# Perspectives in Mathematical Biology

## Abstracts of the talks

**Gheorghe Craciun ***University of Wisconsin, USA***Mathematical methods for analyzing biological interaction networks**

Mathematical models of biological interaction networks give rise to a very large family of nonlinear dynamical systems. Some of the most common such models are based on mass-action kinetics and give rise to polynomial dynamical systems.

We will discuss how the analysis of these models over the last few decades has resulted in a wealth of new ideas and open problems. For example: (i) the study of uniqueness of equilibria in reaction network models led to general theorems on global injectivity of polynomial functions; (ii) the study of persistence properties of these models (i.e., finding conditions that imply that no species in an ecosystem goes extinct) led to the introduction of new mathematical tools, such as toric differential inclusions; and (iii) the construction of convex invariant regions for these models led to theorems on existence of solutions for nonlinear reaction-diffusions PDEs.

We will also mention some connections to the Jacobian Conjecture, and to classical results in thermodynamics, such as Boltzmann's H-theorem.

**Josef Hofbauer ***University Vienna, Austria***Permanence of ecological systems**

Given a mathematical model of an ecological/biological community, can one ﬁnd out whether the whole community will survive? Is there is a global attractor in which all species/types of the system coexist? If the state space is the nonnegative orthant ℝ_{+}^{n} this means that its boundary (together with ∞) is a repeller for the dynamics. I will describe some methods and results about this problem. Important tools are Lyapunov exponents and average Lyapunov functions.

**Matteo Smerlak ***Max Planck Institute for Mathematics in the Sciences, Germany***Evolutionary metastability**

From the molecular to the geological scale, evolving populations are often seen to change by leaps and bounds rather than “steadily, slowly, and continuously”, as Darwin imagined. Such “punctuated equilibria” can be traced back to the irregular structure of fitness landscapes: depending on the mutation rate, it can take a long time for a population to cross a fitness valley (genotypic stasis) or exit a large neutral network (phenotypic stasis). In this talk I will show that punctuated equilibria may be viewed as an instance of Markovian metastability—as fluctuation-activated jumps over potential barriers. Importantly, the relevant potential is not the fitness landscape itself, but rather a mollified version of it that also includes mutational robustness as an evolutionary optimand. Evolutionary potentials enable a coarse-grained description of evolutionary dynamics in complex fitness landscapes—a prerequisite to any meaningful notion of evolutionary prediction.

**Angela Stevens ***Universität Münster, Germany***Regeneration in planaria: boundary dynamics and signaling gradients**

We introduce and analyze a mathematical model for the regeneration of planarian flatworms. This system of differential equations incorporates dynamics of head and tail cells which express positional control genes that in turn translate into localized signals that guide stem cell differentiation. Orientation and positional information is encoded in the dynamics of a long range wnt-related signaling gradient. We motivate our model in relation to experimental data and demonstrate how it correctly reproduces cut and graft experiments. In particular, our system improves on previous models by preserving polarity in regeneration, over orders of magnitude in body size during cutting experiments and growth phases. Our model relies on tristability in cell density dynamics, between head, central body parts, and tail. In addition, key to polarity preservation in regeneration, our system includes sensitivity of cell differentiation to gradients of wnt-related signals measured relative to the tissue surface. This process is particularly relevant in a small tissue layer close to wounds during their healing, and modeled here in a robust fashion through dynamic boundary conditions. The mechanism proposed here resolves a conundrum in modeling efforts. In fact, many models of spontaneous formation of finite-size structure in unstructured tissue allude to a activator-inhibitor mechanism to select a finite wavelength (e.g. models for regeneration in hydra). Such Turing type mechanisms however do not scale across several orders of magnitude as seen in the patterning of planarians and hydra, nor do they incorporate robust selection of polarity. (Joint work with Arnd Scheel and Christoph Tenbrock)

**Reidun Twarock ***University of York, United Kingdom***Mathematical virology**

Viruses encapsulate their genetic material into protein containers that act akin to molecular Trojan horses, protecting viral genomes between rounds of infection and facilitating their release into the host cell environment. In the majority of viruses, including major human pathogens, these containers have icosahedral symmetry. Many open questions in virology can therefore be addressed through the lens of viral geometry, and novel mathematical techniques from group, graph, tiling and lattice theory, in partnership with biophysical modelling, bioinformatics, and stochastic simulations, can act as drivers of discovery in virology. Topics of interest include the combinatorics of virus assembly, the functional roles of symmetry breaking in viral life cycles, and mathematical techniques, e.g. from percolation or lattice transition theory, to model genome release. Viruses are also ideal model systems to study the laws of evolution in a laboratory environment, where mathematical insights from viral geometry enable a novel perspective on evolutionary fitness landscapes, and can also be used to construct alternative types of function-based phylogenies. The talk will finish with applications of the geometric and mechanistic insights in the context of multi-scale models of viral infections, and discuss potential applications in anti-viral therapy, vaccination and gene therapy.

## Date and Location

**November 05, 2019**

MPI für Mathematik in den Naturwissenschaften Leipzig

Inselstr. 22

04103 Leipzig

## Scientific Organizers

**Jürgen Jost**

MPI for Mathematics in the Sciences

Leipzig**Peter Stadler**

Leipzig University

Interdisciplinary Centre for Bioinformatics**Bernd Sturmfels**

MPI for Mathematics in the Sciences

Leipzig

## Administrative Contact

**Antje Vandenberg**

MPI for Mathematics in the Sciences

Leipzig

Contact by Email