BIOMAT 2009

The following sections are included:

• Preface

• Editorial Board of the BIOMAT Consortium

• Contents

Hypoxia or low oxygen is a characteristic feature of many solid tumours, associated with poor drug delivery and low rate of cell proliferation, factors that limit the efficacy of therapies designed to target proliferating cells. Since macrophages localise within hypoxic tumour regions, a promising way to target them involves engineering macrophages to express therapeutic genes under hypoxia. In this paper we present three mathematical models of increasing complexity that have been used to investigate the feasibility of using genetically-engineered macrophages to target hypoxic tumour regions. A robust prediction of the models is that the macrophage therapy is unable to eliminate the tumour. However it can reduce significantly the proportion of hypoxic tumour cells and, thereby, increase the sensitivity to standard chemotherapy. Thus we conclude that maximum therapeutic benefit will be achieved by using the macrophage-based therapy in combination with other drugs.

It is well known that tumor microenvironment affects tumor growth and metastasis: Tumor cells may proliferate at different rates and migrate at different patterns depending on the microenvironment in which they are embedded. There is a huge literature that deals with mathematical models of tumor growth and proliferation, in both the avascular and vascular phases. In particular, a review of the literature of avascular tumor growth (up to 2006) can be found in Lolas ¹. In this article we report on some of our recent work. We consider two aspects, of proliferation and of migration, and describe mathematical models based on in vitro experiments. Simulations of the models show tight fit with experimental results. The models can be used to generate hypotheses regarding the development of drugs which will confine tumor growth.

The 2D growth of Vero cell colonies grown from both quasi-straight-edges and radial expanding fronts is investigated. The latter are followed starting from either a 3D cluster of cells or from a few basal cells. The quasi-straight-edge strategy is first selected for the dynamic scaling of rough growth fronts to derive the critical exponents related to roughness characteristics and dynamics of the biological processes at the interface. In the range 0 ≤ t ≤ 12000 min, the average unidirectional colony front propagation velocity is v_h = 0.24 ± 0.04 µms^-1. The fractal dimension of propagation front, determined by the box-counting method, is D_F = 1.3±0.1, and the growth exponent from the dynamic scaling analysis is β = 0.50 ±0.05 without roughness saturation. From the dynamic behaviour of growth patterns, changes in the order of cell domains related to cell size and shape modifications are observed. At constant growth front velocity, comparable results are obtained from radially expanding growth fronts of confluent cell colonies. The Vero cell colony growth dynamics is likely determined by the random birth of cells at the interface, a process that is compatible with a random distribution of cells at different cell-cycle stage, yielding non-equilibrium colony contours.

In this paper, we study the behavior of immune memory against antigenic mutation. Using a dynamic model proposed by one of the authors in a previous study, we have performed simulations of several inoculations, where in each virtual sample the viral population undergo mutations. Our results suggest that the sustainability of the immunizations is dependent on viral variability and that the memory lifetimes are not random, what condradicts what was suggested by Tarlinton et al.. We show that what may cause an apparent random behavior of the immune memory is the antigenic variability.

The progress achieved in biology during last century has led to development of mathematical and computer modelling control mechanisms in molecular-genetic processes in living systems. In this work we consider one of the possible methods for mathematical and computer modelling control mechanisms of hierarchical molecular-genetic systems. The corresponding equations (in the class of delay functional-differential equations) are developed, using the approaches by B.Goodwin, Bl. Sendov, M. Eigen, B.A. Ratner, and taking into account temporary relations, presence of multifunctional feedback and cooperative nature of processes in cell's regulatory loops. Results of using the considered method for analyzing cell's molecular-genetic systems in the presence of the alien genes (on the example of hepatic cell infections by hepatitis B virus) are given.

Systems of atoms or molecules with complex free energy landscapes are common for quite diverse systems in nature ranging from magnetic "glasses" to proteins undergoing folding. Although Monte Carlo methods often represent the best approach to the study of suitable models for such systems, the complexity of the resultant rough energy landscape presents particular problems for "standard" Monte Carlo algorithms because of the long time scales that result at low temperatures where behavior is "interesting". We shall first review several inventive sampling algorithms that have proven to be useful for such systems and attempt to describe the advantages and disadvantages of each. Then we shall prevent results for wide ranges of temperature, obtained primarily using Wang-Landau sampling, for three models that are physically quite distinct. For pedagogical reasons we begin with spin glasses in condensed matter physics and then consider HP "lattice proteins" in which interest comes from disciplines as diverse as statistical mechanics, statistics, and biology. We shall then close with a "realistic" model for membrane protein dimerization in the continuum. All of these results will demonstrate advances in our understanding of the behavior of diverse systems that possess rough free energy landscapes.

