Cell and cell-aggregate mechanics: remodelling, growth and interaction with the extracellular environment.

The scope of this dissertation is to give a contribution to the understanding of the mathematical description of cell and cellular aggregate mechanics, focusing on remodelling and growth processes which occur inside living structures and on the interactions among cells and the extracellular environment, during the process of cell migration

Branching instability in expanding bacterial colonies

International audienceSelf-organization in developing living organisms relies on the capability of cells to duplicate and perform a collective motion inside the surrounding environment. Chemical and mechanical interactions coordinate such a cooperative behaviour, driving the dynamical evolution of the macroscopic system. In this work, we perform an analytical and computational analysis to study pattern formation during the spreading of an initially circular bacterial colony on a Petri dish. The continuous mathematical model addresses the growth and the chemotactic migration of the living monolayer, together with the diffusion and consumption of nutrients in the agar. The governing equations contain four dimensionless parameters, accounting for the interplay among the chemotactic response, the bacteria-substrate interaction and the experimental geometry. The spreading colony is found to be always linearly unstable to perturbations of the interface, whereas branching instability arises in finite-element numerical simulations. The typical length scales of such fingers, which align in the radial direction and later undergo further branching, are controlled by the size parameters of the problem, whereas the emergence of branching is favoured if the diffusion is dominant on the chemotaxis. The model is able to predict the experimental morphologies, confirming that compact (resp. branched) patterns arise for fast (resp. slow) expanding colonies. Such results, while providing new insights into pattern selection in bacterial colonies, may finally have important applications for designing controlled patterns

Influence of nucleus deformability on cell entry into cylindrical structures

The mechanical properties of cell nuclei have been demonstrated to play a fundamental role in cell movement across extracellular networks and micro-channels. In this work, we focus on a mathematical description of a cell entering a cylindrical channel composed of extracellular matrix. An energetic approach is derived in order to obtain a necessary condition for which cells enter cylindrical structures. The nucleus of the cell is treated either (i) as an elastic membrane surrounding a liquid droplet or (ii) as an incompressible elastic material with Neo-Hookean constitutive equation. The results obtained highlight the importance of the interplay between mechanical deformability of the nucleus and the capability of the cell to establish adhesive bonds and generate active forces in the cytoskeleton due to myosin action

Cell orientation under stretch: Stability of a linear viscoelastic model

The sensitivity of cells to alterations in the microenvironment and in particular to external mechanical stimuli is significant in many biological and physiological circumstances. In this regard, experimental assays demonstrated that, when a monolayer of cells cultured on an elastic substrate is subject to an external cyclic stretch with a sufficiently high frequency, a reorganization of actin stress fibres and focal adhesions happens in order to reach a stable equilibrium orientation, characterized by a precise angle between the cell major axis and the largest strain direction. To examine the frequency effect on the orientation dynamics, we propose a linear viscoelastic model that describes the coupled evolution of the cellular stress and the orientation angle. We find that cell orientation oscillates tending to an angle that is predicted by the minimization of a very general orthotropic elastic energy, as confirmed by a bifurcation analysis. Moreover, simulations show that the speed of convergence towards the predicted equilibrium orientation presents a changeover related to the viscous–elastic transition for viscoelastic materials. In particular, when the imposed oscillation period is lower than the characteristic turnover rate of the cytoskeleton and of adhesion molecules such as integrins, reorientation is significantly faster

Foreword to the Special Issue in honour of Prof. Luigi Preziosi “Nonlinear mechanics: The driving force of modern applied and industrial mathematics”

