Stochastic lattice model of synaptic membrane protein domains

Neurotransmitter receptor molecules, concentrated in synaptic membrane domains along with scaffolds and other kinds of proteins, are crucial for signal transmission across chemical synapses. In common with other membrane protein domains, synaptic domains are characterized by low protein copy numbers and protein crowding, with rapid stochastic turnover of individual molecules. We study here in detail a stochastic lattice model of the receptor-scaffold reaction-diffusion dynamics at synaptic domains that was found previously to capture, at the mean-field level, the self-assembly, stability, and characteristic size of synaptic domains observed in experiments. We show that our stochastic lattice model yields quantitative agreement with mean-field models of nonlinear diffusion in crowded membranes. Through a combination of analytic and numerical solutions of the master equation governing the reaction dynamics at synaptic domains, together with kinetic Monte Carlo simulations, we find substantial discrepancies between mean-field and stochastic models for the reaction dynamics at synaptic domains. Based on the reaction and diffusion properties of synaptic receptors and scaffolds suggested by previous experiments and mean-field calculations, we show that the stochastic reaction-diffusion dynamics of synaptic receptors and scaffolds provide a simple physical mechanism for collective fluctuations in synaptic domains, the molecular turnover observed at synaptic domains, key features of the observed single-molecule trajectories, and spatial heterogeneity in the effective rates at which receptors and scaffolds are recycled at the cell membrane. Our work sheds light on the physical mechanisms and principles linking the collective properties of membrane protein domains to the stochastic dynamics that rule their molecular components.

Figure 1

Schematic of the stochastic lattice model of the reaction dynamics of synaptic receptor and scaffold molecules [36, 37, 62] considered in this article. To model molecular crowding, the rates of all reaction and diffusion processes increasing the molecule number at a given lattice site $i$, delineated by vertical ticks, are taken to be $\propto \left(1-N_i^r-N_i^s\right)$. The transition rates associated with the indicated reaction processes are given by Eq. (6) with Eqs. (15, 16, 17, 18, 19, 20, 21, 22, 23).

Figure 2

Receptor and scaffold diffusion with ${\nu }_r={\nu }_s\equiv \nu =0.01\mu {\mathrm{m}}^2/\mathrm{s}$. (a) Molecule occupancies of receptors and scaffolds at $t=10$ s starting from the initial conditions shown in the inset, obtained from KMC simulations of the ME Eq. (1) with $\epsilon =1/40$ and $a=0.05\mu \mathrm{m}$, the mean-field Eqs. (24) and (25) modeling diffusion under exclusion constraints, and the standard (Fickian) diffusion equations for receptors and scaffolds, which are given by the linear terms in Eqs. (24) and (25). We set $L=50\mu \mathrm{m}$. (b) Receptor profiles as in panel (a), but using $\epsilon =1$, 1/2, 1/8, and 1/40 for the KMC simulations of the ME Eq. (1). The KMC simulations were averaged over 2000 independent realizations each.

Figure 3

MSD in diffusion-only systems with ${\nu }_r={\nu }_s\equiv \nu =0.01\mu {\mathrm{m}}^2/\mathrm{s}$. (a) Receptor MSD obtained from KMC simulations as in Fig. 2 but using $L=5\mu \mathrm{m}$, for receptors located initially at the positions marked by circles and squares in the inset. The black dashed line shows the MSD implied by standard (Fickian) diffusion with no steric constraints, and the red and magenta dashed lines show the MSD for free diffusion scaled by $\left(1-\langle N_i^r+N_i^s\rangle \right)$, where $\langle N_i^r+N_i^s\rangle =0.2$ is the average molecule occupancy in the system. (b) Receptor MSD obtained as in panel (b), but for a system composed of only receptors and starting from homogeneous receptor distributions with $\langle N_i^r\rangle =0.1,0.3$, and 0.6. The MSD curves were obtained by averaging over ${10}^4$ molecule trajectories each.

