July 30-August 2, 2008
Society for Mathematical Biology Conference

hosted by the Centre for Mathematical Medicine, Fields Institute
held at University of Toronto, Medical Sciences Bldg.


Accepted Contributed Talks
Contributed talks are 15 minutes, plus 5 minutes for questions.

Modeling glioblastoma with cell-cell and cell-substrate interactions
Mathilde Badoual

Laboratoire IMNC, 15 rue Georges Clemenceau, bat 104, 91406 Orsay Cedex, France
Coauthors: Christophe Deroulers (Laboratoire IMNC) Marine Aubert (Laboratoire IMNC) Basile Grammaticos (Laboratoire IMNC)

Glioblastoma are malignant tumors associated with a very poor prognosis, due to the capacity of individual glioma cells to invade surrounding normal brain tissue, far from the tumor focal area. This infiltration results in the inability to completely resect this tumor and is responsible for the almost inevitable recurrence after treatment.
Cell-cell (homotype) as well as cell-substrate (heterotype) interactions are key events in the migration process.
We have developed a cellular automaton where the strength of each type of interaction is ajustable, to describe the migration of glioma cells [1,2].
From this automaton, we were able to derive a macroscopic equation of diffusion, where the diffusion coefficient is original compared to other classical models[3]. First, it is nonlinear as it depends on the cell density. Second, it depends on the two parameters measuring the strength of homotype and heterotype interactions.
Here, we use this nonlinear diffusion coefficient in a diffusion-proliferation equation to model the growth of glioblastoma. We define two cell populations, characterized by different homotype and heterotype interaction parameters and a proliferation rate that depends on the strength of heterotype interactions. First, we study the interplay between cell-cell and cell-matrix interactions during cell migration in vitro on different substrates, and we reproduce some experimental results [4].

We also compare the effects of classical treatments (surgery, radiotherapy) for different values of homotype and heterotype interaction parameters and we show that inhibing heterotype (or increasing homotype) interactions (by inhibiting gap-junctions or intergrins for example) in the margin of an operated tumor could have a clinical interest, by reducing the chances of recurrence.
[1] Aubert M, Badoual M, Fereol S, Christov C and Grammaticos B, 2006, A cellular automaton model for the migration of glioma cells, Phys. Biol. 3 93.

[2] Aubert M, Badoual M, Christov C and Grammaticos B, 2008, A model for glioma cell migration on collagen and astrocytes, J. R. Soc. Interface, 5,75-83.

[3] Tracqui P, Cruywagen GC, Woodward DE, Bartoo GT, Murray JD and Alvord EC Jr, A mathematical model of glioma growth: the effect of chemotherapy on spatio-temporal growth, Cell Prolif, 1995, 28, 17-31.

[4] Giese A, Loo MA, Tran N, Haskett D, Coons SW, Berens ME, 1996, Dichotomy of astrocytoma migration and proliferation, Int. J. Cancer 67, 275-282.

Weak solutions for a two-sidedly degenerate chemotaxis model with volume-filling effect related to the p-laplacian operator
Ricardo Ruiz Baier

Departamento de Ingenieria Matematica, Universidad de Concepcion, CHILE
Coauthors: Mostafa Bendahmane, Departamento de Ingenieria Matematica, Universidad de Concepcion, mostafab@ing-mat.udec.cl Raimund Bürger, Departamento de Ingenieria Matematica, Universidad de Concepcion, rburger@ing-mat.udec.cl José Miguel Urbano, Departamento de Matemática, Universidade de Coimbra, Portugal, jmurb@mat.uc.pt

We address the question of existence and Hölder regularity of weak solutions for a fully parabolic model for chemotaxis with volume-filling effect, that degenerates in a two-sided fashion, including a p-Laplacian diffusion term. The relevant system is suplemented with nonlinear Neumann boundary conditions. For the proof of existence of weak solutions we use a Schauder fixed-point argument on a regularized problem and the compactness method, and for the regularity, we use the rescaling method.

**Simulated Two-Dimensional Red Blood Cell Motion, Deformation, and Partitioning in Microvessel Bifurcations
Jared Barber

Program in Applied Mathematics, University of Arizona
Coauthors: Jonathan P. Alberding, Juan M. Restrepo, Timothy W. Secomb

Movement, deformation, and partitioning of mammalian red blood cells (RBCs) in diverging microvessel bifurcations are simulated using a two-dimensional, flexible-membrane model. A set of viscoelastic elements represents the RBC membrane and the cytoplasm. These elements are coupled to finite elements that represent the surrounding fluid and the coupled system is numerically solved. Simulated isolated RBC trajectories deviate from background flow streamlines primarily because of cell migration towards vessel centerlines and cell obstruction of downstream vessels. Estimates of RBC distributions at a bifurcation are determined as a function of total blood fluxes into the two branches and upstream RBC spatial and velocity distributions. RBCs preferentially enter the higher-flow branch, leading to unequal RBC fluxes in the downstream branches. Cell migration gives cells a stronger tendency to enter the high flow branch. Cell obstruction, on the other hand, counteracts this tendency. In unequally-sized daughter vessels, partitioning is asymmetric, with RBCs tending to enter the smaller vessel. Partitioning is not significantly affected by the daughter vessel orientations. Significant differences are found between rigid particle and flexible cell distributions. Predicted distributions with flexible cells are consistent with experimental observations, showing that membrane flexibility is an important factor determining realistic RBC distributions in bifurcations.

Modelling the effects of disease on the speed of ecological invasions.
Sally S Bell

Heriot-Watt University, Edinburgh
Coauthors: Dr. Andy White (Heriot-Watt University, Edinburgh) and Prof. Mike Boots (University of Sheffield, Sheffield)

We use mathematical models to analyse how a shared disease affects the spread of an invasive species. We examine the temporal dynamics and compare the speed of invasion with and without disease. This work highlights the importance of the relative effects of different disease parameters when considering the speed of invasion and the speed of replacement of the native species. A spatial extension of the model shows a wave of disease, instigated by the initial introduction of the invading species, spreads through the native population in advance of the wave of invasion. The presence of the disease allows the invading species to extend the range over which it replaces the native species. These results therefore have important conservation implications.

The effects of behavioural patterns on the potential success of the HPV vaccine.
Victoria Brown

University of Bath, UK
Coauthors: Dr. K. A. J. White (University of Bath)

In recent years, links between the human papillomavirus (HPV) and cervical cancer in women has led to the development of a vaccine to protect against HPV as a preventative measure for cervical cancer (Arbyn and Dillner 2007). A national vaccination pilot in the UK targets vaccination at pre-teenage girls, the majority of whom are not yet sexually active (JCVI HPV Sub-group minutes, September 2006); discussion of other target groups, such as the equivalent male cohort, is also taking place.

The vaccine is administered on three separate occasions and information from a clinician suggests that if one of these is missed, then the vaccine will not be an effective barrier to HPV. The cost of the triple dose is £240.

We are interested in exploring the impact of the vaccine in relation to behavioural traits demonstrated at the population level. In particular, given that the vaccine may only be effective for 5-10 years, how does the onset of sexual activity in the population affect the efficacy of the vaccine in eradicating the virus?

To address such questions, we develop a model system which describes the spread of the virus in a heterosexual population (males and females are explicitly modelled), where individuals may either be sexually active or not. We focus the analysis of the model on the interaction between three parameters: the proportion of the female population that is initially vaccinated, the rate at which females become sexually active and the proportion of vaccinated individuals that are still sexually inactive when the vaccine loses its efficacy. Model parameters are estimated using published datasets.

We discuss the results of our analysis; in particular, we describe the conditions under which HPV might be eradicated from a population with different behavioural patterns. We will discuss extensions to this work; in particular, the extension to consider an explicit age dependent model.

Arbyn and Dillner 2007: Marc Arbyn and Joakim Dillner, Review of current knowledge on HPV vaccination: An Appendix to the European Guidelines for Quality Assurance in Cervical Cancer Screening, Journal of Clinical Virology, 38 (2007).
JCVI HPV Sub-group minutes: Department of Health, Joint Committee on Vaccination and Immunisation, Minutes of the HPV Sub-group meeting, Wednesday 22 September.


Quantifying the advantages of early intervention on Lupus Nephritis flares
Paula Budu-Grajdeanu

Mathematical Biosciences Institute
Coauthors: Richard C. Schugart, Avner Friedman, Daniel J. Birmingham and Brad H. Rovin

Although the prognosis for lupus nephritis (LN) has dramatically improved with aggressive immunosuppressive therapies, the current approach to treatment remains inadequate. It is likely that combinations of biomarkers will be needed to accurately describe the evolution of LN flare and to forecast impending LN flares. To demonstrate the utility and feasibility of lupus biomarker development, we develop a mathematical model of lupus renal flare that quantifies reduction in immunosuppressive medication as a function of how early flare is detected. The results suggest that early initiation of treatment is advantageous. Further refinement of the model, incorporating clinical and biomarker data, will benefit patients by allowing physicians to individualize immunosuppressive therapies.

Phase Models with Time Delay
Sue Ann Campbell

University of Waterloo
Coauthors: Ilya Kobelevskiy and Andrew Smith

We consider a network of inherently oscillatory neurons with time delayed connections. We reduce the system of delay differential equations to a phase model representation and show how the time delay enters into the reduced model. For the case of two neurons, we show how the time delay may affect the stability of the periodic solution leading to stability switching between synchronous and antiphase solutions as the delay is increased. The results of the phase model analysis are compared with numerical bifurcation analysis of the full system of delay differential equations. Both type I and type II oscillators are considered.


Modeling of cancer virotherapy with recombinant measles viruses
Thomas W. Carr

Southern Methodist University
Coauthors: Zeljko Bajzer (Mayo Clinic College of Medicine) Kresimir Josic (University of Houston) Stephen J. Russell (Mayo Clinic College of Medicine) David Dingli (Mayo Clinic College of Medicine)

The Edmonston vaccine strain of measles virus has potent and selective activity against a wide range of tumors. Tumor cells infected by this virus or genetically modified strains express viral proteins that allow them to fuse with neighboring cells to form syncytia that ultimately die. Moreover, infected cells may produce new virus particles that proceed to infect additional tumor cells. We present a model of tumor and virus interactions based on established biology and with proper accounting of the free virus population. The range of model parameters is estimated by fitting to available experimental data. The stability of equilibrium states corresponding to complete tumor eradication, therapy failure and partial tumor reduction is discussed. We use numerical simulations to explore conditions for which the model predicts successful therapy and tumor eradication. The model exhibits damped, as well as stable oscillations in a range of parameter values. These oscillatory states are organized by a Hopf bifurcation.


An epidemic model with host cross-immunity to a continuum of strains.
Farida Chamchod

Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, UK
Coauthors: Nicholas F Britton

Several models in multi-strain diseases have included cross-immunity. Lots of them are complicated to deal with when the number n of strains increases. A history-based model for example contains 2^n+n*2^(n-1) variables. However, in a status-based model the number of variables is 2^n+n-1 and by the reduced-transmission and polarized immunity assumptions the numbers of variables in total is drastically reduced to 2n. In this work, the model is based on the status-based formulation. We use a real line with each point representing a strain with a particular antigenic make-up to represent a phylogeny of the diseases. The number of equations is now 2. We then study a travelling wave of the system and how it depends on other parameters such as the mutation rate, the basic reproductive ratio, and the cross-immunity coefficient. From the result, we conclude that the travelling wave represents an antigenic drift process with strains present in the population dying out and being replaced by new ones at new points in the antigenic space.

A “Go or Rest” model for cell migration. A step forward toward the “Go or Grow” modelling
Dr. Arnaud Chauvière

Technische Universitaet Dresden - Germany
Coauthors: Haralambos Hatzikirou and Andreas Deutsch

Cell migration is an essential feature of, either physiologic or pathologic, phenomena in biology, such as embryonic development, wound healing or tumor invasion. According to the local microenvironment and the cell function, the characteristics of the migration may vary considerably.
Here we look closer at the influence of the cell density on the migration dynamics, and we assume two different regimes: when cells are isolated, the corresponding motion is essentially characterized by a sequence of “runs” separated by random reorientations of the velocity; in denser areas, migrating cells interact with other cells and “collision effects” become relevant.
Additionally a “resting” regime is included in the migration modelling. This can either result from environmental conditions or relates to a strategy of cells to fulfil efficiently their function. As an example, cells undergo mitosis only under favourable environmental conditions, and an immotile state is then required. A second illustration is the “Go or Grow” hypothesis currently accepted in the biology of brain tumor invasion.
A kinetic (mesoscopic) model is first derived and a continuous (macroscopic) model is deduced as its diffusive limit. This so-called “Go or Rest” model provides anomalous diffusion which is furthermore analyzed.

The model is then extended to include proliferating phenomena. The study of the invasive front will aim to understand heterogeneous patterns observed in tumor invasion.

Turing Pattern Formation in Stochastic Reaction-Subdiffusion Systems
Jiawei Chiu

A*STAR Institute of High Performance Computing
Coauthors: K.-H. Chiam

We investigate the formation of spatial Turing patterns in stochastic reaction-diffusion systems that arise commonly in biology, such as the motion of proteins in a crowded cytoplasm, or the migration of epithelial cells driven by active biological processes. Here, we consider the case of two species of "particles, " be they proteins or cells, performing continuous random time walks specified by the probability distribution function yi(x, t) = m(x)Wi(t), where the Laplace transform of the Wi's is ~ 1 - (hi s)a and a < 1 denotes subdiffusion. We seek to understand under what conditions there is Turing instability by performing linear stability analysis. We find that, if we fix the ratios between the diffusive constants, there is a critical value of a below which there is no pattern. We discuss the relevance of this critical value to several biological processes. In addition, we carry out simulations based on Gillespie's algorithm to study the effect of noise, induced by the low copy number of particles, on the conditions for Turing pattern formation. We find that Turing patterns can survive in a very noisy system.

Low red cell production may protect against severe anemia during a malaria infection – Insights from modeling.
Deborah Cromer

Imperial College, London
Coauthors: Jaroslav Stark, Miles P Davenport

The malaria parasite causes lysis of red blood cells, resulting in anemia, a major cause of mortality and morbidity. Intuitively, one would expect the production of red blood cells to increase in order to compensate for this loss. However, it has been observed that this response is weaker than would be expected. Furthermore iron supplementation for iron deficient children in malaria endemic regions can paradoxically adversely affect the clinical outcome of malaria infection. A possible explanation may lie in the preference that some malaria parasites show for infecting immature red blood cells (reticulocytes). In the presence of a parasite preference for immature red cells, a rise in red cell production can ‘fuel the fire’ of infection by increasing the availability of the parasite’s preferred target cell.
We present a mathematical model of red blood cell production and infection in order to explore this hypothesis. We assess the effect of varying the reticulocyte replacement rate and preference of the parasite for reticulocytes on four key outcome measures assessing anemia and parasitemia.
For a given level of parasite preference for reticulocytes we uncover an optimal erythropoietic response which minimizes disease severity. Increasing red blood cell production much above this optimum confers no benefit to the patient, and in fact can increase the degree of anemia and parasitemia. These conclusions are consistent with epidemiological studies demonstrating that both iron deficiency and anemia are protective against severe malaria, whilst iron supplementation in malaria endemic regions is with an increased number of malaria related adverse effects. Thus, suppression of red blood cell production, rather than being an unfortunate side effect of inflammation, may be a host protective effect against severe malarial anemia.


