Lifetime of a structure evolving by cluster aggregation and particle loss, and application to postsynaptic scaffold domains

Vincent Hakim and Jonas Ranft

Phys. Rev. E 101, 012411 – Published 23 January 2020

Abstract

The dynamics of several mesoscopic biological structures depend on the interplay of growth through the incorporation of components of different sizes laterally diffusing along the cell membrane, and loss by component turnover. In particular, a model of such an out-of-equilibrium dynamics has recently been proposed for postsynaptic scaffold domains, which are key structures of neuronal synapses. It is of interest to estimate the lifetime of these mesoscopic structures, especially in the context of synapses where this time is related to memory retention. The lifetime of a structure can be very long as compared to the turnover time of its components and it can be difficult to estimate it by direct numerical simulations. Here, in the context of the model proposed for postsynaptic scaffold domains, we approximate the aggregation-turnover dynamics by a shot-noise process. This enables us to analytically compute the quasistationary distribution describing the sizes of the surviving structures as well as their characteristic lifetime. We show that our analytical estimate agrees with numerical simulations of a full spatial model, in a regime of parameters where a direct assessment is computationally feasible. We then use our approach to estimate the lifetime of mesoscopic structures in parameter regimes where computer simulations would be prohibitively long. For gephyrin, the scaffolding protein specific to inhibitory synapses, we estimate a lifetime longer than several months for a scaffold domain when the single gephyrin protein turnover time is about half an hour, as experimentally measured. While our focus is on postsynaptic domains, our formalism and techniques should be applicable to other biological structures that are also formed by a balance of condensation and turnover.

Received 29 October 2019

DOI:https://doi.org/10.1103/PhysRevE.101.012411

Physics Subject Headings (PhySH)

Memory Nonequilibrium statistical mechanics Stochastic processes Synapses

Physics of Living SystemsStatistical Physics & ThermodynamicsInterdisciplinary Physics

Authors & Affiliations

Vincent Hakim ^1,* and Jonas Ranft ^2,†

¹Laboratoire de Physique de l'Ecole Normale Supérieure, CNRS, Ecole Normale Supérieure, PSL University, Sorbonne Université, Université Paris-Diderot, Paris, France
²Institut de Biologie de l'ENS (IBENS), Ecole Normale Supérieure, PSL University, CNRS, INSERM, Paris, France

^*vincent.hakim@ens.fr
^†jonas.ranft@ens.fr

Article Text (Subscription Required)

Click to Expand

References (Subscription Required)

Click to Expand

Issue

Vol. 101, Iss. 1 — January 2020

Reuse & Permissions

Access Options

Author publication services for translation and copyediting assistance advertisement

