Noise control for molecular computing

Chemical Systems with Limit Cycles

The dynamics of a chemical reaction network (CRN) is often modeled under the assumption of mass action kinetics by a system of ordinary differential equations (ODEs) with polynomial right-hand sides that describe the time evolution of concentrations of chemical species involved. Given an arbitrarily large integer [Formula: see text], we show that there exists a CRN such that its ODE model has at least K stable limit cycles. Such a CRN can be constructed with reactions of at most second-order provided that the number of chemical species grows linearly with K. Bounds on the minimal number of chemical species and the minimal number of chemical reactions are presented for CRNs with K stable limit cycles and at most second order or seventh-order kinetics. We also show that CRNs with only two chemical species can have K stable limit cycles, when the order of chemical reactions grows linearly with K.

Stochastic approximations of higher-molecular by bi-molecular reactions

Reactions involving three or more reactants, called higher-molecular reactions, play an important role in mathematical modelling in systems and synthetic biology. In particular, such reactions underpin a variety of important bio-dynamical phenomena, such as multi-stability/multi-modality, oscillations, bifurcations, and noise-induced effects. However, as opposed to reactions involving at most two reactants, called bi-molecular reactions, higher-molecular reactions are biochemically improbable. To bridge the gap, in this paper we put forward an algorithm for systematically approximating arbitrary higher-molecular reactions with bi-molecular ones, while preserving the underlying stochastic dynamics. Properties of the algorithm and convergence are established via singular perturbation theory. The algorithm is applied to a variety of higher-molecular biochemical networks, and is shown to play an important role in synthetic biology.

Quasi-robust control of biochemical reaction networks via stochastic morphing

One of the main objectives of synthetic biology is the development of molecular controllers that can manipulate the dynamics of a given biochemical network that is at most partially known. When integrated into smaller compartments, such as living or synthetic cells, controllers have to be calibrated to factor in the intrinsic noise. In this context, biochemical controllers put forward in the literature have focused on manipulating the mean (first moment) and reducing the variance (second moment) of the target molecular species. However, many critical biochemical processes are realized via higher-order moments, particularly the number and configuration of the probability distribution modes (maxima). To bridge the gap, we put forward the stochastic morpher controller that can, under suitable timescale separations, morph the probability distribution of the target molecular species into a predefined form. The morphing can be performed at a lower-resolution, allowing one to achieve desired multi-modality/multi-stability, and at a higher-resolution, allowing one to achieve arbitrary probability distributions. Properties of the controller, such as robustness and convergence, are rigorously established, and demonstrated on various examples. Also proposed is a blueprint for an experimental implementation of stochastic morpher.

Stationary distributions of systems with discreteness-induced transitions

We provide a theoretical analysis of some autocatalytic reaction networks exhibiting the phenomenon of discreteness-induced transitions. The family of networks that we address includes the celebrated Togashi and Kaneko model. We prove positive recurrence, finiteness of all moments and geometric ergodicity of the models in the family. For some parameter values, we find the analytic expression for the stationary distribution and discuss the effect of volume scaling on the stationary behaviour of the chain. We find the exact critical value of the volume for which discreteness-induced transitions disappear.