Modeling viral evolutionary dynamics after telaprevir-based treatment.

For patients infected with hepatitis C virus (HCV), the combination of the direct-acting antiviral agent telaprevir, pegylated-interferon alfa (Peg-IFN), and ribavirin (RBV) significantly increases the chances of sustained virologic response (SVR) over treatment with Peg-IFN and RBV alone. If patients do not achieve SVR with telaprevir-based treatment, their viral population is often significantly enriched with telaprevir-resistant variants at the end of treatment. We sought to quantify the evolutionary dynamics of these post-treatment resistant variant populations. Previous estimates of these dynamics were limited by analyzing only population sequence data (20% sensitivity, qualitative resistance information) from 388 patients enrolled in Phase 3 clinical studies. Here we add clonal sequence analysis (5% sensitivity, quantitative) for a subset of these patients. We developed a computational model which integrates both the qualitative and quantitative sequence data, and which forms a framework for future analyses of drug resistance. The model was qualified by showing that deep-sequence data (1% sensitivity) from a subset of these patients are consistent with model predictions. When determining the median time for viral populations to revert to 20% resistance in these patients, the model predicts 8.3 (95% CI: 7.6, 8.4) months versus 10.7 (9.9, 12.8) months estimated using solely population sequence data for genotype 1a, and 1.0 (0.0, 1.4) months versus 0.9 (0.0, 2.7) months for genotype 1b. For each individual patient, the time to revert to 20% resistance predicted by the model was typically comparable to or faster than that estimated using solely population sequence data. Furthermore, the model predicts a median of 11.0 and 2.1 months after treatment failure for viral populations to revert to 99% wild-type in patients with HCV genotypes 1a or 1b, respectively. Our modeling approach provides a framework for projecting accurate, quantitative assessment of HCV resistance dynamics from a data set consisting of largely qualitative information.

Modeling population heterogeneity in viral dynamics for chronic hepatitis C infection: Insights from Phase 3 telaprevir clinical studies.

Viral dynamic modelling has proven useful for designing clinical studies and predicting treatment outcomes for patients infected with the hepatitis C virus. Generally these models aim to capture and predict the on-treatment viral load dynamics from a small study of individual patients. Here, we explored extending these models (1) to clinical studies with numerous patients and (2) by incorporating additional data types, including sequence data and prior response to interferon. Data from Phase 3 clinical studies of the direct-acting antiviral telaprevir (T; total daily dose of 2250 mg) combined with pegylated-interferon alfa and ribavirin (PR) were used for the analysis. The following data in the treatment-naïve population were reserved to verify the model: (1) a T/PR regimen where T was dosed every 8 h for 8 weeks (T8(q8h)/PR) and (2) a T/PR regimen where T was dosed twice daily for 12 weeks (T12(b.i.d.)/PR). The resulting model accurately predicted (1) sustained virologic response rates for both of these dosing regimens and (2) viral breakthrough characteristics of the T8(q8h)/PR regimen. Since the observed viral variants depend on the T exposure, the second verification suggested that the model was correctly sensitive to the different T regimen even though the model was developed using data from another T regimen. Furthermore, the model predicted that b.i.d. T dosing was comparable to q8h T dosing in the PR-experienced population, a comparison that has not been made in a controlled clinical study. The methods developed in this work to estimate the variability occurring below the limit of detection for the viral load were critical for making accurate predictions.

The stochastic quasi-steady-state assumption: reducing the model but not the noise.

Highly reactive species at small copy numbers play an important role in many biological reaction networks. We have described previously how these species can be removed from reaction networks using stochastic quasi-steady-state singular perturbation analysis (sQSPA). In this paper we apply sQSPA to three published biological models: the pap operon regulation, a biochemical oscillator, and an intracellular viral infection. These examples demonstrate three different potential benefits of sQSPA. First, rare state probabilities can be accurately estimated from simulation. Second, the method typically results in fewer and better scaled parameters that can be more readily estimated from experiments. Finally, the simulation time can be significantly reduced without sacrificing the accuracy of the solution.

Implications of rewiring bacterial quorum sensing.

Bacteria employ quorum sensing, a form of cell-cell communication, to sense changes in population density and regulate gene expression accordingly. This work investigated the rewiring of one quorum-sensing module, the lux circuit from the marine bacterium Vibrio fischeri. Steady-state experiments demonstrate that rewiring the network architecture of this module can yield graded, threshold, and bistable gene expression as predicted by a mathematical model. The experiments also show that the native lux operon is most consistent with a threshold, as opposed to a bistable, response. Each of the rewired networks yielded functional population sensors at biologically relevant conditions, suggesting that this operon is particularly robust. These findings (i) permit prediction of the behaviors of quorum-sensing operons in bacterial pathogens and (ii) facilitate forward engineering of synthetic gene circuits.

Implications of decoupling the intracellular and extracellular levels in multi-level models of virus growth.

Virus infections are characterized by two distinct levels of detail: the intracellular level describing how viruses hijack the host machinery to replicate, and the extracellular level describing how populations of virus and host cells interact. Deterministic, population balance models for viral infections permit incorporation of both the intracellular and extracellular levels of information. In this work, we identify assumptions that lead to exact, selective decoupling of the interaction between the intracellular and extracellular levels, effectively permitting solution of first the intracellular level, and subsequently the extracellular level. This decoupling leads to (1) intracellular and extracellular models of viral infections that have been previously reported and (2) a significant reduction in the computational expense required to solve the model. However, the decoupling restricts the behaviors that can be modeled. Simulation of a previously reported multi-level model demonstrates this decomposition when the intracellular level of description consists of numerous reaction events. Additionally, examples demonstrate that viruses can persist even when the intracellular level of description cannot sustain a steady-state production of virus (i.e., has only a trivial equilibrium). We expect the combination of this modeling framework with experimental data to result in a quantitative, systems-level understanding of viral infections and cellular antiviral strategies that will facilitate controlling both these infections and antiviral strategies.

