A Model for Membrane Degradation Using a Gelatin Invadopodia Assay.

One of the most crucial and lethal characteristics of solid tumors is represented by the increased ability of cancer cells to migrate and invade other organs during the so-called metastatic spread. This is allowed thanks to the production of matrix metalloproteinases (MMPs), enzymes capable of degrading a type of collagen abundant in the basal membrane separating the epithelial tissue from the connective one. In this work, we employ a synergistic experimental and mathematical modelling approach to explore the invasion process of tumor cells. A mathematical model composed of reaction-diffusion equations describing the evolution of the tumor cells density on a gelatin substrate, MMPs enzymes concentration and the degradation of the gelatin is proposed. This is completed with a calibration strategy. We perform a sensitivity analysis and explore a parameter estimation technique both on synthetic and experimental data in order to find the optimal parameters that describe the in vitro experiments. A comparison between numerical and experimental solutions ends the work.

Adaptive dynamics of hematopoietic stem cells and their supporting stroma: a model and mathematical analysis.

We propose a mathematical model to describe the evolution of hematopoietic stem cells (HSCs) and stromal cells in considering the bi-directional interaction between them. Cancerous cells are also taken into account in our model. HSCs are structured by a continuous phenotype characterising the population heterogeneity in a way relevant to the question at stake while stromal cells are structured by another continuous phenotype representing their capacity of support to HSCs. We then analyse the model in the framework of adaptive dynamics. More precisely, we study single Dirac mass steady states, their linear stability and we investigate the role of parameters in the model on the nature of the evolutionary stable distributions (ESDs) such as monomorphism, dimorphism and the uniqueness properties. We also study the dominant phenotypes by an asymptotic approach and we obtain the equation for dominant phenotypes. Numerical simulations are employed to illustrate our analytical results. In particular, we represent the case of the invasion of malignant cells as well as the case of co-existence of cancerous cells and healthy HSCs.

A chemotaxis-based explanation of spheroid formation in 3D cultures of breast cancer cells.

Three-dimensional cultures of cells are gaining popularity as an in vitro improvement over 2D Petri dishes. In many such experiments, cells have been found to organize in aggregates. We present new results of three-dimensional in vitro cultures of breast cancer cells exhibiting patterns. Understanding their formation is of particular interest in the context of cancer since metastases have been shown to be created by cells moving in clusters. In this paper, we propose that the main mechanism which leads to the emergence of patterns is chemotaxis, i.e., oriented movement of cells towards high concentration zones of a signal emitted by the cells themselves. Studying a Keller-Segel PDE system to model chemotactical auto-organization of cells, we prove that it admits Turing unstable solutions under a time-dependent condition. This result is illustrated by two-dimensional simulations of the model showing spheroidal patterns. They are qualitatively compared to the biological results and their variability is discussed both theoretically and numerically.

Oscillatory regimes in a mosquito population model with larval feedback on egg hatching.

Understanding mosquitoes life cycle is of great interest presently because of the increasing impact of vector borne diseases in several countries. There is evidence of oscillations in mosquito populations independent of seasonality, still unexplained, based on observations both in laboratories and in nature. We propose a simple mathematical model of egg hatching enhancement by larvae which produces such oscillations that conveys a possible explanation. We propose both a theoretical analysis, based on slow-fast dynamics and Hopf bifurcation, and numerical investigations in order to shed some light on the mechanisms at work in this model.

Derivation of the bacterial run-and-tumble kinetic equation from a model with biochemical pathway.

Kinetic-transport equations are, by now, standard models to describe the dynamics of populations of bacteria moving by run-and-tumble. Experimental observations show that bacteria increase their run duration when encountering an increasing gradient of chemotactic molecules. This led to a first class of models which heuristically include tumbling frequencies depending on the path-wise gradient of chemotactic signal. More recently, the biochemical pathways regulating the flagellar motors were uncovered. This knowledge gave rise to a second class of kinetic-transport equations, that takes into account an intra-cellular molecular content and which relates the tumbling frequency to this information. It turns out that the tumbling frequency depends on the chemotactic signal, and not on its gradient. For these two classes of models, macroscopic equations of Keller-Segel type, have been derived using diffusion or hyperbolic rescaling. We complete this program by showing how the first class of equations can be derived from the second class with molecular content after appropriate rescaling. The main difficulty is to explain why the path-wise gradient of chemotactic signal can arise in this asymptotic process. Randomness of receptor methylation events can be included, and our approach can be used to compute the tumbling frequency in presence of such a noise.

