Jakob Löber

Collisions of deformable cells lead to collective migration

Collective migration of eukaryotic cells plays a fundamental role in tissue growth, wound healing and immune response. The motion, arising spontaneously or in response to chemical and mechanical stimuli, is also important for understanding life-threatening pathologies, such as cancer and metastasis formation. We present a phase-field model to describe the movement of many self-organized, interacting cells. The model takes into account the main mechanisms of cell motility – acto-myosin dynamics, as well as substrate-mediated and cell-cell adhesion. It predicts that collective cell migration emerges spontaneously as a result of inelastic collisions between neighboring cells: collisions lead to a mutual alignment of the cell velocities and to the formation of coherently-moving multi-cellular clusters. Small cell-to-cell adhesion, in turn, reduces the propensity for large-scale collective migration, while higher adhesion leads to the formation of moving bands. Our study provides valuable insight into biological processes associated with collective cell motility.

Controlling the position of traveling waves in reaction-diffusion systems.

We present a method to control the position as a function of time of one-dimensional traveling wave solutions to reaction-diffusion systems according to a prespecified protocol of motion. Given this protocol, the control function is found as the solution of a perturbatively derived integral equation. Two cases are considered. First, we derive an analytical expression for the space (x) and time (t) dependent control function f(x,t) that is valid for arbitrary protocols and many reaction-diffusion systems. These results are close to numerically computed optimal controls. Second, for stationary control of traveling waves in one-component systems, the integral equation reduces to a Fredholm integral equation of the first kind. In both cases, the control can be expressed in terms of the uncontrolled wave profile and its propagation velocity, rendering detailed knowledge of the reaction kinetics unnecessary.

Shaping wave patterns in reaction-diffusion systems.

We present a method to control the two-dimensional shape of traveling wave solutions to reaction-diffusion systems, such as, interfaces and excitation pulses. Control signals that realize a pregiven wave shape are determined analytically from nonlinear evolution equation for isoconcentration lines as the perturbed nonlinear phase diffusion equation or the perturbed linear eikonal equation. While the control enforces a desired wave shape perpendicular to the local propagation direction, the wave profile along the propagation direction itself remains almost unaffected. Provided that the one-dimensional wave profile of all state variables and its propagation velocity can be measured experimentally, and the diffusion coefficients of the reacting species are given, the new approach can be applied even if the underlying nonlinear reaction kinetics are unknown.

Front propagation in channels with spatially modulated cross section.

Propagation of traveling fronts in three-dimensional reaction-diffusion media with spatially modulated cross-section is studied using the Schlögl model as a representative example. Applying appropriate perturbation techniques leads first to a reduction of dimensionality in which the spatially dependent Neumann boundary condition translate into a boundary-induced advection term and, secondly, to an equation of motion for the traveling wave position in weakly corrugated confinements. Comparisons with numerical simulations demonstrate that our analytical results properly predicts the nonlinear dependence of the propagation velocity on ratio of the spatial period of the confinement to the intrinsic width of the front; including the peculiarity of propagation failure. Based on the eikonal equation, we obtain an analytical estimate for the finite interval of propagation failure. Lastly, we demonstrate that the front velocity is determined by the suppressed diffusivity of the reactants if the intrinsic width of the front is much larger than the spatial variation of the medium.

Front propagation in one-dimensional spatially periodic bistable media.

Wave propagation in one-dimensional heterogeneous bistab le media is studied using the Schlögl model as a representative example. Starting from the analytically k nown traveling wave solution for the homogeneous medium, infinitely extended, spatially periodic variation s i kinetic parameters as the excitation threshold, for example, are taken into account perturbatively. Two differ ent multiple scale perturbation methods are applied to derive a differential equation for the position of the fro nt under perturbations. This equation allows the computation of a time independent average velocity, depend ing on the spatial period length and the amplitude of the heterogeneities. The projection method reveals to be applicable in the range of intermediate and large period lengths but fails when the spatial period becomes sma ller than the front width. Then, a second order averaging method must be applied. These analytical results are capable to predict propagation failure, velocity overshoot, and the asymptotic value for the front velocity i n the limit of large period lengths in qualitative, often quantitative agreement with the results of numerical simul ations of the underlying reaction-diffusion equation. Very good agreement between numerical and analytical resul ts has been obtained for waves propagating through a medium with periodically varied excitation threshold.