A simple model is set forth in which discrete changes in the statistical distribution of various features are treated by means of stochastic matrices, which transform one probability distribution into another one. A few examples of application of this method, like phyllotaxis and heredity of blood groups in humans, are presented and discussed.

In recent years, it has been used time-series analysis to characterize complex biochemical oscillations. The aim of this work is to show that fractal analysis has the ability to provide further information on biochemical mechanisms leading to complex biochemical oscillations. Analysis of simulated and experimental time-series data for two systems, intracellular calcium and circadian oscillations, is made in order to illustrate the relation between fractal parameters and biochemical mechanisms leading to complex oscillations. For completeness, standard non-linear analysis, such as delayed phase-plane reconstruction and lyapunov exponents, is also performed.

Control and synchronization of HH neurons is a central topic in understanding the rhythmicity of living organisms in neuroscience. By using a feedback control approach, we provide a method for both control and synchronization of single and coupled HH neurons. Our main purpose is to elucidate how the synchronization between coupled HH neurons is achieved by a plausible external stimuli. Numerical simulations shown the effectiveness of our control approach.

The consideration of generalized Fermat problems for modelling the localization of atom sites in biomolecules is a successful approach for studying their structures. In the present note, the generic methods of this kind of modelling are introduced and an application is made to protein structures.

Protecting the rain forest in Madagascar is essential to conserve biodiversity on the island. However, regulations must take into account the development of the local population. The aim of this paper is to determine if there are viable solutions which lead to the conservation of the forest and the development of the inhabitants of the forest corridor of Fianarantsoa. We will also determine what decisions must be taken in accordance with the two aspects of the problem, by elaborating a model and studying it with the mathematical tools of the viability theory. This theory deals with controlled dynamical systems and consists in finding controls which keep the evolutions governed by these systems in a defined constraint set. The set which gathers all the situations for which such controls exist is called the viability kernel. The computation of the viability kernel with a condition on equity between the different generations excludes a large part of situations. However, adding monetary transfers enables to increase the size of the viability kernel.

Food webs are typically complex ecological networks describing who eats whom in a community of species. Presumably the fundamental question of ecology is how species diversity is maintained in nature or, alternatively, why food webs of different complexity are found in different ecosystems. Today, facing negative impacts of anthropogenic activities in virtually every corner of the world, the question of utmost importance for assessing current threats to biodiversity is how fragile or robust are species communities or, alternatively, how do food webs respond to species extinctions and/or to alien species invasions. In the quest for answers to these questions, mathematical models play an inevitable role, since experimenting on or observing (long-term) changes in species communities is risky and often in fact logistically infeasible. While food web complexity shapes and constrains dynamics of the involved populations, these dynamics feed back to shape and constrain food web complexity, so that the two need to be studied simultaneously. Research on food webs is currently among the topical ones and the number of studies that deal with population dynamics on complex food webs via mathematical models steadily increases. Some of these studies explore how food web complexity increases with considering various ecological mechanisms known to stabilize simpler consumer-resource systems. Other studies address the question of how complex food webs arise in the first place, via adding species one by one and looking at whether these new species persist in a growing community or not. The goal of this paper is to show these two types of modelling approaches, discuss their pros and cons, and suggest some avenues for further, systematic research in this exciting and relatively novel branch of ecological science. We also present a specific model of each type. Using the top-down approach, we study effects of simultaneous operation of several stabilizing mechanisms, showing, somewhat surprisingly, that they might suppress one another. Through the bottom-up approach, we examine an extent to which food webs formed via speciation vs. invasion events differ, showing that they differ in size, structure as well as propensity to collapse.

Following Atholl Anderson, there has been a growing acceptance of a late (12th century) first settlement date for New Zealand. This is sometimes advanced to AD 1280. For a modest initial number of settlers, a very rapid and sustained population growth is then required to match the population extant around AD 1800. The most detailed New Zealand palæodemographic study to date, that of Brewis, Molloy and Sutton, does not support this scenario. Their data analysis hangs on several significant assumptions and approximations. We show how to proceed from weaker assumptions and also provide confidence intervals for our estimates. Our analysis does not depend on radiocarbon or other dating. We conclude by considering the implications for New Zealand prehistory.

