Active poroelastic two-phase model for the motion of physarum microplasmodia.

The onset of self-organized motion is studied in a poroelastic two-phase model with free boundaries for Physarum microplasmodia (MP). In the model, an active gel phase is assumed to be interpenetrated by a passive fluid phase on small length scales. A feedback loop between calcium kinetics, mechanical deformations, and induced fluid flow gives rise to pattern formation and the establishment of an axis of polarity. Altogether, we find that the calcium kinetics that breaks the conservation of the total calcium concentration in the model and a nonlinear friction between MP and substrate are both necessary ingredients to obtain an oscillatory movement with net motion of the MP. By numerical simulations in one spatial dimension, we find two different types of oscillations with net motion as well as modes with time-periodic or irregular switching of the axis of polarity. The more frequent type of net motion is characterized by mechano-chemical waves traveling from the front towards the rear. The second type is characterized by mechano-chemical waves that appear alternating from the front and the back. While both types exhibit oscillatory forward and backward movement with net motion in each cycle, the trajectory and gel flow pattern of the second type are also similar to recent experimental measurements of peristaltic MP motion. We found moving MPs in extended regions of experimentally accessible parameters, such as length, period and substrate friction strength. Simulations of the model show that the net speed increases with the length, provided that MPs are longer than a critical length of ≈ 120 µm. Both predictions are in line with recent experimental observations.

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.

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

We consider the stability of position control of traveling waves in reaction-diffusion systems as proposed in Löber and Engel [Phys. Rev. Lett. 112, 148305 (2014)]. Instead of analyzing the controlled reaction-diffusion 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.

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.

Modeling crawling cell movement on soft engineered substrates.

Self-propelled motion, emerging spontaneously or in response to external cues, is a hallmark of living organisms. Systems of self-propelled synthetic particles are also relevant for multiple applications, from targeted drug delivery to the design of self-healing materials. Self-propulsion relies on the force transfer to the surrounding. While self-propelled swimming in the bulk of liquids is fairly well characterized, many open questions remain in our understanding of self-propelled motion along substrates, such as in the case of crawling cells or related biomimetic objects. How is the force transfer organized and how does it interplay with the deformability of the moving object and the substrate? How do the spatially dependent traction distribution and adhesion dynamics give rise to complex cell behavior? How can we engineer a specific cell response on synthetic compliant substrates? Here we generalize our recently developed model for a crawling cell by incorporating locally resolved traction forces and substrate deformations. The model captures the generic structure of the traction force distribution and faithfully reproduces experimental observations, like the response of a cell on a gradient in substrate elasticity (durotaxis). It also exhibits complex modes of cell movement such as "bipedal" motion. Our work may guide experiments on cell traction force microscopy and substrate-based cell sorting and can be helpful for the design of biomimetic "crawlers" and active and reconfigurable self-healing materials.

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.

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.

Front propagation in one-dimensional spatially periodic bistable media.

Front propagation in heterogeneous bistable media is studied using the Schlögl model as a representative example. Spatially periodic modulations in the parameters of the bistable kinetics are taken into account perturbatively. Depending on the ratio L/l (L is the spatial period of the heterogeneity, l is the front width), appropriate singular perturbation techniques are applied to derive an ordinary differential equation for the position of the front in the presence of the heterogeneities. From this equation, the dependence of the average propagation speed on L/l as well as on the modulation amplitude is calculated. The analytical results obtained predict velocity overshoot, different cases of propagation failure, and the propagation speed for very large spatial periods in quantitative agreement with the results of direct numerical simulations of the underlying reaction-diffusion equation.