Stochastic Turing Pattern Formation in a Model with Active and Passive Transport
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Stochastic Turing Pattern Formation in a Model with Active and Passive Transport Hyunjoong Kim1 · Paul C. Bressloff1 Received: 18 June 2020 / Accepted: 20 October 2020 © Society for Mathematical Biology 2020
Abstract We investigate Turing pattern formation in a stochastic and spatially discretized version of a reaction–diffusion–advection (RDA) equation, which was previously introduced to model synaptogenesis in C. elegans. The model describes the interactions between a passively diffusing molecular species and an advecting species that switches between anterograde and retrograde motor-driven transport (bidirectional transport). Within the context of synaptogenesis, the diffusing molecules can be identified with the protein kinase CaMKII and the advecting molecules as glutamate receptors. The stochastic dynamics evolves according to an RDA master equation, in which advection and diffusion are both modeled as hopping reactions along a one-dimensional array of chemical compartments. Carrying out a linear noise approximation of the RDA master equation leads to an effective Langevin equation, whose power spectrum provides a means of extending the definition of a Turing instability to stochastic systems, namely in terms of the existence of a peak in the power spectrum at a nonzero spatial frequency. We thus show how noise can significantly extend the range over which spontaneous patterns occur, which is consistent with previous studies of RD systems.
1 Introduction One major mechanism for self-organization within cells and between cells is the interplay between diffusion and nonlinear chemical reactions. Historically speaking, the idea that a reaction–diffusion (RD) system can spontaneously generate spatiotemporal patterns was first introduced by Turing in his seminal 1952 paper (Turing 1952). Turing considered the general problem of how organisms develop their structures during the growth from embryos to adults. He established the principle that two nonlinearly interacting chemical species differing significantly in their rates of diffusion can amplify spatially periodic fluctuations in their concentrations, resulting in the formation of a
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Paul C. Bressloff [email protected] Department of Mathematics, University of Utah, Salt Lake City, UT 84112, USA 0123456789().: V,-vol
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stable periodic pattern. The Turing mechanism for morphogenesis was subsequently refined by Gierer and Meinhardt (1972), who showed that one way to generate a Turing instability is to have an antagonistic pair of molecular species known as an activator– inhibitor system, which consists of a slowly diffusing chemical activator and a quickly diffusing chemical inhibitor. Over the years, the range of models and applications of the Turing mechanism has expanded dramatically (Murray 2008; Cross and Greenside 2009; Walgraef 1997). Motivated by experimental studies of synaptogenesis in Caenorhabditis elegans (Rongo and Kaplan 1999; Hoerndli et al. 2013, 2015), we recently introduced a reaction–d
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