RNA Pol. Thus, RNA Polis part of a negative-feedback loop that ultimately regulates its own expression. Whileit is challenging to anticipate how a drug that targets a component of an earlyregulatory loop will ultimately affect the production of virus progeny, modeling offersa useful way to expand our intuition. Our study suggested how drug targeting ofregulatory loops could offer benefits by enabling one to block established pathways ofdrug escape or drug resistance. However, taking a still broader perspective, it isimportant that novel antiviral strategies should be viewed as simply defining modifiedcriteria for the selection of new viral escape strategies. In other terms, mutations infunctions that are not directly related to the targeted function may still alter how virusgrowth responds to antiviral drug treatment in ways that would be challenging toanticipate. In this context, the model may serve as a foundation to test other escapestrategies.




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From Transcriptome to Proteome

By accounting for the synthesis of the phage mRNAs and proteins that are essentialintermediates for the production of progeny particles, we simulate as a by-product theconcentration-versus-time trajectory of all phage mRNAs and proteins. This informationprovides an idealized data set with which one may test data mining or mechanisticinference tools. For example, one may consider the simplest way to express therelationship between the concentration of a protein and its corresponding mRNAwithin the cell, as follows: d[Pi]/dt � ktrans[mRNAi], where [Pi] and [mRNAi] are freeconcentrations of the ith protein and ith mRNA, respectively, and ktrans is the rate atwhich the ith mRNA is translated into protein. For a fixed translation rate, plottingd[Pi]/dt versus [mRNAi] yields a line with the slope ktrans and an intercept of zero.Alternatively, one may view the free concentration of the ith protein as a process thatintegrates the level of the ith mRNA over time.

When the free concentration of a protein can be influenced by other processes, suchas protein degradation, posttranslational modifications, or formation of complexes withother proteins or nucleic acids, additional terms appear on the right-hand side, asfollows: d[Pi]/dt � ktrans[mRNAi] � kd[Pi] � f(modifications) � g(interactions), where kd

is a rate constant for first-order degradation of the ith protein and f and g are functionsrepresenting other processes that may influence the concentration of the ith protein.In general, these additional processes will cause trajectories of d[Pi]/dt versus [mRNAi]to deviate from a straight line. In the context of our phage T7 model, we simulated plotsof d[Pi]/dt versus [mRNAi] and found that some phage proteins produced loop-liketrajectories that reflected their regulatory roles (69). Moreover, the model was used to

FIG 5 Drug targeting of viral gene regulation. The model of phage T7 intracellular development was expanded toincorporate effects of antisense drugs targeting different virus-specific functions. (A) Drug targeting of mRNA thatencodes the major capsid (coat) protein of T7, a major protein component of the progeny phage. As the potency(or affinity) of the drug for this target increased, coat protein production was correspondingly reduced, ultimatelyinhibiting the production of virus progeny. Viruses that spontaneously generate mutations can attenuate the drugpotency (or affinity between the drug and its target) and thereby reduce the inhibitory effects of drug on virusgrowth, as shown, for example, by the path for virus populations moving from point 1 to point 2. More rapidgrowth by drug-resistant viruses than by wild-type viruses allows such viruses to become enriched and enables theresulting virus population to escape from the drug. (B) When a drug targets mRNA that encodes the RNApolymerase of T7 (RNA Pol), a different behavior ensues owing to the negative feedback of RNA Pol on its ownsynthesis. In this case, more potent drugs do not necessarily have greater inhibitory effects on virus growth.Instead, drugs of intermediate potency can have larger inhibitory effects on virus production because they exploitthe contributions of the feedback to the overall growth behavior. Mutations that reduce drug potency enable viruspopulations to move from point 1 to point 2, creating viruses that are more growth inhibited than the wild type.Such mutants would not be expected to become enriched over the wild-type virus. By accounting for the overalleffects of drugs on such regulatory loops, the model provides an opportunity to identify potential “evolutionarytraps” that select against established mechanisms of drug escape (68).




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quantify for each phage protein how it deviated from linearity in its plot of d[Pi]/dtversus [mRNAi] over the course of infection. Such deviant proteins are interestingbecause they represent proteins that are modified, where f(modifications) is nonzero, orproteins that interact with other proteins, where g(interactions) is nonzero. If two ormore proteins interact with each other, then they would be expected to be depleted(and to deviate) in a correlated manner. We calculated extents of such correlation forall phage protein pairs and found that, indeed, the protein pairs that were computa-tionally modeled to form complexes also shared highly correlated deviations. It will beinteresting to use highly quantitative measures of mRNA and proteins from the samecells over time to explore whether such a “correlated deviation” approach can be usedto infer physical interactions and formation of multiprotein complexes during infection.There is not yet a consensus on how global transcript and protein data should betreated to gain mechanistic insight. While much of the field has focused on character-izing the extent of correlation between protein and mRNA levels (70, 71), alternativeapproaches that combine mechanistic models of translation with mRNA and proteinmeasurements have begun to highlight challenges in characterizing how limited andchanging translation resources influence quantitative links between message andprotein (69, 72–75).

Genome Organization Affects Virus Fitness

Given that we have a simulation for phage T7 one-step growth, one can begin toexplore alternative genome designs by changing the way the simulation is imple-mented. This may be done by changing the order of genes or the parameters of themodel. In the case of phage T7, entry of the genome into the cell is mediated by theaction of the host RNA polymerase and later by the phage RNA polymerase, withthe processivity of the polymerase corresponding to the entry of phage genes. In short,genes at the left end of the genome are transcribed before genes at the right end.Moreover, because the strength of promoters increases as one moves from the left tothe right end, later genes are also expressed at higher levels. By altering the positionsof genes within the phage genome, one can alter their timing and level and therebychange the fitness of the phage in subtle or not-so-subtle ways. One of the not-so-subtle ways is to move the T7 RNA polymerase gene to positions where its transcriptionis put under the control of a T7 promoter, creating a positive-feedback loop of T7 RNAPol on its own transcription. Predicted enhancements in phage protein synthesis andphage production were not supported by quantitative experiments on protein pro-duction by pulse-chase methods or by yields of progeny phage by one-step growthexperiments (76). One reason for the discrepancy is that the simulation did not accountfor the contribution of nonessential genes, such as phage gene 1.7. Although it isknown that 1.7 is nonessential for phage growth, control experiments established thatthe absence of gene 1.7 does have a detrimental effect on phage progeny yields. Abetter accounting of the finite resources provided by the host cell was not able toexplain the differences observed between experiments and simulations (77). Thesignificant observed differences between predicted phage with the alternative geneorder and three experimentally generated and tested phages highlight the substantialwork that still needs to be done to better understand the nature of the discrepancies.

Synthetic Biology Test of Model Assumptions

Current modeling of virus growth builds on simplifying assumptions that may ormay not be valid. Quantitative experiments can be used to test and modify assump-tions that may subsequently allow the model to account for more observations or makenew predictions. An alternative approach to gaining biological insights is to alter thebiology to match the simplifying assumptions of the model. For example, the genomeof phage T7 has multiple overlapping genes that are not essential for phage growth. Inour model, we assumed that T7 genes were not overlapping, an assumption thatenabled us to model the kinetics of transcription of these genes (67). It was unknownat the time how the presence of overlapping genes might affect phage growth. To




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address these effects, Endy and coworkers redesigned and synthesized 30% of the T7genome to eliminate overlapping regions. The resulting chimeric phage was able toproduce viable phage progeny, though plaque sizes indicated that the growth prop-erties of these variants were attenuated relative to those of the wild type (78). Thisexercise was useful in showing how simplifying modeling assumptions might be testedby “changing the biology to fit the model” instead of the more common approach of“changing the model to fit the biology.” In the specific case of phage T7, the chimericphage showed that overlapping genes are not essential for T7 growth but likely havean impact on the overall growth or fitness of the phage.

