Stochastic vs. deterministic modeling of intracellular viral kinetics

Dynamic behaviors of a stochastic virus infection model with Beddington-DeAngelis incidence function, eclipse-stage and Ornstein-Uhlenbeck process

In this paper, we present a virus infection model that incorporates eclipse-stage and Beddington-DeAngelis function, along with perturbation in infection rate using logarithmic Ornstein-Uhlenbeck process. Rigorous analysis demonstrates that the stochastic model has a unique global solution. Through construction of appropriate Lyapunov functions and a compact set, combined with the strong law of numbers and Fatou's lemma, we obtain the existence of the stationary distribution under a critical condition, which indicates the long-term persistence of T-cells and virions. Moreover, a precise probability density function is derived around the quasi-equilibrium of the model, and spectral radius analysis is employed to identify critical condition for elimination of the virus. Finally, numerical simulations are presented to validate theoretical results, and the impact of some key parameters such as the speed of reversion, volatility intensity and mean infection rate are investigated.

Infectious Virus Tracking by Fluorescent Live Cell Imaging in Primary Cells

To successfully infect a cell, HIV-1 has to overcome several host barriers while exploiting cellular cofactors. HIV-1 infection is highly inefficient with the great majority of viral particles not being able to successfully integrate into the target cell genome. Nonproductive HIV-1 particles are degraded or accumulated in cellular compartments. Thus, it becomes hard to distinguish between viral behaviors that lead to effectively infecting the cell from the ones that do not by using traditional methods. Here, we describe the infectious virus tracking method that detects and quantifies individual fluorescent viral particles over time and links viral particle behavior to its infectivity. This method employs live-cell imaging at ultra-low MOIs to detect the outcome of infection for every HIV-1 particle.

Comparison Theorems for Stochastic Chemical Reaction Networks

Continuous-time Markov chains are frequently used as stochastic models for chemical reaction networks, especially in the growing field of systems biology. A fundamental problem for these Stochastic Chemical Reaction Networks (SCRNs) is to understand the dependence of the stochastic behavior of these systems on the chemical reaction rate parameters. Towards solving this problem, in this paper we develop theoretical tools called comparison theorems that provide stochastic ordering results for SCRNs. These theorems give sufficient conditions for monotonic dependence on parameters in these network models, which allow us to obtain, under suitable conditions, information about transient and steady-state behavior. These theorems exploit structural properties of SCRNs, beyond those of general continuous-time Markov chains. Furthermore, we derive two theorems to compare stationary distributions and mean first passage times for SCRNs with different parameter values, or with the same parameters and different initial conditions. These tools are developed for SCRNs taking values in a generic (finite or countably infinite) state space and can also be applied for non-mass-action kinetics models. When propensity functions are bounded, our method of proof gives an explicit method for coupling two comparable SCRNs, which can be used to simultaneously simulate their sample paths in a comparable manner. We illustrate our results with applications to models of enzymatic kinetics and epigenetic regulation by chromatin modifications.

Mechanistic modeling of viral particle production

Viral systems such as wild-type viruses, viral vectors, and virus-like particles are essential components of modern biotechnology and medicine. Despite their importance, the commercial-scale production of viral systems remains highly inefficient for multiple reasons. Computational strategies are a promising avenue for improving process development, optimization, and control, but require a mathematical description of the system. This article reviews mechanistic modeling strategies for the production of viral particles, both at the cellular and bioreactor scales. In many cases, techniques and models from adjacent fields such as epidemiology and wild-type viral infection kinetics can be adapted to construct a suitable process model. These process models can then be employed for various purposes such as in-silico testing of novel process operating strategies and/or advanced process control.

Interplay of Cellular mRNA, miRNA and Viral miRNA during Infection of a Cell

The understanding of the kinetics of gene expression in cells infected by viruses is currently limited. As a rule, the corresponding models do not take viral microRNAs (miRNAs) into account. Such RNAs are, however, operative during the replication of some viruses, including, e.g., herpesvirus. To clarify the kinetics of this category (with emphasis on the information available for herpesvirus), I introduce a generic model describing the transient interplay of cellular mRNA, protein, miRNA and viral miRNA. In the absence of viral miRNA, the cellular miRNA is considered to suppress the populations of mRNA and protein due to association with mRNA and subsequent degradation. During infection, the viral miRNA suppresses the population of cellular miRNA and via this pathway makes the mRNA and protein populations larger. This effect becomes appreciable with the progress of intracellular viral replication. Using biologically reasonable parameters, I investigate the corresponding mean-field kinetics and show the scale of the effect of viral miRNAs on cellular miRNA and mRNA. The scale of fluctuations of the populations of these species is illustrated as well by employing Monte Carlo simulations.

Virus-like Particles: Measures and Biological Functions

Virus-like particles resemble infectious virus particles in size, shape, and molecular composition; however, they fail to productively infect host cells. Historically, the presence of virus-like particles has been inferred from total particle counts by microscopy, and infectious particle counts or plaque-forming-units (PFUs) by plaque assay; the resulting ratio of particles-to-PFUs is often greater than one, easily 10 or 100, indicating that most particles are non-infectious. Despite their inability to hijack cells for their reproduction, virus-like particles and the defective genomes they carry can exhibit a broad range of behaviors: interference with normal virus growth during co-infections, cell killing, and activation or inhibition of innate immune signaling. In addition, some virus-like particles become productive as their multiplicities of infection increase, a sign of cooperation between particles. Here, we review established and emerging methods to count virus-like particles and characterize their biological functions. We take a critical look at evidence for defective interfering virus genomes in natural and clinical isolates, and we review their potential as antiviral therapeutics. In short, we highlight an urgent need to better understand how virus-like genomes and particles interact with intact functional viruses during co-infection of their hosts, and their impacts on the transmission, severity, and persistence of virus-associated diseases.

