Modeling and simulation of non-muscle actomyosin bundles

The aim of the project is to gain better understanding of the structure and the functional interplay of the inner components of several specific types of non-muscle actomyosin bundles, namely in-vitro reconstituted actomyosin bundles, cytokinesis constriction rings in budding yeast and C. elegans and stress fibers in fibroblasts. As a tool mathematical modeling by continuum models of PDE-type (partial differential equation), which are derived as macroscopic limits of microscopic models, and time-dependent simulations will be used. These fluid models in one spatial dimension will take into account the two possible orientations of actin filaments and therefore couple continuum equations for the concentrations of each of them. They will be complemented by equations for the associated fluxes assuming an over-damped regime. This modeling approach is very flexible and allows to simulate an arbitrarily large number of filaments which may be highly interconnected by cross-linkers. It can be derived from a class of microscopic models where the endpoints of every single actin filament is tracked and forces due to myosin-II thick filaments as well as drag forces caused by bundling and cross-linking proteins are taken into account. These effects are recovered in the continuum models. For example drag friction caused by bundling proteins appears as a viscosity in the continuum model propagating locally generated contractile forces to the tips of the bundle. A number of additional effects can be modeled. Amongst them are friction with respect to the cytoplasm or the cortex, inhomogeneous distributions of myosin-II thick filaments and cross-linker/bundling proteins, a complex force-velocity relationship of the molecular motors, a spatially inhomogeneous balance of polymerization and depolymerization, insertion, nucleation as well as severing and degradation of actin filaments in the bundle. In the typical cases where the length of the actomyosin bundle is dynamic, the model is posed on a domain with a free boundary. The specific choice of boundary data and free boundary conditions, respectively, has to be formulated as a mathematical model of the mechanical structures at the tips of the bundle. With respect to the three aforementioned experimental systems, the present proposal suggests to frame a complete quantitative, dynamic picture of all the processes and structures involved by mathematical modeling and simulation. I propose to undertake this project in collaboration with members of the host institution who have a long term expertise in modeling motile assemblies. It will allow us to arrive at conclusions concerning phenomena such as the minimal conditions for actomyosin contractility, the origin of contractile force in cytokinesis constriction rings, upregulation mechanisms for stress fiber contractility and mechanisms leading to stress fiber disassembly.

In this project we investigated the effect of molecular motor proteins in intra-cellular structures which contain fibre bundles such as the constriction rings of the cell division process and axons and dendrites of neuron cells. The following sub-projects have been a part of this Schroedinger fellowship research project.a) The most important contribution of this Schroedinger project was to identify a minimal mechanism which explains the contraction of disordered actomyosin bundles in living cells. We focused on the constriction ring of the cell division process having in mind that the mechanism we identified could as well be effective in other contractile structures such as stress fibres. Using computer simulations of a ring filled with actin filaments and myosin motor proteins we found that treadmilling, i.e. biased elongation of filaments at one tip and shortening at the other tip in combination with mechanical cross-linking through specialised proteins promotes the constriction of cell division ring. We also found that with time a pattern formation takes place in the ring worsening the contractile process. The more random actin dynamics is, the longer the actomyosin ring stays disorganised and the higher the contractile force and rate it generates.b) We also derived a fluid-type model which on a coarser level describes the same dynamics as the detailed agent-based model simulations. The model features highly nontrivial pattern formation and explains the aggregation of actin and myosin predicted by the microscopic model. In addition it allows us to explain key experimental observations such as the fact that the duration of cell division does not depend on the cell size.c) One sub-project of this Schroedinger Project also dealt with the cargo transport through motor proteins in filament bundles with polarity graded in space. We derive a drift-diffusion model which reveals that drift dominates in unidirectional bundles while diffusion dominates in isotropic bundles. In general, however, those two modes of transport are balanced according to the polarity and thickness of the filament bundle.d) In a project on the growth of microtubule filled cellular processes we focussed on the question how molecular motors of the kinesin and dynein families organise the microtubule network in Drosophila S2 cells and establish and elongate cellular extensions. Using modelling and Brownian Dynamics simulations of microtubule dynamics and process growth we found that cortex bound dynein promotes polarity sorting of microtubules and further process elongation once a given process is longer than a threshold length. Furthermore we show that the mechanism of process elongation depends critically on microtubule dynamic instability, i.e. the alternating growing and shrinking at the plus ends of microtubules.