Kinetic Modeling of Virus Growth in Cells

The first models of intracellular virus replication, which were published by Barricelli and 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 recombination when two or more parent phage strains coinfect the same host cell and create a progeny phage that is a cross of the parent strains. The models used number strings to represent locations and characteristics of experimentally determined markers on the genomes of the different strains (41), and they showed that one could feasibly store such strings in the available memory of an early computer, enable stored strings to interact (recombine), store the resulting progeny strings, and provide progeny genomes as the readout. In short, these works showed that it was possible to simulate recombination between virus genomes by using an early scientific computer (IBM model 7094).

The models were mechanistic to the extent that they accounted for how different genomes might physically interact to create crossover events. They were also mechanistic in the implementation of rules that allowed genomes to amplify during growth and permitted such amplification processes to continue until only a viable number or experimentally observed burst size of phage progeny was obtained (41). However, the models were nonmechanistic in the sense that arbitrary rules were applied for cases where mechanisms were unknown, such as the regulation of DNA synthesis. Experimental data incorporated into these models included genomic locations of markers from classical genetic crosses and mutagenesis experiments compiled from the phage T4 literature.

In 1975, Knijnenburg and Kreisher initiated development of a computational model to describe the intracellular development of an RNA phage. Their work was motivated by an emerging mechanistic understanding from experiments on the highly coupled roles that the synthesis of viral RNA and viral proteins play in the life cycle of single-stranded RNA bacteriophages MS2 and Qβ (42, 43). Both MS2 and Qβ carry positive-sense RNA genomes which can serve as mRNA templates to synthesize phage proteins or to produce negative-sense RNA templates, which are essential intermediates in phage genome replication. First, Knijnenburg and Kreisher explored, through simulation, how the probability of ribosome binding to sites for translation of the phage coat and RNA polymerase could influence the kinetics of coat and polymerase production during infection. They accounted for the role of coat protein synthesis and accumulation in the subsequent suppression of polymerase synthesis in perhaps the earliest computational study of stochastic gene expression (44). Unfortunately, insufficient details were provided to fully understand how the simulation was implemented. In subsequent work, they expanded their initial model to account for synthesis of essential RNA and proteins of the phage (45). Their efforts were notable because the aim of their model—to better understand how the information content of the viral genome and its interactions with viral proteins contribute to the manufacture of viral progeny—defined a central long-term goal for the field. This work marked the earliest mechanistic 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 independent of the processes of the host cell. This independence or uncoupling was achieved by assuming that all essential cellular resources, including ribosomal subunits, replication factors, tRNAs, amino acids, and nucleoside triphosphates (NTPs), were present in excess throughout the infection cycle. Today, this assumption remains a useful simplifying step to enable one to define an initial set of equations focused on the essential functions encoded by the viral genome. A second key assumption was based on the separation of time scales between fast molecular binding release events and slow processes of template-directed polymerization. In short, the binding release processes were assumed to be sufficiently fast that the participating molecules could be treated as equilibrated with their complexes and characterized by a single equilibrium binding constant. This assumption had the effect of making the dynamics of virus intracellular development be governed primarily by the rates of viral transcription, genome replication, and protein synthesis, which also continues to be a useful starting point for the development of virus simulations.

Detailed in vitro kinetic studies of RNA replication by the Qβ replicase, along with analysis and computer simulations, revealed features of the dynamics that established a foundation for a model of phage Qβ growth. With an excess of high-energy monomers (NTPs), the level of RNA increases either exponentially or linearly, depending on the relative level of replicase (49, 50). When replicase concentrations exceed template concentrations, RNA growth depends only on RNA levels, so RNA levels increase exponentially. Alternatively, in the presence of excess templates (or limiting replicase), 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 model systems have been useful for highlighting similarities between replication of short RNA templates and that of full-length phage RNA genomes. For example, formation of double-stranded RNA through the annealing of cRNA templates can deplete both RNAs serving 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 at the stage of replicase release, requiring hundreds of seconds in vitro, as opposed to in vitro elongation, which occurs in tens of seconds (51). In contrast, for genome-length RNA, the rate-limiting process is elongation (50). For simplicity, these models omit consideration of RNA templates that simultaneously engage more than one replicase molecule, an assumption that may need to be relaxed for genomic templates.

