Workshop on Dynamical Stochastic Modeling in Biology

Abstracts:

Bo Martin Bibby (The Royal Veterinary and Agricultural University, Copenhagen):

Modelling lipid oxidation

A new approach for evaluating lipid oxidation was developed by modelling data obtained by the oxygen consumption method. Based on the generalised scheme for lipid autoxidation, a compartment model involving the concentration of the four oxidation specimens of the unsaturated fatty acid, RH, R., ROO., and ROOH as well as the concentration of oxygen and the rate constants for initiation (a), formation of peroxyl radicals (b), and formation of alkyl radicals (c) was constructed. As all rates of reaction were considered to be of second order the dynamic part of the model could be described by five coupled differential equations expressing the overall reaction rate for both the lag phase and the propagation phase of lipid oxidation.

Dennis Bray (University of Cambridge):

Physical networks in cellular signalling

The interior of living cells is a strange environment - very different to anything usually considered by physical chemists. The solutions are not really aqueous and there is a great deal of organization and inhomogeneity in the chemical compositions. Macromolecules are densely packed together and their chemical reactions are strongly influenced by mechanical effects. Many processes are driven by numbers of molecules small enough that the thermal fluctuations in reaction rates become significant. In order to understand and make predictions about events in this strange domain we believe we must take quantitative data at many different scales, obtained by biological, chemical and physical techniques, and integrate them into large-scale computer models. I will illustrate this approach by the work of our group on the chemotaxis signalling pathway of the bacterium Escherichia coli, focusing especially on a cluster of receptors on the bacterial surface. We recently proposed an atomic level structure for this extended network of proteins and are currently using computational methods to examine the diffusive, biochemical, and conformational events occurring in the lattice and its subjacent cytoplasm.

For more information on our work please see our group web page http://www.zoo.cam.ac.uk/comp-cell.

Susanne Ditlevsen (University of Copenhagen):

A model of the uptake of alternative fatty acids by isolated rat liver based on stochastic differential equations

The uptake of dodecanedioic acid (C12) has been studied in the isolated perfused rat liver, and modeled with a deterministic two-compartment model. However, the experimental data show a larger variation around the fitted curve than can be accounted for by measurement errors alone. Such a model is an idealization and does not consider deficiencies of assumed ideal physical conditions, nor include the accumulated effect of neglected factors, which often occur in physiological descriptions as the systems can rarely be isolated from influences from the surroundings. A generalization of the deterministic model based on a stochastic extension is achieved by randomizing the elimination rate constant from the interstitial space. Estimation of parameters in the model is considered. It becomes more difficult because only incomplete observations are available (only one compartment is observed), and measurement errors have to be accounted for.

Bryan T. Grenfell (University of Cambridge):

Waves and sparks in the dynamics of infectious diseases

With their relatively simple natudal history and rich data sources, viral and bacterial childhood infections such as measles and pertussis offer an unusual opportunity for dissecting the interactions between noise and nonlinearity in spatio-temporal population dynamics. Here, we review recent work which reveals hierarchcial waves in measles dynamics and a markedly different dynamical response of measles and pertussis to the perturbation of vaccination. We also consider analogies with the dynamics of foot and mouth disease.

Michael Höhle (The Royal Veterinary and Agricultural University, Copenhagen):

Waves and sparks in the dynamics of infectious diseases

Understanding the spread of infectious disease is an important issue in order to prevent major outbreaks. In this talk mathematical modelling is used to gain insight into the dynamics of an epidemic. A process model, the Stochastic-Infected-Recovered model, exploiting knowledge about population dynamics serves as framework. Key interest is in adapting the stochastic model to observed data - especially from animal production. Observing all events of an epidemic is not feasible in practice; hence estimation of model parameters has to be done from missing data. The basic SIR model is extended in order to handle two common situations in animal production: interaction into the course of the epidemic and population heterogeneity due to the spatial layout of confinement. Handling partially observed epidemics in these contexts is done by extending an existing technique to handle estimation in partially observed epidemics using Markov Chain Monte Carlo. This paves the way to analyse treatment data from pig farms or disease transmission experiments performed under more controlled circumstances.

Marianne Huebner (Michigan State University):

Daphnia, parasites and lake bottom dynamics

The ecology of parasitism in the plankton differs from terrestrial systems, given the fluid, three-dimensional nature of lakes. Aquatic parasites rely on the physical mixing to remain suspended in the water column. Mixing processes mediate population dynamics and ecological interactions. Daphnia dentifera are common in shallow and deep lakes throughout the midwestern USA. Daphnia are infected after ingesting fungal asci. The infection is typically fatal. We found that the epidemics do not occur until Fall, although the Daphnia populations achieve their highest densities in midsummer. Our model describes this host-parasite interaction and includes the dynamics of spores in the water column.

