BIOMAT 2007

The following sections are included:

• PREFACE

• EDITORIAL BOARD OF THE BIOMAT CONSORTIUM

• CONTENTS

We model protein folding as a physical stochastic process as follows. The unfolded protein chain is treated as a random coil described by SAW (self-avoiding walk). Folding is induced by hydrophobic forces and other interactions, such as hydrogen bonding, which can be taken into account by imposing conditions on SAW. The resulting model is termed CSAW (conditioned self-avoiding walk. Conceptually, the mathematical basis is a generalized Langevin equation. In practice, the model is implemented on a computer by combining SAW and Monte Carlo. To illustrate the flexibility and capabilities of the model, we consider a number of examples, including folding pathways, elastic properties, helix formation, and collective modes.

The analysis of large-scale data sets via clustering techniques is utilized in a number of applications. Many of the methods developed employ local search or heuristic strategies for identifying the "best" arrangement of features according to some metric. In this article, we present rigorous clustering methods based on the optimal re-ordering of data matrices. Distinct mixed-integer linear programming (MILP) models are utilized for the clustering of (a) dense data matrices, such as gene expression data, and (b) sparse data matrices, which are commonly encountered in the field of drug discovery. Both methods can be used in an iterative framework to bicluster data and assist in the synthesis of drug compounds, respectively. We demonstrate the capability of the proposed optimal re-ordering methods on several data sets from both systems biology and molecular discovery studies and compare our results to other clustering techniques when applicable.

A well-known problem in protein modeling is the determination of the structure of a protein with a given set of inter-atomic or inter-residue distances obtained from either physical experiments or theoretical estimates. A general form of the problem is known as the distance geometry problem in mathematics, the graph embedding problem in computer science, and the multidimensional scaling problem in statistics. The problem has applications in many other scientific and engineering fields as well such as sensor network localization, image recognition, and protein classification. We describe the formulations and complexities of the problem in its various forms, and introduce a geometric buildup approach to the problem. Central to this approach is the idea that the coordinates of the atoms in a protein can be determined one atom at a time, with the distances from the determined atoms to the undetermined ones. It can determine a structure more efficiently than other conventional approaches, yet without requiring more distance constraints than necessary. We present the general algorithm and its theory and review the recent development of the algorithm for controlling the propagation of the numerical errors in the buildup process, for determining rigid vs. unique structures, and for handling problems with inexact distances (distances with errors). We show the results from applying the algorithm to some of the model problems and justify the potential use of the algorithm in protein modeling.

Understanding the three-dimensional structure of proteins is critical to understand their function. While great progress is being made in understanding the structures of soluble proteins, large classes of proteins such as membrane proteins, large macromolecular assemblies, and partially organized or heterogeneous structures are being comparatively neglected. Part of the difficulty is that the coordinate models we use to represent protein structure are discrete and static, whereas the molecules themselves are flexible and dynamic. In this article, we review methods to develop a continuous description of proteins more general than the traditional coordinate models and which can describe smooth changes in form. This description can be shown to be strictly equivalent to the traditional atomic coordinate description. We apply the method to three major areas. The first is structure determination. The second is structural modeling, where the capability of smoothly deforming structures allows the exploration of possible forms to create models for classes of proteins. The third is to use coarse-grained mechanical models of proteins to predict the structures, motions, and conformational rearrangements in large oligomeric protein complexes.

Using optical coherence imaging, it is possible to visualize seizure progression intraoperatively. However, it is difficult to pinpoint an exact epileptic focus. This is crucial in attempts to minimize the amount of resection necessary during surgical therapeutic interventions for epilepsy and is typically done approximately from visual inspection of optical coherence imaging stills. In this paper, we create an algorithm with the potential to pinpoint the source of a seizure from an optical coherence imaging still. To accomplish this, a grid is overlaid on optical coherence imaging stills. This then serves as a grid for a two-dimensional cellular automation. Each cell is associated with a Riemannian curvature tensor representing the curvature of the brain's surface in all directions for a cell. Cells which overlay portions of the image which show neurons that are firing are considered "depolarized". The cellular automation is then run with the following rules:

(1) At each step all squares in contact with a depolarized square become depolarized if | ∇_uv| * t ≤ c. ∇_u is the covariant derivative in the direction of vector u. The vector u is a unit vector in the direction the depolarization from the original depolarized square. v is a tangent vector to the brain manifold at a touching square. t is the total time in terms of time steps that has been spent at a square. c is a constant given by the speed of the propagation of the wavefront.

