Descriptions of Sample Research Projects
Note: If the title of a project carries a link, click on it to find out the student(s) who have worked (and possibly are still working) on the project. This list represents only a sample of projects, and is not exhaustive. We plan to update this list periodically.
Resistance to cancer chemotherapeutic drugs
Cancer therapeutics is often thwarted by rise of drug resistant mutations. Guanylate kinase is an enzyme involved in the conversion of the cancer chemotherapeutic drug, 6-thioguanine (6TG), to its toxic product. Resistance to 6TG has been implicated in treatment failure for patients receiving this drug. To assess the nature of this drug resistance, mutations were randomly introduced into the guanylate kinase gene. Functional variants displaying drug resistance were then selected and the particular mutation(s) identified by DNA sequencing. A mathematical analysis of the data combined with mapping where these mutations lie on the structure of guanylate kinase would serve to expand our understanding of how these mutations alter the activity of the enzyme towards the drug but not towards the normal substrate.
| Faculty Mentor(s): | Judi McDonald, Math; Margaret Black, Pharmaceutical Sciences. |
| Math Topics: | Linear algebra. |
| Biology Topics: | Resistance to drugs, mutagenesis. |
| Math Work: | Analyze mutation data using MATLAB. |
| Biology Work: | Use data from previous studies. |
Matrix population projection
Matrices have long been used to project population growth when vital rates (i.e., birth and survival rates) depend on an individual's age, stage (e.g., size, developmental state), or both. Yet it can be difficult or expensive to estimate vital rates for many species and therefore a question of practical importance is: Can age or stage be ignored when projecting population growth? This question will be addressed with a project that includes both theory and data analysis. The theory will consider how closely age-only or stage-only population projections match full age-stage projection, for different methods of averaging stages within an age or ages within a stage. The different averaging methods will employ different weighting schemes. Students will address these issues using analytical methods and with computer simulations. The most promising methods identified by the theoretical analyses will then be tested on available data sets and on data from an ongoing experiment on beetle population growth. The data sets will contain age- and stage-dependent vital rate information and population time series. The result of this project will be a set of practical guidelines for biologists to determine the most valuable vital rate information for projecting population growth in their study system.
| Faculty Mentor(s): | Richard Gomulkiewicz, Math and Biology. |
| Math Topics: | Statistics, calculus. |
| Biology Topics: | Population growth and vital rates. |
| Math Work: | Develop theoretical models for population projections using single variables, and various averaging schemes for each model, test these models on experimental data (some use of MATLAB). |
| Biology Work: | Literature review, use data from previous studies. |
Evolution of form and function in labrid fish skulls
The central goal of my research program is to understand evolutionarily important factors that influence the morphological, mechanical, functional, and ecological richness of a group. Why are some groups morphologically diverse? Does morphological diversity always signal mechanical, functional or ecological diversity? To address these questions I draw upon an interdisciplinary approach that crosses traditional boundaries between functional morphology, molecular phylogenetics, and theoretical evolution. Students will have the opportunity to develop biomechanical models of fish skulls and fins and to develop quantitative theoretical frameworks for exploring how these functional traits evolve.
| Faculty Mentor(s): | Michael Alfaro, Biology. |
| Math Topics: | Mechanics |
| Biology Topics: | Phylogenetics, evolution |
| Math Work: | Regression and other model fitting, some use of MATLAB |
| Biology Work: | field work involving fishes |
How does diversity evolve?
One of the most striking patterns in biology is that some groups, like cichlid fishes show a great deal of variation in form, while others, like lampreys, are highly conserved. UBM students working in my lab will answer questions about the evolution of diversity using coral reef fish as a model system. Potential projects include programming evolutionary simulations of trait evolution, comparing fins, skulls and bodies of the diverse reef fish groups like puffers and wrasses, with computer imaging, and phylogenetic-computational studies of evolutionary rates. See http://alfarolab.sbs.wsu.edu for more info about various projects undertaken by the Alfaro lab.
| Faculty Mentor(s): | Michael Alfaro, Biology. |
| Math Topics: | Geometry, force/moment calculations. |
| Biology Topics: | Form of fish species, evolution. |
| Math Work: | Modeling fish forms using rods etc. solving systems of equations to calculate forces/moments, working with MATLAB or Mathematica, some coding involved. |
| Biology Work: | Literature review, use data from previous studies. |
Analysis of gene expression using a gene network model
Gene array technology enables tracking differential expression of thousands of genes on a genomic scale in a single experiment. The rate of data generation via microarrays currently outstrips the pace of unraveling the underlying biological significance of these changes. Mathematical models provide a promising route to elucidate biologically meaningful information from gene array data. Using microarray data, mathematical models can be used to analyze which genes are affecting which other genes and thus to determine the gene network. This approach will dramatically enhance efforts to utilize microarray data for increased understanding of biological systems.
