Abstract
Studies of stochastic biological dynamics typically compare observed fluctuations to theoretically predicted variances, sometimes after separating the intrinsic randomness of the system from the enslaving influence of changing environments. But variances have been shown to discriminate surprisingly poorly between alternative mechanisms, while for other system properties no approaches exist that rigorously disentangle environmental influences from intrinsic effects. Here, we apply the theory of generalized random walks in random environments to derive exact rules for decomposing time series and higher statistics, rather than just variances. We show for which properties and for which classes of systems intrinsic fluctuations can be analyzed without accounting for extrinsic stochasticity and vice versa. We derive two independent experimental methods to measure the separate noise contributions and show how to use the additional information in temporal correlations to detect multiplicative effects in dynamical systems.
Highlights
Studies of stochastic biological dynamics typically compare observed fluctuations to theoretically predicted variances, sometimes after separating the intrinsic randomness of the system from the enslaving influence of changing environments
Relating intrinsic and extrinsic contributions to separately defined Áx and x" allows us to design alternative experimental strategies to measure the same quantities (Fig. 3): assuming independence between reporters, the sum of many independent copies of the same type of reporter provides a direct estimate for the time series of the conditional average x", while any intrinsic statistical property can be observed with just one additional distinct reporter, whose deviations from x" give Áx
; . . .g, for k where 1⁄4 1; 2; the ..., system which undergoes is true for many models of stochastic gene expression, the dynamics of the ensemble average x" ðtÞ follows exactly from a classic rate equation approach, employing ordinary differential equations subject to time-varying rate constants set by the extrinsic processes [9]
Summary
Studies of stochastic biological dynamics typically compare observed fluctuations to theoretically predicted variances, sometimes after separating the intrinsic randomness of the system from the enslaving influence of changing environments. We identify intrinsic and extrinsic contributions to time correlations and higher moments, show how they can be determined from either of two types of experimental approaches, and establish how they relate to mechanistic properties.
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