(2) If a square depolarizes its neighboring squares, it becomes repolarized.

(3) A repolarized square cannot be depolarized again for t_r time steps, given by a the neuronal refractory period.

While the simulation is running, the depolarizing "wavefront" of cells converges on to a few specific cells which we hypothesis correspond to the epileptic focus. Simulations on several parts of the brain are run, comparisons are made to actual optical coherence imaging visualizations, and a tool is proposed for use intraoperatively during therapeutic epilepsy surgery.

As with all mathematical modeling, the scope of the question to be explored determines the scope of the most appropriate model. The case is no different for the modeling of primary brain tumors (gliomas), ranging from too simple, not accounting for the major feature of gliomas (extensive invasion), to too complicated, with too many variables and no easy way to translate from culture media in vitro to brain tissue in vivo. We settle on a "just right" approach which utilizes currently available magnetic resonance imaging (MRI) to estimate two defining characteristics, net rates of proliferation (ρ) and diffusion (D). Most importantly, these parameters are predictive of clinical behavior, and can be tailored to individual patients in vivo and in real time. These two rates combine to generate a linear radial growth pattern of the MRI visible portion of each glioma. Further, we introduce a novel method for the calculation of glioma invasion through grey and white matter.

This article analyzes a mathematical model for some aeroelastic oscillators in physiology, based on a previous representation for the vocal folds at phonation. The model characterizes the oscillation as superficial wave propagating through the tissues in the direction of the flow, and consists of a functional differential equation with advanced and delay arguments. The analysis shows that the oscillation occurs at a Hopf bifurcation, at which the energy absorbed from the flow overcomes the energy dissipated in the tissues. The bifurcation value of the flow pressure increases linearly with the tissue damping and the oscillation frequency. Also, it is minimum when the phase delay of the superficial wave to travel along the tissues is π, and increases indefinitely when the delay tends to 0 and to 2π.

Conduction of action potentials throughout the complex morphology of neurons may be modulated in an activity-dependent manner. Among modulatory mechanisms, afterhyperpolarization (AHP) plays an important role. To investigate how the AHP modulatory capabilities on transmission were dependent on the axonal geometry as well as on membrane properties such as channel kinetics, channel density distribution and membrane noise, multi-compartment computational neural models were built, using the neurosimulator SNNAP. Two kinetic schema for the sodium and potassium channels were compared. The simulations suggest that channel kinetics profoundly inuence the AHP-dependent modulation of action potential conduction through points of impedance mismatch in the highly branched neurites of neurons.

Allee effects are broadly defined as a decline in individual fitness at low population sizes or densities. Although the roots of the concept go back at least to 1920's, until recently, Allee effects eked out on the periphery of ecological theory, in the shade of the prominently discussed negative density dependence. The situation has changed dramatically in the last ten years or so, and we can find an ever increasing number of studies considering Allee effects from an ever increasing range of disciplines. Mathematical models have always been an important tool by which to assess impacts of Allee effects for population and community dynamics. Actually, much of what we know about Allee effects comes from mathematical models. Up to now, Allee effects have been examined in the context of most existing model structures, and significantly altered our picture of population and community dynamics based on assuming negative density dependence only. This essay concerns modelling Allee effects and presenting their major implications for population and community dynamics. It shows what types of model are available, how they can be used, and, most importantly, how they have contributed to our understanding of the dynamical consequences of Allee effects. This essay is a modified extract from the book "Allee effects in ecology and evolution", co-authored by Franck Courchamp, Luděk Berec and Joanna Gascoigne, which is going to be published by the Oxford University Press early in 2008.