A computational model for phage Qβ intracellular growth was published in 1991 (47). In contrast with the earlier in vitro models and experiments on Qβ RNA replication, in which the replicase enzyme was readily available as an initial condition, replicase levels for the phage model were initially zero, as in actual phage Qβ infections. In the phage model, an initial phage genomic template is translated by host ribosomes to produce phage proteins. Further, the phage model accounts for the diverse competing processes that can engage the phage genomic RNA. It may be bound by ribosomes to serve as mRNA to make protein, or it may be bound by the replicase enzyme to serve as a template to make antigenomic RNA. Further, the genomic RNA may be bound by phage coat proteins during the process of phage particle assembly. However, the timing and sequence of these processes are governed by the initial conditions. In the initial absence of phage replicase and coat protein, only translation of the genomic template can take place. The synthesis of replicase coupled with the amplification of the RNA genome that encodes replicase defines an explosive positive-feedback loop that amplifies both replicase and RNA with an initially hyperbolic dependence, as follows: dx/dt = xⁿ, where 1.5 < n < 2 and x represents the intracellular concentration of the replicase or the phage genome. This amplification is more sensitive to protein and RNA levels and more rapid than exponential amplification (n = 1). Because this amplification depends on the availability of ribosomes to make replicase, and because ribosome levels are finite, rapid amplification of RNA eventually reaches a steady state where the RNA level is balanced by the combined levels of ribosomes and replicases. Beyond the equivalence point, RNA levels increase with linear dependence on these levels. This hyperbolic-to-linear growth transition in the phage infection model mirrors the exponential-to-linear growth transition observed during the in vitro amplification of small RNA templates by replicase. Finally, accumulation of coat protein enables the coat to successfully compete against the ribosomes and replicases for binding to the genomic RNA, allowing packaging of the genomes and production of phage progeny. Over the course of the simulation, NTP resources are described as “buffered,” meaning that these building blocks for transcription and RNA replication are assumed to be unchanging. They are assumed to be supplied at the same rate as that at which they are consumed. Ribosome levels, however, are set at a finite level.

The 1991 model of phage Qβ growth by Eigen and coworkers (47) focuses on the information flows associated with transcription, translation, and genome replication. The initial condition for the model is a single-phage genomic template in a cell-sized volume with a finite number of ribosomes and a fixed supply rate of substrate resources for RNA synthesis. Substrate resources for translation, such as pools of amino acid-charged tRNAs, were not specified but appear to be unbounded. The simulations terminate following packaging of genomic templates with coat proteins at a ratio of 1:180. It was noted that the overall sequence of events and behavior “proved in exhaustive trials to be rather insensitive to moderate changes in the rate constant values,” supporting the idea that the processes and interactions do not need to be finely tuned to a narrow set of parameter ranges in order to exhibit phage-like behavior. 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 that coat protein binding to RNA should be balanced, i.e., favorable enough to repress translation by competing with ribosomes or replicase for RNA but not so high as to shut down replicase production and RNA synthesis. In this manner, the coat protein plays a pivotal role in mediating early functions of genomic RNA as the template for transcription and replication, with a later purpose as a packaged product of infection. The coupling of reactions during phage growth creates other compensatory effects that contribute to robustness. For example, reducing the rate of ribosome binding to phage RNA can have a detrimental effect on phage protein synthesis, but at the same time it may permit greater access of the replicase to the genomic template, an essential step for synthesis of the antigenome template (52).

Virus infections consume energy at every step of their intracellular growth. Kim and Yin sought to estimate how biochemical energy was utilized by accounting for its consumption over the course of a simulated phage infection cycle (52). An energy equivalent was defined as the energy released by hydrolysis of a phosphate group from ATP, the common currency in bioenergetics. In this work, Eigen's Qβ model was extended 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 25,000 phage particles, of which 1 to 10% are typically infectious, requires 3 × 10⁹ energy equivalents, 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 biosynthesis of phage progeny particles. Tenfold changes in parameter values of the model in most cases had relatively little impact on the efficiency of energy expenditures, which were directly related to 2-fold (at most) changes in phage yields. Together these results suggest that the phage has evolved to efficiently utilize available energy resources during growth, and this approach has been extended to other systems, including phage T4, influenza virus (53), and poliovirus (54).

