Diffusion through complex confining environments: fluctuating triply periodic minimal surfaces

TL;DR

This work investigates diffusion through fluctuating three-dimensional triply periodic minimal-surface membranes using a phase-field framework that couples a diffusing spherical tracer to the membrane. The model includes bending, area, and volume constraints plus an interfacial penalty, and it reproduces the correct fluctuation statistics while enabling dynamic pore opening. A key finding is a two-regime diffusion with an intermediate MSD plateau that grows with tracer size, made possible by pore fluctuations that intermittently widen bottlenecks; moreover, coupling between the particle and membrane deformation enhances long-time diffusivity by promoting pore openings. The study provides a mechanistic picture relevant to protein transport in biological cubic membranes and suggests extensions to fluid hydrodynamics to capture solvent effects in realistic cellular environments.

Abstract

The transport of individual entities through interconnected structures is a process of practical relevance both in biology and technology. Examples are given by diffusive dynamics of molecules in porous structures. In soft environments, this transport can be strongly influenced by fluctuations of the porous structure itself. Here, we focus on triply periodic membrane structures found both in cell organelles and in synthetic amphiphilic systems. We theoretically study the effect of a complex three-dimensional fluctuating environment on the diffusive motion of a test object, using a phase field approach. The rigid spherical test object is energetically forced to not penetrate the membrane. Generally, the pores of the membrane structure can be smaller than the diffusing object. Yet, fluctuations of the membrane can intermittently widen its pores, still allowing for the motion of the larger particles through them. Thus, the object stays trapped for a while inside one cavity formed by the membrane, before an appropriate fluctuation event widens a membrane pore in the right moment so that the object can jump into the next cavity. The process is reflected by a pronounced plateau in the time evolution of the mean squared displacement. Moreover, we investigate the impact of the diffusing object on the deformation of the membrane, which leads to an additional increase in the diffusivity. We think that the described scenario should be directly observable, for instance, in protein diffusion through biological environments.

Figures (8)

• Figure 1: A visualization of the P-surface, of $3\times 3\times 3$ unit cells. The surface separates space into two equal volumes that are intertwined channels. A spherical test particle (bright red) can move freely on one side of the surface, but cannot penetrate it. Thermal fluctuations have undulated the surface, so that some pores are open wider than others, and, therefore, at some time, allow the test particle to pass easily.
• Figure 2: Fluctuations $\langle h_\mathbf{k}h_{-\mathbf{k}}\rangle$ of the height of the membrane around a flat quadratic patch of membrane of length $L=60$ at temperature $k_BT=0.033$ plotted over the wave number $|\mathbf{k}|$. The analytical expression in Eq. \ref{['eq:spectrum']} (blue line) is compared to the results from a simulation at $k_BT=0.033$ (red dots), averaged over a time of $t=10000$. For wavelengths above the width of the interface, $|\mathbf{k}|< \epsilon^{-1}$, the results from the analytical expression match those from the numerical simulation.
• Figure 3: Averaged mean squared displacement (MSD) of the diffusing spherical object over the time $t$. The three different lines are for different particle radii $l$, while $M_\mathbf{R}=1$ is constant. Shaded areas indicate standard deviations of the means over 64 independent realizations. Two appropriately diffusive regimes of the MSD (black line) are separated by a region of reduced slope, which is caused by trapping in the cavities of the membrane structure. With increasing particle radius, this region becomes more pronounced, for $l=8$ it is rather plateau-like.
• Figure 4: The energy barrier at the narrowest point of a channel for different particle radii $l$, calculated for the static, initial surface. For a unit cell size of 40, the narrowest channel formed by a P-surface has a radius of 10. However, due to the additional width of the transition layer, which is of order $\sqrt{2}$ for both the membrane and the particle, the effective radius is between 5 and 6.
• Figure 5: Averaged mean squared displacement (MSD) of the particle over the normalized time $tM_\mathbf{R}$. Shaded areas indicate standard deviations of the means over 64 independent realizations. The three different lines are for different mobilities $M_\mathbf{R}$, while $l=7$ is constant.