ABSTRACTS
Abstracts are listed in
alphabetic order by the last name of the presenting author.
Competitive Exclusion and Coexistence for Pathogens
in an Epidemic Model with Variable Population Size
Ackleh, Azmy* and Linda Allen
DEPARTMENT OF MATHEMATICS AND STATISTICS,
TEXAS TECH UNIVERSITY, LUBBOCK, TX
79409-1042
We study an SIR epidemic model with a variable host
population size. We prove that if the model parameters satisfy certain inequalities
then competition between n pathogens for a single host leads to
exclusion of all pathogens except the one with the largest basic reproduction
number. It is shown that the basic reproduction numbers are necessary but not
sufficient for determining competitive exclusion. Numerical results
corroborating the theoretical ones are presented. An example is given which
shows that if such inequalities are not satisfied then coexistence occurs.
Introduction of multiple pathogen strains into a
host population results in either no disease outbreak, where all pathogen
strains are elinminated or a disease outbreak, where either a single strain
dominates or multiple pathogen strains coexist. Many intrinsic and extrinsic factors affecting the host and
pathogen determine whether a disease outbreak will occur and whether one or
more pathogen strains will coexist.
Some of these factors include the reproduction and mortality rate of the
host, the disease-related death rates, and the horizontal and vertical
transmission rates for each strain.
Vertical transmission is the direct transfer of a disease from an
infective parent to its offspring whereas horizontal transmission is the
passage of infection from one individual to another. The effect these factors have on a disease outbreak are
investigated in some basic deterministic and stochastic epidemic models. Analytical and numerical results are
presented and applications to human and wildlife diseases are discussed.
An Interdisciplinary
Course for Biology and Mathematics Majors
DEPARTMENT OF MATHEMATICS,
HOPE COLLEGE, HOLLAND, MI 49422-9000
Mathematical
techniques from linear algebra, differential equations, and probability have
become an integral part of many areas of biological research. Yet few undergraduate biology majors take
more than the minimum number of required mathematics courses and so have little
or no understanding of these techniques. At the same time, mathematics majors
are often unaware of the wide array of biological problems to which their
expertise can be applied. Furthermore, mathematics majors often lack a
fundamental understanding of the differences between biology and physics (where
mathematical applications are typically given) and so do not understand the
idiosyncrasies involved in biological experimentation. To address both of these
problems, we have recently designed and taught a course for a mixed audience of
biology and mathematics students. This
course is co-taught by a biologist and mathematician to a mixture of biology
and mathematics major, with different prerequisites for each group. The central
theme of the course is for both groups of students to develop the ability to
speak a common language. Rather than a standard textbook, we are using biology
research papers that highlight the interplay of mathematics and biology and how
this gives new insight into biological problems. Students work in mixed groups
where biology and math majors help one another to understand the concepts,
techniques, and limitations of their respective fields. The course includes a
laboratory component focusing on demographic studies of several invertebrate
and plant species. We review the development of the course, some of the
materials we've used, and the results of its first offering during the Spring
2002 semester. This course has been
developed with the support of NSF grant DUE-0089021.
1DEPARTMENT OF BASIC MEDICAL
SCIENCES, PURDUE UNIVERSITY, 1246 LYNN HALL, WEST LAFAYETTE, IN 47907-1246.
2OAK RIDGE NATIONAL
LABORATORY
Problem: Valveless
pumping refers to nonzero mean flow of fluid within a closed loop of elastic
tubing without valves, when a compliant section of the loop is rhythmically
compressed at particular frequencies.
Valveless pumping is thought to play a role in blood circulation in
embryos and in cardiopulmonary resuscitation (CPR). Heretofore, the physical mechanism causing valveless pumping has
remained a mystery.
Approach: We consider closed loops composed of one length of soft,
compliant tubing and one length of stiff, non-compliant tubing. The loop is filled with water and is
compressed near one end of the soft section.
To model such pumps we characterize pulse (pressure) waves in the soft
section using the classical wave equation with reflection of the pulse waves at
the soft/stiff boundaries, and exponential damping of pulse amplitude with
distance traveled by the wave. Pulse
wave velocity is specified by the Moens-Korteweg equation. The resulting instantaneous pressure
difference across the fluid in the stiff segment is computed, and Newton’s
second law is used to describe the instantaneous and average movement of fluid
in the stiff section.
Results: Mean flow equals zero if compression is performed at the midpoint
of the soft segment. However, when
changes in pulse wave velocity caused by expansion of the uncompressed regions
of the soft segment are properly accounted for, nonzero mean flow develops
during asymmetrical compression of the soft segment at particular
frequencies. The direction of flow
depends upon the compression frequency, and flow can be reversed simply by
changing from one compression frequency to another, as was found in earlier
physical and computational models.
Other parameters, such stiffness constants and tubing length also act to
determine the direction of flow.