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

Models of phage Qβ growth, to date, have neglected processes of initial binding of the virus to its host cell, virus entry and release of the phage genome into the cytoplasm 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 an optimization owing to ecological factors (57, 58). While it is known that the phage lysis protein prevents cell wall biosynthesis by inhibiting an essential enzyme in the process (59), how inhibition of cell wall biosynthesis ultimately affects the timing of phage progeny release remains to be elucidated mechanistically. For other aspects of phage growth, refinements in the mechanisms of other processes, such as Qβ RNA replication (26) and particle assembly (60), may also enable refinements or improvements to current models.

Many current antiviral drugs are directed to inhibit specific viral functions, but viruses can develop mutations that enable them to resist such treatments. One alternative strategy may be to develop defective viruses that parasitize normal virus infections 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 Qβ infection, we employed our model to computationally explore the potential effectiveness of such a parasitic antiviral strategy (62). We simulated the behavior of minivariant 11 (MNV-11), an 87-nucleotide RNA derived from the Qβ genomic RNA that can compete with the wild-type template for the Qβ replicase. The ability of this molecular parasite to very rapidly amplify itself and compete for Qβ replicase allowed it to prevent amplification of the wild-type phage genome, based on our simulations. Other strategies against positive-strand viruses, based on RNA silencing, have also been explored by simulation and found to yield potentially diverse responses, including inhibition of virus growth, oscillatory behavior, or complete clearance of infection (63).

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

By accounting for the synthesis of the phage mRNAs and proteins that are essential intermediates for the production of progeny particles, we simulate as a by-product the concentration-versus-time trajectory of all phage mRNAs and proteins. This information provides an idealized data set with which one may test data mining or mechanistic inference tools. For example, one may consider the simplest way to express the relationship between the concentration of a protein and its corresponding mRNA within the cell, as follows: d[P_i]/dt = k_trans[mRNA_i], where [P_i] and [mRNA_i] are free concentrations of the ith protein and ith mRNA, respectively, and k_trans is the rate at which the ith mRNA is translated into protein. For a fixed translation rate, plotting d[P_i]/dt versus [mRNA_i] yields a line with the slope k_trans and an intercept of zero. Alternatively, one may view the free concentration of the ith protein as a process that integrates the level of the ith mRNA over time.

When the free concentration of a protein can be influenced by other processes, such as protein degradation, posttranslational modifications, or formation of complexes with other proteins or nucleic acids, additional terms appear on the right-hand side, as follows: d[P_i]/dt = k_trans[mRNA_i] − k_d[P_i] + f(modifications) + g(interactions), where k_d is a rate constant for first-order degradation of the ith protein and f and g are functions representing other processes that may influence the concentration of the ith protein. In general, these additional processes will cause trajectories of d[P_i]/dt versus [mRNA_i] to deviate from a straight line. In the context of our phage T7 model, we simulated plots of d[P_i]/dt versus [mRNA_i] and found that some phage proteins produced loop-like trajectories that reflected their regulatory roles (69). Moreover, the model was used to quantify for each phage protein how it deviated from linearity in its plot of d[P_i]/dt versus [mRNA_i] over the course of infection. Such deviant proteins are interesting because they represent proteins that are modified, where f(modifications) is nonzero, or proteins that interact with other proteins, where g(interactions) is nonzero. If two or more 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 for all phage protein pairs and found that, indeed, the protein pairs that were computationally modeled to form complexes also shared highly correlated deviations. It will be interesting to use highly quantitative measures of mRNA and proteins from the same cells over time to explore whether such a “correlated deviation” approach can be used to infer physical interactions and formation of multiprotein complexes during infection. There is not yet a consensus on how global transcript and protein data should be treated to gain mechanistic insight. While much of the field has focused on characterizing the extent of correlation between protein and mRNA levels (70, 71), alternative approaches that combine mechanistic models of translation with mRNA and protein measurements have begun to highlight challenges in characterizing how limited and changing translation resources influence quantitative links between message and protein (69, 72–75).

