Selected aspects of avascular tumor growth reproduced by a hybrid model of cell dynamics and chemical kinetics.

We here propose a hybrid computational framework to reproduce and analyze aspects of the avascular progression of a generic solid tumor. Our method first employs an individual-based approach to represent the population of tumor cells, which are distinguished in viable and necrotic agents. The active part of the disease is in turn differentiated according to a set of metabolic states. We then describe the spatio-temporal evolution of the concentration of oxygen and of tumor-secreted proteolytic enzymes using partial differential equations (PDEs). A differential equation finally governs the local degradation of the extracellular matrix (ECM) by the malignant mass. Numerical realizations of the model are run to reproduce tumor growth and invasion in a number scenarios that differ for cell properties (adhesiveness, duplication potential, proteolytic activity) and/or environmental conditions (level of tissue oxygenation and matrix density pattern). In particular, our simulations suggest that tumor aggressiveness, in terms of invasive depth and extension of necrotic tissue, can be reduced by (i) stable cell-cell contact interactions, (ii) poor tendency of malignant agents to chemotactically move upon oxygen gradients, and (iii) presence of an overdense matrix, if coupled by a disrupted proteolytic activity of the disease.

Modelling Cell Orientation Under Stretch: The Effect of Substrate Elasticity.

When cells are seeded on a cyclically deformed substrate like silicon, they tend to reorient their major axis in two ways: either perpendicular to the main stretching direction, or forming an oblique angle with it. However, when the substrate is very soft such as a collagen gel, the oblique orientation is no longer observed, and the cells align either along the stretching direction, or perpendicularly to it. To explain this switch, we propose a simplified model of the cell, consisting of two elastic elements representing the stress fiber/focal adhesion complexes in the main and transverse directions. These elements are connected by a torsional spring that mimics the effect of crosslinking molecules among the stress fibers, which resist shear forces. Our model, consistent with experimental observations, predicts that there is a switch in the asymptotic behaviour of the orientation of the cell determined by the stiffness of the substratum, related to a change from a supercritical bifurcation scenario, whereby the oblique configuration is stable for a sufficiently large stiffness, to a subcritical bifurcation scenario at a lower stiffness. Furthermore, we investigate the effect of cell elongation and find that the region of the parameter space leading to an oblique orientation decreases as the cell becomes more elongated. This implies that elongated cells, such as fibroblasts and smooth muscle cells, are more likely to maintain an oblique orientation with respect to the main stretching direction. Conversely, rounder cells, such as those of epithelial or endothelial origin, are more likely to switch to a perpendicular or parallel orientation on soft substrates.

A phenotype-structured model to reproduce the avascular growth of a tumor and its interaction with the surrounding environment.

We here propose a one-dimensional spatially explicit phenotype-structured model to analyze selected aspects of avascular tumor progression. In particular, our approach distinguishes viable and necrotic cell fractions. The metabolically active part of the disease is, in turn, differentiated according to a continuous trait, that identifies cell variants with different degrees of motility and proliferation potential. A parabolic partial differential equation (PDE) then governs the spatio-temporal evolution of the phenotypic distribution of active cells within the host tissue. In this respect, active tumor agents are allowed to duplicate, move upon haptotactic and pressure stimuli, and eventually undergo necrosis. The mutual influence between the emerging malignancy and its environment (in terms of molecular landscape) is implemented by coupling the evolution law of the viable tumor mass with a parabolic PDE for oxygen kinetics and a differential equation that accounts for local consumption of extracellular matrix (ECM) elements. The resulting numerical realizations reproduce tumor growth and invasion in a number scenarios that differ for cell properties (i.e., individual migratory behavior, duplication, and mutation potential) and environmental conditions (i.e., level of tissue oxygenation and homogeneity in the initial matrix profile). In particular, our simulations show that, in all cases, more mobile cell variants occupy the front edge of the tumor, whereas more proliferative clones are selected at more internal regions. A necrotic core constantly occupies the bulk of the mass due to nutrient deprivation. This work may eventually suggest some biomedical strategies to partially reduce tumor aggressiveness, i.e., to enhance necrosis of malignant tissue and to promote the presence of more proliferative cell phenotypes over more invasive ones.