Conclusion: Reflected pulse waves explain the existence and major features of
valveless pumping on the basis of classical Newtonian physics
Bajaria, Seema H.*1, Glenn
Webb2, Denise E. Kirschner1
1DEPARTMENT OF MICROBIOLOGY
AND IMMUNOLOGY AND DEPARTMENT OF BIOMEDICAL ENGINEERING, THE UNIVERSITY OF
MICHIGAN, 6730 MEDICAL SCIENCE BUILDING II, ANN ARBOR, MI 48109-0620
2VANDERBILT UNIVERSITY
Highly
Active Antiretroviral Therapy (HAART) is presently the only available antiviral
therapy for HIV-infected patients.
HAART has been used clinically in various administration schemes for
several years. However, due to the development
of drug resistance, evolution of viral strains, or poor patient compliance,
this combination of drugs fail to effectively contain the virus long-term in
the majority of patients. Our group and
others have suggested a change to the usual regimen of continuous HAART through
the use of structured treatment interruptions (STIs). STIs are believed to provide the same benefits as continuous
treatment, such as reduced viral loads and return of CD4+ T cells,
while allowing the patient a drug holiday.
We explore the use of STIs using a previously published model that
accurately represents CD4+ T-cell counts and viral loads both during
HIV-1 infection and HAART therapy. We
simulate the effects of different STI regimens including weekly and monthly
interruptions together with variations in treatment initiation time. We show that treatment outcomes depend upon
duration of the interruption, CD4+ T-cell counts and plasma HIV-1
RNA levels at initiation of treatment, and the patient's immune response to
therapy. If a patient exhibits a strong
response to continuous treatment, an STI strategy could prove to be a major
benefit. This approach may be
particularly relevant for underdeveloped countries, where AIDS prevalence and
medical costs are high but resources are lacking.
Banks, John
INTERDISCIPLINARY ARTS &
SCIENCES, UNIVERSITY OF WASHINGTON, TACOMA, 1900 COMMERCE STREET, TACOMA, WA
98402
Recent
congressional legislation aimed at reducing the widespread use of harmful
chemical pesticides has generated renewed interest in the development of viable
alternatives to insect pest control in agroecosystems. The development of a new suite of selective
pesticides designed to target pests while sparing “beneficial” arthropods
provides new opportunities to explore the compatibility of chemical and natural
controls. I describe field experiments
performed in Washington State designed to assess the potential for
incorporating these selective pesticides into an integrated pest management
(IPM) regime for controlling herbivorous aphid pests in a broccoli
agroecosystem. I discuss insect movement models focused on designing better IPM
strategies, as well as an optimization modeling approach aimed at elucidating
mechanisms underlying insect distribution patterns recorded in the field.
The
described research effort is concerned with the central role of dredging
operations in the development and expansion of trade in containerized freight
at the Port of New York/New Jersey over the past quarter century and discovery
of policy implications revealed through exercise of an ecological economic
model capturing dynamic spatial-temporal change of this extended port system . Dredging
practices are recognized as needed to service the operational requirement of
ocean-going vessel access to interior parts of the Ports’ navigation network
and the dredged sediment thus produced are treated as a time series or
disturbance frequency driving succession dynamics of selected immotile species
associated with the benthic community resident at the federally designated
sediment disposal site know as the Mud Dump Site. The disturbance event time
series is analyzed using spectral methods and succession dynamics modeled
according to birth, death, emigration, immigration rates for early and late
colonizers. Some interpretive spatial-temporal trade economic data is also
provided.
Modeling the dynamics of
natural rotifer populations
1*Berezovsky,
Faina, 2Georgy Karev, 3Mark
Borodovsky , 3Terry W. Snell
1DEPARTMENT OF
MATHEMATICS, HOWARD UNIVERSITY, WASHINGTON, DC 20059,
2NATIONAL INSTITUTE OF
HEALTH, BETHESDA MD 20894
3GEORGIA INSTITUTE OF
TECHNOLOGY, ATLANTA, GA 30332-0145
A model of the dynamics of natural rotifer
populations, based on real data for nine species, is described as a discrete
nonlinear map depending on three parameters which reflect characteristics of
the population (a species g-index) and environment. Model dynamics and their change by variation
of these parameters were investigated by methods of bifurcation theory. It was shown that the phase–parametric
portrait consists of 7 domains of qualitatively different behavior of the
model. The portrait allow to predict the principal characteristics of the
population dynamics of other rotifer species by calculating a species g-index for given values of the environmental parameter
and placing it in the parametrical portrait.
The principal
contribution was to show parameter domains where population persistence is
possible (in stable equilibrium, periodic or a- periodic oscillations of
population size) as well as to analyze
domains of population getting to
extinct. Five of the nine rotifer species investigated had population dynamical
behaviors that lead to steady-states in stable environments; for the remaining
species the parametric points lie near domain boundaries. In this case, small
reductions in environmental quality due to toxicant exposure can dramatically
alter the dynamical behavior of a population. A population formerly in
steady-state oscillation may enter a domain of a complex behavior and even
chaotic dynamics, which is likely to drive the population to extinction.
