Spatial Wave Patterns

Self-organization in space is a striking process that can be observed in many areas of the natural world. It is especially relevant in the development of an organism.

The emergence of spatial order results when the underlying interactions extend in space affecting the surroundings. Commonly spatial coupling is mediated by diffusion of the constituents. However, long-range interactions are often found to play an important role in spatial self-organization, such as mechanical forces between microtubules or actin filaments. Multiple dynamical behaviors have been found depending on the local dynamics, such as traveling waves and spatially extended patterns, which can have a big impact for the biological function.

Oscillatory dynamics may lead to traveling waves, which can serve to spatially coordinate different events in an organism. Different waves can emerge, such as target patterns, which radially emerge from inhomogeneities, or spiral waves, which are driven by the presence of topological defects. A different type of waves, also named fronts, emerge as a result of bistability. When multiple stable states coexist, an interface separating those different states may propagate in different directions depending on the relative stability between both.

Spatial coupling can lead to different extended patterns, such as stripes or hexagonal arrangements, which create impressive patterns of spatial organization. Turing patterns emerge in reaction-diffusion systems and represent the most prominent example in the literature. In general, these patterns appear when the fastest diffusing reactants inhibit the other. Moreover, not only reaction-diffusion systems can create patterns, but other non-local interactions can lead to self-organization, such as those produced by mechanical forces.

Often waves and extended patterns can occur simultaneously. This scenario leads to highly complex spatio-temporal dynamics involving oscillatory patterns, waves in patterns, isolated pulses or even turbulence.

In the lab, we especially investigate spatially-extended behavior using a combination of theory and experiments with frog egg extracts. With our work, we try to identify the leading mechanisms driving different types of spatio-temporal dynamics.