Cell interaction, cell motion and self-organization

Population biology and ecology

Goals

The intention of this workshop is to bring together two groups of scientists: mathematicians whose research is relevant for or inspired by models from biology, and biologists who are interested in the scope of mathematical models for their field. We hope for a lively exchange of ideas on a selected number of topics. Special attention will be paid to the relevance of stochastic and deterministic modeling and to the interplay between model and experiment in various biological fields.

The coalescent process of Kingman (1982) in evolutionary genetics describes the ancestry of a sample of genes back in time. In the simplest formulation the coalescence rate while k ancestors is , a rate of 1 for each unordered pair of edges in the tree. There is a connection with classical population genetics models and many earlier results can be rederived from a coalescent model, rather than from diffusion processes. This talk will describe recent research on general binary exchangeable coalescent trees and subtrees where times between coalescences have an arbitary distribution. Included are results about the age of mutations in gene trees, the distribution of mutations on trees and the characteristics of subtrees under a mutation.

We consider the evolution of populations under the joint action of mutation and selection. To this end, mutation and reproduction are modelled as a multi-type branching process, of which we consider both the forward and the backward direction of time. The stationary state of the reversed process is the ancestral distribution, which turns out as a key for the study of mutation-selection balance. For the single-step mutation model in the continuous limit, the equilibrium properties of both the present and the ancestral population are obtained from a maximum principle. In particular, exhaustive criteria are available for the presence and absence of the error-threshold phenomenon.

Empirical models of amino acid substitution involve the estimation of a matrix describing the relative rates of instantaneous change between the different amino acids from a limited set of data under the assumption that the recovered model is applicable to all future sets of observed data. Such models are useful for a variety of purposes, for example phylogenetic inference, detection of homologous sequences from protein databases and protein structure prediction. Current maximum likelihood approaches based on a complete evolutionary model of the observed data provide a relatively accurate estimate of the empirical model but are limited by the amount of sequence data that may be analysed and consequently by the generality of the resulting model. Procedures based on naive counts of the number of amino acid replacements between large numbers of pairs of related sequences suffer from errors inherent in the counting process and discard a large amount of information present in the sequences. We have recently introduced an approximation to the maximum likelihood method that can estimate an accurate model from large amounts of sequence data. This model has been been compared to other widely used models of amino acid substitution for phylogenetic inference and has been found to provide a better description of the evolutionary process in the majority of cases. We have also recently assessed the impact of modelling rate variation when estimating amino acid substitution models.

Tumour cords grow around blood vessels and therefore they are characterized by the fact that oxygen and nutrients are supplied through their inner surface, differently from the so-called spheroidal tumours that have instead an inner necrotic core. Here we consider a fully developed system of cords, schematized as a regular array of identical elements, each one having approximately a rotational symetry around its blood vessel. A mathematical model for the evolution of the cord is presented in which the following facts are taken into account: the influence of a limiting nutrient on the proliferation rate of viable cells, the presence of totally quiescent regions and of a necrotic region, the presence of a death rate possibly induced by chemical or radiative treatments, the volume reduction rate of the necrotic material due to fluid loss from the cord, the role of the interstitial fluid in the overall mass balance. We point out in which conditions it is not necessary to include in the model the study of the flow of interstitial fluids, without neglecting its important effect. Both the steady state and the evolution problem are considered, showing existence and uniqueness of a solution.

Under starvation conditions myxobacteria aggregate into large, flat colonies that coalesce into many fruiting bodies. During this aggregation phase the colony develops a cooperative phenomenon called 'rippling'. This appears as a pattern of waves on the surface of the colony that propagate continuously for long periods. The wave crests are 10-20 body lengths apart, and appear to pass through one another with no interference. The waves can persist with no net mass transport, analogous to water waves. Development of the ripple phase requires both the social (S) and asocial (A) motility systems, as well as the C-signalling system of intercellular communication. This signalling system operates by direct cell contact, without any diffusible chemotactic ligands. I will present a model for the ripple phenomenon based on the observation that individual cells appear to posses an internal cycle that manifests itself in individuals as periodic reversals in direction with no net progress in either direction. Understanding the mechanism of ripple formation reveals how intracellular dynamics, intercellular communication, and cell motility can coordinate to produce collective behavior. This pattern of waves is quite different from that observed in other social bacteria, especially Dictyostelium discoidum, that depend on diffusible morphogens.

The understanding of the principles underlying tumor growth is essential in order to optimize treatment strategies. To identify the dominant mechanisms during tumor growth mathematical and computer-based models can be very useful since they allow to test competing hypotheses in caricatures of well-defined biological experiments free from unknown or uncontrolled influences. Clearly it is neither possible nor desirable to model an experimental situation in minute detail since (i) in general many details of the biological system under study are not known and (ii) incorporating too many and most often irrelevant details usually completely hides the view on essential ingredients and mechanisms. Therefore a major question is to which degree details must be incorporated into a model and which quantities in model and experiment should be compared in order to obtain an appropriate description of a biological system. One class of candidates are quantities that depend on the very specific biological situation, e.g. the growth law and morphology of a particular type of tumor in a particular situation. Another class of candidates are robust system properties as generic growth regimes which do neither depend on every detail of the model nor on every detail of the biological system. We here present a (i) cellular automaton approach, (ii) a lattice-free, fluid-like single-cell based approach (in which the model parameters are directly linked to kinetic and biomechanical quantities), and (iii) a number of heuristic approaches, all capable to describe the growth law of avascular tumors quantitatively. We suggest to evaluate the competing models based on a classification by their generic growth properties on which in a further step predictions, the hallmark of a model, can be based on. We show that this line of argument favors the direct, mechanistic model approaches (i.e., the cellular automaton and the fluid-like-model) over heuristic approaches concerning the system properties. We argue that the fluid-like model is an appropriate choice in the case of a quantitative description and motivate our line of argument by applications of this model type on further biological multicellular systems.