Mathematical modelling is a discipline pledged to identify problems, which may arise from virtually any branch of the human knowledge, and formalise them in the language of mathematics by developing suitable methodologies of investigation. To pursue its goals, modelling must build connections with other mathematical sciences and, in particular, with numerics. Three major examples of the efficiency of such combination are industrial mathematics, mathematical biology and biomechanics. At first sight, industrial mathematics is a branch of applied mathematics focusing on problems that come from industry and it aims at determining solutions relevant to manufacturing. Some relevant examples are petroleum engineering, hydrogeology, and the description of sand dynamics in the neighbourhood of railways in desert zones. On the other hand, the adoption of mathematics to formalise problems of biological relevance has attracted scientists working on population dynamics, epidemiology and related fields. Moreover, a strong impact has been given by the combination of modelling with the mechanics of biological tissues, thereby giving rise to biomechanics. Few examples in this respect are the mechanics of cell motion and migration, which relate to kinetic theories, the mechanics of the interactions between cells and the extracellular matrix, the conversion of mechanical signals into chemical stimuli, and "mathematical oncology". Since it is not possible to present a theoretical corpus of all that, the aim of the present special issue is to put together a list of outstanding scientific papers giving clear connections among nonlinear mechanics, industrial mathematics, biomathematics, biomechanics and kinetic theories, in different fields of interest. This special issue of IJNLM is the Festschrift celebrating the 60th birthday of Luigi Preziosi, whose research is a recognised example of how mechanics may be the fuel for interesting applied mathematics

A non local model for cell migration in response to mechanical stimuli

Cell migration is one of the most studied phenomena in biology since it plays a fundamental role in many physiological and pathological processes such as morphogenesis, wound healing and tumorigenesis. In recent years, researchers have performed experiments showing that cells can migrate in response to mechanical stimuli of the substrate they adhere to. Motion toward regions of the substrate with higher stiffness is called durotaxis, while motion guided by the stress or the deformation of the substrate itself is called tensotaxis. Unlike chemotaxis (i.e. the motion in response to a chemical stimulus), these migratory processes are not yet fully understood from a biological point of view. In this respect, we present a mathematical model of single-cell migration in response to mechanical stimuli, in order to simulate these two processes. Specifically, the cell moves by changing its direction of polarization and its motility according to material properties of the substrate (e.g., stiffness) or in response to proper scalar measures of the substrate strain or stress. The equations of motion of the cell are non-local integro-differential equations, with the addition of a stochastic term to account for random Brownian motion. The mechanical stimulus to be integrated in the equations of motion is defined according to experimental measurements found in literature, in the case of durotaxis. Conversely, in the case of tensotaxis, substrate strain and stress are given by the solution of the mechanical problem, assuming that the extracellular matrix behaves as a hyperelastic Yeoh's solid. In both cases, the proposed model is validated through numerical simulations that qualitatively reproduce different experimental scenarios

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

Coupling solid and fluid stresses with brain tumour growth and white matter tract deformations in a neuroimaging-informed model

Brain tumours are among the deadliest types of cancer, since they display a strong ability to invade the surrounding tissues and an extensive resistance to common therapeutic treatments. It is therefore important to reproduce the heterogeneity of brain microstructure through mathematical and computational models, that can provide powerful instruments to investigate cancer progression. However, only a few models include a proper mechanical and constitutive description of brain tissue, which instead may be relevant to predict the progression of the pathology and to analyse the reorganization of healthy tissues occurring during tumour growth and, possibly, after surgical resection. Motivated by the need to enrich the description of brain cancer growth through mechanics, in this paper we present a mathematical multiphase model that explicitly includes brain hyperelasticity. We find that our mechanical description allows to evaluate the impact of the growing tumour mass on the surrounding healthy tissue, quantifying the displacements, deformations, and stresses induced by its proliferation. At the same time, the knowledge of the mechanical variables may be used to model the stress-induced inhibition of growth, as well as to properly modify the preferential directions of white matter tracts as a consequence of deformations caused by the tumour. Finally, the simulations of our model are implemented in a personalized framework, which allows to incorporate the realistic brain geometry, the patient-specific diffusion and permeability tensors reconstructed from imaging data and to modify them as a consequence of the mechanical deformation due to cancer growth