A Statistical Mechanics Approach to Describe Cell Reorientation Under Stretch.

Experiments show that when a monolayer of cells cultured on an elastic substratum is subject to a cyclic stretch, cells tend to reorient either perpendicularly or at an oblique angle with respect to the main stretching direction. Due to stochastic effects, however, the distribution of angles achieved by the cells is broader and, experimentally, histograms over the interval [Formula: see text] are usually reported. Here we will determine the evolution and the stationary state of probability density functions describing the statistical distribution of the orientations of the cells using Fokker-Planck equations derived from microscopic rules for describing the reorientation process of the cell. As a first attempt, we shall use a stochastic differential equation related to a very general elastic energy that the cell tries to minimize and, we will show that the results of the time integration and of the stationary state of the related forward Fokker-Planck equation compare very well with experimental results obtained by different researchers. Then, in order to model more accurately the microscopic process of cell reorientation and to shed light on the mechanisms performed by cells that are subject to cyclic stretch, we consider discrete in time random processes that allow to recover Fokker-Planck equations through classical tools of kinetic theory. In particular, we shall introduce a model of reorientation as a function of the rotation angle as a result of an optimal control problem. Also in this latter case the results match very well with experiments.

Modeling hypoxia-related inflammation scenarios.

Cells respond to hypoxia via the activation of three isoforms of Hypoxia Inducible Factors (HIFs), that are characterized by different activation times. HIF overexpression has many effects on cell behavior, such as change in metabolism, promotion of angiogenic processes and elicitation of a pro-inflammatory response. These effects are driving forces of malignant progression in cancer cells. In this work we study in detail hypoxia-induced dynamics of HIF1α and HIF2α, which are the most studied isoforms, comparing available experimental data on their evolution in tumor cells with the results obtained integrating the deduced mathematical model. Then, we examine the possible scenarios that characterize the link between hypoxia and inflammation via the activation of NFkB (Nuclear Factor k-light-chain-enhancer of activated B cells) when the dimensionless groups of parameters of the mathematical model change. In this way we are able to discuss why and when hypoxic conditions lead to acute or chronic inflammatory states.

Modelling chase-and-run migration in heterogeneous populations.

Cell migration is crucial for many physiological and pathological processes. During embryogenesis, neural crest cells undergo coordinated epithelial to mesenchymal transformations and migrate towards various forming organs. Here we develop a computational model to understand how mutual interactions between migrating neural crest cells (NCs) and the surrounding population of placode cells (PCs) generate coordinated migration. According to experimental findings, we implement a minimal set of hypotheses, based on a coupling between chemotactic movement of NCs in response to a placode-secreted chemoattractant (Sdf1) and repulsion induced from contact inhibition of locomotion (CIL), triggered by heterotypic NC-PC contacts. This basic set of assumptions is able to semi-quantitatively recapitulate experimental observations of the characteristic multispecies phenomenon of "chase-and-run", where the colony of NCs chases an evasive PC aggregate. The model further reproduces a number of in vitro manipulations, including full or partial disruption of NC chemotactic migration and selected mechanisms coordinating the CIL phenomenon. Finally, we provide various predictions based on altering other key components of the model mechanisms.

Coherent modelling switch between pointwise and distributed representations of cell aggregates.

Biological systems are typically formed by different cell phenotypes, characterized by specific biophysical properties and behaviors. Moreover, cells are able to undergo differentiation or phenotypic transitions upon internal or external stimuli. In order to take these phenomena into account, we here propose a modelling framework in which cells can be described either as pointwise/concentrated particles or as distributed masses, according to their biological determinants. A set of suitable rules then defines a coherent procedure to switch between the two mathematical representations. The theoretical environment describing cell transition is then enriched by including cell migratory dynamics and duplication/apoptotic processes, as well as the kinetics of selected diffusing chemicals influencing the system evolution. Finally, biologically relevant numerical realizations are presented: in particular, they deal with the growth of a tumor spheroid and with the initial differentiation stages of the formation of the zebrafish posterior lateral line. Both phenomena mainly rely on cell phenotypic transition and differentiated behaviour, thereby constituting biological systems particularly suitable to assess the advantages of the proposed model.

A Cellular Potts Model of single cell migration in presence of durotaxis.