It seems prudent to explore how much
toxicant exposure it takes thrust rotifer populations into a new dynamical
regime. If these exposures are lower than those causing classical toxicity,
this would have important implications for setting safe thresholds for toxicant
exposures in the environment as well as ecological risk assessment.
DEPARTMENT
OF MATHEMATICS, CORNELL UNIVERSITY, MALOTT HALL, ITHACA, NY 14853 USA
A considerable amount of ecological literature has addressed
formulations of competitive exclusion principles; that is, when is it true that
n species cannot persist on fewer than n resources? A well known mechanism for
avoiding such principles is via predator-mediated coexistence: a predator
species can facilitate coexistence between prey species that would otherwise
exhibit mutual exclusion. In the simplest ODE example, a maximum of two prey
species can persist in this manner with the predator, and this bound is shared
by a number of models. We analyze types of predators and feeding behaviors,
finding sufficient conditions for a single predator species to mediate uniform
persistence of more than two prey species. We also consider the number of prey
species that may be supported and describe the population dynamics.
Dynamics of the 'echo' bloom
in a plankton system with nitrogen fixation
Boushaba, Khalid and Mercedes Pascual
DEPARTMENT OF ECOLOGY AND EVOLUTIONARY
BIOLOGY, THE UNIVERSITY OF MICHIGAN, NATURAL SCIENCE BUILDING (KRAUS), 830
NORTH UNIVERSITY, ANN ARBOR, MI 48109-1048
Consideration of nitrogen fixation adds a
positive nonlinear feedback to plankton ecosystem models. We investigate the
consequences of this feedback for secondary phytoplankton blooms and the
response of phytoplankton dynamics to physical forcing. The dynamics of phytoplankton, Trichodesmium
(the nitrogen fixer), and nutrients is modelled with a system of three
differential equations. The model
includes two types of nonlinear interactions: the competition of phytoplankton
and Trichodesmium for light, and the positive feedback resulting from Trichodesmium
recycling.
A typical simulation of the model in
time, with forcing by a varying mixed-layer depth, reveals a clear successional
sequence including a secondary phytoplankton bloom known as an `echo' bloom. We
explain this sequence of events through the stability analysis of three different
steady states of the model. Our
analysis shows the existence of a critical biological parameter, the ratio of
normalized growth rates, determining the occurrence of `echo' blooms and the
specific sequence of events following a physical perturbation. The interplay of positive and negative
feedbacks appears essential to the timing and the type of events following such
a perturbation.
Undergraduate Applied
Mathematics Projects in a Biology Context
Burke, Meghan
DEPARTMENT OF MATHEMATICS,
KENNESAW STATE UNIVERSITY, BOX #0409, 1000 CHASTAIN ROAD, KENNESAW, GA 30144
This presentation will
discuss several project ideas to be used for discovery learning or
reinforcement of mathematical concepts from precalculus, differential and
integral calculus, and differential equations.
The source of these applied projects will be biology, and specifically,
epidemiology. The projects have been
used in the mathematics classroom, but could also be made available to biology
students interested in brushing up on mathematics skills within a biology
context. The projects would be
appropriate for use in courses aimed
specifically at life-science majors, or in courses for all science (including
mathematics) majors. The intuition and
comfort many students have in biology that they may not feel with physics or
other traditional sources for projects can be used to excite their learning.
Including Habitat Quality
and Density-dependent Migration in Spatially Explicit Metapopulation Models
Diego Ruiz Moreno, Graciela Canziani*, Paula Federico
GRUPO DE ECOLOGIA MATEMATICA, FACULTAD DE CIENCIAS
EXACTAS, UNIVERSIDAD NACIONAL DEL CENTRO DE LA PROVINCIA DE BUENOS AIRES,
CAMPUS PARAJE ARROYO SECO, 7000 TANDIL, ARGENTINA
Spatially explicit models are becoming very important
in Ecology. The spatial distribution of different populations have a deep effect on their dynamics as well
as on the landscape.
Patch occupancy models include two different spatial
scales: that of the local neighborhood and that of the global landscape. Among
these models are found Cellular Autamata, which permit to visualize the
evolution of the spatial dynamics of populations and analyze the strategies of
the species in regard to environmental changes. Cellular Aurtomata permit to
include spatial heterogeneity is a simple manner.
A standard tool to link spatial heterogeneity to
population dynamics is the definition of Habitat Quality Indices, usually
respecting the philosophy of Habitat Suitability Evaluation Procedures (HEP).
Here we introduce a methodology for the utilization of remote sensing information for the
production of dynamic maps for landscape classification. The corresponding
habitat quality indices are then defined and used to feed the discrete
mathematical population models that run within the framework of Cellular
Automata. These local population models permit to calculate population
densities depending on the quality of the patch and to define migration
processes.
