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Mathematical Modelling and Simulation

Working Group 8.41

Active Fluids

Active fluids are self-sustained fluids that permanently transform energy from their surrounding into a directed motion. Such a system is always out of thermodynamic equilibrium even without external forces. Representative examples for active fluids are bacterial suspensions, chemical/optical driven colloids or moving cells. Investigations of active fluids give at a better insight into non-equilibrium systems and may have the potential for novel applications.

Bacterial suspensions


In the past decade, bacterial like Bacillus subtilis an Escherichia coli have emerged as important biophysical model systems for investigations of collective phenomena of microswimmers. Such bacteria are swimming in water by the rotational motion of flagella. The flagella motion gives rise to a characteristic flow of the surrounding fluid. The solvent flow generated by the bacterium averaged over a swimming stroke is approximately given by a force dipole. Thereby, it is assumed that forces acts locally at the flagella and the bacteria’s head (see Fig. 1). 

Figure 1: Sketch of a bacteria with flagella. Forces act on the ambient fluid in both directions (swimmer is force-free).

The collective motion of a large number of such bacteria is determined by the motion of the individuals and the dynamics of the ambient solvent fluid. For a simple microscopic model overdamped Langevin equations describing the position and orientation of the bacteria were coupled to the Stokes equations for the solvent fluid. For a large number of bacteria, continuum equations can be derived from the microscopic model. Thereby, the averaged orientations of the microswimmer are described by a polar order parameter P. The collective dynamics of the bacterial suspension is than modeled by a relaxation equation for the polar order parameter P coupled to the Stokes-equations for the ambient fluid u. By simulations, we found that the coupling between self-propulsion, orientation and the dynamics of u is responsible for a transition to turbulence in bacterial suspensions.

Mesocale turbulence

Figure 2: Snap shot of a simulation of a dense bacterial suspension in 3D. Streamlines show the velocity of mean bacteria. The color-code depicts the vorticity.

Experimental investigations of dense Bacillus subtilis suspensions and numerical simulations of bacterial turbulence show a remarkable feature. In contrast to turbulence found in ordinary passive fluids, the vortex dynamics is restricted to a certain range (to a meso-scale). As a result, vortices of a specific size are emergent in the dynamics of dense bacterial suspensions. This phenomenon can be described by continuum equations that we derived from a simple microscopic swimmer model. Here the typical vortex size can be related with details of the individual swimmers. According to this the typical vortex size results from the interplay between local orientation, self-propulsion and the generated flow field.  Furthermore, the vortex size depends on the force-dipole distance. Due to these features turbulence of active fluids is to be distinguished from ordinary turbulence (Fig. 2).


Fig. 3: Mesoskalenturbulenz (links) und Wirbelgitter (rechts) in einer Simulation selbst-angetriebener Teilchen mit konkurrierenden Wechselwirkungen.

Active viscoelastic models of flows in cells


 Biophysicists and cell biologists have taken an interest for some time now in how spatial structures form spontaneously in cells and tissues and which physical, chemical and biological mechanisms are decisive for this process. Whereas biochemical and genetical processes have long been investigated as "driving forces", increasingly, mechanical forces and the flow processes they create inside the cell have recently moved into the focus of research. Knowing the dynamics of such mechanisms and of chemical processes in cells is decisive to understand important biological aspects such as cell motion and the development of cell tissues.

Mechanical forces are generated by the contraction of clusters of the biopolymer "actin". This motion, in turn, is generated by the activity of myosin molecules that belong to the molecular motors. The myosin-regulating calcium ions inside the cell have to redistribute to enable this process to create biological structures; this causes a feedback reaction towards the mechanics of the actin clusters

Modell and Simulation

Mathematical models of these processes developed within the scope of a cooperation project between PTB and the AG Engel at  TU Berlin, describe the content of a cell  as active viscoelastic medium.  In particular, an approach to consider the cell as a porous and elastic (also "poroelastic") medium has  been applied successfully to the biophysical model organism Physarum polycephalum. Physarum cells contain a great number of cell nuclei and reach macroscopic dimensions. Hereby, they take on complex shapes. If a small amount of cell plasma is taken from larger cells, cylindrical microdrops form. In these drops, the most diverse deformation waves can be observed and PTB's model has now allowed them to be reproduced successfully. In addition, the model predicts a strong coupling of the mechanical processes to the intracellular fluid dynamics and the local calcium concentration.

These simulations are also dedicated to the quantitative assessment of dynamic processes in cells and are to initiate new measurements of the intracellular flows and spatial distributions of calcium and actin (e.g. by means of fluorescence methods) as well as of the mechanical and elastic properties of the cell material. 

These numerical simulations of the poroelastic model show the microdrops with different structures. From top right (clockwise): turbulent structures, rotating spiral waves, an antiphase oscillation, and an "even" wave running from right to left. The colours indicate regions of mechanical contraction (blue) and relaxation (red) which cause flow motions with velocities in the range from 0.1 µm to 1 µm per second.

