The Influence of Nucleus Mechanics in Modelling Adhesion-independent Cell Migration in Structured and Confined Environments.

Recent biological experiments (Lämmermann et al. in Nature 453(7191):51-55, 2008; Reversat et al. in Nature 7813:582-585, 2020; Balzer et al. in ASEB J Off Publ Fed Am Soc Exp Biol 26(10):4045-4056, 2012) have shown that certain types of cells are able to move in structured and confined environments even without the activation of focal adhesion. Focusing on this particular phenomenon and based on previous works (Jankowiak et al. in Math Models Methods Appl Sci 30(03):513-537, 2020), we derive a novel two-dimensional mechanical model, which relies on the following physical ingredients: the asymmetrical renewal of the actin cortex supporting the membrane, resulting in a backward flow of material; the mechanical description of the nuclear membrane and the inner nuclear material; the microtubule network guiding nucleus location; the contact interactions between the cell and the external environment. The resulting fourth order system of partial differential equations is then solved numerically to conduct a study of the qualitative effects of the model parameters, mainly those governing the mechanical properties of the nucleus and the geometry of the confining structure. Coherently with biological observations, we find that cells characterized by a stiff nucleus are unable to migrate in channels that can be crossed by cells with a softer nucleus. Regarding the geometry, cell velocity and ability to migrate are influenced by the width of the channel and the wavelength of the external structure. Even though still preliminary, these results may be potentially useful in determining the physical limit of cell migration in confined environments and in designing scaffolds for tissue engineering.

A mathematical model of p62-ubiquitin aggregates in autophagy.

Aggregation of ubiquitinated cargo by oligomers of the protein p62 is an important preparatory step in cellular autophagy. In this work a mathematical model for the dynamics of these heterogeneous aggregates in the form of a system of ordinary differential equations is derived and analyzed. Three different parameter regimes are identified, where either aggregates are unstable, or their size saturates at a finite value, or their size grows indefinitely as long as free particles are abundant. The boundaries of these regimes as well as the finite size in the second case can be computed explicitly. The growth in the third case (quadratic in time) can also be made explicit by formal asymptotic methods. In the absence of rigorous results the dynamic stability of these structures has been investigated by numerical simulations. A comparison with recent experimental results permits a partial parametrization of the model.

Hypocoercivity and Fast Reaction Limit for Linear Reaction Networks with Kinetic Transport.

The long time behavior of a model for a first order, weakly reversible chemical reaction network is considered, where the movement of the reacting species is described by kinetic transport. The reactions are triggered by collisions with a nonmoving background with constant temperature, determining the post-reactional equilibrium velocity distributions. Species with different particle masses are considered, with a strong separation between two groups of light and heavy particles. As an approximation, the heavy species are modeled as nonmoving. Under the assumption of at least one moving species, long time convergence is proven by hypocoercivity methods for the cases of positions in a flat torus and in whole space. In the former case the result is exponential convergence to a spatially constant equilibrium, and in the latter it is algebraic decay to zero, at the same rate as solutions of parabolic equations. This is no surprise since it is also shown that the macroscopic (or reaction dominated) behavior is governed by the diffusion equation.

Load Adaptation of Lamellipodial Actin Networks.

Actin filaments polymerizing against membranes power endocytosis, vesicular traffic, and cell motility. In vitro reconstitution studies suggest that the structure and the dynamics of actin networks respond to mechanical forces. We demonstrate that lamellipodial actin of migrating cells responds to mechanical load when membrane tension is modulated. In a steady state, migrating cell filaments assume the canonical dendritic geometry, defined by Arp2/3-generated 70° branch points. Increased tension triggers a dense network with a broadened range of angles, whereas decreased tension causes a shift to a sparse configuration dominated by filaments growing perpendicularly to the plasma membrane. We show that these responses emerge from the geometry of branched actin: when load per filament decreases, elongation speed increases and perpendicular filaments gradually outcompete others because they polymerize the shortest distance to the membrane, where they are protected from capping. This network-intrinsic geometrical adaptation mechanism tunes protrusive force in response to mechanical load.

