IGTC Graduate Summer School in Mathematical Biology
A PIMS/Accelerate BC sponsored Summer School in Mathematical Biology
University of British Columbia, Vancouver, May 11-June 11, 2008
Abstracts for Talks
This page provides abstracts for many of the lectures and seminars.
For a detailed daily schedule click HERE.
For a list of events by category and/or lecturer(s) click HERE.
For details of the main Course Content click HERE.
For an overview of the month-long schedule click HERE.
Unless otherwise indicated, events will take place at the UBC campus of PIMS.
Directions for PIMS can be found HERE.
Sunday May 11, Opening Lecture:
Prof Michael Doebeli (Depts of Mathematics and Biology, UBC)
Adaptive diversification: theory and experiments
Understanding the origin of diversity is a fundamental problem in
biology. Traditional evolutionary theory predicts uniformity: natural
selection, acting on organisms under given environmental conditions
and developmental constraints, produces a unique, optimally adapted
phenotype. According to this view, different types only come about
through a change in conditions over space or time. In particular, the
process of diversification, that is, the split of an ancestral
population into distinct descendent lineages, is a by-product of
geographical separation. This traditional view misses out on the
important perspective that diversification itself can be an adaptive
process. In this talk I will review theoretical work showing that
diversification as an adaptive response to ecological interactions is
a plausible evolutionary process. This work is based on the
mathematical framework of adaptive dynamics, and in particular on the
phenomenon of evolutionary branching due to frequency-dependent
selection. I will describe basic models for evolutionary branching
based on resource competition. I will then describe ongoing efforts
to test the theory of evolutionary branching in evolving Escherichia
coli populations, which provide promising experimental model systems
for studying adaptive diversification at the genetic, physiological
and ecological level.
Special Mathematical-Biology Guest Lectures
These Seminars will be held at PIMS. Pizza and drinks will be served at 12:30PM.
Friday May 16, 1:00PM
Prof Eirikur Palsson (Dept. of Biology, SFU)
Excitability of Dictyostelium discoideum is regulated by the ratio of
membrane bound to secreted phosphodiesterase
After onset of starvation randomly distributed Dictyostelium discoideum
cells initiate formation of cooperative aggregation territories over a
wide range of initial cells densities. Recruitment of a high number of
cells is important, and it appears that formation of large territories
has been selected for. Propagating cAMP waves enable the formation of
territories larger than 1 cm diameter. These cAMP waves are generated by
the means of an elaborate cAMP signaling system that makes the whole
field of cells excitable. Dictyostelium cells respond chemotactically to
these waves, guiding cell aggregation towards a signaling center. An
important component of the signaling system is the PdsA
phosphodiesterase (PDE), that breaks down the external cAMP. PdsA can be
either membrane bound or secreted. Here I show that by utilizing both of
those two forms of PDE and by fine tuning the ratio, Dictyostelium
extends the range of cell densities where cAMP waves can propagate and
thus where aggregation can be successful. The membrane bound PDE reduces
the likelihood that the aggregation territory breaks up into many
smaller territories, as the cell density increases, while the secreted
PDE is important for wave propagation at low cell densities. An
interesting interpretation of these findings is that for some excitable
systems, with discrete point sources located far apart, wave propagation
is not possible if the sink is in the same location as the source.
However, when the source and sink are in a different location wave
propagation is possible. An example would be Ca++ propagation in cardiac
Friday May 23, 1:00PM
Prof Daniel Coombs (Dept. of Mathematics, UBC)
Models of T cell activation
T cells are activated via their T cell receptors (TCR) binding to specific
antigen on other cells, in the form of antigenic peptide-MHC complexes. T
cells can be activated by very few presented peptide-MHC of the right type.
The serial engagement hypothesis suggests that each peptide-MHC binds to
multiple TCR, amplifying its effect. In this talk, I will talk about serial
engagement and related things at an introductory level.
Tuesday May 27, 1:00PM
Prof Yue Xian Li (Dept. of Mathematics, UBC)
Tango Waves in Inhomogeneous Calcium Excitable Media
Oscillations and travelling waves are often observed in intracellular levels of
calcium when egg cells are either activated by a sperm or by other kinds of
stimuli. This is often due to the existence of calcium stores that are capable
of dynamically releasing calcium
making the intracellular medium a calcium excitable medium (Li et al, Am J
Physiol 296:C1079, 1995). Tango waves are wave fronts that propagate in a
back-and-forth manner and was found in a bidomain model of calcium waves in
frog eggs (Li, Physica D, 186:27, 2003). We show that such waves can occur in
the presence of a time-dependent spatial inhomogeneity in an excitable medium.