Incompressible limit of a mechanical model of tumour growth with viscosity.

Various models of tumour growth are available in the literature. The first type describe the evolution of the cell number density when considered as a continuous visco-elastic material with growth. The second type describe the tumour as a set, and rules for the free boundary are given related to the classical Hele-Shaw model of fluid dynamics. Following previous papers where the material is described by a purely elastic material, or when active cell motion is included, we make the link between the two types of description considering the 'stiff pressure law' limit. Even though viscosity is a regularizing effect, new mathematical difficulties arise in the visco-elastic case because estimates on the pressure field are weaker and do not immediately imply compactness. For instance, travelling wave solutions and numerical simulations show that the pressure is discontinuous in space, which is not the case for an elastic material.

Modeling the effects of space structure and combination therapies on phenotypic heterogeneity and drug resistance in solid tumors.

Histopathological evidence supports the idea that the emergence of phenotypic heterogeneity and resistance to cytotoxic drugs can be considered as a process of selection in tumor cell populations. In this framework, can we explain intra-tumor heterogeneity in terms of selection driven by the local cell environment? Can we overcome the emergence of resistance and favor the eradication of cancer cells by using combination therapies? Bearing these questions in mind, we develop a model describing cell dynamics inside a tumor spheroid under the effects of cytotoxic and cytostatic drugs. Cancer cells are assumed to be structured as a population by two real variables standing for space position and the expression level of a phenotype of resistance to cytotoxic drugs. The model takes explicitly into account the dynamics of resources and anticancer drugs as well as their interactions with the cell population under treatment. We analyze the effects of space structure and combination therapies on phenotypic heterogeneity and chemotherapeutic resistance. Furthermore, we study the efficacy of combined therapy protocols based on constant infusion and bang-bang delivery of cytotoxic and cytostatic drugs.

Adaptation and fatigue model for neuron networks and large time asymptotics in a nonlinear fragmentation equation.

Motivated by a model for neural networks with adaptation and fatigue, we study a conservative fragmentation equation that describes the density probability of neurons with an elapsed time s after its last discharge. In the linear setting, we extend an argument by Laurençot and Perthame to prove exponential decay to the steady state. This extension allows us to handle coefficients that have a large variation rather than constant coefficients. In another extension of the argument, we treat a weakly nonlinear case and prove total desynchronization in the network. For greater nonlinearities, we present a numerical study of the impact of the fragmentation term on the appearance of synchronization of neurons in the network using two "extreme" cases. Mathematics Subject Classification (2000)2010: 35B40, 35F20, 35R09, 92B20.

Beyond blow-up in excitatory integrate and fire neuronal networks: Refractory period and spontaneous activity.

The Network Noisy Leaky Integrate and Fire equation is among the simplest model allowing for a self-consistent description of neural networks and gives a rule to determine the probability to find a neuron at the potential v. However, its mathematical structure is still poorly understood and, concerning its solutions, very few results are available. In the midst of them, a recent result shows blow-up in finite time for fully excitatory networks. The intuitive explanation is that each firing neuron induces a discharge of the others; thus increases the activity and consequently the discharge rate of the full network. In order to better understand the details of the phenomena and show that the equation is more complex and fruitful than expected, we analyze further the model. We extend the finite time blow-up result to the case when neurons, after firing, enter a refractory state for a given period of time. We also show that spontaneous activity may occur when, additionally, randomness is included on the firing potential VF in regimes where blow-up occurs for a fixed value of VF.

Direct competition results from strong competition for limited resource.

We study a model of competition for resource through a chemostat-type model where species consume the common resource that is constantly supplied. We assume that the species and resources are characterized by a continuous trait. As already proved, this model, although more complicated than the usual Lotka-Volterra direct competition model, describes competitive interactions leading to concentrated distributions of species in continuous trait space. Here we assume a very fast dynamics for the supply of the resource and a fast dynamics for death and uptake rates. In this regime we show that factors that are independent of the resource competition become as important as the competition efficiency and that the direct competition model is a good approximation of the chemostat. Assuming these two timescales allows us to establish a mathematically rigorous proof showing that our resource-competition model with continuous traits converges to a direct competition model. We also show that the two timescales assumption is required to mathematically justify the corresponding classic result on a model consisting of only finite number of species and resources (MacArthur in, Theor Popul Biol 1:1-11, 1970). This is performed through asymptotic analysis, introducing different scales for the resource renewal rate and the uptake rate. The mathematical difficulty relies in a possible initial layer for the resource dynamics. The chemostat model comes with a global convex Lyapunov functional. We show that the particular form of the competition kernel derived from the uptake kernel, satisfies a positivity property which is known to be necessary for the direct competition model to enjoy the related Lyapunov functional.