Life cycle process dependencies of positive-sense RNA viruses suggest strategies for inhibiting productive cellular infection

Life cycle processes of positive-strand (+)RNA viruses are broadly conserved across families, yet they employ different strategies to grow in the cell. Using a generalized dynamical model for intracellular (+)RNA virus growth, we decipher these life cycle determinants and their dependencies for several viruses and parse the effects of viral mutations, drugs and host cell permissivity. We show that poliovirus employs rapid replication and virus assembly, whereas the Japanese encephalitis virus leverages its higher rate of translation and efficient cellular reorganization compared to the hepatitis C virus. Stochastic simulations demonstrate infection extinction if all seeding (inoculating) viral RNA degrade before establishing robust replication critical for infection. The probability of this productive cellular infection, 'cellular infectivity', is affected by virus-host processes and defined by early life cycle events and viral seeding. An increase in cytoplasmic RNA degradation and delay in vesicular compartment formation reduces infectivity, more so when combined. Synergy among these parameters in limiting (+)RNA virus infection as predicted by our model suggests new avenues for inhibiting infections by targeting the early life cycle bottlenecks.

Competitive exclusion during co-infection as a strategy to prevent the spread of a virus: A computational perspective

Inspired by the competition exclusion principle, this work aims at providing a computational framework to explore the theoretical feasibility of viral co-infection as a possible strategy to reduce the spread of a fatal strain in a population. We propose a stochastic-based model—called Co-Wish—to understand how competition between two viruses over a shared niche can affect the spread of each virus in infected tissue. To demonstrate the co-infection of two viruses, we first simulate the characteristics of two virus growth processes separately. Then, we examine their interactions until one can dominate the other. We use Co-Wish to explore how the model varies as the parameters of each virus growth process change when two viruses infect the host simultaneously. We will also investigate the effect of the delayed initiation of each infection. Moreover, Co-Wish not only examines the co-infection at the cell level but also includes the innate immune response during viral infection. The results highlight that the waiting times in the five stages of the viral infection of a cell in the model—namely attachment, penetration, eclipse, replication, and release—play an essential role in the competition between the two viruses. While it could prove challenging to fully understand the therapeutic potentials of viral co-infection, we discuss that our theoretical framework hints at an intriguing research direction in applying co-infection dynamics in controlling any viral outbreak’s speed.

Steady-State Fluorescence Signatures of Intramolecular Singlet Fission from Stochastic Predictions

The advent of new multichromophoric systems capable of undergoing efficient intramolecular singlet fission (iSF) has greatly expanded the range of possible motifs for multiexciton generation approaches for organic light energy harvesting materials. Transient absorption (TA) spectroscopic probes are typically used to characterize singlet fission processes that may place limitations on sensitivity and time resolution on scales comparable to the full lifespan of spin-forbidden triplets and interactions. Here, we investigate the ability of fluorescence-based spectroscopic probes to detect iSF activity in isolated dyads based on large substituted conjugated acenes (e.g., tetracene and pentacene derivatives). Photophysical models are simulated from several iSF-active dyad systems reported in the literature using a stochastic approach to assess the sensitivity of steady-state fluorescence to the presence of triplet excitons. The results demonstrate large fluctuations in expected fluorescence yields with varying excitation rate constants for systems with Φ_iSF > 0.5 (assuming weak interchromophore coupling). Exciton-exciton interactions are also investigated, and we further demonstrate how treating iSF dyads stochastically (i.e., finite number of chromophores) accentuates dependences of photophysical yields with excitation rates. Last, our approach reveals the potential ability of single molecule level fluorescence spectroscopy to detect iSF activity that can aid efforts to design and optimize candidate iSF systems.

Single-Cell Analysis Uncovers a Vast Diversity in Intracellular Viral Defective Interfering RNA Content Affecting the Large Cell-to-Cell Heterogeneity in Influenza A Virus Replication

Virus replication displays a large cell-to-cell heterogeneity; yet, not all sources of this variability are known. Here, we study the effect of defective interfering (DI) particle (DIP) co-infection on cell-to-cell variability in influenza A virus (IAV) replication. DIPs contain a large internal deletion in one of their eight viral RNAs (vRNA) and are, thus, defective in virus replication. Moreover, they interfere with virus replication. Using single-cell isolation and reverse transcription polymerase chain reaction, we uncovered a large between-cell heterogeneity in the DI vRNA content of infected cells, which was confirmed for DI mRNAs by single-cell RNA sequencing. A high load of intracellular DI vRNAs and DI mRNAs was found in low-productive cells, indicating their contribution to the large cell-to-cell variability in virus release. Furthermore, we show that the magnitude of host cell mRNA expression (some factors may inhibit virus replication), but not the ribosome content, may further affect the strength of single-cell virus replication. Finally, we show that the load of viral mRNAs (facilitating viral protein production) and the DI mRNA content are, independently from one another, connected with single-cell virus production. Together, these insights advance single-cell virology research toward the elucidation of the complex multi-parametric origin of the large cell-to-cell heterogeneity in virus infections.

Algebraic expressions of conditional expectations in gene regulatory networks

Gene Regulatory Networks are powerful models for describing the mechanisms and dynamics inside a cell. These networks are generally large in dimension and seldom yield analytical formulations. It was shown that studying the conditional expectations between dimensions (interactions or species) of a network could lead to drastic dimension reduction. These conditional expectations were classically given by solving equations of motions derived from the Chemical Master Equation. In this paper we deviate from this convention and take an Algebraic approach instead. That is, we explore the consequences of conditional expectations being described by a polynomial function. There are two main results in this work. Firstly, if the conditional expectation can be described by a polynomial function, then coefficients of this polynomial function can be reconstructed using the classical moments. And secondly, there are dimensions in Gene Regulatory Networks which inherently have conditional expectations with algebraic forms. We demonstrate through examples, that the theory derived in this work can be used to develop new and effective numerical schemes for forward simulation and parameter inference. The algebraic line of investigation of conditional expectations has considerable scope to be applied to many different aspects of Gene Regulatory Networks; this paper serves as a preliminary commentary in this direction.

