Download What is Mathematical Biology?

Florida Tech’s BioMath Faculty
What is Mathematical Biology?
•Mathematical Biology is a highly interdisciplinary
area that lies at the intersection of significant
mathematical problems and fundamental questions
in biology.
•The value of mathematics in biology comes partly
from applications of statistics and calculus to
quantifying life science phenomena.
What is Mathematical Biology?
•Biomathematics plays a role in organizing
information and identifying and studying emergent
structures.
•Novel mathematical and computational methods are
needed to make sense of all the information coming
from modern biology (human genome project,
computerized acquisition of data, etc.).
BioMath Program at FIT
• Education and research program supported by the NSF and
co-hosted by Mathematical Sciences and Biological Sciences
Departments.
• Program faculty are Drs. Semen Koksal and Eugene
Dshalalow from mathematics and Drs. David Carroll, Richard
Sinden and Robert van Woesik from biology.
• Both undergraduate and graduate students from these two
departments conduct cutting edge research at the
intersection of biology and mathematics.
BioMath Program at FIT
• As of Fall 2009, an undergraduate major in BioMath has
been initiated under the leadership of the biomath
faculty. Every year, six undergraduate (three from each
department) students are financially supported by NSF
for a year long research and training activities in
mathematical biology.
• This program fosters interactions among the
undergraduate and graduate students as well as the
faculty from two departments.
Descriptions of the Current Projects
Population Dynamics of Coral Reefs:
We know little about vital coral population rates and
how they vary spatially, seasonally, and under different
environmental circumstances. Yet these vital rates are
the agents driving the population structures,
community composition, and will ultimately determine
the reef state. We are interested in obtaining universal
functions and probability distributions of vital rates that
can be utilized to predict future population trajectories.
The objectives of this project are two folds:
1) To quantify coral colony growth, partial mortality,
and whole colony mortality and derive functional
relationships that will allow us to develop a
comprehensive population model to predict future
population trajectories;
2) To develop discrete and continuous population
models that will include vital parameters.
Station1
Nishihama
Site 1
Station 2
Station1
Aka Jima
Kushibaru
Site 2
Station 2
0-1 m
3-4 m
6-7 m
0-1 m
3-4 m
6-7 m
0-1 m
3-4 m
6-7 m
0-1 m
3-4 m
6-7 m
Fig. 1 Study site: Akajima , an island of Southern Japan.
In this study, corymbose Acropora coral colonies were tracked through time to
determine growth, partial mortality and mortality rates.
Analysis of data collected during 1996-98 at the sites shown in Fig.1 produced the
patterns these rates follow. Sample graphs are given below.
GROWTH
GROWTH
Size (cm)
PARTIAL COLONY MORTALITY
Size (cm)
WHOLE_COLONY MORTALITY
Currently, our students are in the process of developing and testing two models
that use two different approaches:
Model for the expectation approach
Model for the Boolean approach
We define:
Neural Network Model for PLCγ Signaling
Pathway:
The fertilization signaling pathway that occurs in starfish is
initiated by contact between the sperm and the egg membrane.
Fusion of sperm and egg triggers a cascade of events that leads
to release of intracellular free calcium. The activation of PLCγ is
important in cleaving its substrate PIP2 into molecules IP3 and
DAG. IP3 then binds to its receptor on the endoplasmic
reticulum and allows for an
wave of calcium to propagate through the egg. This
calcium wave is necessary reinitiating the cell cycle and
embryonic development.
Neural Network Model for PLCγ Signaling
Pathway:
The specific goals of this project are to
• Develop an artificial neural network (ANN) to model
the fertilization signaling pathway;
• Use the net to predict the amount of PLCγ activity
required to initiate a Calcium release;
• Test the ANN in living starfish eggs at fertilization.
An artificial neural network has been constructed to
model the PLCγ – dependent calcium release and
growth after fertilization in starfish Asterina miniata.
The neural network whose architecture shown below
processes PLCγ concentration as input and
produces the growth level of the fertilized starfish egg
as its output. This is a multilayer network that uses the
combination of Hebbian learning and backprobagation
algorithm for training.
Figure 3. Neural Net Architecture for Starfish Egg Fertilization. PLCγ is an input
node and Growth is an output node. Intermediate molecules in pathway are
represented as nodes in the hidden layers. Each connection has its own weight,
wi, and its own activation function.
Mathematical formulation and error correction formulas for training are given as:
The initial training results indicate that a certain
“threshold” level of PLCγ activity required for calcium
release. Once the training is complete the NN will be
able to determine this threshold level.
The estimations of the unknown parameters in the
signaling pathway will then be used in a differential
equation model to study the dynamics of the enzyme
activities.
One such model has been already
developed to analyze the MAPK
pathway in starfish oocytes. A brief
summary of the model is given
below.
Modeling the MAPK pathway in starfish
oocytes:
• MAPK is a mitogen - activated protein kinase
and a component of MAPK pathway.
• The MAPK pathway is one of the most
important and intensely studied signaling
pathway that governs growth, proliferation,
cell differentiation and survival.
• It plays a pivotal role during oocyte
maturation, meiosis re-initiation and
fertilization in eggs of various species.
A nonlinear system of differential equations was developed to analyze the enzyme
MAPK activity in a single starfish oocyte. Several steps in this process are still
unknown.
1-MA
…
Phosphatase1
Raf
Raf*
Phosphatase2
MEK
Phosphatase2
MEK-P
MEK-PP
Phosphatase3
MAPK
Phosphatase3
MAPK-P
MAPK-PP
The reactions involved in MAPK Pathway shown above are
The nonlinear system of ODE’s together with the initial conditions are given as
r' c1x(t)m0 
r(0)  0.08M.
k3mp
k1r
k6m


