Abstract
In order to analyse large complex stochastic dynamical models such as those studied in systems biology there is currently a great need for both analytical tools and also algorithms for accurate and fast simulation and estimation. We present a new stochastic approximation of biological oscillators that addresses these needs. Our method, called phase-corrected LNA (pcLNA) overcomes the main limitations of the standard Linear Noise Approximation (LNA) to remain uniformly accurate for long times, still maintaining the speed and analytically tractability of the LNA. As part of this, we develop analytical expressions for key probability distributions and associated quantities, such as the Fisher Information Matrix and Kullback-Leibler divergence and we introduce a new approach to system-global sensitivity analysis. We also present algorithms for statistical inference and for long-term simulation of oscillating systems that are shown to be as accurate but much faster than leaping algorithms and algorithms for integration of diffusion equations. Stochastic versions of published models of the circadian clock and NF-κB system are used to illustrate our results.
Highlights
Dynamic cellular oscillating systems such as the cell cycle, circadian clock and other signaling and regulatory systems have complex structures, highly nonlinear dynamics and are subject to both intrinsic and extrinsic stochasticity
There is an extensive theory for perfectly noise-free dynamical systems and very effective algorithms for simulating their temporal behaviour
In this article we describe how to accurately approximate such systems in a way that facilitates fast simulation, parameter estimation and new approaches to analysis, such as calculating probability distributions that describe the system’s stochastic behaviour and describing how these distributions change when the parameters of the system are varied
Summary
Dynamic cellular oscillating systems such as the cell cycle, circadian clock and other signaling and regulatory systems have complex structures, highly nonlinear dynamics and are subject to both intrinsic and extrinsic stochasticity. The stochastic kinetics that arise due to random births, deaths and interactions of individual species give rise to Markov jump processes that, in principle, can be analyzed by means of master equations These are rarely tractable and an exact numerical simulation algorithm is available [1], for the large systems we are interested in, this is very slow. The large system size validity of the LNA has been shown in [6], in the sense that the distribution of the Markov jump process at a fixed finite time converges, as the system size O tends to 1, to the LNA probability distribution The latter distribution is analytically tractable allowing for fast estimation and simulation algorithms. The LNA has significant limitations, in approximating long-term behaviour of oscillatory systems
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