Kinetic Modeling of Virus Growth in Cells

When a virus infects a host cell, it hijacks the biosynthetic capacity of the cell to produce virus progeny, a process that may take less than an hour or more than a week. The overall time required for a virus to reproduce depends collectively on the rates of multiple steps in the infection process, including initial binding of the virus particle to the surface of the cell, virus internalization and release of the 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 these particles into the extracellular environment. For a large number of virus types, much 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 been made-using mathematical modeling-to account for how the different steps contribute to the overall timing and productivity of the infection cycle in a cell. Here we review the initial case studies, which include studies of the one-step growth behavior of viruses that infect bacteria (Qβ, T7, and M13), human immunodeficiency virus, influenza A virus, poliovirus, vesicular stomatitis virus, baculovirus, hepatitis B and C viruses, and herpes simplex virus. Further, we consider how such models enable one to explore how cellular resources are utilized and how antiviral strategies might be designed to resist escape. Finally, we highlight challenges and opportunities at the frontiers of cell-level modeling of virus infections.

How Computational Models Enable Mechanistic Insights into Virus Infection

An implicit aim in cellular infection biology is to understand the mechanisms how viruses, microbes, eukaryotic parasites, and fungi usurp the functions of host cells and cause disease. Mechanistic insight is a deep understanding of the biophysical and biochemical processes that give rise to an observable phenomenon. It is typically subject to falsification, that is, it is accessible to experimentation and empirical data acquisition. This is different from logic and mathematics, which are not empirical, but built on systems of inherently consistent axioms. Here, we argue that modeling and computer simulation, combined with mechanistic insights, yields unprecedented deep understanding of phenomena in biology and especially in virus infections by providing a way of showing sufficiency of a hypothetical mechanism. This ideally complements the necessity statements accessible to empirical falsification by additional positive evidence. We discuss how computational implementations of mathematical models can assist and enhance the quantitative measurements of infection dynamics of enveloped and non-enveloped viruses and thereby help generating causal insights into virus infection biology.

Kinetics of lipid-nanoparticle-mediated intracellular mRNA delivery and function

mRNA delivery into cells forms the basis for one of the new and promising ways to treat various diseases. Among suitable carriers, lipid nanoparticles (LNPs) with a size of about 100 nm are now often employed. Despite high current interest in this area, the understanding of the basic details of LNP-mediated mRNA delivery and function is limited. To clarify the kinetics of mRNA release from LNPs, the author uses three generic models implying (i) exponential, (ii) diffusion-controlled, and (iii) detachment-controlled kinetic regimes, respectively. Despite the distinct differences in these kinetics, the associated transient kinetics of mRNA translation to the corresponding protein and its degradation are shown to be not too sensitive to the details of the mRNA delivery by LNPs (or other nanocarriers). In addition, the author illustrates how this protein may temporarily influence the expression of one gene or a few equivalent genes. The analysis includes positive or negative regulation of the gene transcription via the attachment of the protein without or with positive or negative feedback in the gene expression. Stable, bistable, and oscillatory schemes have been scrutinized in this context.

Snoopy's hybrid simulator: a tool to construct and simulate hybrid biological models

BACKGROUND

Hybrid simulation of (computational) biochemical reaction networks, which combines stochastic and deterministic dynamics, is an important direction to tackle future challenges due to complex and multi-scale models. Inherently hybrid computational models of biochemical networks entail two time scales: fast and slow. Therefore, it is intricate to efficiently and accurately analyse them using only either deterministic or stochastic simulation. However, there are only a few software tools that support such an approach. These tools are often limited with respect to the number as well as the functionalities of the provided hybrid simulation algorithms.

RESULTS

We present Snoopy's hybrid simulator, an efficient hybrid simulation software which builds on Snoopy, a tool to construct and simulate Petri nets. Snoopy's hybrid simulator provides a wide range of state-of-the-art hybrid simulation algorithms. Using this tool, a computational model of biochemical networks can be constructed using a (coloured) hybrid Petri net's graphical notations, or imported from other compatible formats (e.g. SBML), and afterwards executed via dynamic or static hybrid simulation.

CONCLUSION

Snoopy's hybrid simulator is a platform-independent tool providing an accurate and efficient simulation of hybrid (biological) models. It can be downloaded free of charge as part of Snoopy from http://www-dssz.informatik.tu-cottbus.de/DSSZ/Software/Snoopy .

Corticosterone oscillations during mania induction in the lateral hypothalamic kindled rat-Experimental observations and mathematical modeling

Changes in the hypothalamic-pituitary-adrenal (HPA) axis activity constitute a key component of bipolar mania, but the extent and nature of these alterations are not fully understood. We use here the lateral hypothalamic-kindled (LHK) rat model to deliberately induce an acute manic-like episode and measure serum corticosterone concentrations to assess changes in HPA axis activity. A mathematical model is developed to succinctly describe the entwined biochemical transformations that underlay the HPA axis and emulate by numerical simulations the considerable increase in serum corticosterone concentration induced by LHK. Synergistic combination of the LHK rat model and dynamical systems theory allows us to quantitatively characterize changes in HPA axis activity under controlled induction of acute manic-like states and provides a framework to study in silico how the dynamic integration of neurochemical transformations underlying the HPA axis is disrupted in these states.

Simulation of the M13 life cycle I: Assembly of a genetically-structured deterministic chemical kinetic simulation

To expand the quantitative, systems level understanding and foster the expansion of the biotechnological applications of the filamentous bacteriophage M13, we have unified the accumulated quantitative information on M13 biology into a genetically-structured, experimentally-based computational simulation of the entire phage life cycle. The deterministic chemical kinetic simulation explicitly includes the molecular details of DNA replication, mRNA transcription, protein translation and particle assembly, as well as the competing protein-protein and protein-nucleic acid interactions that control the timing and extent of phage production. The simulation reproduces the holistic behavior of M13, closely matching experimentally reported values of the intracellular levels of phage species and the timing of events in the M13 life cycle. The computational model provides a quantitative description of phage biology, highlights gaps in the present understanding of M13, and offers a framework for exploring alternative mechanisms of regulation in the context of the complete M13 life cycle.

A Simple, Realistic Stochastic Model of Gastric Emptying

Several models of Gastric Emptying (GE) have been employed in the past to represent the rate of delivery of stomach contents to the duodenum and jejunum. These models have all used a deterministic form (algebraic equations or ordinary differential equations), considering GE as a continuous, smooth process in time. However, GE is known to occur as a sequence of spurts, irregular both in size and in timing. Hence, we formulate a simple stochastic process model, able to represent the irregular decrements of gastric contents after a meal. The model is calibrated on existing literature data and provides consistent predictions of the observed variability in the emptying trajectories. This approach may be useful in metabolic modeling, since it describes well and explains the apparently heterogeneous GE experimental results in situations where common gastric mechanics across subjects would be expected.

