Path integrals and fluctuations in irreversible thermodynamics

Abstract

We express the set of stochastic differential equations which describe fluctuations in linear irreversible thermodynamics in terms of path integrals. The stochastic terms which are added to the linearized macroscopic equations have a correlation matrix that is singular, which implies that the straightforward formulation of the problem in terms of path integrals fails. We therefore begin by constructing a path-integral representation which is valid whether or not the correlation matrix is singular. We apply this to linearized irreversible thermodynamics, but the technique is designed to be applicable to more general versions of the theory. The approach emphasizes the role of the response and correlation functions as basic elements of the theory, and we calculate these quantities explicitly for the case of density fluctuations in a fluid.

Stochastic single-molecule dynamics of synaptic membrane protein domains

“…In this letter we show that key features of the observed stochastic dynamics of synaptic domains can be understood in terms of a simple stochastic lattice model of receptor and scaffold reaction-diffusion processes at the membrane [30,31], and thereby demonstrate emergence of synaptic domains in the presence of rapid stochastic turnover of individual molecules. Our stochastic lattice model yields excellent agreement with mean-field models [30,31,[39][40][41][42][43] of nonlinear diffusion in crowded membranes, but we find substantial discrepancies between mean-field and stochastic models for the reaction dynamics at synaptic domains. Kinetic Monte Carlo (KMC) simulations of our stochastic lattice model yield, in agreement with previous experiments and mean-field calculations [21,[23][24][25][26][27][28][29][30][31], spontaneous formation of synaptic domains, and demonstrate that the molecular noise inp-1 arXiv:1610.09536v2 [q-bio.SC] 7 Nov 2016 duced by the underlying reaction and diffusion dynamics of synaptic receptors and scaffolds can produce collective fluctuations in synaptic domains [22,33].…”

“…(4) and (5) result [30,31] from crowding of distinct protein species. Mathematically equivalent terms arise in population biology [39][40][41] and general models of non-Fickian diffusion [42,43]. In line with experiments and large-scale computer simulations of crowded membranes [51,52], the nonlinear diffusion terms in eqs.…”

“…(2) and (3), where r(x, t) and s(x, t) are the deterministic continuum fields associated with N α , the receptor and scaffold diffusion coefficients ν α = a 2 / (2τ α ), and the polynomials F α (r, s) describe the mean-field reaction dynamics [55][56][57]. Table 1 summarizes the additive contributions to F α (r, s) implied by the standard formalism of chemical dynamics [40,41,[55][56][57] for the reaction kinetics considered here [30,31]. Unless indicated otherwise we use, consistent with experiments on glycine receptors and gephyrin scaffolds [5,11,17,18,21,25,30,31], the diffusion coefficients ν r = 10 2 ν s = 10 −2 µm 2 /s, with the corresponding hopping rates 1/τ α = 2ν α /a 2 in eq.…”

“…-Membrane protein domains are characterized [1-5, 8, 9, 11, 17, 18, 22, 33] by low protein copy numbers (≈ 10-1000) while also providing highly crowded environments for reaction-diffusion processes to occur. We employ [30,31,[39][40][41][42][43][46][47][48][49] a stochastic lattice model of synaptic domains in which we divide the cell membrane into equal-sized patches (lattice sites) with reaction processes only occurring between receptors (R) and scaffolds (S) occupying the same membrane patch. For the sake of conceptual and computational simplicity, we focus here on the most straightforward scenario of a 1D system of length L with periodic boundary conditions and K patches of size a = L/K, and allow receptors and scaffolds to hop randomly to nearestneighbor patches with hopping rates 1/τ α , where α = r, s for receptors and scaffolds, respectively.…”

“…Cell membranes provide highly crowded and heterogeneous molecular environments, which can strongly affect protein reaction kinetics and give rise to anomalous diffusion of membrane proteins [51,52]. Based on previous work [30,31,[39][40][41][42][43] on reaction and diffusion processes in crowded environments, we use here a phenomenological model of crowding and assume that the rates of reaction and diffusion processes locally increasing the receptor or scaffold number are ∝ (1 − N r i − N s i ) for each site i, where N α i / are the occupation numbers of receptors and scaffolds at site i so that 0 ≤ N r i +N s i ≤ 1 and each membrane patch can accommodate up to 1/ receptors or scaffolds. The physically relevant values of the normalization constant are coupled to the patch size a and the size of the molecules under consideration, with decreasing with increasing a and decreasing molecule size.…”