Ecosystem engineering in predator-prey interactions
Kim Cuddington

Ohio University (University of Waterloo as of August 2008)
Coauthors: Alan Hastings (UCDavis) Theresa Talley (UCDavis)

Predator-prey interactions are described as having a negative impact on the prey species and a positive impact on the predator species. We develop a general model to describe how ecosystem engineering may alter the net effect of trophic species interactions. We modify a standard predator (P) - prey (N) population model to include the effects of an environmental state (E). We describe the modification of the environmental state as a linear function of predator or prey density. The environment has a normal state, E0, to which it will return in the absence of environmental modification by either of the two species, and either or both of these species can move the environment away from this state. Throughout we make the simple assumption that the modified environmental state could linearly modify all population parameters as:

dN/dt=[(a0 + a1 E) - (b0 + b1 E)N - (c0+c1E)P]N

dP/dt = [(f0+f1E)-(g0+g1E)P+(h0+h1E)N]P

dE/dt = -k(E-E0)+mP+nN.

We apply this model to a conservation problem in a crayfish-dragonfly system. We derive the conditions where the crayfish predator may benefit the endangered dragonfly species through the building of burrows. The net impact of the crayfish on the dragonfly will be positive where the product of the environmental modification rate and the benefit of the burrow to the dragonfly species is greater than the product of the rate at which the environment returns to its unmodified state and the predation rate (a1 m > c0 k). As a result, we predict that the relationship between the predator and prey species could swing back and forth between a net positive and a net negative interaction with environmental variation.

Using Phylogenetics and Mutual Information to Identify Coevolving Sites in Protein Families
Christopher DeHaan

The University of Western Ontario
Coauthors: Lindi Wahl, Andrew Fernandes

Proteins with similar functions found in different organisms can be aligned in a multiple sequence alignment (MSA), and standard techniques are now available to infer the phylogenetic tree which relates these sequences in evolutionary history. A number of recent papers have elucidated the use of Mutual Information (MI) in identifying positions within such a protein family which co-evolve or are in contact in the folded structure. However, to date, none of these MI methods have made use of the phylogenetic history of the proteins in the MSA. I have used the inferred phylogeny of an MSA to simulate realistic random protein sequences, but with no interdependence on any of the individual positions. Initial amino acid distributions and their mutation probabilities are based on observed data (a contingency table). This allows the computation of a distribution of MI in the absence of interdependence, which serves as a null distribution which is specific to a particular MSA. Positions in the real MSA which share significantly more MI than the simulated null distribution can then be identified with some certainty. This method leads to more accurate identification of protein active sites and sites which are in contact, both of which are critical in determining the protein mechanism and structure.

Modeling the migration of cancer cells: from microscopic to macroscopic models
Christophe Deroulers

Laboratory IMNC, Campus d'Orsay, 91406 Orsay Cedex, France
Coauthors: Mathilde BADOUAL, Marine AUBERT, Basile GRAMMATICOS

It is well known that the migration of cancer cells plays a key role in the development of some brain tumors, such as gliomas. Because cancer cells invade tissues far from the tumor center, the tumors have no sharp boundary. A surgeon cannot remove all cancer cells, and cancer reccurs invariably.

Therefore, it is crucial to take cell migration into account in the modelling of glioma. This is easy in a cellular automaton-like model, where some stochastic rules tell how the individual cells move, duplicate and die, and some of us used such a model to reproduce in vitro experiments of cancer cells migration and to show that cells interact while migrating [1]. However, a cellular automaton is not so convenient as partial differential equations (PDEs) that are often used to model the density of cancer cells in the brain. Especially, it is not suited for the study of real-size tumors with several millions of cells. Therefore, it would be nice to derive some PDE that takes the migration and interaction of cancer cells into account.

We give one analytic technique to go from the definition of a microscopic model (a cellular automaton) to a macroscopic model (a PDE). We apply this technique to the situation of [1] and we show that our PDE reproduces both simulation and in vitro experimental results [2]. We notice that the PDE we obtain is, because of the interaction of cancer cells, a nonlinear diffusion equation (belonging to the family of the porous media equations), whereas it is often postulated that diffusion of cancer cells is linear. Interestingly, we notice that our model is closely related to some kinetically constrained models introduced for the study of the physics of glasses, supercooled liquids and jamming systems.

We also give some results on the effect of taking into account the shape of cancer cells, which are not point-like as assumed in [2] but elongated, and on dealing with statistical correlations of the position of cancer cells.

[1] M. Aubert, M. Badoual, S. Fereol, C. Christov and B. Grammaticos, A cellular automaton model for the migration of glioma cells, Phys. Biol. vol. 3 p. 93 (2006).

[2] C. Deroulers, M. Badoual, M. Aubert and B. Grammaticos, Modelling tumour cell migration: from microscopic to macroscopic, submitted (2008).

Stoichiometric network analysis and graph theoretic methods for studying spatial models of chemical reaction networks
Mirela Domijan

Unversity of Warwick

Chemical reaction networks play an important role in understanding the biological processes that take part on a cellular level. However, these networks still continue to be a challenge to model and analyse. If we take a deterministic approach and we assume that the chemicals involved cannot diffuse, the chemical reaction networks are modelled by systems of ODEs. Modelling of chemical interactions can involve many chemicals and complex interactions, resulting in large ODE systems with nonlinear terms. If there is lack of quantitative information about the interactions, there will also be parameter uncertainty in the models. These issues can make it impossible to numerically simulate the system dynamics or to perform bifurcation analysis. In such cases, analysis can be successfully performed via graph theory and stoichiometric network analysis. Analysis via these methods does not depend on the unknown parameters. Instead, they relate system's dynamic properties to network structure and more specifically, to reaction stoichiometry.

Yet, in certain cellular processes reactions may not proceed in "well-mixed" environments and chemicals may diffuse. Then, spatial effects on the chemical dynamics need to be considered and analysis of appropriately-constructed systems of reaction-diffusion PDEs needs to be performed. Diffusion can lead to new behaviors in the PDE systems, such as diffusion-driven (Turing) instability. This instability constitutes the basis of Turing's mechanism for pattern formation. Turing instability has been well described for systems of two chemicals, however, theory is still lacking for large systems. We will present concepts from stoichiometric network analysis and related graph theory that can be applied to analysis of spatial reaction networks. The conditions will address the occurrence of Turing instability. A Turing unstable two-chemical system relies on the activation and inhibition between the two chemicals. The conditions that we will give generalise the conditions of positive/negative feedback cycles for Turing instability.

Random Spatial Networks: A Biological Solution to the Structure/Transport/Connection Problem
Donald A. Drew

Rensselaer Polytechnic Institute
Coauthors: Yanthe Pearson

Spatial networks are collections of fibers occupying a spatial region. The network is called random if we are interested in an ensemble of equivalent such networks, where the positions of the individual fibers are not identical but have some statistical “sameness.” Examples of random statistical networks include microtubule structures, capillaries, neurons, and trees. Each of these networks has a function that it can fulfill by producing a realization out of an ensemble of such networks having certain properties. For example, a microtubule network that is responsible for cell integrity must support the forces that maintain the cell shape; capillaries must deliver and/or absorb chemical species to the surrounding matrix. We discuss microscale (individual fiber) and structural (probability density function) models to describe random spatial networks. so as to relate network statistical structure to assembly dynamics of individual network fibers, and to determine the biological functionality of the network from network statistical properties. In addition, network assembly, disassembly, and interactions with the surroundings during network formation and structure can add to the understanding of the biochemistry and biophysics of fiber formation and guidance.

We shall focus on axonogenesis. Axons are the propagation elements in neurostructures in all higher species. During formation of the brain and nervous system, axons are generated by extension of processes from neural cells with dynamics determined at the growth cone. The progress of the growth cone is determined by the response of surface receptors to gradients of signaling molecules. Surface structures bind the signaling molecules, leading to changes in the assembly of the actin/microtubule structure that drives axonal growth cone motion. In this paper we introduce a two-dimensional stochastic model which captures the random behavior of axon growth to simulate axonal trajectories for cells in a homogeneous medium. We use data to evaluate the standard deviation of the angle changes on the axonal trajectories and to verify the validity of the structure of the stochastic differential equations for the axonal trajectories. For the model of growth of axons, we analyze trajectory data consisting of measurements of the position of the axon tip at different frames in the time sequence of micrographs. This data shows that the axon tip changes direction randomly, but the data is noisy due to the data collection procedures. We develop algorithms to filter out noise while maintaining the underlying dynamics of the axon growth process. We perform statistical analyses on relevant variables generated from our filtered data. We present Monte Carlo simulations of stochastic differential equation systems.

Analysis of serial engagement and peptide-MHC transport in T cell receptor microclusters
Omer Dushek

University of British Columbia
Coauthors: Daniel Coombs

During stimulation of a T cell by an antigen-presenting-cell (APC) bearing cognate peptide-major-histocompatibility complexes (pMHC), T cell receptors (TCR) have been shown to form stable micrometer-scale clusters in the contact region. pMHC molecules diffusing in the APC membrane may bind and unbind from multiple TCR in a cluster. Such serial engagements of multiple TCR by a single pMHC have been hypothesized to be important in T cell signal amplification. We use mathematical modeling to characterize the number of clustered TCR bound by a single pMHC. We show that the TCR-pMHC bond kinetics alone do not allow substantial serial engagement of TCR and suggest molecules that could enhance TCR engagements. Mathematical tools: MFP calculations, asymptotic analysis, numerical solutions of PDEs.


Epidemic control in weighted social networks
Ken Eames

University of Cambridge

Social networks provide a valuable tool for understanding the link between population mixing behaviour and epidemic dynamics. This understanding motivates the design and development of targeted epidemic control through the identification of the most high-risk individuals. Social mixing surveys demonstrate clearly that not all interactions are of equal strength and suggest the use of weighted networks to capture variations in contact intensity. Here, we present the results of simulation models used to investigate targeted interventions in weighted networks; we compare the use of different individual-based measures of risk to prioritise individuals for intervention and conclude that the existence of weighted social networks offers new challenges and new opportunities for disease control. We discuss the data requirements of proposed intervention strategies and present some new data on the mixing patterns of one of the most epidemiologically significant population subgroups: school children from the ages of 5 to 11.

A Mathematical Model for the Effects of HER2 Over expression on Cell Cycle Progression in Breast Cancer
Amina Eladdadi

Department of Mathematical Sciences, Rensselaer Polytechnic Institute, Troy, NY 12180, USA
Coauthors: David Isaacson

We present a mathematical model to study the effects of HER2 over-expression on cell-cycle progression in breast cancer. The model addresses the following question: How do changes in the number of HER2 and EGFR receptors during the cell-cycle affect the cell proliferation rate? In order to characterize the effects of HER2 over-expression on the cell cycle progression, we use a three-compartment cell cycle model with non-constant transition rates. Our new hypothesis is that the transition rates depend on the number of the cell surface HER2 receptors and their signaling properties. The model relates the different phases of the cell cycle transition rates to the signaling properties of the EGFR-HER2 receptors (through their binding kinetics), and the population dynamics of cells in the corresponding cell-cycle phase.


Modeling early tumor development dynamics - implications for treatment design
Heiko Enderling

Center of Cancer Systems Biology, Caritas St. Elizabeth's Medical Center, Tufts University School of Medicine
Coauthors: Afshin Beheshti, Lynn Hlatky, Philip Hahnfeldt

Cancer development may be considered an evolutionary process whereby genetically unstable cell clones compete under selective influences of the local environment and succeed in accordance with the relative fitnesses of their expressed phenotypes. The competition process involves traversal through a number of bottleneck challenges at all phases of tumor development. One factor limiting tumor cell proliferation is space to grow, a condition which may be alleviated by cell death within the mass. We show theoretically how tumor populations devoid of stem cells could still persist as long-term dormant lesions, and offer a possible explanation for the incidence of dormant tumors observed in recent autopsy studies. This finding questions the notion that tumors escaping dormancy will necessarily become symptomatic. Finally, if the tumor population is assumed to contain cancer stem cells, we show 1) that certain conditions may paradoxically limit the growth of the lesion, even if it escapes dormancy, and 2) the number of stem cells can be amplified through adjustments in other parameters that reduce the local density of progeny cells. The latter observation lends support to the theory that tumors grow in part through the creation and merging of local metastases. From the presented model we derive implications for treatment.

Synchronization Of Insulin Secretion Through Intrapancreatic Ganglia
Bernard Fendler

Florida State University

ß-cells are cells located in the human pancreas and are known to produce electrical activity. When these cells are electrically active, they secrete a hormone necessary for maintaining glucose homeostasis in the blood called insulin. The ß-cells are located in the pancreas in small micro-organs called islets of Langerhans. There are thousands of islets in the pancreas which are known to produce oscillatory insulin secretion. Measurements of insulin have shown that oscillatory secretion also occurs in the blood. Since plasma insulin reflects the secretion from the entire islet population, oscillations in plasma insulin levels suggest that islet oscillations must be largely synchronized. Bertram et al. “Bio. Phys. Jour., 92, 1544-1555, 2007” has developed a mathematical model of the ß-cell which reproduces many of the measured electrical and calcium properties of the ß-cell. We use this model to investigate methods of synchronization of the islet population. The islet is known to be innervated by neurons, in ganglia, interspersed throughout the pancreas. We investigate the viability of islet synchronization by coordinated action of the intrapancreatic ganglia.

Myotome formation and the importance of shape
Edward Flach

Innovative Methods of Computing, Technical University Dresden
Coauthors: Lutz Brusch, Andy Oates, Andreas Deutsch

Dramatic cell shape change and relative movement are readily observable in the differentiation of muscle cells during early embryonic development of zebrafish. An initial population of cuboidal “adaxial” cells specified in the early embryo differentiate into rod-shaped cells that form the slow-twitch muscle fibres used by the fish to swim through the water. There is a complex and stereotypical shape change and spatial rearrangement of the cells at the collective level.
We consider the underlying dynamics of the cell required to effect this change. We produce a minimal model which can reproduce the behaviour. Using a simulation we investigate the balance of forces required.

Modeling Mammalian Circadian Rhythms
Daniel B Forger

University of Michigan

Biological circadian (~24-hour) clocks time many biological processes that must occur at specific times of the day. Circadian behavior in mammals is co-ordinated by a group of ~20,000 neurons in the suprachiasmatic nucleus (SCN). The molecular basis for these clock within each SCN neuron is a complex network of genetic feedback loops.

Several large-scale detailed mathematical models of the molecular biology and electrophysiology of the SCN will be presented. Predictions from these models have large implications for sleep disorders, the effects of genetic mutations, and how timekeeping in encoded in firing rates.

I will also present several experimental studies which have confirmed model predictions.

Channel mechanisms for structural basis for the Hodgkin and Huxley relation
Charles M. Fortmann

Stony Brook university
Coauthors: Yeona Kang

Neural channel transport was analyzed using a previously reported relation for charged particle transport in two energy-type gradients. One energy type gradient is the electric field, expressible as a concentration gradient along the axis of transport, the second results from the transporting cation coupling with water and with a neural channel deformation. Neural channels are lined with alpha helix protein secondary structure that have near neutral charge and are filled with water vapor and sequestered hydrophobic amino acids arranged to present minimum water vapor and water-hydrophobic interface. Cation point charges generate enormous electric fields on sub-nanometer distances. Electrostatic energy reduction is characterized by water, a strong dielectric, being pulled toward the transporting ion, thereby deforming the neural channel structure.
An energy gradient results whenever the ion-water-structure coupling energy is modified by changes in channel diameter and/or channel deformation in the axial direction. The resultant two energy gradient relation for cation transport: reduces to the Hodgkin-Huxley relation, explains channel selectivity and environmental sensitivity, and predicts fast non-dispersive transport under a narrow range of conditions. The transporting cation-water-deformation model produces current-voltage characteristics consistent with observation.


The effect of antioxidant supplementation on the viral load of HIV
R D van Gaalen

Dept. of Applied Mathematics, University of Western Ontario
Coauthors: L M Wahl

A byproduct of cellular respiration, small highly reactive molecules called reactive oxygen species (ROS) play a crucial role in cell signalling and infection control. However, high levels of ROS can cause significant damage to cell structure and function. Under normal conditions, a healthy diet supplying adequate quantities of antioxidants helps to maintain a safe level of ROS. Studies have shown that infection with the human immunodeficiency virus (HIV) results in an increase in oxidative stress. Acting like a catalyst in nuclear factor kB activation, ROS in turn lead to faster progression of HIV infection, and cause CD4+ T-cell apoptosis. Clinical studies have explored the possibility of raising antioxidant levels with mixed results. In this talk, a mathematical model is used to explore this contested therapy.