Image-guided modeling of virus growth and spread.

Although many tools of cellular and molecular biology have been used to characterize single intracellular cycles of virus growth, few culture methods exist to study the dynamics of spatially spreading viruses over multiple generations. We have previously developed a method that addresses this need by tracking the spread of focal infections using immunocytochemical labeling and digital imaging. Here, we build reaction-diffusion models to account for spatio-temporal patterns formed by the spreading viral infection front as well as data from a single cycle of virus growth (one-step growth). Systems with and without the interferon-mediated antiviral response of the host cells are considered. Dynamic images of the spreading infections guide iterative model refinement steps that lead to reproduction of all of the salient features contained in the images, not just the velocity of the infection front. The optimal fits provide estimates for key parameters such as virus-host binding and the production rate of interferon. For the examined data, highly-lumped infection models that ignore the one-step growth dynamics provide a comparable fit to models that more accurately account for these dynamics, highlighting the fact that increased model complexity does not necessarily translate to improved fit. This work demonstrates how model building can facilitate the interpretation of experiments by highlighting contributions from both biological and methodological factors.

Synthetic gene circuits: design with directed evolution.

Synthetic circuits offer great promise for generating insights into nature's underlying design principles or forward engineering novel biotechnology applications. However, construction of these circuits is not straightforward. Synthetic circuits generally consist of components optimized to function in their natural context, not in the context of the synthetic circuit. Combining mathematical modeling with directed evolution offers one promising means for addressing this problem. Modeling identifies mutational targets and limits the evolutionary search space for directed evolution, which alters circuit performance without the need for detailed biophysical information. This review examines strategies for integrating modeling and directed evolution and discusses the utility and limitations of available methods.

Two classes of quasi-steady-state model reductions for stochastic kinetics.

The quasi-steady-state approximation (QSSA) is a model reduction technique used to remove highly reactive species from deterministic models of reaction mechanisms. In many reaction networks the highly reactive intermediates (QSSA species) have populations small enough to require a stochastic representation. In this work we apply singular perturbation analysis to remove the QSSA species from the chemical master equation for two classes of problems. The first class occurs in reaction networks where all the species have small populations and the QSSA species sample zero the majority of the time. The perturbation analysis provides a reduced master equation in which the highly reactive species can sample only zero, and are effectively removed from the model. The reduced master equation can be sampled with the Gillespie algorithm. This first stochastic QSSA reduction is applied to several example reaction mechanisms (including Michaelis-Menten kinetics) [Biochem. Z. 49, 333 (1913)]. A general framework for applying the first QSSA reduction technique to new reaction mechanisms is derived. The second class of QSSA model reductions is derived for reaction networks where non-QSSA species have large populations and QSSA species numbers are small and stochastic. We derive this second QSSA reduction from a combination of singular perturbation analysis and the Omega expansion. In some cases the reduced mechanisms and reaction rates from these two stochastic QSSA models and the classical deterministic QSSA reduction are equivalent; however, this is not usually the case.

Stochastic simulation of catalytic surface reactions in the fast diffusion limit.

The master equation of a lattice gas reaction tracks the probability of visiting all spatial configurations. The large number of unique spatial configurations on a lattice renders master equation simulations infeasible for even small lattices. In this work, a reduced master equation is derived for the probability distribution of the coverages in the infinite diffusion limit. This derivation justifies the widely used assumption that the adlayer is in equilibrium for the current coverages and temperature when all reactants are highly mobile. Given the reduced master equation, two novel and efficient simulation methods of lattice gas reactions in the infinite diffusion limit are derived. The first method involves solving the reduced master equation directly for small lattices, which is intractable in configuration space. The second method involves reducing the master equation further in the large lattice limit to a set of differential equations that tracks only the species coverages. Solution of the reduced master equation and differential equations requires information that can be obtained through short, diffusion-only kinetic Monte Carlo simulation runs at each coverage. These simulations need to be run only once because the data can be stored and used for simulations with any set of kinetic parameters, gas-phase concentrations, and initial conditions. An idealized CO oxidation reaction mechanism with strong lateral interactions is used as an example system for demonstrating the reduced master equation and deterministic simulation techniques.

On the origins of approximations for stochastic chemical kinetics.

This paper considers the derivation of approximations for stochastic chemical kinetics governed by the discrete master equation. Here, the concepts of (1) partitioning on the basis of fast and slow reactions as opposed to fast and slow species and (2) conditional probability densities are used to derive approximate, partitioned master equations, which are Markovian in nature, from the original master equation. Under different conditions dictated by relaxation time arguments, such approximations give rise to both the equilibrium and hybrid (deterministic or Langevin equations coupled with discrete stochastic simulation) approximations previously reported. In addition, the derivation points out several weaknesses in previous justifications of both the hybrid and equilibrium systems and demonstrates the connection between the original and approximate master equations. Two simple examples illustrate situations in which these two approximate methods are applicable and demonstrate the two methods' efficiencies.