A hybrid integro-differential model for the early development of the zebrafish posterior lateral line.

The aim of this work is to provide a mathematical model to describe the early stages of the embryonic development of zebrafish posterior lateral line (PLL). In particular, we focus on evolution of PLL proto-organ (said primordium), from its formation to the beginning of the cyclical behavior that amounts in the assembly of immature proto-neuromasts towards its caudal edge accompanied by the deposition of mature proto-neuromasts at its rostral region. Our approach has an hybrid integro-differential nature, since it integrates a microscopic/discrete particle-based description for cell dynamics and a continuous description for the evolution of the spatial distribution of chemical substances (i.e., the stromal-derived factor SDF1a and the fibroblast growth factor FGF10). Boolean variables instead implement the expression of molecular receptors (i.e., Cxcr4/Cxcr7 and fgfr1). Cell phenotypic transitions and proliferation are included as well. The resulting numerical simulations show that the model is able to qualitatively and quantitatively capture the evolution of the wild-type (i.e., normal) embryos as well as the effect of known experimental manipulations. In particular, it is shown that cell proliferation, intercellular adhesion, FGF10-driven dynamics, and a polarized expression of SDF1a receptors are all fundamental for the correct development of the zebrafish posterior lateral line.

An agent-based approach for modelling collective dynamics in animal groups distinguishing individual speed and orientation.

Collective dynamics in animal groups is a challenging theme for the modelling community, being treated with a wide range of approaches. This topic is here tackled by a discrete model. Entering in more details, each agent, represented by a material point, is assumed to move following a first-order Newtonian law, which distinguishes speed and orientation. In particular, the latter results from the balance of a given set of behavioural stimuli, each of them defined by a direction and a weight, that quantifies its relative importance. A constraint on the sum of the weights then avoids implausible simultaneous maximization/minimization of all movement traits. Our framework is based on a minimal set of rules and parameters and is able to capture and classify a number of collective group dynamics emerging from different individual preferred behaviour, which possibly includes attractive, repulsive and alignment stimuli. In the case of a system of animals subjected only to the first two behavioural inputs, we also show how analytical arguments allow us to a priori relate the equilibrium interparticle spacing to critical model coefficients. Our approach is then extended to account for the presence of predators with different hunting strategies, which impact on the behaviour of a prey population. Hints for model refinement and applications are finally given in the conclusive part of the article. This article is part of the theme issue 'Multi-scale analysis and modelling of collective migration in biological systems'.

Collective migration and patterning during early development of zebrafish posterior lateral line.

The morphogenesis of zebrafish posterior lateral line (PLL) is a good predictive model largely used in biology to study cell coordinated reorganization and collective migration regulating pathologies and human embryonic processes. PLL development involves the formation of a placode formed by epithelial cells with mesenchymal characteristics which migrates within the animal myoseptum while cyclically assembling and depositing rosette-like clusters (progenitors of neuromast structures). The overall process mainly relies on the activity of specific diffusive chemicals, which trigger collective directional migration and patterning. Cell proliferation and cascade of phenotypic transitions play a fundamental role as well. The investigation on the mechanisms regulating such a complex morphogenesis has become a research topic, in the last decades, also for the mathematical community. In this respect, we present a multiscale hybrid model integrating a discrete approach for the cellular level and a continuous description for the molecular scale. The resulting numerical simulations are then able to reproduce both the evolution of wild-type (i.e. normal) embryos and the pathological behaviour resulting form experimental manipulations involving laser ablation. A qualitative analysis of the dependence of these model outcomes from cell-cell mutual interactions, cell chemical sensitivity and internalization rates is included. The aim is first to validate the model, as well as the estimated parameter values, and then to predict what happens in situations not tested yet experimentally. This article is part of the theme issue 'Multi-scale analysis and modelling of collective migration in biological systems'.

Kinesin-2 Controls the Motility of RAB5 Endosomes and Their Association with the Spindle in Mitosis.