Images

Figure 1
Dynamics of diffusing clusters and of an immobile domain of fluctuating size. (a) Schematic description of the dynamics of postsynaptic scaffold clusters and of an immobile scaffold domain. Clusters of scaffolding proteins (orange) diffuse laterally under the membrane, complexed with neurotransmitter receptors (dashed ellipses). Upon encounter, the diffusing clusters can aggregate and grow. They can also encounter an immobile domain (blue) of fluctuating size meant to represent the postsynaptic scaffold domain of a synapse. The growth of the diffusing clusters and of the immobile domain is counterbalanced by the turnover of scaffolding proteins into the cytoplasm. (b) Snapshot of a simulation of diffusing and aggregating particles (orange) subject to turnover [3], with an immobile domain of fluctuating size (blue). For illustrative purposes, the cluster and domain sizes are scaled by a factor of 5 with respect to their actual sizes. The diameter $a$ of a single particle is taken as length unit. Taking $a$ as the distance between two neighboring vertices in a hexagonal lattice of gephyrin trimers gives $a ≃ 20 nm$ . (c) Stationary concentrations $c_{n}$ of diffusing clusters of size $n$ ( $n$ being the number of constituent particles) resulting from the simulated aggregation-turnover dynamics. (d) Example of the evolution over time of the sizes of an immobile domain (blue) and of a freely diffusing cluster (orange) that grow by stochastic encounters with other clusters and shrink by particle desorption. Fusion with impinging clusters of different sizes lead to instantaneous size increases, while Poissonian particle loss produces slower size decreases. (e) The probability distributions $A_{n}$ of the sizes $n$ of an immobile domain (solid blue line) and of long-lived diffusing clusters (solid orange line) appear stationary after a few turnover times $1 / k_{d}$ . The panel also shows the corresponding quasistationary distributions (dashed lines) obtained from the master equations (ME) for immobile domains [Eq. (42)] and diffusing clusters [Eq. (51)], corresponding to the shot-noise model proposed in the present work, where the size dynamics are based on the rates of impingement shown in panel (f). (f) Encounter rates $r_{m}$ of the immobile domain with diffusing clusters of size $m$ as measured in the simulation (blue dots). The encounter rates with diffusing clusters of different sizes $m$ are also shown (orange) for a freely diffusing cluster followed over time. In this case, the encounter rates as well as the diffusion constant [Eq. (45)] of the followed cluster depends on its size $n$ (here $n = 10, σ = 0.5$ ).
Reuse & Permissions
Figure 2
Shot-noise model for the stochastic size dynamics of an immobile domain in the case of monodisperse impinging clusters. (a) Example of size evolution for a single domain described by the simplified shot-noise model. The randomly impinging clusters contain 10 particles, with an encounter rate relative to particle loss $r / k_{d} = 2$ (the simulation shown was performed with $r = 2, k_{d} = 1$ ). (b) The distribution of sizes explored by clusters after a transient time of $5 / k_{d}$ is shown. The distribution (solid blue line) represents the average over many ( $n = 2000$ ) realizations of the stochastic shot-noise process for a duration of $T = 500 / k_{d}$ each. The theoretical quasistationary distribution $A_{n}$ is also shown (dashed orange line). Its determination is explained in panels (c) and (d). (c) The solution $A_{n}$ to the master equations (13) and (14) that govern the domain size distribution depends on the parameter $ν$ . The slow $n^{- (1 + ν)}$ large- $n$ asymptotics of the $A_{n}$ is proportional to $C (ν)$ [Eq. (26)] shown here. The sought-for solution of the master equations is obtained for $ν = ν^{*}$ , when $C (ν^{*}) = 0$ . (d) Three solutions of the master equations with different values of $ν$ are shown, $ν = ν^{*} = 5.69 \cdot 10^{- 3}$ (orange), $ν_{-} = 0.9 ν^{*}$ (blue), and $ν_{+} = 1.1 ν^{*}$ (green). Inset: The distributions corresponding to $ν_{-}$ and $ν_{+}$ have a slow decay for large $n$ which agrees with the predicted scaling $n^{- (1 + ν)}$ . The distribution for $ν = ν^{*}$ decays much faster. All distributions are normalized to one. The contributions of the slow positive tail for $ν_{-}$ and slow negative tail for $ν_{+}$ explain their apparent differences in normalization. (e) The number of surviving clusters (solid blue line) in the simulations agrees well with the predicted exponential decay $exp (- ν^{*} k_{d} t)$ (dashed black curve). (f) The characteristic lifetime $τ_{d}$ of the clusters increases exponentially with the rate of supply $r_{10}$ . The theoretical prediction of the lifetime with the numerically determined $ν^{*}$ (blue line) is well approximated by the analytical expression [Eq. (20)] for $r_{10} / k_{d} ≳ 1$ (orange dashed line).
Reuse & Permissions
Figure 3
Shot-noise description vs particle-based simulations of cluster-cluster aggregation for an immobile domain. Simulation parameters are given in Table 1, case I. (a) The stationary cluster size distribution (average concentrations of clusters of a given size on the membrane) observed in the simulations (black dots) is shown together with the result from the rate-model description (red line). The latter requires to determine a single free parameter $K$ (see text) that is obtained by fitting the cumulative distribution of cluster sizes shown in the inset. (b) The observed rates $r_{m}$ of encounters with clusters of size $m$ for an immobile cluster (blue dots) are shown together with the rate-model approximation (brown line) using the previously determined parameter $K$ . (c) Quasistationary probability distribution $A_{n}$ of explored domain sizes for an immobile domain as observed in simulations (blue line) and predicted by the theory with the previously fitted value of $K$ (brown line). (d) Clusters disappear due to stochastic fluctuations with a rate $1 / τ_{d}$ , where $τ_{d}$ is the typical cluster lifetime. The fraction of surviving clusters $S$ observed in simulation (blue) is well fitted by a decreasing exponential with a time constant $τ_{d} = 49.2 / k_{d}$ (thin black line). The theoretical prediction based on the rate equations is $τ_{d} = 46.0 / k_{d}$ (brown line). The approximate analytical expression of the typical lifetime is about $7 %$ shorter than the simulation result.
Reuse & Permissions
Figure 4