Ketill Ingolfsson (Temple University, Philadelphia):

Singular perturbations in the Mendelean dynamical system

In this study is presented a systematic treatment of the influence of small random perturbations on a dynamical system evolving in compliance with the Weinberg Hardy laws. The subject is given in the vocabulary of statistical thermodynamics and is so on the border between dynamical systems and stochastic processes. The calculational results for Mendelian entropy in solutions of the necessary partial differential equations with small parameters for very long time durations have direct applications in heridirary models.

Valerie Isham (University College London):

The effect of spatial scale and spatial clumping in the infection process on the spread of macroparasites

For mathematical modellers, understanding the effects of spatial structure on the transmission dynamics of infectious diseases and making appropriate allowance for this structure in their models represents an important challenge. In this talk, I will describe some joint work with Stephen Cornell and Bryan Grenfell (Department of Zoology, University of Cambridge) in which the focus is on macroparasitic infections within a managed animal population, illustrated by gastrointestinal nematodes in a herd of sheep. In this context, spatial effects are caused by the spatial scale of the system (represented by the size of the host population) which has significant effects over and above host density, the spatial clumping of the infecting parasites, and the need for the parasite to mate within the host in order to reproduce. Such spatial structure will be shown to have important influences on the persistence/extinction of a parasite population and the enhanced invasion of treatment-resistant strains. These questions will be addressed through the use of a stochastic model representing the physical processes involved, as well as two simplified generic metapopulation models that seek to focus on particular aspects of the process, using a combination of analytic techniqes and simulation.

Mogens Høgh Jensen (University of Copenhagen):

Time-delay modelling of gene expressions

A number of genes change their expression pattern dynamically by displaying oscillations. In a few important cases these oscillations are sustained and can work as molecular clocks, as for instance the recently studied Hes1 protein system [1] and the regulation of transcription factors [2]. It is also observed that there is a delay between the transcription of the mRNA and the production of the corresponding protein of the order of 20 mins [1]. We therefore propose non-linear coupled rate equations for the productions of mRNA and protein by including a delay between the interaction of mRNA and protein [3]. The biological reason for this delay is related to transcription and translation times and to transport between cellular compartments. From analysis and simulations of the model we obtain excellent agreement with experimental data [3].

[1] H. Hirata et al. Science 298, 840 (2002).
[2] A. Hoffmann, A. Levchenko, M.L. Scott and D. Baltimore, Science 298, 1241 (2002).
[3] M.H. Jensen, K. Sneppen and G. Tiana, submitted to Science.

Hidde de Jong (INRIA, France):

Qualitative simulation of gene regulatory networks

The study of genetic regulatory networks has received a major impetus from the recent development of experimental techniques allowing the measurement of patterns of gene expression in a massively parallel way. This experimental progress calls for the development of appropriate mathematical methods and computer tools for the modeling and simulation of gene regulation processes. We present Genetic Network Analyzer (GNA), a computer tool for the qualitative modeling and simulation of genetic regulatory networks. The use of GNA is illustrated by a case study of the network of genes and interactions regulating the initiation of sporulation in Bacillus subtilis.

Anders Krogh (University of Copenhagen):

Hidden Markov models of proteins and DNA

At the primary level of analysis both proteins and DNA are one dimensional sequences of symbols from a finite alphabet. Many secondary properties, such as gene structure, have a grammatical structure, and therefore methods from language modelling can often be applied to biological sequences. A hidden Markov model (HMM) is a probabilistic model originally applied mostly in speech recognition research, but more recently it has proven very useful also for biological sequence analysis. In this talk I will give a couple of examples of such HMM applications.

Catherine Laredo (INRA/Paris 6-7):

Mechanistic models and field experiments for studying pollen dispersal in homogeneous and inhomogeneous environments

To make quantitative predictions about the pollen dispersal of a plant species under different environmental conditions, it is necessary to determine its individual dispersal function, i.e. the 2-dimensional density function describing the probability that a pollen grain emitted in (0,0) fertilizes an ovule in (x,y). This function depends on biological and climatic parameters. We present models for the individual dispersion function of corn, which has airborne pollen. These models are based on hitting times of diffusion processes, and integrate biological (difference of height between male and female flowers) and aerodynamical parameters (settling velocity, wind speed, air turbulence) parameters. The models presented differ according to (i) the importance of vegetation in stopping the paths of pollen grains, (ii) the presence or absence of discontinuities in the environment. The models are fitted on the data from several large field experiments of corn using the color of kernels as a phenotypic marker for pollen dispersal. The resulting estimations for the parameters of the models and comparisons between models indicate that (i) these models can provide good predictions of the observed data, (ii) there is a benefit in considering the difference of height between male and female flowers. Further more, values of the parameters estimated from dispersal data appear consistent with meteorogical and biological data acquired independently. This talk is based on a joint work between E.K. Klein, C. Lavigne, X. Foueillassar, P-H. Gouyon, A. Grimaud, and C.Laredo.