Figure 4

Profiles of (a) receptor and scaffold occupancies and (b) the total receptor and scaffold occupancy, obtained by adding up the curves in panel (a), at $t=10$ s for ${\nu }_r/{\nu }_s=2,8,32$ for the left, middle, and right panels, respectively, calculated from KMC simulations of the ME Eq. (1) and the mean-field Eqs. (24) and (25) as in Fig. 2 using ${\nu }_s=0.01\mu {\mathrm{m}}^2/\mathrm{s}$ and $\epsilon =1/40$. The KMC simulations were averaged over 2000 independent realizations each.

Figure 5

MSD curves of (a) diffusing receptors and (b) diffusing scaffolds obtained from KMC simulations as in Fig. 2 with $\epsilon =1/40$ but using $L=5\mu \mathrm{m}$ with ${\nu }_r=0.08\mu {\mathrm{m}}^2/\mathrm{s}$ and ${\nu }_s=0.01\mu {\mathrm{m}}^2/\mathrm{s}$, for receptors and scaffolds located initially at the positions marked by triangles, squares, and circles in the initial molecule distributions shown in the lower-right insets. The upper-left insets show the long-term evolution of the MSD curves. As in Fig. 3, the black dashed lines indicate the MSDs implied by standard (Fickian) diffusion with no steric constraints, and the magenta dashed lines show the MSDs for free diffusion scaled by $\left(1-\langle N_i^r+N_i^s\rangle \right)$, where $\langle N_i^r+N_i^s\rangle =0.2$ is the average molecule occupancy in the system. The MSD curves were obtained by averaging over ${10}^4$ molecule trajectories each.

Figure 6

Average scaffold occupancies for (a) $S_b\stackrel{k_7}{\to }S$ with $N_0^s=0$, (b) $S_b+S\stackrel{{\overline{k}}_8}{\to }2S$ with $N_0^s=0.2$, and (c) $S_b+2S\stackrel{k_9}{\to }3S$ with $N_0^s=0.1$ obtained from the mean-field Eqs. (29, 30, 31), the general analytic solutions of the MEs Eqs. (32)–(34) for the mean jump times in Eq. (40) with Eqs. (36)–(38), and KMC simulations of the MEs Eqs. (32)–(34). We set $k_7={\overline{k}}_8=k_9=1{\mathrm{s}}^{-1}$. The KMC simulations were averaged over ${10}^5$ independent realizations each.

Figure 7

(a) Average scaffold occupancies for the reaction scheme $S\stackrel{k_6}{\to }{S}_b$ and $S_b+S\stackrel{k_9}{\to }2S$ obtained from the mean-field Eq. (41) and KMC simulations of the ME Eq. (42). The inset shows the average negative slopes of the average scaffold occupancies obtained from the ME Eq. (42) for $\epsilon =1/400$, 1/300, 1/200, and 1/100, estimated from KMC data starting at the global maxima of the scaffold occupancies. The KMC simulations were averaged over 2000 independent realizations each. (b) Selected individual KMC trajectories for $\epsilon =1/100$ and corresponding mean-field solution as in panel (a). We set $k_6=1{\mathrm{s}}^{-1}$ and $k_9=1.3{\mathrm{s}}^{-1}$, and used an initial scaffold occupancy $N_0^s=0.04$.

Figure 8

Scaffold occupancies for the reaction scheme $M_b+S\stackrel{k_8}{\to }{M}_b+S_b$ and $S_b+S\stackrel{{\overline{k}}_8}{\to }2S$ obtained from the mean-field Eq. (43) and KMC simulations or direct numerical solutions of the ME Eq. (44) for (a) $k_8={\overline{k}}_8=1.0{\mathrm{s}}^{-1}$ and (b) $k_8=1.0{\mathrm{s}}^{-1}$ and ${\overline{k}}_8=1.1{\mathrm{s}}^{-1}$. The left panels show mean scaffold occupancies, with the KMC simulations averaged over $2\times {10}^4$ independent realizations each, and the right panels show selected individual KMC trajectories of the system, together with the corresponding mean-field results. We use the same labeling conventions in panel (b) as in panel (a).

