Florida Tech’s BioMath Faculty

1. Florida Tech’s BioMath Faculty

2. 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.

3. 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.).

4. 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.

5. 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.

6. Descriptions of the Current Projects

7. 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.

9. The objectives of this project are two folds: • 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; • To develop discrete and continuous population models that will include vital parameters.

10. 0-1 m 0-1 m Station1 Station1 3-4 m 3-4 m 6-7 m 6-7 m • Fig. 1 Study site: Akajima , an island of Southern Japan. Nishihama Site 1 0-1 m Station 2 3-4 m Aka Jima 6-7 m Kushibaru Site 2 0-1 m Station 2 3-4 m 6-7 m

11. 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

12. GROWTH

13. PARTIAL COLONY MORTALITY Size (cm)

14. WHOLE_COLONY MORTALITY Size (cm)

15. 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

16. We define:

17. 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.

19. Neural Network Model for PLCγ Signaling Pathway:

20. 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.

22. 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.

23. 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.

24. Mathematical formulation and error correction formulas for training are given as:

27. 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.

28. One such model has been already developed to analyze the MAPK pathway in starfish oocytes. A brief summary of the model is given below.

29. 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.

30. 1-MA … Phosphatase1 Raf Raf* Phosphatase2 Phosphatase2 MEK MEK-P MEK-PP Phosphatase3 Phosphatase3 MAPK MAPK-P MAPK-PP 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.

31. The reactions involved in MAPK Pathway shown above are

32. The nonlinear system of ODE’s together with the initial conditions are given as 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.

33. 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

34. Raf* vs. MEKpp vs. MAPKpp • Raf* is red • MEKpp is green • MAPKpp is blue

35. 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.

37. 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.