In this talk I will present a statistical model for the analysis of tumor specific gene expression profiles. I will demonstrate the use of this method in the context of a gene expression profiling study of 49 human breast cancers. The study is aimed at defining molecular characteristics of tumors that reflect estrogen receptor status. In addition to good predictive performance with respect to pure classification of the expression profiles, the model also uncovers conflicts in the data with respect to the classification of some of the tumors, highlighting them as critical cases for which additional investigations are appropriate.

In this talk I will describe a class discovery method for microarray gene expression data. Based on a collection of gene expression profiles from different tissue samples, we search for binary class distinctions in the set of samples that show clear separation in the expression levels of specific subsets of genes. The analysis of data sets from cancer gene expression studies shows that the method can identify cancer types in an unsupervised fashion.

Population genetics applications are often interested in the
descendants and the ancestry of a sample of genes or individuals.
This leads to so-called descendant processes X, which characterize
the evolution of the considered population forward
in time, and ancestral processes Y describing the population
backward in time. The processes X and Y are in many cases
related in a certain sense, called duality, which turns out
to be a powerful tool to study the behaviour of the population.
This talk will give some examples of dual processes arising in
interacting particle systems and mathematical population genetics.
The presentation will focus on a class of haploid population models
with exchangeable reproduction law. The corresponding processes X
and Y are dual for fixed population size N and
for the diffusion limits as N tends to infinity.
The Wright-Fisher diffusion X and its dual
counterpart, the coalescent process Y (Kingman 1982),
appear if and only if
If this condition is relaxed then the corresponding
asymptotic coalescent process Y allows for simultaneous and
multiple mergers of ancestral lines. This reflects the presence of
individuals with large offspring sizes. The talk will finish with
an outlook on recent research for diploid models and models
with varying environment.

Linkage disequilbrium is a widely used measure of association between two polymorphic loci. Classical population genetics theory suggests that for two loci with recombinaton any amount of linkage disequilibrium will be eliminated after sufficient number of generations. This result is sometimes misinterpreted particularly when equilibrium is observed. I will discuss the properties of linkage disequilibrium between two segregating sites (or single nucleotide polymorphism [SNPs]) in a sample, utilizing coalescent theory.

Due to recent advances in statistical methodology and to the explosion of genomic data, hierarchical models of molecular evolution and population genetics are increasingly being utilized. We have applied hierarchical models to a variety of evolutionary topics and we discuss these applications here with an emphasis on the important role played by hyperparameters. One use of hierarchical models is to study the stochastic process of changing evolutionary rates over time. By doing so, we can explore the extent of change in evolutionary rate, associations between selection pressure and functional diversification, and correlated patterns of evolution among genes. By bridging techniques from phylogenetics and population genetics, hierarchical models can also be applied to the analysis of serially collected viral sequences. Although population parameters themselves are already hyperparameters, modeling the distribution of adaptation processes among viral hosts necessitates an additional hierarchical level. In addition, hierarchical models are valuable when using genomic data to address ecological topics. In contrast to paternity analysis in humans where the goal is to identify an individual, the pertinent issue in conservation biology is often to study population parameters such as the effective distance of pollen dispersal.

The distribution function of the optimal local gapped alignment
score of two
long unrelated DNA sequences of length n is widely believed to be
of the Gumbel type
. Whereas this is well known
for gapless alignment
(Dembo et al., 1994), a rigorous justification of this belief in the
gapped case has only been given in special cases (e.g. Siegmund/Yakir, 2000).
The two parameters K and of course depend on the scoring scheme,
but numerically useable representations are available in these cases.
We look at
in the large
deviations regime, that is where . We give a connection
between this large deviations rate and the growth constant of the
expectation of in the logarithmic regime (Arratia/Waterman,
1994). We also provide a new representation for this limit via a characteristic
equation for moment-generating functions of optimal global
alignment scores. This representation is similar to the one known
from gapless alignment.
Assuming the correctness of the Gumbel form, this gives a new representation
for the parameter .

I will give an overview of several mathematical results and applications of a general model for the evolution of a quantitative trait under selection, with mutations drawn from an arbitrary distribution. For stabilizing selection, the properties of the equilibrium distribution are derived, in particular, the problem of the maintenance of genetic variation will be discussed briefly. For a form of directional selection, convergence to an asymptotic distribution that proceeds at a constant rate is proved. Applications to the evolution of recombination will be outlined.

High score alignments of DNA sequences give evidence of a common ancestry or function. It is therefore natural to ask whether an observed high score could have arisen by pure chance, and to explore what the high score alignments look like under the null hypothesis of unrelated sequences. We introduce and investigate a simple Poisson model which reflects important features of high score gapped local alignments of independent sequences.

In this talk I will describe some systems of partial differential equations that in a suitable asymptotic limit describe the motion of "spora-like" solutions. The considered system of equations becomes in a suitable limit case the well studied Keller-Segel system of chemotaxis. Is is known that such system of equations exhibits singular solutions that develop singularities in a finite time. The problem addressed in this talk is that of the continuation of the solutions after singularity formation. Asymptotic analysis indicates that the corresponding limit solutions are made of a singular part and a regular part. The singular part consists on a set of "sporae" containing a positive mass of microorganism concentrated in a small region. Evolution equation for these sporae as well as their limit behaviour for long times will be also discussed.

This talk summarizes known results for a model by E. F. Keller and L. A. Segel describing the aggregation of the Dictyostelium discoideum myxamoebae caused by chemotactic attraction. A short description of the model and its mathematical aspects will be given and discussed.