Sexual Moran Model: Theory and Simulation
J M Grant

Applied Mathematics, University of Western Ontario
Coauthors: L Wahl (Western) and G Wild (Western)

We describe an extension of the birth-death Moran model, incorporating sexual reproduction and a finite population size. In this model, one male and one female parent are chosen to give birth, and neither can be displaced by their offspring (in other words, neither parent can be chosen for death in that time step). If the sex ratio is fixed, we demonstrate analytically that the limiting distribution for the number of females (or males) in the population is binomial. We confirm this result numerically, by considering the eigenvalues of the associated Markov transition matrix, and by individual-based simulation. We also investigate social evolutionary questions in the context of this model.


A Stochastic Survival Analysis for Phase II Clinical Trials in Relapsed Ovarian Cancer - Non-Stationary Poisson Process -
by R Gunawan

University of Waterloo
Coauthors: Tenti, G; Oza, A; Sivaloganathan, S

Ovarian cancer has long been known to be one of the leading causes of cancer death. It commonly strikes women who are 50 years of age or older and it has a poor prognosis. Our collaborators at Princess Margaret Hospital in Toronto, Ontario, have conducted various extensive clinical trials investigating molecularly targeted agents in relapsed ovarian cancer patients. Our data were obtain from four groups of phase II clinical trials: PHL019 (UCN-01 and Topotecan), PHL025 (Sorafenib and Gemcitabine), PHL037 (AZD2171), and PHL041 (PXD101). Patients participating in the clinical trials were monitored regularly* according to the Response Evaluation Criteria in Solid Tumor (RECIST) and the Gynecologic Cancer Intergroup (GCIG). During each follow-up, the check-up date, the longest diameter (LD) of the target lesion and the level of Cancer Antigen-125 serum, a surrogate marker for drug efficacy, were recorded. Survival data were collected at the end of the study. If a patient was still alive at the end of the study, she was assigned a ‘0’; whereas, if a patient died by the end of the study, she was assigned a ‘1’. From the data, we have demonstrated that the survival time of the patients follows a non-stationary Poisson process. This stochastic formulation leads us to a variety of useful information such as average survival rates in each clinical trial group. This could help determine the effectiveness of a particular drug or combination of drugs. Additionally, we verified whether CA125 was truly a marker for drug efficacy in these clinical trial groups.

"Go or Grow": the key to the emergence of invasion in tumor progression?
Haralambos Hatzikirou

TU Dresden
Coauthors: D. Basanta, M. Simon, C. Schaller, A. Deustch

Uncontrolled proliferation and abnormal cell migration are two of the main characteristics of tumor growth. Of ultimate importance is the question: what are the mechanisms that trigger the progression from benign neoplasms (high proliferation) to malignant invasive tumors (high migration)?. We show with a lattice-gas cellular automaton that the transition to invasive tumor phenotypes can be explained solely on the basis of the microscopic "Go or Grow" mechanism (migration/proliferation dichotomy) and the oxygen shortage, i.e. hypoxia, in the tumor environment. This result challenges the currently prevailing view that the emergence of invasiveness is mainly the consequence of acquired cancer cell mutations. Moreover, we provide a theoretical explanation of our results by means of a cut-off mean-field approach. Finally, we suggest possible therapies that could help prevent the progression towards malignancy and invasiveness of benign tumors.

The Inseparability of Spatial and Temporal Clustering in a Population Model with Spatially Correlated Disturbances
David Hiebeler

Dept. of Mathematics and Statistics, University of Maine
Coauthors: Isaac Michaud (University of Maine) Nicholas Millett (University of Maine)

Our prior work studied the effects of large-scale disturbance events in a locally-dispersing spatial patch-occupancy population model, where contiguous blocks of sites were simultaneously disturbed in such a way that the per-site disturbance rate was kept fixed. Results indicated that increasing the spatial scale of disturbance events had a negative effect on equilibrium population density. However, the reason for this effect was speculated to be possibly due to different factors, such as slow recolonization of disturbed regions via only local dispersal, or the increased temporal variability that accompanied the increased spatial scale of disturbances. Here, several variations of the model are explored via simulations, to further explore why spatially correlated disturbances adversely affect a locally-dispersing population, and to try to separate the effects of spatial and temporal clustering in such disturbances. Some issues related to efficient simulation of such models will also be discussed.

Feedback, ratios and robustness in Hedgehog signaling
David Irons

University of Sheffield, UK
Coauthors: Nick Monk (University of Nottingham, UK)

The secreted protein Hedgehog (Hh) acts a morphogen, forming a concentration gradient and controlling cell fate decisions in various developmental stages in many animals. Here we consider Hh gradient formation in one such developmental context, the Drosophila wing disc.

One evolutionarily conserved component of this pathway, of particular interest, is a feedback loop where Hedgehog signalling up-regulates its own receptor, Patched (Ptc). It has been suggested that this feedback loop could enhance the robustness of the steady-state Hh gradient against variability in Hh production levels. However, previous models have failed to take into account two important issues in the biological system. Firstly, interpretation of the gradient by responsive cells (and hence Ptc production) is dependent on the 'ratio' of unbound Ptc to bound Hh-Ptc in the cell. Here, the Hh-Ptc complex 'dilutes' the inhibitive effect that Ptc has on its production, altering the very nature of the feedback loop. Secondly, total Hh levels and the expression range of downstream target genes, such as Decapentaplegic (Dpp), are growing over time, implying that both the formation and interpretation of the gradient are dynamic throughout wing development.

In order to re-examine the formation and interpretation of the Hh gradient, we present a new multi-cellular differential equation model that represents the core logic of the feedback loop. The model is centred on the three primary factors, Hh, Ptc and Hh-Ptc complex; other components of the signaling pathway (such as Smo, Ci and Cos2) are grouped together into a single 'pathway activity' variable. We use this model to investigate the two biological issues discussed above. In particular, we show that making Hh gradient interpretation dependent on Hh-Ptc levels (as well as on Ptc levels) results in a number of beneficial changes, including:

A refinement in signal interpretation, so that cells receiving Hh levels just above / below a threshold are more likely to respond in the appropriate way,

A boost in Ptc production in response to high levels of Hh, strengthening the effect of the feedback loop,

Enhanced robustness of the expression ranges of downstream target genes, in response to variable Hh levels.

We also use the model to examine how the expression ranges of downstream target genes (e.g. Dpp) expand as the wing disc grows. We show that making interpretation dependent on Hh-Ptc levels (as well as on Ptc levels) has a limiting effect on this expansion, indicating that the 'dilution' of Ptc function by Hh-Ptc also plays a role in size regulation.

Evaluation of screening strategies for pre-malignant lesions using a biomathematical approach
Jihyoun Jeon

Program in Biostatistics and Biomathematics, Fred Hutchinson Cancer Research Center, Seattle, USA
Coauthors: 1. Rafael Meza, Division of Mathematical Modeling, UBC Centre for Disease Control, Vancouver, Canada & Program in Biostatistics and Biomathematics, Fred Hutchinson Cancer Research Center, Seattle, USA 2. Suresh H. Moolgavkar, Program in Biostatistics and Biomathematics, Fred Hutchinson Cancer Research Center, Seattle, USA 3. E. Georg Luebeck, Program in Computational Biology, Fred Hutchinson Cancer Research Center, Seattle, USA

We present mathematical expressions for the size distribution of screen-detectable pre-malignant lesions, conditional on no prior detection of cancer in the tissue of interest, based on a general multistage clonal expansion model of carcinogenesis. We apply these expressions to simulate the natural history of colorectal cancer and to evaluate the effect of a screen for adenomatous polyps and concomitant intervention on cancer risk. Our approach allows the efficient simulation of multiple screens and interventions and determination of the optimal timing of the screens. We further demonstrate the utility of our approach by computing the benefits of up to two colonoscopies on the lifetime risk of colorectal cancer. If time permits, I will present some preliminary results of the analysis of a screening trial using our methodology.

Can delays replace Phosphorylation – dephosphorylation cycles in signaling cascades?
Srividhya Jeyaraman

Indiana University School of Informatics and Biocomplexity Institute, Bloomington, IN, USA
Coauthors: M. S. Gopinathan (Indian Institute of Information Technology and Management, Kerala, India) Schnell, Santiago (Indiana University School of Informatics and Biocomplexity Institute)

Many biochemical pathways operate on signaling cascades consisting of a series of phosphorylation–dephosphorylation (PD) cycles. These cycles are coupled with several positive and negative regulations causing inherent delays during signal propagation. Modeling these complex signaling cascades with ordinary differential equations (ODE) often requires a large number of variables and parameters. Simplifying these ODE models with delays could prevent spending time on redundant mechanisms and focus on the key regulators of the dynamics.

Delay differential equation (DDE) models have been helpful in the description of inherent time delays and in the reduction of the number of variables [1]. However the consequences of model reduction via DDEs have not been fully explored. We have systematically examined the effect of delays in a complex network of PD cycles [2], which commonly occur in many biochemical pathways [3]. By introducing delays in the positive and negative regulatory interactions, we show that a delay model can indeed reduce the number of PD cycles and still describe the dynamics of the network effectively. In my presentation I will show the effects of the delays and how the results of this study can be extended to model complex biochemical pathways [4] .

1. Smolen, P., D.A. Baxter, and J.H. Byrne, Biophys. J., 2002. 83(5): p. 2349-2359.
2. Gonze, D. and A. Goldbeter, Journal of Theoretical Biology, 2001. 210(2): p. 167-186.
3. Srividhya, J., M.S. Gopinathan, and S. Schnell, Biophysical Chemistry, 2007. 125(2-3): p. 286-297.
4. Srividhya, J. and M.S. Gopinathan, Journal of Theoretical Biology, 2006. 241(3): p. 617-627.

A mathematical model for the formation of feather germs
Charlotte Jupp

Centre for Mathematical Biology, Mathematical Institute, University of Oxford
Coauthors: Dr. Ruth Baker, Centre for Mathematical Biology, Mathematical Institute, University of Oxford. baker@maths.ox.ac.uk; Prof. Philip Maini, Centre for Mathematical Biology, Mathematical Institute, University of Oxford. maini@maths.ox.ac.uk

In developmental biology, a number of theories, none of which fully explain biological results, have been suggested for the mechanisms involved in the generation of spatial patterns. We develop a cell-chemotaxis model to describe the pattern of forming feather germs in the skin of chicken embryos. In our model the cells are responding to a chemical gradient created by two morphogens which react and diffuse according to a Turing mechanism. The model will be used to analyse the effects of an increase in cell density and morphogen concentration upon the system. We will show that these results match the findings of biological experimentation with regards to the number, size and spacing of feather germs. We shall also demonstrate that our model is capable of replicating both the simultaneous and sequential generation of feather germs as occurs in vitro and in vivo, respectively.

Synchronization in inhibitory networks
Abdoul Kane

University of Toronto
Coauthors: Jonathan O. Dostrovsky, Frances K. Skinner

There is increasing evidence that large scale brain rhythms play an important role in the execution and regulation of cognitive and behavioral functions. Recent studies also highlight the importance of inhibitory subnetworks in rhythm generation. This has motivated the development of many experimental and computational models seeking insights into the mechanisms underlying rhythms and pattern generation. However very few analytical results are available.

We consider a biophysical model describing an inhibitory network of neurons and investigate conditions under which certain specific modes of activity can be observed. By applying techniques from dynamical systems theory we derive a pared-down model that captures the essential features of the model interneuron and also allows an analytical treatment. We then consider a pair of such neurons coupled through GABA-A type synapses and describe how the synaptic time scales interact with the intrinsic dynamics to generate various stable configurations depending on initial conditions and parameters.

Static Representation of Epidemics on Well-characterised Dynamic Networks
Rowland R. Kao

University of Glasgow

Social network representations of populations are becoming an increasingly useful tool when analysing epidemic structures. Most analyses of social networks assume that contacts between nodes (individuals) are fixed, i.e. a static network. In reality, most links between individuals are dynamic, appearing and disappearing over time. Here, I discuss a class of dynamic networks where epidemic properties can be defined by an equivalent "static snapshot" taken over the infectious periods of the individuals in the network, using small-world networks and real datasets where contacts structure and timing are precisely known, as examples.

Mathematical Modeling of the Blood-Atheroma Plaque Interaction
Nader El Khatib

University of Lyon, France
Coauthors: Stephane Genieys genieys@math.univ-lyon1.fr and Vitaly Volpert volpert@math.univ-lyon1.fr

The inflammatory reaction of atherosclerosis leads to the formation of an atheroma plaque in the blood vessel. The interaction between the blood and the plaque may have very dangerous consequences such as the rupture of the plaque liberating solid parts in the blood flow that can lead to a heart attack. The blood-plaque interaction also produces some recirculations downstream of the plaque, and these recirculations can give rise to the coagulation of the blood and the formation of a clot that can block the blood flow too. In this paper we study the interaction between the blood flow and the atheroma plaque using a fluid-structure interaction model. The blood is considered as a non-Newtonian fluid with a variable viscosity defined by the Carreau's law. We investigate the influence of this Non-newtonian variable viscosity on the plaque displacement (and hence the risk of plaque rupture) and on the blood flow recirculations (and hence the risk of blood clot formation). The atheroma plaque is composed of a lipid pool and a fibrous cap and both are considered as hyper elastic materials. The parameters of these materials are taken from experimental data, as well as the parameters of Carreau's law for the blood.

The simulations show that the usual Newtonian models significantly underestimate the recirculations and overestimate the plaque displacement.

Analytical Study of Blood Flow with Periodic Body Acceleration in the Presence of Magnetic Field
Dr Anil Kumar

Mathematics , Dornacharya College of Engineering , Greater Noida UP India
Coauthors: Dr CL Varshney and Veer Pal Singh, Mathematics , SV College Aligarh, UP India

In this paper we consider a human body which is quite often subjected to acceleration under magnetic effects. The induced magnetic field is neglected. Such acceleration can generate significant effects on blood circulation depending on the configuration and geometry of the blood arteries. The current study is concerned with a mathematical model of the study of blood flow in the presence of magnetic effects subject to externally imposed periodic body acceleration. By using the finite Hankel Transform, an exact solution of the steady flow of blood considered as an incompressible, couple stress fluid has been obtained. The results have been compared with other existing models. The effect of body acceleration on blood flow in the presence of magnetic fields is analyzed.

Keywords: Magnetic Field; Couple Stress; Blood Flow; Periodic Body Acceleration; Hypertension.

Tubuloglomerular Feedback Signal Transduction In a Model of a Compliant Thick Ascending Limb
Anita T. Layton

Duke University, Department of Mathematics

We used a mathematical model to predict the impact of tubular compliance on tubuloglomerular feedback (TGF) signal transduction in the thick ascending limb (TAL). In several previous studies, we used a mathematical model that represented the TAL as a rigid tube. That model predicts that TGF signal transduction by the TAL is a generator of nonlinearities: if a sinusoidal oscillation is added to constant intratubular flow, the NaCl concentration alongside the macula densa will be nonsinusoidal owing to an accumulation of harmonics. We have hypothesized that complexity found in power spectra based on in vivo time series of key TGF variables arises in part from those harmonics and that nonlinearities in TGF-mediated oscillations may result in increased NaCl delivery to the distal nephron. To address the concern that a more realistic TAL would damp harmonics, we have conducted simulations in a model TAL that has compliant walls and thus a tubular radius that depends on transmural pressure. These simulations predict that compliant TAL walls do not damp, but instead, intensify the harmonics.