RAB5 is a small GTPase that belongs to the wide family of Rab proteins and localizes on early endosomes. In its active GTP-bound form, RAB5 recruits downstream effectors that, in turn, are responsible for distinct aspects of early endosome function, including their movement along microtubules. We previously reported that, at the onset of mitosis, RAB5positive vesicles cluster around the spindle poles and, during metaphase, move along spindle microtubules. RNAi-mediated depletion of the three RAB5 isoforms delays nuclear envelope breakdown at prophase and severely affects chromosome alignment and segregation. Here we show that depletion of the Kinesin-2 motor complex impairs long-range movement of RAB5 endosomes in interphase cells and prevents localization of these vesicles at the spindle during metaphase. Similarly to the effect caused by RAB5 depletion, functional ablation of Kinesin-2 delays nuclear envelope breakdown resulting in prolonged prophase. Altogether these findings suggest that endosomal transport at the onset of mitosis is required to control timing of nuclear envelope breakdown.

A particle model analysing the behavioural rules underlying the collective flight of a bee swarm towards the new nest.

The swarming of a bee colony is guided by a small group of scout individuals, which are informed of the target destination (the new nest). However, little is known on the underlying mechanisms, i.e. on how the information is passed within the population. In this respect, we here present a discrete mathematical model to investigate these aspects. In particular, each bee, represented by a material point, is assigned its status within the colony and set to move according to individual strategies and social interactions. More specifically, we propose alternative assumptions on the flight synchronization mechanism of uninformed individuals and on the characteristic dynamics of the scout insects. Numerical realizations then point out the combinations of behavioural hypotheses resulting in collective productive movement. An analysis of the role of the scout bee percentage and of the phenomenology of the swarm in domains with structural elements is finally performed.

Adhesion and volume constraints via nonlocal interactions determine cell organisation and migration profiles.

The description of the cell spatial pattern and characteristic distances is fundamental in a wide range of physio-pathological biological phenomena, from morphogenesis to cancer growth. Discrete particle models are widely used in this field, since they are focused on the cell-level of abstraction and are able to preserve the identity of single individuals reproducing their behavior. In particular, a fundamental role in determining the usefulness and the realism of a particle mathematical approach is played by the choice of the intercellular pairwise interaction kernel and by the estimate of its parameters. The aim of the paper is to demonstrate how the concept of H-stability, deriving from statistical mechanics, can have important implications in this respect. For any given interaction kernel, it in fact allows to a priori predict the regions of the free parameter space that result in stable configurations of the system characterized by a finite and strictly positive minimal interparticle distance, which is fundamental when dealing with biological phenomena. The proposed analytical arguments are indeed able to restrict the range of possible variations of selected model coefficients, whose exact estimate however requires further investigations (e.g., fitting with empirical data), as illustrated in this paper by series of representative simulations dealing with cell colony reorganization, sorting phenomena and zebrafish embryonic development.

A discrete particle model reproducing collective dynamics of a bee swarm.

In this article, we present a microscopic discrete mathematical model describing collective dynamics of a bee swarm. More specifically, each bee is set to move according to individual strategies and social interactions, the former involving the desire to reach a target destination, the latter accounting for repulsive/attractive stimuli and for alignment processes. The insects tend in fact to remain sufficiently close to the rest of the population, while avoiding collisions, and they are able to track and synchronize their movement to the flight of a given set of neighbors within their visual field. The resulting collective behavior of the bee cloud therefore emerges from non-local short/long-range interactions. Differently from similar approaches present in the literature, we here test different alignment mechanisms (i.e., based either on an Euclidean or on a topological neighborhood metric), which have an impact also on the other social components characterizing insect behavior. A series of numerical realizations then shows the phenomenology of the swarm (in terms of pattern configuration, collective productive movement, and flight synchronization) in different regions of the space of free model parameters (i.e., strength of attractive/repulsive forces, extension of the interaction regions). In this respect, constraints in the possible variations of such coefficients are here given both by reasonable empirical observations and by analytical results on some stability characteristics of the defined pairwise interaction kernels, which have to assure a realistic crystalline configuration of the swarm. An analysis of the effect of unconscious random fluctuations of bee dynamics is also provided.

TRPM8 inhibits endothelial cell migration via a non-channel function by trapping the small GTPase Rap1.