Figure 9

Fluctuating steady states. Average scaffold occupancies for (a) $S\stackrel{k_6}{\to }{S}_b$ and $S_b\stackrel{k_7}{\to }S$ obtained from the mean-field Eq. (45), the steady-state distribution of the ME Eq. (46) in Eq. (54), and KMC simulations of the ME Eq. (46), and (b,c) $2S\stackrel{k}{\to }2S_b$ and $S_b\stackrel{k_7}{\to }S$ obtained from the mean-field Eq. (55), direct numerical (steady-state) solutions of the ME Eq. (57), and KMC simulations of the ME Eq. (57). In panel (c), the steady-state scaffold occupancy in Eq. (56) implied by the mean-field Eq. (55), $s\approx 0.095$, is indicated by a horizontal dashed line. For panel (a) we set $k_6=1{\mathrm{s}}^{-1}$ and $k_7=2{\mathrm{s}}^{-1}$, and for panels (b) and (c) we set $k=100{\mathrm{s}}^{-1}$ and $k_7=1{\mathrm{s}}^{-1}$. For all panels we used an initial scaffold occupancy $N_0^s=s_0=0$. All KMC simulations were averaged over $2\times {10}^4$ independent realizations each.

Figure 10

Average (a) receptor and (b) scaffold occupancies in a reaction-only system with the reaction kinetics at synaptic domains described in Sec. 2 (see Table 1) [36, 37, 62], obtained from direct numerical solutions of the ME Eq. (1), KMC simulations of the ME Eq. (1), and the mean-field Eqs. (10) and (11) versus scaled time $\tilde{t}=t/\tau$, with $\tau =1.0\times {10}^4$ s and $\tau =5.0\times {10}^5$ s for the stochastic and mean-field models, respectively. The inset in panel (b) shows how the average scaffold occupancies implied by the ME Eq. (1) change with $\epsilon$. All KMC simulations were averaged over $2\times {10}^4$ independent realizations each.

Figure 11

Individual stochastic trajectories of (a) receptor and (b) scaffold occupancies in a reaction-only system with the reaction kinetics at synaptic domains described in Sec. 2 (see Table 1) [36, 37, 62], obtained from the KMC data in Fig. 10.

Figure 12

Receptor-scaffold correlation function in Eq. (58) for (a) $\epsilon =1/100$, (b) $\epsilon =1/200$, and (c) $\epsilon =1/500$ in a reaction-only system with the reaction kinetics at synaptic domains described in Sec. 2 (see Table 1) [36, 37, 62], obtained from KMC simulations of the ME Eq. (1) as in Fig. 10. The yellow and green curves show $C_{r,s}\left(\overline{t}\right)$ computed for single KMC trajectories, while the black curves show the average $C_{r,s}\left(\overline{t}\right)$ obtained, as in Fig. 10, from $2\times {10}^4$ independent realizations each. The vertical dashed lines in the main panels and insets indicate the locations of the global maxima of $C_{r,s}\left(\overline{t}\right)$ at $\overline{t}=T_{\max }$.

Figure 13

Statistical properties of the steady-state receptor and scaffold occupancies implied by the stochastic reaction dynamics at synaptic domains described in Sec. 2 (see Table 1) [36, 37, 62]. (a) Marginal steady-state probability distributions of the receptor occupancy for the indicated values of $\epsilon$. (b) Average (left panel) and mode (right panel) of the steady-state receptor and scaffold occupancies versus $1/\epsilon$, obtained from the respective marginal steady-state probability distributions. The gray regions show the approximate range of $\epsilon$ for which the receptor and scaffold probability distributions are bistable. For such values of $\epsilon$, we use the global maximum as the mode. The steady-state values of $r$ and $s$ implied by the mean-field Eqs. (10) and (11), $r=s=0.05$, are indicated by a vertical dashed line in panel (a) and by horizontal dashed lines in panel (b). All KMC simulations were averaged, from $t_{\min }=1.0\times {10}^5$ s to $t_{\max }=4.5\times {10}^5$ s, over ${10}^4$ independent realizations each.