Gesine Reinert (University of Oxford):

Small world networks

Small world models are networks consisting of many local links and fewer long range `shortcuts'. They are used to model social interactions, metabolic networks, and epidemics, to name just a few examples. In this talk, we consider some particular instances for continuous as well as discrete models, and rigorously investigate the distribution of their inter-point network distances. Our results are framed in terms of approximations, whose accuracy increases with the size of the network. The limiting behaviour changes considerably dependent on the probabilities of shortcuts. This is joint work with Andrew Barbour, Zurich.

Michael S. Samoilov (University of California, Berkeley):

Stochastic Effects in Enzymatic Biomolecular Systems

Enzymatic reactions represent a ubiquitous class of biochemical mechanisms. Their dynamics within broader biomolecular networks provides the chemical basis for many types of cellular behaviors while subnetworks of enzymatic reactions often form recognizable control motif topologies making better understanding of these mechanisms an increasingly important subject. The characteristic feature of many such systems is a type of a mesoscopic property, whereby typically high concentrations of the reaction substrates are contrasted with frequently low concentrations of the enzymes driving these processes, which could be present in quantities as low as single digit molecular copy numbers. While this feature of enzymatic biomolecular systems has been extensively studied within the scope of classical deterministic chemistry, e.g. to obtain the various Michaelis-Menten type approximations to such systems' mass-action kinetics description, its stochastic properties are generally not as well understood. This is often the case in spite of the fact that the low molecular enzyme counts make stochastic treatment essential for accurate modeling of even the most basic among such systems. By looking at some of these mechanisms we show that, in addition to generally improving accuracy, stochastic analysis may in fact suggest fundamentally different resultant behavior patterns as compared to what the well-known deterministic treatment would predict. This in turn may have a significant impact on the overall functionality of the larger biomolecular systems these mechanisms are imbedded in.

Eugene van Someren (Technical University of Delft):

Multicriterion optimization for genetic network modeling

A major problem associated with the reverse engineering of genetic networks from micro-array data is how to reliably find genetic interactions when faced with a relatively small number of arrays compared to the number of genes. To cope with this dimensionality problem, it is imperative to employ additional (biological) knowledge about real genetic networks, such as limited connectivity, redundancy, stability and robustness, to sensibly constrain the modeling process. In previous work, we have shown that by applying single constraints, the inference of genetic interactions under realistic conditions can be significantly improved. This presentation covers the problem of how multiple constraints can be combined by formulating genetic network modeling as a multi-criterion optimization problem.

David Steinsaltz (University of California, Berkeley):

Stochastic models of aging and mortality

Two ancient facts about aging and mortality: they are inevitable, and they are random. Not only humans, most organisms show unwavering trends of senescence with increasing age, and yet most also show enormous variability and plasticity. These two facts have attracted the attention of the more mathematically-minded researchers to some simple Markov-process models of the aging process, hoping to explain the conspicuous regularities in the demographic data. This talk will present some of the salient demographic facts, and give an overview over the major stochastic models. Markov models do seem to offer insights into the widespread development of "mortality plateaus" - the levelling off of mortality rates after an initial rise - which did not immediately occur to those intimate with the data. The models have also led not a few researchers into embarrassing blunders, in their vain attempts to squeeze out the "Gompertz curve", the exponential rise of mortality rates with age.

Fengzhu Sun (University of Southern California):

Understanding the mutation mechanism during the polymerase chain reaction

We develop a mathematical model for point mutations and for the mutation process of microsatellites during polymerase chain reaction (PCR) using the theory of branching processes. We propose methods to estimate the point mutation rate during PCR. For microsatellites, we develop a method to estimate the mutation rate of microsatellites with j repeat units per PCR cycle and the probability of expansion by maximizing a quasi-likelihood of the observed data. We show by simulations that the proposed estimation method can accurately recover the relationship between the mutation rate and number of repeat units. The theoretical basis for the proposed method is also given. We apply the method to experimental data on poly-A and poly-CA repeats and discuss the biological implications of our findings.

Fengzhu Sun (University of Southern California):

Prediction of protein function using protein-protein interaction data

Assigning functions to novel proteins is one of the most important problems in the post-genomic era. Several approaches have been applied to this problem, including analyzing gene expression patterns, phylogenetic profiles, protein fusions and protein-protein interactions. We develop a novel approach that applies the theory of Markov random fields to infer a protein's functions using protein-protein interaction data and the functional annotations of its interaction protein partners. For each function of interest and a protein, we predict the probability that the protein has that function using Bayesian approaches. Unlike in other available approaches for protein annotation where a protein has or does not have a function of interest, we give a probability for having the function. This probability indicates how confident we are about the prediction. We apply our method to predict protein functions based on ``Biochemical function", ``Subcellular location", and ``Cellular role" for yeast proteins defined in the Yeast Proteome Database (YPD, http://www.incyte.com/sequence/proteome/databases/YPD.shtml), using the protein-protein interaction data from the Munich Information Center for Protein Sequences (MIPS, http://mips.gsf.de). We show that our approach outperforms other available methods for function prediction based on protein interaction data.