Models that have these characteristics permit, as a
first approach, to monitor the impact of environmental changes on the dynamics
of the population under study as well as on the landscape. Moreover, these
models are appropriate for the analysis of different strategies that could help
improve the habitat conditions of species that suffer the stress of
negative environmental changes.
Spatial
modeling to evaluate the effects of habitat loss on mammal populations in
tallgrass prairie
Carr*, Eric1,2,
Henriette Jager2, Tanya Kostova3, Rebecca Efroymson2,
Tina Carlsen3, Tom Ashwood2, and Jim Kercher3
1UNIVERSITY OF TENNESSEE,
KNOXVILLE, TN
2OAK RIDGE NATIONAL
LABORATORY (ORNL), OAK RIDGE, TN
3LAWRENCE LIVERMORE NATIONAL
LABORATORY (LLNL), LIVERMORE, CA
We
developed spatial models to quantify the effects of petroleum-related habitat
loss on mammal populations in the Tallgrass Prairie Preserve in Oklahoma. We focus mainly on a demographic model,
developed at ORNL, for the American badger (Taxodea
taxus). The badger is a voracious
predator on small burrowing mammals.
Males and females have large, overlapping home ranges, and mate during
summer when the probability of encountering a mate is high compared with the
winter, when the animals become dormant.
Implantation and fetal development are delayed until early spring in
anticipation of more-abundant resources.
Spatial variation in the badger model is mediated by a habitat
suitability index that reflects the availability of prey in different
vegetation types. Native grasses, the
dominant category of vegetation in the Preserve, are preferred over other categories. Habitat suitability influences the
acquisition of a home ranges and survival probabilities of simulated
individuals. A
second model, developed at LLNL, simulates the trophic dynamics of the prairie
vole (Microtus ochrogaster). The trophic model includes seasonal dynamics
of vegetation and its depletion by the herbivorous vole. The model reflects the territorial behavior
of the vole, which is predominantly monogamous. Prairie voles breed throughout the year, but breeding activity
and life expectancy have seasonal dependence. The vole model incorporates
bioenergetics and movement that depends on population density, availability of
resources, and mating opportunity. For
each model, we simulated population size on landscapes with increasing areas of
habitat and differing distributions of habitat subjected to brine spills. In addition to the amount of habitat removed
by spills, we compared population-level effects of one large spill with the
effects of many small spills summing to the same area. Simulated population responses to habitat loss
were related to the trophic position and spatial life histories of the two
species.
Spatio-Temporal and long
term Mathematical Models on Methane Emission from rice fields
**DEPARTMENT OF PURE
MATHEMATICS, CALCUTTA UNIVERSITY, 35, BALLYGANGE CIRCULAR ROAD, WEST BENGAL,
INDIA
Mathematically
the phenomenon of methane emission is a continuous process varying with time.
Naturally in this connection, attempt has been made towards developing a
continuos dynamic model of methane emission from rice fields expressed by a
system of ordinary differential equations with time as the independent
variable. As much as possible, all factors affecting the growth of methane has
been taken into considerations. These factors are taken as depending variables
and their growths are expressed by ordinary differential equations. These
differential equations taken together form the required dynamic model.
Stability analysis of this dynamical system ascertains the long-term behavior
of the model. In this connection, there arises an optimal control problem.
Secondly, the growth of methane depending on various factors considered above,
also changes significantly with change of places. In other words the growth of
methane is also spatio-temporal in nature. The change of different factors
responsible for the growth of methane with respect to space co-ordinates
separately resulting again as three systems of ordinary first order
differential equations, each system has been solved separately. Each such
solution has given an estimate of how growth of methane changes as a particular
co-ordinate of the points of the space (other two co-ordinates remaining the
same) changes. A net change with respect to space co-ordinate has determined by
total changes in directions taking two of them fixed. Lastly the
spatio-temporal and long term results have been combined with those obtained in
separate cases.
Drug-induced
amplification of resistant tuberculosis strains
DEPARTMENT OF
BIOMATHEMATICS, UCLA
We
develop a model that describes the epidemic dynamics of an arbitrary number of drug resistant strains of Mycobacterium
tuberculosis. The model tracks the dynamics of susceptible individuals,
individuals latently infected with M. tuberculosis and diseased
individuals. The stability criteria R0 corresponding to the growth
of each of the drug-resistant strains is found to be analogous to the stability
criterion for the growth of a single strain of M. tuberculosis.
Numerical simulations of the differential equations using reasonable parameter
values reveals a series of sequential epidemics caused by the drug-resistant
strains. These epidemics are predicted to continue to increase for hundreds of
years (despite continued treatment of tuberculosis) if the sequential treatment
of cases of tuberculosis (due to inadequate treatment regimens) leads to the
selection of cases of acquired resistance that have progressively lower cure
rates..
