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Walter Zimmermann

Publications and source records attributed to Walter Zimmermann.

8 recordsLinked to original sources
PubMed2006

Traveling ion channel density waves affected by a conservation law.

A model of mobile, charged ion channels embedded in a biomembrane is investigated. The ion channels fluctuate between an opened and a closed state according to a simple two-state reaction scheme whereas the total number of ion channels is a conserved quantity. Local transport mechanisms suggest that the ion channel densities are governed by electrodiffusionlike equations that have to be supplemented by a cable-type equation describing the dynamics of the transmembrane voltage. It is shown that the homogeneous distribution of ion channels may become unstable to either a stationary or an oscillatory instability. The nonlinear behavior immediately above threshold of an oscillatory bifurcation occurring at finite wave number is analyzed in terms of amplitude equations. Due to the conservation law imposed on ion channels, large-scale modes couple to the finite-wave-number instability and have thus to be included in the asymptotic analysis near the onset of pattern formation. A modified Ginzburg-Landau equation extended by long-wavelength stationary excitations is established, and it is highlighted how the global conservation law affects the stability of traveling ion channel density waves.

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PubMed2006

Particles held by springs in a linear shear flow exhibit oscillatory motion.

The dynamics of small spheres that are held by linear springs in a low Reynolds number shear flow at neighboring locations is investigated. The flow elongates the beads, and the interplay of the shear gradient with the nonlinear behavior of the hydrodynamic interaction among the spheres causes in a large range of parameters a bifurcation to a surprising oscillatory bead motion. The parameter ranges wherein this bifurcation is either super- or subcritical are determined.

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PubMed2006

Harmonic versus subharmonic patterns in a spatially forced oscillating chemical reaction.

The effects of a spatially periodic forcing on an oscillating chemical reaction as described by the Lengyel-Epstein model are investigated. We find a surprising competition between two oscillating patterns, where one is harmonic and the other subharmonic with respect to the spatially periodic forcing. The occurrence of a subharmonic pattern is remarkable as well as its preference up to rather large values of the modulation amplitude. For small modulation amplitudes we derive from the model system a generic equation for the envelope of the oscillating reaction that includes an additional forcing contribution, compared to the amplitude equations known from previous studies in other systems. The analysis of this amplitude equation allows the derivation of analytical expressions even for the forcing corrections to the threshold and to the oscillation frequency, which are in a wide range of parameters in good agreement with the numerical analysis of the complete reaction equations. In the nonlinear regime beyond threshold, the subharmonic solutions exist in a finite range of the control parameter that has been determined by solving the reaction equations numerically for various sets of parameters.

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PubMed2004

Pattern formation driven by nematic ordering of assembling biopolymers.

The biopolymers actin and microtubules are often in an ongoing assembling-disassembling state far from thermal equilibrium. Above a critical density this leads to spatially periodic patterns, as shown by a scaling argument and in terms of a phenomenological continuum model, which meets also Onsager's statistical theory of the nematic-to-isotropic transition in the absence of reaction kinetics. This pattern forming process depends much on nonlinear effects and a common linear stability analysis of the isotropic distribution of the filaments is often misleading. The wave number of the pattern decreases with the assembling-disassembling rate and there is an uncommon discontinuous transition between the nematic and periodic states.

Actins↗
PubMed2003

Static dynamics approach to relaxation modes and times for deformed polymers.

We discuss relaxation of conformational fluctuations around deformed polymer states. To this end, Brownian dynamics simulations of bead-spring models including a finite extensibility of the springs as well as excluded volume and hydrodynamic interactions between the beads have been performed. Complete spectra of relaxation times as well as corresponding relaxation modes are obtained from the simulation data by applying the static dynamics formalism, which rigorously describes the initial decay of correlations between the bead positions. As shown here, this procedure amounts to using a generalized Rouse-Zimm-like model which is governed by linear effective equations of motion having the same initial decay of correlations as the full nonlinear bead-spring model used in the simulations. In thermal equilibrium, the well-known scaling laws in the presence of excluded volume and hydrodynamic interactions between the beads are recovered, but, in addition, the static dynamics method also yields numeric values for the nonuniversal prefactors of the respective laws. The method is equally applicable to a broad range of problems, where the polymer is deformed by the action of flows or forces. Two examples of recent interest are considered: a tethered polymer pulled at its free end and one which is stretched by a uniform flow. It is shown that in both cases, the relaxation process is dominated by a finite extensibility of the springs.

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PubMed2003

Modeling oscillatory microtubule polymerization.

Polymerization of microtubules is ubiquitous in biological cells and under certain conditions it becomes oscillatory in time. Here, simple reaction models are analyzed that capture such oscillations as well as the length distribution of microtubules. We assume reaction conditions that are stationary over many oscillation periods, and it is a Hopf bifurcation that leads to a persistent oscillatory microtubule polymerization in these models. Analytical expressions are derived for the threshold of the bifurcation and the oscillation frequency in terms of reaction rates, and typical trends of their parameter dependence are presented. Both, a catastrophe rate that depends on the density of guanosine triphosphate liganded tubulin dimers and a delay reaction, such as the depolymerization of shrinking microtubules or the decay of oligomers, support oscillations. For a tubulin dimer concentration below the threshold, oscillatory microtubule polymerization occurs transiently on the route to a stationary state, as shown by numerical solutions of the model equations. Close to threshold, a so-called amplitude equation is derived and it is shown that the bifurcation to microtubule oscillations is supercritical.

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PubMed2002

Spiral-defect chaos: Swift-Hohenberg model versus Boussinesq equations.

Spiral-defect chaos (SDC) in Rayleigh-Bénard convection is a well-established spatio-temporal complex pattern, which competes with stationary rolls near the onset of convection. The characteristic properties of SDC are accurately described on the basis of the standard three-dimensional Boussinesq equations. As a much simpler and attractive two-dimensional model for SDC generalized Swift-Hohenberg (SH) equations have been extensively used in the literature from the early beginning. Here, we show that the description of SDC by SH models has to be considered with care, especially regarding its long-time dynamics. For parameters used in previous SH simulations, SDC occurs only as a transient in contrast to the experiments and the rigorous solutions of the Boussinesq equations. The small-scale structure of the vorticity field at the spiral cores, which might be crucial for persistent SDC, is presumably not perfectly captured in the SH model.

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