Given that we have a simulation for phage T7 one-step growth, one can begin to explore alternative genome designs by changing the way the simulation is implemented. This may be done by changing the order of genes or the parameters of the model. In the case of phage T7, entry of the genome into the cell is mediated by the action of the host RNA polymerase and later by the phage RNA polymerase, with the 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 to the right end, later genes are also expressed at higher levels. By altering the positions of genes within the phage genome, one can alter their timing and level and thereby change 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 transcription is put under the control of a T7 promoter, creating a positive-feedback loop of T7 RNA Pol on its own transcription. Predicted enhancements in phage protein synthesis and phage production were not supported by quantitative experiments on protein production by pulse-chase methods or by yields of progeny phage by one-step growth experiments (76). One reason for the discrepancy is that the simulation did not account for the contribution of nonessential genes, such as phage gene 1.7. Although it is known that 1.7 is nonessential for phage growth, control experiments established that the absence of gene 1.7 does have a detrimental effect on phage progeny yields. A better accounting of the finite resources provided by the host cell was not able to explain the differences observed between experiments and simulations (77). The significant observed differences between predicted phage with the alternative gene order and three experimentally generated and tested phages highlight the substantial work that still needs to be done to better understand the nature of the discrepancies.

Current modeling of virus growth builds on simplifying assumptions that may or may not be valid. Quantitative experiments can be used to test and modify assumptions that may subsequently allow the model to account for more observations or make new predictions. An alternative approach to gaining biological insights is to alter the biology to match the simplifying assumptions of the model. For example, the genome of phage T7 has multiple overlapping genes that are not essential for phage growth. In our model, we assumed that T7 genes were not overlapping, an assumption that enabled us to model the kinetics of transcription of these genes (67). It was unknown at the time how the presence of overlapping genes might affect phage growth. To address these effects, Endy and coworkers redesigned and synthesized 30% of the T7 genome to eliminate overlapping regions. The resulting chimeric phage was able to produce viable phage progeny, though plaque sizes indicated that the growth properties of these variants were attenuated relative to those of the wild type (78). This exercise was useful in showing how simplifying modeling assumptions might be tested by “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 chimeric phage showed that overlapping genes are not essential for T7 growth but likely have an impact on the overall growth or fitness of the phage.

A key assumption that enables development of initial models of virus intracellular growth is that host resources are infinite. Using this assumption, one may then simulate virus growth based on the dynamics of the virus-encoded processes. These minimally include transcription of viral mRNA, translation of viral proteins, synthesis of viral genomes, and assembly of virus progeny particles. The assumption of infinite host resources is most likely to be reasonable at the earliest stages of infection, when the demand by the virus infection is defined for a small number of initial viral genomes (the ones that initiated infection). To better account for the effects of host resources on growth, phage T7 infections were performed on host cells cultured under different conditions that were set up to provide different host resources. In general, rapidly growing 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 controlled their growth by adjusting the dilution rate of the chemostat (79). Cells at different growth rates, spanning from 0.7 to 1.7 doublings/h, were used as hosts for synchronized one-step growth cultures of phage T7, and the resulting rise rates and eclipse times (characteristics of one-step growth) were determined. To predict how altered growth rates of host cells would influence phage growth, we employed empirical correlations established by others to indicate how cellular resources and properties, such as host RNA polymerase levels and elongation kinetics, ribosome levels and elongation rates, NTP and amino acid pools, and cell size, correlated with host growth rate. Thus, cellular growth rates set by the experimental conditions were used as inputs to the correlations to estimate cellular resource levels and kinetics, which were used as initial conditions for the T7 simulation. In experiments, faster cell growth reduced the eclipse time (time to production of initial phage progeny) and increased the rise rates of progeny production, and these behaviors were captured by the simulation. The simulation then provided an opportunity to uncouple the effects of different host resource conditions on phage growth in cases where such uncoupling would be difficult or impossible to implement in experiments. Through such studies, we identified the processing of the translation machinery, specifically the ribosome elongation rates, to be most limiting for phage production. Moreover, independent changes in simulated host resources indicated how resources of host transcription or translation would need to change in order for phage infections to become limited or “bottlenecked” by host RNA polymerase levels or ribosome levels. An assumption of the model was that host cell resources for the entire infection process could be defined adequately based on their state at the time of sampling from the chemostats. A more refined perspective will need to account for the consequences of infection on the host physiological state, accounting for the effects of potential changes in energy metabolism on levels of biosynthetic resources or, for example, the synthesis or decay of NTP or amino acid pools during infection.

