Mathematical Modeling of Myosin Induced Bistability of Lamellipodial Fragments

For various cell types and for lamellipodial fragments on flat surfaces, externally induced and spontaneous transitions between symmetric nonmoving states and polarized migration have been observed. This behavior is indicative of bistability of the cytoskeleton dynamics. In this work, the Filament Based Lamellipodium Model (FBLM), a two-dimensional, anisotropic, two-phase continuum model for the dynamics of the actin filament network in lamellipodia, is extended by a new description of actin-myosin interaction. For appropriately chosen parameter values, the resulting model has bistable dynamics with stable states showing the qualitative features observed in experiments. This is demonstrated by numerical simulations and by an analysis of a strongly simplified version of the FBLM with rigid filaments and planar lamellipodia at the cell front and rear

An Extended Filament Based Lamellipodium Model Produces Various Moving Cell Shapes in the Presence of Chemotactic Signals

The Filament Based Lamellipodium Model (FBLM) is a two-phase two-dimensional continuum model, describing the dynamcis of two interacting families of locally parallel actin filaments (C.Schmeiser and D.Oelz, How do cells move? Mathematical modeling of cytoskeleton dynamics and cell migration. Cell mechanics: from single scale-based models to multiscale modeling. Chapman and Hall, 2010). It contains accounts of the filaments' bending stiffness, of adhesion to the substrate, and of cross-links connecting the two families. An extension of the model is presented with contributions from nucleation of filaments by branching, from capping, from contraction by actin-myosin interaction, and from a pressure-like repulsion between parallel filaments due to Coulomb interaction. The effect of a chemoattractant is described by a simple signal transduction model influencing the polymerization speed. Simulations with the extended model show its potential for describing various moving cell shapes, depending on the signal transduction procedure, and for predicting transients between nonmoving and moving states as well as changes of direction

A mathematical model of actin-myosin interaction and its application to keratocyte movement

Zielsetzung dieser Diplomarbeit ist das bessere Verständnis von Prozessen, die der eigenständigen Bewegung von Zellen zugrunde liegen. Ein detailliertes mathematisches Model für die Wechselwirkungen zwischen den filament-bildenden Proteinen Aktin und Myosin wird hergeleitet, welches versucht die wesentlichen physikalischen Kräfte zu beschreiben, die auf einen gebundenen Myosinkopf wirken. Es führt auf eine nicht-lineare kinetische Transportgleichung für die Verteilungsfunktion der gebundenen Köpfe. Ungewöhnlich an dieser Gleichung ist einerseits die Verwendung von Delta-Distributionen, andererseits die Tatsache, dass sich für die Momente der Verteilungsfunktion ein geschlossenes, autonomes System nicht-linearer, gewöhnlicher Differenzialgleichungen ergibt. Es zeigt sich, dass die Gleichgewichtspunkte des Ode-Systems global stabil sind. Nachdem das Moment-System gelöst ist, kann die Transportgleichung durch die Charakteristiken-Methode gelöst werden. Es zeigt sich, dass der Einfluss einer vorgegebenen Anfangsverteilung mit der Zeit abnimmt und der restliche Anteil zu jedem Zeitpunkt auf einer einzelnen Kurve konzentriert ist. In Kapitel 3 schließlich werden die Überlegungen der Aktin-Myosin Interaktionen auf die konkrete Situation von Keratozyten (bewegliche Fischzellen) angewendet, welche sich mit Hilfe eines dünnen, blattartigen Fortsatz, welcher Lamellipodium genannt wird, fortbewegen. Im Zuge der Modelliering wird ein Variationsproblem mit zugehöriger Euler-Lagrange Gleichung hergeleitet.This work aims at a better understanding of processes, which form the basis of cell movement. A detailed mathematical model for the interaction between the filament forming proteins actin and myosin is derived, which tries to take into account all essential forces which act on attached myosin heads. It leads to a non-linear kinetic transport equation for a distribution function of attached heads. Two things are unusual about this equation: First the usage of Delta distributions and secondly, it is possible to derive a system of closed ordinary differential equations for the moments of the distribution function. It can be shown that their (unique) steady states are globally stable. After solving the moment system, the transport equation can be solved by the methods of characteristics. It turns out that the influence of a given initial distribution decays with time and that the remaining part is concentrated on a single curve in phase space. Then the considerations about actin-myosin interactions are applied to the concrete situation of keratocytes (motile fish cells) which move by thin sheet-like protrusions called lamellipodia. In the course of the modeling a variational problem and a corresponding Euler-Lagrange equation is derived

