Computer simulation of T3/T7 phage infection using lag times

Small binding-site clearance delays are not negligible in gene expression modeling

During the templated biopolymerization processes of transcription and translation, a macromolecular machine, either an RNA polymerase or a ribosome, binds to a specific site on the template. Due to the sizes of these enzymes, there is a waiting time before one clears the binding site and another can bind. These clearance delays are relatively short, and one might think that they could be neglected. However, in the case of transcription, these clearance delays are associated with conservation laws, resulting in surprisingly large effects on the bifurcation diagrams in models of gene expression networks. We study an example of this phenomenon in a model of a gene regulated by a non-coding RNA displaying bistability. Neglecting the binding-site clearance delays in this model can only be compensated for by making ad hoc, unphysical adjustments to the model's kinetic constants.

Mathematical Analysis of Viral Replication Dynamics and Antiviral Treatment Strategies: From Basic Models to Age-Based Multi-Scale Modeling

Viral infectious diseases are a global health concern, as is evident by recent outbreaks of the middle east respiratory syndrome, Ebola virus disease, and re-emerging zika, dengue, and chikungunya fevers. Viral epidemics are a socio-economic burden that causes short- and long-term costs for disease diagnosis and treatment as well as a loss in productivity by absenteeism. These outbreaks and their socio-economic costs underline the necessity for a precise analysis of virus-host interactions, which would help to understand disease mechanisms and to develop therapeutic interventions. The combination of quantitative measurements and dynamic mathematical modeling has increased our understanding of the within-host infection dynamics and has led to important insights into viral pathogenesis, transmission, and disease progression. Furthermore, virus-host models helped to identify drug targets, to predict the treatment duration to achieve cure, and to reduce treatment costs. In this article, we review important achievements made by mathematical modeling of viral kinetics on the extracellular, intracellular, and multi-scale level for Human Immunodeficiency Virus, Hepatitis C Virus, Influenza A Virus, Ebola Virus, Dengue Virus, and Zika Virus. Herein, we focus on basic mathematical models on the population scale (so-called target cell-limited models), detailed models regarding the most important steps in the viral life cycle, and the combination of both. For this purpose, we review how mathematical modeling of viral dynamics helped to understand the virus-host interactions and disease progression or clearance. Additionally, we review different types and effects of therapeutic strategies and how mathematical modeling has been used to predict new treatment regimens.

Parametric Sensitivity Analysis of Oscillatory Delay Systems with an Application to Gene Regulation

A parametric sensitivity analysis for periodic solutions of delay-differential equations is developed. Because phase shifts cause the sensitivity coefficients of a periodic orbit to diverge, we focus on sensitivities of the extrema, from which amplitude sensitivities are computed, and of the period. Delay-differential equations are often used to model gene expression networks. In these models, the parametric sensitivities of a particular genotype define the local geometry of the evolutionary landscape. Thus, sensitivities can be used to investigate directions of gradual evolutionary change. An oscillatory protein synthesis model whose properties are modulated by RNA interference is used as an example. This model consists of a set of coupled delay-differential equations involving three delays. Sensitivity analyses are carried out at several operating points. Comments on the evolutionary implications of the results are offered.

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.

Intracellular kinetics of a growing virus: a genetically structured simulation for bacteriophage T7

Viruses have evolved to efficiently direct the resources of their hosts toward their own reproduction. A quantitative understanding of viral growth will help researchers develop antiviral strategies, design metabolic pathways, construct vectors for gene therapy, and engineer molecular systems that self-assemble. As a model system we examine here the growth of bacteriophage T7 in Escherichia coli using a chemical-kinetic framework. Data published over the last three decades on the genetics, physiology, and biophysics of phage T7 are incorporated into a genetically structured simulation that accounts for entry of the T7 genome into its host, expression of T7 genes, replication of T7 DNA, assembly of T7 procapsids, and packaging of T7 DNA to finally produce intact T7 progeny. Good agreement is found between the simulated behavior and experimental observations for the shift in transcription capacity from the host to the phage, the initiation times of phage protein synthesis, and the intracellular assembly of both wild-type phage and a fast-growing deletion mutant. The simulation is utilized to predict the effect of antisense molecules targeted to different T7 mRNA. Further, a postulated mechanism for the down regulation of T7 transcription in vivo is quantitatively examined and shown to agree with available data. The simulation is found to be a useful tool for exploring and understanding the dynamics of virus growth at the molecular level. (c) 1997 John Wiley & Sons, Inc. Biotechnol Bioeng 55: 375-389, 1997.

Modeling the fitness consequences of a cyanophage-encoded photosynthesis gene

Background

Phages infecting marine picocyanobacteria often carry a psbA gene, which encodes a homolog to the photosynthetic reaction center protein, D1. Host encoded D1 decays during phage infection in the light. Phage encoded D1 may help to maintain photosynthesis during the lytic cycle, which in turn could bolster the production of deoxynucleoside triphosphates (dNTPs) for phage genome replication.

Methodology / Principal Findings

To explore the consequences to a phage of encoding and expressing psbA, we derive a simple model of infection for a cyanophage/host pair — cyanophage P-SSP7 and Prochlorococcus MED4— for which pertinent laboratory data are available. We first use the model to describe phage genome replication and the kinetics of psbA expression by host and phage. We then examine the contribution of phage psbA expression to phage genome replication under constant low irradiance (25 µE m⁻² s⁻¹). We predict that while phage psbA expression could lead to an increase in the number of phage genomes produced during a lytic cycle of between 2.5 and 4.5% (depending on parameter values), this advantage can be nearly negated by the cost of psbA in elongating the phage genome. Under higher irradiance conditions that promote D1 degradation, however, phage psbA confers a greater advantage to phage genome replication.

