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.

中文翻译：

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具有主动和被动传输的模型中的随机图灵模式形成

我们在反应-扩散-平流 (RDA) 方程的随机和空间离散版本中研究图灵模式的形成，该方程之前被引入线虫中的突触发生模型。该模型描述了被动扩散的分子物种与在顺行和逆行电机驱动传输（双向传输）之间切换的平流物种之间的相互作用。在突触发生的背景下，扩散分子可以与蛋白激酶 CaMKII 和平流分子识别为谷氨酸受体。随机动力学根据 RDA 主方程发展，其中对流和扩散都被建模为沿着一维化学隔室阵列的跳跃反应。对 RDA 主方程进行线性噪声近似会得到一个有效的朗之万方程，其功率谱提供了一种将图灵不稳定性的定义扩展到随机系统的方法，即在功率谱中存在峰值非零空间频率。因此，我们展示了噪声如何显着扩展自发模式发生的范围，这与先前对 RD 系统的研究一致。