In this work, a bioeconomic model of an open access single-species fishery is analyzed, using a catch-rate function suggested by W. C. Clark and considering Allee effect in the exploited resource. The harvesting effort is considered to be a dynamic variable (a function of time) and also it is assumed that the exploitation of the fishery is regulated by an agency by imposing a tax per unit of landed biomass. The main objectivesare to establish the maximization of the monetary social benefit as well as to prevent the extinction of the resource. i.e., an optimal control problems is obtained, which is solved by means of the Pontryagin's Maximum Pinciple.

A surprising feature of Axelrod's model for culture dissemination or social influence is the existence of many multicultural absorbing states, despite the fact that the local rules that specify the agents interactions are explicitly designed to decrease the cultural differences between agents. In particular, Axelrod's model has two control parameters, namely, the number F of culture features that characterize the culture of an agent, and the number q of values that each feature can take on. The agents are placed on the sites of a 2-dimensional lattice of linear size L. For F > 2 the model exhibits a discontinuous non-equilibrium phase transition between a homogeneous regime for which all agents share the same culture and a completely disordered regime for which the number of cultures is maximum, q^F. Here we re-examine the problem of introducing a global interaction - the mass media effect - in the interaction rules of Axelrod's model: in addition to their nearest-neighbors, each agent has a certain probability p to interact with a virtual neighbor whose cultural features are the average cultural features of the entire population. Most surprisingly, this apparently homogenizing effect actually increases the cultural diversity of the population. We show that, contrary to previous claims in the literature, even a vanishingly small value of p is sufficient to destabilize the homogeneous regime, so there is no phase transition when a global mass media effect is taken into account in Axelrod's model.

By mean of numerical simulations we demonstrated here that Leslie matrices contain information about the way as population dissipate energy and pump entropy outside the system. We established two trade-off axes between fertility and survival and writing the corresponding matrices for each case. We found that r populations have higher entropy costs that K populations, whereas populations having Type I survival curves (high survival of larvae and low survival of adults) shows lower entropy costs than populations with Type IV survival curves. This study demonstrates that both, survival curve and projection matrix, have information about thermodynamic characteristics of populations.

This paper presents graph partitioning methods for identifying clusters in biological networks. Integer programming formulations are proposed for graph partitioning and several relaxations of these formulations are introduced for solving the problems. These relaxations include spectral methods, quadratic programming and semidefinite programming. In addition, the results of some numerical examples on biological networks are reported.

In this paper, we consider the problem of predicting protein-protein interactions. The motivation is the prediction of interacting proteins can give greater insight in the study of many diseases like cancer, and it provides valuable information in the study of active small molecules. Here we formulate the problem as a binary classification problem and apply k-Nearest Neighbors classification technique to the classes of interacting and noninteracting proteins. A case study is analyzed to show it is possible to reconstruct a real network of thousands interacting proteins with high prediction accuracy in cross validation.

Chemosystematics is the classification of plants according to their chemical composition. This paper describes a clustering algorithm, based on minimal spanning trees, that uses the penalty concept to determine the dissimilarity among objects. The algorithm is applied to a dataset of compounds from the essential oil of plants acquired for classification purpose. The results achieved are compared with the ones that consider the proposed algorithm without the penalty concept and the ones that consider a preprocessing of the original data using the principal component analysis method.

Clustering involves the task of dividing data into homogeneous clusters so that items in the same cluster are as similar as possible and items in different clusters are dissimilar. The Fuzzy C-Means Clustering (FCM) algorithm is one of the most widely used fuzzy clustering algorithms. Using a combination of fuzzy clustering, resampling bootstrapping) and cluster stability analysis for all possible numbers of clusters of the dataset, it is possible to obtain the correct number of clusters. Real datasets present samples which may have some attribute values inconsistent within the same cluster. Using these samples can insert an error that interferes with the quality of classification. This can be solved by modifying the FCM algorithm to accept a degree of reliability for each attribute of each sample. Adapting this method to work with datasets with a large number of samples is computationally intensive. We use Python for the implementation of the proposed method. Python is a dynamic object-oriented programming language that offers strong support for integration with other languages and comes with extensive standard libraries. Because we use the MPI parallel routines with Python we developed a classification method based on FCM and resampling, which has excellent computing performance and greatly reduced implementation costs.

Endogenous infection and exogenous reinfection are two mechanisms responsible for the reactivation or regeneration of active tuberculosis (TB) in individuals who have experienced prior active TB infections. We provide a brief review of a classical reinfection model, introduce a more general model, and include some new results. We conclude with a snapshot on the use of reinfection models in the study of the evolution of TB.