Reduction methods for multiple-time-scale biochemical reaction networks
Chang Hyeong Lee

Worcester Polytechnic Institute

In this talk, we consider deterministic and stochastic descriptions of reaction networks in which different reactions occur on at least two distinct time scales. In the deterministic description, based on perturbation analysis and complex reaction network theory, we derive a necessary and sufficient condition under which there is a complete separation of slow and fast variables, and we discuss network topological properties which guarantee that the condition is satisfied. Given this condition, we obtain an explicit expression for the reduced equation on the slow time scale and we clarify the geometric meaning of the reduction. In the stochastic description, by applying a state space decomposition method, we rigorously obtain the reduced master equation from the full master equation of the system, which enables us to implement an efficient simulation algorithm. Lastly, we illustrate the numerical accuracy and efficiency of the reduction method by simulating several multiple-time-scale deterministic and stochastic models including a stochastic reaction-diffusion model of gene expression.

Rare event simulation for T cells recognising foreign antigens
Florian Lipsmeier

Bielefeld University
Coauthors: Ellen Baake (Bielefeld University)

We reconsider the problem of foreign-self distinction in immunobiology, namely, the discrimination of foreign antigens against a background of the body's own molecules. As is well known, the precise mechanism, though one of the major tasks of the immune system, continues to be a fundamental puzzle. We reconsider it here as a problem of statistical recognition as recently formulated by van den Berg, Rand and Burroughs [1], who modelled the probabilistic nature of the interaction between T cells and antigen-presenting cells (APC's). Here, the stochasticity is due to the random sample of antigens present on the surface of every APC, and to the random receptor type that characterises individual T cells. It has been shown previously [1, 2] that this model, though highly idealised, is capable of reproducing important aspects of the recognition phenomenon, and of explaining them on the basis of stochastic rare events. The `rare events' come into play here because the probability that a randomly chosen T cell will be activated by a randomly chosen APC is very low, whether the APC carries foreign antigens or not. It is therefore adequate to use large deviations theory, which characterises tail events. However, the results obtained so far are asymptotic in nature; simulations have been restricted to the straightforward simple sampling approach, which does not allow for sample sizes large enough to address more detailed questions. Building on the available large deviation results, we develop an importance sampling technique here that allows for a convenient exploration of the relevant tail events by means of simulation [3]. With its help, we investigate the mechanism of statistical recognition in some depth. In particular, we illustrate how a foreign antigen can `stand out' against the self background if it is present in sufficiently many copies, although no a priori difference between self and nonself is built into the model. This method will also allow to tackle models that are more realistic than the basic caricature considered so far.

[1] Van Den Berg, H.A., Rand, D.A., Burroughs, N.J.: A reliable and safe T cell repertoire based on low-affinity T cell receptors. J Theor Biol 209(4), 465-486 (2001)

[2] Zint, N., Baake, E., den Hollander, F.: How T-cells use large deviations to recognize foreign antigens. J Math Biol., in press

[3] Lipsmeier, F. , Baake, E. Rare event simulation for T cells, in preparation

Spatially-localized scaffold proteins may simultaneously boost and supress signaling
Xinfeng Liu

University of California at Irvine
Coauthors: Bardwell Lee and Qing Nie

During cell signaling, scaffold proteins are thought to promote both signal transmission and specificity by binding to multiple componnents of a given pathway, but the mechanisms by which they accomplish this are unclear. In this talk, we develop a mathematical model of generic, spatially localized scaffold protein. The model indicates that a scaffold protein could boost signaling locally (i.e. in and near the region where it was localized) while simultaneously supressing at a distance. Furthermore, localization could switch a scaffold from a global inhibitor to a local enhancer to distant supressor. Distant supression was found to be due to reactant sequestration. Thus, spatial localization increases the versatility of scaffold proteins, and creates a novel mechanism by which they can augment signaling specificity.

Elucidating the HPA axis stress response via computational inverse analysis
James Lu

Johann Radon Institute for Computational and Applied Mathematics (RICAM)
Coauthors: Clemens Zarzer, RICAM; Rainer Machne, Theoretical Biochemistry Group, University of Vienna; Gottfried Koehler, Max F. Perutz Laboratories, University of Vienna

The hypothalamic pituitary adrenal (HPA) axis represents a feedback system that plays an important role in maintaining the body homeostasis in response to various stresses. When stress is encountered, the hypothalamus releases the corticotropin releasing hormone (CRH) as a central neuro-transmitter in the HPA axis. There exist diverse differential equation models, which account for induction of ACTH synthesis in the pituitary by CRH, leading to adrenal activation and release of cortisol, which in turn inhibits the synthesis of ACTH.

Starting from these basic models, several additional feedback mechanisms could be included. One is the incorporation of an additional membrane bound glucocorticoid receptor (GR) in the inhibition of the ACTH release in the pituitary, responsible for fast feedback effects. Including such model extensions, computational inverse analysis is crucial in identifying the possible dynamical behaviors, such as oscillations,modulated by circadian rhythms and switching between multiple steady states.

To identify factors controlling the qualitative nature of the stress responses, we apply the method of inverse bifurcation analysis, using a hierarchical identification strategy based upon a sparse-promoting regularization method. In particular, diseased phenotypes as represented mathematically by the respective bifurcation diagrams are computationally mapped to the underlying regulation mechanisms. For instance, the identified mechanisms underlying the delayed activation of the stress response include the degradation rate of GR as well as the rate of up-regulation in the GR synthesis via its dimer. In addition to mapping diseased phenotypes to possible underlying mechanisms, inverse analysis can also point to mechanistic details of the model that should be elucidated via experimental studies.

Noise-based rules govern neural circuit assembly
Victor Luria

Columbia University, Department of Genetics and Development

Sensory-motor circuits are assembled by neurons whose axons execute discrete, binary decisions at sequential trajectory selection points. Motor axon trajectory selection is controlled by guidance cues composed of ligands and receptors whose expression levels are variable. Some cues direct axons to opposite trajectories. Genetic inactivation of cues results in inaccurate trajectories that are also variable, suggesting genetic variability is translated into phenotypic variability. Quantitative modeling shows the total number of cues is limited by expression noise and energetic cost constraints. I propose this model applies to trajectory choices and generally to discrete decisions controlled by noisy and competing cues.

Existence and uniqueness of the total quasi-steady-state approximation for coupled systems of enzyme kinetics
Shev MacNamara

The Institute for Molecular Biosciences, The University Of Queensland, Australia
Coauthors: Alberto M. Bersani, Kevin Burrage, Roger B. Sidje

The total quasi-steady-state approximation (tQSSA) is obtained merely by introducing a simple change of variable into the conventional QSSA and has the benefit of being valid over a wider parameter range. The original work on the tQSSA (Segel et al., Bulletin of Mathematical Biology, 1996) demonstrated this for the quintessential example of Michaelis-Menten enzyme kinetics and recently interest has focused on being able to generalize the approach to more complicated networks of coupled enzymatic reactions (Ciliberto et al., PLoS Computational Biology, 2007). In special cases explicit formulae for the approximation may be derived but for more complicated systems these are not available. We provide a theorem guaranteeing the existence and uniqueness of the approximation for networks of coupled enzymatic reactions, as well as an accompanying numerical method. These results are applied to the Goldbeter-Koshland switch and the mitogen-activated-protein kinase cascade. One novel aspect of this work is the application to the chemical master equation to understand the dynamics of discrete and stochastic biochemical kinetics. Previously this has been felt infeasible because of difficulties involved with the computation of the exponential of a matrix of very high dimension but by using Krylov methods and the extra structure present in the reactions arising in enzymatic networks we show that significant progress can be made.

Parameter sensitivity investigation of a mathematical model of glioma tumorigenesis mediated by platelet-derived growth factor
Susan Christine Massey

University of Washington.edu
Coauthors: Peter Canoll, MD, PhD; Kristin R. Swanson, PhD

Gliomas are the most prevalent form of primary brain tumor in adults. Despite all possible treatment attempts, including aggressive surgical resection, these tumors are uniformly fatal. Dr. Peter Canoll at Columbia University has demonstrated that rats develop brain tumors closely resembling human gliomas when their glial progenitor cells are injected with a retrovirus expressing platelet-derived growth factor (PDGF). Most notably, at 17 days post infection only 30infected progenitor cells—the other 70progenitor cells, presumably recruited to the tumor by interactions with PDGF. Using the empirical data collected by his lab, we have developed a mathematical model to describe the observed tumor growth in this rat experiment. We used a sensitivity analysis technique incorporating latin hypercube sampling (LHS) and partial rank correlation coefficients (PRCC) to vary parameters against each other and determine which parameters in the model are most influential upon the ratio of uninfected progenitor cells to total (infected and uninfected progenitors) in the tumor at day 17. Our investigation revealed that the two most influential parameters affecting the observed tumor growth pattern are the max proliferation rate of infected progenitors (and thus the amount of extra cellular PDGF available) and the max rate of consumption of PDGF by nearby uninfected progenitors. Specifically, even when controlling for variability in the other unknown model parameters, an increase in the max proliferation rate of the infected cell population results in an increase in the percentage of the cells at the core of the tumor that are recruited (rather than infected). This may suggest that more aggressive (highly proliferative) gliomas would have the most recruited cells within the tumor and may benefit most from PDGF targeted therapies (e.g., Gleevec). This is a novel insight that may help in patient selection for such targeted therapies. We are exploring the potential impact of PDGF targeted therapies on tumors with differing affinities for PDGF within the context of the current model and parallel experimentation. Our initial results suggest that this model may lead to a better understanding of what drugs may help glioma patients, by quantifying the relative importance of PDGF. Future work will also focus more on analyzing the model to look for PDGF influenced spatial migration patterns of the respective cells in the growing tumor to compare with observational studies tracking the migration of individual cells over several hours.

Estimation of scaling index on space records of cell proliferation in the developing central nervous system
Jorge Mazzeo

Institute of Biomedical Engineering, Buenos Aires University and Interdisciplinary Group in Theoretical Biology, Favaloro University, Argentina
Coauthors: Melina Rapacioli (Interdisciplinary Group in Theoretical Biology, Favaloro University, Argentina) Santiagoo Duarte (IBCYN, School of Medicine, Buenos Aires University, Argentina) Carlos D’Attellis (Interdisciplinary Group in Theoretical Biology, Favaloro University, Argentina) Vladimir Flores (Interdisciplinary Group in Theoretical Biology, Favaloro University and IBCYN, School of Medicine, Buenos Aires University, Argentina) vflores@favaloro.edu.ar

The dynamics of neuroepithelial cell proliferation in the chicken tectum opticum is analyzed using a model within the framework provided by the theory of stochastic point processes. Spatial signals of cell proliferation consisting of numerical sequences of intermitotic intervals were recorded under microscopic observation. The main goal of this work is to determine the possible existence of some kind of correlation or dependency between proliferating cells. The central hypothesis is that, if proliferating cells behave interactively, such interactions should impart some kind of dependency or memory on the signals representing the spatial organization of the proliferative activity. Additionally, appropriate methods of signal analysis should provide information about the spatial range of such interactions.

The analyses were performed by means of standardized algorithms designed to characterize the dynamics of numerical sequences by computing the scaling index of the stochastic processes. Among these methods, the Hurst index (one of the earliest proposed), the Detrended Fluctuation Analysis, the Fano Factor, the Power Spectral Density and the Dispersional Analysis were applied in this study.

Many authors [1],[2],[3] have pointed out that some of these methods are sensitive to non-stationarities. To overcome these pitfalls, elimination (remotion) of global trends is recommended [4] to transform the signal into a stationary sequence.

Here we analyze inaccuracies in scaling index estimation that, paradoxically, appear because of global trends removal.
This study was realized on artificial series, synthesized with defined statistical parameters, and also on spatial records of cell proliferation. We propose some methodological guidelines which are relevant to our specific field of research.

[1] Kun Hu, Plamen Ch. Ivanov1, Zhi Chen, Pedro Carpena, H. Eugene Stanley, “Effects of Trends on Detrended Fluctuation Analysis”, arXiv:physics/0103018 v4 14 May 2001
[2] M. Ignaccolo, P. Allegrini, P. Grigolini, P. Hamilton, B. J. West, “Scaling in Non-stationary time series I”, arXiv:physics/0301057 v1 22 Jan 2003
[3] Zhi Chen, Plamen Ch. Ivanov, Kun Hu, H. Eugene Stanley, “Effect of Nonstationarities on Detrended Fuctuation Analysis”, arXiv:physics/0111103 v2 15 Apr 2002
[4] Trang Dinh Dang and Sándor Molnár, “On the Effects of Non-Stationarity in Long-Range Dependence Tests”, El. Eng. Vol. 43, No. 4, Pp. 227–250, 1999

Modeling Natural Killer Cell Development and Repertoire Formation
Ramit Mehr

The Mina and Everard Goodman Faculty of Life Sciences, Bar-Ilan University, Ramat-Gan, Israel
Coauthors: Mali Salmon-Divon, Sofia Johansson, Maria Johansson, Yishai Pickman, Marjet Elemans, Petter Höglund, and Ramit Mehr

NK cells are able to recognize and reject cells lacking expression of self- MHC class I molecules. Inhibition of lysis is mediated by inhibitory receptors expressed by NK cells, such as the murine Ly49 receptors, which bind to MHC class I molecules. NK cells adapt to the self-MHC environment by a process ensuring that each cell expresses at least one self-specific inhibitory receptor but not too many. Two models have been proposed to account for the development of a useful Ly49 repertoire. The two-step selection model proposes a stochastic initial receptor expression combined with selection of cells expressing appropriate receptor compositions. The sequential model proposes that NK cells sequentially express Ly49 receptors and continue to do so until an interaction of sufficient magnitude between a Ly49 receptor and self-class I MHC occurs. These two models predict different repertoire compositions under various conditions. The complexity of experimental observations on NK cell repertoire development necessitates the application of theoretical techniques in order to elucidate the principles underlying this development and evaluate the proposed models. We conducted mathematical modeling and computer simulation studies of each NK cell education model, fitting them to published (Salmon-Divon et al, 2003a, b; 2004) and newly generated (Salmon-Divon et al, in preparation) experimental data. Our results favor the two-step selection model over the sequential model, and raise several questions, which will be addressed in our future studies combining experiments, mathematical modeling and computer simulations.

Antigen-Driven Selection In Germinal Centers As Reflected By The Shape Characteristics Of Immunoglobulin Gene Lineage Trees: A Large-Scale Simulation Study
Ramit Mehr

The Mina and Everard Goodman Faculty of Life Sciences, Bar-Ilan University, Ramat-Gan, Israel
Coauthors: Gitit Shahaf, Michal Barak, Neta S. Zuckerman, Naamah Swerdlin, Malka Gorfine and Ramit Mehr

Lineage trees of somatically hypermutated immunoglobulin (Ig) genes from B lymphocytes often serve to qualitatively illustrate claims concerning the dynamics of affinity maturation in germinal centers (GC). Using a novel method for graphical quantification of lineage tree properties, we have in past studies demonstrated that lineage tree analysis detects fine differences in Ig gene intraclonal diversity between B cell clones generated under different conditions. We found age- and tissue-related differences in the dynamics of the normal humoral immune response in humans, unique features of Ig gene diversification in B cell malignancies and autoimmune responses, and B cell diversification in other species which utilize gene conversion rather than rearrangement as the main primary diversification mechanism.

In order to test quantitative claims regarding the GC response and affinity maturation, we created a computer simulation which combines mathematical models for all mature B cell populations, stochastic models of hypermutation and selection, and lineage tree generation and measurement. We ran this program many times varying the values of dynamical parameters (such as the proliferation, differentiation and mutation rates, initial affinity of the Ig to the antigen, and selection thresholds), creating almost a million simulated lineage trees. We analyzed the data in order to identify the ranges of dynamic parameters that yield biologically correct results based on experimental data regarding germinal center responses, obtaining interesting insights regarding response dynamics. We found statistically significant correlations between quite a few tree characteristics and the initial affinity and selection threshold, which seem to be the main parameters that affect lineage tree shapes, in both primary and secondary response trees. We found that GC cells may be divided into a subset possessing low values of selection threshold and mutation rate, and a second, small subset with high values of these parameters. The results also confirmed that recycling from centrocytes back to centroblasts is highly likely. Finally, analysis of correlations between tree properties removed redundant properties, improving the statistical power of this method.