How the Host Cell Environment Can Affect Virus Fitness

A key assumption that enables development of initial models of virus intracellulargrowth is that host resources are infinite. Using this assumption, one may then simulatevirus growth based on the dynamics of the virus-encoded processes. These minimallyinclude transcription of viral mRNA, translation of viral proteins, synthesis of viralgenomes, and assembly of virus progeny particles. The assumption of infinite hostresources is most likely to be reasonable at the earliest stages of infection, when thedemand by the virus infection is defined for a small number of initial viral genomes (theones that initiated infection). To better account for the effects of host resources ongrowth, phage T7 infections were performed on host cells cultured under differentconditions that were set up to provide different host resources. In general, rapidlygrowing cells will have more biosynthetic resources, such as ribosomes for translation,than more slowly growing cells. We cultured E. coli hosts in a chemostat and controlledtheir growth by adjusting the dilution rate of the chemostat (79). Cells at differentgrowth rates, spanning from 0.7 to 1.7 doublings/h, were used as hosts for synchro-nized one-step growth cultures of phage T7, and the resulting rise rates and eclipsetimes (characteristics of one-step growth) were determined. To predict how alteredgrowth rates of host cells would influence phage growth, we employed empiricalcorrelations established by others to indicate how cellular resources and properties,such as host RNA polymerase levels and elongation kinetics, ribosome levels andelongation rates, NTP and amino acid pools, and cell size, correlated with host growthrate. Thus, cellular growth rates set by the experimental conditions were used as inputsto the correlations to estimate cellular resource levels and kinetics, which were used asinitial conditions for the T7 simulation. In experiments, faster cell growth reduced theeclipse time (time to production of initial phage progeny) and increased the rise ratesof progeny production, and these behaviors were captured by the simulation. Thesimulation then provided an opportunity to uncouple the effects of different hostresource conditions on phage growth in cases where such uncoupling would bedifficult or impossible to implement in experiments. Through such studies, we identi-fied the processing of the translation machinery, specifically the ribosome elongationrates, to be most limiting for phage production. Moreover, independent changes insimulated host resources indicated how resources of host transcription or translationwould need to change in order for phage infections to become limited or “bottle-necked” by host RNA polymerase levels or ribosome levels. An assumption of the modelwas that host cell resources for the entire infection process could be defined ade-quately based on their state at the time of sampling from the chemostats. A morerefined perspective will need to account for the consequences of infection on the hostphysiological state, accounting for the effects of potential changes in energy metab-olism on levels of biosynthetic resources or, for example, the synthesis or decay of NTPor amino acid pools during infection.

Epistasis: Quantitative Assessment of Genetic Interactions

A fundamental challenge in quantitative genetics is to better understand howinteractions between genes quantitatively affect the fitness of an organism. Simulationsof virus growth are potentially useful here because they provide a quantifiable linkbetween functional molecular characteristics of gene products, such as binding affin-




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ities or kinetic parameters, and the growth rate, infection productivity, or fitness of thecorresponding virus. To help to understand the terminology, it is useful to consider asimple quantitative example of how mutations in two genes, call them A and B, mayinteract. If the wild type has a fitness of 1.0 and mutations in genes A and B give riseto mutants that have fitness levels of 0.80 and 0.70, respectively, then one may ask,what is the fitness of a mutant phage containing both mutations? If these mutations donot interact, then the double mutant will have a fitness of 0.56, just the product of thefitness values for the single variants. If these deleterious mutations interact synergisti-cally, then the fitness of the double mutant will be �0.56, and if they interactantagonistically or by buffering their deleterious effects, then the fitness of the doublemutant will be closer to that of the wild type (�0.56). In quantitative genetics one seeksto answer the following question: on average, to what extent do deleterious mutationsinteract synergistically, antagonistically, or not at all? This can be a challenging exper-iment because it requires that one be able to generate mutants having well-definedmutations and to accurately quantify the effects of these mutations on some measureof fitness. In simulations of virus growth, it is feasible to create such mutations byaltering parameters that correspond to molecular functions, such as a binding constantfor complex formation or a rate constant corresponding to the elongation rate of anRNA polymerase, and then to calculate the effects of such alterations on the yield ofvirus progeny from a cell, one measure of fitness. The rationale here is that somemutations can alter quantitative characteristics of molecules and that the correspond-ing parameters in the models can be changed to simulate the effects of such mutationson molecular function. If one seeks to simulate deleterious mutations, then one justneeds to check that the alteration in a parameter reduces the calculated fitness. Weused this approach to generate and test interactions among diverse computationallygenerated deleterious mutations to examine the effects on the calculated fitness ofphage T7 (80). In this study, two metrics for fitness were employed: one for a resource-rich environment and one for a resource-poor one. For the rich environment, weassumed that host cells were available in unlimited supply, so the fitness of the virusdepended on maximizing its production rate within a given cell. For the poor environ-ment, we assumed that only the infected cell’s resources were available, so fitness wasbased on maximizing the yield, without limitations on the rate. It was found that mildlydeleterious mutations tended to act synergistically in resource-poor environments butantagonistically in resource-rich environments. Moreover, severely deleterious muta-tions tended to buffer themselves, acting antagonistically in both rich and poorenvironments. The work was relevant to population genetics in suggesting that theeffects of synergistic interactions on fitness, while important for theory, may bechallenging to detect in practice owing to their emergence when the quantitativeeffects of mutation on fitness are minimal. Future studies would benefit by exploringhow quantitative rather than qualitative changes in environmental resources affectwhether epistasis is synergistic or antagonistic. This may be achieved, for example, byusing the metric for fitness based on production rate (e.g., the rich environment, asdescribed above) and altering the growth rate of the host, creating richer or poorerenvironments based on higher or lower rates of host cell growth (79). Still more realisticassessments of epistasis might define the average fitness of the virus over multiplecycles of infection where one allows for fluctuations in host resources, conditions thatcould readily be implemented in virus growth simulations.

Robustness versus Fragility

Biological systems tend to perform robustly with respect to conditions under whichthey evolved but are more sensitive to environmental changes that they have notencountered (81, 82). These ideas were examined in the context of the bacteriophageT7 system by testing the effects of simulated natural mutations, implemented bychanging parameters of the model over plausible ranges for natural mutations, andsimulated unnatural mutations, using previously described genomic rearrangements(76, 77). In general, the simulated phage growth was robust with respect to parameter




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changes but relatively sensitive or fragile with respect to genomic rearrangements,which are not known to occur in nature for phage T7. Such observations wereconsistent with theory (83). The robust behavior of growth was also a feature of thephage Q� growth models (47). The effects of natural and unnatural perturbations to thesimulated one-step growth of phage T7 were also evaluated in rich and poor hostresource environments, and it was interesting that the fitness of wild-type phage wasnearly optimal under resource-poor conditions but average under resource-rich con-ditions (83). This finding suggests that limited host resources served as a constraint onprocesses of phage T7 evolution. The extent to which such findings are general remainsto be tested.