Central to the dynamics of population biology are various versions of the Lotka-Volterra equations. Particular cases may be used to model competitive, commensal, predatory and other behaviour. Similar equations describe macro-economic interactions, epidemics and other processes of mass action. Refinements of many versions of these equations have been exhibited in the BIOMAT meetings to describe new biological features. The solutions to such equations may display a variety of forms. Broadly speaking, quite a lot of qualitative information may often be obtained about the solutions. In many cases it is convenient to eliminate time from the equations, so obtaining equations for the joint values of population sizes. By contrast, determining explicit expressions for the evolution with time of the component population sizes frequently appears to be infeasible, although numerical procedures may be available. This is less than fully satisfactory, as numerical work with many joint choices of the driving parameters, involving a number of dimensions, may be needed to obtain any sort of collective overview of the population dynamics. Perhaps surprisingly, in a number of general situations it is in fact possible to derive explicit solutions for the population sizes as functions of time. We may indeed decouple the basic equations and perform explicit quadratures. Several techniques are available for our purposes, such as Euler substitution and separation and the methods of Gambier and Painlevé. The working is sometimes heavy but often manageable. We illustrate these techniques by considering a number of well-known classical biological models from the literature.

In this work we analyze a predator-prey model proposed by Kent et. al. in¹⁶, in which two aspect of the model are considered: an effect of emigration or immigration on prey population to constant rate and a prey threshold level for predators. We prove that the system when the immigration effect is introduced in the model has a dynamics that is similar to the Rosenzweig-MacArthur model. Also, when emigration is considered in the model, we show that the behavior of the system is strongly dependent on this phenomenon, this due to the fact that trajectories are highly sensitive to the initial conditions, in similar way as when Allee effect is assumed on prey. Furthermore, we determine constraints in the parameters space for which two stable attractor exist, indicating that the extinction of both population is possible in addition with the coexistence of oscillating of populations size in a unique stable limit cycle. We also show that the consideration of a threshold level of prey population for the predator is not essential in the dynamics of the model.

The spread of epidemics is inevitably entangled with human behavior, social contacts, and population flows among different geographical regions. The collection and analysis of datasets which trace the activities and interactions of individuals, social patterns, transportation infrastructures and travel fluxes, have unveiled the presence of connectivity patterns characterized by complex features encoded in large-scale heterogeneities and unbounded statistical fluctuations. These features dramatically affect the behavior of dynamical processes occurring on networks, and are responsible for the observed statistical properties of the processes' dynamics and evolution patterns. In the context of large-scale propagation of emerging infectious diseases, the air transportation network is known to play a major role in shrinking distances around the globe, by connecting far apart regions and allowing infectious travelers to potentially spread the disease to different geographic areas in a relatively short time. Here we will present a large-scale stochastic computational approach for the study of the global spread of emergent infectious diseases which explicitly incorporates real world transportation networks and census data. The simulated spatio-temporal pattern of epidemic propagation is analyzed in relation to the heterogeneous properties of the underlying complex architecture. Specific quantitative indicators are introduced to evaluate the predictive capability of the computational approach with respect to the intrinsic stochasticity of the disease transmission and of human interactions and movements. The interplay of the complex properties of the transportation infrastructure with the disease dynamics leads to the emergence of epidemic pathways as the most probable routes of propagation of the disease, selected out of the huge number of possible paths the disease could take by following airline connections. A case study for risk assessment analysis and comparison with historical epidemics is analyzed.

In this work we study a spatial model for the West Nile Virus (WNV) propagation across the USA from the east to the west. WNV is an arthropod-borne flavivirus that appeared at first time in New York city in the summer of 1999 and then spread prolifically within birds. Mammals, as human and horse, do not develop sufficiently high bloodstream titers to play a significant role in transmission, which is the reason to consider the mosquito-bird cycle. The proposed model aims to study this propagation in a system of partial differential reaction-diffusion equations considering the mosquito and the avian populations. The diffusion is allowed to both populations, being greater in avian than in the mosquito. When a threshold value R₀, depending on the model's parameters, is greater than one, the disease remains endemic and could propagate to regions previously free of disease. The travelling wave solutions of the model are studied to determine the speed of the disease propagation. This wave speed is obtained as a function of the model's parameters, for instance, vertical transmission rate and avian diffusion coefficient.