Conclusions / Significance

These analyses illustrate how psbA may benefit phage in the dynamic ocean surface mixed layer.

Complex dynamics in the Oregonator model with linear delayed feedback

The Belousov-Zhabotinsky (BZ) reaction can display a rich dynamics when a delayed feedback is applied. We used the Oregonator model of the oscillating BZ reaction to explore the dynamics brought about by a linear delayed feedback. The time-delayed feedback can generate a succession of complex dynamics: period-doubling bifurcation route to chaos; amplitude death; fat, wrinkled, fractal, and broken tori; and mixed-mode oscillations. We observed that this dynamics arises due to a delay-driven transition, or toggling of the system between large and small amplitude oscillations, through a canard bifurcation. We used a combination of numerical bifurcation continuation techniques and other numerical methods to explore the dynamics in the strength of feedback-delay space. We observed that the period-doubling and quasiperiodic route to chaos span a low-dimensional subspace, perhaps due to the trapping of the trajectories in the small amplitude regime near the canard; and the trapped chaotic trajectories get ejected from the small amplitude regime due to a crowding effect to generate chaotic-excitable spikes. We also qualitatively explained the observed dynamics by projecting a three-dimensional phase portrait of the delayed dynamics on the two-dimensional nullclines. This is the first instance in which it is shown that the interaction of delay and canard can bring about complex dynamics.

TABASCO: A single molecule, base-pair resolved gene expression simulator

Background

Experimental studies of gene expression have identified some of the individual molecular components and elementary reactions that comprise and control cellular behavior. Given our current understanding of gene expression, and the goals of biotechnology research, both scientists and engineers would benefit from detailed simulators that can explicitly compute genome-wide expression levels as a function of individual molecular events, including the activities and interactions of molecules on DNA at single base pair resolution. However, for practical reasons including computational tractability, available simulators have not been able to represent genome-scale models of gene expression at this level of detail.

Results

Here we develop a simulator, TABASCO , which enables the precise representation of individual molecules and events in gene expression for genome-scale systems. We use a single molecule computational engine to track individual molecules interacting with and along nucleic acid polymers at single base resolution. Tabasco uses logical rules to automatically update and delimit the set of species and reactions that comprise a system during simulation, thereby avoiding the need for a priori specification of all possible combinations of molecules and reaction events. We confirm that single molecule, base-pair resolved simulation using TABASCO (Tabasco) can accurately compute gene expression dynamics and, moving beyond previous simulators, provide for the direct representation of intermolecular events such as polymerase collisions and promoter occlusion. We demonstrate the computational capacity of Tabasco by simulating the entirety of gene expression during bacteriophage T7 infection; for reference, the 39,937 base pair T7 genome encodes 56 genes that are transcribed by two types of RNA polymerases active across 22 promoters.

Conclusion

Tabasco enables genome-scale simulation of transcription and translation at individual molecule and single base-pair resolution. By directly representing the position and activity of individual molecules on DNA, Tabasco can directly test the effects of detailed molecular processes on system-wide gene expression. Tabasco would also be useful for studying the complex regulatory mechanisms controlling eukaryotic gene expression. The computational engine underlying Tabasco could also be adapted to represent other types of processive systems in which individual reaction events are organized across a single spatial dimension (e.g., polysaccharide synthesis).

Graph-theoretic methods for the analysis of chemical and biochemical networks. II. Oscillations in networks with delays

Delay-differential equations are commonly used to model genetic regulatory systems with the delays representing transcription and translation times. Equations with delayed terms can also be used to represent other types of chemical processes. Here we analyze delayed mass-action systems, i.e. systems in which the rates of reaction are given by mass-action kinetics, but where the appearance of products may be delayed. Necessary conditions for delay-induced instability are presented in terms both of a directed graph (digraph) constructed from the Jacobian matrix of the corresponding ODE model and of a species-reaction bipartite graph which directly represents a chemical mechanism. Methods based on the bipartite graph are particularly convenient and powerful. The condition for a delay-induced instability in this case is the existence of a subgraph of the bipartite graph containing an odd number of cycles of which an odd number are negative.

Stochastic kinetics description of a simple transcription model

We study a stochastic model of transcription kinetics in order to characterize the distributions of transcriptional delay and of elongation rates. Transcriptional delay is the time which elapses between the binding of RNA polymerase to a promoter sequence and its dissociation from the DNA template strand with consequent release of the transcript. Transcription elongation is the process by which the RNA polymerase slides along the template strand. The model considers a DNA template strand with one promoter site and n nucleotide sites, and five types of reaction processes, which we think are key ones in transcription. The chemical master equation is a set of ordinary differential equations in 3(n) variables, where n is the number of bases in the template. This model is too huge to be handled if n is large. We manage to get a reduced Markov model which has only 2n independent variables and can well approximate the original dynamics. We obtain a number of analytical and numerical results for this model, including delay and transcript elongation rate distributions. Recent studies of single-RNA polymerase transcription by using optical-trapping techniques raise an issue of whether the elongation rates measured in a population are heterogeneous or not. Our model implies that in the cases studied, different RNA polymerase molecules move at different characteristic rates along the template strand. We also discuss the implications of this work for the mathematical modeling of genetic regulatory circuits.