Shot-noise description vs particle-based simulations of cluster-cluster aggregation for diffusing clusters. Simulation parameters are given in Table 1, case I. (a) The observed rates $r_{m, n}$ of encounters of a cluster of size $n$ (orange triangles: $n = 5$ ; orange squares: $n = 20$ ) with clusters of size $m$ are shown together with the rate-model approximation using the previously determined parameter $K$ [Fig. 3]. (b) The quasistationary probability distribution of explored cluster sizes for freely diffusing clusters as observed in the simulations (orange line) and predicted (solid green line) from the rate equations with full size dependence of encounter rates $r_{m, n}$ [Eq. (50)] and choosing the value $ν = ν^{*}$ which gives a fast decay of the $A_{n}$ for large $n$ . The theoretically predicted probability distribution obtained with a self-consistently determined effective cluster size $n_{eff}$ and corresponding encounter rates $r_{m}$ that do not depend on the size $n$ of the tracked cluster is also shown (dashed green line). The respective predicted characteristic lifetimes are given in the plot legend. The given standard deviation is obtained using the bootstrap method explained in Appendix pp3-s3. (c) Determination of the self-consistent cluster size $n_{eff}$ . The predicted quasistationary probability distribution $A_{n}$ of observing a diffusing cluster of size $n$ depends on the effective cluster size $n_{eff}$ via the effective encounter rates; distributions are shown for $n_{eff} = 5$ (solid gray line), $n_{eff} = 10$ (dash-dotted gray line), and $n_{eff} = 20$ (dotted gray line); the distribution obtained from the simulation is also plotted [orange line, same as in panel (b)]. For larger $n_{eff}$ , the rates are lower and the size distribution tends to lower $n$ . Inset: graphical explanation of the self-consistent determination of $n_{eff}$ . The expected average cluster diffusion constant $〈 D_{n} 〉 = \sum_{n} A_{n} (n_{eff}) D_{n}$ depends on $n_{eff}$ via the $A_{n}$ and increases with $n_{eff}$ . A self-consistent solution for the effective size is then given by $n_{eff} = {(〈 D_{n} 〉 / D_{0})}^{- 1 / σ}$ , obtained as the crossing of the curve representing the expression of the right-hand side (solid line) with the diagonal (dashed line).
Reuse & Permissions
Figure 5
Predicted distribution of explored cluster sizes for synaptic gephyrin domains, which allow us to infer the typical domain lifetime. Simulation parameters are given in Table 1, case II. (a) Particle-based simulations with the parameter values obtained in a previous work [3] give rise to a stationary distribution of diffusing clusters in the membrane (black dots). The stationary cluster size distribution (average concentrations of clusters of a given size on the membrane) is shown together with the result from the rate-model description (red line). The latter requires us to determine a single free parameter $K$ (see text). Inset: The best fit to the cumulative distribution of cluster sizes is obtained for $K = 1.81$ . (b) The observed rates of encounter $r_{m}$ for domains of size $n$ with clusters of size $m$ . For immobile domains (blue dots), the rates do not depend on the size $n$ ; for diffusing clusters that are followed, the rates are shown for $n = 10$ (orange triangles) and $n = 30$ (orange crosses). The theoretical predictions for the $r_{m}$ based on the fitted $K$ [panel (a)] are shown as lines (immobile: solid brown, diffusing $n = 10$ : solid green, diffusing $n = 30$ : dashed green). (c) The quasistationary distribution of sizes explored by immobile domains subject to encounter with diffusing clusters and turnover of constituent particles for particle-based simulations (blue line) and the prediction from the rate-equation description with fitted $K$ (brown line). The expected lifetime of immobile domains is given in the plot legend, where the uncertainty in the simulation result reflects limited sampling of occurrences where the domain size is 1. (d) Theoretically predicted lifetimes for immobile domains for different values of $c_{0}, k_{d}$ , and $σ$ . For given $σ$ , the predicted lifetime depends only on the average domain size $〈 n 〉$ . The parameters corresponding to the simulations and theory shown in panels (a)–(c) and (f) are surrounded by the black circle; for all parameters, we used the value $K = 1.81$ for the mean-field equations as determined in panel (a). (e) The averaged rescaled quasistationary size distributions of immobile clusters for different combinations of $c_{0}, k_{d}$ , and $σ$ show a trend toward narrower size distributions with increasing $σ$ , implying larger domain lifetimes. Error bars represent the standard deviation over the averaged quasistationary size distributions for different $k_{d}$ and $c_{0}$ . (f) For diffusing domains, the distribution of explored cluster sizes tends to larger values as the rates of encounter with other clusters are larger on average than for immobile domains; see panel (b). The self-consistent solution (see text for details) captures the overall shape of the distribution but can be expected to overestimate the probability of observing clusters of size 1; see text for details. Because the corresponding cluster lifetimes are extremely long, the quantitative accuracy of the predictions is difficult to assess by direct numerical simulations.
Reuse & Permissions
Figure 6
Determination and influence on predicted lifetimes of the Smoluchowski kinetic coefficient $K$ . Left panels correspond to case I parameters, right panels to case II parameters. The optimal parameter $K$ in the Smoluchowski equations is determined by best fitting the results of full numerical equations. (a) Sum of least-squared errors for the three objective functions indicated on the graph, as a function of the kinetic coefficient $K$ in the mean-field description. Fitted values of $K$ correspond to the respective minima of the errors. [(b), (c)] Predicted cluster lifetime $τ_{d}$ based on the Smoluchowski rate equations as a function of $K$ (thick solid gray line) and the value of $τ_{d}$ for the three optimal values of $K$ determined according to the fits shown in panel (a), for immobile (b) and diffusing (c) domains. The expected true cluster lifetime can be inferred from the value $A_{1}$ of the quasistationary size distribution observed in the simulations and is shown (dashed black line) together with the 95% confidence interval obtained from bootstrap resampling; see Appendix pp3 for details. In the case of fast particle turnover and for immobile domains [panel (b1)], the cluster lifetime can be measured directly from the particle-based simulations (thin solid black line).
Reuse & Permissions

Physical Review E

covering statistical, nonlinear, biological, and soft matter physics