A space-jump derivation for non-local models of cell-cell adhesion and non-local chemotaxis

How Cells Stay Together: A Mechanism for Maintenance of a Robust Cluster Explored by Local and Non-local Continuum Models

Formation of organs and specialized tissues in embryonic development requires migration of cells to specific targets. In some instances, such cells migrate as a robust cluster. We here explore a recent local approximation of non-local continuum models by Falcó et al. (SIAM J Appl Math 84:17-42, 2023). We apply their theoretical results by specifying biologically-based cell-cell interactions, showing how such cell communication results in an effective attraction-repulsion Morse potential. We then explore the clustering instability, the existence and size of the cluster, and its stability. For attractant-repellent chemotaxis, we derive an explicit condition on cell and chemical properties that guarantee the existence of robust clusters. We also extend their work by investigating the accuracy of the local approximation relative to the full non-local model.

Modelling non-local cell-cell adhesion: a multiscale approach

Cell-cell adhesion plays a vital role in the development and maintenance of multicellular organisms. One of its functions is regulation of cell migration, such as occurs, e.g. during embryogenesis or in cancer. In this work, we develop a versatile multiscale approach to modelling a moving self-adhesive cell population that combines a careful microscopic description of a deterministic adhesion-driven motion component with an efficient mesoscopic representation of a stochastic velocity-jump process. This approach gives rise to mesoscopic models in the form of kinetic transport equations featuring multiple non-localities. Subsequent parabolic and hyperbolic scalings produce general classes of equations with non-local adhesion and myopic diffusion, a special case being the classical macroscopic model proposed in Armstrong et al. (J Theoret Biol 243(1): 98-113, 2006). Our simulations show how the combination of the two motion effects can unfold. Cell-cell adhesion relies on the subcellular cell adhesion molecule binding. Our approach lends itself conveniently to capturing this microscopic effect. On the macroscale, this results in an additional non-linear integral equation of a novel type that is coupled to the cell density equation.

A novel nonlocal partial differential equation model of endothelial progenitor cell cluster formation during the early stages of vasculogenesis

The formation of new vascular networks is essential for tissue development and regeneration, in addition to playing a key role in pathological settings such as ischemia and tumour development. Experimental findings in the past two decades have led to the identification of a new mechanism of neovascularisation, known as cluster-based vasculogenesis, during which endothelial progenitor cells (EPCs) mobilised from the bone marrow are capable of bridging distant vascular beds in a variety of hypoxic settings in vivo. This process is characterised by the formation of EPC clusters during its early stages and, while much progress has been made in identifying various mechanisms underlying cluster formation, we are still far from a comprehensive description of such spatio-temporal dynamics. In order to achieve this, we propose a novel mathematical model of the early stages of cluster-based vasculogenesis, comprising of a system of nonlocal partial differential equations including key mechanisms such as endogenous chemotaxis, matrix degradation, cell proliferation and cell-to-cell adhesion. We conduct a linear stability analysis on the system and solve the equations numerically. We then conduct a parametric analysis of the numerical solutions of the one-dimensional problem to investigate the role of underlying dynamics on the speed of cluster formation and the size of clusters, measured via appropriate metrics for the cluster width and compactness. We verify the key results of the parametric analysis with simulations of the two-dimensional problem. Our results, which qualitatively compare with data from in vitro experiments, elucidate the complementary role played by endogenous chemotaxis and matrix degradation in the formation of clusters, suggesting chemotaxis is responsible for the cluster topology while matrix degradation is responsible for the speed of cluster formation. Our results also indicate that the nonlocal cell-to-cell adhesion term in our model, even though it initially causes cells to aggregate, is not sufficient to ensure clusters are stable over long time periods. Consequently, new modelling strategies for cell-to-cell adhesion are required to stabilise in silico clusters. We end the paper with a thorough discussion of promising, fruitful future modelling and experimental research perspectives.

Derivation of continuum models from discrete models of mechanical forces in cell populations

In certain discrete models of populations of biological cells, the mechanical forces between the cells are center based or vertex based on the microscopic level where each cell is individually represented. The cells are circular or spherical in a center based model and polygonal or polyhedral in a vertex based model. On a higher, macroscopic level, the time evolution of the density of the cells is described by partial differential equations (PDEs). We derive relations between the modelling on the micro and macro levels in one, two, and three dimensions by regarding the micro model as a discretization of a PDE for conservation of mass on the macro level. The forces in the micro model correspond on the macro level to a gradient of the pressure scaled by quantities depending on the cell geometry. The two levels of modelling are compared in numerical experiments in one and two dimensions.