Existence of and decay to equilibrium of the filament end density along the leading edge of the lamellipodium.

A model for the dynamics of actin filament ends along the leading edge of the lamellipodium is analyzed. It contains accounts of nucleation by branching, of deactivation by capping, and of lateral flow along the leading edge by polymerization. A nonlinearity arises from a Michaelis-Menten type modeling of the branching process. For branching rates large enough compared to capping rates, the existence and stability of nontrivial steady states is investigated. The main result is exponential convergence to nontrivial steady states, proven by investigating the decay of an appropriate Lyapunov functional.

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 dynamics of two interacting families of locally parallel actin filaments (Oelz and Schmeiser, 2010b). 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 non-moving and moving states as well as changes of direction.

The flatness of Lamellipodia explained by the interaction between actin dynamics and membrane deformation.

The crawling motility of many cell types relies on lamellipodia, flat protrusions spreading on flat substrates but (on cells in suspension) also growing into three-dimensional space. Lamellipodia consist of a plasma membrane wrapped around an oriented actin filament meshwork. It is well known that the actin density is controlled by coordinated polymerization, branching, and capping processes, but the mechanisms producing the small aspect ratios of lamellipodia (hundreds of nm thickness vs. several µm lateral and inward extension) remain unclear. The main hypothesis of this work is a strong influence of the local geometry of the plasma membrane on the actin dynamics. This is motivated by observations of co-localization of proteins with I-BAR domains (like IRSp53) with polymerization and branching agents along the membrane. The I-BAR domains are known to bind to the membrane and to prefer and promote membrane curvature. This hypothesis is translated into a stochastic mathematical model where branching and capping rates, and polymerization speeds depend on the local membrane geometry and branching directions are influenced by the principal curvature directions. This requires the knowledge of the deformation of the membrane, being described in a quasi-stationary approximation by minimization of a modified Helfrich energy, subject to the actin filaments acting as obstacles. Simulations with this model predict pieces of flat lamellipodia without any prescribed geometric restrictions.

Electron tomography and simulation of baculovirus actin comet tails support a tethered filament model of pathogen propulsion.

Several pathogens induce propulsive actin comet tails in cells they invade to disseminate their infection. They achieve this by recruiting factors for actin nucleation, the Arp2/3 complex, and polymerization regulators from the host cytoplasm. Owing to limited information on the structural organization of actin comets and in particular the spatial arrangement of filaments engaged in propulsion, the underlying mechanism of pathogen movement is currently speculative and controversial. Using electron tomography we have resolved the three-dimensional architecture of actin comet tails propelling baculovirus, the smallest pathogen yet known to hijack the actin motile machinery. Comet tail geometry was also mimicked in mixtures of virus capsids with purified actin and a minimal inventory of actin regulators. We demonstrate that propulsion is based on the assembly of a fishbone-like array of actin filaments organized in subsets linked by branch junctions, with an average of four filaments pushing the virus at any one time. Using an energy-minimizing function we have simulated the structure of actin comet tails as well as the tracks adopted by baculovirus in infected cells in vivo. The results from the simulations rule out gel squeezing models of propulsion and support those in which actin filaments are continuously tethered during branch nucleation and polymerization. Since Listeria monocytogenes, Shigella flexneri, and Vaccinia virus among other pathogens use the same common toolbox of components as baculovirus to move, we suggest they share the same principles of actin organization and mode of propulsion.

Arp2/3 complex is essential for actin network treadmilling as well as for targeting of capping protein and cofilin.

Lamellipodia are sheet-like protrusions formed during migration or phagocytosis and comprise a network of actin filaments. Filament formation in this network is initiated by nucleation/branching through the actin-related protein 2/3 (Arp2/3) complex downstream of its activator, suppressor of cAMP receptor/WASP-family verprolin homologous (Scar/WAVE), but the relative relevance of Arp2/3-mediated branching versus actin filament elongation is unknown. Here we use instantaneous interference with Arp2/3 complex function in live fibroblasts with established lamellipodia. This allows direct examination of both the fate of elongating filaments upon instantaneous suppression of Arp2/3 complex activity and the consequences of this treatment on the dynamics of other lamellipodial regulators. We show that Arp2/3 complex is an essential organizer of treadmilling actin filament arrays but has little effect on the net rate of actin filament turnover at the cell periphery. In addition, Arp2/3 complex serves as key upstream factor for the recruitment of modulators of lamellipodia formation such as capping protein or cofilin. Arp2/3 complex is thus decisive for filament organization and geometry within the network not only by generating branches and novel filament ends, but also by directing capping or severing activities to the lamellipodium. Arp2/3 complex is also crucial to lamellipodia-based migration of keratocytes.

