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Phylogenetic comparative methods: applying modern probabilistic methods to evolutionary biology

Since Darwin biologists have recognized the need for a specialized mathematical apparatus to study inter-species measurements. However it was only Felsenstein [1985]’s independent contrasts method [although similar models were discussed earlier, e.g. Edwards, 1970] that proposed an appropriate framework for correctly analyzing trait data obtained from different taxonomic units. What makes such data different from the usual independent, identically distributed sample setting is that species level measurements are not independent. Due to a shared evolutionary history one can easily observe that more recently diverged species tend to be similar. Therefore using standard statistical methodologies, will make it impossible to distinguish between similarities due to shared environment or trait function (selection and adaptation) and phylogenetic inertia.

To remedy this situation Felsenstein [1985] assumed that a trait evolves as a Brownian motion on top of a phylogenetic tree. Along a branch we have usual Brownian motion evolution until a speciation point is reached. After speciation along the daughter branches two independent Brownian motions start running. Their initial values are the value of the process just before the speciation event. Very quickly this method (called independent contrasts in the biological community) became a field standard.

However Felsenstein [1988], Hansen [1997] pointed out that a Brownian motion model is not appropriate for traits under stabilizing selection – a pure Brownian motion has no stationary distribution. Therefore they proposed an Ornstein–Uhlenbeck model. Only now due to the increase of computational power is this process taking its place in an evolutionary biologist’s toolbox [Butler and King, 2004, Labra et al., 2009, Hansen and Orzack, 2005, Hansen et al., 2008]. This framework is currently undergoing rapid expansion allowing for measurement error [Felsenstein, 2008, Hansen and Bartoszek, 2012, Rohlfs et al., 2013], multiple interacting traits [Bartoszek et al., 2012] or different drift and diffusion parameters for different clades [Beaulieu et al.,2012].

Usually a phylogenetic comparative method will assume that the tree describing the relationships between species is fully resolved. This however need not always be the case. We do not know all the currently alive species, they are often reclassified or they are undergoing speciation. What may happen also in many insect orders is that molecular based tree inference methods can resolve the tree on the family level but there is not enough information to say much about the clades’ subtrees. Hence there is also a need for tree-free methods. These methods only assume a branching process model conditioned on the number of contemporary tips for the phylogenetic tree. Of course with an unknown tree we can infer much less about the process driving the phenotype’s evolution. However we still may infer a remarkable amount of information and also gain new insights into the species’ evolution.

Using this tree-free framework we may study the expected similarity of species and how quickly they loose it via the so-called interspecies correlation coefficient [Bartoszek, 2014, Bartoszek and Sagitov, 2015, Mulder and Crawford, 2015, Sagitov and Bartoszek, 2012]. We may also derive phylogenetic confidence intervals for the optimal trait values and effectively estimate the stationary variance of an adapting trait [Bartoszek and Sagitov, 2015] or diffusion parameter of a Brownian motion one [Bartoszek and Sagitov, 2014, Crawford and Suchard, 2013]. These methods also allow one to observe a phase transition in phylogenetic inertia. If the speciation process is fast enough then ancestral dependencies have an effect for long stretches [Adamczak and Miłoś, 2011, in press, An\’e et al., 2014, Bartoszek and Sagitov, 2015].

In the first lecture hour I will introduce the field of phylogenetic comparative methods, discuss the biological motivation and mathematical approach of using diffusion type stochastic differential equations to model inter-species data. I will discuss multivariate and measurement error extensions to these methods and illustrate why these are important. The second lecture hour will be more theoretical concentrating on tree-free methods. I will discuss their probabilistic background and what model parameters can be inferred using them.