Aktive Stäbchen und Myxobakterien


Synchronization of coupled oscillators

A basic collective phenomenon of several oscillators is synchronization where the phase difference between oscillators is constant in time. We investigate various scenarios in which synchronization of coupled oscillators is prominent. In a minimal model of beating cilia, hydrodynamic interactions are sufficient to explain synchronization and the observed metachronal waves [Niedermayer et.al., Chaos 18, 2008]. In some biological systems, the motion of individual units is also coupled to the synchronization of their internal degrees of freedom. In this context, an interesting theoretical problem is the influence of oscillator motion on synchronization. This issue has been studied by mean of a particle based model in conjunction with a field theory. It turns out that pattern formation relies on particle transport. In particular, active motion facilitates the formation of synchronized states [Großmann et.al., Phys.Rev.E., 2016].


Diffusion of active particles with directional reversal

Various bacterial species, such as Myxococcus xanthus and Pseudomonas putida, exhibit a so-called "run-and-reverse" motion: Persistent motion is interrupted by sudden directional reversal. Our analysis of the "run-and-reverse" motion pattern suggests the existence of finite parameter values which maximize locomotion efficiency [Großmann et.al., New J.Phys., 2016].


Self-propelled rods and myxobacteria

Motivated by the diverse dynamical structures which are observed during the life cycle of Myxococcus xanthus, various models of active ("self-propelled") rods have been developed, simulated, analyzed, and compared with measurements. Extended self-propelled rods in two dimensions with excluded volume interactions exhibit a strong tendency for cluster formation [Peruani et.al., Phys. Rev. E, 2006]. These model predictions have been confirmed in experiments with myxobacteria [Peruani et.al., Phys. Rev. Lett., 2012]. Simplified models allow for the simulation of millions of self-propelled rods and exhibit fascinating macroscopic patterns, such as nematic bands and turbulent fragments [Ginelli et.al., Phys. Rev. Lett., 2010]. The analysis of the nonlinear dynamics of these bands and their description via effective hydrodynamic equations are subject to current research.



Fig. 3: Mesoscale turbulence (left) and vortex array (right) in a simulation of self-propelled particles with competing interactions.

Diffusion aktiver Teilchen mit Bewegungsumkehr

Die Charakterisierung der Bewegung von Mikroorganismen ist zentral für das Verständnis, wie Bakterienkolonien wachsen sowie für die Nahrungssuche dieser Lebewesen und ihr kollektives Verhalten. Verschiedene Spezies von Bakterien, beispielsweise Myxococcus xanthus oder Pseudomonas putida, bewegen sich gemäß einer "run-and-reverse"-Strategie, d.h. persistente Bewegung ("runs") sind unterbrochen durch plötzliche Umkehr der Bewegungsrichtung ("reversal"). Unsere Analyse der "run-and-reverse"-Bewegungsform legt nahe, dass optimale Parameterwerte im Sinne einer maximalen Ausbreitungseffizienz existieren [Großmann et.al., New J.Phys., 2016] -- eine wichtige Voraussetzung für eine effiziente Nahrungssuche.

Synchronisation gekoppelter Oszillatoren

Ein elementares kollektives Phänomen mehrerer Oszillatoren ist die Synchronisation, bei dem die Phasendifferenz der Oszillatoren konstant wird. Wir untersuchen verschieden Situationen in denen die Synchronisation gekoppelter Oszillatoren eine Rolle spielt. Es zeigt sich zum Beispiel, dass in einem minimalen Modell [Niedermayer et.al., Chaos 18, 2008] die hydrodynamischen Wechselwirkungen ausreichen um die beobachtete Synchronisation von Zilien zu erklären. Eine Erweiterung des Modells ermöglicht darüber hinaus weitere kollektive Phänomene. In vielen biologischen Systemen spielt zusätzlich die Bewegung einzelner Entitäten und die Synchronisation innerer Freiheitsgrade entscheidend zusammen. Aus theoretischer Sicht stellt sich die Frage, welchen Einfluss die Bewegung der Oszillatoren auf das Synchronisationsverhalten des Gesamtsystems hat. Anhand eines teilchenbasierten Modells in Verbindung mit einer Feldtheorie wurde eben dieses Wechselspiel von Synchronisation und Teilchentransport studiert. Es stellt sich heraus, dass die Musterbildung entscheidend vom Teilchentransport abhängt und insbesondere aktive Bewegung die Ausbildung synchronisierter Zustände fördert [Großmann et.al., Phys.Rev.E., 2016].


Publication single view


Title: Oscillations and uniaxial mechanochemical waves in a model of an active poroelastic medium: Application to deformation patterns in protoplasmic droplets of Physarum polycephalum
Author(s): S. Alonso, U. Strachauer, M. Radszuweit, M. Bär and M. Hauser
Journal: Physica D
Year: 2016
Volume: 318
Pages: 58-69
DOI: 10.1016/j.physd.2015.09.017
Tags: 8.41, Spatio-Diff, ActFluid

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