public 01:14:47

Benoit Perthame : Adaptive evolution and concentrations in parabolic PDEs

  -   CGTP Group Meeting Seminar ( 152 Views )

Living systems are characterized by variability; they are subject to constant evolution through the three processes of population growth, selection and mutations, a principle established by C. Darwin. In a very simple, general and idealized description, their environment can be considered as a nutrient shared by all the population. This allows certain individuals, characterized by a 'phenotypical trait', to expand faster because they are better adapted to use the environment. This leads to select the'fittest trait' in the population (singular point of the system). On the other hand, the new-born individuals undergo small variations of the trait under the effect of genetic mutations. In these circumstances, is it possible to describe the dynamical evolution of the current trait? An area of population biology that aims at describing mathematically these processes is born in the 1980's under the name of 'adaptive dynamics' and, compared to population genetics, considers usually asexual reproduction, a continuous phenotypical trait and population growth. We will give a self-contained mathematical model of such dynamics, based on parabolic equations, and show that an asymptotic method allows us to formalize precisely the concepts of monomorphic or polymorphic population. Then, we can describe the evolution of the 'fittest trait' and eventually to compute various forms of branching points which represent the cohabitation of two different populations. The concepts are based on the asymptotic analysis of the above mentioned parabolic equations once appropriately rescaled. This leads to concentrations of the solutions and the difficulty is to evaluate the weight and position of the moving Dirac masses that describe the population. We will show that a new type of Hamilton-Jacobi equation, with constraints, naturally describes this asymptotic. Some additional theoretical questions as uniqueness for the limiting H.-J. equation will also be addressed. Several other modeling methods have been proposed, stochastic individual based models, evolutionary game theory, dynamical systems. Connections will be presented too. This talk is based on collaborations with G. Barles, J. Carrillo, S. Cuadrado, O. Diekmann, M. Gauduchon, S. Genieys, P.-E. Jabin, S. Mirahimmi, S. Mischler and P. E. Souganidis.

public 01:14:43

... : No seminar

  -   CGTP Group Meeting Seminar ( 153 Views )

public 01:29:53

(open) : Mathematical Biology Colloquium

  -   CGTP Group Meeting Seminar ( 162 Views )

public 01:14:47

Scott McKinley : Characterizing Antibody-Mucin Interactions That Produce a Dynamic Molecular Shield Against Viral Invasion

  -   CGTP Group Meeting Seminar ( 139 Views )

Given the difficulty in finding a cure for HIV/AIDS, a promising prevention
strategy to reduce HIV transmission is to directly block infection at the
portal of entry. The recent Thai RV144 vaccine trial offered the first
evidence that a vaccine may provide location protection and block HIV
transmission in the vagina. Unfortunately, the underlying mechanisms for
protection remain unclear. In this talk, we examine theoretically a
hypothesis that builds on Sam Lai's recent laboratory observation that
virus-specific antibodies (Ab) may be capable of trapping individual
virions in genital mucus secretions. Ab are known to have a weak –
previously considered inconsequential – binding affinity with the mucin
fibers that constitute cervicovaginal mucus (CVM). However, several Ab
may be bound to a single virion at the same time, multiplying the Ab-mucin
binding effect, thereby creating an indirect virion-mucin affinity. Our
model takes into account biologically relevant length and time scales,
while incorporating known HIV-Ab affinity and the respective diffusivities
of viruses and Ab in semen and CVM. The model predicts that HIV-specific
Ab in CVM can effectively immobilize HIV in a shock-like front near the
semen-CVM interface, far from the vaginal epithelium. The robustness of
the result implies that even weak Ab-mucin affinity can markedly reduce the
flux of virions reaching target cells. Beyond this specific application,
the model developed here is adaptable to other pathogens, mucosal barriers,
geometries, kinetic and diffusional effects, providing a tool for hypothesis
testing and producing quantitative insights into dynamics of immune-
mediated protection.