A stochastic spatio-temporal (SST) model to study cell-to-cell variability in HIV-1 infection

Although HIV viremia in infected patients proceeds in a manner that may be accounted for by deterministic mathematical models, single virus-cell encounters following initial HIV exposure result in a variety of outcomes, only one of which results in a productive infection. The development of single molecule tracking techniques in living cells allows studies of intracellular transport of HIV, but it remains less clear what its impact may be on viral integration efficiency. Here, we present a stochastic intracellular mathematical model of HIV replication that incorporates microtubule transport of viral components. Using this model, we could study single round infections and observe how viruses entering cells reach one of three potential fates - degradation of the viral RNA genome, formation of LTR circles, or successful integration and establishment of a provirus. Our model predicts global trafficking properties, such as the probability and the mean time for a HIV viral particle to reach the nuclear pore. Interestingly, our model predicts that trafficking determines neither the probability or time of provirus establishment - instead, they are a function of vRNA degradation and reverse transcription reactions. Thus, our spatio-temporal model provides novel insights into the HIV infection process and may constitute a useful tool for the identification of promising drug targets.

Tools for Single-Cell Kinetic Analysis of Virus-Host Interactions

Measures of cellular gene expression or behavior, when performed on individual cells, inevitably reveal a diversity of behaviors and outcomes that can correlate with normal or diseased states. For virus infections, the potential diversity of outcomes are pushed to an extreme, where measures of infection reflect features of the specific infecting virus particle, the individual host cell, as well as interactions between viral and cellular components. Single-cell measures, while revealing, still often rely on specialized fluid handling capabilities, employ end-point measures, and remain labor-intensive to perform. To address these limitations, we consider a new microwell-based device that uses simple pipette-based fluid handling to isolate individual cells. Our design allows different experimental conditions to be implemented in a single device, permitting easier and more standardized protocols. Further, we utilize a recently reported dual-color fluorescent reporter system that provides dynamic readouts of viral and cellular gene expression during single-cell infections by vesicular stomatitis virus. In addition, we develop and show how free, open-source software can enable streamlined data management and batch image analysis. Here we validate the integration of the device and software using the reporter system to demonstrate unique single-cell dynamic measures of cellular responses to viral infection.

Single-cell analysis and stochastic modelling unveil large cell-to-cell variability in influenza A virus infection

Biochemical reactions are subject to stochastic fluctuations that can give rise to cell-to-cell variability. Yet, how this variability affects viral infections, which themselves involve noisy reactions, remains largely elusive. Here we present single-cell experiments and stochastic simulations that reveal a large heterogeneity between influenza A virus (IAV)-infected cells. In particular, experimental data show that progeny virus titres range from 1 to 970 plaque-forming units and intracellular viral RNA (vRNA) levels span three orders of magnitude. Moreover, the segmentation of IAV genomes seems to increase the susceptibility of their replication to noise, since the level of different genome segments can vary substantially within a cell. In addition, simulations suggest that the abortion of virus entry and random degradation of vRNAs can result in a large fraction of non-productive cells after single-hit infection. These results challenge current beliefs that cell population measurements and deterministic simulations are an accurate representation of viral infections.

Experimentally guided models reveal replication principles that shape the mutation distribution of RNA viruses

Life history theory posits that the sequence and timing of events in an organism's lifespan are fine-tuned by evolution to maximize the production of viable offspring. In a virus, a life history strategy is largely manifested in its replication mode. Here, we develop a stochastic mathematical model to infer the replication mode shaping the structure and mutation distribution of a poliovirus population in an intact single infected cell. We measure production of RNA and poliovirus particles through the infection cycle, and use these data to infer the parameters of our model. We find that on average the viral progeny produced from each cell are approximately five generations removed from the infecting virus. Multiple generations within a single cell infection provide opportunities for significant accumulation of mutations per viral genome and for intracellular selection.

DOI:http://dx.doi.org/10.7554/eLife.03753.001

Viruses with genetic information made up of molecules of RNA can multiply quickly, but not very accurately. This means that many errors, or mutations, occur when the RNA is copied to create new viruses. The advantage of this rapid, but mistake-filled, RNA replication process is that some of the mutations will be beneficial to the virus. This allows viruses to rapidly evolve, for example, to develop resistance against drugs.

The poliovirus is an RNA virus that can cause paralysis and death in humans. To prevent such infections, scientists have extensively studied the poliovirus and have developed effective vaccines against it that have eliminated the virus from all but a few countries. Because so much is known about the poliovirus and because it has a very simple structure, scientists continue to use the poliovirus as a model to study virus behavior.

One unknown aspect of the poliovirus' behavior is how it replicates after invading a cell. Are all of its RNA copies made from the original viral RNA that first infected the cell, in what is known as a ‘stamping machine’ model? Or do the new copies of the RNA also get copied themselves in a ‘geometric replication mode’ that increases the likelihood of mutations and enables the virus to evolve more rapidly?

Viral RNA molecules are copied by one of the virus's own proteins and so before the viral RNA can be replicated, it must first be translated to form viral proteins. When and where replication begins depends on the concentration of translated proteins around the RNA and so replication tends to begin in particular areas of the cell at different times. Schulte, Draghi et al. used mathematical modeling to create computer simulations of the number of polioviruses in a cell that take into account these time and space constraints. By including random elements in the model, the simulated behavior more accurately follows experimentally recorded data than previously used models.

The results of the model led Schulte, Draghi et al. to conclude that the poliovirus replicates by the ‘geometric mode’; as new copies of the poliovirus RNA are made, each copy goes on to make more copies. This means that in a single infected cell there are multiple generations of RNA, and each generation may undergo distinct mutations that are passed on to the next set of RNA copies. In fact, Schulte, Draghi et al. found that the average virus released from an infected cell is the great-great-great-granddaughter of the original virus that infected the cell. With so many different generations of virus coexisting in a cell, there are a lot of opportunities for new genetic combinations to occur and for viruses to evolve new abilities.

