A Review of Mathematical Models for the Formation of\ud Vascular Networks

Mainly two mechanisms are involved in the formation of blood vasculature: vasculogenesis and angiogenesis. The former consists of 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 consists of the sprouting of new vessels from an existing capillary or post-capillary venule. Similar phenomena are also involved in the formation of the lymphatic system through a process generally called lymphangiogenesis.\ud \ud A number of mathematical approaches have analysed these phenomena. This paper reviews the different modelling procedures, with a special emphasis on their ability to reproduce the biological system and to predict measured quantities which describe the overall processes. A comparison between the different methods is also made, highlighting their specific features

MODELLING THE INFLUENCE OF NUCLEUS ELASTICITY ON CELL INVASION IN FIBER NETWORKS AND MICROCHANNELS

Cell migration in highly constrained extracellular matrices is exploited in scaffold-based tissue engineering and is fundamental in a wide variety of physiological and pathological phenomena, among others in cancer invasion and development. Research into the critical processes involved in cell migration has mainly focused on cell adhesion and proteolytic degradation of the external environment. However, rising evidence has recently shown that a number of cell-derived biophysical and mechanical parameters, among others nucleus stiffness and cell deformability, plays a major role in cell motility, especially in the ameboid-like migration mode in 3D confined tissue structures. We here present an extended cellular Potts model (CPM) first used to simulate a micro-fabricated migration chip, which tests the active invasive behavior of cancer cells into narrow channels. As distinct features of our approach, cells are modeled as compartmentalized discrete objects, differentiated in the nucleus and in the cytosolic region, while the migration chamber is composed of channels of different widths. We find that cell motile phenotype and velocity in open spaces (i.e., 2D flat surfaces or large channels) are not significantly influenced by cell elastic properties. On the contrary, the migratory behavior of cells within subcellular and subnuclear structures strongly relies on the deformability of the cytosol and of the nuclear cluster, respectively. Further, we characterize two migration dynamics: a stepwise way, characterized by fluctuations in cell length, within channels smaller than nucleus dimensions and a smooth sliding (i.e., maintaining constant cell length) behavior within channels larger than the nuclear cluster. These resulting observations are then extended looking at cell migration in an artificial fiber network, which mimics cell invasion in a 3D extracellular matrix. In particular, in this case, we analyze the effect of variations in elasticity of the nucleus on cell movement. In order to summarize, with our simulated migration assays, we demonstrate that the dimensionality of the environment strongly affects the migration phenotype and we suggest that the cytoskeletal and nuclear elastic characteristics correlate with the tumor cell's invasive potentia

A multiscale hybrid model for pro-angiogenic calcium signals in a vascular endothelial cell

Cytosolic calcium machinery is one of the principal signaling mechanisms by which endothelial cells (ECs) respond to external stimuli during several biological processes, including vascular progression in both physiological and pathological conditions. Low concentrations of angiogenic factors (such as VEGF) activate in fact complex pathways involving, among others, second messengers arachidonic acid (AA) and nitric oxide (NO), which in turn control the activity of plasma membrane calcium channels. The subsequent increase in the intracellular level of the ion regulates fundamental biophysical properties of ECs (such as elasticity, intrinsic motility, and chemical strength), enhancing their migratory capacity. Previously, a number of continuous models have represented cytosolic calcium dynamics, while EC migration in angiogenesis has been separately approached with discrete, lattice-based techniques. These two components are here integrated and interfaced to provide a multiscale and hybrid Cellular Potts Model (CPM), where the phenomenology of a motile EC is realistically mediated by its calcium-dependent subcellular events. The model, based on a realistic 3-D cell morphology with a nuclear and a cytosolic region, is set with known biochemical and electrophysiological data. In particular, the resulting simulations are able to reproduce and describe the polarization process, typical of stimulated vascular cells, in various experimental conditions.Moreover, by analyzing the mutual interactions between multilevel biochemical and biomechanical aspects, our study investigates ways to inhibit cell migration: such strategies have in fact the potential to result in pharmacological interventions useful to disrupt malignant vascular progressio

Hierarchical nanomechanics of collagen microfibrils

Collagen constitutes one third of the human proteome, providing mechanical stability, elasticity and strength to connective tissues. Collagen is also the dominating material in the extracellular matrix (ECM) and is thus crucial for cell differentiation, growth and pathology. However, fundamental questions remain with respect to the origin of the unique mechanical properties of collagenous tissues, and in particular its stiffness, extensibility and nonlinear mechanical response. By using x-ray diffraction data of a collagen fibril reported by Orgel et al. (Proceedings of the National Academy of Sciences USA, 2006) in combination with protein structure identification methods, here we present an experimentally validated model of the nanomechanics of a collagen microfibril that incorporates the full biochemical details of the amino acid sequence of the constituting molecules. We report the analysis of its mechanical properties under different levels of stress and solvent conditions, using a full-atomistic force field including explicit water solvent. Mechanical testing of hydrated collagen microfibrils yields a Young’s modulus of ≈300 MPa at small and ≈1.2 GPa at larger deformation in excess of 10% strain, in excellent agreement with experimental data. Dehydrated, dry collagen microfibrils show a significantly increased Young’s modulus of ≈1.8 to 2.25 GPa (or ≈6.75 times the modulus in the wet state) owing to a much tighter molecular packing, in good agreement with experimental measurements (where an increase of the modulus by ≈9 times was found). Our model demonstrates that the unique mechanical properties of collagen microfibrils can be explained based on their hierarchical structure, where deformation is mediated through mechanisms that operate at different hierarchical levels. Key mechanisms involve straightening of initially disordered and helically twisted molecules at small strains, followed by axial stretching of molecules, and eventual molecular uncoiling at extreme deformation. These mechanisms explain the striking difference of the modulus of collagen fibrils compared with single molecules, which is found in the range of 4.8±2 GPa or ≈10-20 times greater. These findings corroborate the notion that collagen tissue properties are highly scale dependent and nonlinear elastic, an issue that must be considered in the development of models that describe the interaction of cells with collagen in the extracellular matrix. A key impact the atomistic model of collagen microfibril mechanics reported here is that it enables the bottom-up elucidation of structure-property relationships in the broader class of collagen materials such as tendon or bone, including studies in the context of genetic disease where the incorporation of biochemical, genetic details in material models of connective tissue is essential