Controlling periodic long-range signalling to drive a morphogenetic transition

Cells use signal relay to transmit information across tissue scales. However, the production of information carried by signal relay remains poorly characterised. To determine how the coding features of signal relay are generated, we used the classic system for long-range signalling: the periodic cAMP waves that drive Dictyostelium collective migration. Combining imaging and optogenetic perturbation of cell signalling states, we find that migration is triggered by an increase in wave frequency generated at the signalling centre. Wave frequency is regulated by cAMP wave circulation, which organises the long-range signal. To determine the mechanisms modulating wave circulation, we combined mathematical modelling, the general theory of excitable media and mechanical perturbations to test competing models. Models in which cell density and spatial patterning modulate the wave frequency cannot explain the temporal evolution of signalling waves. Instead, our evidence leads to a model where wave circulation increases the ability for cells to relay the signal, causing further increase in the circulation rate. This positive feedback between cell state and signalling pattern regulates the long-range signal coding that drives morphogenesis

Chaos in synthetic microbial communities

Predictability is a fundamental requirement in biological engineering. As we move to building coordinated multicellular systems, the potential for such systems to display chaotic behaviour becomes a concern. Therefore understanding which systems show chaos is an important design consideration. We developed a methodology to explore the potential for chaotic dynamics in small microbial communities governed by resource competition, intercellular communication and competitive bacteriocin interactions. Our model selection pipeline uses Approximate Bayesian Computation to first identify oscillatory behaviours as a route to finding chaotic behaviour. We have shown that we can expect to find chaotic states in relatively small synthetic microbial systems, understand the governing dynamics and provide insights into how to control such systems. This work is the first to query the existence of chaotic behaviour in synthetic microbial communities and has important ramifications for the fields of biotechnology, bioprocessing and synthetic biology

Centering and symmetry breaking in confined contracting actomyosin networks

Centering and decentering of cellular components is essential for internal organization of cells and their ability to perform basic cellular functions such as division and motility. How cells achieve proper localization of their components is still not well-understood, especially in large cells such as oocytes. Here, we study actin-based positioning mechanisms in artificial cells with persistently contracting actomyosin networks, generated by encapsulating cytoplasmic Xenopus egg extracts into cell-sized water-in-oil droplets. We observe size-dependent localization of the contraction center, with a symmetric configuration in larger cells and a polar one in smaller cells. In the symmetric state, the contraction center is actively centered, via a hydrodynamic mechanism based on Darcy friction between the contracting network and the surrounding cytoplasm. During symmetry breaking, transient attachments to the cell boundary drive the contraction center to a polar location near the droplet boundary. Our findings demonstrate a robust, yet tunable, mechanism for subcellular localization

Analyzing Collective Motion with Machine Learning and Topology

We use topological data analysis and machine learning to study a seminal model of collective motion in biology [D'Orsogna et al., Phys. Rev. Lett. 96 (2006)]. This model describes agents interacting nonlinearly via attractive-repulsive social forces and gives rise to collective behaviors such as flocking and milling. To classify the emergent collective motion in a large library of numerical simulations and to recover model parameters from the simulation data, we apply machine learning techniques to two different types of input. First, we input time series of order parameters traditionally used in studies of collective motion. Second, we input measures based in topology that summarize the time-varying persistent homology of simulation data over multiple scales. This topological approach does not require prior knowledge of the expected patterns. For both unsupervised and supervised machine learning methods, the topological approach outperforms the one that is based on traditional order parameters.Comment: Published in Chaos 29, 123125 (2019), DOI: 10.1063/1.512549