DOI:http://dx.doi.org/10.7554/eLife.03753.002

Stochastic population switch may explain the latent reservoir stability and intermittent viral blips in HIV patients on suppressive therapy

Highly active antiretroviral therapy can suppress plasma viral loads of HIV-1 infected individuals to below the detection limit of standard clinical assays. However, low-level viremia still persists. Many patients also have transient viral load measurements above the detection limit (the so-called "viral blips"). The latent reservoir consisting of latently infected CD4+ T cells represents a major obstacle to HIV-1 eradication. These cells can be activated to produce virions but the size of the latent reservoir is relatively stable. The mechanisms underlying low viral load persistence, emergence of intermittent viral blips and stability of the latent reservoir are not well understood. Cellular and viral transcription factors play an important role in the establishment and maintenance of HIV-1 latency. Infected cells with intermediate transcriptional activities may either revert to a latent state or become highly activated and produce virions due to intracellular perturbations. Here we develop a mathematical model that includes such stochastic population switch. We demonstrate that the model can generate a stable latent reservoir, intermittent viral blips, as well as low-level viremia persistence. Latently infected cells with intermediate transcription activities may maintain their size through a high level of homeostatic proliferation, while cells with low transcriptional activities are likely to be maintained through the reversion from cells with intermediate transcription activities. Simulations also suggest that treatment intensification or activation therapy may not help to eradicate the latent reservoir. Blocking the proliferation of latently infected cells might be a good strategy. These results provide more insights into the long-term dynamics of virus and latently infected cells in HIV patients on suppressive therapy and may help to develop novel treatment strategies.

Intracellular viral infection kinetics using a stochastic approach

Stochastic simulation is carried out to investigate intracellular viral reaction kinetics and the time evolution of the average particle number ( N̄) and coefficient variation (CV) for genome, template, and structural protein. The coefficient variation of these components is found to be ordered as: CVtemplate > CVstructural protein > CVgenome. The average particle number is also calculated via a deterministic approach. The magnitude value of the difference between the stochastic and deterministic approaches is found to be N̄template ≈ N̄structural protein > N̄genome. The Poisson algorithm has been used to investigate the number of particles in the dynamics of intracellular viral reaction kinetics. Our results show that the average particle number ( N̄) obtained using the Poisson algorithm for both dynamics and stationary states is le ss than that obtained using the mean field approach relative to the quantity ( N) found via the Gillespie algorithm. Also, the standard deviation of N̄ calculated via the Poisson algorithm is more than that obtained via the Gillespie algorithm. The equilibrium time for the population of particles via the Poisson algorithm is less than that via the Gillespie algorithm and the mean field approach. In the present study, the Gillespie algorithm is more reliable than the Poisson algorithm.

Computational modeling and multilevel cancer control interventions

This chapter presents an overview of computational modeling as a tool for multilevel cancer care and intervention research. Model-based analyses have been conducted at various "beneath the skin" or biological scales as well as at various "above the skin" or socioecological levels of cancer care delivery. We review the basic elements of computational modeling and illustrate its applications in four cancer control intervention areas: tobacco use, colorectal cancer screening, cervical cancer screening, and racial disparities in access to breast cancer care. Most of these models have examined cancer processes and outcomes at only one or two levels. We suggest ways these models can be expanded to consider interactions involving three or more levels. Looking forward, a number of methodological, structural, and communication barriers must be overcome to create useful computational models of multilevel cancer interventions and population health.

Interaction of virions with membrane glycolipids

Cellular membranes contain various lipids including glycolipids (GLs). The hydrophilic head groups of GLs extend from the membrane into the aqueous environment outside the cell where they act as recognition sites for specific interactions. The first steps of interaction of virions with cells often include contacts with GLs. To clarify the details of such contacts, we have used the total internal reflection fluorescence microscopy to explore the interaction of individual unlabelled virus-like particles (or, more specifically, norovirus protein capsids), which are firmly bound to a lipid bilayer, and fluorescent vesicles containing glycosphingolipids (these lipids form a subclass of GLs). The corresponding binding kinetics were earlier found to be kinetically limited, while the detachment kinetics were logarithmic over a wide range of time. Here, the detachment rate is observed to dramatically decrease with increasing concentration of glycosphingolipids from 1% to 8%. This effect has been analytically explained by using a generic model describing the statistics of bonds in the contact area between a virion and a lipid membrane. Among other factors, the model takes the formation of GL domains into account. Our analysis indicates that in the system under consideration, such domains, if present, have a characteristic size smaller than the contact area between the vesicle and the virus-like particle.

MODELING AND SIMULATION OF MOLECULAR BIOLOGY SYSTEMS USING PETRI NETS: MODELING GOALS OF VARIOUS APPROACHES

Petri nets are a discrete event simulation approach developed for system representation, in particular for their concurrency and synchronization properties. Various extensions to the original theory of Petri nets have been used for modeling molecular biology systems and metabolic networks. These extensions are stochastic, colored, hybrid and functional. This paper carries out an initial review of the various modeling approaches based on Petri net found in the literature, and of the biological systems that have been successfully modeled with these approaches. Moreover, the modeling goals and possibilities of qualitative analysis and system simulation of each approach are discussed.

ON EXPLORING EFFECTS OF MOLECULAR NOISE IN A SIMPLE VIRAL INFECTION MODEL

Intrinsic and extrinsic noises are all believed to be important in the development and function of many living organisms. In this study, we investigate the sources of the intrinsic noise and the influence of the extrinsic noise on an intracellular viral infection system. The contribution of the intrinsic noise from each reaction is measured by means of a special form of stochastic differential equations (SDEs), chemical Langevin equation. The intrinsic noise of the system is a linear sum of the noise in each of the reactions. The intrinsic noise mainly arises from the degradation of mRNA and the transcription processes. We then study the effects of extrinsic noise by the means of a general form of SDE. It is found that the noise of the viral components grows logarithmically with the increasing noise intensities. The system is most susceptible to the noise in the virus assembly process. A high level of noise in this process can even inhibit the growth of the viruses. This study also demonstrates the utility of SDEs in analyzing genetic regulatory networks perturbed by either inherent or parametric stochasticity.

