## Periodic Reporting for period 4 - MESOPROBIO (Mesoscopic models for propagation in biology)

Reporting period: 2020-03-01 to 2020-08-31

This project is aimed at unraveling propagation phenomena arising in the life sciences by means of multi-scale models. These models incorporate several biologically relevant variables at once (position of individuals, age, dispersal ability, life-history traits to name but a few). It is a great mathematical challenge to investigate these models without separating the various scales. This is of particular relevance when evolutionary processes are at play during the propagation phenomena. We are working on several case studies (concentration waves of bacteria, invasion of the cane toads in Northern Australia). The mathematical models we are studying belong to the wide class of kinetic equations. However, they depart significantly from the equations which are found usually in mathematical physics such as the Boltzmann equation for the kinetic theory of gases. Indeed, biological problems offer new research highlights to put mathematical efforts on.

We investigated collective motion of swimming bacteria Escherichia coli, a primer to the formation of biofilms. We focussed on kinetic models at the mesoscopic scale, that were previously calibrated on microbiology experiments. We exhibited special solutions for these models which propagate like waves (constant shape at constant speed). These waves are the result of communication between cells and the directed motion searching for food. In addition, we pushed the standard approximation of geometric optics to accommodate kinetic equations in the regime of short wavelength, and we developed numerical schemes with good accuracy in that regime. Using similar ideas, we investigated species' invasion with a rich phenotypic structure, and in particular the issue of dispersal evolution in the course of range expansion. It may happen after many generations that selection processes yield a faster propagation. It was reported previously that the invasion of cane toads is accelerating due to the natural selection of faster individuals at the edge of the front. We calculated the exact rate of acceleration in a minimal equation for this process. Finally, we investigated the influence of the mode of reproduction (asexual vs. sexual) on the adaptation of species to a gradual (slow) environmental change.

As a main conclusion, we developed mathematical tools and theories to address relevant problems in biology. These theories are rooted in mathematical physics (kinetic theory of gases, semi-classical analysis). The extension to biological models reveals new challenges and needs innovative solutions. These theories are quantitative in the sense that they provide explicit formulas to the biologists who are studying propagation phenomena and maladaptation of species. Hence, they are truly dedicated to interdisciplinarity.

We investigated collective motion of swimming bacteria Escherichia coli, a primer to the formation of biofilms. We focussed on kinetic models at the mesoscopic scale, that were previously calibrated on microbiology experiments. We exhibited special solutions for these models which propagate like waves (constant shape at constant speed). These waves are the result of communication between cells and the directed motion searching for food. In addition, we pushed the standard approximation of geometric optics to accommodate kinetic equations in the regime of short wavelength, and we developed numerical schemes with good accuracy in that regime. Using similar ideas, we investigated species' invasion with a rich phenotypic structure, and in particular the issue of dispersal evolution in the course of range expansion. It may happen after many generations that selection processes yield a faster propagation. It was reported previously that the invasion of cane toads is accelerating due to the natural selection of faster individuals at the edge of the front. We calculated the exact rate of acceleration in a minimal equation for this process. Finally, we investigated the influence of the mode of reproduction (asexual vs. sexual) on the adaptation of species to a gradual (slow) environmental change.

As a main conclusion, we developed mathematical tools and theories to address relevant problems in biology. These theories are rooted in mathematical physics (kinetic theory of gases, semi-classical analysis). The extension to biological models reveals new challenges and needs innovative solutions. These theories are quantitative in the sense that they provide explicit formulas to the biologists who are studying propagation phenomena and maladaptation of species. Hence, they are truly dedicated to interdisciplinarity.

"The construction of traveling wave solutions for concentration waves of chemotactic bacteria E. coli results from a delicate interplay between several scales: a macroscopic view where bacteria navigate in a heterogeneous environment, searching for food and for relatives, and a microscopic view about the details of individual locomotion. The mesoscopic scale is suitable to encompass both viewpoints. Hundred of thousands of bacteria are described as a cloud with statistical properties. Bacteria move collectively while modifying their chemical environment (communication signals, food). The main mathematical challenge consists in making the cloud of bacteria and the chemical environment moving at the same speed which is a good candidate for being the collective speed. This work concludes a long term research program initiated 10 years ago to decipher collective motion of bacteria.

This analysis initiated a long term study about front propagation in kinetic equations. The standard approximation of geometric optics for reaction-diffusion equations can be extended to kinetic models, leading to a new type of ""kinetic"" Hamilton-Jacobi equations in the limit of short wavelength. This enables to compute the exact rate of front propagation, and possibly front acceleration. Appropriate numerical schemes can be designed to capture the various regimes at once.

There is a unique case study of front acceleration in the field: the invasion of cane toads in Northern Australia (1930-today). Mathematical tools dedicated to kinetic equations can be adapted to cope with this biological problem. The exact rate of expansion in a minimal model can be calculated with good accuracy, revealing the subtle role of competition in the range expansion, on the contrary to standard results in this theory where competition can be safely neglected to calculte the rate of propagation in most cases.

