Elimination of fast variables in chemical Langevin equations

Elimination of fast variables in stochastic nonlinear kinetics

A reduced chemical scheme involving a small number of variables is often sufficient to account for the deterministic evolution of the concentration of the main species contributing to a reaction. However, its predictions are questionable in small systems used, for example in fluorescence correlation spectroscopy (FCS) or in explosive systems involving strong nonlinearities such as autocatalytic steps. We make precise dynamical criteria defining the validity domain of the quasi-steady-state approximation and the elimination of a fast concentration in deterministic dynamics. Designing two different three-variable models converging toward the same two-variable model, we show that the variances and covariance of the fluctuations of the slow variables are not correctly predicted using the two-variable model, even in the limit of a large system size. The more striking weaknesses of the reduced scheme are figured out in mesoscaled systems containing a small number of molecules. The results of two stochastic approaches are compared and the shortcomings of the Langevin equations with respect to the master equation are pointed out. We conclude that the description of the fluctuations and their coupling with nonlinearities of deterministic dynamics escape reduced chemical schemes.

Channel based generating function approach to the stochastic Hodgkin-Huxley neuronal system

Internal and external fluctuations, such as channel noise and synaptic noise, contribute to the generation of spontaneous action potentials in neurons. Many different Langevin approaches have been proposed to speed up the computation but with waning accuracy especially at small channel numbers. We apply a generating function approach to the master equation for the ion channel dynamics and further propose two accelerating algorithms, with an accuracy close to the Gillespie algorithm but with much higher efficiency, opening the door for expedited simulation of noisy action potential propagating along axons or other types of noisy signal transduction.

Equilibrium distributions of simple biochemical reaction systems for time-scale separation in stochastic reaction networks

Many biochemical reaction networks are inherently multiscale in time and in the counts of participating molecular species. A standard technique to treat different time scales in the stochastic kinetics framework is averaging or quasi-steady-state analysis: it is assumed that the fast dynamics reaches its equilibrium (stationary) distribution on a time scale where the slowly varying molecular counts are unlikely to have changed. We derive analytic equilibrium distributions for various simple biochemical systems, such as enzymatic reactions and gene regulation models. These can be directly inserted into simulations of the slow time-scale dynamics. They also provide insight into the stimulus–response of these systems. An important model for which we derive the analytic equilibrium distribution is the binding of dimer transcription factors (TFs) that first have to form from monomers. This gene regulation mechanism is compared to the cases of the binding of simple monomer TFs to one gene or to multiple copies of a gene, and to the cases of the cooperative binding of two or multiple TFs to a gene. The results apply equally to ligands binding to enzyme molecules.

Modeling synthetic gene oscillators

Genetic oscillators have long held the fascination of experimental and theoretical synthetic biologists alike. From an experimental standpoint, the creation of synthetic gene oscillators represents a yardstick by which our ability to engineer synthetic gene circuits can be measured. For theorists, synthetic gene oscillators are a playground in which to test mathematical models for the dynamics of gene regulation. Historically, mathematical models of synthetic gene circuits have varied greatly. Often, the differences are determined by the level of biological detail included within each model, or which approximation scheme is used. In this review, we examine, in detail, how mathematical models of synthetic gene oscillators are derived and the biological processes that affect the dynamics of gene regulation.

Stochastic delay accelerates signaling in gene networks

The creation of protein from DNA is a dynamic process consisting of numerous reactions, such as transcription, translation and protein folding. Each of these reactions is further comprised of hundreds or thousands of sub-steps that must be completed before a protein is fully mature. Consequently, the time it takes to create a single protein depends on the number of steps in the reaction chain and the nature of each step. One way to account for these reactions in models of gene regulatory networks is to incorporate dynamical delay. However, the stochastic nature of the reactions necessary to produce protein leads to a waiting time that is randomly distributed. Here, we use queueing theory to examine the effects of such distributed delay on the propagation of information through transcriptionally regulated genetic networks. In an analytically tractable model we find that increasing the randomness in protein production delay can increase signaling speed in transcriptional networks. The effect is confirmed in stochastic simulations, and we demonstrate its impact in several common transcriptional motifs. In particular, we show that in feedforward loops signaling time and magnitude are significantly affected by distributed delay. In addition, delay has previously been shown to cause stable oscillations in circuits with negative feedback. We show that the period and the amplitude of the oscillations monotonically decrease as the variability of the delay time increases.

Delay in gene regulatory networks often arises from the numerous sequential reactions necessary to create fully functional protein from DNA. While the molecular mechanisms behind protein production and maturation are known, it is still unknown to what extent the resulting delay affects signaling in transcriptional networks. In contrast to previous studies that have examined the consequences of fixed delay in gene networks, here we investigate how the variability of the delay time influences the resulting dynamics. The exact distribution of “transcriptional delay” is still unknown, and most likely greatly depends on both intrinsic and extrinsic factors. Nevertheless, we are able to deduce specific effects of distributed delay on transcriptional signaling that are independent of the underlying distribution. We find that the time it takes for a gene encoding a transcription factor to signal its downstream target decreases as the delay variability increases. We use queueing theory to derive a simple relationship describing this result, and use stochastic simulations to confirm it. The consequences of distributed delay for several common transcriptional motifs are also discussed.

Adiabatic coarse-graining and simulations of stochastic biochemical networks

We propose a universal approach for analysis and fast simulations of stiff stochastic biochemical networks, which rests on elimination of fast chemical species without a loss of information about mesoscopic, non-Poissonian fluctuations of the slow ones. Our approach is similar to the Born-Oppenheimer approximation in quantum mechanics and follows from the stochastic path integral representation of the cumulant generating function of reaction events. In applications with a small number of chemical reactions, it produces analytical expressions for cumulants of chemical fluxes between the slow variables. This allows for a low-dimensional, interpretable representation and can be used for high-accuracy, low-complexity coarse-grained numerical simulations. As an example, we derive the coarse-grained description for a chain of biochemical reactions and show that the coarse-grained and the microscopic simulations agree, but the former is 3 orders of magnitude faster.