Dynamics of alternative modes of RNA replication for positive-sense RNA viruses

We propose and study nonlinear mathematical models describing the intracellular time dynamics of viral RNA accumulation for positive-sense single-stranded RNA viruses. Our models consider different replication modes ranging between two extremes represented by the geometric replication (GR) and the linear stamping machine replication (SMR). We first analyse a model that quantitatively reproduced experimental data for the accumulation dynamics of both polarities of turnip mosaic potyvirus RNAs. We identify a non-degenerate transcritical bifurcation governing the extinction of both strands depending on three key parameters: the mode of replication (α), the replication rate (r) and the degradation rate (δ) of viral strands. Our results indicate that the bifurcation associated with α generically takes place when the replication mode is closer to the SMR, thus suggesting that GR may provide viral strands with an increased robustness against degradation. This transcritical bifurcation, which is responsible for the switching from an active to an absorbing regime, suggests a smooth (i.e. second-order), absorbing-state phase transition. Finally, we also analyse a simplified model that only incorporates asymmetry in replication tied to differential replication modes.

Mapping the architecture of the HIV-1 Tat circuit: A decision-making circuit that lacks bistability and exploits stochastic noise

Upon infection of a CD4(+) T cell, HIV-1 appears to 'choose' between two alternate fates: active replication or a long-lived dormant state termed proviral latency. A transcriptional positive-feedback loop generated by the HIV-1 Tat protein appears sufficient to mediate this decision. Here, we describe a coupled wet-lab and computational approach that uses mathematical modeling and live-cell time-lapse microscopy to map the architecture of the HIV-1 Tat transcriptional regulatory circuit and generate predictive models of HIV-1 latency. This approach provided the first characterization of a 'decision-making' circuit that lacks bistability and instead exploits stochastic fluctuations in cellular molecules (i.e. noise) to generate a decision between an on or off transcriptional state.

Stochastic E2F activation and reconciliation of phenomenological cell-cycle models

A new, stochastic model of entry into the mammalian cell cycle provides a mechanistic understanding of the temporal variability observed across populations of cells and reconciles previously proposed phenomenological cell-cycle models.

The transition of the mammalian cell from quiescence to proliferation is a highly variable process. Over the last four decades, two lines of apparently contradictory, phenomenological models have been proposed to account for such temporal variability. These include various forms of the transition probability (TP) model and the growth control (GC) model, which lack mechanistic details. The GC model was further proposed as an alternative explanation for the concept of the restriction point, which we recently demonstrated as being controlled by a bistable Rb-E2F switch. Here, through a combination of modeling and experiments, we show that these different lines of models in essence reflect different aspects of stochastic dynamics in cell cycle entry. In particular, we show that the variable activation of E2F can be described by stochastic activation of the bistable Rb-E2F switch, which in turn may account for the temporal variability in cell cycle entry. Moreover, we show that temporal dynamics of E2F activation can be recast into the frameworks of both the TP model and the GC model via parameter mapping. This mapping suggests that the two lines of phenomenological models can be reconciled through the stochastic dynamics of the Rb-E2F switch. It also suggests a potential utility of the TP or GC models in defining concise, quantitative phenotypes of cell physiology. This may have implications in classifying cell types or states.

Mammalian cells enter the division cycle in response to appropriate growth signals. For each cell, the decision to do so is critically dependent on the interplay between environmental cues and the internal state of the cell and is influenced by random fluctuations in cellular processes. Indeed, experimental evidence indicates that cell cycle entry is highly variable from cell to cell, even within a clonal population. To account for such variability, a number of phenomenological models have been previously proposed. These models primarily fall into two types depending on their fundamental assumptions on the origin of the variability. “Transition probability” models presume that variability in cell cycle entry originates from the fact that entry in each individual cell is random but also governed by a fixed probability. In contrast, “growth-controlled” models assume that the growth rates across a population are variable and result in cells that are out of phase developmentally. While both kinds of models provide a good fit to experimental data, their lack of mechanistic details limits their predictive power and has led to unresolved debate between their practitioners. In this study, we developed a mechanistically based stochastic model of the temporal dynamics of activation of the E2F transcription factor, which is used here as a marker of the transition of cells from quiescence to active cell cycling. Using this model, we show that “transition probability” and “growth-controlled” models can be reconciled by incorporation of a small number of basic cellular parameters related to protein synthesis and turnover, protein modification, stochasticity, and the like. Essentially our work shows that each kind of phenomenological model holds true for describing a particular aspect of the cell cycle transition. We suggest that incorporation of basic cellular parameters in this manner into phenomenological models may constitute a broadly applicable approach to defining concise, quantitative phenotypes of cell physiology.

Comparison of gene regulatory networks via steady-state trajectories

The modeling of genetic regulatory networks is becoming increasingly widespread in the study of biological systems. In the abstract, one would prefer quantitatively comprehensive models, such as a differential-equation model, to coarse models; however, in practice, detailed models require more accurate measurements for inference and more computational power to analyze than coarse-scale models. It is crucial to address the issue of model complexity in the framework of a basic scientific paradigm: the model should be of minimal complexity to provide the necessary predictive power. Addressing this issue requires a metric by which to compare networks. This paper proposes the use of a classical measure of difference between amplitude distributions for periodic signals to compare two networks according to the differences of their trajectories in the steady state. The metric is applicable to networks with both continuous and discrete values for both time and state, and it possesses the critical property that it allows the comparison of networks of different natures. We demonstrate application of the metric by comparing a continuous-valued reference network against simplified versions obtained via quantization.

Optimal viral strategies for bypassing RNA silencing

The RNA silencing pathway constitutes a defence mechanism highly conserved in eukaryotes, especially in plants, where the underlying working principle relies on the repressive action triggered by the intracellular presence of double-stranded RNAs. This immune system performs a post-transcriptional suppression of aberrant mRNAs or viral RNAs by small interfering RNAs (siRNAs) that are directed towards their target in a sequence-specific manner. However, viruses have evolved strategies to escape from silencing surveillance while promoting their own replication. Several viruses encode suppressor proteins that interact with different elements of the RNA silencing pathway and block it. The different suppressors are not phylogenetically nor structurally related and also differ in their mechanism of action. Here, we adopt a model-driven forward-engineering approach to understand the evolution of suppressor proteins and, in particular, why viral suppressors preferentially target some components of the silencing pathway. We analysed three strategies characterized by different design principles: replication in the absence of a suppressor, suppressors targeting the first protein component of the pathway and suppressors targeting the siRNAs. Our results shed light on the question of whether a virus must opt for devoting more time into transcription or into translation and on which would be the optimal step of the silencing pathway to be targeted by suppressors. In addition, we discussed the evolutionary implications of such designing principles.

