Juan Soler

MULTISCALE BIOLOGICAL TISSUE MODELS AND FLUX-LIMITED CHEMOTAXIS FOR MULTICELLULAR GROWING SYSTEMS

This paper deals with the derivation of macroscopic tissue models from the underlying description delivered by a class of equations that models binary mixtures of multicellular systems by methods of the kinetic theory for active particles. Cellular interactions generate both modification of the biological functions and proliferative and destructive events. The asymptotic analysis deals with suitable parabolic and hyperbolic limits, and is specifically focused on the modeling of the chemotaxis phenomena.

ON THE MATHEMATICAL THEORY OF THE DYNAMICS OF SWARMS VIEWED AS COMPLEX SYSTEMS

This paper deals with the modeling and simulation of swarms viewed as a living, hence complex, system. The approach is based on methods of kinetic theory and statistical mechanics, where interactions at the microscopic scale are nonlinearly additive and modeled by stochastic games.

MULTICELLULAR BIOLOGICAL GROWING SYSTEMS: HYPERBOLIC LIMITS TOWARDS MACROSCOPIC DESCRIPTION

This paper deals with the analysis of the asymptotic limit towards the derivation of hyperbolic macroscopic equations for a class of equations modeling complex multicellular systems. Cellular interactions generate both modification of biological functions and proliferating destructive events related to growth of tumor cells in competition with the immune system. The asymptotic analysis refers to the hyperbolic limit to show how the macroscopic tissue behavior can be described by linear and nonlinear hyperbolic systems which seem the most natural in this context.

ON THE DIFFICULT INTERPLAY BETWEEN LIFE, "COMPLEXITY", AND MATHEMATICAL SCIENCES

This paper presents a revisiting, with developments, of the so-called kinetic theory for active particles, with the main focus on the modeling of nonlinearly additive interactions. The approach is based on a suitable generalization of methods of kinetic theory, where interactions are depicted by stochastic games. The basic idea consists in looking for a general mathematical structure suitable to capture the main features of living, hence complex, systems. Hopefully, this structure is a candidate towards the challenging objective of designing a mathematical theory of living systems. These topics are treated in the first part of the paper, while the second one applies it to specific case studies, namely to the modeling of crowd dynamics and of the immune competition.

Long-time dynamics of the Schrödinger-Poisson-Slater system

In this paper we analyze the asymptotic behaviour of solutions to the Schrödinger–Poisson–Slater (SPS) system in the frame of semiconductor modeling. Depending on the potential energy and on the physical constants associated with the model, the repulsive SPS system develops stationary or periodic solutions. These solutions preserve the Lp(ℝ3) norm or exhibit dispersion properties. In comparison with the Schrödinger–Poisson (SP) system, only the last kind of solutions appear.

ON THE ASYMPTOTIC THEORY FROM MICROSCOPIC TO MACROSCOPIC GROWING TISSUE MODELS: AN OVERVIEW WITH PERSPECTIVES

This paper proposes a review and critical analysis of the asymptotic limit methods focused on the derivation of macroscopic equations for a class of equations modeling complex multicellular systems by methods of the kinetic theory for active particles. Cellular interactions generate both modification of biological functions and proliferative/destructive events. The asymptotic analysis deals with suitable parabolic, hyperbolic, and mixed limits. The review includes the derivation of the classical Keller–Segel model and flux limited models that prevent non-physical blow up of solutions.

An Analysis of Quantum Fokker-Planck Models: A Wigner Function Approach

The analysis of dissipative transport equations within the framework of open quantum systems with Fokker-Planck-type scattering is carried out from the perspective of a Wigner function approach. In particular, the well-posedness of the self-consistent whole-space problem in 3D is analyzed: existence of solutions, uniqueness and asymptotic behavior in time, where we adopt the viewpoint of mild solutions in this paper. Also, the admissibility of a density matrix formulation in Lindblad form with Fokker-Planck dissipation mechanisms is discussed. We remark that our solution concept allows to carry out the analysis directly on the level of the kinetic equation instead of on the level of the density operator.

Asymptotic behavior of an initial-boundary value problem for the Vlasov-Poisson-Fokker-Planck system

The asymptotic behavior for the Vlasov--Poisson--Fokker--Planck system in bounded domains is analyzed in this paper. Boundary conditions defined by a scattering kernel are considered. It is proven that the distribution of particles tends for large time to a Maxwellian determined by the solution of the Poisson--Boltzmann equation with Dirichlet boundary condition. In the proof of the main result, the conservation law of mass and the balance of energy and entropy identities are rigorously derived. An important argument in the proof is to use a Lyapunov-type functional related to these physical quantities.

Morphogenetic action through flux-limited spreading

A central question in biology is how secreted morphogens act to induce different cellular responses within a group of cells in a concentration-dependent manner. Modeling morphogenetic output in multicellular systems has so far employed linear diffusion, which is the normal type of diffusion associated with Brownian processes. However, there is evidence that at least some morphogens, such as Hedgehog (Hh) molecules, may not freely diffuse. Moreover, the mathematical analysis of such models necessarily implies unrealistic instantaneous spreading of morphogen molecules, which are derived from the assumptions of Brownian motion in its continuous formulation. A strict mathematical model considering Ficks diffusion law predicts morphogen exposure of the whole tissue at the same time. Such a strict model thus does not describe true biological patterns, even if similar and attractive patterns appear as results of applying such simple model. To eliminate non-biological behaviors from diffusion models we introduce flux-limited spreading (FLS), which implies a restricted velocity for morphogen propagation and a nonlinear mechanism of transport. Using FLS and focusing on intercellular Hh-Gli signaling, we model a morphogen gradient and highlight the propagation velocity of morphogen particles as a new key biological parameter. This model is then applied to the formation and action of the Sonic Hh (Shh) gradient in the vertebrate embryonic neural tube using our experimental data on Hh spreading in heterologous systems together with published data. Unlike linear diffusion models, FLS modeling predicts concentration fronts and the evolution of gradient dynamics and responses over time. In addition to spreading restrictions by extracellular binding partners, we suggest that the constraints imposed by direct bridges of information transfer such as nanotubes or cytonemes underlie FLS. Indeed, we detect and measure morphogen particle velocity in such cell extensions in different systems.

Complexity and mathematical tools toward the modelling of multicellular growing systems

This paper deals with a multiscale modelling approach to complex biological systems constituted by several interacting entities. The methodology is based on mathematical kinetic theory for active particles and is focused on the modelling of complex multicellular systems under therapeutic actions at the cellular level and mutations with onset of new populations. Asymptotic hyperbolic methods are developed to derive models at the macroscopic scale of tissues from the underlying description at the level of cells for a open system with variable number of populations.