Analysis of such a phenomenon was carried out using the FitzHugh-Nagumo (F-N)
model with a spatially varying but time-independent term, z(x), added to the
first equation. Inhomogeneities cause pinning and oscillations of the front.
This is best shown when z(x) is a linear ramp. When the slope is large, it
stabilizes the front. The front becomes less stable as the slope decreases. At
a critical slope, the front becomes unstable through a Hopf bifurcation beyond
which oscillations occur (Prat and Li, Physica D, 186:50, 2003). These results
were generalized to a wider class of excitable media including systems of PDEs
and IDEs (integral-differential equations) (Prat et al, Physica D, 202:177,
2005). In this talk, I will present a summary of these results.
Monday June 2, 1:00PM
Prof Eric Cytrynbaum (Dept. of Mathematics, UBC)
Finding the center - how to solve simple geometry problems at
the cellular scale
Fragments of fish melanophore cells can form and center
aggregates of pigment granules by dynein-motor-driven transport along
a self-organized radial array of microtubules (MTs). This system has
been extensively studied recently as a model for cytoskeleton-driven
organization of the intracellular space. I will present a
quantitative model that describes pigment aggregation and MT-aster
self-organization and the subsequent centering of both structures.
The model is based on the observations that MTs are immobile and
treadmill, while dynein-motor-covered granules have the ability to
nucleate MTs. From assumptions based on experimental observations,
I'll derive partial integro-differential equations describing the
coupled granule-MT interaction. Scaling arguments and perturbation
theory allow for analysis in two limiting cases. This analysis
explains the mechanism of aster self-organization as a positive
feedback loop between motor aggregation at the MT minus ends and MT
nucleation by motors. Furthermore, the centering mechanism is
explained as a global geometric bias in the cell established by
self-nucleated microtubules. Numerical simulations lend additional
support to the analysis. The model sheds light on role of polymer
dynamics and polymer-motor interactions in cytoskeletal
Student Research Seminars on Thursday May 15, 2:00PM
Department of Mathematics, Indiana University
A 3-D model of the passive cochlea
The cochlea is the primary sound processing mechanism of the (inner) ear.
The aim of this talk is to present a 3-D model of the passive
(non-energetic) cochlear mechanism. A brief overview of the function of
the ear will be included.
INRIA de Rocquencourt, BANG Project, France
Modeling The Pharmacokinetics and Pharmacodynamics of the anticancer drug
Irinotecan at the intracellular level
Irinotecan is an anticancer drug which is currently used in chemother-
apy against colorectal cancer. Here we are interested in its action on
a cell population, at a molecular level. We model its Pharmacokinet-
ics(PK),which is what the cells do to the drug (e.g. metabolization, trans-
port), and its Pharmacodynamics(PD), which is what the drug does to
the cells(e.g. DNA damages).
The PK-PD of Irinotecan is largely influenced by the circadian rhythms:
each living being has an internal clock that is responsible for rhythms over
a 24-hour period. Its most obvious manifestation is the regular cycle of
sleeping and waking, but body temperature and the concentration of en-
zymes also vary over the day.
Our mathematical model is based on ODEs and simulates the PK-PD
of Irinotecan, taking into account the circadian variation of the chemi-
cals. An aim of this modelization could be to optimize the schedule of
administration of the drug.
Mathematics, Rice University
Models for Robust Protein Gradients in the Drosophila Embryo
In early Drosophila development, one of the first specifications made in
determining cell location is the formation of the anterior-posterior axis,
and this event is carried out with incredible precision, despite many
variations of the genetic regulatory network and environmental
fluctuations. My focus is on the first step in this process, where the
protein, Hunchback (Hb), forms a very sharp boundary at the midpoint of
the embryo, despite receiving noisy information from its known
upstream regulator, Bicoid (Bcd).
Using systems of partial differential equations to model the gene network,
we seek an answer to the following inverse problem: given the robust
Hb boundary, what conditions does this place on the regulatory network?