| Faculty Mentor(s): | Judith McDonald, Mathematics, Derek McLean, Animal Sciences, and Nairanjana Dasgupta, Statistics. |
| Math Topics: | Matrix analysis, statistics. |
| Biology Topics: | Gene expression networks, microarray data. |
| Math Work: | Search for patterns in gene array data using matrix analysis and statistics techniques (work in MATLAB or a similar package). |
| Biology Work: | Literature review, use data from previous studies. |
Optimization models for wildlife habitat management
A major challenge for wildlife management is the design of habitat reserve areas to support endangered species. The establishment of reserve areas generally removes a resource from other uses and consequently has an economic cost. Techniques of optimization can be employed to maximize species benefit for a specific level of economic cost or to minimize economic cost for a specified level of benefit. For territorial dispersers, like the Northern Spotted Owl (NSO), researchers have developed an index function for comparing the value of various habitat configurations to the NSO. The use of this index function as the objective in an optimization program has only been briefly explored. A thorough examination of the practical (time) limits is of interest. Students participating in this project would explore this issue while becoming familiar with the index function and optimization software. They will design multiple sets of habitat landscapes and related optimization problems to test the limits of practical use of the NSO index in reserve area design. Depending on the success of results obtained for simulated data, the model will be applied in the future to study real-life data collected by biologists at WSU.
| Faculty Mentor(s): | Bala Krishnamoorthy and David Allen, Mathematics. |
| Math Topics: | Cost/benefit analysis, integer optimization. |
| Biology Topics: | Habitat reserves, species that disperse territorially. |
| Math Work: | Simulating habitat scenarios using Excel or using MATLAB or C, finding optimal dispersals using a software package (AMPL/CPLEX). Some coding involved. |
| Biology Work: | Literature review. |
Evolution of heritable genetic variation of function-valued traits
Function-valued traits are traits that can be described by mathematical functions of a continuous index. Some examples of function-valued-traits include performance as a function of temperature and body size as a function of age (growth trajectories). This study seeks to understand how heritable genetic variation is generated and maintained for a class of complex characters known as function-valued traits. This student project is based upon the development and analysis of mathematical models of genetic, ecological, and evolutionary processes, hence it lies at the interface between biology and mathematics. Heritable genetic variation is the fuel of evolutionary change and understanding how genetic variation itself evolves over time is crucial to the development of accurate models of natural selection. Currently models have been developed for univariate traits (such as height). Models for multivariate traits are also being developed (such as stature which is height and weight) through extensions of the models of univariate traits. The models of multivariate traits must take into account synergistic or antagonistic effects between characters from the fitness of the organism as well as pleiotropy (where a single gene affects multiple traits). The infinite extensions of multivariate traits are function-valued traits. The resulting model will give a clearer picture of how the heritable variation observed for function-valued traits evolves which in turn will give a clearer picture of how organisms evolve over time.
| Faculty Mentor(s): | Patrick Carter, Biology and Richard Gomulkiewicz, Biology and Mathematics. |
| Math Topics: | Rates of change, differential equations. |
| Biology Topics: | Genetic variation of traits, larval amphibians. |
| Math Work: | Solving systems of differential equations (MATLAB), analyzing experimental data (MS Excel). |
| Biology Work: | Literature review, use data from previous studies. |
Diffusion of drugs through human skin via transdermal patches
Transdermal drug delivery is a useful way to delivery a variety of drugs and hormones. The permeability of human skin has been studied for many years and yet it is still not completely understood. The skin acts as a shield to the body protecting it from environmental toxins and is made up of three layers; the stratum corneum, or horny layer, the epidermis and the dermis. The project involves modeling the diffusion of various chemicals through the skin under various conditions. The work involved a mathematical model in the form of a nonlinear partial differential equation that can be applied to several different types of drugs, such as nicotine, herbal supplements for weight loss and hormones for contraceptive purposes. The goal is to get a better understanding of transdermal drug delivery that can lead to the development of a transdermal insulin patch to release insulin at precise rates to diabetic patients.