Wavetrain selection following predator invasions in oscillatory reaction-diffusion systems.
Sandra Merchant

University of British Columbia
Coauthors: Wayne Nagata (University of British Columbia)

Periodic travelling waves, also known as wavetrains, are known to evolve behind invasion fronts in oscillatory reaction-diffusion models for predator-prey systems. Mathematical theory predicts that for a given set of parameter values there is in fact a family of possible wavetrain solutions and in a particular predator invasion a single member of this family is somehow selected. Sherratt (1998) has studied this selection mechanism, using the Normal Form approximation that is valid for such models near the Hopf bifurcation in the local system. However, away from this Hopf bifurcation the predictions from the Normal Form lose accuracy. We conjecture a more general selection criterion that retrieves the prediction from the Normal Form system, but that applies to the full (non-reduced) predator-prey system and that depends on the properties of the wavetrains for the full system and hence retains accuracy away from the Hopf bifurcation. We illustrate how to apply this selection criterion using three sample oscillatory reaction-diffusion models from the literature on predator invasions. The selection criterion does indeed provide more accurate predictions for these models than the criterion based on the Normal Form, but does eventually lose accuracy as well. We therefore conclude with future directions for work on this problem.

Neutral Stability Manifolds for Reaction-Diffusion Systems
R. P. Mondaini

Federal University of Rio de Janeiro
Coauthors: Mariano Rodriguez-Ricard (Havana University)

(abstract to be included shortly)

Model identification from noisy data: solving ill-posed inverse problems using regularization
Stefan Mueller

Radon Institute for Computational and Applied Mathematics, Austrian Academy of Sciences
Coauthors: James Lu (Radon Institute for Computational and Applied Mathematics, Austrian Academy of Science) Rainer Machne (Theoretical Biochemistry Group, University of Vienna) Lukas Endler (Theoretical Biochemistry Group, University of Vienna)

The quality of a mathematical model for a biological system depends - aside from its explanatory value - on its consistency with the data available. From a data-driven viewpoint, modeling is an "inverse problem": given a certain class of models, one tries to identify unknown parameters or even functions which give rise to the observed data or a desired qualitative dynamics. In the presence of data noise, however, model identification is an ill-posed inverse problem in the sense that its solution lacks stability properties: a small amount of data noise can be considerably amplified and may lead to unreliable solutions. To overcome this problem, we suggest the use of so-called regularization methods.
One of the systems we study is an ODE model of a metabolic pathway, which has been used as a benchmark problem for parameter identification. The ODE model contains 36 parameters all of which are identified from noisy data. Using simple least squares minimization (without regularization), a few percent of data noise leads to more than 100% relative error in some of the identified parameters, thus highlighting the ill-posedness of the inverse problem. Using regularization, the relative parameter error is comparable with the data noise. More specifically, we use Tikhonov regularization to counter the instability of the problem. By choosing the regularization parameter appropriately (based on the knowledge of the data noise), the model parameters can be identified in a stable and accurate manner.


A Computational Framework for Simulating Multiphase Models of Tissue Growth.
Mr James Osborne

Oxford University Computing Laboratory, Oxford, OX1 3QD, UK
Coauthors: Helen M Byrne: Centre for Mathematical Medicine, Division of Theoretical Mechanics, School of Mathematical Sciences, University of Nottingham, University Park, Nottingham, NG7 2RD, UK. email: Helen.Byrne@maths.nottingham.ac.uk Sarah Waters: Oxford Center for Industrial & Applied Mathematics, Mathematical Institute, 24-29 St. Giles', Oxford, OX1 3LB, UK, email: waters@maths.ox.ac.uk Jonathan Whiteley: Oxford University Computing Laboratory, Oxford University, Oxford, OX1 3QD, UK, email: Jonathan.Whiteley@comlab.ox.ac.uk

Multiphase modelling is a natural framework for studying many biological systems, for example tissue engineering and cancer development, where different phases represent the constituents of the tissue of interest (e.g. extracellular matrix, cancer cells and interstitial fluid when studying solid tumour growth). The resulting models comprise non-standard mixed systems of nonlinear PDEs. For example, multiphase models used to describe tissue engineering applications and solid tumour growth may generate equations that consist of: (i) viscous fluid flow equations for each phase: (ii) hyperbolic PDEs for mass conservation; and (iii) elliptic or parabolic PDEs for chemical concentrations. Analytical progress with such systems is usually only possible if additional model assumptions are made (e.g. radial symmetry, or small aspect ratio). A complementary approach is to seek a numerical solution of the governing equations without making any such simplifications. The numerical solution of these equations presents numerous challenges: the numerical methods for solving fluid flow equations and hyperbolic PDEs are notoriously prone to complications such as instability and computational time. Further complexity may be introduced if the problem is posed on a growing domain (e.g. a growing tumour). Advanced numerical algorithms are required in order to guarantee an accurate and efficient solution.

We have developed a numerical and computational framework based upon the Galerkin Finite Element Method that allows the numerical solution of coupled systems of parabolic, elliptic and hyperbolic PDEs described above in two or three dimensions. This enables us to investigate the effect of interactions between constitutive phases. We have used this framework to investigate tissue growth in a bioreactor and also the development of a solid tumour, under non-uniform environmental conditions. We illustrate the versatility of our numerical method by presenting results for these two case studies.

Sensitive dependence of fixation probability on life history: The lytic virus case.
Zaheerabbas Patwa

University of Western Ontario
Coauthors: Dr. Lindi Wahl (University of Western Ontario)

The fixation probability of a beneficial mutation is the probability with which the allele takes over the entire population. This probability is extremely sensitive to assumptions regarding the organism's life history. We compute the fixation probability using a life-history model for lytic viruses. The model assumes exponentially distributed attachment times and a constant time between attachment and host cell lysis (lysis time). We derive a partial differential equation, including a delay term, which describes the time evolution of the probability generating function (p.g.f.) for the number of individuals in the mutant lineage. By finding the fixed point of this p.g.f., we compute the fixation probability for mutations that increase attachment rate, increase burst size, decrease the lysis time or reduce the probability of clearance. These four mechanisms of mutation give widely varying fixation probabilities. It was found that in all cases, the fixation probability of beneficial mutations was sensitive to the time between population bottlenecks.


Optimal movement and motor capacity in a short-range aquatic predator
Claire Postlethwaite

University of Houston
Coauthors: Tiffany M. Psemeneki, Jangir Selimkhanov, Mary Silber, Malcolm A. MacIver

The black ghost knifefish `Apteronotus albifrons' makes fast reactive strikes at prey that it detects with its weak electric field and electrosensory system. Prey are detected at short range throughout an omnidirectional sensory volume around the body. Following detection, the body is rapidly reoriented to bring the mouth to the position of the prey. Because of the short range of detection and need for rapid movements in many directions, constraints due to the dynamics and mechanics of fish swimming are likely to play a significant role in determining the behaviour. Here we examine mechanically optimal trajectories for an idealised fish body moving in an inviscid fluid, and compare these with measured prey capture trajectories. We present evidence that the measured fish trajectories are close to those of the idealised fish moving to minimise effort. This is particularly notable given that the idealised fish is able to move with all possible linear and rotational degrees of freedom while the real fish is not able to do so.

Mathematical Modeling of Brain Tumor and Related Therapeutic Strategies
Gibin Powathil

University of Waterloo, Waterloo, Ontario,Canada
Coauthors: M Kohandel, S Sivaloganathan

Gliomas are the most common and aggressive primary brain tumors. The most common treatment protocols for these brain tumors are combinations of surgery, chemotherapy and radiotherapy. However, even with the most aggressive combination of surgery and radiotherapy and/or chemotherapy schedules, gliomas almost always recur resulting in a median survival time for patients of not more than 12 months. This highly diffusive and invasive nature of brain tumors makes it very important to study the effects of these combined therapeutic strategies in an effort to improve the survival time of patients. It is also important to study the tumor micro- environment, since the complex nature of the cerebral vasculature, including the blood brain barrier and several other tumors induced conditions such as hypoxia, high interstitial pressure, and cerebral edema affect drug delivery as well as the effectiveness of radiotherapy. Recently, a novel strategy using antiangiogenic therapy has been studied for the treatment of brain tumors. Antiangiogenic therapy interferes with the development of tumor vasculature and indirectly helps in the control of tumor growth. Recent clinical trials suggest that anti-angiogenic therapy is usually more effective when given in combination with other therapeutic strategies. In an effort to study the effects of the above mentioned therapeutic strategies, we consider a simple spatio-temporal model that incorporates the tumor cell growth and the effects of radiotherapy and chemotherapy. We study the effects of different schedules of radiation therapy, using a generalized linear quadratic model, and compare the results with published clinical data. The model is then extended to include the interactions of tumor vasculature and oxygen concentration, in an effort to explain tumor hypoxia, cerebral edema and high interstitial pressure (as well as their changes during antiangiogenic therapy). We also discuss the optimum way of sequencing these therapeutic strategies so as to maximize patient survival time.

Evolution of asymmetric division: an in-silico model
Armin Rashidi

Institute for Ageing and Health, Newcastle University, UK
Coauthors: Daryl Shanley

Symmetric reproduction precludes ageing; all individuals would be affected by any deterioration and the lineage would vanish. Segregation of damaged macromolecules, by asymmetric division, to one progeny cell results in an ageing parent and a rejuvenated daughter. Asymmetric reproduction is also a precursor for germ-soma specialization, a prerequisite for the evolution of multicellularity. However, surprisingly little work has been done on the evolution of asymmetry. Using in-silico experiments, we here determine circumstances under which selection forces drive the evolution of asymmetry. Assumptions used in the model are: (i) Limited resource availability creates a trade-off between investment in growth/reproduction and in maintenance/repair. (ii) The population is near its carrying capacity and the optimum investment strategy has already evolved. (iii) The rate of damage accumulation (r) is inversely related to the level of investment in maintenance/repair (m). (iv) The doubling time (T) is shortened by larger reproductive investments (b). (v) The likelihood of survival up to a certain time decreases with the time-integrated damage up to that time. (vi) The mode of damage distribution at division is an evolvable trait. The damage segregation coefficient, s, was allowed to evolve between zero (full symmetry) and one (full asymmetry), and was averaged over the population at any given time. The outcome of each run was defined as (a) asymmetry: s > 0.8 (more than 90% of damage segregating to one progeny cell) for at least 50% of the monitoring period or (b) symmetry: s > 0.8 for less than 25% of the monitoring period. Two stochastic parameters (random mutations and the likelihood of survival to next division) and three fundamental constants (T, C1, C2) are potential determinants of the dynamics of the system. C1 determines the shape of the relationship between r and T (and also the one between m and b if r and T are assumed linear functions of m and b, respectively), and C2 is the level of time-integrated damage above which the chance of survival is negligible. For a given T, the selection pressure for asymmetry is lower for more concave m/b trade-offs and and larger C2 values. Asymmetry evolves if damage-related mortality makes survival to the age at reproduction sufficiently unlikely. Of particular note, convex m/b trade-offs promote the evolution of asymmetry. These results have important implications to the evolution of multicellularity, ageing, and division of labour.


Three dimensional simulation of glioma growth and response to radiation therapy: a case study
Russell Rockne

University of Washington Department of Pathology
Coauthors: Dr. Kristin R. Swanson, Julia L. Moore

Gliomas are human brain tumors that are unique in their extensive invasion of the surrounding tissue without metastasis outside the brain. Using tumor volume data collected from a human glioma patient, we investigate a reaction diffusion model for both untreated glioma growth (Swanson 1999, Swanson et al 2002), and response to radiation therapy (Swanson et al 2007, Rockne et al 2008) on the brainweb 3D virtual brain. Three dimensional spatial analysis of model predicted disease distribution and that observed on magnetic resonance imaging reveals a mean point-wise error of 0.51mm, within measurement error. Using the linear-quadratic model for radiation efficacy, in conjunction with [18F]-Flouromisonidazole positron emission tomography, we observe a radiation resistance effect in regions of the tumor with increased levels of hypoxia and demonstrate a 3 fold oxygen enhancement ratio for hypoxic cell radio-resistance. Heterogeneity in radiation dose per cell reveal regions of insufficient dose and suggests avenues for treatment field modification.


A Mathematical Model of the Human Sleep/Wake Cycle
Lisa Rogers

Rensselaer Polytechnic Institute
Coauthors: Mark Holmes (Rensselaer Polytechnic Institute)

Most living organisms exhibit daily biological rhythms. One of the most curious and difficult to understand is the sleep/wake cycle. Although there is little discussion about the importance of the wake phase, the reasons for sleep are not well understood. In this talk a model of the human sleep/wake cycle will be presented that consists of a system of nonlinear integro-differential equations that describe the behavior of inhibitory and excitatory neurons contributing to sleep and wake. Specifically, the model is based on recent experimental studies identifying the brain circuitry and neurotransmitters that regulate the sleep and wake states. We utilize inherent properties of the REM/NREM cycle as well as stability theory and computational methods to analyze the system. A correlation is shown between the model’s output and experimental data taken from sleep studies. The consequences of qualitative and quantitative analysis are utilized to shed light on the various physiological implications that arise from the mathematical structure of the human sleep and wake processes.

Network development in biological gels: Role in lymphatic vessel development.
Dr. Tiina Roose

OCIAM and CMB, Mathematical Institute, University of Oxford

Even though the existence of lymphatic vessels has been known since the 17th century, until very recently not very much was known about their functioning and development. This was due to a failure to understand their importance in the proper functioning of tissues. However, in last the 10 years lymphatics have come to the forefront of biomedical research, largely due to findings highlighting their importance to cancer growth and metastasis Stacker et al. (2002). Thus, there are now a large number of experimental studies on the molecular and micromechanical factors that control lymphatic function and development.

We present a model that explains the prepatterning of lymphatic vessel morphology during development. This model is derived using the theory of two phase rubber material due to Flory and co-workers and it consists of two coupled fourth order partial differential equations describing the evolution of the collagen volume fraction, and the evolution of the proton concentration in a collagen implant; as described in experiments of Boardmand&Swartz (2003). Using linear stability analysis we find that, above a critical level of proton concentration, spatial patterns form due to small perturbations in the initially uniform steady state. Using a long wavelength reduction we can reduce the two coupled partial differential equations to one fourth order equation that is very similar to the Cahn-Hilliard equation; however, it has more complex non-linearities and degeneracies. We present the results of numerical simulations and discuss the biological implications of our model.


A Coalescent Theory Analysis of the Population Structure Statistic Fst
Sivan Rottenstreich

Georgetown University

Populations are often divided into subpopulations. Fst is a statistic used to experimentally test for population subdivision. We consider a stochastic model of evolution for subdivided populations. We analyze Fst under different scaling limits for parameters of this model. We show that the distribution of Fst depends on mutation rate and we characterize the distribution in the different scaling limits. Our results are the first to describe the distribution of Fst and demonstrate its dependence on mutation rate. We use a coalescent theory approach to derive our results.


A tunable biochemical band-pass filter robust to internal noise
Marc R. Roussel

Department of Chemistry and Biochemistry, University of Lethbridge

Some biochemical systems respond differently to the same signal presented at different frequencies, implying a signal filtering capability. Calcium signals, for instance, can be interpreted differently in a cell depending on their temporal characteristics. The ingredients for a minimal, tunable band-pass filter turn out to be remarkably simple: A competitively inhibited enzyme operating in the slow, tight-binding inhibition regime is able to accomplish this function. The central frequency of the filter can be tuned by varying the concentration of inhibitor. Since biochemical components are frequently present in very small numbers, we examine the effect of decreasing the volume of the system holding the concentrations constant, which corresponds to increasing the level of internal noise. Remarkably, the band-pass filtering property is retained, even at very small extensivities corresponding to a few dozen enzyme molecules.