While a simple answer does not likely exist, we provide examples of such
networks, as the result of both numeric searches and analytic techniques,
and an interpretation of the conditions in terms of the biological system.
Department of Applied Mathematics, University of
Model for the dynamics of Malaria
The transmission and maintenance of malaria is considered
between human being hosts and mosquitio vectors that are infected by
protozoan parasites. On the basis of the simple Ross-Macdonald model,
it considers a model both the incubation period in the disease circle
of hosts and vectors, especially on the latency of the malaria in
mosquito vectors. For the new model with an exposed class , the basic
reproduction number R0 is identified and its threshold property is
discussed. Numerical simulations re given to illustrate the dynamics
of the population.
Dept of Mathematics and Institute of Applied Mathematics, UBC
Receptor clustering is required for accurate detection of ligands
T cells discriminate between pMHC ligands using their T cell receptors.
Accurate pMHC discrimination is critical to defend against forgein pathogens
while avoiding autoimmunity. The mechanism of accurate detection has
remained elusive. T cell receptors have recently been shown to cluster on
the surface of T cells. The role of T cell receptor clusters is discussed
controversially. I will use physical arguments and mathematical modeling to
show that T cell receptor clustering is critical for accurate pMHC
SKILL: Stochastic simulation of reaction-diffusion system
For this work I needed to perform accurate simulations of diffusing and
reacting proteins on a 2D membrane. I also needed to relate all parameters
in the stochastic model to the relevant PDE model since experiments were
carried out in the PDE regime (i.e. high concentration). I can use a
simplified example to show how one can simulate diffusion and reaction and
how to pick the parameters so that the simulation converges to a PDE.
Dept of Mathematics, UBC Okanagan
The effects of fragmentation of habitat on cyclic population dynamics
We investigate how fragmentation of habitat affects cyclic population dynamics by constructing a spatially explicit predator-prey model in which predators and prey orient their movement toward high quality patches. For our reaction terms, we use four different pairs of functions found in the literature. Our study is motivated by cyclic mammalian populations such as the Snowshoe Hare and Canada Lynx that exhibit high amplitude, 8-11 year population cycles. We investigate whether increasing anthropogenic fragmentation of habitat could affect the population dynamics and persistence of cyclic populations. This fragmentation of habitat could be attributed to temporary disturbances such as forest harvesting, or more permanent disturbances such as roads and agricultural or urban development. We use a Partial Differential Equation (PDE) model to describe the dispersal of predators and prey in a heterogeneous landscape made of high quality and low quality habitat patches. We show that habitat fragmentation affects the amplitude and average densities of both predator and prey in high quality patches. This result may be important to conservation efforts for species with cyclic population dynamics in fragmented habitats.
SKILL: Determining critical patch size for multi-species population dynamics with dispersal
In my research, I had to determine critical patch size for predators and prey for my spatially-explicit model. I could present an example on how
to determine critical patch size for one of the models I used.
Student Research Seminars on Thursday May 22, 2:00PM
Centre for Nonlinear Dynamics in Physiology and Medicine, McGill
Galactosemia: a mathematical approach based on the impairment of
galactose metabolism in Saccharomyces cerevisiae
In organisms ranging from E. coli to mammals, galactose is metabolized
through a series of sequentially reactions known as the Leloir pathway. In
humans, the deficiency in any of the Leloir enzymes leads to a potentially
lethal disorder called galactosemia. Despite decades of study, the
underlying bases of galactosemia pathophysiology remain unknown, limiting
the development of novel and potentially more effective treatments. The
difficulties in understanding this inborn error of metabolism created an
impetus for a model organism susceptible to metabolic changes and genetic
engineering, and for an investigation based on advanced computational
methods. We address the problem of galactosemia by employing a model system
with high relevance for human health (S. cerevisiae) and through
mathematical modeling. Our main goal is to propose a comprehensive kinetic
model of galactose signaling pathway in yeast and to investigate the
mechanisms underlying galactosemia. We plan to examine the effect of the
Leloir enzymes impairment on the cell ability to metabolize galactose, to
follow the flux distribution through the biochemical routes, and to define
the role of the transferase-independent routes of galactose metabolism in
this disorder. The potential physiological impact of metabolites
accumulation, questions of great importance from a therapeutic point of view
(e.g. the maximum amount of exogenous galactose tolerated by a galactosemic
patient) as well as the logical connection between the metabolic abnormality
and the long-term complications in galactosemia are further research goals.