Again, the standard approximation of geometric optics can be adapted to quantitative genetics models, when the adaptation of species to evolutionary forces can be described via a continuous trait. Interestingly, this provides formulas for measuring the maladaptation of a species to a gradual change of environment. This method is so robust that it can encompass several modes of reproduction, and various details about the species life-cycle (such as age-dependent selection).

New equations emerge from this project. Our results also contain solid mathematical grounds for some of them. Also, the methodogy performed here is of considerable interest for interdisciplinary purposes because it gives access to quantitative results easily transferable to the biologists. Hence, the outcomes and the hypotheses of the models can be thoroughly discussed.

The main results were presented in several seminars, conferences and research schools, including prestigious ones (Collège de France, ECM Berlin, AIMS plenary)."

This analysis initiated a long term study about front propagation in kinetic equations. The standard approximation of geometric optics for reaction-diffusion equations can be extended to kinetic models, leading to a new type of ""kinetic"" Hamilton-Jacobi equations in the limit of short wavelength. This enables to compute the exact rate of front propagation, and possibly front acceleration. Appropriate numerical schemes can be designed to capture the various regimes at once.

There is a unique case study of front acceleration in the field: the invasion of cane toads in Northern Australia (1930-today). Mathematical tools dedicated to kinetic equations can be adapted to cope with this biological problem. The exact rate of expansion in a minimal model can be calculated with good accuracy, revealing the subtle role of competition in the range expansion, on the contrary to standard results in this theory where competition can be safely neglected to calculte the rate of propagation in most cases.

Again, the standard approximation of geometric optics can be adapted to quantitative genetics models, when the adaptation of species to evolutionary forces can be described via a continuous trait. Interestingly, this provides formulas for measuring the maladaptation of a species to a gradual change of environment. This method is so robust that it can encompass several modes of reproduction, and various details about the species life-cycle (such as age-dependent selection).

New equations emerge from this project. Our results also contain solid mathematical grounds for some of them. Also, the methodogy performed here is of considerable interest for interdisciplinary purposes because it gives access to quantitative results easily transferable to the biologists. Hence, the outcomes and the hypotheses of the models can be thoroughly discussed.

The main results were presented in several seminars, conferences and research schools, including prestigious ones (Collège de France, ECM Berlin, AIMS plenary)."

"The main achievements beyond the state-of-the-art are:

1/ the construction of concentration waves of chemotactic bacteria in a coupled kinetic-parabolic model for a population of bacteria in a dynamic environment. The problem was mathematically very challenging, and the solution came after a long analytical work. This opens a wide range of problems in the mathematical community. Furthermore, some counter-intuitive phenomena were discovered thanks to a close collaboration with experts in numerical analysis;

2/ the approximation of geometric optics for a kinetic equations, revealing a new ""kinetic"" non-local Hamilton-Jacobi equation with original viscosity solutions.

3/ the approximation of geometric optics for genetical segregation processes via the Fisher infinitesimal model. This opens a new theory which make possible a comparison analysis between several modes of reproduction in quantitative genetics.

Some unexpected results were also obtained:

1/ the exact rate of acceleration for a minimal model of dispersal evolution in the course of range expansion. The initial guess, published before the beginning of the project was not correct. Much efforts, building on several progresses made during the project, resulted in the calculation of the exact rate. On the contrary to standard reaction-diffusion equations, competition plays a role at the edge of the front, in a very subtle way;

2/ the uniqueness of solutions to some constrained Hamilton-Jacobi equations arising in quantitative genetics in the regime of small phenotypic variance. This closes the well-posedness issue for this class of equations."

1/ the construction of concentration waves of chemotactic bacteria in a coupled kinetic-parabolic model for a population of bacteria in a dynamic environment. The problem was mathematically very challenging, and the solution came after a long analytical work. This opens a wide range of problems in the mathematical community. Furthermore, some counter-intuitive phenomena were discovered thanks to a close collaboration with experts in numerical analysis;

2/ the approximation of geometric optics for a kinetic equations, revealing a new ""kinetic"" non-local Hamilton-Jacobi equation with original viscosity solutions.

3/ the approximation of geometric optics for genetical segregation processes via the Fisher infinitesimal model. This opens a new theory which make possible a comparison analysis between several modes of reproduction in quantitative genetics.

Some unexpected results were also obtained:

1/ the exact rate of acceleration for a minimal model of dispersal evolution in the course of range expansion. The initial guess, published before the beginning of the project was not correct. Much efforts, building on several progresses made during the project, resulted in the calculation of the exact rate. On the contrary to standard reaction-diffusion equations, competition plays a role at the edge of the front, in a very subtle way;

2/ the uniqueness of solutions to some constrained Hamilton-Jacobi equations arising in quantitative genetics in the regime of small phenotypic variance. This closes the well-posedness issue for this class of equations."