Surface reaction-diffusion kinetics on lattice at the microscopic scale

Open Access

Surface reaction-diffusion kinetics on lattice at the microscopic scale

Wei-Xiang Chew, Kazunari Kaizu, Masaki Watabe, Sithi V. Muniandy, Koichi Takahashi, and Satya N. V. Arjunan

Phys. Rev. E 99, 042411 – Published 19 April 2019

Abstract

Microscopic models of reaction-diffusion processes on the cell membrane can link local spatiotemporal effects to macroscopic self-organized patterns often observed on the membrane. Simulation schemes based on the microscopic lattice method (MLM) can model these processes at the microscopic scale by tracking individual molecules, represented as hard spheres, on fine lattice voxels. Although MLM is simple to implement and is generally less computationally demanding than off-lattice approaches, its accuracy and consistency in modeling surface reactions have not been fully verified. Using the Spatiocyte scheme, we study the accuracy of MLM in diffusion-influenced surface reactions. We derive the lattice-based bimolecular association rates for two-dimensional (2D) surface-surface reaction and one-dimensional (1D) volume-surface adsorption according to the Smoluchowski-Collins-Kimball model and random walk theory. We match the time-dependent rates on lattice with off-lattice counterparts to obtain the correct expressions for MLM parameters in terms of physical constants. The expressions indicate that the voxel size needs to be at least $0.6 %$ larger than the molecule to accurately simulate surface reactions on triangular lattice. On square lattice, the minimum voxel size should be even larger, at $5 %$ . We also demonstrate the ability of MLM-based schemes such as Spatiocyte to simulate a reaction-diffusion model that involves all dimensions: three-dimensional (3D) diffusion in the cytoplasm, 2D diffusion on the cell membrane, and 1D cytoplasm-membrane adsorption. With the model, we examine the contribution of the 2D reaction pathway to the overall reaction rate at different reactant diffusivity, reactivity, and concentrations.

Received 17 November 2018
Revised 30 January 2019

DOI:https://doi.org/10.1103/PhysRevE.99.042411

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Adsorbed biomolecules Cellular automata Chemical Physics & Physical Chemistry Chemical kinetics, dynamics & catalysis Diffusion Protein-ligand interactions Protein-membrane interactions Protein-protein interactions

Physics of Living Systems

Authors & Affiliations

Wei-Xiang Chew^1,2, Kazunari Kaizu¹, Masaki Watabe¹, Sithi V. Muniandy², Koichi Takahashi¹, and Satya N. V. Arjunan^1,*

¹Laboratory for Biologically Inspired Computing, RIKEN Center for Biosystems Dynamics Research, Suita, Osaka, Japan
²Department of Physics, Faculty of Science, University of Malaya, 50603, Kuala Lumpur, Malaysia