Endothelial cell adhesion and migration are critical steps of the angiogenic process, whose dysfunction is associated with tumor growth and metastasis. The TRPM8 channel has recently been proposed to play a protective role in prostate cancer by impairing cell motility. However, the mechanisms by which it could influence vascular behavior are unknown. Here, we reveal a novel non-channel function for TRPM8 that unexpectedly acts as a Rap1 GTPase inhibitor, thereby inhibiting endothelial cell motility, independently of pore function. TRPM8 retains Rap1 intracellularly through direct protein-protein interaction, thus preventing its cytoplasm-plasma membrane trafficking. In turn, this mechanism impairs the activation of a major inside-out signaling pathway that triggers the conformational activation of integrin and, consequently, cell adhesion, migration, in vitro endothelial tube formation, and spheroid sprouting. Our results bring to light a novel, pore-independent molecular mechanism by which endogenous TRPM8 expression inhibits Rap1 GTPase and thus plays a critical role in the behavior of vascular endothelial cells by inhibiting migration.

An Innovative Assay for the Analysis of In Vitro Endothelial Remodeling: Experimental and Computational Evidence.

The molecular and cellular mechanisms underlying vascular remodeling are currently investigated by experimental strategies which aim to mimic the complex environmental conditions found in vivo. Some of them focus on the tubulogenic activity of dispersed endothelial cell populations, while others evaluate vascular sprouting. Here we propose a new method to assess matrigel invasion starting from confluent or subconfluent monolayers of human microvascular ECs (HMVEC) seeded on different substrates. The experimental setting is also validated by an improved hybrid multiscale mathematical approach, which integrates a mesoscopic grid-based cellular Potts model, that describes HMVEC phenomenology, with a continuous one, accounting for the kinetics of diffusing growth factors. Both experimental and theoretical approaches show that the endothelial potential to invade, migrate, and organize in tubule structures is a function of selected environmental parameters. The present methodology is intended to be simple to use, standardized for rapid screening and suitable for mechanistic studies. J. Cell. Physiol. 232: 243-248, 2017. © 2016 Wiley Periodicals, Inc.

Computational Approaches for Translational Oncology: Concepts and Patents.

BACKGROUND: Cancer is a heterogeneous disease, which is based on an intricate network of processes at different spatiotemporal scales, from the genome to the tissue level. Hence the necessity for the biomedical and pharmaceutical research to work in a multiscale fashion. In this respect, a significant help derives from the collaboration with theoretical sciences. Mathematical models can in fact provide insights into tumor-related processes and support clinical oncologists in the design of treatment regime, dosage, schedule and toxicity. OBJECTIVE AND METHOD: The main objective of this article is to review the recent computational-based patents which tackle some relevant aspects of tumor treatment. We first analyze a series of patents concerning the purposing the purposing or repurposing of anti-tumor compounds. These approaches rely on pharmacokinetics and pharmacodynamics modules, that incorporate data obtained in the different phases of clinical trials. Similar methods are also at the basis of other patents included in this paper, which deal with treatment optimization, in terms of maximizing therapy efficacy while minimizing side effects on the host. A group of patents predicting drug response and tumor evolution by the use of kinetics graphs are commented as well. We finally focus on patents that implement informatics tools to map and screen biological, medical, and pharmaceutical knowledge. RESULTS AND CONCLUSIONS: Despite promising aspects (and an increasing amount of the relative literature), we found few computational-based patents: there is still a significant effort to do for allowing modelling approaches to become an integral component of the pharmaceutical research.

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.

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.

An extended Cellular Potts Model analyzing a wound healing assay.

A suitable Cellular Potts Model is developed to reproduce and analyze an in vitro wound-healing assay. The proposed approach is able both to quantify the invasive capacity of the overall cell population and to evaluate selected determinants of single cell movement (velocity, directional movement, and final displacement). In this respect, the present CPM allows us to capture differences and correlations in the migratory behavior of cells initially located at different distances from the wound edge. In the case of an undifferentiated extracellular matrix, the model then predicts that a maximal healing can be obtained by a chemically induced increment of cell elasticity and not by a chemically induced downregulation of intercellular adhesive contacts. Moreover, in the case of two-component substrates (formed by a mesh of collagenous-like threads and by a homogeneous medium), CPM simulations show that both fiber number and cell-fiber adhesiveness influence cell speed and wound closure rate in a biphasic fashion. On the contrary, the topology of the fibrous network affects the healing process by mediating the productive directional cell movement. The paper, also equipped with comments on the computational cost of the CPM algorithm, ends with a throughout discussion of the pertinent experimental and theoretical literature.