BACTERIOPHAGE M13

Phage M13 is a single-stranded DNA virus with a 6.4-kb genome that encodes 11proteins, 5 of which combine to encase a single genome in a well-defined filamentousstructure. The structure is composed of four minor coat proteins (p3, p6, p7, and p9),each incorporated at 5 copies, and a major coat protein (p8) that is present at about2,700 copies per phage particle. The well-resolved structures of the proteins and theirarrangement in defining the surface of the particle, along with the facile tools foraltering the gene for each protein, have in the last 15 years made phage M13 popularfor controlled engineering at the nanometer scale, with diverse applications, includingbiosensors, batteries, and memory devices (84). Prior to these technological develop-ments, fundamental studies established many of the molecular mechanisms associatedwith M13 DNA replication, mRNA processing, and mRNA degradation (85, 86). Usingthis and an extensive literature review, a kinetic model for M13 intracellular growth wasdeveloped that employed 81 differential equations and 64 kinetic parameters, of which43 parameters could be estimated from experimental data (37). The model assumedunlimited resources in pools of amino acids, nucleic acids, and energy, but it avoidedinfinite virus growth owing to limitations on ribosomal availability. If specific data werenot available, alternative approaches were found in some cases. For example, thedistribution of a limited pool of ribosomes to phage mRNA should depend on thenature of the interaction of each mRNA species with the ribosome, as reflected in partby the ribosome binding sequence (RBS), but strengths of RBS sequences for M13mRNA have not been measured experimentally. To address this limitation, an RBScalculator was used; the calculator employs an equilibrium statistical thermodynamicmodel to account for the effects of the RBS sequence on translation initiation rates (87).

The overall M13 intracellular growth model was able to reproduce diverse aspectsof the phage’s biology, including the timing and extent of phage DNA replication, thelevels of mRNA and protein production, and the timing and levels of progeny phageproduction. Further, by observing how different parameter values could be combinedto produce higher or lower simulated progeny levels, the work suggested that changesin rates of phage progeny assembly are tightly linked to levels of phage DNA andprotein production. Because the phage does not kill its host cell after a single cycle ofphage progeny production, phage production can continue over multiple cell divisions.Extension of the intracellular model to account for multiple cell cycles allowed fortesting of the sensitivities of phage processing in establishing a persistently infectedstate or a cured state (38), conditions that have both been observed experimentally.More specifically, simulations suggested that p5, a protein that binds single-strandedDNA, may be involved in previously undocumented feedback loops between p1, p3,and p8, working at the level of translational attenuation. Ultimately, because thesemodels quantitatively and mechanistically account for production of each phagecomponent (DNA, mRNA, and protein) in the broader production of phage progeny,they provide an opportunity to explore diverse perturbations, such as mechanisms oftranslational control, that would be experimentally challenging to modify or elucidateindependently. Here such simulations suggested that dynamic control of the amountof p5 in the infected cell plays a key role in the allocation of biosynthetic resourcesduring M13 infection.




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HIV-1

HIV-1, the virus that causes AIDS, is of great interest in human health owing to the33 million people worldwide who suffer from AIDS and the 2 million annually who diefrom the disease. As a retrovirus, it carries two single-stranded positive-sense RNAgenomes that serve as templates for the packaged viral reverse transcriptase tosynthesize a double-stranded DNA molecule, which integrates into the host genome aspart of its infection cycle.

To what extent can the dynamics of HIV-1 growth within its mammalian host cell, aCD4� T lymphocyte, be explained or accounted for by the kinetics of its underlyingprocesses? In the case of HIV-1, following binding to the host cell and particle entry, theviral genomic RNA is released into the cytoplasm, where it is reverse transcribed tomake double-stranded proviral DNA, which is then brought into the nucleus by the viralintegrase enzyme and integrated into the genome of the host cell. The proviral DNAremains in a nonproductive or latent state until the cell and viral transcription processesare activated by external factors. The model of Reddy and Yin neglected the latentphase by assuming that the cell was activated, so transcription of viral messages couldimmediately proceed. The model accounted for splicing of mRNA and its transport fromthe nucleus to the cytoplasm, translation of viral proteins in the cytoplasm, feedbacksof regulatory proteins Tat and Rev on transcription, transport of viral proteins to the cellmembrane, and particle assembly, budding, and maturation (88). Simulated levels ofHIV-1 DNA and entering genomic RNA, which are synthesized and degraded duringreverse transcription, matched well with experimental observations. Further, the kinet-ics of production of viral RNA genomes (full-length RNA) and translation of viralproteins was consistent with the sparse available data at the time. In the absence ofmechanisms of virus particle assembly, it was assumed that virions assembled instan-taneously, with the only constraint being that each virus particle must satisfy theestablished particle stoichiometry (e.g., 1 genome and 1,200 Gag, 80 Gag-Pol, and 280Env proteins). This assumption resulted in a simulated production of virus progeny thatpreceded the observed production by about 6 h, providing an estimate for the timerequired for the assembly process. The model also enabled study of how perturbationsto individual virus functions might influence the overall growth. If growth is particularlysensitive to small changes in a specific parameter or function, then there is a rationalefor targeting drugs to that function. The simulation highlights a need to exercisecaution in targeting regulatory proteins, such as Rev. When the effects of inhibiting Revwere tested, simulations suggested that doubly spliced transcripts would be enhanced,activating Tat, which would activate viral transcription overall and lead to an enhance-ment of viral growth. However, simulations exploring the effects of directly inhibitingTat suggested that such interventions would always have a detrimental effect on virusgrowth.

More-detailed simulations enabled studies of the effects of transcript splicing ongrowth. Inefficient splicing of HIV-1 mRNA was generally beneficial for HIV-1 growth,but an extreme reduction in the splicing efficiency could be detrimental, suggesting theexistence of a splicing efficiency that optimizes HIV-1 growth (20). When splicing causesan increase in the fraction of either Rev or Tat mRNA relative to that of the other viralmRNA pool, the outcomes are generally beneficial for HIV-1 growth. However, simula-tions indicated that when mutations cause either Rev or Tat mRNA to be favored overthe other, the imbalance is amplified, suggesting that a balance of Rev and Tat isneeded in order for HIV-1 to optimize its growth (19). Further, interactions between twofeedback loops, the negative feedback of Rev on nuclear export of fully splicedtranscripts and the positive feedback of Tat on overall transcription of viral mRNA,create a robust regulatory network that is able to compensate for mutations that mightalter functions of components within the network (89). Such interactions betweenpositive- and negative-feedback loops may have more general relevance to the robust-ness and evolvability of developmental processes in higher organisms (90).

Other approaches to modeling HIV intracellular growth have employed agent-based




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modeling, where different states of the cell or stages of intracellular infection areinitially defined and transitions between states are expressed as rules, with probabilitiesof transition related to the magnitudes of experimentally determined parameters (91).Such rule-based approaches may enable accounting for compartments within the cell,such as the nucleus or assembly sites, in a manner that explicitly incorporates infor-mation about the position, size, or movement of compartments.