A model of calcium transport along the rat nephron.

We developed a mathematical model of calcium (Ca(2+)) transport along the rat nephron to investigate the factors that promote hypercalciuria. The model is an extension of the flat medullary model of Hervy and Thomas (Am J Physiol Renal Physiol 284: F65-F81, 2003). It explicitly represents all the nephron segments beyond the proximal tubules and distinguishes between superficial and deep nephrons. It solves dynamic conservation equations to determine NaCl, urea, and Ca(2+) concentration profiles in tubules, vasa recta, and the interstitium. Calcium is known to be reabsorbed passively in the thick ascending limbs and actively in the distal convoluted (DCT) and connecting (CNT) tubules. Our model predicts that the passive diffusion of Ca(2+) from the vasa recta and loops of Henle generates a significant axial Ca(2+) concentration gradient in the medullary interstitium. In the base case, the urinary Ca(2+) concentration and fractional excretion are predicted as 2.7 mM and 0.32%, respectively. Urinary Ca(2+) excretion is found to be strongly modulated by water and NaCl reabsorption along the nephron. Our simulations also suggest that Ca(2+) molar flow and concentration profiles differ significantly between superficial and deep nephrons, such that the latter deliver less Ca(2+) to the collecting duct. Finally, our results suggest that the DCT and CNT can act to counteract upstream variations in Ca(2+) transport but not always sufficiently to prevent hypercalciuria.

Applying ecological and evolutionary theory to cancer: a long and winding road.

Since the mid 1970s, cancer has been described as a process of Darwinian evolution, with somatic cellular selection and evolution being the fundamental processes leading to malignancy and its many manifestations (neoangiogenesis, evasion of the immune system, metastasis, and resistance to therapies). Historically, little attention has been placed on applications of evolutionary biology to understanding and controlling neoplastic progression and to prevent therapeutic failures. This is now beginning to change, and there is a growing international interest in the interface between cancer and evolutionary biology. The objective of this introduction is first to describe the basic ideas and concepts linking evolutionary biology to cancer. We then present four major fronts where the evolutionary perspective is most developed, namely laboratory and clinical models, mathematical models, databases, and techniques and assays. Finally, we discuss several of the most promising challenges and future prospects in this interdisciplinary research direction in the war against cancer.

Evolution of species trait through resource competition.

To understand the evolution of diverse species, theoretical studies using a Lotka-Volterra type direct competition model had shown that concentrated distributions of species in continuous trait space often occurs. However, a more mechanistic approach is preferred because the competitive interaction of species usually occurs not directly but through competition for resource. We consider a chemostat-type model where species consume resource that are constantly supplied. Continuous traits in both consumer species and resource are incorporated. Consumers utilize resource whose trait values are similar with their own. We show that, even when resource-supply has a continuous distribution in trait space, a positive continuous distribution of consumer trait is impossible. Self-organized generation of distinct species occurs. We also prove global convergence to the evolutionarily stable distribution.

Analysis of nonlinear noisy integrate & fire neuron models: blow-up and steady states.

Nonlinear Noisy Leaky Integrate and Fire (NNLIF) models for neurons networks can be written as Fokker-Planck-Kolmogorov equations on the probability density of neurons, the main parameters in the model being the connectivity of the network and the noise. We analyse several aspects of the NNLIF model: the number of steady states, a priori estimates, blow-up issues and convergence toward equilibrium in the linear case. In particular, for excitatory networks, blow-up always occurs for initial data concentrated close to the firing potential. These results show how critical is the balance between noise and excitatory/inhibitory interactions to the connectivity parameter.AMS Subject Classification: 35K60, 82C31, 92B20.

Mathematical description of bacterial traveling pulses.

The Keller-Segel system has been widely proposed as a model for bacterial waves driven by chemotactic processes. Current experiments on Escherichia coli have shown the precise structure of traveling pulses. We present here an alternative mathematical description of traveling pulses at the macroscopic scale. This modeling task is complemented with numerical simulations in accordance with the experimental observations. Our model is derived from an accurate kinetic description of the mesoscopic run-and-tumble process performed by bacteria. This can account for recent experimental observations with E. coli. Qualitative agreements include the asymmetry of the pulse and transition in the collective behaviour (clustered motion versus dispersion). In addition, we can capture quantitatively the traveling speed of the pulse as well as its characteristic length. This work opens several experimental and theoretical perspectives since coefficients at the macroscopic level are derived from considerations at the cellular scale. For instance, the particular response of a single cell to chemical cues turns out to have a strong effect on collective motion. Furthermore, the bottom-up scaling allows us to perform preliminary mathematical analysis and write efficient numerical schemes. This model is intended as a predictive tool for the investigation of bacterial collective motion.

