Non-Local Cell Adhesion Models: Steady States and Bifurcations

Published in arXiv, 2020

Abstract: In this manuscript, we consider the modelling of cellular adhesions, which is a key interaction between biological cells. Continuum models of the diffusion-advection-reaction type have long been used in tissue modelling. In 2006, Armstrong, Painter, and Sherratt proposed an extension to take adhesion effects into account. The resulting equation is a non-local advection-diffusion equation. While immensely successful in applications, the development of mathematical theory pertaining to steady states and pattern formation is lacking. The mathematical analysis of the non-local adhesion model is challenging. In this monograph, we contribute to the analysis of steady states and their bifurcation structure. The importance of steady-states is that these are the patterns observed in nature and tissues (e.g. cell-sorting experiments). In the case of periodic boundary conditions, we combine global bifurcation results pioneered by Rabinowitz, equivariant bifurcation theory, and the mathematical properties (maximum principle) of the non-local term to obtain a global bifurcation result for the branches of non-trivial solutions. Read more

Download here

Cell size, mechanical tension, and GTPase signaling in the Single Cell

Published in arXiv, 2019

Abstract: Cell polarization requires redistribution of specific proteins to the nascent front and back of a eukarytotic cell. Among these proteins are Rac and Rho, members of the small GTPase family that regulate the actin cytoskeleton. Rac promotes actin assembly and protrusion of the front edge, whereas Rho activates myosin-driven contraction at the back. Mathematical models of cell polarization at many levels of detail have appeared. One of the simplest based on” wave-pinning”, consists of a pair of reaction-diffusion equations for a single GTPase. Mathematical analysis of wave-pinning so far is largely restricted to static domains in one spatial dimension. Here we extend the analysis to cells that change in size, showing that both shrinking and growing cells can lose polarity. We further consider the feedback between mechanical tension, GTPase activation, and cell deformation in both static, growing, shrinking, and moving cells. Special cases (spatially uniform cell chemistry, absence or presence of mechanical feedback) are analyzed, and the full model is explored by simulations in 1D. We find a variety of novel behaviors, including” dilution-induced” oscillations of Rac activity and cell size, as well as gain or loss of polarization and motility in the model cell. Read more

Download here

Correlated random walks inside a cell: actin branching and microtubule dynamics

Published in Journal of Mathematical Biology, 2019

Abstract: Correlated random walks (CRW) have been explored in many settings, most notably in the motion of individuals in a swarm or flock. But some subcellular systems such as growth or disassembly of bio-polymers can also be described with similar models and understood using related mathematical methods. Here we consider two examples of growing cytoskeletal elements, actin and microtubules. We use CRW or generalized CRW-like PDEs to model their spatial distributions. In each case, the linear models can be reduced to a Telegrapher’s equation. A combination of explicit solutions (in one case) and numerical solutions (in the other) demonstrates that the approach to steady state can be accompanied by (decaying) waves. Read more

Recommended citation: Buttenschön, A. & Edelstein-Keshet, L. J. Math. Biol. (2019)

Nonlocal Adhesion Models for Microorganisms on Bounded Domains

Published in arXiv, 2019

Abstract: In 2006 Armstrong, Painter and Sherratt formulated a non-local differential equation model for cell-cell adhesion. For the one dimensional case we derive various types of adhesive, repulsive, and no-flux boundary bonditions. We prove local and global existence and uniqueness for the resulting integro-differential equations. In numerical simulations we consider adhesive, repulsive and neutral boundary conditions and we show that the solutions mimic known behavior of fluid adhesion to boundaries. In addition, we observe interior pattern formation due to cell-cell adhesion. Read more

Download here

Cops-on-the-Dots: The Linear Stability of Crime Hotspots for a 1-D Reaction-Diffusion Model of Urban Crime

Published in European Journal of Applied Mathematics, 2018

Abstract: In a singularly perturbed limit, we analyze the existence and linear stability of steady-state hotspot solutions foran extension of the 1-D three-component reaction-diffusion (RD) system formulated and studied numerically in Joneset. al. [Math. Models. Meth. Appl. Sci.,20, Suppl., (2010)], which models urban crime with police intervention. In ourextended RD model, the field variables are the attractivenessfield for burglary, the criminal density, and the police density,and it includes a scalar parameter that determines the strength of the police drift towards maxima of the attractivenessfield. For a special choice of this parameter, we recover the “cops-on-the-dots” policing strategy of Jones et. al., wherethepolice mimic the drift of the criminals towards maxima of the attractiveness field. For our extended model, the methodof matched asymptotic expansions is used to construct 1-D steady-state hotspot patterns as well as to derive nonlocaleigenvalue problems (NLEPs), each having three distinct nonlocal terms, that characterize the linear stability of thesehotspot steady-states toO(1) time-scale instabilities. For a cops-on-the-dots policing strategy, where some key identitiescan be used to recast these NLEPs into equivalent NLEPs with only one nonlocal term, we prove that a multi-hotspotsteady-state is linearly stable to synchronous perturbations of the hotspot amplitudes. Moreover, for asynchronousperturbations of the hotspot amplitudes, a hybrid analytical-numerical method is used to construct linear stability phasediagrams in the police versus criminal diffusivity parameterspace. In one particular region of these phase diagrams, thehotspot steady-states are shown to be unstable to asynchronous oscillatory instabilities in the hotspot amplitudes thatarise from a Hopf bifurcation. Within the context of our model,this provides a parameter range where the effect of acops-on-the-dots policing strategy is to only displace crime temporally between neighboring spatial regions. Our hybridapproach to study the equivalent NLEP combines rigorous spectral results with a numerical parameterization of anyHopf bifurcation threshold. For the cops-on-the-dots policing strategy, our linear stability predictions for steady-statehotspot patterns are confirmed from full numerical PDE simulations of the three-component RD system. Read more

Recommended citation: Buttenschoen, A., Kolokolnikov, T., Ward, M.J. and Wei, J., 2019. Cops-on-the-dots: The linear stability of crime hotspots for a 1-D reaction-diffusion model of urban crime. European Journal of Applied Mathematics, pp.1-47.

Off-Lattice Agent-Based Models for Cell and Tumor Growth: Numerical Methods, Implementation, and Applications

Published in Numerical Methods and Advanced Simulation in Biomechanics and Biological Processes, 2017

Book contribution about off-lattice based models for biological processes and tissues Read more

Recommended citation: Van Liedekerke, P., Buttenschön, A. and Drasdo, D., 2018. Off-lattice agent-based models for cell and tumor growth: numerical methods, implementation, and applications. In Numerical Methods and Advanced Simulation in Biomechanics and Biological Processes (pp. 245-267). Academic Press.

Agent-Based Lattice Models of Multicellular Systems: Numerical Methods, Implementation, and Applications

Published in Numerical Methods and Advanced Simulation in Biomechanics and Biological Processes, 2017

Book contribution about lattice based models for biological processes and tissues Read more

Recommended citation: Drasdo, D., Buttenschön, A. and Van Liedekerke, P., 2018. Agent-based lattice models of multicellular systems: numerical methods, implementation, and applications. In Numerical Methods and Advanced Simulation in Biomechanics and Biological Processes (pp. 223-238). Academic Press.