An Anti-HIV Strategy

As noted earlier, one may implement antiviral strategies within a model of virusgrowth by including additional reactions and parameters to simulate the action of anantiviral “drug” on a specific viral target. Moreover, one can test drug escape strategiesby simulating how changes in virus parameters, corresponding to virus mutations, mayenable virus variants to grow better than the original wild-type virus in the presence ofdrug. For HIV-1, an RNA interference (RNAi) strategy was proposed to computationallyexplore factors that would influence targeting of the Tat-mediated positive-feedbackloop (92). This example is interesting because of the sequence-specific targeting byRNAi of the trans-acting responsive (TAR) element, a highly conserved RNA structurethat is essential for Tat transactivation. One can imagine that mutations that wouldenable the virus to escape from such RNAi might also have fitness penalties on virusgrowth. Empirically based rules were applied to account for the effects of specific basechanges on the contribution of this structure to the transcriptional feedback, andultimately the fitness of the virus. When this strategy was carried out in experiments, itwas found that no base changes were detected in the TAR element. Instead, mutationsoccurred in nontargeted regions that enabled indirect upregulation of transcription(93). These results, which would have been improbable to be anticipated by currentmethods, highlight the limitations that even quite comprehensive models have inanticipating multiple ways that viruses may find to evade antiviral strategies.

INFLUENZA A VIRUS

In 2004, Sidorenko and Reichl developed the first kinetic model for influenza A virusgrowth in animal cell culture with an aim to identify resource-limiting steps that mightbe addressed to increase virus yields for vaccine production (94). The genome ofinfluenza A virus has eight segments that encode at least 10 virus proteins, nine ofwhich are incorporated into the virus particle. The kinetic model accounted for steps forviral particle attachment to the cell surface, receptor-mediated entry or endocytosis ofthe viral particle, release of the viral ribonucleoprotein (vRNP) (RNA-protein complexes)into the cytoplasm and its transport to the nucleus, transcription of viral mRNA and itsexport to the cytoplasm, translation of viral proteins, replication of viral genomes,formation of viral ribonucleoprotein complexes, and budding and release of progenyvirus particles from the cell surface. It was initially assumed that translation resources(ribosomes and precursors for protein synthesis) were present in excess, so levels ofthese components were not explicitly included in the model. Virus particles were alsoassumed to assemble with correct segregation of the eight RNA species and proteinstoichiometries. Based on experimental observations of cell death 12 h following theinitiation of infection, simulations were carried out by integrating the model to 12 h.

The model enabled one to identify potential bottlenecks in virus production andsuggested strategies for increasing virus yields for vaccine production. The simulationssuggested that M1, the viral matrix protein, becomes a limiting factor in the productionof vRNP complexes, which was apparent as levels of M1 initially accumulated and thenfell to zero as vRNP complexes were formed. Subsequently, vRNPs become a limitingfactor in the budding and release of progeny virus from the infected cell. Moreover,according to the simulation, rates of virus production could be enhanced by increasingthe activity of the viral polymerase, which could be achieved in practice by usingstronger promoters. Likewise, simulations suggested that increasing the efficiency oftranslation of viral proteins would also increase the production of virus progeny. Suchefficiencies could conceivably be enhanced in practice by using virus mutants that




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more efficiently inhibited the utilization of cellular translation resources by cellularmRNAs.

Subsequent measurements of two viral proteins (NP and M1) and virus productionover the course of an infection cycle provided constraints for a population balancemodel of influenza virus infection (95). In this model, levels of these essential viralproteins, quantified by flow cytometry, were used to define an internal coordinate forprogression of infected cells to production of virus. For an excess of added virusparticles to cells (multiplicity of infection [MOI] of 3), the model was able to capture theaccumulation of these proteins in infected cells during earlier stages of infection, up toabout 10 h, but deviated at later times, reflecting potential limitations in amino acidresources or the onset of virus-induced apoptosis. A more-detailed rendering of theviral replication process did not reduce deviations or provide further insight intounderlying mechanisms for deviations (96).

A mathematical model of influenza virus growth in cells was also published byBazhan et al. (97). Their work describes (in Russian) the regulation of transcription,translation, replication, and assembly of the virus. An analysis of the model indicates ahigh sensitivity of model behavior to parameters associated with the binding of viralpolymerase to virus-specific RNAs, which are essential processes for transcription andgenome replication. Such essential processes might be effective targets for the devel-opment of potent antiviral drugs.

Control of Viral RNA Synthesis

In a reformulation of the Sidorenko and Reichl model, the set of original kineticequations was reduced and further data sets from the literature were incorporated toestimate key parameters (98). In addition, mechanisms of RNA stabilization and nuclearexport of RNA species were incorporated to better resolve the dynamics of viral RNAtranscription and genome replication. Constraints on the model included experimentalmeasures of virus entry, i.e., absolute and relative levels of viral messenger, replicativeintermediates, and genomic RNA per cell, as well as average levels of viral progenyreleased per cell. The resulting model provided support for early regulation of genomereplication by stabilization of viral RNA replicative intermediates. It also suggested howthe viral matrix protein 1 (M1), which normally mediates export of viral genome copiesfrom the nucleus, also might control viral RNA levels in the late phase of infection (98).Finally, the model predicted an intracellular accumulation of viral proteins and RNAtoward the end of infection, providing evidence that transport processes or particlebudding limits the process of virus progeny release.

Kinetics of Defective Interfering Particles

It has long been known that influenza A virus infections can produce defective virusparticles that can interfere with the production of infectious virus (von Magnus phe-nomenon). Such particles carry deletions in one or more essential genes needed forgrowth, making them unable to reproduce alone. However, during coinfection withinfectious virus, defective genomes divert resources to their own replication andpackaging, which interferes with the production of infectious particles. Such defectiveinterfering particles (DIPs), described above in “Testing Antiviral Strategies,” have beenfound in clinical isolates of influenza virus (99) and can negatively affect the manufac-ture of live vaccines (100). To better understand mechanistically how DIPs reproduce,the intracellular kinetic model for influenza virus was extended to include defectiveinterfering RNAs that replicate more rapidly than full-length RNAs owing to theirreduced length (101). The extended model was able to account for observed effects ofDIPs on infectious virus production. Moreover, the model suggests that DIPs thatspecifically carry deletions in RNA segments encoding the viral polymerase can becomeenriched, in agreement with experimental observations. Further, the model and exper-imental observations suggest that other mechanisms, such as competition for viralproteins (polymerase and nucleoprotein), can also contribute to interference and DIPenrichment.




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POLIOVIRUS

An interest in better understanding viral evolutionary mechanisms motivated thedevelopment of models that couple intracellular virus growth and release with extra-cellular spread and infection of other susceptible cells. In 2003, Krakauer and Komarovaexplored how selection pressures acting on virus growth within cells differed fromselection pressures acting at the population level, where viral persistence depends onreplication as well as transmission between susceptible host cells (102). They chose todevelop a model for poliovirus owing to its relatively simple regulation, where a singlepositive-sense RNA genome serves as mRNA. Further, they assumed that viral fitnesscould be defined directly by the growth kinetics of the virus within its host cell. Thismodel for poliovirus intracellular growth accounted for the following three main steps:first, translation of the entering viral RNA genome, producing viral proteins that includethe viral RNA polymerase; second, synthesis of the positive-sense RNA genome tem-plate by employing a negative-sense RNA intermediate; and third, encapsidation of theRNA genome by viral proteins for release from the infected cell. To facilitate themathematical analysis, it was assumed that the system could establish a steady state (ordynamic equilibrium) within the cell, where rates of production and decay of viralintermediates are balanced, resulting in constant levels of such intermediates in thecell. Further, the analysis allowed for an extracellular level of infection spread wherevirus released from infected cells could infect other susceptible cells. By consideringonly the extracellular level, higher rates of encapsidation appeared to be favored byevolution owing to higher rates of production of viral progeny. However, by alsoconsidering the intracellular level, it was found that the productive equilibrium couldbe established only for a bounded rate of encapsidation. Beyond a particular threshold,a productive intracellular steady state would not exist because genomic templatesneeded for viral replication would be removed too rapidly by encapsidation.