Comparative analysis of continuum angiogenesis models

Although discrete approaches are increasingly employed to model biological phenomena, it remains unclear how complex, population-level behaviours in such frameworks arise from the rules used to represent interactions between individuals. Discrete-to-continuum approaches, which are used to derive systems of coarse-grained equations describing the mean-field dynamics of a microscopic model, can provide insight into such emergent behaviour. Coarse-grained models often contain nonlinear terms that depend on the microscopic rules of the discrete framework, however, and such nonlinearities can make a model difficult to mathematically analyse. By contrast, models developed using phenomenological approaches are typically easier to investigate but have a more obscure connection to the underlying microscopic system. To our knowledge, there has been little work done to compare solutions of phenomenological and coarse-grained models. Here we address this problem in the context of angiogenesis (the creation of new blood vessels from existing vasculature). We compare asymptotic solutions of a classical, phenomenological “snail-trail” model for angiogenesis to solutions of a nonlinear system of partial differential equations (PDEs) derived via a systematic coarse-graining procedure (Pillay et al. in Phys Rev E 95(1):012410, 2017. https://doi.org/10.1103/PhysRevE.95.012410). For distinguished parameter regimes corresponding to chemotaxis-dominated cell movement and low branching rates, both continuum models reduce at leading order to identical PDEs within the domain interior. Numerical and analytical results confirm that pointwise differences between solutions to the two continuum models are small if these conditions hold, and demonstrate how perturbation methods can be used to determine when a phenomenological model provides a good approximation to a more detailed coarse-grained system for the same biological process.

Multi-bit Boolean model for chemotactic drift of Escherichia coli

Dynamic biological systems can be modelled to an equivalent modular structure using Boolean networks (BNs) due to their simple construction and relative ease of integration. The chemotaxis network of the bacterium Escherichia coli (E. coli) is one of the most investigated biological systems. In this study, the authors developed a multi-bit Boolean approach to model the drifting behaviour of the E. coli chemotaxis system. Their approach, which is slightly different than the conventional BNs, is designed to provide finer resolution to mimic high-level functional behaviour. Using this approach, they simulated the transient and steady-state responses of the chemoreceptor sensory module. Furthermore, they estimated the drift velocity under conditions of the exponential nutrient gradient. Their predictions on chemotactic drifting are in good agreement with the experimental measurements under similar input conditions. Taken together, by simulating chemotactic drifting, they propose that multi-bit Boolean methodology can be used for modelling complex biological networks. Application of the method towards designing bio-inspired systems such as nano-bots is discussed.

Nonlocal and local models for taxis in cell migration: a rigorous limit procedure

A rigorous limit procedure is presented which links nonlocal models involving adhesion or nonlocal chemotaxis to their local counterparts featuring haptotaxis and classical chemotaxis, respectively. It relies on a novel reformulation of the involved nonlocalities in terms of integral operators applied directly to the gradients of signal-dependent quantities. The proposed approach handles both model types in a unified way and extends the previous mathematical framework to settings that allow for general solution-dependent coefficient functions. The previous forms of nonlocal operators are compared with the new ones introduced in this paper and the advantages of the latter are highlighted by concrete examples. Numerical simulations in 1D provide an illustration of some of the theoretical findings.

Mathematical models for cell migration: a non-local perspective

We provide a review of recent advancements in non-local continuous models for migration, mainly from the perspective of its involvement in embryonal development and cancer invasion. Particular emphasis is placed on spatial non-locality occurring in advection terms, used to characterize a cell's motility bias according to its interactions with other cellular and acellular components in its vicinity (e.g. cell-cell and cell-tissue adhesions, non-local chemotaxis), but we also briefly address spatially non-local source terms. Following a short introduction and description of applications, we give a systematic classification of available PDE models with respect to the type of featured non-localities and review some of the mathematical challenges arising from such models, with a focus on analytical aspects. This article is part of the theme issue 'Multi-scale analysis and modelling of collective migration in biological systems'.

On the macroscopic limit of Brownian particles with local interaction

An interacting particle system made of diffusion processes with local interaction is considered and the macroscopic limit to a nonlinear PDE is investigated. Few rigorous results exists on this problem and in particular the explicit form of the nonlinearity is not known. This paper reviews this subject, some of the main ideas to get the limit nonlinear PDE and provides both heuristic and numerical informations on the precise form of the nonlinearity which are new with respect to the literature and coherent with the few known informations.

From a discrete model of chemotaxis with volume-filling to a generalized Patlak-Keller-Segel model

We present a discrete model of chemotaxis whereby cells responding to a chemoattractant are seen as individual agents whose movement is described through a set of rules that result in a biased random walk. In order to take into account possible alterations in cellular motility observed at high cell densities (i.e. volume-filling), we let the probabilities of cell movement be modulated by a decaying function of the cell density. We formally show that a general form of the celebrated Patlak-Keller-Segel (PKS) model of chemotaxis can be formally derived as the appropriate continuum limit of this discrete model. The family of steady-state solutions of such a generalized PKS model are characterized and the conditions for the emergence of spatial patterns are studied via linear stability analysis. Moreover, we carry out a systematic quantitative comparison between numerical simulations of the discrete model and numerical solutions of the corresponding PKS model, both in one and in two spatial dimensions. The results obtained indicate that there is excellent quantitative agreement between the spatial patterns produced by the two models. Finally, we numerically show that the outcomes of the two models faithfully replicate those of the classical PKS model in a suitable asymptotic regime.