Cell migration is a fundamental biological phenomenon during which cells sense their surroundings and respond to different types of signals. In presence of durotaxis, cells preferentially crawl from soft to stiff substrates by reorganizing their cytoskeleton from an isotropic to an anisotropic distribution of actin filaments. In the present paper, we propose a Cellular Potts Model to simulate single cell migration over flat substrates with variable stiffness. We have tested five configurations: (i) a substrate including a soft and a stiff region, (ii) a soft substrate including two parallel stiff stripes, (iii) a substrate made of successive stripes with increasing stiffness to create a gradient and (iv) a stiff substrate with four embedded soft squares. For each simulation, we have evaluated the morphology of the cell, the distance covered, the spreading area and the migration speed. We have then compared the numerical results to specific experimental observations showing a consistent agreement.

A hybrid mathematical model for self-organizing cell migration in the zebrafish lateral line.

In this paper we propose a discrete in continuous mathematical model for the morphogenesis of the posterior lateral line system in zebrafish. Our model follows closely the results obtained in recent biological experiments. We rely on a hybrid description: discrete for the cellular level and continuous for the molecular level. We prove the existence of steady solutions consistent with the formation of particular biological structure, the neuromasts. Dynamical numerical simulations are performed to show the behavior of the model and its qualitative and quantitative accuracy to describe the evolution of the cell aggregate.

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.

A review of mathematical models for the formation of vascular networks.

Two major mechanisms are involved in the formation of blood vasculature: vasculogenesis and angiogenesis. The former term describes the formation of a capillary-like network from either a dispersed or a monolayered population of endothelial cells, reproducible also in vitro by specific experimental assays. The latter term describes the sprouting of new vessels from an existing capillary or post-capillary venule. Similar mechanisms are also involved in the formation of the lymphatic system through a process generally called lymphangiogenesis. A number of mathematical approaches have been used to analyze these phenomena. In this paper, we review the different types of models, with special emphasis on their ability to reproduce different biological systems and to predict measurable quantities which describe the overall processes. Finally, we highlight the advantages specific to each of the different modelling approaches.

Mechano-transduction in tumour growth modelling.

The evolution of biological systems is strongly influenced by physical factors, such as applied forces, geometry or the stiffness of the micro-environment. Mechanical changes are particularly important in solid tumour development, as altered stromal-epithelial interactions can provoke a persistent increase in cytoskeletal tension, driving the gene expression of a malignant phenotype. In this work, we propose a novel multi-scale treatment of mechano-transduction in cancer growth. The avascular tumour is modelled as an expanding elastic spheroid, whilst growth may occur both as a volume increase and as a mass production within a cell rim. Considering the physical constraints of an outer healthy tissue, we derive the thermo-dynamical requirements for coupling growth rate, solid stress and diffusing biomolecules inside a heterogeneous tumour. The theoretical predictions successfully reproduce the stress-dependent growth curves observed by in vitro experiments on multicellular spheroids.

Modelling the compression and reorganization of cell aggregates.

In this paper, we study the mechanical behaviour of multicellular aggregates using the notion of multiple natural configurations. In particular, we extend the elasto-visco-plastic model proposed in Preziosi et al. (2010, An elasto-visco-plastic model of cell aggregates. J. Theor. Biol., 262, 35-47) taking into account the liquid constituent present in cellular spheroids. Aggregates are treated as porous materials, composed of cells and filled with water. The cellular constituent is responsible for the elastic and the plastic behaviour of the material. The plastic component is due to the rearrangement of adhesion bonds between cells and it is translated into the existence of a yield stress in the macroscopic constitutive equation. On the other hand, the liquid constituent is responsible for the viscous-like response during deformation. The general framework is then applied to describe uniaxial homogeneous compression both when a constant load is applied and when a fixed deformation is imposed and subsequently released. We compare the results of the model with the dynamics observed during the experiments in Forgacs et al. (1998, Viscoelastic properties of living embryonic tissues: a quantitative study. Biophys. J., 74, 2227-2234).

A model of cell migration within the extracellular matrix based on a phenotypic switching mechanism.

