Deterministic characterization of stochastic genetic circuits

Abstract

For cellular biochemical reaction systems where the numbers of molecules is small, significant noise is associated with chemical reaction events. This molecular noise can give rise to behavior that is very different from the predictions of deterministic rate equation models. Unfortunately, there are few analytic methods for examining the qualitative behavior of stochastic systems. Here we describe such a method that extends deterministic analysis to include leading-order corrections due to the molecular noise. The method allows the steady-state behavior of the stochastic model to be easily computed, facilitates the mapping of stability phase diagrams that include stochastic effects, and reveals how model parameters affect noise susceptibility in a manner not accessible to numerical simulation. By way of illustration we consider two genetic circuits: a bistable positive-feedback loop and a negativefeedback oscillator. We find in the positive feedback circuit that translational activation leads to a far more stable system than transcriptional control. Conversely, in a negative-feedback loop triggered by a positive-feedback switch, the stochasticity of transcriptional control is harnessed to generate reproducible oscillations.intrinsic noise ͉ phase diagram ͉ synthetic biology ͉ oscillations P arallel advances in the conceptual understanding of gene regulation along with technological advances in molecular biology have given rise to the possibility of system-level quantitative kinetic measurements of living organisms (1) and synthetic genetic circuit designs (2, 3). Interpretation of time-series data from complex networks and reliable forward-design of gene circuits depend on detailed quantitative mathematical models (2, 4). These models generally take one of two largely exclusive forms: either deterministic formulations with reactant concentration varying continuously in time and governed by a system of rate equations or stochastic formulations that explicitly include the discrete and probabilistic change in reactant molecule numbers as each subsequent reaction occurs (5). Both approaches have benefits and associated limitations.The great practical advantage of rate equation models is the ease with which the qualitative behavior of the system can be extracted. By focusing on the long-term behavior, the model dynamics are simplified, and one is able to gain insight into the expected response of the system (6). Rate equation models, however, neglect the fact that chemical reaction networks are composed of species that evolve on discrete space, jumping from some number of molecules to another as each reaction occurs (7). The resulting deviation from the deterministic formulation is called the ''intrinsic noise'' in the system (because the fluctuations arise from the reaction dynamics themselves and not from some external source) (8, 9). In cellular systems with small numbers of reactant molecules, the relative magnitude of the intrinsic noise can be large, and it can give rise to qualitatively different behavior than...

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