Alejandro J. Garza S01163018 Department of Chemistry, Rice University, Houston, TX email: ajg7@rice.edu, ext. 2657 Submitted December 12, 2011 Abstract Spontaneous antispiral wave formation was observed during the simulation of typical reaction-diffusion systems using differential equations models. The equations for the FitzHugh-Nagumo (FHN), complex Ginzburg-Landau (CGL) and the two-variable Oregonator models were integrated using Runge-Kutta methods. It was found that antispiral waves can occur only near the Hopf bifurcation, i.e. the point at which the system starts to oscillate. The specific conditions required for antispiral wave formation were established through theoretical analysis and numerical simulations. The results explain a number of phenomena observed in oscillatory reactions such as the Belousov-Zhabotinsky (BZ) reaction. Introduction This paper is intended to describe and explain the findings of Gong and Christini in Antispiral Waves in Reaction-Diffusion systems [1]. With this purpose, this section will give an introduction on reaction diffusion-systems and their connection to chemistry. Excitable Media An excitable medium is a nonlinear dynamical system which has the capacity to propagate a wave, and which cannot support the passing of another wave until a so-called refractory amount of time has passed [2]. Arguably, excitable media are constitute one of the foremost important, and fascinating, nonlinear dynamical systems; excitable media can show selforganization, a requirement for the existence of life. Excitable media can be analyzed/simulated by means of three models; (1) stochastic models, (2) cellular automata models, and (3) partial differential equations. Theses models have been able to describe the most notable qualitative features observed in experiments (e.g formation of waves and self-organization). Gong and Christini have utilized the method of partial differential equations in their study (see the methods section). Reaction-diffusion systems have been show to act as excitable media under certain conditions. The most famous example of such a class of systems in chemistry is the unstirred Belousov-Zhbotinsky (BZ) reaction. The BZ-Reaction The BZ-reaction was discovered by Belousov in 1950 while trying to find and inorganic analogous to the Krebs cycle [3]. However, his findings were dismissed by the scientific community until Zhabotinsky confirmed his results [4]. Today, the BZ-reaction is the most studied, and arguably most important, nonbiological oscillatory reaction. The overall BZ-reaction is the oxidation of malonic acid by BrO 3 to form CO 2 and H 2 O 3 CH 2 (CO 2 H) 2 + 4 BrO 3 4 Br + 9 CO 2 + 6 H 2 O (1) However, this equation is deceptively simple. The mechanism for this reaction is extremely complicated and many models of varying complexity have been proposed [5]. The simplest model that is able to retain the qualitative features of the BZ-reaction is the Oregonator model [6]. This model can be explained as follows; consider a reaction described by the mechanism
Concentration 0.1 0.05 X Y Z P impossibility of an oscillatory overall reaction (i.e. the reaction will be moving away from equilibrium). When coupled with diffusion, which is governed by the diffusion equation c t D c 2 c (6) the Oregonator forms a model for an excitable medium. 0 0 5 10 Time Figure 1: Time series for the solutions to the Field and Noyes equations. The parameters used where ɛ 9.9 10 3, ɛ 1.98 10 5, q 7.62 10 5, f 1, x(t 0) 0, y(t 0) 0.001 and z(t 0) 0. A + Y k1 X + P X + Y k2 2 P A + X k3 2 X + 2 Z X + X k4 A + P B + Z k5 1 2 fy (O1) (O2) (O3) (O4) (O5) (i.e. the Oregonator mechanism). Assuming steadystate for A and B dx qy xy + x(1 x) dτ ɛ (2) dy qy xy + fz dτ ɛ (3) dz dτ x z (4) where x 2k 4 X/(k 3 A), y k 2 Y/(k 3 A), z k c k 4 BZ/(k 3 A) 2, τ k c Bt, ɛ k c B/(k 3 A) y ɛ 2k c k 4 B/(k 2 k 3 A), q 2k 1 k 4 /(k 2 k 3 ). These equations are known as the Field and Noyes equations. The solutions to these equations are oscillatory for certain sets of values of ɛ, ɛ, q and f. Figure 1 shows an example of these solutions for a set of parameters that produce oscillations. Note that, in this model, the concentration of products is always increasing in