Host-Parasitoid
Systems: Modeling the Forest Tent
Caterpillar
Cobbold, Christina*, Lele
Subhash, Lewis Mark, Lutscher Frithjof, Roland Jens
DEPARTMENT OF MATHEMATICS
AND STATISTICAL SCIENCES, UNIVERSITY OF ALBERTA, EDMONTON, T6G 2G1, ALBERTA,
CANADA
Host-Parasitoid
systems are often ideal candidates for integro-difference models, with
dispersal and dytnamics occurring as disinct events with non-overlapping
generations. The Forest Tent
Caterpillar has a life cycle which follows this behavior. The principle parasites of the caterpillar
are fly species which also adopt the required life cycle for and
integro-difference model.
The
caterpillar population is regarded as a pest causing major forest defoliation
during outbreak years. Although not
killing the trees they do slow wood growth.
By using an integro-difference model of the host-parasitoid system we
hope to examine how habitat fragmentation can effect host-parasitoid
populations and duration of the outbreak period. From a management perspective the goal is not necessarily
eradicate the caterpillar population but minimize the number of years at which
we see outbreak densities.
The Roles of Serial Engagement and Kinetic Proofreading in
Peptide-Induced
T-Cell Activation
T-10, THEORETICAL BIOLOGY
AND BIOPHYSICS, LOS ALAMOS NATIONAL LABORATORY, LOS ALAMOS, NM 87545
The
activation of a T-cell requires the formation of a long-lived attachment to an
antigen-presenting cell (APC). APCs
present peptide on their surfaces, held by a major-histocompatibility complex
(MHC). A given T-cell carries receptors (TCRs) specific for a particular
MHC-peptide group, along with other less specific adhesion and costimulatory
molecules. The stable region of close apposition (immunological synapse) may
facilitate signal transduction, by concentrating the TCR and MHC-peptide
together and allowing long-lived bond formation. A TCR will become activated if it can form a sufficiently
long-lived bond to allow multiple biochemical changes to occur (the kinetic
proofreading hypothesis). We have
developed a mathematical model to examine the TCR-MHC-peptide interaction
within the stable region and use it to study (1) the competing effects of serial engagement (sequential activation
of TCR by one MHC-peptide) and kinetic proofreading, and (2) the possible role
of TCR oligomerization in activation of T-cells. In conjunction with the model,
recent experimental data indicates that activated TCR must remain active for a
period after dissociation from MHC-peptide. Recent extensions of the model to
deal with other situations will also be presented.
*Crowley, Philip H.,
Christopher Stieha, and D. Nicholas McLetchie
DEPARTMENT OF BIOLOGY AND
CENTER FOR ECOLOGY, EVOLUTION & BEHAVIOR, UNIVERSITY OF KENTUCKY,
LEXINGTON, KY 40506
Some
sessile organisms are capable of growing over their neighbors or parts of
themselves and thereby blocking access to essential resources—resulting in
mortality, abscission of covered distal parts, or fragmentation into
functionally separate individuals.
Bryophytes, such as the liverwort Marchantia, fragment into
separate ramets by sloughing overgrown tissue within mats on substrate
patches. Simulating this process in
MATLAB yields insights into the competitive determinants of sex ratio in these
organisms inhabiting tropical rainforests.
It emerges that between-sex overgrowth competition eventually leads to
elimination of one sex or the other (often males) from the patch in a fashion
resembling unstable Volterra (logistic) dynamics, as anticipated in simpler
models of this system. But the role of
within-patch reproduction and disturbance in determining the outcome seems to
be reduced in this more realistic representation.
Culshaw,
Rebecca Veronica and Ruan, Shigui (research advisor)
DEPARTMENT OF MATHEMATICS AND STATISTICS,
DALHOUSIE UNIVERSITY, HALIFAX, NOVA SCOTIA,
CANADA
The role of the specific immune
response to HIV infection has received much attention in recent years. In this
paper, we attempt to determine the qualitative nature of optimal chemotherapies
for HIV when immune response is specifically considered. Our model is a
three-dimensional system of ordinary differential equations based upon a model
presented by Wodarz and Nowak (). We approach the problem from an optimal
control perspective. We seek to maximise levels of healthy CD4+ cells and
immune response cells while minimising drug cost. Existence of an optimal
control is proven, and it is characterised in terms of the state and the
adjoint variables. Numerical simulations of the optimality system reveal the
optimal treatments are linearly decreasing from their maximum value, and that
at the final time, we retain a portion of the initial pool of immune response
cells, while reducing infection to very low levels.
Non-equilibrium
Competition
Cushing, J. M.
DEPARTMENT OF MATHEMATICS,
UNIVERSITY OF ARIZONA, 617 N. SANTA RITA, TUCSON, AZ 85721
The
fundamentals of competition theory in ecology (ecological niche, competitive
exclusion, limiting similarity, etc.) are founded on the well-known
Lotka/Volterra two species competition model. The Lotka/Volterra model permits
only a limited number of dynamic scenarios, all of which involve only
equilibrium asymptotics. While it is true that many other models also support
such a dynamically restricted, equilibrium theory of competition, there are
exceptions -- exceptions that often contradict some of the basic tenets of
classical theory. In this talk I will discuss such a competition model. The
model is based on a single species model (the LPA model) that has an
outstanding track record of successful application to real population data. The
LPA model has been extensively used during the last decade to study the
dynamics of an insect species (flour beetles) that, as it turns out, played a
significant role in the early history and development of competition theory.