Clonal expansion determines CD8+ T cell phenotype in vivo.
Timothy Schlub

University of New South Wales, Australia
Coauthors: Vanessa Venturi (University of New South Wales, Australia) Katherine Kedzierska (University of Melbourne, Australia) Cameron Wellard (The Walter and Eliza Hall Institute, Australia) Peter Doherty (University of Melbourne, Australia) Stephen Turner (University of Melbourne, Australia) Ruy Ribeiro (Los Alamos National Laboratory, USA) Philip Hodgkin (The Walter and Eliza Hall Institute, Australia) Miles Davenport (University of New South Wales, Australia)

The CD8+ “killer” T cell response to infection involves extensive T cell division and differentiation. Expression of the adhesion molecule CD62L is high on naïve cells that have not seen virus, and rapidly down regulated on the surface of the majority ( ~ 90%) of cells present in the ‘effector’ phase of acute infection. Various models have been proposed to explain the progression of the cellular differentiation of this system. We demonstrate that the extent of CD62L down-regulation is positively correlated with clone size in vivo, suggesting that the number of divisions a T cell has undergone may determine its levels of CD62L expression (phenotype). We develop a mathematical model of division-linked CD62L differentiation that reproduces the experimental population kinetics and phenotype during the acute infection. The model is subsequently used to simulate a heterogeneous clonal population responding to influenza virus infection and generating a repertoire of responding clonotypes (T cell receptor sequences that enable recognition of viral peptides). The model demonstrates that many of the features of the CD62Lhi and CD62Llo T cell receptor repertoire observed in vivo can be explained with a simple mechanism of ‘division-linked differentiation’. We further demonstrate that division-linked CD62L differentiation adequately describes our experimental kinetic and repertoire data. Moreover, this is robust in terms of the mathematical description used for cell division, or the parameters used to distinguish different clonotypes.

A Feline Leukemia Virus Model with Potential Vector
Matthew Schuette

William Jewell College

Many microparasitic diseases are spread horizontally, i.e. person-to-person or animal-to-animal. Certain diseases, such as malaria, are spread by means of a vector, e.g. mosquito.

Feline leukemia is a retroviral disease among cats, caused by feline leukemia virus (FeLV). Virus is shed in very high quantities in saliva and nasal secretions, but also in urine, feces, and milk from infected cats. Cat-to-cat transfer of FeLV may occur from a bite wound as well as during mutual grooming, and possibly through the shared use of litter boxes and feeding dishes. Until recently, no vectors had been reported as possible sources of transmission. In 2003, Vobis, et al. published a paper suggesting the cat flea (Ctenocephalides felis) as a potential vector for this disease.

For this talk, we investigate a deterministic model of FeLV transmission that includes the spread of the virus by the cat flea. We will focus primarily on the development of the model and some of the analytical results.

Population Behaviour towards Voluntary BCG Vaccination Policies
Schehrazad Selmane

University of Sciences and Technology of Algiers

Mathematical models that take into account the interplay between human behaviour, economics, and disease ecology may be more useful in understanding the dynamics and spread of diseases than models that rely on epidemiology alone. The transmission and control of infectious disease is strongly influenced by how people make choices, both individually and collectively, when presented with opportunities to engage in preventive actions or to use preventive health care services. The effectiveness of the TB vaccine in preventing TB is controversial; studies have shown variable efficacy. Game theory is used to analyze population behaviour towards voluntary BCG vaccination policies. Such an approach allows quantifying how risk perception influences expected vaccine uptake and coverage levels. The threshold in perceived relative risk is computed from the basic reproduction, computed from a simplified mathematical model for the dynamics of tuberculosis with vaccination.

Diversity and shape of peripheral effector and regulatory T cell repertoires: validating the Crossregulation model with experimental data
Nuno Sepúlveda

Instituto Gulbenkian de Ciência
Coauthors: Jorge Carneiro, Instituto Gulbenkian de Ciência

A healthy immune system involves a fine balance between effector T cells (Teffs) that mount immune responses, and regulatory T cells (Tregs) that suppress them. When this balance is perturbed, immunopathologies arise. Understanding this balance requires to know how diverse are the repertoires of Teffs and Tregs and how they relate to each other. A too large intersection between the repertoires could lead to deleterious inhibition of specific immune responses against harmfull microorganisms, while a too small overlap may open the way to autoimmune responses. Here we address this issue by a Crossregulation model (Immunol. Rev. 216:48-68, 2007) that describes the peripheral dynamics of a large number of clones with both Tregs and Teffs competing for antigen-presenting cells. The model produces different, but testable, predictions for the shape and diversity of peripheral Teff and Treg repertoires: (1) a higher diversity of Teffs than of Tregs, (2) a Lognormal distribution for the clonal size distributions, and (3) a negligible correlation between clonal size distributions. Here we confront these predictions with available experimental data.

A "Vuggy" Medium Approach to Fluid and Drug Transport in Tumours
Rebecca Shipley

Mathematical Institute, University of Oxford
Coauthors: Prof. S. J. Chapman

The neoplastic vasculature of solid tumours comprises a network of capillaries with highly permeable walls. Understanding the flow of blood through this vasculature and the surrounding porous interstitium of the tumour is important for two main reasons. Firstly, it is a key ingredient when predicting the oxygen distribution within the tumour, which is crucial for predicting the micro-environment and growth rate of cancer cells. Secondly, it is vital for predicting the treatment of cancer by therapeutic drugs administered intraveneously. For example, the success of treatment by chemotherapy drugs has been limited by low transport rates across the vasculature into the main tumour body. Increased convection induced by elevating the systemic blood pressure or applying intratumoural infusion has been shown to improve drug delivery by 40% and several orders of magnitude respectively.

Here we develop a multiscale model of the tumour vasculature, using a similar approach to that used in the petrochemical industry to model groundwater aquifiers and petroleum reservoirs. We assume that the tumour has a locally periodic structure, comprised of a network of capillaries embedded in the surrounding interstitium (a porous medium). The capillaries are small compared to the size of the tumour itself, but much larger than the pore size of the tissue, and so these disparate length-scales can be exploited to describe fluid and drug transport in a tumour. On the local scale, we describe the flow of blood through the capillaries and across the vascular boundary into the interstitium. A multiple-scales technique is then used to move from the local to the global descriptions and so determine the equations describing the effective fluid transport on the tumour-scale. This approach is extended to describe drug transport, and finally numerical simulations are presented.

Simulating invasion: Macroscopic and microscopic approaches
Matthew J Simpson

Department of Mathematics and Statistics, University of Melbourne, Parkville 3010, Victoria, Australia
Coauthors: Kerry A Landman, k.landman@ms.unimelb.edu.au Barry D Hughes, b.hughes@ms.unimelb.edu.au

Cell invasion is fundamental to many biological processes ranging from developmental morphogenesis to disease progression. Experimental data describing invasion is typically collected across a range of scales encompassing population-level and individual-level data. Interpretive and predictive tools capable of replicating and connecting these kinds of data are needed. Continuum population-level and agent-based individual-level methods are two modelling approaches that can replicate and connect multiscale experimental data. We are using agent-based and continuum modelling approaches in parallel to identify and explore various similarities and inconsistencies that arise when considering the same problem using different modelling scales.


Propagation of extrinsic perturbation in a multi-step biochemical pathway
Somdatta Sinha

Centre for Cellular & Molecular Biology, Uppal Road, Hyderabad 500007, AP, India
Coauthors: R. Maithreye

Biochemical pathways underlie cellular processes. These pathways are inter-connected chemical reactions forming an intricate network of functional and physical interactions between molecular species in the cell. Many of the steps in a pathway take place at different cellular compartments and hence are subjected to different environmental milieus. It is thus both interesting and surprising as to how such an interacting dynamical system can faithfully transmit signals in spite of perturbations of different types acting at different steps of the multi-step process. Using a simple three-step negatively auto-regulated model pathway, we show that the effect of perturbation at different steps of the pathway and its transmission through the network is dependent on the context (i.e., the position) of the particular reaction step in relation to the topology of the regulatory network, stoichiometry of reactions, type of nonlinearity involved in the reactions and also on the intrinsic dynamical state of the pathway variables. We delineate the qualitative and quantitative changes in the pathway dynamics for constant (‘bias’) and random external perturbations acting on the pathway steps locally or globally to all steps. We show that constant perturbation induces qualitative change in dynamics, whereas random fluctuations cause significant quantitative variations in the concentrations of the different variables. Thus, the dynamic response of multi-step biochemical pathways to external perturbation depends on their biochemical, topological and dynamical features.


Toward an understanding of electrically coupled inhibitory networks in hippocampus
Frances Skinner

Toronto Western Research Institute (TWRI), University Health Network and University of Toronto
Coauthors: Tariq Zahid (TWRI), Fernanda Saraga (University of Toronto)

Direct electrical communication is well-established in the hippocampus, a brain region known to be important for learning and memory. This form of communication is mediated by gap junctions and it is known that this coupling is important for brain rhythms such as gamma (20-80 Hz) which occur during active behavioural states. It is also known that gap junctions are present at several locations along the dendrites of inhibitory cells in hippocampus, so that spatially extended models need to be considered. Weakly coupled oscillator theory, which uses phase response curves, has been used to predict network dynamics of electrically coupled networks. I will describe this work in the context of our use of a quantification of phase response curves to determine whether synchronous or asynchronous modes (as predicted from the theory) occur in inhibitory networks coupled with gap junctions at dendritic locations.

Life or Death: Complement Activation and Response to a Streptococcus pneumoniae Infection
Amber M. Smith

University of Utah
Coauthors: Frederick R. Adler, University of Utah, Jonathan A. McCullers, St. Jude Children's Research Hospital, Ruy M. Ribeiro, Los Alamos National Laboratory, Alan S. Perelson, Los Alamos National Laboratory

A Streptococcus pneumoniae infection in mice has three possible outcomes: (i) rapid clearance, (ii) acute infection with clearance, or (iii) acute infection ending in death. Which of these occurs depends on the initial dose of bacteria and their interaction with the complement system, one of the first lines of defense in the immune system. There are three biochemical pathways of complement activation: classical, alternative and mannose-binding lectin (MBL). Initiated by antigen-antibody complexes, bacterial surfaces and MBL, respectively, the complement cascade involves more than 30 proteins which act as opsonizers, anaphylatoxins, and initiators of other components of both the innate and adaptive immune system. The final product, called the membrane attack complex, can lyse and kill pathogens. The classical pathway initiates the response to S. pneumoniae infections, but is amplified by the positive feedback in the alternative pathway. We use differential equations to model the kinetics of initiation and amplification of the cascade, and to identify the factors which determine the outcome of a bacterial infection.


One and Two Compartment Stochastic Integrate-and-Fire Neural Models
Charles E. Smith

Biomathematics Program, Dept. of Statistics, North Carolina State Univ
Coauthors: Mamiko Arai, Biomathematics Program, N C State Univ.

One and two compartment stochastic integrate-and-fire neural models are investigated by simulation and by analytic approximation methods. The models used are motivated primarily by the papers of Lansky and Rodriguez (1999 a, b). One main difference is that the output of our model is a renewal process rather than a correlated point process. Biophysically this corresponds to antidromic invasion of the action potential into the dendrite to reset the membrane voltage following an action potential.

We concentrate on two neurons, both with the same compartment(membrane electrical properties) at the site of action potential initiation, however the two compartment model includes the dendritic partition of the neuron by a second compartment. The two compartment model is to contrast spatial effects in neurons with longer thinner dendrites to those with short thicker dendrites that can be modeled as a one compartment equivalent circuit. Biophysically this means that the voltage is roughly the same in the spike initiation site and the proximal dendritic processes.

Euler forward method is used to simulate the Ito version of the stochastic differential equations corresponding to these equivalent circuits. A Wiener process is used as the noise term to represent many smaller synaptic inputs and paralleling the approach of Lansky and Rodriguez. The approximation methods for first passage times outlined in Smith(1991) were computed and compared to moments of simulated output of the neurons.

The shapes of the simulated histograms were fit by a normal, gamma and third and fourth order Laguerre series approximations using the method of moments. The corresponding fits and moment plots (skew vs CV; excess vs. skewness (Pearson plot)) were examined. The single gamma seemed adequate in most cases.

The simulation was done in a 3 factor experimental design and blocked on type of model (one vs. two compartment). The factors were: (1) value of voltage threshold for firing “ S “; (2) synaptic input strength “u “; and (3) intensity of the noise input “k”.

For smaller noise variation and when the mean voltage crosses the threshold a heuristic explanation explains the systematic variation in the mean and standard deviation of the first passage time, namely the mean interval of the output point process and its standard deviation. The approximation is simply that expected from the delta method. The standard deviation of the firing time is approximately the standard deviation of the voltage divided by the slope the mean voltage trajectory. Both voltage terms are evaluated at time t* which is when the mean voltage trajectory reaches the threshold S.

For equal firing rates or output mean intervals, the two compartment model shows a pronounced reduction of variability in firing times. Said differently, it can more effectively code input intensity levels using a mean rate neural coding scheme since it has less variability about the mean firing time.

Finally some suggestions for further work are presented.

A poroelastic model of transcapillary flow
Sean Speziale

University of Waterloo
Coauthors: S. Sivaloganathan G. Tenti

Transcapillary exchange is the movement of fluid and molecules across the porous capillary wall, and plays an important role in maintaining homeostasis in tissues. To reach the cells of a given tissue, molecules must traverse a porous matrix known as the interstitial space, whose main function is to mediate exchange of oxygen, nutrients and waste products between the vascular and cellular compartments. The classical picture of transcapillary exchange was suggested by Starling in 1896, namely that the forces determining fluid flow were the hydrostatic and osmotic pressure differences between the capillary and surrounding interstitial space. However, experimental observations indicated that this view must be revised, and subsequently Michel and Weinbaum put forward the idea that the Starling principle should be applied not across the entire capillary wall but instead across a structure lining the wall known as the endothelial glycocalyx. Existing ultrastructural models are quite complicated, so our aim is to model transcapillary flow using a simpler approach, without losing the essential characteristics. We adopt the Michel-Weinbaum hypothesis, but instead of looking at the microstructure we idealize the capillary wall as a homogenized porous media, and introduce a slight modification to the theory of Biot. Due to the presence of solutes, a modified version of Darcy's law is used, in which fluid flow is driven by both hydrostatic and osmotic gradients. A unique feature of the present work is to be able to predict the stress and strain distributions in the capillary wall, which had not been attempted previously. This work may have implications in understanding edema formation, as well as in explaining the elevated interstitial fluid pressure in tumours.

A Computational Model of Cell-Substrate Interaction in Three Dimensions
Magdalena Stolarska

University of St. Thomas, Saint Paul, MN, 55105
Coauthors: Hans G. Othmer, University of Minnesota, Minneapolis, MN 55455, othmer@math.umn.edu

Mechanical interactions between a cell and the substrate are vital for cell migration and are involved in various cellular processes, such as wound healing, embryonic development, and metastasis of cancerous tumors. In addition, experiments have shown that inter-cellular and cell-substrate mechanical interactions affect signal transduction pathways within the cell (see for example [1, 2, 3]). As a result, understanding the nature of force generation by single cells and mechanical interaction of a cell with the substrate is extremely important.