Analytical, Optimal, and Sparse Optimal Control of Traveling Wave Solutions to Reaction-Diffusion Systems

This work deals with the position control of selected patterns in reaction-diffusion systems. Exemplarily, the Schlogl and FitzHugh-Nagumo model are discussed using three different approaches. First, an analytical solution is proposed. Second, the standard optimal control procedure is applied. The third approach extends standard optimal control to so-called sparse optimal control that results in very localized control signals and allows the analysis of second order optimality conditions.

Stability of position control of traveling waves in reaction-diffusion systems.

We consider the stability of position control of traveling waves in reaction-diffusion system as proposed in [J. Löber, H. Engel, arXiv:1304.2327]. Instead of analyzing the controlled reactiondiffusion system, stability is studied on the reduced level of the equation of motion for the position over time of perturbed traveling waves. We find an interval of perturbations of initial conditions for which position control is stable. This interval can be interpreted as a localized region where traveling waves are susceptible to perturbations. For stationary solutions of reaction-diffusion systems with reflection symmetry, this region does not exist. Analytical results are in qualitative agreement with numerical simulations of the controlled Schlögl model.

Macroscopic Model of Substrate-Based Cell Motility

Eukaryotic cells moving in response to chemical or mechanical stimuli play a fundamental role in tissue growth, wound healing and the immune response. In addition, cell migration is essential for understanding several life-threatening pathologies. At the developmental stage, dysfunction of motility can result in certain disabilities, while in the developed organism it may result in cardiovascular diseases. Motility dysfunction is also involved in cancer growth, especially during metastasis. Beyond such obvious biological and medical relevance, cell motility is also a fascinating example of a self-organized and self-propelled system within the realm of physics.

Analytical approximations for spiral waves

We propose a non-perturbative attempt to solve the kinematic equations for spiral waves in excitable media. From the eikonal equation for the wave front we derive an implicit analytical relation between rotation frequency Ω and core radius R(0). For free, rigidly rotating spiral waves our analytical prediction is in good agreement with numerical solutions of the linear eikonal equation not only for very large but also for intermediate and small values of the core radius. An equivalent Ω(R(+)) dependence improves the result by Keener and Tyson for spiral waves pinned to a circular defect of radius R(+) with Neumann boundaries at the periphery. Simultaneously, analytical approximations for the shape of free and pinned spirals are given. We discuss the reasons why the ansatz fails to correctly describe the dependence of the rotation frequency on the excitability of the medium.

Oscillatory motion of a droplet in an active poroelastic two-phase model

We investigate flow-driven amoeboid motility as exhibited by microplasmodia of Physarum polycephalum. A poroelastic two-phase model with rigid boundaries is extended to the case of free boundaries and substrate friction. The cytoskeleton is modeled as an active viscoelastic solid permeated by a fluid phase describing the cytosol. A feedback loop between a chemical regulator, active mechanical deformations, and induced flows gives rise to oscillatory and irregular motion accompanied by spatio-temporal contraction patterns. We cover extended parameter regimes of active tension and substrate friction by numerical simulations in one spatial dimension and reproduce experimentally observed oscillation periods and amplitudes. In line with experiments, the model predicts alternating forward and backward ectoplasmatic flow at the boundaries with reversed flow in the center. However, for all cases of periodic and irregular motion, we observe practically no net motion. A simple theoretical argument shows that directed motion is not possible with a spatially independent substrate friction.