Anomalous versus Slowed-Down Brownian Diffusion in the Ligand-Binding Equilibrium

Abstract

Measurements of protein motion in living cells and membranes consistently report transient anomalous diffusion (subdiffusion) that converges back to a Brownian motion with reduced diffusion coefficient at long times after the anomalous diffusion regime. Therefore, slowed-down Brownian motion could be considered the macroscopic limit of transient anomalous diffusion. On the other hand, membranes are also heterogeneous media in which Brownian motion may be locally slowed down due to variations in lipid composition. Here, we investigate whether both situations lead to a similar behavior for the reversible ligand-binding reaction in two dimensions. We compare the (long-time) equilibrium properties obtained with transient anomalous diffusion due to obstacle hindrance or power-law-distributed residence times (continuous-time random walks) to those obtained with space-dependent slowed-down Brownian motion. Using theoretical arguments and Monte Carlo simulations, we show that these three scenarios have distinctive effects on the apparent affinity of the reaction. Whereas continuous-time random walks decrease the apparent affinity of the reaction, locally slowed-down Brownian motion and local hindrance by obstacles both improve it. However, only in the case of slowed-down Brownian motion is the affinity maximal when the slowdown is restricted to a subregion of the available space. Hence, even at long times (equilibrium), these processes are different and exhibit irreconcilable behaviors when the area fraction of reduced mobility changes.

Spatial distributions at equilibrium under heterogeneous transient subdiffusion

“…In terms of mean-squared displacement 〈 R 2 〉, NHB thus preserves the long-time behavior of NHC, but does not feature the initial anomalous transient. In a previous work (Soula et al, 2013), we showed that decreasing the ratio of diffusion coefficients D in / D out in NHB, leads to accumulation at equilibrium inside the patch. We obtained a very good theoretical approximation of this accumulation with a simple expression: $N_{\text{in}}/N_{\text{T}}=\frac{\varphi }{\varphi +\left(1-\varphi \right)\frac{D_{\text{in}}}{D_{\text{out}}}}$…”

Explicit Spatiotemporal Simulation of Receptor-G Protein Coupling in Rod Cell Disk Membranes

“…This setup reduces the diffusional mobility of R and G by a factor of 2 more strongly than in the free-diffusion case (to 0.26D 0 ), a behavior that has already been suggested in theoretical studies (61). If the single R Ã is considered to be not part of a rack, the encounter rate between R Ã and G is increased due to confinement compared to the free-diffusion case (also compare to Soula et al (71)). This increase is compensated by the decreased long-range diffusion through racks.…”

“…It has been shown that obstacles can increase educt encounter rates (71). In our case, the R Ã G encounter rate FIGURE 5 Microscopic ReaDDy simulation of the pre-complex scenario.…”

“…The effects of immobile clustered obstacles on diffusion have been investigated in several theoretical studies (44,57,58,71). It was shown by Saxton (43) that a percolation threshold, c p , exists when the occupied area fraction of obstacles, c, exceeds 0.332.…”

Fractional Diffusion in Gaussian Noisy Environment

“…Let us mention a few recent publications concerning the applications of subdiffusive fractional equations. The work in [3] studied the applications to the transport in biological cells. The work in [6,7] studied the fractional chemotaxis diffusion equation.…”