Stochastic Pattern Formation and Spontaneous Polarisation: The Linear Noise Approximation and Beyond

Abstract

We review the mathematical formalism underlying the modelling of stochasticity in biological systems. Beginning with a description of the system in terms of its basic constituents, we derive the mesoscopic equations governing the dynamics which generalise the more familiar macroscopic equations. We apply this formalism to the analysis of two specific noiseinduced phenomena observed in biologically-inspired models. In the first example, we show how the stochastic amplification of a Turing instability gives rise to spatial and temporal patterns which may be understood within the linear noise approximation. The second example concerns the spontaneous emergence of cell polarity, where we make analytic progress by exploiting a separation of time-scales.

Motif analysis for small-number effects in chemical reaction dynamics

“…26 As is discussed in Introduction, recent numerical studies support the validity of chemical Fokker-Planck equation up to smaller value of Ω than the range adopted for LNA. 7,31,32 In fact, we also demonstrated that the direct numerical simulations agree quite well with the predictions of chemical Fokker-Planck equation for all motifs marked by "+" or "++" in Fig. 1, where small-number effects are well reproduced by the chemical Fokker-Planck equation, rather than by the LNA.…”

Exploiting Fast-Variables to Understand Population Dynamics and Evolution

“…For further background on how to arrive precisely at the forms given by Eq. (3) the reader is referred to previous discussions in the literature [1,6,43]. If W (1) and W (2) are independent of n, then the selection is known as frequency independent selection.…”

“…In a previous paper [43], we have reviewed the process of setting-up a description of this class of biological systems in terms of its basic constituents, and from this deriving the mesoscopic equations governing the dynamics which generalise what might be the more familiar macroscopic equations. In particular, in Ref.…”

“…In particular, in Ref. [43], we give formulae for writing down the form of the mesoscopic dynamics in terms of quantities which appear in the microscopic formulation. This essentially is the first point of our methodology described above, and so while we will discuss it here, we will refer the reader to this earlier paper for more details.…”

Modes of competition and the fitness of evolved populations

“…For instance, new effective extrema may be formed through the effects of noise [36], as well as the effective location of extrema being moved. In the present case, the effect is more closely related to the situation found in some kinds of stochastic pattern formation [37,38], where the region of stability of patterns can be increased by the stochastic effects giving a new effective condition for stability which differs from that found through a purely deterministic analysis.…”