^{1,2,3}, A. Lemarchand

^{2,3}and B. Nowakowski

^{1,4,a)}

### Abstract

The direct simulation Monte Carlo method is used to reproduce Turing patterns at the microscopic level in reaction-diffusion systems. In order to satisfy the basic condition for the development of such a spatial structure, we propose a model involving a solvent, which allows for disparate diffusivities of individual reactive species. One-dimensional structures are simulated in systems of various lengths. Simulation results agree with the macroscopic predictions obtained by integration of the reaction-diffusion equations. Additional effects due to internal fluctuations are observed, such as temporal transitions between structures of different wavelengths in a confined system. For a structure developing behind a propagating wave front, the fluctuations suppress the induction period and accelerate the formation of the Turing pattern. These results support the ability of reaction-diffusion models to robustly reproduce axial segmentation including the formation of early vertebrae or somites in noisy biological environments.

The work was realized within the International Ph.D. Projects Program cofinanced by the Foundation for Polish Science and the European Regional Development Fund within the Innovative Economy Operational Program, “Grants for Innovation.” We thank Université Pierre et Marie Curie and PICS-2011 Program of PAN/CNRS for support.

I. INTRODUCTION

II. MODEL

III. MICROSCOPIC SIMULATIONS

IV. RESULTS

V. CONCLUSIONS

### Key Topics

- Diffusion
- 24.0
- Boltzmann equations
- 8.0
- Monte Carlo methods
- 8.0
- Eigenvalues
- 7.0
- Reaction rate constants
- 7.0

## Figures

Snapshot of the concentrations *A* (solid lines) and *B* (dashed lines) obtained from DSMC (orange) and the numerical integration of Eqs. (4) and (5) (black). System length is 2*L* _{0} and there are 50 cells per *L* _{0}. The horizontal axis gives the scaled space coordinate *x*/*L* _{0} and vertical axis, species concentrations.

Snapshot of the concentrations *A* (solid lines) and *B* (dashed lines) obtained from DSMC (orange) and the numerical integration of Eqs. (4) and (5) (black). System length is 2*L* _{0} and there are 50 cells per *L* _{0}. The horizontal axis gives the scaled space coordinate *x*/*L* _{0} and vertical axis, species concentrations.

Time evolution of a stable 1.5-wavelength structure. System length is 2*L* _{0}. There are 50 cells per *L* _{0}. The horizontal axis gives scaled space coordinate *x*/*L* _{0}, vertical axis, scaled time *t*/τ and color gradation, the concentration of A species.

Time evolution of a stable 1.5-wavelength structure. System length is 2*L* _{0}. There are 50 cells per *L* _{0}. The horizontal axis gives scaled space coordinate *x*/*L* _{0}, vertical axis, scaled time *t*/τ and color gradation, the concentration of A species.

As in Figure 2, but with the emergence of a 2-wavelength structure.

As in Figure 2, but with the emergence of a 2-wavelength structure.

As in Figure 2, but with a temporal transition from a 2-wavelength structure to a 1.5-wavelength structure.

As in Figure 2, but with a temporal transition from a 2-wavelength structure to a 1.5-wavelength structure.

Concentration of species A versus time in two example cells located at *x* = 0.7*L* _{0} (solid lines) and *x* = 1.2*L* _{0} (dashed lines) in the case of the history presented in Figure 2. The orange lines are results of DSMC, the black lines represent the function *A* _{+} + *c* _{1}(exp (λ(*q* _{0})*t*) − 1), where *c* _{1} is used as a fitting parameter.

Concentration of species A versus time in two example cells located at *x* = 0.7*L* _{0} (solid lines) and *x* = 1.2*L* _{0} (dashed lines) in the case of the history presented in Figure 2. The orange lines are results of DSMC, the black lines represent the function *A* _{+} + *c* _{1}(exp (λ(*q* _{0})*t*) − 1), where *c* _{1} is used as a fitting parameter.

As in Figure 2, but for a large system of 40*L* _{0} length. There are 20 cells per *L* _{0}.

As in Figure 2, but for a large system of 40*L* _{0} length. There are 20 cells per *L* _{0}.

As in Figure 2, but for step function initial conditions with a Turing pattern emerging behind a moving wave front. System of 20*L* _{0} length and with 20 cells per *L* _{0}.

As in Figure 2, but for step function initial conditions with a Turing pattern emerging behind a moving wave front. System of 20*L* _{0} length and with 20 cells per *L* _{0}.

As in Figure 7, but for a system of 20*L* _{0} length, with 30 cells per *L* _{0} and for the following set of parameters close to the bifurcation: *k* _{1}/*k* _{2} = 2.92 × 10^{4}, *k* _{3}/*k* _{2} = 0.73 × 10^{4}, *k* _{−3}/*k* _{2} = 7.3 × 10^{6}, *k* _{2} = 7.17 × 10^{−7}, σ_{ R } = 0.0238, σ_{ A } = 2.2σ_{ R }, *R* = 30*A* _{+}.

As in Figure 7, but for a system of 20*L* _{0} length, with 30 cells per *L* _{0} and for the following set of parameters close to the bifurcation: *k* _{1}/*k* _{2} = 2.92 × 10^{4}, *k* _{3}/*k* _{2} = 0.73 × 10^{4}, *k* _{−3}/*k* _{2} = 7.3 × 10^{6}, *k* _{2} = 7.17 × 10^{−7}, σ_{ R } = 0.0238, σ_{ A } = 2.2σ_{ R }, *R* = 30*A* _{+}.

Instantaneous concentration *A* versus scaled space *x*/*L* _{0} for an inhomogeneous initial condition and for the same set of parameters as in Figure 8, close to the bifurcation. The black line gives the result of the numerical integration of Eqs. (4) and (5) and the orange line, the result of DSMC.

Instantaneous concentration *A* versus scaled space *x*/*L* _{0} for an inhomogeneous initial condition and for the same set of parameters as in Figure 8, close to the bifurcation. The black line gives the result of the numerical integration of Eqs. (4) and (5) and the orange line, the result of DSMC.

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