Cell migration involves different mechanisms in different cell types and tissue environments. Changes in migratory behaviour have been observed experimentally and associated with phenotypic switching in various situations, such as the migration-proliferation dichotomy of glioma cells, the epithelial-mesenchymal transition or the mesenchymal-amoeboid transition of fibrosarcoma cells in the extracellular matrix (ECM). In the present study, we develop a modelling framework to account for changes in migratory behaviour associated with phenotypic switching. We take into account the influence of the ECM on cell motion and more particularly the alignment process along the fibers. We use a mesoscopic description to model two cell populations with different migratory properties. We derive the corresponding continuum (macroscopic) model by appropriate rescaling, which leads to a generic reaction-diffusion system for the two cell phenotypes. We investigate phenotypic adaptation to dense and sparse environments and propose two complementary transition mechanisms. We study these mechanisms by using a combination of linear stability analysis and numerical simulations. Our investigations reveal that when the cell migratory ability is reduced by a crowded environment, a diffusive instability may appear and lead to the formation of aggregates of cells of the same phenotype. Finally, we discuss the importance of the results from a biological perspective.

An elasto-visco-plastic model of cell aggregates.

Concentrated cell suspensions exhibit different mechanical behavior depending on the mechanical stress or deformation they undergo. They have a mixed rheological nature: cells behave elastically or viscoelastically, they can adhere to each other whereas the carrying fluid is usually Newtonian. We report here on a new elasto-visco-plastic model which is able to describe the mechanical properties of a concentrated cell suspension or aggregate. It is based on the idea that the rearrangement of adhesion bonds during the deformation of the aggregate is related to the existence of a yield stress in the macroscopic constitutive equation. We compare the predictions of this new model with five experimental tests: steady shear rate, oscillatory shearing tests, stress relaxation, elastic recovery after steady prescribed deformation, and uniaxial compression tests. All of the predictions of the model are shown to agree with these experiments.

Mechanics and chemotaxis in the morphogenesis of vascular networks.

The formation of vascular networks in vitro develops along two rather distinct stages: during the early migration-dominated stage the main features of the pattern emerge, later the mechanical interaction of the cells with the substratum stretches the network. Mathematical models in the relevant literature have been focusing just on either of the aspects of this complex system. In this paper, a unified view of the morphogenetic process is provided in terms of physical mechanisms and mathematical modeling.

Mathematical modelling of the loss of tissue compression responsiveness and its role in solid tumour development.

This paper presents a mathematical model of normal and abnormal tissue growth. The modelling focuses on the potential role that stress responsiveness may play in causing proliferative disorders which are at the basis of the development of avascular tumours. In particular, we study how an incorrect sensing of its compression state by a cell population can represent a clonal advantage and can generate hyperplasia and tumour growth with well-known characteristics such as compression of the tissue, structural changes in the extracellular matrix, change in the percentage of cell type (normal or abnormal), extracellular matrix and extracellular liquid. A spatially independent description of the phenomenon is given initially by a system of non-linear ordinary differential equations which is explicitly solved in some cases of biological interest showing a first phase in which some abnormal cells simply replace the normal ones, a second phase in which the hyper-proliferation of the abnormal cells causes a progressive compression within the tissue itself and a third phase in which the tissue reaches a compressed state, which presses on the surrounding environment. A travelling wave analysis is also performed which gives an estimate of the velocity of the growing mass.

Percolation, morphogenesis, and burgers dynamics in blood vessels formation.

Experiments of in vitro formation of blood vessels show that cells randomly spread on a gel matrix autonomously organize to form a connected vascular network. We propose a simple model which reproduces many features of the biological system. We show that both the model and the real system exhibit a fractal behavior at small scales, due to the process of migration and dynamical aggregation, followed at large scale by a random percolation behavior due to the coalescence of aggregates. The results are in good agreement with the analysis performed on the experimental data.

Cloning and sequencing of the sacA gene: characterization of a sucrase from Zymomonas mobilis.

The Zymomonas mobilis gene (sacA) encoding a protein with sucrase activity has been cloned in Escherichia coli and its nucleotide sequence has been determined. Potential ribosome-binding site and promoter sequences were identified in the region upstream of the gene which were homologous to E. coli and Z. mobilis consensus sequences. Extracts from E. coli cells, containing the sacA gene, displayed a sucrose-hydrolyzing activity. However, no transfructosylation activity (exchange reaction or levan formation) could be detected. This sucrase activity was different from that observed with the purified extracellular protein B46 from Z. mobilis. These two proteins showed different electrophoretic mobilities and molecular masses and shared no immunological similarity. Thus, the product of sacA (a polypeptide of 58.4-kDa molecular mass) is a new sucrase from Z. mobilis. The amino acid sequence, deduced from the nucleotide sequence of sacA, showed strong homologies with the sucrases from Bacillus subtilis, Salmonella typhimurium, and Vibrio alginolyticus.