Stochastic modelling of the eukaryotic heat shock response

The heat shock response (HSR) is a highly evolutionarily conserved defence mechanism allowing the cell to promptly react to elevated temperature conditions and other forms of stress. It has been subject to intense research for at least two main reasons. First, it is considered a promising candidate for deciphering the engineering principles underlying regulatory networks. Second, heat shock proteins (main actors of the HSR) play crucial role in many fundamental cellular processes. Therefore, profound understanding of the heat shock response would have far-reaching ramifications for the cell biology. Recently, a new deterministic model of the eukaryotic heat shock response has been proposed in the literature. It is very attractive since it consists of only the minimum number of components required by any functional regulatory network, while yet being capable of biological validation. However, it admits small molecule populations of some of the considered metabolites. In this paper a stochastic model corresponding to the deterministic one is constructed and the outcomes of these two models are confronted. The aim with this comparison is to show that, in the case of the heat shock response, the approximation of a discrete system with a continuous model is a reasonable approach. This is not always the truth, especially when the numbers of molecules of the considered species are small. By making the effort of performing and analysing 1000 stochastic simulations, we investigate the range of behaviour the stochastic model is likely to exhibit. We demonstrate that the obtained results agree well with the dynamics displayed by the continuous model, which strengthens the trust in the deterministic description. A proof of the existence and uniqueness of the stationary distribution of the Markov chain underlying the stochastic model is given. Moreover, the obtained view of the stochastic dynamics and the performed comparison to the outcome of the continuous formulation provide more insight into the dynamics of the heat shock response mechanism.

Solving the chemical master equation using sliding windows

BACKGROUND

The chemical master equation (CME) is a system of ordinary differential equations that describes the evolution of a network of chemical reactions as a stochastic process. Its solution yields the probability density vector of the system at each point in time. Solving the CME numerically is in many cases computationally expensive or even infeasible as the number of reachable states can be very large or infinite. We introduce the sliding window method, which computes an approximate solution of the CME by performing a sequence of local analysis steps. In each step, only a manageable subset of states is considered, representing a "window" into the state space. In subsequent steps, the window follows the direction in which the probability mass moves, until the time period of interest has elapsed. We construct the window based on a deterministic approximation of the future behavior of the system by estimating upper and lower bounds on the populations of the chemical species.

RESULTS

In order to show the effectiveness of our approach, we apply it to several examples previously described in the literature. The experimental results show that the proposed method speeds up the analysis considerably, compared to a global analysis, while still providing high accuracy.

CONCLUSIONS

The sliding window method is a novel approach to address the performance problems of numerical algorithms for the solution of the chemical master equation. The method efficiently approximates the probability distributions at the time points of interest for a variety of chemically reacting systems, including systems for which no upper bound on the population sizes of the chemical species is known a priori.

Solving the chemical master equation using sliding windows

Background

The chemical master equation (CME) is a system of ordinary differential equations that describes the evolution of a network of chemical reactions as a stochastic process. Its solution yields the probability density vector of the system at each point in time. Solving the CME numerically is in many cases computationally expensive or even infeasible as the number of reachable states can be very large or infinite. We introduce the sliding window method, which computes an approximate solution of the CME by performing a sequence of local analysis steps. In each step, only a manageable subset of states is considered, representing a "window" into the state space. In subsequent steps, the window follows the direction in which the probability mass moves, until the time period of interest has elapsed. We construct the window based on a deterministic approximation of the future behavior of the system by estimating upper and lower bounds on the populations of the chemical species.

Results

In order to show the effectiveness of our approach, we apply it to several examples previously described in the literature. The experimental results show that the proposed method speeds up the analysis considerably, compared to a global analysis, while still providing high accuracy.

Conclusions

The sliding window method is a novel approach to address the performance problems of numerical algorithms for the solution of the chemical master equation. The method efficiently approximates the probability distributions at the time points of interest for a variety of chemically reacting systems, including systems for which no upper bound on the population sizes of the chemical species is known a priori.

The role of asymmetric binding in ligand-receptor systems with 1:2 interaction ratio

Dynamical models for cellular ligand-receptor systems are among the most successful examples of mathematical approaches in systems biology. Here we present a general kinetic and mechanistic model for systems with asymmetric 1:2 ligand-receptor interaction ratio, such as erythropoietin and growth hormone systems. In these systems, the ligand presents two very different binding affinities to its receptor, and the weak interaction being often neglected for modeling purposes. Here, we demonstrate that the weak binding is the one tightly regulating the signaling, while the strong binding sets the threshold for the auto-inhibition effect characteristic of 1:2 asymmetric ligand-receptor systems. The model constitutes an improved mathematical framework for erythropoietin activation and equivalent biological processes, which are, due to their widespread use and relevance, on the forefront of pharmacological systems biology.

Stochastic kinetic modeling of vesicular stomatitis virus intracellular growth

By building kinetic models of biological networks one may advance the development of new modeling approaches while gaining insights into the biology. We focus here on building a stochastic kinetic model for the intracellular growth of vesicular stomatitis virus (VSV), a well-studied virus that encodes five genes. The essential network of VSV reactions creates challenges to stochastic simulation owing to (i) delayed reactions associated with transcription and genome replication, (ii) production of large numbers of intermediate proteins by translation, and (iii) the presence of highly reactive intermediates that rapidly fluctuate in their intracellular levels. We address these issues by developing a hybrid implementation of the model that combines a delayed stochastic simulation algorithm (DSSA) with Langevin equations to simulate the reactions that produce species in high numbers. Further, we employ a quasi-steady-state approximation (QSSA) to overcome the computational burden of small time steps caused by highly reactive species. The simulation is able to capture experimentally observed patterns of viral gene expression. Moreover, the simulation suggests that early levels of a low-abundance species, VSV L mRNA, play a key role in determining the production level of VSV genomes, transcripts, and proteins within an infected cell. Ultimately, these results suggest that stochastic gene expression contribute to the distribution of virus progeny yields from infected cells.