| Faculty Mentor(s): | Valipurum Manoranjan, Mathematics. |
| Math Topics: | Partial differential equations (PDE). |
| Biology Topics: | Drug diffusion rates through skin layers. |
| Math Work: | Study/solve models of drug diffusion through skin layers using PDE techniques (some use of MATLAB). |
| Biology Work: | Use data from previous studies. |
Computational geometry and protein structure
The primary sequence of amino acids determine unique proteins, yet the correct three-dimensional structure of a protein currently cannot easily be predicted from its primary sequence. One of the key steps involved in ab initio protein structure prediction is the discrimination of native and native-like (correct) conformations from non-native (incorrect) conformations of a protein. This task is usually achieved by the use of statistical potentials or empirical scoring functions, with the success of the prediction method depending largely on the efficiency and accuracy of the scoring function used. Recently, we developed a modified a computational geometry data structure known as Delaunay tessellation-based scoring function which successfully discriminates native from non-native structure for several classes of proteins. More interestingly, method successfully discriminates between pre- and post-transition state conformations and the native structure in the folding simulation trajectories of a few proteins. The undergraduate project will involve basic concepts from computational geometry that are used in the analysis of protein structures and necessary background material from biochemistry. The students will learn to use the software programs developed by the mentor. The undergraduate students working on this project will learn the use of these programs on CASP 6 predictions, and on other publicly available data sets. Once they become facile with the use of the programs, they will use it to tackle related problems on the definition and development of the 4B scoring function.
| Faculty Mentor(s): | Bala Krishnamoorthy. |
| Math Topics: | Geometry, statistics. |
| Biology Topics: | Protein structure and function. |
| Math Work: | Work with C programs to analyze geometric properties of proteins. |
| Biology Work: | Literature review, use data from public databases. |
Structure of multimer proteins
A protein chain is composed of varying number of amino acids connected by chemical bonds. The chain of amino acids twists around and forms non-random geometric shapes in three dimensions. UBM project students have been investigating relationships between sequence (identities of amino acids) and structure of proteins, which in turn is important to determine function. While many proteins exist as a single chain (monomers), there are a substantial number of proteins that are composed of two or more chains (multimers). Many multimers have distinct chains, and their functions are effected by all the chains working together. Students working on this project will examine relationships between the sequence and structure of multimer proteins. The project will use concepts from geometry and other mathematical techniques that can be understood by students with no formal training in the subjects. Tasks include assembling data sets of multimer proteins for investigation from the Protein Data Bank (www.pdb.org), and reading relevant publications. The investigator has already developed certain programs (in C/C++) that can be used to analyze the geometry of proteins. The student(s) working on the project can use the programs just as executables, but can also contribute to writing new programs if desired. It may be possible to collaborate with molecular biologists on campus who are working with multimers.
| Faculty Mentor(s): | Bala Krishnamoorthy, Mathematics. |
| Math Topics: | Geometry of a set of points and tetrahedra. |
| Biology Topics: | Protein sequence, structure, and function; multimer proteins. |
| Math Work: | Analyze properties of tetrahedra formed in proteins (programs in C/C++). Extract relevant information from protein files. Some coding involved. |
| Biology Work: | Collect appropriate sets of protein files for analysis, literature review. |
Mathematical modeling of motor protein function
We study the biophysical and biomechanical mechanisms that result in the generation of force and motion by living organisms for biologically useful movement. We want to understand the fundamental biological processes that allow each of us to move about our daily life. We use mathematical modeling approaches to study the myosin family of motor proteins, which result in muscle contraction, and kinesin-family motors, which help power intracellular organelle movements and mitosis. A multi-scale approach is employed. Systems modeling techniques are employed to analyze the behavior of the large ensembles of myosin motors that are working in functioning muscle. Atomic level, molecular dynamics simulation approaches are used to understand better the detailed conformational changes that occur in the motor proteins to generate force and motion. Computer-based, computational approaches are generally used for analysis. We additionally have extensive collaborations with experimental laboratories. The collaborative data forms the underlying basis of our analyses and includes muscle fiber mechanics and other motility assays, along with spectroscopic and x-ray crystallographic structural data.