The aim of this paper is to study the cropping system as complex one, applying methods from theory of dynamic systems and from the control theory to the mathematical modeling of the biological pest control. The complex system can be described by different mathematical models. Based on three models of the pest control, the various scenarios have been simulated in order to obtain the pest control strategy only through natural enemies' introduction.

We examine the appearance of Turing instabilities of spatially homogeneous periodic solutions in reaction-diffusion equations when such periodic solutions are consequence of Hopf bifurcations. First, we asymptotically develop limit cycle solutions associated to the appearance of Hopf bifurcations in reaction systems. Particularly, we will show conditions to the appearance of multiple limit cycles after Hopf bifurcation. Then, we propose expansions to normal modes associated with Turing instabilities from spatially homogeneous periodic solutions associated to limit cycles which appear as a consequence of a Hopf bifurcation. Finally, we discuss examples of reaction-diffusion systems arising in biology and chemistry in which can be observed spatial and time-periodic patterning.

Satellite images of the grassland area in Durango México were obtained of altitude, slope, average annual temperature, annual precipitation, type of vegetation, type of soil, normal vegetation index, percentage of herbaceous and percentage of bares soil, in order to relate them with grasshopper density population (GDP) surveyed in 35 sampling sites from June to November in 2003 in the study area. A stepwise regression analysis was performed with the most abundant grasshopper species Phoetaliotes nebrascensis (Thomas), Melanoplus lakinus (Scudder) and Boope-don nubilum (Say) with data extracted from the satellite images. Results showed R > 0.798, F(4, 27) > 9.86 and P > 0.000016. The significant variables were normal vegetation index, type of vegetation, altitude and precipitation. GDP raster maps were interpolated using the stepwise regression equations. Then, classification neural networks models were used in order to classify GDP maps. Analysis of percentage of classification error showed that the adequate number of hidden neurons was between six and twelve. Results of error classification were 21% for P. nebrascensis, 5% for M. lakinus and 20% for B. nubilum. Neural networks are practical tools to classify grasshopper population and it will help to take control measurements in overpopulated areas.

High-throughput experiments have produced convicing evidence for an extensive contribution of diverse classes of RNAs in the expression of genetic information. Instead of a simple arrangement of mostly protein-coding genes, the human transcriptome features a complex arrangement of overlapping transcripts, many of which do not code for proteins at all, while others "sample" exons from several different "genes". The complexity of the transcriptome and the prevalence of non-coding transcripts forces us to reconsider both the concept of the "gene" itself and our understanding of the mechanisms that regulate "gene expression".

The present literature has several contributions to the understanding of biomolecular structure by adopting helix curves through the atom sites as a paradigm. In the present contribution we introduce the weaker paradigm of geodesic curves of the manifold along ordered sets of points. This alternative allows us to use many interesting curves with some basic properties of helices in the modelling of protein and DNA structures.

The germinal centre reaction selects B cells from a large diversity of clones in order to optimise the efficiency of an immune response. We discuss two rather different approaches to tackle the puzzle of selection mechanisms in germinal centres with mathematical methods. A space-averaged differential equation approach is compared to a space-resolved agent-based approach. The same two novel selection mechanisms could be localised with both methods which increases the predictive power of the result. In addition, the comparison of both methods allows interesting conclusions about the suitability of diverse approaches in Theoretical Biology.

The following sections are included:

• Introduction

• Surface Reaction Models of Protocells

• Synchronization in Linear Surface–reaction Models

• Eigenvalues and Eigenvectors

• Conclusions

• Appendix A. Several Jordan Blocks

• Appendix B. Parameters and Initial Values Used to Perform the Numerical Simulations

• Acknowledgments

• References

The following sections are included:

• INDEX