This talk is about fronts and propagation phenomena for reaction-diffusion equations in non-homogeneous media. I will discuss some specific models arising in population dynamics or in medicine where the medium imposes a direction of propagation.
Cells migrate on surfaces by protruding their front through growth of actin networks, retracting the rear by myosin-driven contraction and adhering to the substrate. Recent experimental and modeling efforts elucidated specific molecular and mechanical processes that allow motile cells to maintain constant distances from front to rear and from side to side while maintaining steady locomotion.
Remarkably, these processes are multiple and redundant, and one of the future modeling challenges is a synthesis of these processes (operating on multiple scales) within a computational framework. Necessarily, such framework have to treat the cell as an object with a free boundary leading to a very nontrivial mathematical problem. I will describe initial successes in modeling the simplest motile cell, fish keratocyte, and discuss future challenges in simulating more complex cells.
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Stochastic generalizations of moving boundary problems appear quite naturally in the continuum description of e.g. solidification problems. Perhaps the simplest example is provided by a so-called one-sided solidification problem in which a condensed (solid) non-diffusing phase grows at the expense of a diluted diffusing phase (vapor or liquid). In this context, noise terms can be introduced to account for fluctuations in the interface kinetics leading to irreversible growth, and in the diffusive currents in the diluted phase. Thus, an effective closed evolution equation for the interface profile can be derived in a systematic way, carrying both deterministic and stochastic contributions, with parameters that can be related to those of the full moving boundary problem. This effective equation provides an interesting instance in which one can study the interplay between noise and non-local effects induced by diffusive interactions. Going beyond the approximations made in this process requires, e.g., formulation of a (stochastic) phase-field description that is equivalent to the original moving boundary problem. In turn, phase-field simulations allow to explore the rich morphological diagram that ensues. Applications will be discussed in the context both of non-living and biological systems subject to diffusion-limited growth, such as surfaces of thin films produced by Chemical Vapor Deposition or by Electrochemical deposition, or bacterial colonies. We will describe work done in collaboration with Mario Castro and Matteo Nicoli.
Blood coagulation is an extremely complex process which is the result of the action of platelets and of a large number of chemicals going through a chemical cascade. Its aim is the formation of a clot, sealing a wound The clot evolution leads to a free boundary problem. It goes in parallel with the process of clot dissolution (fibrinolysis), taking place with a slower time scale. Due to its complexity, the process may fail in various ways because of pathological conditions, leading to thrombosis or bleeding disorders of various types, that have also been the subject of mathematical models. The classical 3-pathway cascade model for blood coagulation, that was formulated in 1964, has been questioned after forty years. Though it is now ascertained to be wrong, its influence has been so strong that many new publications still refer to it. During the last few years a new model has been proposed in the medical literature (the so called cell-based model) and new mathematical papers have been written accordingly. Recently two opposite trends have been observed in mathematical models: on one side a tendency towards "completeness" with an incredible number of pde's describing the biochemistry in great detail (but sometimes ignoring platelets!); on the other side a tendency to focus just on the role of platelets. Those ways of approaching the problem have their own advantages and drawbacks. The "complete models" fail in any case to consider elements of great importance, that, very surprisingly, have been systematically ignored in the huge literature on the subject. The models considering just platelets can be used only for some very early stage of the process. A basic feature of any realistic coagulation model is the coupling between the biochemistry, the evolution of platelets population, and the flow of blood (in turn influenced by the growing clot). Thus blood rheology has a basic role. Blood rheology is known to be a very complicated subject and many different options have been offered. Nevertheless, the main point here is not which rheological model is preferable for blood, but the boundary conditions for blood flow. All models on blood coagulation use a no-slip condition. We prove that even a modest slip can have a dominant influence, depending on the geometry of the growing clot. We will also make a general discussion on the strategy to approach the problem (How many ingredients should be included? How to simplify the description of the chemistry? What targets can be considered realistic? etc.). New perspectives should also account for the most recent discoveries, suggesting that the cell-based model too may need some revision.
We study the regularity of almost minimizers for the types of functionals analyzed by Alt, Caffarelli and Freidman. Although almost minimizers do not satisfy an equation using appropriate comparison functions we prove several regularity results. For example in the one phase situation we show that almost minimizers are Lipschitz. Our approach reminiscent of the one used in geometric measure theory to study the regularity of almost minimizers for area. This project is joint work with Guy David.
There are a number of interesting and important biological processes that are best modelled as two-phase material mixtures. These include mucin exocytosis and transport, blood clot formation and biofilm formation. These all involve the interplay between flow, physical structure, mechanics and chemistry in a environment with complex dynamic geometry. The mathematical description of these processes requires equations describing multiphase flow, the evolution of composition and structure, and the relationship between stresses and composition/ structure (i.e., constitutive relations). Additionally, these equations of motion must properly account for interactions of the complex materials with dynamic physical boundaries, moving interfaces between materials with markedly different physical properties, and typically include strongly nonlinear constitutive relations or rate expressions.
In this talk, I will describe two features of mucus: the dynamics of mucus vesicle exocytosis and its transport of acid against an acid gradient.
The short story is as follows: Mucin is packaged into vesicles at very high concentration (volume fraction) and when the vesicle is released to the extracellular environment, the mucin expands in volume by two orders of magnitude in a matter of seconds. This rapid expansion is mediated by the rapid exchange of calcium and sodium that changes the crosslinking structure of the mucin polymers, thereby causing it to swell. I will describe a model of gel swelling and deswelling that accounts for these features, and is an interesting free boundary problem.
One of the important functions of the mucus lining of the stomach is to allow digestion of food to take place without the lining of the stomach being digested. An intriguing question is how acid can be released into the lumen of the stomach while maintaining a low concentration of hydrogen ions near the epithelial lining. A possible answer is that the flow of acid against its gradient is mediated by buffering by mucin. When mucin is secreted it rapidly binds hydrogen, but when it reaches the lumen where the pH is low, mucin is degraded by pH-activated pepsin, releasing its acid. The model associated with this process includes a free boundary problem to determine the thickness of the mucus layer and its acid-protective ability.