In this talk, we present a continuum model of single cell motility in which the stresses that result from the active deformation of the cell are transmitted to a substrate via controlled adhesion sites. We propose to use large strain viscoelasticity to describe this mechanism and study cell-substrate interactions. Both the cell and the substrate are treated as three-dimensional deformable continua. A finite element implementation of this model is used to numerically examine the nature of the stresses generated by the cell and the resulting traction patterns that occur at the substrate. The simulations are compared to experimental results where predictions about the stresses in the cell are based on measured deformations of the substrate on which the cell is crawling [4, 5, 6].
[1] P.A. Janmey and D.A. Wietz. Dealing with mechanics: mechanisms of force transduction in cells. Trends in Biochemical Sciences, 29:364–370, 2004.
[2] V. Lecausey and D. Gilmour. Organizing moving groups during morphogenesis. Current Opinions in Cell Biology, 18:102–107, 2006.
[3] A. Bershadsky, M. Kozlov, and B. Geiger. Adhesion-mediated mechanosensitivity: a time to experiment, and a time to theorize. Current Opinions in Cell Biology, 18:472–481, 2006.
[4] J. Lee, M. Leonard, T. Oliver, A. Ishihara, and K. Jacobson. Traction forces generated by locomoting keratocytes. The Journal of Cell Biology, 127:1957–1964, 1994.
[5] S. Munevar, Y.-L. Wang, and M. Dembo. Traction force microscopy of migrating normal and h-ras transformed 3T3 fibroblasts. Biophysical Journal, 80:1744–1757, 2001.
[6] K.S.K Uchida, T. Kitanishi-Yumura, and S. Yumura. Myosin II contributes to the posterior contraction and the anterior extension during the retraction phase in migrating dictyostelium cells. Journal of Cell Science, 116:51–60, 2003.

Detecting spill-over: A dynamical systems modeling approach to glutamatergic synaptic signaling
Emily Stone

Dept. of Mathematical Sciences, University of Montana-Missoula
Coauthors: Greg Leary, Katie Hoffman, Micheal Kavanaugh, Dept. of Biomedical and Pharmaceutical Science, University of Montana-Missoula

The connectivity of neurons in the hippocampus depends in part on whether neurotransmitter from one release site can leak out and activate receptors in another synapse or extrasynaptic patch. The existence of such "spill-over" is under debate in the neuroscience community, since direct measurements of neurotranmitter in such detail cannot, as of yet, be made. Experimental evidence of spillover is thus indirect, and should be sifted through as many different filters as possible. In this talk I will present the contributions of dynamical systems modeling to this effort.

Inhibition of breast cancer growth by GM-CSF: A mathematical model
Barbara Szomolay

Mathematical Biosciences Institute, The Ohio State University
Coauthors: Tim D. Eubank, Ryan D. Roberts, Clay B. Marsh, Avner Friedman

GM-CSF is a drug that enhances the ability of macrophages to present antigen and initiate immune response. GM-CSF also stimulates monocytes to secrete soluble VEGF receptor (sVEGFR-1) which binds to and inactivates VEGF. Eubank and colleagues in a recent work discovered that GM-CSF treatment locally in murine breast cancers reduced tumor growth and metastasis; moreover, GM-CSF lowered oxygen level and reduced blood vessel density within the tumor. We developed a mathematical model which addresses the effect of GM-CSF on the growth of breast cancer in mice. The model takes into account the experimentally established interactions among cancer cells, macrophages, endothelial cells, VEGF and M-CSF. The model simulates the growth of tumor as a function of the local GM-CSF dose. The model simulations were validated against in vivo data and show a good fit with experimental results. We used the model to compare the efficacy of different dosing protocols of injection of GM-CSF, as well as to suggest new hypotheses for slowing the progression of breast tumor. For example, the model suggests that injecting the drug daily, twice or 3 times a week are comparably effective. In contrast, reducing or over increasing the frequency of dosing is counterproductive. We hope to further refine the model in future work, by including the interactions of macrophages and other immune cells, fibroblasts and cytokines that communicate between the tumor and its micro-environment.

The Interaction of Swimming Microorganisms With Flow: Modelling Motile Phytoplankton in Turbulence
Graeme Thorn

Department of Mathematical Sciences, University of Liverpool

The interaction of swimming phytoplankton and typical ocean flow conditions is important to study in order to understand how the spatial distribution evolves due to the dispersive properties of turbulence. Many harmful algal bloom-forming species are motile: one plausible mechanism for the formation of such a bloom is that cells swim into a lower-salinity layer at the top of the water column becoming trapped due to the stratification. Eutrophication of this layer from pollutants in river run-off can cause a population explosion leading to a bloom. A study of the interaction of turbulence with motility will therefore provide insights into how these blooms develop once they are formed. For a gyrotactic microorganism, whose preferred motile behaviour is to swim upwards, the interaction is non-trivial, as turbulence can alter this behaviour due to relatively rapid (compared to the cell’s intrinsic reorientation to the vertical) along its Lagrangian particle path.

As phytoplankton swimming typically occurs on length scales (of the order 1-100 µm) some orders of magnitude below the Kolmogorov scale for ocean turbulence (of the order 1 cm), a model for the swimming behaviour in arbitrarily-oriented simple flows can be used to parameterise the effects of the smallest eddies on the mean swimming velocity. This result can then be incorporated into a population-level equation which then describes the time-evolution of the spatial distribution of a patch of phytoplankton. This advection diffusion model extends previous work which has concentrated on developing models for population dispersal in linear homogeneous flows using the macroscopic generalised Taylor dispersion method.
This talk will begin with a description of the advection-diffusion model, by showing how it is built up from the simple flow model, and show comparisons of this population-level model with simulations of individuals in turbulent flows. Finally, an application to the modelling of the effects of turbulence on the recruitment of gyrotactic cells into a pre-existing stratified fluid.

Spatiotemporal Modelling of Intracellular Signalling in Bacterial Chemotaxis
Marcus Tindall

Centre for Mathematical Biology, Mathematical Institute, University of Oxford.
Coauthors: Steven L. Porter, Philip K. Maini and J.P. Armitage

The role that spatial protein localisation plays in altering the expression of flagellar motor driving proteins in bacterial chemotaxis has to date largely been ignored. The work presented here focuses on two spatiotemporal reaction-diffusion models of signal transduction developed to describe phosphotransfer within E. coli and R. sphaeroides. R. sphaeroides is a bacterial species whose phosphotransfer pathway is considerably more complex than E. coli. The mathematical model developed is used to understand the role that spatial protein localisation has on affecting the motor protein expression, both dynamically and in the steady-state. The model is used to elucidate the role that cytoplasmic and receptor clusters play in describing the overall bacterial response.

Modeling cancer stem cells: Implications for novel therapeutic strategies
Colin Turner

University of Waterloo
Coauthors: Mohammad Kohandel, University of Waterloo and Centre for Mathematical Medicine, kohandel@math.uwaterloo.ca Siv Sivaloganathan, University of Waterloo and Centre for Mathematical Medicine, ssivalog@math.uwaterloo.ca Sheila Singh, McMaster Stem Cell and Cancer Research Institute and Department of Surgery, McMaster University

In recent years, support has increased for the cancer stem cell hypothesis, which states that a subpopulation of cancer cells in possession of properties typically associated with stem cells is responsible for initiating and maintaining tumour growth. Such “cancer stem cells” were first identified in leukaemias, and have since been been implicated in solid tumours including those of the breast, brain and colon. Unravelling the details of the cancer stem cell hierarchy, as well as the interactions of these cells with various therapies, will be essential in the design of optimal treatment strategies. We develop a mathematical model of the cancer stem cell hypothesis that may aid in characterizing tumours, as well as in predicting effective therapeutic strategies. This model is stochastic when small numbers of cells are under consideration; for larger populations of cancer cells, we adopt a deterministic approach. The importance of certain parameters in dictating properties of the tumour is discussed, as well as how the dependence of the tumour on these parameters may be exploited in therapy.

A mean field Ising model for cortical rotation in amphibian one cell stage embryos
Jack A. Tuszynski

University of Alberta
Coauthors: Richard Gordon (University of Manitoba, gordonr@cc.umanitoba.ca)

The fertilized amphibian egg is ideal for mathematical analysis, with its apparently spherically symmetric cortex (membrane and a few microns of attached cytoskeleton), and initially axially symmetric, bottom heavy stratified yolk and cytoplasm inside. Before the first cell division, the cortex rotates 30deg. It is believed that the rotation is driven by microtubule motors and/or polymerization of microtubules attached to the inner surface of the cortex. While these microtubules are initially randomly oriented, they take on a common orientation by the end of the cortical rotation. The basic interaction appears to be two way: the microtubules drive the cortical rotation, leading to sloshing of the yolk and cytoplasm, and the fluid motion aligns the microtubules in such a way as to enhance the rotation. The startup is stochastic in nature. We model this interaction by a mean field Ising model, but due to the coupling between the microtubules, the actual “field” is equal to the mean field, giving the model an unusual precision. Here we show what can be gleaned from observed stochastic rotational trajectories using this model, and how the model is altered if nearest neighbor microtubule-microtubule interactions are included.

Neurodynamics of epileptiform activity: advances towards prevention of seizures
Jose Luis Perez Velazquez

University of Toronto and Hospital for Sick Children

Indications of some specific dynamical regimes have been found in studies of epileptiform activity in brain in vivo and in vitro. These analyses, in turn, may be of help in the current efforts to prevent paroxysmal discharges (seizures) from occurring, and in the solution of the related question of the anticipation, or prediction, of seizure events. However, so far, a general method for seizure prediction or control has not been found. In this presentation, it will be argued that, according to the present methodologies and concepts, seizure prediction and control is certainly feasible for individual cases, but a general approach will be difficult to find considering the multifactorial nature of brain dynamics.

A simulation approach to understanding the role of production frequency in the sharing of T cell receptors between macaques in immune responses to simian immunodeficiency virus
Vanessa Venturi

Centre For Vascular Research, University of New South Wales, Australia
Coauthors: Hui Yee Chin (University of New South Wales, Australia) David A Price (Cardiff University School of Medicine, UK) Daniel C Douek (Vaccine Research Center, NIAID/NIH, USA) Miles P Davenport (University of New South Wales, Australia)

The effectiveness of T cell immune responses to viruses depends largely on the diversity of T cell receptors (TCRs) expressed on the surface of individual T cells that detect viral peptides (epitopes). These TCRs are produced by a process of germline gene recombination. The enormous potential diversity of TCRs produced in the thymus greatly exceeds the number of T cells found in an individual at any given time (eg. > 1018 vs. 1012 in a human). Thus, it is considered surprising that T cells expressing identical TCRs in different individuals are observed in a wide variety of immune responses. However, variation in the production frequencies of different TCRs is rarely taken into account. We used simulations of a random gene recombination process to demonstrate that, even with unbiased gene recombination, some TCRs can be produced more frequently than others. For example, in a random generation of 107 in-frame TCR sequences some TCRs are produced 103 times while others are rarely produced. The simulations also demonstrate that the variety of different ways that a TCR sequence can be made from the germline genes plays an important role in how efficiently it can be produced (as opposed to being produced by one or a few frequently occurring gene recombination events). We term this process ‘convergent recombination’. To test whether convergent recombination and TCR production frequency play a role in vivo, we have investigated the experimentally observed sharing of TCRs in CD8+ (or ‘killer’) T cell responses specific for simian immunodeficiency virus (SIV) in an outbred population of rhesus macaques. The in silico TCR production frequency was found to be a good predictor of the observed extent of sharing of TCRs between macaques in these immune responses to SIV, suggesting that the hierarchy of TCR production frequencies in the simulated repertoire is predictive of that in the observed TCR repertoire in vivo.

Modeling the relationship between radiotherapy delay and cancer outcomes
Jon-Paul Voroney

Queen's University Cancer Research Centre, Division of Cancer Care and Epidemiology
Coauthors: William J Mackillop Sarah J Rauth

Background: Worldwide, delay in radiotherapy for cancer allows progression of untreated tumours. The current average delay in Ontario is over 4 weeks. Meta-analyses of retrospective cohorts relate delay of a one month for radiotherapy to a relative risk of 1.15 for local control or survival from head & neck cancer. Paradoxically, in studies that do not adjust for confounding, delay can be associated with better prognosis: more advanced tumours are often preferentially treated first. Treating poor-prognosis patients first is a triaging strategy appropriate for an emergency department, and does not provide a population with as high a cure rate as would a triaging strategy based on priority for those cancer patients who will be most harmed by delay. Objectives: To model the effect of delay in radiotherapy on cancer outcomes, including local control and overall survival, with particular emphasis on data from published series in head & neck cancer. Methods: (1) Meta-analysis of the head & neck cancer literature on the prognostic effect of tumour volume on local control and overall survival in patients treated with radiotherapy, including modeling the functional relationship between volume and radiotherapy outcome. (2) Meta-analysis of radiologic and serologic markers of tumour volume and their time-dependence in untreated patients. (3) Mathematical modeling of the effect of radiotherapy delay on outcomes through combining the prognostic effect of tumour volume and tumour volume doubling times. Results: (1) We pooled results from 55 studies in head & neck cancer, to demonstrate relationships between initial tumour volume, V, and radiotherapy outcomes. The relationship is exponential, exp(-kV), with the rate constant k depending on site, whether primary tumour volume or total tumour volume (including lymph node spread) was measured, and choice of radiotherapy outcome (local control or survival). This is consistent with a Poisson distribution for the fraction of surviving cells after treatment depending linearly on the initial tumour volume. The rate k is also dependent on treatment, surrogates of radioresistance, and the choice of the model relating volume to outcomes e.g. a one-parameter model exp(-kV) versus two-parameter model Aexp(-kV). The prognostic effect of volume is robust- it persists in studies when additional therapy is given to patients with higher tumour volumes. (2) We abstracted data from 100 studies (5702 tumours) on tumour volume doubling times. Doubling time distribution is lognormal within each site. Summary statistics show considerable variation of medians and inter-quartile ranges for doubling times, depending on the primary tumour site, the histologic subtype, and whether the tumour is metastatic or recurrent. For example, primary prostate cancer has a median tumour volume doubling time of 7 years, while a head & neck cancer recurrence has a doubling time of 1 week. (3) Calculation of the effect of treatment delay, using the relationship between outcomes and volume, and tumour volume doubling times is thus possible: for example, for an average head and neck tumour with a volume of 12 cc, volume doubling time of 8 weeks, and dependence of volume on prognosis of exp(-0.03V), the relative risk of 1 month delay is 1.16. An initial cancer local control rate of 70% drops to 60%. Incorporating distributions in population values of model parameters and incorporating a Gompertz or logistic function into modelling tumour growth are refinements that improve modelling for large tumours and long delays. Conclusions: The increased risk of poor outcomes with RT delay is predictable using tumour growth and knowledge of the association between outcomes and tumour volume. We have validated our model of delay in radiotherapy for head & neck cancer, and this model can be tailored to data from particular treatment centres, particular populations, and particular tumours.

Rigid and Nonrigid Registration Methods for Medical Images
Justin Wan

University of Waterloo
Coauthors: Lin Xu and Zhao Yi

In image guided procedures such as radiation therapies and computer-assisted surgeries, physicians often need to align images that are taken at different times or by different modalities. Typically, a rigid registration is performed first, followed by a nonrigid registration. We are interested in 2D-3D registration which align digitally reconstructed radiographs generated from 3D datasets (e.g. CT volume from pre-operative planning) with 2D portal image slices (e.g. X-ray image collected in real-time). It is a very computationally intensive procedure. In this talk, we propse fast models that can achieve real-time performance. Afterwards, nonrigid registration needs to be used to enhance the results. Elastic and fluid models were usually used but edges and small details often appear smeared in the transformed templates. In this talk, we also propose a new inviscid model formulated in a particle framework. We will derive the corresponding nonlinear partial differential equations for computing the spatial transformation. Our idea is to simulate the template image as a set of free particles moving toward the target positions under applied forces. Our model can accommodate both small and large deformations, with sharper edges and clear texture achieved at less computational cost. We demonstrate the performance of our model on a variety of images including 2D and 3D, mono-modal and multi-modal, synthetic and clinical data.