A key point of this work is its accounting for ways that population-level selection (atthe level of cell and virus population interactions) can influence within-cell kinetics ofvirus production. While the assumption that steady states can arise at both theintracellular and extracellular levels enabled analysis of the model, the validity of suchassumptions should be validated by experiments. Nevertheless, the work importantlyshows how selection acting at multiple levels may have outcomes different from thosefor selection at a single level.

Optimal Resource Use within Cells

A model of poliovirus growth focusing on intracellular processes was developed toexplore how principles of evolutionary ecology could be extended to virus growth (54).Life-history theory aims to reveal how different quantifiable traits of an organism mustbe balanced across its life span, from birth to death, in order to optimize some measureor correlate of its fitness (103). In the case of humans, material or energy resources thatare spent on reproduction may not be available for nurturing or protecting offspring.How should such resources then best be allocated? In the context of a growing virus,this study of poliovirus sought to quantify how potentially limited resources of theinfected cell should best be allocated over the course of the virus life cycle in the cell.The model incorporated the synthesis of viral RNA and proteins, accounting for delaysof elongation by polymerase and ribosomes, respectively, as well as the time requiredfor queuing multiple polymerases or ribosomes onto the respective template. By usingthe overall rate of production of viral genomes as a measure of virus growth, the modelsuggested that optimal virus growth would arise from an imbalance in the productionof genomic and antigenomic templates, with about 40 genomic RNA molecules syn-thesized from each antigenomic RNA and about two antigenomic RNA molecules madefrom each genomic RNA, in reasonable agreement with experimental observations. It ispossible that other measures of productivity, such as the total yield of viral genomesrather than their production rate, would yield similar results. The model was extendedto account for potential resource limitations at the levels of ribosomes, amino acids,and nucleotides. The analysis highlighted tradeoffs between time spent on translation




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and that spent on transcription, because RNA synthesis is driven by products oftranslation and use of an RNA template for protein synthesis must be completed beforeit can be used further as a template for RNA synthesis. In addition, it was found thatunder conditions of limiting translation resources, optimal production would favorhigher ratios of genomic to antigenomic RNA, while the inverse would be true underconditions of limiting transcription resources. The magnitude of asymmetries in tem-plate replication for poliovirus in this analysis may also apply for observed asymmetriesin vesicular stomatitis virus (VSV), with its negative-sense genome. In future work, itwould be interesting to see whether the observed trends hold in light of the costlyenergetics of protein synthesis relative to the less costly energetics of RNA synthesis.More broadly, while the analysis was built on assumptions that remain to be substan-tiated, it is useful in showing that optimization of virus growth under different resourcelimitations can potentially shift how resources are allocated.

VSV

VSV is an RNA virus that possesses a single negative-strand RNA with a genomelength of 11 kb, carrying five genes. Other negative-sense single-stranded RNA virusesinclude Ebola, measles, and rabies viruses. Historically, VSV has served as a model virusfor understanding gene regulation in negative-strand RNA viruses (104, 105), andexperimentally, the low fidelity of its RNA-dependent RNA polymerase has enabledlaboratory studies of virus evolution and adaptation (106–108). In the biomedical arena,recombinant VSV may express engineered surface proteins of influenza A virus or HIVas a potential vaccine strategy. VSV also has the ability to discriminate and productivelyinfect cancer cells while leaving healthy cells intact (109), opening opportunities for itsuse as an anticancer therapeutic (110–112). Live viruses have advantages as vaccinesover subunit or killed virus because they tend to elicit more potent immune responses.However, vaccination with live viruses also carries greater risk because the processes forattenuation of their growth and pathogenic properties have historically relied onserially passaged cultures that weaken growth by poorly understood and potentiallyreversible mechanisms. As one learns more about the detailed molecular mechanismsthat influence virus growth, one may be in the position to engineer mutants thatrationally attenuate growth, as described below.

Effects of Genome Organization on Virus Growth

One approach for attenuating growth in VSV has been to change the linear order ofgenes in its genome. The order is important because the genome (3=-N-P-M-G-L-5=) hasonly a single promoter for its polymerase, near its 3= end, with attenuation sequencesbetween genes that cause a fraction of the passing polymerase to leave the template(113). As a result, mRNA levels follow a gradient (N � P � M � G � L), with the highestexpression occurring for the N gene (nucleocapsid), directly adjacent to the 3= end, andthe lowest expression for the L gene (large protein of the polymerase), adjacent to the5= end. In order for the virus to grow, all five proteins must be expressed and availablefor incorporation into progeny virus particles. One might well expect the rate ofprogeny production, which is one measure of virus fitness, to be optimized in someway. For a virus, optimal growth may be obtained by using available host resources tomaximize the production of virus progeny (yield) or the rate at which progeny arecreated. In the case of VSV, high levels of nucleocapsid protein (N) are needed to bringabout a switch in viral RNA synthesis from transcription to genome replication, so it isplausible that the timing and level of N protein are optimized in the wild-type virus.Moreover, altering the timing and level of N protein production from its wild-typeexpression pattern might well move it away from optimal growth and detrimentallyinfluence the virus growth or fitness. To test this idea, positional mutants of VSV werecreated, with the N gene in the N2, N3, and N4 mutants positioned at progressivelygreater distances from the 3= promoter (114). Yields of virus from infected cells werecorrespondingly perturbed, with the highest yields from N1 (wild type) and the lowestyields from N4. Expanding these gene-order variants to consider all permutations of five




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genes would define 5! or 120 possible versions. Additional mutants that retained the N1and L5 positions of the wild type but accounted for all six perturbations of the internalthree genes, P, M, and G, were also created and characterized with respect to geneexpression and growth (115). We sought to better understand how gene order influ-ences virus fitness by developing a kinetic model for VSV intracellular growth (116). Ourmodel accounted for diverse features of VSV, including transcriptional regulation byintergenic attenuation, the switch in viral polymerase activity from transcription togenome replication, effects of different promoter strengths on the balance of full-length genomic and antigenomic RNA templates, diversion of the cellular translationmachinery to synthesis of viral proteins, and the stoichiometry of RNA and proteincomponents in VSV progeny particles. The model correctly predicted the observedexperimental ranking of N gene mutants (N1 � N2 � N3 � N4) and suggested thatoptimal growth depends on a balance of promoter strengths for genome and anti-genome synthesis. Further, we used the model to explore how the expression of othergenes, such as the antigenic surface glycoprotein (encoded by the G gene), may entailtradeoffs with virus fitness. More recently, the model was extended to predict thegrowth of all 120 gene-order permutations, offering a means to study how fitnessmight depend on genome organization (117). The simulated virus fitness was found tobe most sensitive to permutations that changed levels of the L and N genes, which arethe least and most expressed genes of the wild-type virus, respectively. The roles ofgene order and transcriptional regulation were also probed by use of this model. Bycomputationally deleting the intergenic attenuations that regulate how transcriptionalresources are distributed, one could test how the 120 gene-order variants changed.Their growth yields or fitness levels were drastically narrowed, from 6,000- to 20-fold,and many variants produced higher progeny yields than those of the wild type. Amongthe gene-order mutants, the wild type emerged as a fitness winner only in the presenceof intergenic attenuation, suggesting that in the natural evolution of VSV this mode ofregulation preceded or coevolved with the fixation of the wild-type gene order.