time (see Figure 1); only the concentration of intermediaries oscillate. Since the overall reaction is spontaneous, we conclude that S univ > 0 (5) for all time, t. This is very important because, when first discovered, oscillatory reaction where thought to be errors of measurement, due to the thermodynamic The FHN and CGL Model Apart from the oregonator model, two other models for excitable media were examined by Gong and Christini; the FitzHugh-Nagumo (FHN) and the Complex Ginzburg-Landau (CGL) models [7, 8]. The FHN model is a generic model for excitable media described by the system of partial differential equations u/ t (u u 3 /3 v)/ɛ + D u 2 u v/ t u γv + δ + D v 2 v (7) The CGL equation arises in many physical systems, including Rayleigh-Bénard convection, and contains diffusion, non-linear dispersion and amplitudedependent frequency. The CGL equation can be written as W/ t W (1 + iα)w W 2 + (1 + iβ) 2 W (8) Note that the equation has a real and a complex part. Antispiral Waves Soon after the confirmation of the discovery of oscillatory reactions, the formation of patterns in unstirred reaction-diffusion systems was also discovered. Among a wide variety of patterns observed, spiral waves seem to be of particular interest and importance due to their occurrence in biological and inorganic processes. Nevertheless, little attention was paid to the direction of rotation of the spirals (normally observed to rotate outwardly) until the recent observation of inwardly rotating spirals, termed antispirals [9]. Gong and Christini utilized numerical simulations to qualitatively reproduce these observations. Methods The FHN, CGL and oregonator models were analyzed numerically using partial differential equations. For Garza 2
the FHN and CGL models, Equation 7 and Equation 8 were integrated; for the Oregonator, the twovariable Oregonator model was used ( u u 2 fv u q u + q ) + D u 2 u du dt 1 ɛ dv dt u v + D v 2 v (9) This model can be derived by noting that ɛ is considerable larger than ɛ, so that it is reasonable to assume steady state for y y ss fz q + x (10) Making x u and z v, and considering the corresponding diffusion terms, one obtains Equation 9. All the systems were solved in a 100 100 grid with no-flux boundaries and from random initial conditions. The differential equations were integrated using the explicit Euler method. This method can be described by the equations u(t) t f(t, u(t)) (11) u(t) f(t, u(t)) t (12) u(t + t) u(t) + f(t, u(t)) t (13) which is basically a discretization of the differential equation. To calculate the Laplacian, the standard five-point approximation was utilized 2 u 2 u x 2 + 2 u y 2 2 u u(x + h, y) + u(x h, y) 2u(x, y) x2 h 2 (14) 2 u u(x, y + h) + u(x, y h) 2u(x, y) y2 h 2 (15) where h is the size of the step. Here, h 0.5 while t 0.005 for CGL and t 0.01 for the FHN and Oregonator models. The total number of integration steps was 5 10 6. The results were also verified to remain the same using the fourth-order Runge-Kutta integrating method. Results Figure 2 summarizes and illustrates some of the most important results for the FHN and CGL models. Figure 2a shows the time series for the u in the FHN Figure 2: Spiral and antispiral waves in typical excitable (FHN with γ 0.5, δ 0.7, D u 1.0, D v 0.0) and oscillatory (CGL) media. (a) Time history (arbitrary units) of the fast activator variable u of the FHN model. (b) Snapshot of spirals in the FHN model; the arrow indicates outward propagation of waves. (c) The parameter plane (α, β) shows where spirals ( ) and antispirals ( ) exist in the CGL system. (d) Snapshot of a well-developed single antispiral in the CGL system. (e) (f) Snapshots of antispirals in the CGL system. model; the small amplitude oscillations occur for a relatively large ɛ (ɛ 1.39) near the Hopf bifurcation (i.e. the point at which oscillations start occuring as ɛ changes). It is near this point at which antispirals occur; in contrast, spirals are formed when the oscillations have large amplitudes, for relatively small values of ɛ (ɛ0.09). Figure 2b shows a typical spiral in the FHN system. Figure 2c shows the parameter plane (α, β) shows where spirals ( ) and antispirals ( ) exist in the CGL system; that is, the