arXiv subjects

Markus Baer

Publications and source records attributed to Markus Baer.

16 recordsLinked to original sources

arXiv2016

Pattern formation mechanisms in motility mutants of Myxococcus xanthus

Formation of spatial patterns of cells is a recurring theme in biology and often depends on regulated cell motility. Motility of M. xanthus depends on two motility machineries: the S-engine and A-engine. Moving M. xanthus cells can organize into spreading colonies or spore-filled fruiting bodies depending on their nutritional status. To understand these two pattern formation processes and the contributions by the two motility machineries, as well as cell reversal, we analyze spatial self-organization in 3 strains: i) a mutant that moves unidirectionally without reversing by the A-motility system only, ii) a unidirectional mutant that is also equipped with the S-motility system, and iii) the wild-type that, in addition to the two motility systems, reverses its direction of movement. The mutant moving by the A-engine illustrates that collective motion in the form of large moving clusters can arise in gliding bacteria due to steric interactions of the rod-shaped cells, without the need of invoking any biochemical signal regulation. The two-engine strain mutant reveals that the same phenomenon emerges when both motility systems are present, and as long as cells exhibit unidirectional motion only. From the study of these two strains, we conclude that unidirectional cell motion induces the formation of large moving clusters at low and intermediate densities, while it results into vortex formation at very high densities. These findings are consistent with what is known from self-propelled rods which strongly suggests that the combined effect of self-propulsion and volume exclusion interactions is the pattern formation mechanism leading to the observed phenomena. In addition, we learn that when cells reverse, as observed in the wild-type, cells form small but strongly elongated clusters and self-organize into a mesh-like structure at high enough densities.

physics.bio-ph↗

arXiv2014

Spatio-temporal dynamics induced by competing instabilities in two asymmetrically coupled nonlinear evolution equations

Pattern formation often occurs in spatially extended physical, biological and chemical systems due to an instability of the homogeneous steady state. The type of the instability usually prescribes the resulting spatio-temporal patterns and their characteristic length scales. However, patterns resulting from the simultaneous occurrence of instabilities cannot be expected to be simple superposition of the patterns associated with the considered instabilities. To address this issue we design two simple models composed by two asymmetrically coupled equations of non-conserved (Swift-Hohenberg equations) or conserved (Cahn-Hilliard equations) order parameters with different characteristic wave lengths. The patterns arising in these systems range from coexisting static patterns of different wavelengths to traveling waves. A linear stability analysis allows to derive a two parameter phase diagram for the studied models, in particular revealing for the Swift-Hohenberg equations a co-dimension two bifurcation point of Turing and wave instability and a region of coexistence of stationary and traveling patterns. The nonlinear dynamics of the coupled evolution equations is investigated by performing accurate numerical simulations. These reveal more complex patterns, ranging from traveling waves with embedded Turing patterns domains to spatio-temporal chaos, and a wide hysteretic region, where waves or Turing patterns coexist. For the coupled Cahn-Hilliard equations the presence of an weak coupling is sufficient to arrest the coarsening process and to lead to the emergence of purely periodic patterns. The final states are characterized by domains with a characteristic length, which diverges logarithmically with the coupling amplitude.

nlin.PS↗

arXiv2014

Cluster dynamics and cluster size distributions in systems of self-propelled particles

Systems of self-propelled particles (SPP) interacting by a velocity alignment mechanism in the presence of noise exhibit a rich clustering dynamics. It can be argued that clusters are responsible for the distribution of (local) information in these systems. Here, we investigate the statistical properties of single clusters in SPP systems, like the asymmetric spreading of clusters with respect to their moving direction. In addition, we formulate a Smoluchowski-type kinetic model to describe the evolution of the cluster size distribution (CSD). This model predicts the emergence of steady-state CSDs in SPP systems. We test our theoretical predictions in simulations of SPP with nematic interactions and find that our simple kinetic model reproduces qualitatively the transition to aggregation observed in simulations.

physics.bio-ph↗

arXiv2013

A kinetic model and scaling properties for non-equilibrium clustering of self-propelled particles

We demonstrate that the clustering statistics and the corresponding phase transition to non-equilibrium clustering found in many experiments and simulation studies with self-propelled particles (SPPs) with alignment can be obtained from a simple kinetic model. The key elements of this approach are the scaling of the cluster cross-section with the cluster mass -- characterized by an exponent $α$ -- and the scaling of the cluster perimeter with the cluster mass -- described by an exponent $β$. The analysis of the kinetic approach reveals that the SPPs exhibit two phases: i) an individual phase, where the cluster size distribution (CSD) is dominated by an exponential tail that defines a characteristic cluster size, and ii) a collective phase characterized by the presence of non-monotonic CSD with a local maximum at large cluster sizes. At the transition between these two phases the CSD is well described by a power-law with a critical exponent $γ$, which is a function of $α$ and $β$ only. The critical exponent is found to be in the range $0.8 < γ< 1.5$ in line with observations in experiments and simulations.

