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MBW:Timescales in Metabolic Pathways

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Introduction

Waves in a variety of contexts can be modeled under a generic framework. Gaining insight into wave generation and propagation can be done by dividing the analysis into two time scales: an small time scale during the abrupt changes on the wave fronts and backs, and the larger time scale relating the frequency of passing waves to the speed at which they travel. Tyson and Keener [1] show that methods of perturbation analysis applied to a governing singular partial differential equation are adept at offering insight into this division of time scales. The one-dimensional case offers no serious challenges, but analysis in two dimensions presents challenges that have no clear unique solution. In Tyson and Keener[1], several methods are compared and contrasted. Connections are made between the theories and they identify some fundamental relationships that underlie all approaches. They also show how the results can offer insight into the three dimensional case.

Mathematical Model

Generically, a wave can be characterized by two different reagents: a propagator and a controller. What the propagator and controller are depend on the context (Table 1), but their dynamics can be modeled studied by a setting independent of the specific context of interest. These two reagents exist in a medium that nominally allows the two to coexist in a stable relaxed state. However, given a sufficient perturbation to this resting steady state, a cascade of effects characterizing a wave will occur: an abrupt rise in the levels of the propagator triggers a rise in the levels of the controller, which in turn drives the levels of the propagator back to the relaxed steady state.

SystemPropagatorController
Neuromuscular tissue membrane potential ionic conductance
Belousov-Zhabotinskii reaction bromous acid ferroin
Dictyostelium discoideum (slime mold) cyclic AMP membrane receptor
Epidemics infectious agent level of immunity
Spiral galaxies density of molecular cloud temperature of molecular cloud

Table 1: State variables of some representative excitable media


Mathematically, let u(x,t) represent the levels of the propagator and v(x,t) the levels of the controller. They are defined in space (over some domain D as a subset of the n-dimensional reals) and evolve in time (t). The dynamics between the propagator and controller can be framed under a reaction-diffusion model,

Romero eq1.png


where epsilon is a positive small parameter that represents the ratio of rates of reaction in the quantities u and v and delta=D_v/D_u is the ratio of the diffusion coefficients of the two components. The functions f(u,v) and g(u,v) describe the non-linear kinetics of the system. For wave propagation, the nullclines of f and g have a particular shape (Figure 1); f(u,v)=0 is N-shaped and g(u,v) is monotone with only one intersection with f(u,v)=0 (and thus one possible steady state solution).

Romero fig1.png

Figure 1: From [1]

For a detailed example of modeling under the framework of reaction diffusion models, see APPM4390:Traveling Waves in Excitable Media.

Example in 1-D media

In one dimension, there are two possible behaviors: a single solitary pulse or a periodic wave train. In the case of the wave train, the properties are summarized by the dispersion relation,

omega = H(k;epsilon,delta)

which relates the frequency (cycles per second: omega/(2 pi)) to the wave number (k= 2 pi/ wavelength) of the traveling wave train. For k small, this dispersion relation is essentially linear in k: H ~= c k. (For more information on traveling waves, please see APPM4390:Traveling Waves in Excitable Media).

Higher Dimensions: 2D and 3D media

In two dimensions, the principle difference to consider is that the curvature of the wave front affects wave propagation. The effect of curvature leads to difficulties in the study of wave propagation emanating at from a point: near a point, the curvature of a wave front is large. Because of this limitation, the jumping point for the study of waves in two dimensions is in an annular region. The extension of analysis to the whole plane, including every small neighborhood of the origin, is difficult and has no good answer.

There are two canonical forms of wave propagation:

Target patterns An emanating series of concentric circular waves; here as the circular waves emanate outward, the curvature becomes negligible and the problem reduces to the one-dimensional case.

Spiral Waves Rotating waves that spiral out; here curvature plays an important role in limiting possible solutions.

The authors show that their singular perturbation analysis leads them to describing waves in 2-D as subject to two relations:


dispersion relation curvature relation
speed in terms of frequency frequency in terms of speed, curvature
Romero eq2.png Romero eq3.png

where the dispersion relation depends on the specific excitable medium and the curvature relation is universally applicable to all mediums.

Previous Approaches

Zykov [2] parametrizes the wave differently than Tyson and Keener and arrives at a relationship that combines the dispersion relationship with a curvature relation. Greenberg [3]introduces a coordinate system and that allows the two dimensional case to approximately be separated into one dimensional cases. The approximation is valid when not to close to the origin and for large enough wavelengths, but leads to a dispersion relation of a different form than Tyson and Keener's, but qualitatively similar. Fife[4] improves on Greenberg's approach by using his coordinate system and not making simplifying approximations. Then, a dispersion relation identical up to a constant to Tyson and Keener's is derived.

Analysis of Results

Singular perturbation analysis leads to simple relations that summarize the properties of wave propagation. In the one dimensional case, the fundamental relationship is a dispersion relation; that is a relation between the frequency and speed of the traveling waves. In two dimensions, curvature plays a role, and in addition to the dispersion relation, the eikonal equation plays a role,

N = c + DK

where N is the normal velocity of the wave front, K is its curvature and c is the velocity of the plane waves (that is, when the wave front has no curvature: K=0).

The type of possible solutions in each dimension can be summarized as follows. In one dimension, there exists a family of one parameter wave-train solutions.In two dimensions, there exists a one-parameter family of expanding concentric circle waves but only one possible spiral pattern. In three dimensions, the situation is analogous: there exists a one-parameter family of spherical wave solutions, but the number of possible scroll waves is limited.

In all cases, the singular perturbation analysis of Keener and Tyson agrees well with previous theories and numerical results.

References

  1. 1.0 1.1 1.2 John J. Tyson and James P. Keener, “Singular Perturbation Theory of Traveling Waves in Excitable Media (A Review)”. Physica D. 32 (1988) 327-361.
  2. V.S. Zykov, Modeling of Wave Processes in Excitable Media (Manchester Univ. Press, Manchester, 1988.
  3. J.M. Greenberg, Periodic solutions to reaction-diffusion equations, SIAM J. Appl. Math. 30 (1976) 199-205.
  4. P.C. Fife, Propagator-controller systems and chemical patterns, in:Non-Equilibrium Dynamics in Chemical Systems, C. Vidal and A. Pacault, eds. (Springer, Berlin, 1984), pp. 76-88.