Division of Mathematics, University of Dundee
Mathematical Modelling of Cancer Cell Invasion of Tissue
When solid tumour cells are no longer contained in their primary
compartment, they acquire an ability to break out of the compartment
and invade the surrounding tissue locally. The ability of tissue
invasion gives solid tumours a defining deadly characteristic where
they become malignant cancerous cells by growing rapidly and
establishing a new colony in distant organs, a process that is known
as metastasis. A mathematical model of cancer cell invasion of tissue
with overcrowding prevention mechanism is formulated and analysed.
The model includes the role of urokinase plasminogen activation (uPA)
system with a system of 5 taxis-diffusion-reaction partial
differential equations describing the interactions between cancer
cells, uPA, uPA inhibitors, plasmin or matrix degrading enzyme, and
host tissue. In this model, cell motility is expressed by random
motion and local continuum model based on chemotaxis and haptotaxis
terms. Further, the effect of cell-cell and cell-matrix adhesions is
incorporated by replacing local taxis flux terms by nonlocal flux
Institute of Applied Math, UBC
Models of the actin-like MreB helix in prokaryotes
MreB is an actin-like protein that forms a helix running the length of
cylindrical bacterial cells. I will present a model of the helix. Individual
polymers that make up the helical
cables are represented by simple force-dependent polymer models bundled into
a supramolecular array. Boundary conditions and external forces are provided
by a global elasticity model representing the cables as flexible rods
buckled into a helix inside the
confinement of the cell wall. The model produces a relationship between the
pitch of the helix, the thickness of the cables and the total abundance of
MreB, and has implications for cell
growth, macromolecule trafficking and the polarization of Caulobacter
Ehsan Ellahi Ashraf
Department of Mathematics, University of Glasgow
Resistive Force Theory and the swimming of Biflagellated Green Algae
Green Algae plays an important role in the development of the Earth's
atmosphere by photosynthesizing and acting as a sink for carbon dioxide.
They are estimated to make up more than half of the Earth's biomass and
acts as a potential source for Hydrogen gas production. They can be the
most efficient source of feedstock for Biofuel or Biodiesel industry.
The green biflagellated algae Chlamydomonas nivalis has a prolate
spheroidal body and has two long, thin flagella attached at one end which
are propelled to cause the swimming of the organism. The length of the
cell body and flagella is approx. same as 10 micro meter and the flagella
beat at approx. 50 hertz. C. nivalis is usually considered to swim in a
human breast-stroke manner with an effective-recovery style, approximately
in the direction of its axis of symmetry. However, the latest research
proved that this is not exactly the accurate swimming stroke.
The approximation known as Resistive Force Theory (RFT) was established by
Gray & Hancock (1953) and states that the normal and tangential components
of force and torque on an element of a flagellum is directly proportional
to the normal and tangential components of the fluid velocity relative to
that element. RFT was used by Jones et al. (1994) to model the swimming of
a single cell of C. nivalis in a viscous flow of low Reynolds number.
Analytical and numerical techniques were used to calculate the magnitude
and direction of the cell's swimming velocity and angular velocity.
The aim of our work is to extend the Jones et al. model to calculate the
the cell's swimming velocity and angular velocity in the vicinity of a
stationary no-slip boundary or sphere using appropriate image system
techniques. Finally, we will model interactions of two algae cells using
RFT and image systems and calculate the magnitude and direction of both
cell's swimming velocity and angular velocity.
Department of Mathematical Sciences, Carnegie Mellon
Probabilistic Modeling of the Immune System
Our ultimate goal is to obtain a reasonable model of the
gene interactions within the immune system. From the data
we have which contains time plots of the growth of
specific proteins produced by classified immune system
genes, we want to determine the inhibiting and catalyzing
interactions between those genes. We are specifying these
interactions as stochastic in nature. The probabilities
we are attempting to determine are defined by the
likelihood that a gene affects another gene, positively
or negatively. In our attempt to realize these
interactions, we have produced various models that show
protein production. Here we try to solve the forward
problem: given a specified number of probabilities,
create a time plot of the protein productions. These
models consider single genes producing one protein,
simple two gene interactions producing two proteins, and
random n gene interactions producing n proteins.
