Mesoscopic modeling of random walk and reactions in crowded media

We develop a mesoscopic modeling framework for diffusion in a crowded environment, particularly targeting applications in the modeling of living cells. Through homogenization techniques we effectively coarse grain a detailed microscopic description into a previously developed internal state diffusive framework. The observables in the mesoscopic model correspond to solutions of macroscopic partial differential equations driven by stochastically varying diffusion fields in space and time. Analytical solutions and numerical experiments illustrate the framework.

Figure 1

A snapshot from a molecular dynamics simulation: the interior of an E. coli is a highly crowded environment. Picture courtesy of David van der Spoel, Uppsala University.

Figure 2

(a) The cell volume $\mathrm{\Omega }$ discretized by a grid with size $h$ giving rise to the nonperforated subvolumes ${\omega }_{*}$. (b) The circular, perforated domain $\omega$ (pink) of radius $\rho$ where the cutouts represent the obstacles, and the outer $\partial {\omega }_O$ and inner $\partial {\omega }_I$ boundaries. (c) Solution to (2) on $\omega$ with crowders represented as holes with reflective boundary conditions and with high values of $E\left(\mathbf{x}\right)$ in red (in the middle) and low values in blue (along the outer boundary). (d) The excluded volume consists of the volume occupied by the crowding molecule enlarged by the radius $r$ of the diffusing tracer molecule.

Figure 3

Histogram counts of $\gamma /{\gamma }_0$ for different fractions of occupied volume $\varphi$. (a) Detailed resolution in $\varphi$ with $K=10$ bins for $Q=100$ different circular crowder distributions in 2D, where $R/\rho =0.1$ and $r/\rho =0.1$. (b), (c) Less resolution in $\varphi$ and finer resolution in the internal states with $K=15$ bins for $Q=500$ different crowder distributions with circular obstacles (b) and a mix of circular and rectangular obstacles (c).

Figure 4

Simulation of coarse-grained subdiffusion in 2D. (a) The mean-square displacement as a function of time (top solid red). The dashed curves are obtained with the initial and the steady state diffusion, ${\gamma }_0$ and $\overline{\gamma }$, and the slope of the comparison curve $t^{\alpha }$ is $\alpha =0.7$ (bottom solid blue). (b) As above, but with a four times faster diffusion of obstacles and hence faster scaling of time for the switching between internal states, ${\kappa }_0\to 4{\kappa }_0$, resulting in a faster approach to the steady state diffusion (but still such that $\alpha =0.7$ for the comparison slope). These plots are computed with data for $\varphi =0.35$ in Fig. 3.

Figure 5

Results of the bimolecular reaction in (77) presented as the time history of the number of resulting molecules $C$. The solid colored lines represent simulations with $H_{ij}$ in the different cases, from bottom to top: case 1 (blue with dots), case 2 (red with dots), case 3 (smooth red), and case 4 (smooth blue), respectively. The dashed line is the pure diffusion case with $k_0$ as the single rate.

Figure 6

A discretized model of an E. coli bacterium.

Figure 7

Two realizations of MinD oscillations in the membrane of an E. coli bacterium. (a) The number of MinD molecules in the leftmost (dashed red) and rightmost (solid blue) quarters of the bacterium, respectively. Top: ordinary diffusion without internal states. Bottom: our coarse-grained subdiffusion model. (b) The Fourier power spectrum of the pole oscillations of the two models: ordinary diffusion (solid) and coarse-grained subdiffusion (dashed).

Figure 8

Summary of the method. (a) A triangulation (gray edges) between the nodes ${\mathbf{x}}_j$ defines the primal mesh and the dual mesh (blue boundaries) which form the voxels ${\mathcal{V}}_j$. Red and black tracer molecules diffuse (red arrows) between voxels. They react with each other (green arrow) when they are located in the same voxel. (b) The molecules in a voxel are in $K$ internal states ($K=5$ here). In the equation without chemical reactions for the mean values $\overline{\mathbf{y}}$ of the number of molecules in each state in every voxel, ${\gamma }_0$ is the free diffusion coefficient, $\mathbf{\Lambda }$ is the matrix of jump coefficients between the voxels given by a finite element discretization, $\mathbf{T}$ scales the hindered diffusion due to crowding in the $K$ internal states, $1/{\kappa }_0$ determines the time scale of the internal jumps, and $\mathbf{A}$ is the matrix of jump coefficients between the internal states in a voxel. (c) The jump rates between the internal states in $\mathbf{A}$ are determined by computing the mean first exit time from the blue circle for a red tracer molecule with black obstacle molecules. Statistics is collected for many different obstacle configurations. Sampling from this distribution means that the tracer molecule is experiencing different crowder densities and consequently changes its internal state. Summary: Our approach focuses on coupling the internal states model (b), which has previously been used to simulate anomalous diffusion, to the explicit description of crowder molecules by coarse graining the microscopic information to the mesoscopic level (c).