Noise source importance in linear stochastic models of biological systems that grow, shrink, wander, or persist

Abstract

While noise is an important factor in biology, biological processes often involve multiple noise sources, whose relative importance can be unclear. Here we develop tools that quantify the importance of noise sources in a network based on their contributions to variability in a quantity of interest. We generalize the edge importance measures proposed by Schmidt and Thomas [1] for first-order reaction networks whose steady-state variance is a linear combination of variance produced by each directed edge. We show that the same additive property extends to a general family of stochastic processes subject to a set of linearity assumptions, whether in discrete or continuous state or time. Our analysis applies to both expanding and contracting populations, as well as populations obeying a martingale ("wandering") at long times. We show that the original Schmidt-Thomas edge importance measure is a special case of our more general measure, and is recovered when the model satisfies a conservation constraint ("persists"). In the growing and wandering cases we show that the choice of observables (measurements) used to monitor the process does not influence which noise sources are important at long times. In contrast, in the shrinking or persisting case, which noise sources are important depends on what is measured. We also generalize our measures to admit models with affine moment update equations, which admit additional limiting scenarios, and arise naturally after linearization. We illustrate our results using examples from cell biology and ecology: (i) a model for the dynamics of the inositol trisphospate receptor, (ii) a model for an endangered population of white-tailed eagles, and (iii) a model for wood frog dispersal. Author summaryBiological processes are frequently subject to an ensemble of independent noise sources. Noise sources produce fluctuations that propagate through the system, driving fluctuations in quantities of interest such as population size or ion channel configuration. We introduce a measure that quantifies how much variability each noise source contributes to any given quantity of interest. Using these methods, we identify which binding events contribute significantly to fluctuations in the state of a molecular signalling channel, which life history events contribute the most variability to an eagle population before and after a successful conservation effort rescued the population from the brink of extinction, and which dispersal events, at what times, matter most to variability in the recolonization of a series of ponds by wood frogs after a drought.