- University of Gothenburg
- The Faculty of Science
- Theoretical biology
- The 2nd Swedish meeting on Mathematics in Biology
- Abstracts

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# Abstracts

**"Emergence and self-organization in complex systems" (mini-course)**

P. Degond, Institut de Mathematiques, CNRS & Universite Paul Sabatier, Toulouse, France

Complex systems are characterized by the spontaneous formation of spatio-temporal structures as a result of simple local interactions between agents without leaders. In this series of talks, we will review some questions related to the establishment of macroscopic models from the underlying microscopic dynamics. Indeed, usual methods of kinetic theory fail, due to the lack of conservation properties and the build-up of statistical correlations and the design of new methodologies is necessary to bypass these obstacles. We will also investigate the self-organization capabilities of complex systems. In particular the non-overlapping constraint, which prevents particles of finite size to overlap, is a powerful morphogenetic force. We will discuss both theoretical and numerical for this question. The plan of the three one hours lectures is as follows:

1. Overview ; Macroscopic models for self-propelled particles

2. Self-organization vs chaos assumption

3. Morphogenesis driven by non-overlapping constraint

**"Evolutionary Game Theory - An Introduction" (mini-course)**

Arne Traulsen, Max Planck Institute for Evolutionary Biology

The dynamics of evolving systems can be described by evolutionary game theory. In the simplest case, the dynamics is given by a set of (typically nonlinear) differential equations that describe how the abundance of different types change over time. More recently, research has focused on stochastic dynamics, but also on spatially extended populations on lattices and networks. Evolutionary game theory can be applied to a variety of fields, including social dynamics and also population genetics. As the most popular application, I will discuss the evolution of cooperation: How can a behavior evolve that increases the fitness of others at an expense to one self? In addition, it will be shown how the children’s game of rock-paper-scissors can be applied in social dynamics, in ecology as well as in a genetic context.

Suggested reading:

A short review article:

Nowak & Sigmund, Evolutionary Dynamics of Biological Games, Science 303, 793-799 (2004)

A more technical introduction on stochastic game dynamics:

Traulsen & Hauert, Stochastic evolutionary game dynamics, in Reviews of nonlinear dynamics and complexity, Wiley-VCH, p.29 (2009) - available at http://arxiv.org/pdf/0811.3538

A entry-level textbook:

Nowak, Evolutionary Dynamics - Exploring the Equations of Life (Harvard University Press, 2006)

A more mathematical book:

Hofbauer & Sigmund, Evolutionary Games and Population Dynamics (Cambridge University Press, 1998)

**"Inferring speciation and extinction rates under different species sampling schemes" **

Tom Britton, Department of Mathematics, Stockholm University

The birth-death process is widely used in phylogenetics to model speciation and extinction. Recent studies have shown that the inferred rates are sensitive to assumptions about the sampling probability of lineages. Here, we examine the effect of the method used to sample lineages. Whereas previous studies have assumed random sampling, we consider two extreme cases of biased sampling: ``biologist sampling'', where tips are selected to maximize diversity, and ``cluster sampling'', where sample diversity is minimized. Using both simulations and analyses of empirical data, we show that inferred rates may be heavily biased if the sampling strategy is not modeled correctly. In particular, when a biologist sample is treated as if it were a random or complete sample, the extinction rate is severely underestimated, often close to 0

**"Rejuvenation in yeast - How something old can generate something young?"**

Marija Cvijovic, Chalmers

**"Single molecule simulations in complex geometries and mixed dimensions"**

Stefan Hellander, Uppsala University

It is well established and generally accepted that the discrete nature of chemical reactions requires stochastic modeling when low numbers of molecules are involved in biological cells. Sometimes it is even necessary to follow the motion and reactions of single molecules. For example, no accurate mesoscopic description is known for crowding or we may have molecules associating to and diffusing on membranes or polymers.

We have developed an algorithm for simulation of the motion and reactions of single molecules at a microscopic level. The molecules diffuse in a solvent and react with each other, a polymer or a membrane.

The algorithm is similar to the Green’s function reaction dynamics (GFRD) algorithm by van Zon and ten Wolde where longer time steps than in regular Brownian dynamics simulations can be taken by computing the probability density function (PDF) for a pair of molecules. However, the analytic expression for the full PDF is rather complicated and therefore we propose an operator splitting in order to obtain two simpler equations.

By approximating complicated surfaces locally by planes, we simulate molecules binding to, diffusing and reacting on membranes in the cell using the Smoluchowski equation in 1D and 2D. Using operator splitting also for the PDF in 2D we can let the molecules react on the membranes in the same manner as in 3D allowing for much larger time steps than would otherwise be possible.