A fundamental challenge in quantitative genetics is to better understand how interactions between genes quantitatively affect the fitness of an organism. Simulations of virus growth are potentially useful here because they provide a quantifiable link between functional molecular characteristics of gene products, such as binding affinities or kinetic parameters, and the growth rate, infection productivity, or fitness of the corresponding virus. To help to understand the terminology, it is useful to consider a simple quantitative example of how mutations in two genes, call them A and B, may interact. If the wild type has a fitness of 1.0 and mutations in genes A and B give rise to 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 do not interact, then the double mutant will have a fitness of 0.56, just the product of the fitness values for the single variants. If these deleterious mutations interact synergistically, then the fitness of the double mutant will be <0.56, and if they interact antagonistically or by buffering their deleterious effects, then the fitness of the double mutant will be closer to that of the wild type (>0.56). In quantitative genetics one seeks to answer the following question: on average, to what extent do deleterious mutations interact synergistically, antagonistically, or not at all? This can be a challenging experiment because it requires that one be able to generate mutants having well-defined mutations and to accurately quantify the effects of these mutations on some measure of fitness. In simulations of virus growth, it is feasible to create such mutations by altering parameters that correspond to molecular functions, such as a binding constant for complex formation or a rate constant corresponding to the elongation rate of an RNA polymerase, and then to calculate the effects of such alterations on the yield of virus progeny from a cell, one measure of fitness. The rationale here is that some mutations can alter quantitative characteristics of molecules and that the corresponding parameters in the models can be changed to simulate the effects of such mutations on molecular function. If one seeks to simulate deleterious mutations, then one just needs to check that the alteration in a parameter reduces the calculated fitness. We used this approach to generate and test interactions among diverse computationally generated deleterious mutations to examine the effects on the calculated fitness of phage 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, we assumed that host cells were available in unlimited supply, so the fitness of the virus depended on maximizing its production rate within a given cell. For the poor environment, we assumed that only the infected cell's resources were available, so fitness was based on maximizing the yield, without limitations on the rate. It was found that mildly deleterious mutations tended to act synergistically in resource-poor environments but antagonistically in resource-rich environments. Moreover, severely deleterious mutations tended to buffer themselves, acting antagonistically in both rich and poor environments. The work was relevant to population genetics in suggesting that the effects of synergistic interactions on fitness, while important for theory, may be challenging to detect in practice owing to their emergence when the quantitative effects of mutation on fitness are minimal. Future studies would benefit by exploring how quantitative rather than qualitative changes in environmental resources affect whether epistasis is synergistic or antagonistic. This may be achieved, for example, by using the metric for fitness based on production rate (e.g., the rich environment, as described above) and altering the growth rate of the host, creating richer or poorer environments based on higher or lower rates of host cell growth (79). Still more realistic assessments of epistasis might define the average fitness of the virus over multiple cycles of infection where one allows for fluctuations in host resources, conditions that could readily be implemented in virus growth simulations.

Biological systems tend to perform robustly with respect to conditions under which they evolved but are more sensitive to environmental changes that they have not encountered (81, 82). These ideas were examined in the context of the bacteriophage T7 system by testing the effects of simulated natural mutations, implemented by changing parameters of the model over plausible ranges for natural mutations, and simulated unnatural mutations, using previously described genomic rearrangements (76, 77). In general, the simulated phage growth was robust with respect to parameter changes but relatively sensitive or fragile with respect to genomic rearrangements, which are not known to occur in nature for phage T7. Such observations were consistent with theory (83). The robust behavior of growth was also a feature of the phage Qβ growth models (47). The effects of natural and unnatural perturbations to the simulated one-step growth of phage T7 were also evaluated in rich and poor host resource environments, and it was interesting that the fitness of wild-type phage was nearly optimal under resource-poor conditions but average under resource-rich conditions (83). This finding suggests that limited host resources served as a constraint on processes of phage T7 evolution. The extent to which such findings are general remains to be tested.

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

In a reformulation of the Sidorenko and Reichl model, the set of original kinetic equations was reduced and further data sets from the literature were incorporated to estimate key parameters (98). In addition, mechanisms of RNA stabilization and nuclear export of RNA species were incorporated to better resolve the dynamics of viral RNA transcription and genome replication. Constraints on the model included experimental measures of virus entry, i.e., absolute and relative levels of viral messenger, replicative intermediates, and genomic RNA per cell, as well as average levels of viral progeny released per cell. The resulting model provided support for early regulation of genome replication by stabilization of viral RNA replicative intermediates. It also suggested how the viral matrix protein 1 (M1), which normally mediates export of viral genome copies from 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 RNA toward the end of infection, providing evidence that transport processes or particle budding limits the process of virus progeny release.