Indeed, several famous laboratory experiments involving flour beetles carried
out by Thomas Park during the 1940-1960's helped to establish the fundamental
principles of competitive exclusion and ecological niche. Park's experiments
were designed explicitly to test Lotka/Volterra theory and, to this day, are
still cited in most ecology textbooks. The experimental results, however,
contain anomalies that are not mentioned in these citations (although Park and
his collaborator P. H. Leslie wrote about them in several papers). These anomalies
seemingly do not support Lotka/Volterra dynamics or the basic principle of
competitive exclusion (namely, that two species cannot share a single limiting
resource). I will show how the LPA model can provide a possible explanation of
these anomalies. The explanation involves, however, non-equilibrium dynamics
and non-Lotka/Volterra competitive scenarios that "contradict", or
are not encompassed by, the classical theory.
Aggregation and centering in
fish melanophore cells - a quantitative exploration of cytoskeletal dynamics
Cytrynbaum, Eric1,
Alex Mogilner1, Vladimir Rodinov2
1INSTITUTE OF THEORETICAL
DYNAMICS, UNIVERSITY OF CALIFORNIA, DAVIS, 2201 ACADEMIC SURGE, ONE SHIELDS
AVE., DAVIS, CALIFORNIA 95616
2UNIVERSITY OF CONNECTICUT
Fish
melanophore cells demonstrate a self-organizing behaviour that depends on the
same cytoskeletal components that are at work in the centering of chromosomes
during mitosis and therefore provide a good "warmup" problem for that
more complicated and vital process.
When a fragment of the melanophore cell is excised, eliminating the
centrosome (the regular cytoskeletal organizer) and therefore the cytoskeletal
structure, stimulating the cell with adrenaline somehow reintroduces
cytoskeletal organization, leading to the formation of a microtubule aster and
the aggregation of the cell's pigment particles at the center of the
fragment. It is this centering
behaviour that is analogous to the mitotic process of chromosome alignment.
We
derive a system of seven non-linear PDEs (1D) that describes the biological
system. Numerical simulations of the
equations demonstrate certain observed features (aggregation) but not others
(centering). The system can be reduced
so as to facilitate analysis which allow for an understanding of the successes
and failures of the original model.
Finally, we generalize the reduced model to 2D, incorporating a
stochatic element, and present numerical results.
Modeling Biological Control of Scentless Chamomile Using a Optimization
on a
Coupled Map Lattice
de-Camino-Beck, Tomas*1 , Lewis, Mark1, McClay, Alec2
1CANADIAN
CENTER FOR MATHEMATICAL BIOLOGY, DEPARTMENT OF BIOLOGICAL SCIENCES, UNIVERSITY
OF ALBERTA, EDMONTON AB, T6G 2E9, CANADA
2ALBERTA
RESEARCH COUNCIL
Scentless Chamomile (Matricaria
perforata Mérat) is an
invasive annual or short-lived perennial weed that is becoming a pest problem
in Canada. It is a patchily distributed plant which can rapidly increase to
outbreak densities on disturbed sites. Several potential biological control
agents have been tested empirically, but little is known about effect that
these agents have on the persistence and stability of scentless chamomile
populations, and what biological control optimal release strategy would work
best to reduce scentless chamomile populations.
Our goal is to determine
optimal strategies for biocontrol of scentless chamomile. Our first step was to
develop a multi-stage coupled map lattice model with long distance dispersal,
to study the plant-herbivore interactions in a spatial context. This model will
allow us to study the interaction between scentless chamomile and the
biological control agents (the gall midge Rhopalomyia
tripleurospermi and the
seed weevil Omphalapion hookeri), and
provide a solution for the best release strategies for biological control. To
consider spatial heterogeneity we link our model to a GIS landscape information
layer. Future research includes linking the model to a genetic algorithm to
find optimal release strategies.
De Boer, Rob J.*1, Jose A. M. Borghans2,
Michiel van Boven3, Can Kesmir1 & Franz J. Weissing4
1THEORETICAL
BIOLOGY UU, PADUALAAN 8, 3584 CH
UTRECHT, THE NETHERLANDS
2INSTITUT
PASTEUR, PARIS
3INSTITUTE
FOR ANIMAL SCIENCE AND HEALTH, LELYSTAD
4
THEORETICAL BIOLOGY, UNIVERSITY OF GRONINGEN
Molecules from
the major histocompatibility complex (MHC) play a crucial role in vertebrate
immunity and are encoded by the most polymorphic genes known for vertebrates.