After creating these models, we use them to solve the
backward problem: given a time plot of protein
productions, find the inhibiting and catalyzing
interactions between specific genes and their associated
Department of Mathematics, Tulane University
Introduction to Projection Methods
Projection Methods are a numerical method for solving the full
incompressible Navier Stokes equations, based on decoupling the
momentum and continuity equations and solving two separate systems
within a single time step. In the talk, I will give an overview of
the method applied to a staggered grid, and show an example with
periodic boundary conditions.
Student Research Seminars on Tuesday May 27, 2:30PM
University of Alberta
Optimal Cancer Radiotherapy Treatment Schedule under
Cumulative Radiation Effect constraint
Radiotherapy is a very important method to treat malignant cancer. Usher
(1980) derived some optimal fractionated cancer radiotherapy treatment schedules by
minimizing the total survival fraction of cancerous cells. But his method has two restrictions: 1) His
optimization method is not a global optimization, 2) his method can only be used for uniform
treatment, i.e. the treatment schedule with the same dose per fraction and same intertreat-
ment time between two fractions. In this report, first we use both analytic and numerical
methods to get global optimal treatment schedules for uniform treatment, which confirm that
Usher's results are almost right. Furthermore, we generalized the cumulative radiation effect
and total survival fraction model to nonuniform treatment with variable intertreatment time,
use them to rank ten universally used protocols in prostate cancer treatment and compare with
the rank obtained by tumor control probability model, which is another universal model used
to determine the treatment schedule, they are almost the same.
Institute of Applied Mathematics, UBC
Pattern formation on growing domains
Domain growth plays a central role in robustly selecting a pattern in reaction-diffusion systems.
In this talk I will derive the generalized reaction-diffusion equations on growing domains for 1 and 2-D.
I will show an example, and discuss the implications on pattern selection.
Department of Mathematical and Statistical Sciences, University of Alberta
Applying a Correlated Random Walk Model to LFA-1 Protein Receptors
Single Particle Tracking (SPT) and Fluoresence Recovery After Photobleaching (FRAP)
are two methods used to study the diffusion of membrane proteins in cells. SPT involves
tagging proteins with fluorescent labels and observing their individual trajectories,
while FRAP involves photobleaching a subcellular region of the cell membrane, creating
a bleached population of proteins and allowing the observation of the fluorescence recovery
in this region. Both methods provide important information about protein interactions through
the measurements of diffusion. The methods provide estimates of diffusion coefficients
from an ensemble (in the case of FRAP) or from the trajectories of single molecules (SPT).
While these methods generally agree in a qualitative sense, it has been generally observed
that they rarely agree quantitatively. As an early investigation into this problem, I will
introduce a conventional method used to estimate a diffusion coefficient from a series of
SPT trajectories, and try a different approach (measuring and analyzing turning angles)
to uncover characteristics about protein diffusion. In particular, I will analyze a data
set for LFA-1, an integral membrane protein found on cells associated with the immune system.
Dept of Chemical and Biological Engineering, UBC
Smoothed particle hydrodynamics
In several human diseases like cancer and malaria, an external factor (such as malarial parasites)
changes the mechanical behavior of living cells by starting some biochemical reactions. This
single-cell biomechanical response, such as increasing or decreasing the elastic modulus due to
membrane or cytoskeleton reorganization (in the case of red blood cell infection), facilitates
disease progression in the body. We first present a new model of a living cell based on Smoothed
Particle Hydrodynamics (SPH) method. The particle-based nature of SPH method allows us to go
beyond the continuum approaches and move toward micro/nano structural approaches for a mechanical
model of living cells. This model is utilized to investigate two important phenomena: the
interaction between living cells and external factors of disease; the molecular/structural
changes inside the cells and in the cell membranes during the disease development.