points at which spirals or antispirals are formed as a function of α and β. Spirals occur when β > α 0 or 0 α > β; antispirals when α > β 0 or 0 β > α. For α β the medium may either support phase waves or remain static, depending on the parameter choice. Nevertheless, Figure 2c does not shows the expected behavior when αβ < 0; Figure 2c was extended improved in a comment to the paper of Gong and Christini by L. Brusch et al [10]. The improved verison of Figure 2x is shown in Figure 3, where RD denotes a general reaction diffusion system of the form i ũ f(ũ, µ) + D 2 ũ (16) Figures 2d, 2e, and 2f show typical spirals and antispirals observed in the CGL system. Garza 3
Figure 3: Extended parameter space (α, β) of the CGL equation and a general reaction-diffusion (RD) model. The parameters k s and ω s denote the wavenumber and frequency, repsectively, of the (spiral or antispiral) waves. The direction of wave-propagation is governed by the result of the competition between waves (whether spiral or antispiral) and their bulk oscillations [9]. Also, spiral and antispiral waves behave asymptotically as plane waves far from their cores. Because of this, Gong and Christine hypothesize that antispirals waves will be observed when the frequency of the bulk oscillation is larger than the asymptotic frequency of the wave (antispiral in this case). This hypothesis is in agreement with Figure 2c. Similarly, antispiral waves were observed in the two-variable Oregonator model. Figure 4 shows similar antispiral wave patterns in the FHN and Oregonato models. The case is similar to the FHN model; when ɛ is relatively small (e.g ɛ 0.01), the system becomes a typical relaxation oscillator with large amplitude. The small ɛ assumption was widely used in previous theoretical studies of the BZ reaction, which may account for the fact that antispirals waves were not observed. If, however, ɛ is near the Hopf bifurcation (ɛ 0.7782 for q 0.002 and f 0.95), the motion will be sinusoidal with small amplitude oscillations, which in turn lead to the spontaneous formation of antispirals. In addition, in order to form antispirals, the diffusion coefficients in the Oregonator model must be considerably small ( 0.001) in comparison to the large amplitude case ( 0.01). Conclusions The spontaneous formation of antispiral waves in typical reaction-diffusion systems (the CGL, FHN and Oregonator models) was demonstrated by means of numerical simulations. In addition, the condi- Figure 4: Antispiral waves in the FHN (a-d) and the Oregonator (e,f) models, where the arrows indicate inward propagation of waves. For FHN, ɛ 1.95, γ 0.5, δ 0.01, D u D v 0.005. For the Oregonator model, q 0.002, f 0.95, D u D v 0.001. tions for formation of antispiral waves in reactiondiffusion systems were determined. The results qualitatively reproduce phenomena observed in Belousoz- Zhabotisnky-type reactions. The study of pattern formation in reactionsdiffusion is important for the understanding of several physical phenomena. Spiral waves form in cardiac tissue, and it has been speculated that antispiral waves might similarly occur. Reaction-diffusion systems have promising applications in the development of chemical computers. In addition, morphogenesis and self-organization in nature remain in considerable mystery among science and the study of reaction diffusion systems may shed light on these processes. References [1] Gong, Y.; Christini, D. J. Phys. Rev. Lett.. 2003, 90, 088302(4). [2] Karfunkel, H. R.; Seelig, F. F. Journal of Mathematical Biology, 1975, 2, 123. [3] Epstein, I. R.; Pojman, J. A. An Introduction to Nonlinear Chemical Dynamics: Oscillations, Waves, Patterns and Chaos. Oxford University Press, New York, 1998. [4] Zaikin, A. N.; Zhabotinsky, A. M. Nature, 1970, 225, 535. Garza 4
[5] Kalishin, E. Yu.; Gonchareko M. M.; Khavrus, V. A.; Strizhak, P. E. Kinet. Catal.. 2002, 43, 256. [6] Noyes, R. M.J. Chem Ed. 1989, 66, 190. [7] FitzHugh, R. A. Biophysics, 1980, 25, 906. [8] Kuramoto, Y. Chemical Oscillations, Waves, and Turbulence, 1984 Springer-Verlag, Berlin. [9] Vanag, V. K.; Epstein, I. R. Science, 2001, 294, 835. [10] Brusch, L.; Nicola, E. M.; Bär, M. Phys. Rev. Lett. 2004, 92, 089801 (comment). Garza 5