| Faculty Mentor(s): | Edward Pate, Mathematics. |
| Math Topics: | Molecular dynamics, multi-scale systems modeling. |
| Biology Topics: | Motor proteins, muscle physiology. |
| Math Work: | Perform molecular dynamics simulation (special software), analyze protein structure in three dimensions using visualization software. |
| Biology Work: | Literature review, use data from previous studies. |
Mathematical modeling of solid tumor growth
Therapeutic strategies for treating solid tumors must take into account the physiological and physical environments in which these cancers grow. A fluid dynamical perspective is seen as an increasingly important aspect of tumor growth. The growth of solid tumors is constrained by nutrient limitation and by the influence of the surrounding medium. Solid tumors in vivo are typically composed of cells embedded in a deformable extracellular matrix. An experimental model for solid tumor has been developed in order to begin to understand the interaction among the cells and the chemical composition of the media. One classic mathematical model of tumor growth uses partial differential equations (PDE). It considers the tumor as a homogeneous incompressible fluid and does not include a representation of the elastic forces due to tumor cell structure and the extracellular matrix. Our alternative hybrid model treats the cells as discrete entities surrounded by a continuum of fluid. The fluid/mechanical processes are modeled by the immersed boundary method. This approach allows us to model chemical concentration gradients as well as stress and pressure that may arise from interaction with a deformable extracellular matrix. Undergraduates will learn various facets of tumor growth and then study our hybrid model of tumor growth. They will then implement numerical simulations of the classic model and compare the results with numerical simulations of our hybrid model. The results from these simulations will then be compared to data compiled by the students supplemented by available experimental data sets. These studies will illuminate the relative importance of fluid and mechanical processes in solid tumor growth.
| Faculty Mentor(s): | Robert Dillon, Mathematics, and Derek McLean, Animal Sciences. |
| Math Topics: | Partial differential equations, continuum models. |
| Biology Topics: | Solid tumor growth. |
| Math Work: | Do simulations of tumor growth models (use FORTRAN or MATLAB to do calculations involving PDEs). |
| Biology Work: | Literature review, use data from previous studies. |
Mathematical models of muscle undergoing lengthening contractions
Mathematical models of muscle force-generating properties are based on the kinetics of molecular interactions between contractile proteins. These so-called crossbridge models describe the transitions between the different states of attachment and detachment of the myosin head and actin-binding site (i.e., the crossbridge) and assume spring-like mechanics of crossbridge to calculate the forces generated in response to a movement. Imposed lengthening of a muscle is thought to change the kinetics of the contractile process by mechanically breaking formed crossbridges instead of crossbridges normally detaching through the ATP hydrolysis. How to model the effects of lengthening on crossbridge cycling has been a matter of debate. This project will explore the different models of lengthening contractions, especially in the context of differences between slow and fast motor proteins in skeletal muscle.
| Faculty Mentor(s): | David Lin, VCAPP |
| Math Topics: | Mechanics of springs, differential equations. |
| Biology Topics: | Reaction kinetics, motor proteins, muscle physiology. |
| Math Work: | Analyzing mechanics of muscle models (in MATLAB or Mathematica; also using specialized software), calculating reaction rates (MATLAB). |
| Biology Work: | Literature review, use data from previous studies. |
Physical models of the human neuromuscular system
We have been exploring the idea of creating a simple robotic system that "feels" human-like. By implementing a computational model of muscle to control a torque motor, observers interact with a robot that is able to present mechanical impedance similar to human muscle. In order to achieve more realistic human-like interactions, the mechanical impedance should also reflect the influence of spinal reflexes and its role in modulating muscle activation. This project will implement different models of spinal reflexes to augment an existing muscle model that is used to control a torque motor and will test the resulting perceived mechanical impedance among subjects who interact with the robotic system.
| Faculty Mentor(s): | David Lin, VCAPP |
| Math Topics: | Mechanics, force/torque calculations. |
| Biology Topics: | Muscle motions and physiology, spinal reflexes. |
| Math Work: | Analyze models of muscle motions an reflexes, work on special software. |
| Biology Work: | Literature review. |
Mathematical characterization of curved muscle paths in a neck model
Biomechanical models of the musculoskeletal system are used to study human and animal motion in order to improve performance, analyze medical treatments or understand mechanisms of injury. These models rely on accurate representation of muscle paths to predict muscle length, force and moment-generating properties. In our lab, we have developed a graphics-based computational model of the neck musculoskeletal system to analyze head and neck postural control and whiplash injury. We have also obtained magnetic resonance imaging (MRI) scans of the neck musculature with the head and neck in different postures and used these geometric data to guide the representation of neck muscles in the model. The goal of this project is to improve our representation of neck muscles using mathematical descriptions of their paths and to evaluate how modeling curved paths affects the calculation of neck muscle mechanical properties.