  2(m)
k2  r k4  mp  k5mpp k7  m k8mp

m'

m'pp 

e'

e'pp 

c2r
 1(r)
c3  r

k9 mp
k12mpp
k r
k e


 15  17   3(mpp )
k10  mp  k11m k13  mpp  k14mp k16  r k18  e
q1mpp
q3 ep
q6 e


  4(e)
q2  mpp q4  ep  q5 epp q7  e  q8 ep
q9ep
q10  ep  q11e

q12epp
q13  epp  q14ep

q15mpp
q16  mpp
  5(epp )


m(0)  0.01M
mpp(0)  0.063M
e(0)  0.002M
epp(0)  0.0092M

Where r(t) = concentration of Raf; m(t) = concentration of MEK; mpp(t) = concentration of
MEKpp; e(t) = concentration of MAPKp; and epp(t) = concentration of MAPKpp.
Several graphs obtained from the numerical simulations of
the system based on the experimental data are shown
below.
MEKpp
will trigger the
activation of MAPKpp
graph
plotted in a 40
min interval
initial
concentration
(low value) increases till
it reaches the 0.23M,
when it levels off
Raf*
is red
MEKpp is green
MAPKpp is blue
Estimating Mutation Rates:
The genetic stability of quadruplex DNA structures has not
been analyzed in a model mutational analysis system. In
this project, a mutational selection system that allows
measurement of rates of DNA-directed mutation has been
developed. This involves the insertion of DNA repeats into
the chloramphenicol acetyltransferase (CAT) gene. DNA
insertions usually render the gene inactive resulting in a
chloramphenicol sensitive (Cms) phenotype. Reversion to
chloramphenicol resistance (Cmr) occurs by loss (deletion)
of all or part of the inserted DNA repeats. Differences in
deletion rates can occur from orientation differences of the
repeats because alternative DNA secondary structures can
form and these form at different rates in the leading or
lagging strands of replication.
Biological and mathematical objectives
of this project are to:
• Determine the effect of the DinG helicase on the
genetic stability of a quadruplex-forming DNA
sequence from the human RET oncogene.
• Develop a mathematical model to calculate mutation
rates.