cond-mat.stat-mech↗

arXiv2008

A mean-field theory for self-propelled particles interacting by velocity alignment mechanisms

A mean-field approach (MFA) is proposed for the analysis of orientational order in a two-dimensional system of stochastic self-propelled particles interacting by local velocity alignment mechanism. The treatment is applied to the cases of ferromagnetic (F) and liquid-crystal (LC) alignment. In both cases, MFA yields a second order phase transition for a critical noise strength and a scaling exponent of 1/2 for the respective order parameters. We find that the critical noise amplitude $η_c$ at which orientational order emerges in the LC case is smaller than in the F-alignment case, i.e. $η^{LC}_{C}<η^{F}_{C}$. A comparison with simulations of individual-based models with F- resp. LC-alignment shows that the predictions about the critical behavior and the qualitative relation between the respective critical noise amplitudes are correct.

cond-mat.stat-mech↗

arXiv2006

Alternative mechanisms of structuring biomembranes: Self-assembly vs. self-organization

We study two mechanisms for the formation of protein patterns near membranes of living cells by mathematical modelling. Self-assembly of protein domains by electrostatic lipid-protein interactions is contrasted with self-organization due to a nonequilibrium biochemical reaction cycle of proteins near the membrane. While both processes lead eventually to quite similar patterns, their evolution occurs on very different length and time scales. Self-assembly produces periodic protein patterns on a spatial scale below 0.1 micron in a few seconds followed by extremely slow coarsening, whereas self-organization results in a pattern wavelength comparable to the typical cell size of 100 micron within a few minutes suggesting different biological functions for the two processes.

q-bio.CB↗

arXiv2004

Antispiral waves are sources in oscillatory reaction-diffusion media

Spiral and antispiral waves are studied numerically in two examples of oscillatory reaction-diffusion media and analytically in the corresponding complex Ginzburg-Landau equation (CGLE). We argue that both these structures are sources of waves in oscillatory media, which are distinguished only by the sign of the phase velocity of the emitted waves. Using known analytical results in the CGLE, we obtain a criterion for the CGLE coefficients that predicts whether antispirals or spirals will occur in the corresponding reaction-diffusion systems. We apply this criterion to the FitzHugh-Nagumo and Brusselator models by deriving the CGLE near the Hopf bifurcations of the respective equations. Numerical simulations of the full reaction-diffusion equations confirm the validity of our simple criterion near the onset of oscillations. They also reveal that antispirals often occur near the onset and turn into spirals further away from it. The transition from antispirals to spirals is characterized by a divergence in the wavelength. A tentative interpretaion of recent experimental observations of antispiral waves in the Belousov-Zhabotinsky reaction in a microemulsion is given.

nlin.PS↗

arXiv2003

Doppler Effect of Nonlinear Waves and Superspirals in Oscillatory Media

Nonlinear waves emitted from a moving source are studied. A meandering spiral in a reaction-diffusion medium provides an example, where waves originate from a source exhibiting a back-and-forth movement in radial direction. The periodic motion of the source induces a Doppler effect that causes a modulation in wavelength and amplitude of the waves (``superspiral''). Using the complex Ginzburg-Landau equation, we show that waves subject to a convective Eckhaus instability can exhibit monotonous growth or decay as well as saturation of these modulations away from the source depending on the perturbation frequency. Our findings allow a consistent interpretation of recent experimental observations concerning superspirals and their decay to spatio-temporal chaos.

cond-mat.stat-mech↗

arXiv2001

Dewetting of thin films on heterogeneous substrates: Pinning vs. coarsening

We study a model for a thin liquid film dewetting from a periodic heterogeneous substrate (template). The amplitude and periodicity of a striped template heterogeneity necessary to obtain a stable periodic stripe pattern, i.e. pinning, are computed. This requires a stabilization of the longitudinal and transversal modes driving the typical coarsening dynamics during dewetting of a thin film on a homogeneous substrate. If the heterogeneity has a larger spatial period than the critical dewetting mode, weak heterogeneities are sufficient for pinning. A large region of coexistence between coarsening dynamics and pinning is found.

cond-mat.soft↗

arXiv2001

Nonlinear Analysis of the Eckhaus Instability: Modulated Amplitude Waves and Phase Chaos with Non-zero Average Phase Gradient

We analyze the Eckhaus instability of plane waves in the one-dimensional complex Ginzburg-Landau equation (CGLE) and describe the nonlinear effects arising in the Eckhaus unstable regime. Modulated amplitude waves (MAWs) are quasi-periodic solutions of the CGLE that emerge near the Eckhaus instability of plane waves and cease to exist due to saddle-node bifurcations (SN). These MAWs can be characterized by their average phase gradient $ν$ and by the spatial period P of the periodic amplitude modulation. A numerical bifurcation analysis reveals the existence and stability properties of MAWs with arbitrary $ν$ and P. MAWs are found to be stable for large enough $ν$ and intermediate values of P. For different parameter values they are unstable to splitting and attractive interaction between subsequent extrema of the amplitude. Defects form from perturbed plane waves for parameter values above the SN of the corresponding MAWs. The break-down of phase chaos with average phase gradient $ν$ > 0 (``wound-up phase chaos'') is thus related to these SNs. A lower bound for the break-down of wound-up phase chaos is given by the necessary presence of SNs and an upper bound by the absence of the splitting instability of MAWs.