BACULOVIRUS

The large-scale production of recombinant proteins, particularly proteins requiringposttranslational modification, has long been implemented in insect cells infected by arecombinant baculovirus engineered to express heterologous proteins of interest (118).Baculoviruses are double-stranded DNA viruses with genomes of 80 to 180 kb thattypically encode about 150 proteins. An early study highlighted two factors, the timeof infection and the multiplicity of infection (MOI), for defining key tradeoffs in theproduction of virus and heterologous protein (119). Infection of cells during the lateexponential growth phase, before they have reached their culture capacity (maximumcell concentration), can result in lower total yields of virus. However, infection of cellslate, as they approach their culture capacity, can also limit virus production owing tolower biosynthetic capabilities of cells as their growth slows. If the MOI is well below 1,then cells that are not initially infected may continue to grow and become infectedwhen the first generation of virus is released, contributing to overall higher productivityof the culture. In such scenarios, the initial virus production and release need to occurbefore cell growth can progress to stationary phase, when productivity drops. For thesestudies, the kinetics of virus production was not considered mechanistically. Instead,the focus was on three kinetic milestones: the time postinfection for extracellular virionsynthesis, the time postinfection for extracellular protein (recombinant product) syn-thesis, and the time postinfection for cell lysis. The importance of culture time and MOIon the dynamics of cell, infected-cell, and virus populations were subsequently vali-dated experimentally in a study that extended application of the baculovirus expres-sion system for the production of virus-like particles (VLPs) (120). Virus-like particles areoften highly immunogenic, making them potentially useful as vaccines. Infected cellswere immunostained using an antibody against a key protein of the VLPs. Further, itwas shown that a higher MOI could increase the rate of VLP production, reducing thetime to harvest.




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The baculovirus expression system has been harnessed for the production of VLPsof rotavirus, a common cause of severe diarrhea in children. In this application, threerotaviral proteins were coexpressed, and their measured viral DNA, mRNA, and proteinlevels were found to be consistent with models of baculovirus transcription andtranslation (23). As a notable aside, it was found that experimental uncertainty associ-ated with estimating the MOI propagated exponentially in the calculation of viral DNAtemplates. To minimize these effects, experimental measures of early processes (e.g.,binding and entry) were not used in the estimation of parameters. Instead, parametersfor viral DNA replication, transcription, and translation were estimated from levels ofviral DNA, mRNA, and protein measured at least 5 h after the start of infection.

Recent studies on baculovirus production have returned to the classic problemof producing high virus yields when host insect cells are at high density. A usefulapproach has been to explore how cell physiological changes, particularly changes thatare coupled with the transition from exponential-phase cell growth to stationary-phasecell behavior, might limit the availability of resources that are essential for virusproduction. Global metabolic changes associated with such transitions can be esti-mated by combining measurements of metabolite changes with known metabolicnetworks and their analysis. Methods for analysis of metabolic fluxes have becomeincreasingly standardized (121). Application of these approaches to Sf9 insect cells athigh density or after infection by baculovirus suggested a depletion of intermediateswithin the tricarboxylic acid (TCA) cycle (122). More specifically, carbon fluxes throughglycolysis and the TCA cycle were found to decrease as cell density increased, causingsharp drops in ATP production and availability of metabolic energy. Virus infectionswere found to have similar effects on cell metabolism, highlighting the depletion ofATP, the central currency for metabolic energy, as a key factor linking high cell densitywith drops in virus production. Further analysis supporting this result indicated that thedrop in productivity of viruses at high cell density could not be attributed to thedepletion of essential nutrients or the accumulation of inhibitory by-products. Toaddress limitations of metabolic energy on virus production, the cell culture mediumwas supplemented with key depleted intermediates. Specifically, addition of pyruvateor �-ketoglutarate at the time of infection resulted in higher virus yields (up to 7-fold)during high-cell-density culture (123). In this case, metabolic flux analysis showed astrong correlation between the net rate of ATP formation and the generation of redoxequivalents in the form of NADH.

HBV

HBV infection is a major cause of acute and chronic liver disease. Over 350 millionpeople are chronically infected with HBV, and more than 150,000 people die annuallyof liver disease related to hepatitis B (124). The virus may remain relatively unnoticedfor years, until the population of viral progeny in the patient’s liver “explodes.” HBV isa difficult virus to treat, in part because of its high mutation rate. An intracellular kineticmodel was developed to understand how mutation-driven transitions from chronic toacute infection might work mechanistically (125). The model accounted for the kineticsof 10 DNA, RNA, and protein components, including a pregenome-polymerase com-plex, in the virus and progeny viruses, and their mechanisms of production andinteraction defined 18 parameters. Plausible order-of-magnitude parameter valueswere chosen to give simulated levels of virus progeny that approximated experimentalobservations; however, no parameters were estimated by independent wet-lab exper-iments. The model showed that patterns of gene expression that favored packaging ofcore particles to form progeny arrested overall replication, while patterns that favoredautocatalytic amplification of core particles resulted in explosive replication. It wasargued that mutations in the core or precore region of the HBV genome could bringabout switching from arrested to explosive replication, corresponding to the transitionfrom chronic to acute infection. An extension of the model explored how recycling ofreleased virus progeny back into an infected host cell could serve as an additionalsource of intracellular core particles, effectively lowering the threshold for transition to




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explosive replication (126). To test these suggested mechanisms for switching betweenreplication modes, experimental validation of the model parameters will be an impor-tant next step.

HCV

HCV is an enveloped positive-strand RNA virus that can cause chronic liver disease,which can lead to cirrhosis (scarring and impaired function of the liver) and liver cancer.About 130 million individuals worldwide are chronically infected with HCV, and novaccine against HCV infection currently exists. HCV has been very challenging to studyexperimentally, owing in part to the lack of a laboratory system for culturing the virus.A replicon system has enabled the study of HCV genome replication in a specific livercancer cell line (127) and served as the basis for an initial kinetic model of HCVreplication (128). With guidance by the structure of the phage Q� model, this HCVreplicon model accounted for reactions that synthesize genomic and antigenomic RNAstrands, a ribosome-genome complex to synthesize the viral polyprotein, cleavage ofthe polyprotein to produce the viral polymerase, and intermediates in the replicationprocess formed by the polymerase and genomic and antigenomic templates. Themodel also allowed for spatial compartmentalization, with translation occurring in thecytoplasm, replication occurring within a vesicular membrane structure (VMS), andthe viral genomic template and polymerase able to move between the cytoplasmand the VMS. Like the models for other positive-sense RNA viruses, e.g., phage Q� andpoliovirus, an imbalance in the rates of HCV replication favors production of HCVgenomes over antigenomes, at a 10-to-1 ratio. Further, the model was used to explorethe role of replication compartmentalization in the VMS. Specifically, could observedsteady-state levels of HCV RNA be attained without the VMS? This hypothetical ques-tion was addressed by simplifying the model to allow both protein translation andgenome replication to occur in the cytoplasm. The single-compartment model showedthat one could attain steady-state RNA levels that were consistent with experimentalresults. However, the corresponding predicted levels of the viral polymerase (NS5B)were 4 orders of magnitude lower than observed levels. To address this discrepancy,ribosome levels were increased, but this adjustment then caused the steady-state RNAlevels to move significantly out of the observed range. In short, this analysis indicatedthat the VMS may plausibly serve to restrain viral amplification and perhaps limitassociated host cell damage.