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Automatic classification of global dynamics in multi-parameter systems

A dynamical system is a mathematical concept for describing an object varying in time, using a fixed rule that depends on the current state of the object (and not on its past). Dynamical systems can be used to describe a variety of phenomena, such as the growth of a population or spread of an infectious disease. In the first part of the talk, I am going to introduce a framework for automatic classification of global dynamics in a dynamical system depending on a few parameters (such as fertility rates or disease transmission rates). A set-oriented topological approach will be used, based on Conley’s idea of a Morse decomposition (see [3]), combined with rigorous numerics, graph algorithms, and computational algebraic topology. This approach allows to effectively compute outer estimates of all the recurrent dynamical structures encountered in the system (such as equilibria or periodic solutions), as perceived at a prescribed resolution. It thus provides an automatic computational method for concise and comprehensive classification of all the dynamical phenomena found across the given parameter ranges (see [1], [2]). The method is mathematically rigorous (a.k.a. computer-assisted proof), and has a potential for wide applicability thanks to the mild assumptions on the system.

In the second part of the talk, I am going to discuss a few interesting applications. First, I am going to show a population model which contradicts the commonly held belief that the initial state of a population doesn’t matter, because it will eventually stabilize at the equilibrium (see [1]). Next, I am going to show a model which shows that increasing the use of pesticides may sometimes actually increase the amount of pests (see [4]). Finally, I am going to show that incorporating spatial dispersal of individuals into a simple vaccination epidemic model may give rise to a model that exhibits rich dynamical behavior, not normally found in the simple epidemic models (see [5]).

Bibliography:

[1] Z. Arai, W. Kalies, H. Kokubu, K. Mischaikow, H. Oka, P. Pilarczyk, A database schema for the analysis of global dynamics of multiparameter systems, SIAM J. Appl. Dyn. Syst., Vol. 8, No. 3 (2009), 757-789.

[2] J. Bush, M. Gameiro, S. Harker, H. Kokubu, K. Mischaikow, I. Obayashi, P. Pilarczyk, Combinatorial-topological framework for the analysis of global dynamics, Chaos, Vol. 22, No. 4 (2012), 047508.

[3] C. Conley, Isolated invariant sets and the Morse index, CBMS Regional Conference Series in Math., no. 38, Amer. Math. Soc., Providence, RI, 1978.

[4] E. Liz, P. Pilarczyk, Global dynamics in a stage-structured discrete-time population model with harvesting, J. Theoret. Biol., Vol. 297 (2012), 148-165.

[5] D.H. Knipl, P. Pilarczyk, G. Rost, Rich bifurcation structure in a two-patch vaccination model, submitted.


The complex interplay between structure and function in the brain: from experiments to theory

a. Topology of visual cortex: some theoretical thoughts and novel data, towards a theory of topological optimization

b. The role of noise, disorder and heterogeneity in the dynamics of large cortical networks

In these lectures. I will give an overview of two instances in which mathematical analysis lead to a better understanding of cortical function. Both will revolve around collective dynamics in the mammalian brain. In this domain, an ever-increasing amount of data challenges our understanding of cortex and function. At the center of this problem is the inter-relationship between structure and function of cortical networks, on which we will try to shed new light in the course of our lectures.

a. I will start by presenting some works dealing with the structure and function of the visual cortex. In the early visual cortex of higher mammals, information is processed within functional maps whose layout is thought to underlie visual perception. Here, I will preserit a few theoretical thoughts, simulations and experimental data on the possible principles at the basis of their architecture. as well as their possible role in perception. I will present new data on spatial frequency preference representation in cat: we evidence the presence of a continuous map with singularities around which the map organizes as an electric dipole potential. Mathematically, I will show that both architectures are the most parsimonious topologies ensuring local exhaustive representation. Eventually, I will show using computer simulations how these topologies may improve coding capabilities.

b. Another question that is largely open and that may be addressed using mathematics is the role of noise and disorder in the large-scale dynamics of neuronal networks. In order to investigate these questions, I will introduce the main mathematical tools. from the domain of probability theory. that are used in the modeling of large-scale neuronal networks involved at functional scales in the brain. Limits of large networks with complex topologies will be derived, and I will show how levels of noise shape the collective response, in particular I will present a particularly interesting transition to global synchronization in the network as noise is increased.