It has long been known that influenza A virus infections can produce defective virus particles that can interfere with the production of infectious virus (von Magnus phenomenon). Such particles carry deletions in one or more essential genes needed for growth, making them unable to reproduce alone. However, during coinfection with infectious virus, defective genomes divert resources to their own replication and packaging, which interferes with the production of infectious particles. Such defective interfering particles (DIPs), described above in “Testing Antiviral Strategies,” have been found in clinical isolates of influenza virus (99) and can negatively affect the manufacture of live vaccines (100). To better understand mechanistically how DIPs reproduce, the intracellular kinetic model for influenza virus was extended to include defective interfering RNAs that replicate more rapidly than full-length RNAs owing to their reduced length (101). The extended model was able to account for observed effects of DIPs on infectious virus production. Moreover, the model suggests that DIPs that specifically carry deletions in RNA segments encoding the viral polymerase can become enriched, in agreement with experimental observations. Further, the model and experimental observations suggest that other mechanisms, such as competition for viral proteins (polymerase and nucleoprotein), can also contribute to interference and DIP enrichment.

A model of poliovirus growth focusing on intracellular processes was developed to explore 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 must be balanced across its life span, from birth to death, in order to optimize some measure or correlate of its fitness (103). In the case of humans, material or energy resources that are 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 the infected 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 delays of elongation by polymerase and ribosomes, respectively, as well as the time required for queuing multiple polymerases or ribosomes onto the respective template. By using the overall rate of production of viral genomes as a measure of virus growth, the model suggested that optimal virus growth would arise from an imbalance in the production of genomic and antigenomic templates, with about 40 genomic RNA molecules synthesized from each antigenomic RNA and about two antigenomic RNA molecules made from each genomic RNA, in reasonable agreement with experimental observations. It is possible that other measures of productivity, such as the total yield of viral genomes rather than their production rate, would yield similar results. The model was extended to account for potential resource limitations at the levels of ribosomes, amino acids, and nucleotides. The analysis highlighted tradeoffs between time spent on translation and that spent on transcription, because RNA synthesis is driven by products of translation and use of an RNA template for protein synthesis must be completed before it can be used further as a template for RNA synthesis. In addition, it was found that under conditions of limiting translation resources, optimal production would favor higher ratios of genomic to antigenomic RNA, while the inverse would be true under conditions of limiting transcription resources. The magnitude of asymmetries in template replication for poliovirus in this analysis may also apply for observed asymmetries in vesicular stomatitis virus (VSV), with its negative-sense genome. In future work, it would be interesting to see whether the observed trends hold in light of the costly energetics 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 substantiated, it is useful in showing that optimization of virus growth under different resource limitations can potentially shift how resources are allocated.

One approach for attenuating growth in VSV has been to change the linear order of genes in its genome. The order is important because the genome (3′-N-P-M-G-L-5′) has only a single promoter for its polymerase, near its 3′ end, with attenuation sequences between 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 highest expression occurring for the N gene (nucleocapsid), directly adjacent to the 3′ end, and the lowest expression for the L gene (large protein of the polymerase), adjacent to the 5′ end. In order for the virus to grow, all five proteins must be expressed and available for incorporation into progeny virus particles. One might well expect the rate of progeny production, which is one measure of virus fitness, to be optimized in some way. For a virus, optimal growth may be obtained by using available host resources to maximize the production of virus progeny (yield) or the rate at which progeny are created. In the case of VSV, high levels of nucleocapsid protein (N) are needed to bring about a switch in viral RNA synthesis from transcription to genome replication, so it is plausible 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-type expression pattern might well move it away from optimal growth and detrimentally influence the virus growth or fitness. To test this idea, positional mutants of VSV were created, with the N gene in the N2, N3, and N4 mutants positioned at progressively greater distances from the 3′ promoter (114). Yields of virus from infected cells were correspondingly perturbed, with the highest yields from N1 (wild type) and the lowest yields from N4. Expanding these gene-order variants to consider all permutations of five genes would define 5! or 120 possible versions. Additional mutants that retained the N1 and L5 positions of the wild type but accounted for all six perturbations of the internal three genes, P, M, and G, were also created and characterized with respect to gene expression and growth (115). We sought to better understand how gene order influences virus fitness by developing a kinetic model for VSV intracellular growth (116). Our model accounted for diverse features of VSV, including transcriptional regulation by intergenic attenuation, the switch in viral polymerase activity from transcription to genome replication, effects of different promoter strengths on the balance of full-length genomic and antigenomic RNA templates, diversion of the cellular translation machinery to synthesis of viral proteins, and the stoichiometry of RNA and protein components in VSV progeny particles. The model correctly predicted the observed experimental ranking of N gene mutants (N1 > N2 > N3 > N4) and suggested that optimal growth depends on a balance of promoter strengths for genome and antigenome synthesis. Further, we used the model to explore how the expression of other genes, such as the antigenic surface glycoprotein (encoded by the G gene), may entail tradeoffs with virus fitness. More recently, the model was extended to predict the growth of all 120 gene-order permutations, offering a means to study how fitness might depend on genome organization (117). The simulated virus fitness was found to be most sensitive to permutations that changed levels of the L and N genes, which are the least and most expressed genes of the wild-type virus, respectively. The roles of gene order and transcriptional regulation were also probed by use of this model. By computationally deleting the intergenic attenuations that regulate how transcriptional resources 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. Among the gene-order mutants, the wild type emerged as a fitness winner only in the presence of intergenic attenuation, suggesting that in the natural evolution of VSV this mode of regulation preceded or coevolved with the fixation of the wild-type gene order.

