We consider a broad class of systems of nonlinear integro-differential equations posed on the real line that arise as Euler-Lagrange equations to energies involving nonlinear nonlocal interactions. Although these equations are not readily cast as dynamical systems, we develop a calculus that yields a natural Hamiltonian formalism. In particular, we formulate Noether's theorem in this context, identify a degenerate symplectic structure, and derive Hamiltonian differential equations on finite-dimensional center manifolds when those exist. Our formalism yields new natural conserved quantities. For Euler-Lagrange equations arising as traveling-wave equations in gradient flows, we identify Lyapunov functions. We provide several applications to pattern-forming systems including neural field and phase separation problems.Differential equations of gradient form u t "´∇Epuq or of Hamiltonian form u t " J∇Epuq arise throughout mathematical modeling from maximal energy dissipation first principles, or from least action principles, respectively. Mathematically, the energy E is simply a function defined over a finite or infinite-dimensional space. The associated gradient and Hamiltonian flows are differential equations with equilibria given by the critical points of the energy. Energies over infinite-dimensional spaces are usually defined on function spaces through integrals of nonlinear functions of state variables and their derivatives. Critical points then solve Euler-Lagrange equations, commonly of elliptic type. Dependence of the energy on derivatives of the state variables encodes local interactions, often derived in various types of continuum limits. We are interested here in cases where this continuum limit retains nonlocal interaction terms. More specifically, we are interested in the somewhat specific class of energies E that contain nonlocal interaction term of the formmodeling phenomena such as long-range interactions of agents in social interaction, between particles through nonlocal force fields, or of neurons, labeled in a feature space x through synaptic connections. In all those cases, the convolution structure embodies the modeling assumption of translational invariance of physical space. We are

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Adhering cells exert traction forces on the underlying substrate. We numerically investigate the intimate relation between traction forces, the structure of the actin cytoskeleton, and the shape of cells adhering to adhesive micropatterned substrates. By combining the Cellular Potts Model with a model of cytoskeletal contractility, we reproduce prominent anisotropic features in previously published experimental data on fibroblasts, endothelial cells, and epithelial cells on adhesive micropatterned substrates. Our work highlights the role of cytoskeletal anisotropy in the generation of cellular traction forces, and provides a computational strategy for investigating stress fiber anisotropy in dynamical and multicellular settings.Author summaryCells that make up multicellular life perform a variety of mechanical tasks such as pulling on surrounding tissue to close a wound. The mechanisms by which cells perform these tasks are, however, incompletely understood. In order to better understand how they generate forces on their environment, cells are often studied in vitro on compliant substrates, which deform under the so called “traction forces” exerted by the cells. Mathematical models complement these experimental approaches because they help to interpret the experimental data, but most models for traction forces on adhesive substrates assume that cells contract isotropically, i.e., they do not contract in a specific direction. However, many cell types contain organized structures of stress fibers - strong contracting cables inside the cell - that enable cells to exert forces on their environment in specific directions only. Here we present a computational model that predicts both the orientations of these stress fibers as well as the forces that cells exert on the substrates. Our model reproduces both the orientations and magnitudes of previously reported experimental traction forces, and could serve as a starting point for exploring mechanical interactions in multicellular settings.

Travelling waves form a prominent feature in the dynamics of scalar reactiondiffusion equations on unbounded cylinders. The travelling waves can be identified with the bounded solutions of the elliptic PDEwhere c = 0 is the wavespeed, Ω ⊂ R d is a bounded domain, ∆ is the Laplacian on Ω, and B denotes Dirichlet, Neumann, or periodic boundary data. We develop a new homological invariant for the dynamics of the bounded solutions of (1).Restrictions on the nonlinearity f are kept to a minimum, for instance, any nonlinearity exhibiting polynomial growth in u can be considered. In particular, the set of bounded solutions of the travelling wave PDE may not be uniformly bounded. Despite this, the homology is invariant under lower order (but not necessarily small) perturbations of the nonlinearity f , thus making the homology amenable for computation. Using the new invariant we derive lower bounds on the number of bounded solutions of (1), thus obtaining existence and multiplicity results for travelling wave solutions of reaction-diffusion equations on unbounded cylinders.

Cell-based modelling frameworks such as cellular Potts are well-established tools to spatially model cell behavior and tissue morphogenesis. These models generally represent the extracellular matrix (ECM) with mean-field approaches, which assume substrate homogeneity. This assumption breaks down with fibrous ECM, which has non-trivial topology and mechanics. Here, we extend the cellular Potts software library Tissue Simulation Toolkit with the molecular mechanics framework hoomd-blue. We model cells mechanically interacting through discrete focal adhesion-like sites with an ECM fiber network modelled as bead-spring chains. Using a case study of an isolated contractile cell straining a randomly-oriented fiber network, we demonstrate agreement with experimental fiber densification and displacement dynamics. Further, we apply in silico atomic force microscopy to show local cell-induced network stiffening consistent with experiments. Our model overcomes the limitation of mean-field approaches to modelling ECM, and lays the foundation to investigate biomechanical cell-ECM interactions in a multicellular context.