Stochastic single-molecule dynamics of synaptic membrane protein domains

Motivated by single-molecule experiments on synaptic membrane protein domains, we use a stochastic lattice model to study protein reaction and diffusion processes in crowded membranes. We find that the stochastic reaction-diffusion dynamics of synaptic proteins provide a simple physical mechanism for collective fluctuations in synaptic domains, the molecular turnover observed at synaptic domains, key features of the single-molecule trajectories observed for synaptic proteins, and spatially inhomogeneous protein lifetimes at the cell membrane. Our results suggest that central aspects of the single-molecule and collective dynamics observed for membrane protein domains can be understood in terms of stochastic reaction-diffusion processes at the cell membrane. Introduction. – A variety of essential biological functions of cell membranes rely on the organization of membrane proteins into membrane protein domains [1–4]. Superresolution light microscopy [5–7] of membrane protein domains has shown that molecular diffusion can yield rapid stochastic turnover of individual membrane proteins, with complicated diffusion trajectories arising from molecular crowding and interactions between different protein species. A biologically important example of membrane protein domains is provided by synaptic domains [8, 9], which are crucial for signal transmission across chemical synapses. Synaptic domains are crowded with synaptic receptor and scaffold molecules, and mediate synaptic signaling via transient binding of synaptic receptors to neurotransmitter molecules released from the presynaptic terminal. The strength of the transmitted signal depends on the number of receptors localized in synaptic domains [10,11], and regulation of the receptor number in synaptic domains is one mechanism for postsynaptic plasticity [12–14]. Synaptic domains of a well-defined characteristic size can persist over months or even longer periods of time [15, 16]. However, receptor [5, 17, 18] as well as scaffold [19–21] molecules have been observed [14,22] to turn over rapidly, with individual molecules leaving and entering synaptic domains on typical timescales as short as seconds. Experiments [21, 23–30] and theoretical modeling [30, 31] have shown that the reaction and diffusion properties of synaptic receptors and their associated scaffold molecules are sufficient for the spontaneous formation of synaptic domains, and that self-assembly of synaptic domains of the observed characteristic size can be understood in terms of a reaction-diffusion (Turing) instability [32]. Experiments [22,33] and theoretical modeling [34–38] also suggest that synaptic domains undergo collective fluctuations that may affect synaptic signaling. It is largely unknown how the rapid stochastic dynamics of individual synaptic receptors and scaffolds [11, 17, 18, 25–28] relate [8, 9, 22, 33] to the observed collective properties of synaptic 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, 39–43] 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, 23–31], spontaneous formation of synaptic domains, and demonstrate that the molecular noise inp-1 ar X iv :1 61 0. 09 53 6v 2 [ qbi o. SC ] 7 N ov 2 01 6 Osman Kahraman et al. duced by the underlying reaction and diffusion dynamics of synaptic receptors and scaffolds can produce collective fluctuations in synaptic domains [22, 33]. We show that, based on the reaction and diffusion properties of synaptic receptors and scaffolds suggested by previous experiments and mean-field calculations [21, 23–31], our stochastic lattice model can yield the molecular turnover observed at synaptic domains [5, 11, 21], predicts singlemolecule trajectories consistent with experimental observations [5,11,17,18,22,25–28], and provides a simple physical mechanism for spatially inhomogeneous receptor and scaffold lifetimes at the membrane [38, 44, 45]. Thus, our stochastic lattice model links the molecular noise inherent in receptor-scaffold reaction-diffusion dynamics to collective fluctuations in synaptic domains and allows prediction of the stochastic dynamics of individual synaptic receptors and scaffolds, which cannot be achieved via existing mean-field models [30,31]. While we focus here on synaptic domains as a model system, our main results are of broad applicability [1–5] to the stochastic single-molecule dynamics of membrane protein domains. Reaction-diffusion dynamics. – 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, 39–43, 46–49] 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/τα, where α = r, s for receptors and scaffolds, respectively. We find that, consistent with previous work on stochastic reaction-diffusion models in population biology [41], this 1D formulation of the stochastic reaction-diffusion dynamics at synaptic domains [30,31] already captures the basic phenomenology of the observed fluctuations at synaptic domains. Indeed, the 2D formulation of our model [30,31] shows similar stochastic dynamics of synaptic domains as the 1D formulation we focus on here [50]. While not essential for capturing the basic phenomenology of fluctuations at synaptic domains, a 2D formulation would be required to make more detailed and quantitative comparisons with experimental results, which necessarily pertain to 2D systems. 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, 39–43] 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−Nr i −Ns i ) for each site i, where N i / are the occupation numbers of receptors and scaffolds at site i so that 0 ≤ N i +N 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. We employ identical normalization constants for receptors and scaffolds, but distinct could be used for receptors and scaffolds to provide a more detailed model of molecule number in synaptic domains. Our stochastic lattice model is defined mathematically by its master equation (ME) [53,54],