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 with 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.Comment: 25 pages, 6 figure

A particle model reproducing the effect of a conflicting flight information on the honeybee swarm guidance

The honeybee swarming process is steered by few scout individuals, which are the unique informed on the location of the target destination. Theoretical and experimental results suggest that bee coordinated flight arises from visual signals. However, how the information is passed within the population is still debated. Moreover, it has been observed that honeybees are highly sensitive to conflicting directional information. In fact, swarms exposed to fast-moving bees headed in the wrong direction show clear signs of disrupted guidance. In this respect, we here present a discrete mathematical model to investigate different hypotheses on the behaviour both of informed and uninformed bees. In this perspective, numerical realizations, specifically designed to mimic selected experiments, reveal that only one combination of the considered assumptions is able to reproduce the empirical outcomes, resulting thereby the most reliable mechanism underlying the swarm dynamics according to the proposed approach. Specifically, this study suggests that (i) leaders indicate the right flight direction by repeatedly streaking at high speed pointing towards the target and then slowly coming back to the trailing edge of the bee cloud; and (ii) uninformed bees, in turn, gather the route information by adapting their movement to all the bees sufficiently close to their position

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 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 protoorgan (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 stromalderived 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

A particle model analyzing the behavioral 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

A discrete mathematical model for the dynamics of a crowd of gazing pedestrians with and without an evolving environmental awareness

In this article, we present a microscopic-discrete mathematical model describing crowd dynamics in no panic conditions. More specifically, pedestrians are set to move in order to reach a target destination and their movement is influenced by both behavioral strategies and physical forces. Behavioral strategies include individual desire to remain sufficiently far from structural elements (walls and obstacles) and from other walkers, while physical forces account for interpersonal collisions. The resulting pedestrian behavior emerges therefore from non-local, anisotropic and short/long-range interactions. Relevant improvements of our mathematical model with respect to similar microscopic-discrete approaches present in the literature are: (i) each pedestrian has his/her own dynamic gazing direction, which is regarded to as an independent degree of freedom and (ii) each walker is allowed to take dynamic strategic decisions according to his/her environmental awareness, which increases due to new information acquired on the surrounding space through their visual region. The resulting mathematical modeling environment is then applied to specific scenarios that, although simplified, resemble real-word situations. In particular, we focus on pedestrian flow in twodimensional buildings with several structural elements (i.e., corridors, divisors and columns, and exit doors). The noticeable heterogeneity of possible applications demonstrates the potential of our mathematical model in addressing different engineering problems, allowing for optimization issues as well

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

An integro-differential non-local model for cell migration and its efficient numerical solution

Cell migration is fundamental in a wide variety of physiological and pathological phenomena, being exploited in biomedical engineering as well. In this respect, we here present a hybrid non-local integro-differential model where a representative cell, reproduced by a point particle with an orientation, moves on a planar domain upon signals coming from environmental variables. From a numerical point of view, non-locality implies the need to evaluate integral terms which may present non-regular integrand functions because of heterogeneities in the environmental conditions and/or in cell sensing region. Having in mind multicellular applications, we here propose a robust computational method able to handle such non-regularities. The procedure is based on low order Runge–Kutta methods and on an ad hoc application of the Gauss–Legendre quadrature rule. The accuracy and efficiency of the resulting computational method is then tested by selected benchmark settings. In this context, the ad hoc application of the quadrature rule reveals to be crucial to obtain a high accuracy with a remarkably low number of quadrature nodes with respect to the standard Gauss–Legendre quadrature formula, and which thus results in a reduced overall computational cost. Finally, the proposed method is further coupled with the cubic spline interpolation scheme which allows to deal also with possible poor (i.e., point-wise defined) molecular spatial information. The performed simulations (which accounts also for different scenarios) show how the interpolation of the molecular variables affects the efficiency of the overall method and further justify the proposed procedure

A sound understanding of a cropping system model with the global sensitivity analysis

The capability of cropping system models of depicting the crop and soil-related processes implies a high number of parameters. The aim of this work was to detect the key parameters, and the associated processes, of the ARMOSA cropping system model, considering two target outputs, crop yield and nitrogen leaching. A global sensitivity analysis (SA) was carried out in two steps: (1) the Morris method considering the whole set of parameters; (2) Sobol analysis was applied to the Morris outcome. The simulation was run on winter wheat in four soil types in Marchfeld (Austria, 2010–2018). Parameters affecting crop yield was the critical nitrogen concentration, the potential CO2 assimilation rate, and the drought sensitivity parameter. Nitrogen leaching was mainly affected by the decomposition of litter and the early aboveground biomass growth. The parameters ranking did not appreciably change across soil types. This study offers a quick and replicable methodology for model calibration

Non-local hybrid models for collective dynamics

From a mathematical point of view, living systems such as cell aggregates, crowd and swarms, can be regarded as collections of particles characterized by a proper behavior, and by the ability to sense and actively interact with the other individuals and the surrounding environment. In particular, living particles do not respond passively to the rules of inertia but are able to change their individual behavior and affect the collective evolution of the system. In general, there exists several approaches able to describe the collective dynamics of living individuals. First, in microscopic/individual-based models, the single components are represented as localized/discrete units and suitable rules describe their individual behavior and mutual interactions. These approaches thus account for the intrinsic granularity of living systems. However, the usually large amount of individuals involved in interesting self-emergent patterns (such as in morphogenesis and cancer evolution, as well as in pedestrian evacuations) makes it difficult to recover usable synthetic quantitative information about the whole aggregate from the reproduction of the individual behavior. On the other hand, a macroscopic/continuous modelling approach is conversely based on the assimilation of the system to a whole entity distributed in space. The evolution of the collectivity is given by means of non linear conservation laws which directly provide the evolution of average quantities such as density and flux. In this respect, these techniques are able to overcome the above highlighted critical issues posed by an individual-based approach. However, by describing a living system as a whole via phenomenological constitutive relationships rather then by an actual one-to-one interaction basis, the behavior of single entities is not accessible. It thereby results hard to reproduce complex evolutions observable at the aggregate level (for instance, pattern formations) which are generated by microscopic/individual phenomena (such as, cell phenotypic transitions in biological systems, or individual choice of own motion mode in crowd dynamics). In these situations, in order to overcome the difficulties posed by purely micro/macro approaches, it can be convenient to opt for an hybrid modelling approach, i.e., to differentiate the individuals in more groups with specific properties and behavior, and to use different descriptive instances for each subsystem. In particular, the coupling of models based on both localized/discrete and distributed/continuous formulations, might allow to take the advantages of both classical techniques. Taking into account these considerations, the thesis is organized in two parts, respectively dedicated to the illustration of hybrid modelling techniques developed over the course of my Ph.D. to capture the evolution of specific biological systems and pedestrian dynamics. More specifically, we first focus on biological systems whose evolution is regulated by cell phenotypic differentiation processes (e.g., tumor growth and invasion, and zebrafish posterior lateral line development), and then on pedestrian dynamics affected by different types of human perceptions of surrounding individuals. In particular, in both applications, the dynamics of both cells and pedestrians is defined through a phenomenological description of their velocity, which is given by the superposition of a directional contribution and non-local interaction terms accounting for the ability of living particles to perceive and consequently react to the presence of individuals located at a certain distance from them