Since hosts heterozygous at the MHC express a higher diversity of MHC molecules
to present peptides from infectious pathogens, there should be selection for
heterozygous hosts. We have developed a
mathematical model to show that under realistic assumptions heterozygote
advantage yields a small degree of MHC polymorphism. A large degree of polymorphism can only be accounted for if the
different MHC alleles confer unrealistically similar fitnesses. Predicting the immunodominant peptides from
various common viruses we find that different MHC alleles are expected to provide
quite different levels of protection. Heterozygote advantage is therefore insufficient to explain the
high observed polymorphism of the MHC.
Gustavo Carrero, Ellen Crawford, Michael Hendzel, and Gerda de Vries*
DEPARTMENT OF MATHEMATICAL AND STATISTICAL SCIENCES, UNIVERSITY OF
ALBERTA, EDMONTON, AB, T6G 2G1, CANADA
Fluorescence Recovery After
Photobleaching (FRAP) is an imaging technique used to study the mobility of
proteins in the cell nucleus. In FRAP experiments, the protein being studied is
tagged with Green Fluorescent Protein (GFP). An intense laser beam is used to
bleach the fluorophore of the tagged proteins within a small region of the cell
nucleus. Due to diffusional exchange between the bleached and unbleached
proteins, fluorescence in the targeted area recovers. The fluorescence recovery
data can be used to quantify the mobility of the proteins. We propose the use of
compartmental models to interpret FRAP data and estimate dynamical parameters.
In particular, we develop a compartmental model that describes the dynamics of
nuclear GFP-actin. The fluorescence
recovery data for this protein cannot be explained by simple diffusion alone.
We can obtain a good fit of the data if we include in our model terms that
describe association and dissociation of actin monomers with filaments. The
model supports the hypothesis that actin is present in both monomeric and
filamentous forms. We predict the average residency time of GFP-actin within a
filament, the average wandering time of monomers between bindings, and estimate
the proportions of GFP-actin in monomeric and filamentous forms.
Dixon, Kenneth*1, and Lorinda L. Sheeler2
1THE INSTITUTE OF
ENVIRONMENTAL AND HUMAN HEALTH, TEXAS TECH UNIVERSITY, BOX 41163, LUBBOCK, TX
79409-1163
2TENNESSEE DEPARTMENT OF
HEALTH
Individual
animals are exposed to contaminants at different environmental concentrations
depending upon their location in time and space. We developed an individual based, spatial, stochastic model that
simulates exposure to spatially varying environmental concentrations of
contaminants. As the animal moves
around the landscape, it receives a contaminant dose through ingestion of food
items with varying concentrations. The
body burden of each individual then is tracked over time. Many sub-lethal toxic effects of exposure to
contaminants affect the health status of the individual. To model these effects, we need to follow
the status of the animal over time.
Population effects are estimated by simulating many individuals. Modeling health status, and the probability
of a sub-lethal effect, require a modeling approach that maintains a history of
the health status and not just a distribution of the status in the population. Our modeling approach is to use multidimensional
structure arrays to track health status, geographic location, and other
variables for each individual in the population. Mathematical functions, such as arithmetic means and variances,
can be calculated for each variable operating on fields in the arrays using
indexing. A current application of the
model is the simulation of the effects of perchlorate on a raccoon population
at a superfund site.
Duke-Sylvester, Scott M.,
Gross, Louis J., Salinas, Rene A., Purucker, Thomas, Joshi, Hem R.,Harrell,
Susan
DEPARTMENT OF
ECOLOGY AND EVOLUTIONARY BIOLOGY, 569 DABNEY HALL, KNOXVILLE, TN 37919-1610
The development of
resistance by infectious micro-organisms to antibiotics is a matter of global
concern. Resistant pathogens present a significant health risk to humans in
both developed and developing nations. Among the pathogens of concern is Mycobacterium tuberculosis (TB). A number
of different techniques have been proposed to limit the buildup of resistance
including drug cycling. In drug cycling patients are grouped based on when they
became infected and all are treated with a single anti-microbial agent. In this
way a single drug is used for all new infections that occur over some period of
time. Individuals who become infected before or after the given interval fall
into other groups which receive different single drug treatments. Several
studies have reported the successful use of this approach to limit the
development of resistance. We have developed a model of resistance dynamics for
TB that incorporates drug cycling in a number of discrete populations connected
through migration. This model is used in an optimal control framework that
minimizes the total resistance through time and across all populations. This
approach allows us to address two important questions. First, it is known from
empirical work that drug cycling can limit the build up of resistance. Given
this observation, what sequence of drug application provides the smallest
buildup of resistance. Second, how does spatial structure effect the treatment
plan, and is a spatially-structured treatment plan substantially better than a
plan in which all regions receive the same sequence of treatment.