Actin branching in the initiation and maintenance of lamellipodia.

Using correlated live-cell imaging and electron tomography we found that actin branch junctions in protruding and treadmilling lamellipodia are not concentrated at the front as previously supposed, but link actin filament subsets in which there is a continuum of distances from a junction to the filament plus ends, for up to at least 1 µm. When branch sites were observed closely spaced on the same filament their separation was commonly a multiple of the actin helical repeat of 36 nm. Image averaging of branch junctions in the tomograms yielded a model for the in vivo branch at 2.9 nm resolution, which was comparable with that derived for the in vitro actin-Arp2/3 complex. Lamellipodium initiation was monitored in an intracellular wound-healing model and was found to involve branching from the sides of actin filaments oriented parallel to the plasmalemma. Many filament plus ends, presumably capped, terminated behind the lamellipodium tip and localized on the dorsal and ventral surfaces of the actin network. These findings reveal how branching events initiate and maintain a network of actin filaments of variable length, and provide the first structural model of the branch junction in vivo. A possible role of filament capping in generating the lamellipodium leaflet is discussed and a mathematical model of protrusion is also presented.

Actin filament tracking in electron tomograms of negatively stained lamellipodia using the localized radon transform.

The aim of this work was to develop a protocol for automated tracking of actin filaments in electron tomograms of lamellipodia embedded in negative stain. We show that a localized version of the Radon transform for the detection of filament directions enables three-dimensional visualizations of filament network architecture, facilitating extraction of statistical information including orientation profiles. We discuss the requirements for parameter selection set by the raw image data in the context of other, similar tracking protocols.

Simulation of lamellipodial fragments.

A steepest descent approximation scheme is derived for a recently developed model for the dynamics of the actin cytoskeleton in the lamellipodia of living cells. The scheme is used as a numerical method for the simulation of thought experiments, where a lamellipodial fragment is pushed by a pipette, and subsequently changes its shape and position.

Modeling of the actin-cytoskeleton in symmetric lamellipodial fragments.

The pushing structures of cells include laminar sheets, termed lamellipodia, made up of a meshwork of actin filaments that grow at the front and depolymerise at the rear, in a treadmilling mode.We here develop a mathematical model to describe the turnover and the mechanical properties of this network.Our basic modeling assumptions are that the lamellipodium is idealised as a two-dimensional structure, and that the actin network consists of two families of possibly bent, but locally parallel filaments. Instead of dealing with individual polymers, the filaments are assumed to be continuously distributed.The model includes (de)polymerization, of the mechanical effects of cross-linking, cell-substrate adhesion, as well as of the leading edge of the membrane.In the first version presented here, the total amount of F-actin is prescribed by assuming a constant polymerisation speed at the leading edge and a fixed total number and length distribution of filaments. We assume that cross-links at filament crossing points as well as integrin linkages with the matrix break and reform in response to incremental changes in network organization. In this first treatment, the model successfully simulates the persistence of the treadmilling network in radially spread cells.

Multistep navigation of leukocytes: a stochastic model with memory effects.

We present a model for the chemotactically directed migration of neutrophil leukocytes. It reproduces the multistep navigation by memory effects investigated experimentally by E. F. Foxman, J. J. Campbell and E. C. Butcher in 1997. The model consists of a system of stochastic differential equations. The long time behaviour of the corresponding deterministic system is analysed and two approaches for the numerical solution of the full stochastic system are compared. One of them consists in performing direct simulations, the other one is based on a moment approximation of the Fokker-Planck equation and numerical methods for convection-dominated partial differential equations.