To explore antiviral strategies against HCV, Mishchenko and colleagues developedan HCV replicon model which included reactions to target specific HCV functions (129).The model was based on much of the biology of the Dahari model (128) but alsoincluded inhibitors of the HCV protease (NS3), the HCV polymerase (NS5B), and a hostprotein (hVAP-33) that is essential for assembly of the HCV replication complex. Bysimulating the effects of drugs of different potencies on the steady-state level of HCVgenomic RNA, this model showed that direct targeting of the polymerase or the hostfactor would have a greater inhibitory effect on viral replication than that of targetingthe HCV protease. Moreover, testing of combined treatments that target both theprotease and the polymerase showed no enhancement of inhibition, at least for weakinhibitors of these functions.

Taking a similar approach to his modeling of hepatitis B virus, Nakabayashi devel-oped an intracellular kinetic model for HCV. As with HBV, he found two major patternsof replication, one arrested and one explosive, depending on the distribution ofreplication resources (130).

HSV-1

The global prevalence of adult carriers of herpes simplex virus type 1 (HSV-1) wasestimated to be 67% in 2012 (131), reflecting the stability and efficient transmission ofa virus with major impacts on public health. HSV-1, which encodes at least 84 proteinsin productively infected cells (132), expresses its genes based on their timing, which canbe divided into three classes: immediate early, early, and late (133). To better under-




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stand the role of regulatory feedbacks on the temporal order of gene expression,genome replication, and protein synthesis in the production of virus progeny, Naka-bayashi and Sasaki developed an intracellular kinetic model (18). An initial version ofthe model that accounted for viral DNA genomes, mRNAs from each of the three classesof expression, their translation to produce their corresponding proteins, viral assembly,and degradation rates of all nucleic acid, protein, and virus species was simplified bylumping processes of transcription and translation and neglecting rates of speciesdegradation. Analysis of the simplified model revealed two modes of growth, eitherexplosive or arrested, whose characteristics were similarly investigated by Nakabayashiet al. in the models of hepatitis B and hepatitis C. In the case of HSV-1, the explosiveor arrested growth behavior depended on the relative expression of early versus lategene products. Higher expression levels of early gene products supported genomereplication and a positive-feedback loop that explosively amplified viral genomes, whilegreater expression of late gene products yielded more envelope and structural proteinsthat depleted free genomes by packaging them into virus progeny particles. Underconditions for explosive growth, where genome replication is favored over genomepackaging, one could estimate a waiting time for appearance of the explosive growth,which depended inversely on the initial level (or dose) of viral genomes. In short, higherinitial levels of viral infection could result in shorter waiting times for explosive growth.The work further considered scenarios in which a virus with arrested growth behaviorcould accumulate mutations in early or late gene promoters that shift the balance ofgene expression in favor of explosive growth, effectively suggesting how diverse waittimes for explosive growth may arise. Although the model did not explicitly addressissues or mechanisms of viral latency—when the viral genome is maintained in anonreplicative state but poised to transition into a lytic or productive growth state—itis plausible that analysis of the transition from arrested to explosive growth may offerinsights into the transition from latency to lytic growth. Finally, the simplified modelwas extended to account for potential limited intracellular resources needed forsynthesis of viral DNA, RNA, and proteins. Such models, combined with advances in ourmechanistic understanding of viral genes, may help to elucidate how cellular resourcesare distributed to viral functions over the course of infection (134).

FRONTIERS FOR MODELING VIRUS GROWTH IN CELLS

In this final section, we suggest frontiers for modeling virus growth, touching onopportunities and challenges that delve into extending model building to other viruses,the environment of the host cell and its interactions with viral processes, modeling thevariability of single-virus or single-cell behaviors, and the multiscale nature of virusinfections as they propagate over multiple cycles.

Other Viruses

Many viruses that have been studied in great depth at the molecular level, asfeatured, for example, in Fields Virology, define opportunities for development of newkinetic models of virus growth in cells. Studies of more recently emerging viruses, forwhich less may be known, such as Ebola virus or Zika virus, may still benefit fromexisting models. Just as the model for phage Q�, a single-stranded positive-sense RNAphage, served as a scaffold to build the initial replicon model for HCV, anothersingle-stranded positive-sense RNA virus (128), other existing models may serve asscaffolds for related viruses. Ebola virus is a single-stranded negative-sense RNA virusthat likely shares many similarities with VSV, for which a model is available (116). Inaddition, Zika virus, like HCV, is a single-stranded positive-sense RNA virus that encodesa single polyprotein, which subsequently self-cleaves to supply essential replicationand structural proteins for virus growth. Rates or other parameters that quantify themagnitude of molecular interactions or reactions will always be valuable, if not essen-tial, for model building. Databases for modeling, such as the “B10 NUMB3R5” effort(135), are useful for providing estimates of quantitative parameters of cellular functionsessential for virus growth, such as protein synthesis. We hope that interest in modeling




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virus growth will motivate the growth of databases for virus-specific parameters, whichindividually will inform specific virus models but collectively may provide insight intothe diversity of viral kinetics.

Host Cell Physiology and Innate Immune Responses

The availability of biosynthetic resources from a host cell is essential for virusgrowth, so a better understanding of how the cell’s capacity for synthesis of proteins,nucleic acids, lipids, and other components of virus particles changes under differentenvironmental conditions will need to be incorporated into models that seek toaccount for the dependence of virus growth on host physiology. The effects of hostcellular growth rate on the intracellular kinetics of phage growth, which were probedby use of empirical correlations between cell growth and biosynthetic resources (79),have been expanded to incorporate genome-scale metabolic models of the bacterialhost for the effects on phage growth (56, 136). Moreover, models of metabolic fluxesdeveloped for mammalian cell cultures have been applied to identify conditions forhigher glucose uptake rates and more efficient use of ATP, suitable for bioprocessingof influenza A virus vaccines (137, 138), and similar approaches have been applied tobaculovirus growth in insect cell cultures (122).

In addition to supplying biosynthetic resources for virus reproduction, host cells alsoactivate defensive innate immune signaling and responses, typically mediated by typeI interferons (IFNs). Such signaling can trigger both autocrine and paracrine cellularresponses that shut down protein synthesis that is essential for expression of essentialviral functions. Different facets of such responses have been quantified and modeledmechanistically, including initial sensing of the viral dsRNA (139), induction of IFNs thatactivate both positive and negative feedbacks (140), and responses to different degreesof viral antagonism (141). Notably, in the case of infections by the highly pathogenicNipah virus, modeling of the host response along with transcription-level measures ofIFN and inflammatory cytokine activation provided insight by suggesting that the virusmay delay suppression of inflammation and thereby enhance vascular permeability andinfection spread (142).

