Eric L. Haseltine

Approximate simulation of coupled fast and slow reactions for stochastic chemical kinetics

Exact methods are available for the simulation of isothermal, well-mixed stochastic chemical kinetics. As increasingly complex physical systems are modeled, however, these methods become difficult to solve because the computational burden scales with the number of reaction events. This paper addresses one aspect of this problem: the case in which reacting species fluctuate by different orders of magnitude. By partitioning the system into subsets of “fast” and “slow” reactions, it is possible to bound the computational load by approximating “fast” reactions either deterministically or as Langevin equations. This paper provides a theoretical background for such approximations and outlines strategies for computing these approximations. Two motivating examples drawn from the fields of particle technology and biotechnology illustrate the accuracy and computational efficiency of these approximations.

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.

Implications of Rewiring Bacterial Quorum Sensing

ABSTRACT 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.

Dynamics of viral infections: incorporating both the intracellular and extracellular levels

To date, most models of viral infections have focused exclusively on modeling either the intracellular level or the extracellular level. To more realistically model these infections, we propose incorporating both levels of information into the description. One way of performing this task in a deterministic setting is to derive cell population balances from the equation of continuity. We apply such a balance to obtain a two-level model of a viral infection. We then use numerical simulation to demonstrate both cell culture and in vivo responses given a variety of experimental conditions. We compare these responses to those obtained from applying other commonly used models. The results demonstrate that, in contrast to commonly used models, the cell population balance provides a more intuitive and flexible modeling framework for incorporating both the intracellular and extracellular events occurring during viral infections. This improved capability to represent the trends in the biological measurements of interest offers a more systematic and quantitative understanding of how viral infections propagate and how to best control this propagation.

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.

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 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. Biotechnol. Bioeng. 2008;101: 811–820.

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.