Stability analysis of a simplified yet complete model for chronic myelogenous leukemia.

We analyze the asymptotic behavior of a partial differential equation (PDE) model for hematopoiesis. This PDE model is derived from the original agent-based model formulated by Roeder (Nat. Med. 12(10):1181-1184, 2006), and it describes the progression of blood cell development from the stem cell to the terminally differentiated state.To conduct our analysis, we start with the PDE model of Kim et al. (J. Theor. Biol. 246(1):33-69, 2007), which coincides very well with the simulation results obtained by Roeder et al. We simplify the PDE model to make it amenable to analysis and justify our approximations using numerical simulations. An analysis of the simplified PDE model proves to exhibit very similar properties to those of the original agent-based model, even if for slightly different parameters. Hence, the simplified model is of value in understanding the dynamics of hematopoiesis and of chronic myelogenous leukemia, and it presents the advantage of having fewer parameters, which makes comparison with both experimental data and alternative models much easier.

Survival thresholds and mortality rates in adaptive dynamics: conciliating deterministic and stochastic simulations.

Deterministic population models for adaptive dynamics are derived mathematically from individual-centred stochastic models in the limit of large populations. However, it is common that numerical simulations of both models fit poorly and give rather different behaviours in terms of evolution speeds and branching patterns. Stochastic simulations involve extinction phenomenon operating through demographic stochasticity, when the number of individual 'units' is small. Focusing on the class of integro-differential adaptive models, we include a similar notion in the deterministic formulations, a survival threshold, which allows phenotypical traits in the population to vanish when represented by few 'individuals'. Based on numerical simulations, we show that the survival threshold changes drastically the solution; (i) the evolution speed is much slower, (ii) the branching patterns are reduced continuously and (iii) these patterns are comparable to those obtained with stochastic simulations. The rescaled models can also be analysed theoretically. One can recover the concentration phenomena on well-separated Dirac masses through the constrained Hamilton-Jacobi equation in the limit of small mutations and large observation times.

Size distribution dependence of prion aggregates infectivity.

We consider a model for the polymerization (fragmentation) process involved in infectious prion self-replication and study both its dynamics and non-zero steady state. We address several issues. Firstly, we extend a previous study of the nucleated polymerization model [M.L. Greer, L. Pujo-Menjouet, G.F. Webb, A mathematical analysis of the dynamics of prion proliferation, J. Theoret. Biol. 242 (2006) 598; H. Engler, J. Pruss, G.F. Webb, Analysis of a model for the dynamics of prions II, J. Math. Anal. Appl. 324 (2006) 98] to take into account size dependent replicative properties of prion aggregates. This is achieved by a choice of coefficients in the model that are not constant. Secondly, we show stability results for this steady state for general coefficients where reduction to a system of differential equations is not possible. We use a duality method based on recent ideas developed for population models. These results confirm the potential influence of the amyloid precursor production rate in promoting amyloidogenic diseases. Finally, we investigate how the converting factor may depend upon the aggregate size. Besides the confirmation that size-independent parameters are unlikely to occur, the present study suggests that the PrPsc aggregate size repartition is amongst the most relevant experimental data in order to investigate this dependence. In terms of prion strain, our results indicate that the PrPsc aggregate repartition could be a constraint during the adaptation mechanism of the species barrier overcoming, that opens experimental perspectives for prion amyloid polymerization and prion strain investigation.

An age-and-cyclin-structured cell population model for healthy and tumoral tissues.

We present a nonlinear model of the dynamics of a cell population divided into proliferative and quiescent compartments. The proliferative phase represents the complete cell cycle (G (1)-S-G (2)-M) of a population committed to divide at its end. The model is structured by the time spent by a cell in the proliferative phase, and by the amount of Cyclin D/(CDK4 or 6) complexes. Cells can transit from one compartment to the other, following transition rules which differ according to the tissue state: healthy or tumoral. The asymptotic behaviour of solutions of the nonlinear model is analysed in two cases, exhibiting tissue homeostasis or tumour exponential growth. The model is simulated and its analytic predictions are confirmed numerically.