A Coupled Mass Transport and Deformation Theory of Multi-constituent Tumor Growth

We develop a general class of thermodynamically consistent, continuum models based on mixture theory with phase effects that describe the behavior of a mass of multiple interacting constituents. The constituents consist of solid species undergoing large elastic deformations and compressible viscous fluids. The fundamental building blocks framing the mixture theories consist of the mass balance law of diffusing species and microscopic (cellular scale) and macroscopic (tissue scale) force balances, as well as energy balance and the entropy production inequality derived from the first and second laws of thermodynamics. A general phase-field framework is developed by closing the system through postulating constitutive equations (i.e., specific forms of free energy and rate of dissipation potentials) to depict the growth of tumors in a microenvironment. A notable feature of this theory is that it contains a unified continuum mechanics framework for addressing the interactions of multiple species evolving in both space and time and involved in biological growth of soft tissues (e.g., tumor cells and nutrients). The formulation also accounts for the regulating roles of the mechanical deformation on the growth of tumors, through a physically and mathematically consistent coupled diffusion and deformation framework. A new algorithm for numerical approximation of the proposed model using mixed finite elements is presented. The results of numerical experiments indicate that the proposed theory captures critical features of avascular tumor growth in the various microenvironment of living tissue, in agreement with the experimental studies in the literature.

Modelling collective cell migration: neural crest as a model paradigm

A huge variety of mathematical models have been used to investigate collective cell migration. The aim of this brief review is twofold: to present a number of modelling approaches that incorporate the key factors affecting cell migration, including cell-cell and cell-tissue interactions, as well as domain growth, and to showcase their application to model the migration of neural crest cells. We discuss the complementary strengths of microscale and macroscale models, and identify why it can be important to understand how these modelling approaches are related. We consider neural crest cell migration as a model paradigm to illustrate how the application of different mathematical modelling techniques, combined with experimental results, can provide new biological insights. We conclude by highlighting a number of future challenges for the mathematical modelling of neural crest cell migration.

Kinetic models with non-local sensing determining cell polarization and speed according to independent cues

Cells move by run and tumble, a kind of dynamics in which the cell alternates runs over straight lines and re-orientations. This erratic motion may be influenced by external factors, like chemicals, nutrients, the extra-cellular matrix, in the sense that the cell measures the external field and elaborates the signal eventually adapting its dynamics. We propose a kinetic transport equation implementing a velocity-jump process in which the transition probability takes into account a double bias, which acts, respectively, on the choice of the direction of motion and of the speed. The double bias depends on two different non-local sensing cues coming from the external environment. We analyze how the size of the cell and the way of sensing the environment with respect to the variation of the external fields affect the cell population dynamics by recovering an appropriate macroscopic limit and directly integrating the kinetic transport equation. A comparison between the solutions of the transport equation and of the proper macroscopic limit is also performed.

A population dynamics model of cell-cell adhesion incorporating population pressure and density saturation

We discuss several continuum cell-cell adhesion models based on the underlying microscopic assumptions. We propose an improvement on these models leading to sharp fronts and intermingling invasion fronts between different cell type populations. The model is based on basic principles of localized repulsion and nonlocal attraction due to adhesion forces at the microscopic level. The new model is able to capture both qualitatively and quantitatively experiments by Katsunuma et al. (2016). We also review some of the applications of these models in other areas of tissue growth in developmental biology. We finally explore the resulting qualitative behavior due to cell-cell repulsion.

The impact of short- and long-range perception on population movements

Navigation of cells and organisms is typically achieved by detecting and processing orienteering cues. Occasionally, a cue may be assessed over a much larger range than the individual's body size, as in visual scanning for landmarks. In this paper we formulate models that account for orientation in response to short- or long-range cue evaluation. Starting from an underlying random walk movement model, where a generic cue is evaluated locally or nonlocally to determine a preferred direction, we state corresponding macroscopic partial differential equations to describe population movements. Under certain approximations, these models reduce to well-known local and nonlocal biological transport equations, including those of Keller-Segel type. We consider a case-study application: "hilltopping" in Lepidoptera and other insects, a phenomenon in which populations accumulate at summits to improve encounter/mating rates. Nonlocal responses are shown to efficiently filter out the natural noisiness (or roughness) of typical landscapes and allow the population to preferentially accumulate at a subset of hilltopping locations, in line with field studies. Moreover, according to the timescale of movement, optimal responses may occur for different perceptual ranges.