Single-Cell and Single-Virus Tracking

To obtain a measure of viral or cellular function, most biochemical or molecularbiological studies work with millions of cells in a tube or dish, for which average levelsof nucleic acid, protein, intermediates, and infectious virus particles can readily bedetected and quantified. However, in nature, infectious diseases are often transmittedor spread by countable numbers of infectious virus particles or cells. When the behaviorof individual cells was studied in response to viral infection, the single cells exhibitedinfectious particle yields (or burst sizes) that were highly variable (143, 144). Thisvariability among different infected cells can be attributed to genetic heterogeneity ofthe stock virus population (145) or host cells (146), resource differences linked to thestage of the cell cycle (147), or intrinsic stochastic behavior of reactions that involvesmall numbers of molecules (148–150). To account for the stochastic behavior, andassuming a well-mixed environment, Gillespie developed an algorithm for exact sto-chastic simulation of the reaction kinetics (151), which broadly enabled modeling of thelysis-lysogeny decision in phage lambda (152), stochastic versus deterministic intracel-lular infection (153), viral capsid assembly (154), bifurcation of gene expression in HIV-1(16), transcriptional delays and autocatalytic feedbacks in VSV (155), genome amplifi-cation for influenza A virus (156), and infection-mediated activation of innate immunesignaling (157, 158).

Models of virus intracellular kinetics have often applied the following assumptionwithout explicit mention: intracellular environments are spatially homogeneous or wellmixed. This simplifying assumption allows concentrations of reacting components tobe treated as solely time dependent rather than time and position dependent, somathematically the equations are written as sets of ordinary differential equations,which are numerically solved more readily than sets of partial differential equations.




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Future kinetic models will have spatial and temporal dependencies, incorporatingadvances in single-particle tracking by use of dye-labeled or reporter-expressing virusparticles which have enabled direct visualization and quantification of virus-receptorinteractions, internalization, intracellular transport, genomic release, nuclear transport,and cell-to-cell transmission (159–161). Different modeling approaches account forprocesses of intracellular particle transport by diffusion or mediated by active particletransport along microtubules (162), and an approach with relevant parameters foradeno-associated virus (AAV) has been outlined (163). More generally, approaches ofstatistical physics and their application to intracellular transport processes, which havebeen reviewed previously (164, 165), in combination with models for viral geneexpression and function, will provide a more comprehensive accounting of the molec-ular biology and physical facets of virus intracellular growth.

Multiscale Modeling of Virus Infection Spread

To cause disease, virus infections of host cells amplify over multiple generations,requiring the intracellular processes that are the focus of this review to play out overmultiple cycles of susceptible host cell infection. The viral progeny released from aninitial infected host cell go on to further infect other host cells, perhaps in the sametissue of a multicellular host organism.

Models of virus infection in humans, initiated in the 1990s for the dynamics of HIV-1in AIDS patients and informed by the predator-prey models of epidemiology (166, 167),provided the key insight that low levels of HIV-1 in patients were not only activelyreproducing but also developing resistance to antiviral drug treatments (168). Similarmodeling approaches have been extended to other viral infections in humans, includ-ing hepatitis C (169) and influenza A (170); related approaches have also been used tomodel virus infections as they selectively infect and spread in tumors as oncolytictherapies to treat cancer (171). Such within-host models, which accounted for thepredator-prey dynamics of viruses and their host cells, are now serving as foundationsfor multiscale modeling in a top-down manner; the kinetics of processes within infectedcells are of increasing interest owing to their capacity to account for explicit mecha-nisms of drug-target interactions and time delays in the release of virus progeny fromtheir infected host cells (172, 173). At the same time, intracellular models of virusgrowth are serving as starting points for multiscale modeling in a bottom-up manner;the virus particles released from cells are tracked as they bind to and infect a newpopulation of susceptible host cells (174, 175). When virus particles released from aninfected cell are transported to neighboring or distant cells, physical processes ofdiffusion or fluid flow contribute to the dynamics of infection spread. Intracellulargrowth coupled with diffusional spread can be modeled using reaction-diffusionequations (176) or models that computationally simulate the behavior of cells on a gridas “agents” or “automata” that follow preestablished rules, using changes in the localenvironment (e.g., infection or immune signaling status of neighboring cells) to prop-agate infection spread (177–180).

Viruses can readily mutate as they reproduce, so modeling approaches adaptedfrom population genetics may be useful in accounting for how different virus or genevariants become enriched. Moreover, as viral infections spread, they also affect thedynamics of their host populations, whether they are host cells or host organisms; uponinfection, those cells or organisms may recover or die. To unite the combined behaviorof the virus during its transmission and growth with the fall or rise of the hostpopulations, predator-prey models from epidemiology may be appropriate. More than3 decades ago, May and Anderson suggested that models combining the populationgenetics and epidemiology perspectives would potentially account for both essentialfeatures (181). Such models can be viewed as multiscale in the sense that they accountfor local molecular and cellular mechanisms of amplification and genetic diversificationthat originate from within cells, as well as processes of within-host spread betweeninfected and susceptible cells of a multicellular host. Further transmissions betweeninfected and susceptible multicellular host organisms, most notably animals and hu-




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mans, highlight how additional ecological and evolutionary considerations (182), in-cluding the behavior and mobility of animal and human hosts (183, 184), contribute tothe dynamics of epidemics.

ACKNOWLEDGMENTSWe are indebted to the outstanding graduate students, postdocs, and undergrads

in the Yin lab, who over the last 25 years have contributed their ideas and hard worktoward our appreciation and understanding of viruses. We thank Paul Ahlquist, UdoReichl, and Ophelia Venturelli for many thoughtful comments and suggestions on themanuscript.

We gratefully acknowledge support over the years from the National ScienceFoundation (grants BES-0087939, BES-0331337, BES-9896067, EF0313214, EIA-0130874,and EIA-0331337), the National Institutes of Health (grants AI077296, AI071197,AI091646, AI104317, T32-GM08349, and T32-HG002760), the National Library of Medi-cine (grant T15LM007359), the Office of Naval Research (grant N00014-98-1-0226), theTexas-Wisconsin Modeling and Control Consortium, Merck Research Laboratories, theWisconsin Alumni Research Foundation, the Wisconsin Institute for Discovery, andthe University of Wisconsin-Madison (Graduate School, Office of the Vice Chancellor forResearch and Graduate Education, and the William F. Vilas Trust Estate).

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John Yin earned bachelor degrees in theliberal arts and engineering from ColumbiaUniversity and a Ph.D. in chemical engineer-ing from UC-Berkeley. He pursued postdoc-toral studies at the Max Planck Institute forBiophysical Chemistry in Göttingen, Ger-many, advancing the experimental evolutionof viruses. Yin’s academic career started atDartmouth College, and in 1998, he joinedthe Chemical Engineering Department at theUniversity of Wisconsin-Madison. A decadelater, he and colleagues initiated a research thrust in evolutionarysystems biology as a founding theme of the Wisconsin Institute forDiscovery. Today he is the Vilas Distinguished Achievement Professor,pursuing single-cell measures and models of virus-host interactions andquantitative characterization of spreading infections. His research inter-ests also include the chemical origins of information, metabolism,self-replication, and life. Yin is a semiprofessional pianist and cellist, andhe enjoys sous vide methods of quantitative cooking.

Jacob Redovich is an undergraduate at theUniversity of Wisconsin-Madison, where he isstudying chemical engineering and chemis-try. He has held research positions in theUW-Madison Department of Chemistry, theZhejiang University Department of Chemicaland Biological Engineering in Hangzhou,China, and most recently the Yin group inthe Department of Chemical and BiologicalEngineering at UW-Madison. Jacob has con-ducted research in electrocatalytic materialschemistry, biomaterials, and viral kinetics. His main research interestsinclude kinetic modeling, data science, and genetics. He has alsoworked as a chemistry tutor and participated in a process engineeringco-op at Monsanto.




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