Research
How does non-trivial collective behavior of a complex system emerge from the interactions among its components? This question motivated my research on vortices in superconductors, on stability of emulsions in liquid crystals, and, more recently, on evolutionary dynamics. Evolution is a particularly rich example of emergent complex behavior. By studying evolutionary dynamics, we can understand not only the origin and development of life, but also some general features of nonequilibrium processes. A quantitative understanding of evolution and ecology is also necessary to fight cancer and HIV, create a sustainable agriculture, or prevent the spread of antibiotic resistance. My ambition is to understand how various evolutionary and ecological forces prevent or enable adaptation. I am especially interested in connecting my work to the growing number of evolutionary experiments with microbes and the increasing amount of biomedical data. The former enables rigorous tests of the existing theories, while the latter is an exciting opportunity to both learn about evolution and contribute to the improvement of the human condition.
A few highlights from my work are described below.
Motivation
Spatial expansions are a recurrent theme in ecology and evolution. They describe the spread of invasive species, epidemics, and tumor growth among many other phenomena ranging from cellular to continental scales. Since expanding populations often need to adapt to the new environment, there is often a very strong coupling between ecological and evolutionary dynamics. The outcome of this coupling is seemingly unpredictable, which complicates the control of often undesirable growth. While invasion velocity rapidly increase in some species, other species accumulate deleterious mutations and slow down. One explanation for such striking differences is demographic fluctuations (genetic drift), which control the efficacy of natural selection. Deleterious mutations can stochastically reach fixation in populations with strong genetic drift, but only beneficial mutations fix when the genetic drift is weak.
Quantifying genetic drift is an old and well-understood problem in well mixed populations, but it is much more difficult in expanding populations. Indeed, demographic fluctuations are very strong at the leading edge of the expansion front, but these fluctuations are counteracted by constant reseeding from the population bulk. The mathematical formulation of these dynamics leads to nonlinear stochastic partial differential equations that have attracted sustained interest in mathematics and statistical physics because of their simple formulation and yet very rich and hard to understand dynamics. Furthermore, these equations describe many important phenomena from crystal growth to scattering problems in quantum chromodynamics.
Prior work focused mostly on the expansion velocity and established that all moving fronts can be classifies as either “pulled” or “pushed” depending on who wins the competition between the growth at the edge and reseeding from the bulk. The velocity of pulled expansions depends only on the growth rate at low population densities, i.e. the leading edge effectively pulls the expansions. In contrast, the velocity of pushed expansions exceeds the value that one would expect from the growth rate at the leading edge. Thus, the expansion is effectively pushed from the regions of higher growth rate behind the front. The analysis of fixation probabilities further showed that successful mutations occur only at the leading edge in pulled waves, but are spread throughout the expansion from in pushed waves. Given this, one naturally expect that genetic drift is higher in pulled than in pushed waves, but little was known beyond this expectation.
A illustrates that the velocity of pulled waves is given by the Fisher-Kolmogorov velocity, but the velocity is higher than this prediction for pushed waves. B and C illustrate the distribution of growth rate within the expansion front.
A new class of waves: neither pulled nor pushed
We systematically explored how genetic drift changes at the transition from pulled to pushed expansions. Surprisingly, we found not two, but three universality classes with very different pattern of fluctuations. For example, genetic diversity scales logarithmically, as a power law, or linearly with carrying capacity in each of the classes. Similar scalings apply to the effective diffusion constant of the front and other metrics of fluctuations. We computed all scaling exponents analytically and showed that they depend on a single parameter: the ratio of the expansion velocity to the geometric mean of dispersal and growth rates at the leading edge. More recently, we have also shown that the same theory holds for density-dependent dispersal. In such circumstances, expansions are pushed because the dispersal rates peak at higher population densities behind the leading edge.
Expansions create commonly observed, yet often unexplained genealogical structures.
We found that evolutionary outcomes could be very sensitive to the details of the dispersal and growth. A few percent change in the velocity ratio defined above could lead to orders of magnitude difference in genetic diversity. Thus, populations could have dramatically different potential to evolve drug resistance or adapt to other challenges. Furthermore, our work suggests that large variations in the genetic diversity of invasive species could be attributed to subtle density-dependence in their growth and dispersal rates, which in most cases have not been carefully characterized. We also predict that the genealogies of expanding populations will be markedly different between the universality classes. In particular, we expect that the classic Kingman coalescent describes only one of the universality classes while the other two have genealogies with multiple merges that are described by the Lambda coalescent. Numerical work to test this prediction is ongoing.
For more details see the following papers:
Fig. A: Spiraling motion of strain boundaries reveal chiral patterns of growth. Strains are identical except for the color. Segregation into sectors occurs due to genetic drift.
We found that chirality affects the competition between the strains by deforming the shape of the cellular aggregate. Deformations occur at the boundaries between the strains and come in two flavors. At half of the boundaries, the strains move towards each other, which results in the formation of bulge (Fig. D). Dips form at the other half of the boundaries, where the strains move away from each other. Bulges grow much faster than the dips, and, eventually, completely fill the edge of the aggregate. The relative abundance of the strains is then determined by the slopes of the bulged, which are controlled by the relative chirality of the strains.
The dynamics described above are fully captured by a simple analytical theory that includes two equations: one for the shape of the aggregate and one for the relative abundance of the strains. These equations are KPZ and Burger’s equations respectively each with an extra term to describe the two-way coupling between the shape and the composition of the growth front. This effective description follows both from the reaction-diffusion model and from phenomenological considerations. Therefore, it could apply to other out of equilibrium systems with chiral constituents. For example, the deposition of particles with different chirality and homophilic interactions should be analogous to the growth of cellular aggregates that we studied.
To understand whether chirality can affect fitness, we developed a minimal reaction-diffusion model of growth in dense cellular aggregates such as microbial colonies. This model recapitulated spiralling growth dynamics in colonies composed of cells with the same chirality (Fig. A) and made striking predictions about the competition between strains with different chirality. For strains with the same handedness, but different magnitude of chirality we observed competitive exclusion of the less chiral strain. In contrast, strains with the opposite handedness stably coexisted with the relative abundances of the strains determined by their relative chirality (Fig. B).
We also found that, under some conditions, the strains with opposite chirality can completely intermix with each. This mixing can greatly facilitate mutualistic interactions such as cross-feeding. One of our future goals is to understand the mixing transitions in more detail and to explore the role of chirality in social interactions between microbes.
For more details see the following paper:
Microtubules are relatively large and rigid biopolymers within the cell that perform numerous functions. For example, microtubules provide mechanical rigidity and act as highways for intracellular transport. Many diseases are caused by microtubule defects, and a lot of effort went into characterizing mechanical and structural properties of microtubule complexes. The assembly of such complexes, however, is a lot less understood even though microtubule complexes are very dynamic and need to reassemble every time the cell moves or divides. In collaboration with Tim Mitchison at Harvard Medical School, we investigated large microtubule complexes called asters. Asters facilitate cell division and are a very convenient experimental system because they take nearly half an hour to assemble and reach about a millimeter in size in early fish and frog embryos.
Traditionally the growth of asters was attributed to the growth of individual microtubules (Fig. A). Each microtubule switches between an elongating and a shrinking state, so depending on the velocities and switching rates a microtubule can either become infinitely long or vanish after a few cycles of growth and shrinkage. Thus, a simple model for aster formation is nucleation at centrosomes followed by elongation of individual microtubules. The resulting structure should become less dense away from the centrosomes, which contradicts experimental observations. Moreover, Mitchison lab found strong evidence of microtubule nucleation away from the centrosomes suggesting that other growth mechanisms are at play.