Replication mode and landscape topology differentially affect RNA virus mutational load and robustness

Regardless of genome polarity, intermediaries of complementary sense must be synthesized and used as templates for the production of new genomic strands. Depending on whether these new genomic strands become themselves templates for producing extra antigenomic ones, thus giving rise to geometric growth, or only the firstly synthesized antigenomic strands can be used to this end, thus following Luria's stamping machine model, the abundances and distributions of mutant genomes will be different. Here we propose mathematical and bit string models that allow distinguishing between stamping machine and geometric replication. We have observed that, regardless the topology of the fitness landscape, the critical mutation rate at which the master sequence disappears increases as the mechanism of replication switches from purely geometric to stamping machine. We also found that, for a wide range of mutation rates, large-effect mutations do not accumulate regardless the scheme of replication. However, mild mutations accumulate more in the geometric model. Furthermore, at high mutation rates, geometric growth leads to a population collapse for intermediate values of mutational effects at which the stamping machine still produces master genomes. We observed that the critical mutation rate was weakly dependent on the strength of antagonistic epistasis but strongly dependent on synergistic epistasis. In conclusion, we have shown that RNA viruses may increase their robustness against the accumulation of deleterious mutations by replicating as stamping machines and that the magnitude of this benefit depends on the topology of the fitness landscape assumed.

Sequential estimation for prescribed statistical accuracy in stochastic simulation of biological systems

Stochastic simulation of biological systems proceeds by repeatedly generating sample paths or trajectories of the underlying stochastic process, from which many relevant and important system properties can be obtained. While a great deal of research is targeted towards accelerated trajectory generation, issues concerned with the variability across trajectories are often neglected. Advanced methods for properly quantifying the statistical accuracy and determining a reasonable number of trajectories are hardly addressed formally in the context of biological system simulation, though mathematical statistics provides a large body of powerful theory. We invoke this theory and show how mathematically well-founded sequential estimation approaches serve for systematically generating enough but not too many trajectories for achieving a certain prescribed accuracy. The practical applicability is demonstrated and illustrated by numerical examples through simulation studies of an immigration-death process and a gene regulatory network.

A multi-time-scale analysis of chemical reaction networks: I. Deterministic systems

We consider deterministic descriptions of reaction networks in which different reactions occur on at least two distinct time scales. We show that when a certain Jacobian is nonsingular there is a coordinate system in which the evolution equations for slow and fast variables are separated, and we obtain the appropriate initial conditions for the transformed system. We also discuss topological properties which guarantee that the nonsingularity condition is satisfied, and show that in the new coordinate frame the evolution of the slow variables on the slow time scale is independent of the fast variables to lowest order in a small parameter. Several examples that illustrate the numerical accuracy of the reduction are presented, and an extension of the reduction method to three or more time scale networks is discussed.

Hybrid stochastic simulations of intracellular reaction-diffusion systems

With the observation that stochasticity is important in biological systems, chemical kinetics have begun to receive wider interest. While the use of Monte Carlo discrete event simulations most accurately capture the variability of molecular species, they become computationally costly for complex reaction-diffusion systems with large populations of molecules. On the other hand, continuous time models are computationally efficient but they fail to capture any variability in the molecular species. In this study a hybrid stochastic approach is introduced for simulating reaction-diffusion systems. We developed an adaptive partitioning strategy in which processes with high frequency are simulated with deterministic rate-based equations, and those with low frequency using the exact stochastic algorithm of Gillespie. Therefore the stochastic behavior of cellular pathways is preserved while being able to apply it to large populations of molecules. We describe our method and demonstrate its accuracy and efficiency compared with the Gillespie algorithm for two different systems. First, a model of intracellular viral kinetics with two steady states and second, a compartmental model of the postsynaptic spine head for studying the dynamics of Ca+2 and NMDA receptors.

Size-independent differences between the mean of discrete stochastic systems and the corresponding continuous deterministic systems

In this paper, it is shown that for a class of reaction networks, the discrete stochastic nature of the reacting species and reactions results in qualitative and quantitative differences between the mean of exact stochastic simulations and the prediction of the corresponding deterministic system. The differences are independent of the number of molecules of each species in the system under consideration. These reaction networks are open systems of chemical reactions with no zero-order reaction rates. They are characterized by at least two stationary points, one of which is a nonzero stable point, and one unstable trivial solution (stability based on a linear stability analysis of the deterministic system). Starting from a nonzero initial condition, the deterministic system never reaches the zero stationary point due to its unstable nature. In contrast, the result presented here proves that this zero-state is a stable stationary state for the discrete stochastic system, and other finite states have zero probability of existence at large times. This result generalizes previous theoretical studies and simulations of specific systems and provides a theoretical basis for analyzing a class of systems that exhibit such inconsistent behavior. This result has implications in the simulation of infection, apoptosis, and population kinetics, as it can be shown that for certain models the stochastic simulations will always yield different predictions for the mean behavior than the deterministic simulations.

Growth of an RNA virus in single cells reveals a broad fitness distribution

Genetic and environmental factors will influence the growth of an RNA virus, but their relative contributions are challenging to resolve because standard culture methods mask how virus particles interact with individual host cells. Here, single particles of vesicular stomatitis virus, a prototype RNA virus, were used to infect individual BHK cells. Infected cells produced 50 to 8000 progeny virus particles, but these differences were lost upon subsequent culture, suggesting the diversity of yields reflected cell-to-cell differences rather than viral genetic variation. Cells infected at different phases of their cell cycle produced from 1400 (early S) to 8700 (G(2)M) infectious virus particles, coinciding with the middle-to-upper range of the observed distribution. Fluctuations in virus and cell compositions and noisy gene expression may also contribute to the broad distribution of virus yields. These findings take a step toward quantifying how environmental variation can impact the fitness distribution of an RNA virus.