Many viruses that have been studied in great depth at the molecular level, as featured, for example, in Fields Virology, define opportunities for development of new kinetic models of virus growth in cells. Studies of more recently emerging viruses, for which less may be known, such as Ebola virus or Zika virus, may still benefit from existing models. Just as the model for phage Qβ, a single-stranded positive-sense RNA phage, served as a scaffold to build the initial replicon model for HCV, another single-stranded positive-sense RNA virus (128), other existing models may serve as scaffolds for related viruses. Ebola virus is a single-stranded negative-sense RNA virus that likely shares many similarities with VSV, for which a model is available (116). In addition, Zika virus, like HCV, is a single-stranded positive-sense RNA virus that encodes a single polyprotein, which subsequently self-cleaves to supply essential replication and structural proteins for virus growth. Rates or other parameters that quantify the magnitude of molecular interactions or reactions will always be valuable, if not essential, for model building. Databases for modeling, such as the “B10 NUMB3R5” effort (135), are useful for providing estimates of quantitative parameters of cellular functions essential for virus growth, such as protein synthesis. We hope that interest in modeling virus growth will motivate the growth of databases for virus-specific parameters, which individually will inform specific virus models but collectively may provide insight into the diversity of viral kinetics.

The availability of biosynthetic resources from a host cell is essential for virus growth, 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 different environmental conditions will need to be incorporated into models that seek to account for the dependence of virus growth on host physiology. The effects of host cellular growth rate on the intracellular kinetics of phage growth, which were probed by use of empirical correlations between cell growth and biosynthetic resources (79), have been expanded to incorporate genome-scale metabolic models of the bacterial host for the effects on phage growth (56, 136). Moreover, models of metabolic fluxes developed for mammalian cell cultures have been applied to identify conditions for higher glucose uptake rates and more efficient use of ATP, suitable for bioprocessing of influenza A virus vaccines (137, 138), and similar approaches have been applied to baculovirus growth in insect cell cultures (122).

In addition to supplying biosynthetic resources for virus reproduction, host cells also activate defensive innate immune signaling and responses, typically mediated by type I interferons (IFNs). Such signaling can trigger both autocrine and paracrine cellular responses that shut down protein synthesis that is essential for expression of essential viral functions. Different facets of such responses have been quantified and modeled mechanistically, including initial sensing of the viral dsRNA (139), induction of IFNs that activate both positive and negative feedbacks (140), and responses to different degrees of viral antagonism (141). Notably, in the case of infections by the highly pathogenic Nipah virus, modeling of the host response along with transcription-level measures of IFN and inflammatory cytokine activation provided insight by suggesting that the virus may delay suppression of inflammation and thereby enhance vascular permeability and infection spread (142).