Incorporating prior knowledge to the dynamic Bayesian networks modeling of pancreas development gene expression data
Xujing Wang

Max McGee National Research Center for Juvenile Diabetes & Human and Molecular Genetics Center, Medical College of Wisconsin, 8701 Watertown Plank Road, Milwaukee, WI, 53226, USA
Coauthors: Shouguo Gao

The importance of the network structure underlies genes and proteins is gaining increasing appreciation. This is not only fundamental to the understanding of genetic regulation and its functional structure, but also critical to dissect complex diseases. Time series gene expression data offer a rich source for network inference. We have adopted the dynamic Bayesian network (DBYN) approach to model transcription regulatory and co-expression networks, and developed new algorithms to incorporate existing biological information (co-citation, GO (gene ontology) similarity, positional and binding information, etc) in public databases as prior knowledge. We introduced, for the first time, fuzzy theory-based rules to the MCMC learning of DBYN in order to efficiently incorporate the prior biological knowledge, which are often incomplete and plagued with quality issues. Further we defined gene expression (phase) synchronization module and utilized it to assist initial network structure construction. We show that these lead to significantly improved performance. We then applied the algorithm to investigate the pancreatic development. We first compiled a list of pancreas-specific genes by: (1) Manually collecting curated genes that are known to be involved in pancreas development from the literature; (2) Tissue specific gene expression data was downloaded from http://www.t1dbase.org. We then determined for each gene the Z score of its expression in pancreas versus the mean in all tissues. We focus on those with pancreas Z>0.2, and being annotated to GO:0032502 (developmental process) or its descendant categories. Together we obtained a total of 45 genes. Two data sets were obtained from RNA Abundance Database (www.cbil.upenn.edu/RAD) to perform network reconstruction: (1) study id 2, expression of mouse pancreas development at 7 time points: E14.5, E16.5, E18.5, birth, postnatal day 7, and at adulthood; (2) study id 1790, mouse pancreas at 12, 24 and 48 hrs after 50% Ppx or sham operation, which also received Ex-4 or vehicle every 24 hours. We found that with GO and co-citation information our DBYN predicted number of experimental established relationships were improved 1.5 to 2 fold. The improvement is more when we used the experimentally confirmed gene interaction as an initial structure to train the Bayesian network.

Mathematical modelling of the gastrointestinal epithelium stem cell niche.
Sarah Waters (on behalf of Sarah Eastburn)

University of Oxford
Coauthors: Dr Sarah Waters, Dr James Oliver, Prof. Helen Byrne and Dr Felicity Rose

The gastrointestinal tract is lined by a mono-layered epithelium that contains invaginations called the crypts of Lieberkühn. These crypts contain stem cells which are responsible for the regeneration and maintenance of the epithelial lining. The stem cells are thought to reside in a niche at the crypt base. The crypts have a natural hierarchy and transit cells produced by the stem cells, divide, differentiate and migrate towards the intestinal lumen.

Using a number of complementary modelling approaches we investigate the mechanics of the intestinal epithelium and the population kinetics, resulting from, e.g., cell proliferation and cell death. We apply these models to the in vitro experiments carried out in the Tissue Engineering Group at Nottingham, with the aim of determining the spatiotemporal distribution of cell density and stress, and consider the role of different substrate geometries on the morphology of a growing cell aggregate.

We model the tissue mechanics of a growing aggregate of cells using a lattice-free spatial framework. Cells are unrestricted in their position and move in a continuous fashion in response to the resultant force exerted on each cell by its neighbours, with drag balancing the cell-cell interactions which are modelled as springs between neighbouring cells. The model behaviour is captured in one dimensionless parameter, a, the ratio of a typical cell cycle time to the spring relaxation time, with cell cycle times modelled by a simple uniform distribution.

The cell kinetics of the intestinal crypt are modelled via a stochastic branching process model for the evolution of a spatially-homogeneous mixed population of stem, transit and fully differentiated cells. From this model we derive analytic expressions for the expected number and variance of each cell type, and for the total number of crypt cells which are compared with experimental data.

These two models are then coupled together, the kinetic model replacing the original simple cell proliferation assumption in the mechanical model. We investigate how this modified kinetic approach changes the resulting cell densities and stresses. We will present an overview of the modelling techniques used, as well as our key findings from these models.


Crystal aggregation and deposition in the catheterised lower urinary tract
Sarah Waters

University of Oxford
Coauthors: Leah Band, Linda Cummings, Jonathan Wattis

Urethral catheters often become blocked with crystals of magnesium struvite and calcium phosphate. The encrustation can block the catheter, which can cause urine retention in the bladder and reflux of urine into the kidneys. We develop a mathematical model to investigate crystal deposition on the catheter surface, modelling the bladder as a reservoir of fluid and the urethral catheter as a rigid channel. At a constant rate, fluid containing crystal particles of unit size enters the reservoir, and fluid flows from the reservoir through the channel and out of the system. The crystal particles aggregate, which we model using Becker-Doring coagulation theory, and are advected through the channel, where they continue to aggregate and can deposit on the channel's walls. Inhibitor particles also enter the reservoir, and can bind to the crystals, preventing further aggregation and deposition. In the reservoir, we assume the crystal concentrations are spatially homogeneous, whereas in the channel we consider concentrations that vary as a result of advection, diffusion and deposition. We investigate how the aggregation rate and the influx of inhibitor particles affect the amount of deposition. For all parameter values, we find that crystals deposit along the full length of the channel, with maximum deposition at the channel's entrance.

Mathematical modelling of LDL and VLDL endocytosis by HepG2 cells.
Jonathan Wattis

University of Nottingham
Coauthors: Brendan O'Malley (Unilever Corporate Research, UK), Marcus Tindall (University of Oxford, UK), Laura Pickersgill (Unilever Corporate Research, UK), Hannah Blackburn (Unilever Corporate Research, UK), Helen Byrne (University of Nottingham, UK), Kim Jackson (University of Reading, UK), Jasmina Panovska (Unilever Corporate Research, UK).

Individuals with elevated levels of low-density-lipoprotein cholesterol (LDL-C) in their plasma are considered to at risk of developing coronary heart disease. LDL particles are removed from the body mainly by hepatocytes through receptor-mediated endocytosis. Apolipoprotein B-100 present on the surface of LDL particles binds to receptors in pits on the surface of hepatocytes. Upon internalisation of a pit, the bound complex of LDL and receptor is degraded into its constituent parts (cholesterol, fatty acids and amino acids), which are released for use by the cell or are excreted. Very low-density lipoprotein particles (VLDL) are known to inhibit the take-up of LDL [Jackson et al J Lipid Res 47, 393, (2006)].

We formulate a mathematical model of the binding, internalisation and processing of LDL and VLDL particles by cells. The model is calibrated to experimental data of Brown & Goldstein [PNAS 76, 3330, (1979)], and Harwood & Pellarin [Biochem J 323, 649, (1997)]. We find good agreement with in vitro data on LDL-take up, and inhibition of take-up by VLDL. As well as the 'single bolus' scenario where a large dose of lipoprotein is delivered at the start of the experiment, we consider the more realistic (in vivo) case where lipoprotein is continually delivered to the system. We show how the the average occupancy of pits, and the proportions of receptors which are free, bound, or internalised changes over time. We analyse how the cell's flexibility in adapting to changes in its environment depends on the efficiency of receptor-recycling, how different types of VLDL influence LDL takeup, and differences in behaviour exhibited by the single-bolus and continual-delivery models.

Epidemiological interactions between the local and the mean-field
Steven Webb

University of Strathclyde
Coauthors: Mike Boots (Sheffield University) Matt Keeling (Warwick University)

The assumption that populations are completely mixed is reasonable for many populations, but there is likely to be some degree of local interaction whether spatially or socially in many systems. An important question is therefore how strong these local interactions need to be before there are significant effects on the dynamics of the system. Our approach is to use correlation models, namely pair-wise models, to capture the spatial relationships of contacts and interactions between individuals. We first extend previous pairwise models to include immunity and reproduction from infecteds – dynamics that have been largely ignored in such systems. We then derive a multi-scale pair-approximation to move between completely local and completely mixed host-parasite interactions, thereby extending the reproduction processes of hosts and infection, and examine the long term effects of these differing spatial scales on the disease characteristics. Possible evolutionary traits of the pathogen within this multi-scale framework are then derived using adaptive dynamics.

Examining Cerebrospinal Fluid Pulsations as a Causative Mechanism for Hydrocephalus
Kathleen Wilkie

University of Waterloo
Coauthors: Prof. S. Sivaloganathan

Hydrocephalus is a condition characterized by dilated ventricles and compressed brain tissue in the cranial cavity. The ventricles expand due to an increase in cerebrospinal fluid (CSF) with the result that the brain parenchyma is compressed against the skull, often causing neurological impairment. Although the pathophysiology of hydrocephalus has been, and currently is, the focus of much clinical and experimental research, it's etiology is still largely unknown. In the literature, two different causative mechanisms have been proposed: (1) increased intracranial pressure gradient from the ventricles to the subarachnoid space (this fails to explain both communicating hydrocephalus and normal pressure hydrocephalus) and (2) intracranial CSF pulsatility (this begs the question "why don't all subjects develop hydrocephalus?", since CSF flow is always pulsative - even in normal subjects). In this talk, I will present a simple poroelastic model of the brain which allows for space- and time-dependent analytic solutions. Numerical simulations allow us to determine the mechanical effects of the intracranial pulsations on brain parenchyma and whether or not the pulsations have the potential to induce ventricular dilation.

Modelling combined phototaxis and gyrotaxis in suspensions of swimming microorganisms
Rosie Williams

University of Glasgow
Coauthors: Martin Bees

Hydrodynamic instabilities associated with cell aggregations in suspensions of swimming, typically negatively buoyant, microorganisms can lead to the formation of intricate patterns, termed bioconvection. Phototaxis, a response to light, and gyrotaxis, defined as the balance between gravitational (due to bottom heavy cells) and viscous torques, bias the cell swimming direction, cause aggregations and influence the resulting instabilities. We construct a general model to predict stability of suspensions of photo-gyrotactic unicellular microorganisms, specifically looking for the wavenumber and growth rate of the first most unstable mode that grows from an equilibrium state. We extend recent models for purely gyrotactic cells, which couple the Navier-Stokes equation with a cell conservation equation and model cell swimming directions probabilistically using the Fokker-Planck equation. We include the effect of phototaxis in three separate ways. First we include the effect of light in the cell swimming speed, secondly we model the cells as though they change their center of mass offset depending on available light and thirdly we consider torque-like effects due to gradients in light. Equilibrium solutions are found for each model and then perturbed to assess stability for a range of realistic parameter values for the green alga Chlamydomonas nivalis. Results of the stability analysis are presented and the differences between each model are highlighted and their relative viability discussed.

A systematic model of bacterial chemotaxis: from signal transduction to cell motility in Escherichia coli.
Xiangrong Xin

Department of Biomedical Engineering and School of Mathematics, University of Minnesota
Coauthors: David J. Odde, Department of Biomedical Engineering, University of Minnesota, Email: oddex002@umn.edu; Hans G. Othmer, School of Mathematics and Digital Technology Center, University of Minnesota, Email: othmer@math.umn.edu

The movement of bacteria in response to environmental changes of specific metabolites and signaling molecules is called bacterial chemotaxis. Chemotaxis in Escherichia coli (E. coli) is a best studied system. The authors will present a systematic model of E. coli chemotaxis that can capture many features of the system and reproduce a full range of experimental observations from signaling (excitation, perfect adaptation, robustness, high sensitivity, wide dynamic range, etc.) to motor behavior and cellular motility. A remarkable feature of the signaling pathway is its high sensitivity to small relative changes in concentrations of chemical stimuli over a broad range of ambient concentrations. To account for it, the signaling part of the model is based on the structural and functional unit of receptor clusters, ‘trimer of chemoreceptor dimers’, which has been solidly experimentally established but not well quantitatively modeled, so the theoretical work includes more molecular mechanism in modeling and provides a more mechanistically based description of the origin of high sensitivity than the existing models in the field.

A multi-scale model of HIV-1 transmission
Lilit Yeghiazarian

Department of Biostatistics, UCLA
Coauthors: William G. Cumberland (UCLA) Otto O. Yang (UCLA)

Interactions of HIV-1 with the immune system of the host have been extensively studied experimentally and theoretically using mathematical models. On the other end of the spectrum, epidemiological studies provided insight on HIV-1 dynamics within human populations. The importance of the topology of human social networks in disease epidemics became apparent as research on the underlying structure of a variety of technological networks such as the Internet progressed. We have developed a mathematical model that bridges the scales between in-host processes and HIV-1 transmission in human sexual networks. Each individual goes through a sequence of health states that reflect his/her HIV-1 status, treatment stage, and vital information. Associated with each health state is an in-host model describing the interactions between HIV-1 and the immune system. At the same time, each individual is modeled as a node in a sexual network within which the disease spreads as sexual encounters take place. We investigate the effect of a variety of medical care decisions such as timing and type of therapy, on the spread of disease within populations. We have found that early therapy initiation, namely during the acute infection phase, may substantially decrease the spread of disease.

Modeling the evolution of insect phenology: Can insect populations adapt to climate change?
Brian Yurk

Department of Mathematics and Statistics, Utah State University, Logan, Utah
Coauthors: James Powell (Department of Mathematics and Statistics, Utah State University)

Warming temperatures are likely to disrupt insect phenology (the timing of developmental events) to such an extent that some species will face local extinction while others will erupt in new habitats. It is unknown whether phenology can evolve rapidly enough to moderate these effects. Since phenology is a critical determinant of fitness in insect populations, there are strong selective pressures on maintaining appropriate phenology. For example, it is important that development is timed to avoid the coincidence of sensitive life stages with extreme weather. An individual's fitness may also be highly dependent on synchrony between its phenology and the phenology of its biotic resources, as in the case of plant-pollinator interactions. At low population densities, developmental synchrony within a population can also be an important determinant of fitness; the probability of finding mates increases when a large portion of the population reaches reproductive age within a short time period. Synchronized emergence within a population of herbivorous insects may also be necessary to overwhelm resource defenses, as is the case of mountain pine beetles attacking pine trees.

Temperature plays a major role in determining the phenology of insects, since the time it takes for an insect to develop through a life stage is highly dependent on the temperature that it experiences. Previous phenology models have described this plastic response of development time to temperature without considering genetic evolution of the response. Without evolution, these models predict that insect populations may lose developmental synchrony or synchrony with biotic resources. We will present a modeling approach that extends previous phenology models to allow for evolution of the dependence of development time on temperature. Our model results show that evolution may allow populations to adapt to warming temperatures and changing resource phenology, but there are limits to this adaptation. We will also discuss the existence of steady distributions of the evolution model, in which the temporal structure of the mean phenotype and phenotypic variance are invariant under the evolution map with periodic temperatures. Both long term and short term dynamics are controlled by the presence of phenotypes that allow for individuals and their offspring to be oviposited at the same time of year in consecutive years.

A new discrete distribution induced by the Luria-Delbruck mutation model
Qi Zheng

Texas A&M School of Rural Public Health, College Station, Texas 77843

The Luria-Delbruck mutation model has been a subject of mathematical investigation for over six decades. A recent investigation of this celebrated model led to the discovery of a new discrete distribution that can potentially be applied to model data generated by other biological processes. This two-parameter distribution arises as a limiting form of the probability generating function discovered by M.S. Bartlett. We first show that an obvious extension of the limiting form is a valid probability generating function and then present an algorithm for computing the probability mass function. The asymptotic behavior of the probability mass function is revealed by employing the technique of singularity analysis of generating functions. We also suggest likelihood based algorithms for estimating the parameters. The new distribution is found to be infinitely divisible and possess divergent moments.