Evolutionary processes explain the development of populations based upon the mechanisms mutation and selection. Due to mutations and other sources of variability populations consist of distinct phenotypes that differ in one or several trait variables. Long-term survival of a phenotype over several generations depends on its efficiency in resource allocation in competition with other phenotypes. Thus, competition of phenotypes acts as a mechanism of selection. I will present a unified geometric approach for studying competition in mathematical models of ecological interactions and infectious diseases and demonstrate how this theory can be used for gaining a better understanding of evolutionary processes in these systems.
Biotechnology has empirically established that it is easier to construct and evaluate variant genes and proteins than to account for the emergence and function of wild-type macromolecules. Systematizing this constructive approach, synthetic biology now promises to infer and assemble entirely novel genomes, cells and ecosystems. It is argued here that the theoretical and computational tools needed for this endeavor are missing altogether. However, such tools may not be required for diversifying organisms at the basic level of their chemical constitution by adding, substituting or removing elements and molecular components through directed evolution under selection. Most importantly, chemical as well as informational diversification of life forms could be designed to block metabolic cross-feed and genetic cross-talk between synthetic and wild species and hence protect natural habitats and human health through novel types of containment.
First, we will give an introduction to fast-slow systems. The geometric viewpoint of the theory will be emphasized. Then we discuss the three-dimensional FitzHugh-Nagumo (FHN) equation and its bifurcations. The singular limit bifurcation diagram of the FHN equation will be derived. We shall also look at mixed-mode oscillations (MMOs) in the FHN equation and outline the role of MMOs in chemistry, neuroscience and physics in a more general context.
This talk is concerned with infinite median graphs. Many classes of infinite graphs are so rich that one is content to classify just the vertex transitive ones. In this talk examples are given, where regularity (isovalence) and two-endedness suffice, respectively regularity and linear growth.
It is thus shown that regular median graphs of linear growth are the Cartesian product of finite hypercubes by the two-way infinite path. Such graphs are Cayley graphs and have only two ends.
For cubic median graphs G the condition of linear growth can be replaced by the condition that G has two ends. For higher degree the relaxation to two-ended graphs is not possible, which we demonstrate by an example of a median graph of degree four that has two ends, but nonlinear growth.
In this paper, we introduce polynomial time algorithms that generate random $3$-noncrossing partitions and 2-regular, $3$-noncrossing partitions with uniform probability. A $3$-noncrossing partition does not contain any three mutually crossing arcs in its canonical representation and is $2$-regular if the latter does not contain arcs of the form $(i,i+1)$. Using a bijection of Chen et al. [PNAS, 2009, to appear], we interpret $3$-noncrossing partitions and $2$-regular, $3$-noncrossing partitions as restricted generalized vacillating tableaux. Furthermore, we interpret the tableaux as sampling paths of a Markov-processes over shapes and derive their transition probabilities.
Nonlinear dynamical systems are prevalent in systems biology, where they are often used to represent a biological system. In this talk we first focus on the problem of finding experimental setups that allow for full state observability and parameter identifiability of a nonlinear dynamical system. This is important as often observability and identifiability are assumed -- that is, that the values of system states and parameters can be deduced from output data (experimental observations) -- and might lead to extensive, repetitive experiments based only on intuition. We present several novel approaches and use new, state of the art computational tools to implement them. Additionally, we can optimise our experimental setup such that we require the observation of only a few outputs and can still observe all states and identify all parameters. Furthermore, if the observable output function is given then we provide a computational approach to obtain a minimal set of inputs to the system that will provide full state observability and parameter identifiability. Examples from biology are used to further motivate and illustrate our method. In the last part of the talk we show the direct interaction of theoretical analysis and experiment. We present the application of tools from systems and control engineering for designing biological experiments to elucidate signalling pathways in the chemotactic system of Rhodobacter sphaeroides.