To obtain a measure of viral or cellular function, most biochemical or molecular biological studies work with millions of cells in a tube or dish, for which average levels of nucleic acid, protein, intermediates, and infectious virus particles can readily be detected and quantified. However, in nature, infectious diseases are often transmitted or spread by countable numbers of infectious virus particles or cells. When the behavior of individual cells was studied in response to viral infection, the single cells exhibited infectious particle yields (or burst sizes) that were highly variable (143, 144). This variability among different infected cells can be attributed to genetic heterogeneity of the stock virus population (145) or host cells (146), resource differences linked to the stage of the cell cycle (147), or intrinsic stochastic behavior of reactions that involve small numbers of molecules (148–150). To account for the stochastic behavior, and assuming a well-mixed environment, Gillespie developed an algorithm for exact stochastic simulation of the reaction kinetics (151), which broadly enabled modeling of the lysis-lysogeny decision in phage lambda (152), stochastic versus deterministic intracellular infection (153), viral capsid assembly (154), bifurcation of gene expression in HIV-1 (16), transcriptional delays and autocatalytic feedbacks in VSV (155), genome amplification for influenza A virus (156), and infection-mediated activation of innate immune signaling (157, 158).

Models of virus intracellular kinetics have often applied the following assumption without explicit mention: intracellular environments are spatially homogeneous or well mixed. This simplifying assumption allows concentrations of reacting components to be treated as solely time dependent rather than time and position dependent, so mathematically the equations are written as sets of ordinary differential equations, which are numerically solved more readily than sets of partial differential equations. Future kinetic models will have spatial and temporal dependencies, incorporating advances in single-particle tracking by use of dye-labeled or reporter-expressing virus particles which have enabled direct visualization and quantification of virus-receptor interactions, internalization, intracellular transport, genomic release, nuclear transport, and cell-to-cell transmission (159–161). Different modeling approaches account for processes of intracellular particle transport by diffusion or mediated by active particle transport along microtubules (162), and an approach with relevant parameters for adeno-associated virus (AAV) has been outlined (163). More generally, approaches of statistical physics and their application to intracellular transport processes, which have been reviewed previously (164, 165), in combination with models for viral gene expression and function, will provide a more comprehensive accounting of the molecular biology and physical facets of virus intracellular growth.

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 over multiple cycles of susceptible host cell infection. The viral progeny released from an initial infected host cell go on to further infect other host cells, perhaps in the same tissue of a multicellular host organism.

Models of virus infection in humans, initiated in the 1990s for the dynamics of HIV-1 in 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 actively reproducing but also developing resistance to antiviral drug treatments (168). Similar modeling approaches have been extended to other viral infections in humans, including hepatitis C (169) and influenza A (170); related approaches have also been used to model virus infections as they selectively infect and spread in tumors as oncolytic therapies to treat cancer (171). Such within-host models, which accounted for the predator-prey dynamics of viruses and their host cells, are now serving as foundations for multiscale modeling in a top-down manner; the kinetics of processes within infected cells are of increasing interest owing to their capacity to account for explicit mechanisms of drug-target interactions and time delays in the release of virus progeny from their infected host cells (172, 173). At the same time, intracellular models of virus growth 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 new population of susceptible host cells (174, 175). When virus particles released from an infected cell are transported to neighboring or distant cells, physical processes of diffusion or fluid flow contribute to the dynamics of infection spread. Intracellular growth coupled with diffusional spread can be modeled using reaction-diffusion equations (176) or models that computationally simulate the behavior of cells on a grid as “agents” or “automata” that follow preestablished rules, using changes in the local environment (e.g., infection or immune signaling status of neighboring cells) to propagate infection spread (177–180).

Viruses can readily mutate as they reproduce, so modeling approaches adapted from population genetics may be useful in accounting for how different virus or gene variants become enriched. Moreover, as viral infections spread, they also affect the dynamics of their host populations, whether they are host cells or host organisms; upon infection, those cells or organisms may recover or die. To unite the combined behavior of the virus during its transmission and growth with the fall or rise of the host populations, predator-prey models from epidemiology may be appropriate. More than 3 decades ago, May and Anderson suggested that models combining the population genetics and epidemiology perspectives would potentially account for both essential features (181). Such models can be viewed as multiscale in the sense that they account for local molecular and cellular mechanisms of amplification and genetic diversification that originate from within cells, as well as processes of within-host spread between infected and susceptible cells of a multicellular host. Further transmissions between infected and susceptible multicellular host organisms, most notably animals and humans, highlight how additional ecological and evolutionary considerations (182), including the behavior and mobility of animal and human hosts (183, 184), contribute to the dynamics of epidemics.