National University of Singapore
Periodic contractions of the lamellipodium
Lamellipodium dynamics have been shown to be important in the sensing of rigidity of the
extracellular environment to guide in the motility of cells. Sheetz's group have discovered local
contractions in the lamellipodium as the cell moves towards its target (eg an area of higher
rigidity) and this is in contrast to past perception that cell migration is composed only of
protrusion due to actin polymerization. In addition to identifying various factors involved in this
phenomenon, a simple mechanism has also been proposed. In my project, I intend to develop a
1D system of equations to reflect the dynamics of the proteins in the cell and evaluate the model
numerically. The model should be able to predict the effect of rigidity of the substrate on cell
migration, and in 2D, provide some insights into the ruffling phenomenon seen in certain cellular
Dept of Mathematics and Institute of Applied Mathematics, UBC
Modelling Drug Resistance in HIV
Mathematical modelling of HIV, both within-host and between-host, is
a very well studied and still fast moving field. In this talk, I
will develop the pre-existing basic model for within-host HIV,
including drug therapy and resistance, and look at some possible
extensions and improvements of this model. The focus will be on the
best ways to model the emergence of drug resistance in the early
stages of HIV.
Student Research Seminars on Wednesday June 4, 2:00PM
Jennifer (Jen) Lindquist
University of Victoria
Bayesian Markov Chain Monte Carlo Parameter Estimation: Application to
Stochastic Epidemic Models
Maximum likelihood (ML), and least squares (LS) techniques, while incredibly common,
are not the only methods available for parameter estimation. In this talk I will introduce
Markov Chain Monte Carlo (MCMC) methods; these are especially useful in high
dimensional (of order 10,000) models, and situations where data is missing or
incomplete. While I will work exclusively in a Bayesian framework, MCMC is also
applicable in frequentist settings.
I will focus on key ideas in Bayesian and Markov theory, with attention to MCMC
implementation algorithms and a discussion of stochastic epidemic models (SEIR type)
with missing data.
Ritsumeikan University, Kyoto, Japan
Biological Receptor Scheme for the External Synchronization of Mutually Coupled Oscillator Systems.
Recently, we have proposed a generalized theory of the synchronization of rhythms based on the concept
of biological receptors. The theory can achieve both mutual synchronization and external synchronization.
However, previous work has neglected the particular case of the external control of a group
coupled oscillators. In nature, we often encounter complex systems that consist of many subsystems,
e.g., organs that consist of many cells. However, without proper control, it is likely for an external
field to break the synchronous behavior of systems. In this study, we generalize the biological receptor
scheme such that external synchronization can be achieved without breaking the mutual synchronization
of coupled oscillators.
Vasthi Alonso Chavez
School of Mathematics, University of Southampton
Mathematical Studies of conservation and
extinction in inhomogeneous environments.
A fragmented (or patchy) ecosystem is an ecological community constituted by those
organisms who have a preferred living habitat with some emergent discontinuities (frag-
mentation) within. In this work we will model this type of ecosystems using some concepts
from Solid State Physics. Specifically, we will use some diffusion concepts to understand
how individuals of a given species can diffuse out of the preferred habitat toward dangerous
regions around the safe habitat. We also want to find an accurate expression to estimate
the critical patch size in order to assure the endurance of the ecosystem, and extend this
work to discover the ecological properties of extended inhomogeneous regions. This results
are expected to be applied directly in two specific ecosystems in Argentina and Belize.
Department of Physics, Georgia Institute of Technology
Networks in Biology
One of the main goals in theoretical biology is to discover or explain
unifying organizational principles across a wide range of observed
structures and functions. Assigning subunits of these structures and
their interactions to nodes and edges of a graph can often prove
helpful in the development of such theories. In fact, much
understanding can often be directly obtained by studying the
properties of the resulting graph, or network. This mode of
investigation is called "network biology." In this SURVEY I will
highlight several results in network biology that I am familiar with
and discuss some tools that prove useful for studying networks.
Simon Fraser University
Abstract: Many different animal species, such as ants and honeybees, live in colonies
as socialized organisms. These organisms are observed to have intellectual behavior. What is
in charge of their behavior?
How are the ant's duties in the morning allocated? How do they communicate? Are they really smart? If they are not smart,
how do the simple actions of individuals add up to the complex behavior of a group? One of the current theories attributes this
behavior to "swarm intelligence". According to this theory, each organism follows simple rules and acts on local information.
No ant explicitly tells the other ants what to do. They use different optimization strategies to solve various problems,
many of which can be exploited as templates for solving complex human problems such as scheduling airlines or routing trucks.