Schedule- July 21-30                      

Friday, July 21                    1:30 PM    research symposium
biomolecular modeling        H. Mahan and S. Smith

Monday, July 24                  noon lunch 385
Monday, July 24                 1:30 - 4 PM    268    research symposium   
dynamical systems               J. Kalafus, L. Zuchlewski              
modelling cancer part 1       Andrew Matteson   

Tuesday, July 25                 1:30 - 4 PM    268    research symposium   
combinatorial matrix theory N. Chenette, S. Droms 
modelling cancer part 2        Thomas Chen

Wednesday, July 26             1:00 PM    268    research symposium   
matrix stability                      H. Friedlander, M. Parrish, J. Ross             

Thursday, July 27                1:00    268    research symposium                
numerical ananalysis            S. Dupal, M. Yoshizawa
                   
Friday, July 28, 2006           11:30 PM    King Buffet    lunch
check-out day  hand in keys to math department         

Participants

Thomas  Chen                      Princeton University
Nathan Chenette                  Harvey Mudd College
Sean  Droms                         University of Mary Washington
Stephen  Dupal                     Rose-Hulman Institute of Technology
Holley  Friedlander               University of Vermont
James Kalafus                       Swarthmore College
Hannah  Mahan                    Hillsdale College
Andrew Matteson                 Texas A&M University
Max Parrish                           Carleton College
Jessica Ross                         Iowa State University
Steve Smith                           Boston University
Michael Yoshizawa              Pomona College
Laura Zuchlewski                 SUNY Buffalo

Projects 2006

Mathematical Systems Theory 
Prof. Wolfgang Kliemann, Prof. Jiyeon Suh, Morgan Baldwin, James J Kalafus, Laura Zuchlewski.
Dynamically Coupled Linear ODEs and Markov Chains as a Modeling Tool.
Expansions and extensions in modelling random processes: useful theorems for building real world applicable dynamic models with the required properties and characteristics.  We used a feedback loop to model dynamic behavior. We often think of our model in terms of human behavior, but it is applicable in the Biological Sciences, in the Social Sciences, and in Neural Networks among others Solutions in R^3 are projected onto the sphere as shown below.

Numerical Analysis Prof. Roger Alexander, Chris Kurth, Stephen Dupal, Michael Yoshizawa
Design and Optimization of Explicit Runge-Kutta Formulas 
Explicit Runge-Kutta methods have been studied for over a century and have applications in the sciences as well as mathematical software such as Matlab's ode45 solver. We have taken a new look at fourth- and fifth-order Runge-Kutta methods by utilizing techniques based on Grobner bases to design explicit fourth-order Runge-Kutta formulas with step doubling and a family of (4,5) formula pairs that minimize the higher-order truncation error.  Grobner bases, useful tools for eliminating variables, also helped
to reveal patterns among the error terms. A Matlab program based on step doubling was then developed to compare the accuracy
and efficiency of fourth-order Runge-Kutta formulas and ode45.

Combinatorial Matrix Theory  Prof. Leslie Hogben, Rana Mikkelson, Olga Pryporova, Nathan Chenette, Sean Droms
Minimum Rank of a Tree over an Arbitrary Field
 For a field F and  graph G of order n, the minimum rank of G is defined to be the smallest possible rank over all symmetric
n x nmatrices  A in over F whose (i,j)th entry is nonzero whenever i and j are distinct and ij is an edge in G. The minimum rank can vary with the field- an example of a graph for which this happens is shown below.  We extended results that give an explicit computational method for the minimum rank of a tree from the real numbers to an arbitrary field.  A  paper will appear in Electronic Journal of Linear Algebra.

Matrix Stability
 Prof. Leslie Hogben, Olga Pryporova, Holley Friedlander, Max Parrish and Jessica Ross.
Survey of Types of Matrix Stability
Several types of stability are introduced and their potential applications were explored. Types of P-matrices were explored with
respect to their implications for stability. A Venn diagram is presented below to display the relationships found.

Biomolecular Modeling and Simulation Prof. Zhijuan Wu, Di Wu, Hannah Mahan, Stephen Smith
1) Computation of Thermodynamic Fluctuations of Proteins 
Using results from NMR and X-Ray Crystallography, mathematical methods can be used to measure structure fluctuation in protein molecules.  In many cases, there is a strong correlation between predicted and computational values of protein fluctuation.
2) Optimizing Protein Structural Alignment Using Simulated Annealing
The goal of this paper is to explore the possibility of improving protein structure alignment through a method of RMSD optimization which utilizes simulated annealing and methods of local optimization.  Using the alignment software PROfit as our benchmark, we analyze RMSD optimization results.  Our optimization method showed an improvement over PROfit in only 22.2% of samples tested.

Modeling Cancer Mathematically Prof. Khalid Boushaba, Prof. Howard Levine, Prof. Michael Smiley, Ajith Gunaratne,  Shiliang (Thomas) Chen, Andrew Matteson
1) Mathematical Analysis of SELEX against Multiple Targets (Chen)   
     We analyze the SELEX (Systematic Evolution of Ligands by Exponential Enrichment) process against multiple targets. The mathematic analysis visits some of the same principles as a paper by Levine and Nilsen-Hamilton, which analyzed the SELEX process for a single target. We first consider the case of mathematical ideal where we assume perfect partitioning and zero background partitioning. The analysis shows that even in the mathematical ideal case, multiple target SELEX scheme does not select for the best binder in general. One has to control the target concentrations very delicately for selection to occur. This is unlike SELEX with a single target where selection is guaranteed for the ideal case.
     We then introduce the kinetics of the support where we derive an  expression for the background partitioning coefficient. We briefly extend some of the mathematical analysis of the ideal case to a SELEX scheme where we include the support kinetics and partitioning coefficients.

2) A Enzyme Kinetic Model of Tumor Dormancy: Regulation of Secondary Metastases by Plasmin (Matteson)  In this paper we simulate in one dimension the model of tumor dormancy that first appeared in [Boushaba,  Levine, Nislen-Hamilton] based on the ideas of [Zetter], that development of regional metastases can be suppressed by inhibitors released from the primary tumor; also, that
the relative rates of diffusion and decay of growth factors, latent and active inhibitors can explain regional suppression and distant
growth of metastases. Our results align closely with \cite{main paper}, in which surgical removal of a primary tumor reduces the
effective suppression, although does not totally eliminate it. We derive an estimate of the maximum distance a primary tumor can suppress a metastasis as a function of the model's parameter space and initial conditions. This analysis of distant suppression was
motivated by a sensitivity analysis to the coefficients of diffusion, especially diffusion of plasmin.