Multiscale developments of cellular Potts models

Multiscale problems are ubiquitous and fundamental in all biological phenomena that emerge naturally from the complex interaction of processes which occur at various levels. A number of both discrete and continuous mathematical models and methods have been developed to address such an intricate network of organization. One of the most suitable individual cell-based model for this purpose is the well-known cellular Potts model (CPM). The CPM is a discrete, lattice-based, flexible technique that is able to accurately identify and describe the phenomenological mechanisms which are responsible for innumerable biological (and nonbiological) phenomena. In this work, we first give a brief overview of its biophysical basis and discuss its main limitations. We then propose some innovative extensions, focusing on ways of integrating the basic mesoscopic CPM with accurate continuous models of microscopic dynamics of individuals. The aim is to create a multiscale hybrid framework that is able to deal with the typical multilevel organization of biological development, where the behavior of the simulated individuals is realistically driven by their internal state. Our CPM extensions are then tested with sample applications that show a qualitative and quantitative agreement with experimental data. Finally, we conclude by discussing further possible developments of the metho

A cellular Potts model analyzing differentiated cell behavior during in vivo vascularization of a hypoxic tissue

Angiogenesis, the formation of new blood vessel networks from existing capillary or post-capillary venules, is an intrinsically multiscale process occurring in several physio-pathological conditions. In particular, hypoxic tissue cells activate downstream cascades culminating in the secretion of a wide range of angiogenic factors, including VEGF isoforms. Such diffusive chemicals activate the endothelial cells (ECs) forming the external walls of the nearby vessels that chemotactically migrate toward the hypoxic areas of the tissue as multicellular sprouts. A functional network eventually emerges by further branching and anastomosis processes. We here propose a CPM-based approach reproducing selected features of the angiogenic progression necessary for the reoxygenation of a hypoxic tissue. Our model is able to span the different scale involved in the angiogenic progression as it incorporates reaction-diffusion equations for the description of the evolution of microenvironmental variables in a discrete mesoscopic cellular Potts model (CPM) that reproduces the dynamics of the vascular cells. A key feature of this work is the explicit phenotypic differentiation of the ECs themselves, distinguished in quiescent, stalk and tip. The simulation results allow identifying a set of key mechanisms underlying tissue vascularization. Further, we provide evidence that the nascent pattern is characterized by precise topological properties. Finally, we link abnormal sprouting angiogenesis with alteration in selected cell behavior

A node-based version of the cellular Potts model

The cellular Potts model (CPM) is a lattice-based Monte Carlo method that uses an energetic formalism to describe the phenomenological mechanisms underlying the biophysical problem of interest. We here propose a CPM-derived framework that relies on a node-based representation of cell-scale elements. This feature has relevant consequences on the overall simulation environment. First, our model can be implemented on any given domain, provided a proper discretization (which can be regular or irregular, fixed or time evolving). Then, it allowed an explicit representation of cell membranes, whose displacements realistically result in cell movement. Finally, our node-based approach can be easily interfaced with continuous mechanics or fluid dynamics models. The proposed computational environment is here applied to some simple biological phenomena, such as cell sorting and chemotactic migration, also in order to achieve an analysis of the performance of the underlying algorithm. This work is finally equipped with a critical comparison between the advantages and disadvantages of our model with respect to the traditional CPM and to some similar vertex-based approaches

Influence of mechanical forces on the self-organisation of biomolecular systems

Mechanical forces play a crucial role in shaping the development of tissues and organisms. Nature has evolved intricate ways of utilising mechanical cues in order to achieve various objectives. In this dissertation, we examine the role played by mechanical forces in regulating the function of biological systems at the level of many molecules. We employ computational simulations, using minimal coarse-grained models. This allows us to capture the essential information about the investigated systems as well as to derive general phenomenological insights. We begin with the study of mechanosensitive protein channels, which are a class of transmembrane proteins responsible for sensing the osmotic pressure on cells and protecting them against lysis. Recent experiments suggest that such channels separate into liquid-like clusters, but the functional role of this aggregation is still unknown. We examine the collective behaviour of such proteins and we reveal that a dynamic self-assembly of channels, driven by changes in membrane tension, can control the osmotic pressure equilibration and the volume of the whole cell. We then focus on the growth of membrane protrusions, or tubes, which are thin elongated structures used by cells to sense mechanical stimuli. We investigate the influence of proteins linking the membrane to cytoskeletal components on pulling membrane tubes. We find that the force required to extrude a tube has an intriguing non-linear dependence on the concentration of cortex attachments. Subsequently, we turn our attention to the study of the mechanically-induced self-assembly of fibronectin, a structural protein constituent of the extracellular matrix. We examine the emergent fibrillar architectures and show how the morphologies of these networks change depending on various mechanical parameters. Finally, we explore how a nanoparticle adsorbed on a deformable elastic membrane senses the substrate’s mechanical properties through gradients in the membrane’s bending rigidity. We hope that the results presented in this dissertation will spur further discussions and experimental studies related to the functional role played by mechanical forces in regulating the collective macromolecular behaviour at the nanoscale

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 analyse these phenomena. In this article, 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. The research that led to the present paper was partially supported by a grant of the group GNFM of INdA