Trade-offs between retroactivity and noise in connected transcriptional components

At the interconnection of two gene transcriptional components in a biomolecular network, the noise in the downstream component can be reduced by increasing its gene copy number. However, this method of reducing noise increases the load applied to the upstream system, called retroactivity, thereby causing a perturbation in the upstream system. In this work, we quantify the error in the system trajectories caused by perturbations due to retroactivity and noise, and analyze the trade-off between these two perturbations. We model the system as a set of nonlinear chemical Langevin equations and quantify the trade-off by employing contraction theory for stochastic systems.National Science Foundation (U.S.). Division of Computing and Communication Foundations (Award 1058127)United States. Air Force Office of Scientific Research (Award FA9550-12-1-0129

Model reduction for a class of singularly perturbed stochastic differential equations

A class of singularly perturbed stochastic differential equations (SDE) with linear drift and nonlinear diffusion terms is considered. We prove that, on a finite time interval, the trajectories of the slow variables can be well approximated by those of a system with reduced dimension as the singular perturbation parameter becomes small. In particular, we show that when this parameter becomes small the first and second moments of the reduced system's variables closely approximate the first and second moments, respectively, of the slow variables of the singularly perturbed system. Chemical Langevin equations describing the stochastic dynamics of molecular systems with linear propensity functions including both fast and slow reactions fall within the class of SDEs considered here. We therefore illustrate the goodness of our approximation on a simulation example modeling a well known biomolecular system with fast and slow processes.United States. Air Force Office of Scientific Research (Grant FA9550-14-1-0060)National Institute of General Medical Sciences (U.S.) (Grant P50 GMO9879

Deterministic-Like Model Reduction for a Class of Multi-Scale Stochastic Differential Equations with Application to Biomolecular Systems

We consider a class of singularly perturbed stochastic differential equations with linear drift terms, and present a reduced-order model that approximates both slow and fast variable dynamics when the time-scale separation is large. We show that, on a finite time interval, moments of all orders of the slow variables for the original system become closer to those of the reduced-order model as time-scale separation is increased. A similar result holds for the first and second moments of the fast variable approximation. Biomolecular systems with linear propensity functions, modeled by the chemical Langevin equation fit the class of systems considered in this work. Thus, as an application example, we analyze the trade-offs between noise and information transmission in a typical gene regulatory network motif, for which both slow and fast variables are required.United States. Air Force. Office of Scientific Research (grant # FA9550-14-1-0060

Model order reduction for Linear Noise Approximation using time-scale separation

In this paper, we focus on model reduction of biomolecular systems with multiple time-scales, modeled using the Linear Noise Approximation. Considering systems where the Linear Noise Approximation can be written in singular perturbation form, with ε as the singular perturbation parameter, we obtain a reduced order model that approximates the slow variable dynamics of the original system. In particular, we show that, on a finite time-interval, the first and second moments of the reduced system are within an O(ε)-neighborhood of the first and second moments of the slow variable dynamics of the original system. The approach is illustrated on an example of a biomolecular system that exhibits time-scale separation.United States. Air Force. Office of Scientific Research (grant # FA9550-14-1-0060

Reduced Linear Noise Approximation for Biochemical Reaction Networks with Time-Scale Separation: The Stochastic tQSSA⁺

Biochemical reaction networks often involve reactions that take place on different time scales, giving rise to "slow" and "fast" system variables. This property is widely used in the analysis of systems to obtain dynamical models with reduced dimensions. In this paper, we consider stochastic dynamics of biochemical reaction networks modeled using the Linear Noise Approximation (LNA). Under time-scale separation conditions, we obtain a reduced-order LNA that approximates both the slow and fast variables in the system. We mathematically prove that the first and second moments of this reduced-order model converge to those of the full system as the time-scale separation becomes large. These mathematical results, in particular, provide a rigorous justification to the accuracy of LNA models derived using the stochastic total quasi-steady state approximation (tQSSA). Since, in contrast to the stochastic tQSSA, our reduced-order model also provides approximations for the fast variable stochastic properties, we term our method the "stochastic tQSSA⁺". Finally, we demonstrate the application of our approach on two biochemical network motifs found in gene-regulatory and signal transduction networks.United States. Air Force Office of Scientific Research (Grant FA9550-14-1-0060