Ellner, Stephen
DEPARTMENT OF ECOLOGY AND EVOLUTIONARY BIOLOGY, CORNELL UNIVERSITY,
E145 CORSON HALL, ITHACA, NY 14853-2701
Population data, especially
from outside the laboratory, are typically noisy in two ways that complicate
model fitting: process noise and measurement noise. Until recently, methods for
fitting mechanistic models to population time series data could only cope with
these one at a time: deterministic models with inaccurate data, or stochastic
models with accurate data. I will describe two methods that can handle both at
once: gradient matching, and indirect inference using nonlinear forecasting.
Gradient matching is more efficient but makes stringent assumptions about the
data; I will illustrate its use on some laboratory insect populations,
including some non-Nicholson blowflies. Indirect inference is very general but
computationally demanding. I will describe an analysis of long-term data on
outbreak cycles of a forest insect pest (Bupalus piniarius), in which
comparative model-fitting is used to evaluate 3 longstanding hypotheses for the
mechanism underlying the cycles: interactions with food supply, parasitoids,
and delayed self-regulation through maternal effects.
Mathematical Study of Some Immune System
Response Models
Elsady, Zeinab
25 HAMOUDA ST. HELWAN
GARDENS, CAIRO, EGYPT
Nonlinear systems associated with: Normal T cells model and
HIV and cancer immune interaction models adopted with biological theory will be
given. Despite of there are several
studies associated with the solution of these models but there are no
mathematical details about their solution.
In addition, examination of long-term behavior of the dynamical system
by using small time step requires extra computing costs. So the need to use
numerical methods, allows largest possible time step and in the same time
consistent with stability and accuracy criteria.
Stability
and Chaotic Dynamics in Deterministic and Stochastic Structured Epidemic Models
Emmert, K. * and Allen, L. J. S
DEPARTMENT OF MATHEMATICS AND STATISTICS, TEXAS TECH UNIVERSITY,
LUBBOCK, TEXAS
A
discrete-time structured model for the spread of infection in a host population
is developed and analyzed. The host population is subdivided into larvae,
juvenile and adult stages. In the absence of infection, conditions are derived
for existence and stability of a positive equilibrium. In the case of a Ricker
type density-dependence in the larval stage, the model is shown to exhibit
chaotic dynamics. Conditions for
persistence of the infection in the host population are studied. In addition, an analogous stochastic model,
a discrete-time Markov chain model, is formulated. The dynamics of the
deterministic and stochastic models are compared and illustrated with several
numerical simulations.
LIFE AS A QUANTUM STATE:
SOME IMPLICATIONS FOR QUANTITATIVE BIOLOGY
Finkel, Robert
PHYSICS DEPARTMENT, ST.
JOHN’S UNIVERSITY, 8000 UTOPIA PKWY., JAMAICA,
NY 11439
Quantum
physics determines the properties of atoms and molecules—properties not
evidenced by separated electrons and nuclei. The whole exceeds the isolated
parts because mutual interactions are included. Similarly, the integrity and
synchronization of living cells is only partly explained by detailing
biochemical pathways and nucleic acid sequences. The interacting processes of
such elements constitute life. A simple physical model of the living state can
be based on a cooperative quantum effect.1 Here autocatalyic
chemicals in a solution of substrate interact to form a coherent aggregate with
a negative energy spectrum. Two parameters of the theory, presumed to be
universal, are determined from E. Coli data. We show that the aggregate is
stable against thermal assaults and can grow by accretion in sufficient
substrate. The model demonstrates quantitatively how cells can convey solitary
short-range molecules to appropriate locations against any classical
statistical expectations. We derive the venerable Kleiber law relating
metabolic rates to mass as an immediate consequence of the theory.
[1] R.W. Finkel, Nuovo Cimento, 221B, 41 (1978).
Fister, K. Renee
and Jon Ernstberger* (student)
6C FACULTY HALL,
DEPARTMENT OF MATHEMATICS AND STATISTICS, MURRAY STATE UNIVERSITY, MURRAY,
KY 42071
We investigate a mathematical model that studies the dynamics
between tumor cells, immune-effector cells, and cytokine interleukin -2. In
order to determine under what circumstances the tumor can be eliminated, we
implement optimal control theory. Via an objective functional, we maximize the
effector cells and the interleukin-2 cells while minimizing the tumor burden
and the cost of the control. We show that an optimal control for this problem
exists and is unique through the use of an optimality system. Then we analyze
the system graphically using Matlab.
Computational Challenges in Modeling Blood Clotting
Fogelson, Aaron
DEPARTMENT OF MATHEMATICS, UNIVERSITY OF UTAH, SALT LAKE CITY, UT
84112
Fluid dynamics,
biochemistry, and biophysics, in particular cellular adhesion processes,
interact in extremely complex ways to determine whether, where, to what size,
and how stably blood clots form within the vascular system. Modeling these processes poses substantial
computational challenges. We will
describe efforts to build
computational
models of clotting at the cellular level that include immersed-boundary-based
fluid-cell interactions and both fluid-phase and cell-surface
phase-biochemistry. We will also
describe novel computational methods for studying continuum models of clotting
that have important features in common with classical viscoelastic polymer
flow models.