Figure 14

Occurrence probabilities of individual chemical reactions for the reaction dynamics at synaptic domains in Sec. 2 (see Table 1) [36, 37, 62] versus $1/\epsilon$. The occurrence probabilities were computed at steady state using KMC simulations of the ME Eq. (1) by calculating, for the time interval $[t_{\min },t_{\max }]$ with $t_{\min }=1.0\times {10}^5$ s and $t_{\max }=2.0\times {10}^5$ s, the ratio of the occurrence number of a particular reaction and the total occurrence number of all the (receptor and scaffold) reactions in the system. The KMC simulations were averaged over ${10}^4$ independent realizations.

Figure 15

Self-assembly of synaptic domains. (a) Synaptic domains obtained from the ME Eq. (1) via KMC simulation (upper panels) and the mean-field Eqs. (10) and (11) (lower panels) with the reaction-diffusion dynamics described in Sec. 2 using $\epsilon =1/100$. The left and right panels show the receptor and scaffold occupancies with maximum occupancies $(N_i^r,N_i^s)=(0.79,0.55)$ (KMC) and $(r,s)=(0.46,0.15)$ (mean field). The curves in the upper panels delineate the domain boundaries obtained using a threshold ${\overline{N}}^s=0.08$ on the scaffold occupancy of membrane patches. (b) Average receptor and scaffold numbers per synaptic domain and average number of domains in the system in the upper panels of (a) versus ${\overline{N}}^s$. The shaded blue (red) area shows the standard deviation of the receptor (scaffold) number per synaptic domain, while the shaded green area shows the minimum and maximum of the domain number. All data was extracted from the KMC results shown in the upper panels of (a).

Figure 16

Fluctuating receptor and scaffold numbers in synaptic domains. (a) Receptor and scaffold numbers for the synaptic domain delineated by black domain boundaries in Fig. 15 versus time using ${\overline{N}}^s=0.08$. The horizontal dashed lines and shaded areas indicate the averages and standard deviations of the receptor and scaffold numbers per synaptic domain, which we obtained from the domains in the upper panels of Fig. 15. (b) Receptor-scaffold correlation function $C_{r,s}^D\left(\overline{t}\right)$ in Eq. (59) for the domain delineated by black domain boundaries in Fig. 15 versus correlation time $\overline{t}$, using $t_{\min }=1$ h and $t_{\max }=40$ h. The vertical dashed lines in the main panel and inset in panel (b) correspond to $\overline{t}=0$.

Figure 17

Occurrence probabilities of receptor and scaffold reactions across synaptic domains. (a) Average receptor and scaffold profiles for the synaptic domain delineated by black domain boundaries in Fig. 15. (b) Average densities of reactions inserting (removing) receptors and scaffolds into (from) the membrane, ${\varphi }_{R,S}^{+}$ (${\varphi }_{R,S}^{-}$). The average width of the domain is indicated by shaded regions. (c) Average reaction densities as in panel (b), but for each individual reaction in the reaction-diffusion model considered here (see Sec. 2). All the results in panels (b) and (c) were obtained from the domain delineated by black domain boundaries in Fig. 15, by averaging from $t=1$ h to $t=40$ h in Fig. 15. The average reaction densities were normalized by the total number of receptor and scaffold reactions that occurred for the spatial region and time interval considered here.

Figure 18

Fractions of unlabeled receptors and scaffolds versus time, obtained from the synaptic domain delineated by black domain boundaries in Fig. 15 by labeling all the receptors and scaffolds initially localized inside the synaptic domain. Averages were taken over ten realizations corresponding to different times in Fig. 15.

Figure 19

Receptor trafficking along the membrane between the inside and outside of synaptic domains. (a) Representative trajectories of individual receptors (blue and orange curves) inserted inside the synaptic domain delineated by black domain boundaries in Fig. 15. Receptors are tracked at the membrane from insertion until removal. Domain boundaries are indicated by green curves. (b) Receptor switching times between the inside (in) and outside (out) of synaptic domains for the diffusion trajectories along the membrane in panel (a).