Differentiated cell behavior: a multiscale approach using measure theory.

This paper deals with the derivation of a collective model of cell populations out of an individual-based description of the underlying physical particle system. By looking at the spatial distribution of cells in terms of time-evolving measures, rather than at individual cell paths, we obtain an ensemble representation stemming from the phenomenological behavior of the single component cells. In particular, as a key advantage of our approach, the scale of representation of the system, i.e., microscopic/discrete vs. macroscopic/continuous, can be chosen a posteriori according only to the spatial structure given to the aforesaid measures. The paper focuses in particular on the use of different scales based on the specific functions performed by cells. A two-population hybrid system is considered, where cells with a specialized/differentiated phenotype are treated as a discrete population of point masses while unspecialized/undifferentiated cell aggregates are represented by a continuous approximation. Numerical simulations and analytical investigations emphasize the role of some biologically relevant parameters in determining the specific evolution of such a hybrid cell system.

A Cellular Potts Model simulating cell migration on and in matrix environments.

Cell migration on and through extracellular matrix is fundamental in a wide variety of physiological and pathological phenomena, and is exploited in scaffold-based tissue engineering. Migration is regulated by a number of extracellular matrix- or cell-derived biophysical parameters, such as matrix fiber orientation, pore size, and elasticity, or cell deformation, proteolysis, and adhesion. We here present an extended Cellular Potts Model (CPM) able to qualitatively and quantitatively describe cell migration efficiencies and phenotypes both on two-dimensional substrates and within three-dimensional matrices, close to experimental evidence. As distinct features of our approach, cells are modeled as compartmentalized discrete objects, differentiated into nucleus and cytosolic region, while the extracellular matrix is composed of a fibrous mesh and a homogeneous fluid. Our model provides a strong correlation of the directionality of migration with the topological extracellular matrix distribution and a biphasic dependence of migration on the matrix structure, density, adhesion, and stiffness, and, moreover, simulates that cell locomotion in highly constrained fibrillar obstacles requires the deformation of the cell's nucleus and/or the activity of cell-derived proteolysis. In conclusion, we here propose a mathematical modeling approach that serves to characterize cell migration as a biological phenomenon in healthy and diseased tissues and in engineering applications.

Modeling 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 potential.

Multilevel complexity of calcium signaling: Modeling angiogenesis.

Intracellular calcium signaling is a universal, evolutionary conserved and versatile regulator of cell biochemistry. The complexity of calcium signaling and related cell machinery can be investigated by the use of experimental strategies, as well as by computational approaches. Vascular endothelium is a fascinating model to study the specific properties and roles of calcium signals at multiple biological levels. During the past 20 years, live cell imaging, patch clamp and other techniques have allowed us to detect and interfere with calcium signaling in endothelial cells (ECs), providing a huge amount of information on the regulation of vascularization (angiogenesis) in normal and tumoral tissues. These data range from the spatiotemporal dynamics of calcium within different cell microcompartments to those in entire multicellular and organized EC networks. Beside experimental strategies, in silico endothelial models, specifically designed for simulating calcium signaling, are contributing to our knowledge of vascular physiology and pathology. They help to investigate and predict the quantitative features of proangiogenic events moving through subcellular, cellular and supracellular levels. This review focuses on some recent developments of computational approaches for proangiogenic endothelial calcium signaling. In particular, we discuss the creation of hybrid simulation environments, which combine and integrate discrete Cellular Potts Models. They are able to capture the phenomenological mechanisms of cell morphological reorganization, migration, and intercellular adhesion, with single-cell spatiotemporal models, based on reaction-diffusion equations that describe the agonist-induced intracellular calcium events.