Multiscale Modeling of Diffusion in a Crowded Environment

Abstract

We present a multiscale approach to model diffusion in a crowded environment and its effect on the reaction rates. Diffusion in biological systems is often modeled by a discrete space jump process in order to capture the inherent noise of biological systems, which becomes important in the low copy number regime. To model diffusion in the crowded cell environment efficiently, we compute the jump rates in this mesoscopic model from local first exit times, which account for the microscopic positions of the crowding molecules, while the diffusing molecules jump on a coarser Cartesian grid. We then extract a macroscopic description from the resulting jump rates, where the excluded volume effect is modeled by a diffusion equation with space-dependent diffusion coefficient. The crowding molecules can be of arbitrary shape and size, and numerical experiments demonstrate that those factors together with the size of the diffusing molecule play a crucial role on the magnitude of the decrease in diffusive motion. When correcting the reaction rates for the altered diffusion we can show that molecular crowding either enhances or inhibits chemical reactions depending on local fluctuations of the obstacle density.

Keywords: Macromolecular crowding; Stochastic reaction–diffusion simulations.

Fig. 3.1

(a)Excluded volume (grey and red) for the center of mass of the diffusing molecule (blue). (b) Cartesian mesh and protective domain ω_i without crowding. (c) Solution of (3.9) with the effect of molecular crowding.

Fig. 3.2

(a) Boundary treatment. (b) Using the first exit time approach to compute Cartesian jump rates on the boundary of a curved domain Ω. (c) Model error when interpreting molecules as well mixed and jumping between nodes simultaneously.

Fig. 3.3

Not only the occupancy ϕ, but also the microscopic positions and orientations of the crowding molecules affect the jump propensities λ_ij = θ_ijλ_ii. (a) θ_ij = 0. (b) θ_ij > 0.25.

Fig. 4.1

The mean value of the mesoscopic jump coefficients in the crowded environment 𝔼[λ_ii] compared to 𝔼[λ_0,ii] = 4 in dilute media and its dependence on the occupancy ϕ. Averages are taken over M = 100 different crowder distributions. The obstacles are either spheres (blue and red) with radius R or rectangles (orange and green) with ratio of width to length equal to 20. The dashed reference line corresponds to equation (2.4) when only the averaged occupied volume in the target voxel is taken into account. The spherical moving molecule has radius r. In (e) we compare the mesoscopic coefficients with the results from a Brownian dynamics simulation, where we generate 10⁴ trajectories for 10 different crowder distributions with the software Smoldyn.

Fig. 4.2

Standard deviation σ(λ_ii/λ_0,ii) around the mean value of the change in jump coefficients in (a) and (d) in Fig. 4.1.

Fig. 4.3

(a) Diffusion in the crowded cell environment: Initial free diffusion with γ₀ (green). After colliding with the first macromolecules the observed diffusion is slowed down and the molecule’s diffusion coefficient decays (orange) to the long time behavior of slower diffusion with constant γ_∞ in a dense medium (red). (b) MSD curve for diffusion in a crowded medium (solid line) and as reference normal diffusion (dashed line). The pale line corresponds to an ideal well mixed medium and the dark line to a realistic medium with stochastic fluctuations in the positions of the crowders.

Fig. 4.4

The MSD on a mesoscopic grid with h = 0.025 for different distributions of spherical crowders with R = 5 × 10⁻³ and a moving molecule with r = 5 × 10⁻⁴. (a) Two different crowder distributions (red and blue), where we vary the starting position of diffusion from the center [0.5, 0.5] (solid line) to one voxel to the right/left and up/down [0.5 ± h, 0.5 ± h] (dashed lines) to show the sensibility of the MSD plot to the local environment. The crowding coefficient is ϕ = 0.4 and the diffusion constant is γ₀ = 1. (b) By varying the diffusion coefficient γ₀ for the two extreme distributions highlighted in grey in (a) we see that the diffusion coefficient only affects when the molecules undergo anomalous diffusion. (c) Changing the crowding percentage ϕ for γ₀ = 1 for the same two distributions as in (b) affects both the long time reduced diffusion and the duration of the anomalous phase. (d) Varying the space discretization h for the red distribution in (a) for all 5 starting positions. We see that a finer discretization better resolves the transient regime.

Fig. 4.5

Mesoscopic and macroscopic simulations with crowders of different sizes and a moving molecule with radius r = 10⁻³ starting in the center of the domain (red dot). The heat maps show the probability distribution p(0.5) of the location of the diffusing molecule at t = 0.5, which is the solution to equation (4.4) with homogeneous Dirichlet boundary conditions. (a) Rectangles of size 0.0005 × 0.01 and ϕ = 0.01. (b) & (c) The mesoscopic (D) and symmetrized macroscopic (D̃) simulation results with the distribution in (a), respectively. (d) 5 rectangles of size 0.004 × 0.8. (e) & (f) The mesoscopic (D) and symmetrized macroscopic (D̃) simulation results with the distribution in (d), respectively.

Fig. 4.6

(a) Two-dimensional projection of the three-dimensional experimental setting where a B molecule (blue) starts diffusing in x_j = (0.7, 0.5, 0.5) and reacts with A (red) that is confined to voxel 𝒱_i with x_i = (0.5, 0.5, 0.5) (red shaded region). Due to an uneven distribution of crowders we assume that the diffusion rate is γ_i inside 𝒱_i and γ (vertical dashed lines in (b) and (c)) in the rest of the domain. In (b) and (c) we compare the time it takes to react in this crowded environment ${\mathbb{E}}_j^r$ to the time it takes in an uncrowded environment ${\mathbb{E}}_{j,0}^r$ with γ₀ = 1 for different intrinsic reaction rates k_r. (b) γ = 0.7. (c) γ = 0.9.