Understanding the spread of HIV virus in patient HIV-positive and that immune response involvement has motivated a great number of works in Mathematical Immunology. The models representing healthy and infected cell populations show variations of the dynamics of viral infection in some scenes, trying to demonstrate the ways and mechanisms that leading HIV virus invades the target-host cells, unbalancing the immune system. The use of the Basic Reproduction Number concept becomes an important parameter to quantify the viral proliferation and disease evolution, requiring a precise definition in the model of viral transmission for threshold condition calculation and in the continuous viral infection. In this work, the study was performed with the "Next Generation Matrix", methodology used to calculate the ?0 expression from a mathematical model that considers the viral infection in target cells populations that consist of macrophages, dendritic cells, lymphocytes TCD4 and CTL evidencing the most important parameters to obtain the infection threshold. As conclusion, was obtained an expression that relates parameters of HIV infection, correlated with the population of infected macrophages and dendritic cells to lymphocytes TCD4 cells, with control exercised by CTL population, demonstrating the sufficient condition to the establishment of viral proliferation and evolution of the patient of AIDS.

The recent emergence of swine flu outbreak with worldwide confirmed cases indicates that we may be at the brink of an influenza pandemic. Government preparedness policies require the real-time modelling of flu outbreaks and knowledge of possible scenarios of what may happen, to guide decision making. There are various challenges in real-time modelling and forecasting due to the uncertainty in the epidemiological characteristics of the virus, behavioural changes of the population during a pandemic and the difficulties in national surveillance. In this work we develop an appropriate pandemic flu model and integrate it within an adaptive Bayesian framework to predict the outcome of a historic epidemic in the UK(1969) in order to test the performance of this proposed framework. Our results show that with appropriate choices of priors for some of the model parameters it is possible to predict accurately the possible outcome of an ongoing pandemic, implying that good prior information which is based on initial surveillance data analysis is essential to enable reliable predictions.

A simple analytical and simulation framework to study the diffusion of pneumococci in finite population of humans is proposed, using probabilistic cellular automata. Furthermore, this epidemic spatial model permits to reproduce explicitly the interaction of two types of transmission mechanisms in terms of global and local variables, which in turn can be adjusted to simulate respectively the populational mobility and geographical neighborhood contacts. It was possible with this alternative model:

1) to observe the pneumococcal spreading through a population,

2) the study of antibiotics effects in disease's control in relation with effectiveness and percentage of individuals covered by the responsibility analyses of CCC (child care center) in dissemination process, and

3) the understanding of the relationship of hours spent in CCC and pneumococcal transmission.

Toxoplasmosis is a parasitic zoonosis worldwide distributed, infecting a large proportion of human and animal populations, produced by the parasite Toxoplasma gondii. Some individuals are at high risk of serious or fatal illness due to this parasite, including fetuses and newborns with congenital infection and immuneimpaired people. Some epidemiological studies have shown that in most of the world the presence of cats is critical for transmitting the parasite to various intermediary hosts (humans, pets). In addition, an outbreak in Vancouver, Canada, was related to the contamination of reservoir water from the city for a wild cat, and in Brazil, an epidemiological survey also linked the consumption of unfiltered water with infection in disadvantaged socio-economic strata. This paper evaluates the impact of transport by water, through rain, rivers, streams, etc., in the spread of T. gondii. A mathematical model proposed by Duarte and Trejos (2005) for the dispersion of the concentration of parasites from T. gondii in a host population of cats was used. This model combines the transmission model of SIR type with an epidemic spread of the parasite in a rectangular area; the coefficient of dispersion of the parasite includes all factors that influence the transport of the parasite (birds, rodents, insects, etc.). The assessment of the contribution of waterborne transport in the spread of T. gondii, is revealing in the model mentioned a term that describes it, independently with other transmission mechanisms. To do this, a velocity vectorial field on the hydrological map of the Department of Quindío, Colombia, was constructed and was subsequently incorporated into the model. The system simulation was made and the results with and without waterborne transport were compared; concluding that it is fundamental to the spread of the parasite.

An SIS model with pulse vaccination at variable times is presented. The local stability of the disease–free solution is completely studied, whereas some initial results concerning global stability are discussed. By using numerical simulations, the convergence rate toward the disease–free solution is compared with a fixed time vaccination strategy.

INDEX