Leopard Spots, Frog Eggs, and the Spectrum of Nonlinear Diffusion Processes

By Matthew R. Francis

Stripes, spots, or a mix of both appear on the skin of many animals — from tigers to beetles to whale sharks. These patterns are typically unique to individual creatures, and biologists often use them for identification. While distinct patterns may seem random, they obey certain rules that suggest a common underlying description. Striping and spotting occur in many unrelated species, implying that both evolutionary advantages and simple biochemical mechanisms drive such patterns.

As Björn Sandstede of Brown University noted during his invited address at the 2018 SIAM Annual Meeting, held in Portland, Ore., this July, similar patterns appear in certain chemical reactions and granular material under vibration. Nonlinear reactions and diffusion describe biological and non-biological patterns, producing stable concentrations in this space.

Alan Turing—best known for his work in computer science and cryptography—first made the mathematical connection between nonlinear diffusion processes and animal stripes in the 1950s. Many researchers have applied the resulting model to demonstrate how various species get their spots and describe nonlinear waves in chemical reactions.

Sandstede and his colleagues study the mathematical stability of these nonlinear waves and the means by which they might interact with one another. This work necessitates an understanding of the wave spectrum, which describes the nonlinear behavior of many systems quite well.

Sandstede’s talk focused on stable spatial peaks and spiral waves, which have clearly-defined crests that propagate outward from a source. These systems display interesting behavior even when limited to one spatial dimension.

How the Leopard Kept its Spots

Laboratory research on embryos shows that spots and stripes on skin originate very early in development. The stripes of a danio fish arise within three weeks of fertilization, and a leopard’s spots develop while the embryo is still hairless. Even melanistic leopards (called black panthers) carry traces of spots  despite the blackness of their coats.

What processes yield these patterns to begin with? Stripes and spots have distinct boundaries and do not continuously shade into each other. This arrangement indicates the presence of chemical concentrations—which produce discrete wave peaks separated by low concentration regions—during development.

The one-dimensional reaction-diffusion equation describes a wide variety of stable spatial structures:

\[\frac{\partial u}{\partial t}=D\frac{\partial^2u}{\partial^2x}+f(u), \:\:\textrm{where} \:\: u \in R^n.\]

The vector-valued function \(u\) represents the relevant physical quantity: the concentration of chemicals or displacement of materials. The diffusion coefficient \(D\) and reaction function \(f\)—the sources of the system’s nonlinearity—control the dynamics specific to each system.

In linear systems, interactions and perturbations obey the superposition principle: if \(a\) and \(b\) are both solutions to the equation, then \(a+b\) is as well. For example, two interfering linear waves create a new waveform, and traveling waves pass through one another. Nonlinear waves, however, can collide or produce other effects that are not simply additive combinations of the two original waves.

It is the reaction-diffusion equation’s nonlinearity that generates stripes and spots in the first place. Nonetheless, the spots’ distinctness means that one can treat them as independent objects to a certain level of approximation. Another of Sandstede’s examples is particularly useful for visualization: pillars of copper beads produced by vibrations (see Figure 1). These pillars are much higher than the surrounding material and appear to stand independently of each other.

Sandstede simplified the system by beginning with known steady-state (time-independent) concentrations \(q\) and finding solutions to the reaction-diffusion equation of the form

\[u(x,t)=q(x)+e^{\lambda t}v_0(x),\]

where \(|v_0|\) is a small perturbation. This transforms the reaction-diffusion equation into an ordinary differential equation in \(x\), with the eigenvalue \(\lambda\) characterizing the system’s spectrum. These eigenvalues come in two classes: zero (or very close to zero), or complex with real part negative. The spectrum describes decaying and oscillatory perturbations, signifying that the steady-state solutions are largely stable under perturbation. Once the embryonic leopard has its spots, the spots stay.

Similarly, interactions between neighbors along the line are treated perturbatively. Overlap occurs at the tails of the mathematical peaks. Like with stability analysis, the spectrum of the operator governing the perturbations completely describes the interactions. As a result, the spots can exchange concentrations in an oscillatory fashion and even attract each other. However, the interaction’s strength decays exponentially with distance, meaning that steady-state spots interact less when they are further apart.

Of Frogs and Spiral Waves

Though the copper pillars and leopard spots do not move in time, the math that describes them can also describe some nonlinear waves. For instance, certain chemical reactions propagate in spiral waves (see Figure 2), expanding outward in space and time from a source. Each wave peak resembles a closely-packed, moving version of the concentrations in the steady-state example.

One particularly striking example is calcium transport in the oocytes (reproductive cells) of African clawed frogs. The creation of these cells from fertilized eggs releases a wave of calcium into the cell, which forms a clear spiral pattern within the surrounding material (see Figure 3).

Nonlinear waves differ from their linear versions in important ways. A nonlinear wave clearly does not add linearly, but the wave’s frequency also varies nonlinearly with the wavenumber (which is inversely proportional to the wavelength). This means that the velocity of the peaks varies; in contrast, a linear wave’s velocity is fixed.

Sandstede and his collaborators studied one-dimensional spiral waves, which are basically cross-sections of the spiral. In Figure 4, the wave peaks travel outward from the source at varying rates. Sandstede treats the waves one-dimensionally using the same mathematical tools as in the steady-state spot model.

The team found that perturbing a spiral wave changes its pattern, thus shifting the source and affecting the wave’s peak velocity. Sandstede described the disturbance’s propagation as a “shock” that travels along the spiral wave at the speed of the wave itself (see Figure 5). Unlike with the steady-state perturbations, a small disturbance at the wave’s source can therefore affect the entire wave train.

For spiral waves, as with spotting and striping, Sandstede and his colleagues found that the spectrum of the operator defining the system generally described the system’s nonlinear dynamics — at least in one dimension. Real-world spots, stripes, and spiral waves are at minimum two-dimensional phenomena on surfaces, and the second spatial dimension complicates matters.

Nevertheless, researchers continue to study reaction-diffusion processes in higher dimensions, so the one-dimensional case’s tractability is cause for hope. After all, we know that real-world stripes and spots are stable. The spectral description of these two-dimensional phenomena may follow as well. 

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Sandstede’s presentation is available from SIAM either as slides with synchronized audio or a PDF of slides only.

References 
[1] Lechleiter, J., & Clapham, D. (1992). Molecular Mechanisms of Intracellular Calcium Excitability in X. laevis Oocytes. Cell, 69, 283-294.
[2] Sandstede, B., & Scheel, A. (2001). Superspiral Structures of Meandering and Drifting Spiral Waves. Phys. Rev. Lett., 86(1), 171-174.
[3] Sandstede, B., & Scheel, A. (2007). Period-Doubling of Spiral Waves and Defects. SIAM J. Appl. Dynam. Syst., 6(2), 494-547.
[4] Umbanhowar, P.B., Melo, F., & Swinney, H.L. (1996). Localized excitations in a vertically vibrated granular layer. Nature, 382, 793-796.

Matthew R. Francis is a physicist, science writer, public speaker, educator, and frequent wearer of jaunty hats. His website is BowlerHatScience.org.