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 andhow they vary spatially, seasonally, and under differentenvironmental circumstances. Yet these vital rates arethe agents driving the population structures, community composition, and will ultimately determinethe reef state. We are interested in obtaining universalfunctions and probability distributions of vital rates thatcan 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.

6-7 m6-7 m

NishihamaNishihamaSite 1Site 1

Station1Station1 0-1 m0-1 m

6-7 m6-7 m3-4 m3-4 m

Station 2Station 20-1 m0-1 m3-4 m3-4 m

Station1Station1 0-1 m0-1 m

6-7 m6-7 m3-4 m3-4 m

Station 2Station 20-1 m0-1 m

6-7 m6-7 m3-4 m3-4 m

KushibaruKushibaruSite 2Site 2

Aka JimaAka Jima

Fig. 1 Study site: Akajima , an island of Southern Japan.

In this study, corymbose Acropora coral colonies were tracked through time todetermine 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 tomodel the PLCγ – dependent calcium release andgrowth after fertilization in starfish Asterina miniata.The neural network whose architecture shown belowprocesses PLCγ concentration as input andproduces the growth level of the fertilized starfish eggas its output. This is a multilayer network that uses thecombination of Hebbian learning and backprobagationalgorithm 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 alreadydeveloped to analyze the MAPKpathway in starfish oocytes. A briefsummary of the model is givenbelow.

Modeling the MAPK pathway in starfishoocytes:

• 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 enzymeMAPK activity in a single starfish oocyte. Several steps in this process are stillunknown.

Raf Raf*

MEK MEK-P

MAPK MAPK-P MAPK-PP

MEK-PP

1-MA

Phosphatase1

Phosphatase2 Phosphatase2

Phosphatase3 Phosphatase3

…

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 c2rc3 r

1(r)

r(0) 0.08M .

m'k1rk2 r

k3mp

k4 mp k5mpp

k6mk7 m k8mp

2(m)

m(0) 0.01M

m'ppk9mp

k10 mp k11m

k12mppk13 mpp k14mp

k15rk16 r

k17ek18 e

3(mpp)

mpp(0) 0.063M

e'q1mppq2 mpp

q3ep

q4 ep q5epp

q6eq7 e q8ep

4(e)

e(0) 0.002M

e'ppq9ep

q10 ep q11e

q12eppq13 epp q14ep

q15mppq16 mpp

5(epp )

epp(0) 0.0092M

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.

MEKpp will trigger the activation of MAPKppgraph plotted in a 40 min interval initial concentration (low value) increases till it reaches the 0.23M, when it levels off

Several graphs obtained from the numerical simulations of the system based on the experimental data are shown below.

Raf* is redMEKpp is greenMAPKpp is blue

Estimating Mutation Rates:The genetic stability of quadruplex DNA structures has notbeen analyzed in a model mutational analysis system. Inthis project, a mutational selection system that allowsmeasurement of rates of DNA-directed mutation has beendeveloped. This involves the insertion of DNA repeats intothe chloramphenicol acetyltransferase (CAT) gene. DNAinsertions usually render the gene inactive resulting in achloramphenicol sensitive (Cms) phenotype. Reversion tochloramphenicol resistance (Cmr) occurs by loss (deletion)of all or part of the inserted DNA repeats. Differences indeletion rates can occur from orientation differences of therepeats because alternative DNA secondary structures canform and these form at different rates in the leading orlagging strands of replication.

Biological and mathematical objectivesof 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.