nlin.CD↗

arXiv2001

Rippling patterns in aggregates of myxobacteria arise from cell-cell collisions

Experiments with myxobacterial aggregates reveal standing waves called rippling patterns. Here, these structures are modelled with a simple discrete model based on the interplay between migration and collisions of cells. Head-to-head collisions of cells result in cell reversals. To correctly reproduce the rippling patterns, a refractory phase after each cell reversal has to be assumed, during which further reversal is prohibited. The duration of this phase determines the wavelength and period of the ripple patterns as well as the reversal frequency of single cells.

physics.bio-ph↗

arXiv2001

Modulated Amplitude Waves and Defect Formation in the One-Dimensional Complex Ginzburg-Landau Equation

The transition from phase chaos to defect chaos in the complex Ginzburg-Landau equation (CGLE) is related to saddle-node bifurcations of modulated amplitude waves (MAWs). First, the spatial period P of MAWs is shown to be limited by a maximum P_SN which depends on the CGLE coefficients; MAW-like structures with period larger than P_SN evolve to defects. Second, slowly evolving near-MAWs with average phase gradients $ν\approx 0$ and various periods occur naturally in phase chaotic states of the CGLE. As a measure for these periods, we study the distributions of spacings p between neighboring peaks of the phase gradient. A systematic comparison of p and P_SN as a function of coefficients of the CGLE shows that defects are generated at locations where p becomes larger than P_SN. In other words, MAWs with period P_SN represent ``critical nuclei'' for the formation of defects in phase chaos and may trigger the transition to defect chaos. Since rare events where p becomes sufficiently large to lead to defect formation may only occur after a long transient, the coefficients where the transition to defect chaos seems to occur depend on system size and integration time. We conjecture that in the regime where the maximum period P_SN has diverged, phase chaos persists in the thermodynamic limit.

nlin.CD↗

arXiv2000

Drifting Pattern Domains in a Reaction-Diffusion System with Nonlocal Coupling

Drifting pattern domains (DPDs), moving localized patches of traveling waves embedded in a stationary (Turing) pattern background and vice versa, are observed in simulations of a reaction-diffusion model with nonlocal coupling. Within this model, a region of bistability between Turing patterns and traveling waves arises from a codimension-2 Turing-wave bifurcation (TWB). DPDs are found within that region in a substantial distance from the TWB. We investigated the dynamics of single interfaces between Turing and wave patterns. It is found that DPDs exist due to a locking of the interface velocities, which is imposed by the absence of space-time defects near these interfaces.

nlin.PS↗

arXiv2000

Modulated Amplitude Waves and the Transition from Phase to Defect Chaos

The mechanism for transitions from phase to defect chaos in the one-dimensional complex Ginzburg-Landau equation (CGLE) is presented. We introduce and describe periodic coherent structures of the CGLE, called Modulated Amplitude Waves (MAWs). MAWs of various period P occur naturally in phase chaotic states. A bifurcation study of the MAWs reveals that for sufficiently large period P, pairs of MAWs cease to exist via a saddle-node bifurcation. For periods beyond this bifurcation, incoherent near-MAW structures occur which evolve toward defects. This leads to our main result: the transition from phase to defect chaos takes place when the periods of MAWs in phase chaos are driven beyond their saddle-node bifurcation.

nlin.CD↗

arXiv1999

Discrete stochastic modeling of calcium channel dynamics

We propose a simple discrete stochastic model for calcium dynamics in living cells. Specifically, the calcium concentration distribution is assumed to give rise to a set of probabilities for the opening/closing of channels which release calcium thereby changing those probabilities. We study this model in one dimension, analytically in the mean-field limit of large number of channels per site N, and numerically for small N. As the number of channels per site is increased, the transition from a non-propagating region of activity to a propagating one changes in nature from one described by directed percolation to that of deterministic depinning in a spatially discrete system. Also, for a small number of channels a propagating calcium wave can leave behind a novel fluctuation-driven state, in a parameter range where the limiting deterministic model exhibits only single pulse propagation.

patt-sol↗

arXiv1999

Stratified spatiotemporal chaos in anisotropic reaction-diffusion systems

Numerical simulations of two dimensional pattern formation in an anisotropic bistable reaction-diffusion medium reveal a new dynamical state, stratified spatiotemporal chaos, characterized by strong correlations along one of the principal axes. Equations that describe the dependence of front motion on the angle illustrate the mechanism leading to stratified chaos.

patt-sol↗