^*satya@riken.jp

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Issue

Vol. 99, Iss. 4 — April 2019

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Images

Figure 1
Mean squared displacement (msd) of molecules on 2D lattice. Molecules perform random walk on triangular lattice with $ϕ$ fraction of total surface voxels occupied by immobile obstacles. Green solid line represents the expected msd behavior for normal diffusion. Red dashed line denotes the linear scaling for $ϕ = 0.3$ at long time. Simulation was performed with voxel size $l = 0.01 μ m$ in a square compartment of length $L = 5 μ m$ . Diffusion coefficient, $D = 1 μ m^{2} s^{- 1}$ .
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Figure 2
Comparison of on-lattice simulations with on- and off-lattice theories for surface-surface reaction $A + B \overset{}{\to} B$ . (a) Simulated on-lattice time-dependent rate coefficients (solid lines) compared with on-lattice MLM theory in Eq. (41) (dashed lines). For better visualization of the time-dependent behavior of the two extreme cases, the simulated and theoretical lines are normalized by the initial theoretical value. (b) Simulated on-lattice survival probability of $A$ (points) compared with off-lattice SCK theory in Eq. (48) (solid lines). Activation-limited ( $κ = 0.01 \times 4 π$ ) and diffusion-limited ( $κ = 100 \times 4 π$ ) cases are indicated by the top and bottom lines, respectively. Simulations were performed with Spatiocyte and the following parameters: $Area = (6.5 \times 6.5) μ m^{2}, R = 0.01 μ m, l = 0.01 \times 1.0209 μ m, D_{A} = 1, D_{B} = 0 μ m^{2} s^{- 1}, N_{a} = N_{b} = 423$ , duration = 0.2 s, logging $interval = 10 t_{d}$ .
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Figure 3
Survival probability of $A$ in the surface-surface reaction $A + B \underset{k_{d 2 D}}{\overset{k_{a 2 D}}{⇌}} C$ . Dashed curves are the values calculated according to the MPK1 theory given in Eq. (18); solid lines are the simulation results of Spatiocyte. Association rates in the activation-limited ( $κ = 0.1$ ) and diffusion-limited ( $κ = 100$ ) regimes are chosen. Simulation parameters are as follows: $k_{d 2 D} = 10 k_{a 2 D}$ , surface $area = (6.5 \times 6.5) μ m^{2}$ with periodic boundary, $R = 0.01 μ m, l = 0.01 \times 1.0209 μ m, D_{A} = D_{C} = 0 μ m^{2} s^{- 1}, D_{B} = 1 μ m^{2} s^{- 1}, N_{b} = 20, N_{b} = 401$ , duration = 10 s.
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Figure 4
Steady-state probability distribution of dimers from a reversible homodimerization reaction. The reaction is given by $A + A \underset{k_{d 2 D}}{\overset{k_{a 2 D}}{⇌}} B$ , with $k_{a 2 D} = 0.001 μ m^{2} s^{- 1}$ and $k_{d 2 D} = 1 s^{- 1}$ . The histogram on the left is simulated at an uncrowded condition with voxel size $l = 0.01 μ m$ . Dashed line is the analytical solution of the chemical master equation (CME). Histogram on the right is obtained with the same parameters except with a larger voxel size, $l = 0.09 μ m$ , resulting in a crowded compartment. The diffusion coefficient of $A$ is $1 μ m^{2} s^{- 1}$ , the length of the square compartment is $1 μ m$ and the initial number of $A$ is 169.
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Figure 5
(a) Time series of adsorbed molecules simulated with irreversible (IA, triangle and circle markers) and reversible (RA, plus and square markers) adsorptions. In each case, strong ( $k_{s a} = 500 μ m s^{- 1}$ ) and weak ( $k_{s a} = 50 μ m s^{- 1}$ ) adsorption rates were tested. In the reversible adsorption, the membrane dissociation rates are $k_{s d} = 62.5 and 6250 s^{- 1}$ , corresponding to the association rates $k_{s a} = 50 and 500 μ m s^{- 1}$ , respectively. Solid and dashed lines represent the continuum-based values according to the irreversible and reversible reaction formulas in Eqs. (23) and (27), respectively. (b) The concentration profile of cytosolic $A$ along the axis perpendicular to the adsorbing surface at $x = 0$ for the given time points. The adsorption is irreversible with the rate $k_{s a} = 50 μ m s^{- 1}$ . Theoretical lines shown are according to the continuum-based theory in Eq. (26). Simulation parameters are as follows: $l = 0.01 μ m, D_{A} = 1 μ m^{2} s^{- 1}$ , and initial number of cytosolic molecules $N_{a} = 1000$ .
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Figure 6
Contribution of 2D reaction pathway in surface reactions. The fraction of 2D reaction pathway that contributes to the overall surface reaction is indicated by $f_{2 D}$ and is plotted against $D_{c} / D_{m}$ . The fraction is obtained at varying reaction probabilities, $P_{a} = P_{a 2 D} = P_{a 3 D}$ and concentration of the membrane-associated reactant [B] (unit $μ m^{- 2}$ ). Simulation parameters are as follows: $R = 0.01 μ m, l = 0.01 \times 1.0209 μ m, L = 1 μ m, H = 2 L, D_{c} = 10 μ m^{2} s^{- 1}, [A_{c}] = 5 μ M = 3000 μ m^{- 3}, K_{e q} = 0.15 μ m, k_{s d} = 10 s^{- 1}, k_{a 2 D} / k_{r} = 0.001 μ m^{2}$ .
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