| Faculty Mentor(s): | Anita Vasavada, VCAPP, and Bala Krishnamoorthy, Mathematics. |
| Math Topics: | Geometry of curves and surfaces. |
| Biology Topics: | Neck muscle shape and physiology. |
| Math Work: | Study models of neck muscles (using software package SIMM), calculations of curve and surface geometry in MATLAB. |
| Biology Work: | Use data from previous studies. |
Directional tuning of neck muscle reflex responses
Neck muscles are activated to stabilize the head when either the head or body is subject to a postural perturbation. Abrupt changes in head posture or neck muscle length generate a reflex response, activating muscles to restore head position. The neck musculoskeletal system has a complex structure, and most neck muscles have a range of directions over which they are active. However, in static voluntary studies, we have found that the direction in which muscles were most responsive did not correspond to the direction in which they had the greatest mechanical advantage. It is not known whether the directional properties of neck muscle responses are similar for static voluntary activation vs. dynamic reflex activation. Therefore, we have developed an experimental apparatus to apply small force perturbations to the head and measure both head movement and neck muscle activation. We will use this device to characterize the directional properties of neck muscle reflexes. The project involves collecting neck muscle reflex response data from human subjects and analyzing the data using spherical statistics to determine the directional tuning of neck muscles in response to head perturbations.
| Faculty Mentor(s): | Anita Vasavada, VCAPP |
| Math Topics: | Spherical statistics, mechanics. |
| Biology Topics: | Neck muscle responsiveness. |
| Math Work: | Analyze data from experiments using statistical techniques (possible use of MATLAB). |
| Biology Work: | Do/assist in experiments with human subject(s) to study responsiveness to small disturbances to the head/neck. |
Hopf bifurcations in mathematical biology
This project will apply advanced numerical techniques to detect Hopf bifurcations in a higher dimensional (n > 2) mathematical-biology model to be selected over the course of the next semester. We will begin by learning (1) the basics of phase diagram solutions to systems of differential equations and (2) how to identify elementary bifurcations in two-dimensional models. We will then work through the article "Computing Hopf Bifurcations" by Guckenheimer, Myers and Sturmfels in the SIAM (Numerical Analysis). This article makes use of advanced techniques in linear algebra (Professor McDonalds's expertise). We will first apply the techniques to detect Hopf bifurcations in the classical two-dimensional predator-prey model following Kot, Elements of Mathematical Ecology, Chapter 10: Global Bifurcations in Predator-Prey Models. The final step will be to apply the numerical Hopf detection techniques to a higher dimensional mathematical biology model.
| Faculty Mentor(s): | Ray Huffaker, Economic Science, and Judi McDonald, Mathematics. |
| Math Topics: | Hopf bifurcations, linear algebra, differential equations. |
| Biology Topics: | Predator-prey models. |
| Math Work: | Study Hopf bifurcations (literature review), analyze 2D predator-prey models (some use of MATLAB). |
| Biology Work: | Literature review. |
Dynamics of reservoir sedimentation management
Sediment accumulation has significant impact on ecosystem function. Steady decline in reservoir storage capacity due to sediment accumulation is a serious problem worldwide. A recently developed, sediment removal technique called hydrosuction dredging provides a potential solution to this problem. Students will explore the effectiveness of this method in removing sediment accumulation by modeling the dynamics of hydrosuction dredging and will explore ecosystem consequences of alternative sediment removal scenarios.
| Faculty Mentor(s): | Ray Huffaker, Economic Sciences. |
| Math Topics: | Differential equations. |
| Biology Topics: | Ecology of reservoirs. |
| Math Work: | Simulations of sediment removal processes |
| Biology Work: | Literature review. |
Molecular mechanism of photosynthesis
How do plants convert light energy into forms usable for light? One approach is understanding photosynthesis is to develop mathematical models of the molecular mechanism of photosynthesis. Students will develop a random walk model of electron transport in cytochrome b6f, an enzyme that takes part in the transduction of light energy into chemical energy. The model results will be compared with experiments done in the laboratory. The experiments involve shining short pulses of laser light on spinach leaves and interpret the fluorescence which occurs on a time scale of picoseconds to milliseconds. Other, more global model of the biochemistry of photosynthesis, based on a system of ordinary differential equations will be adapted to investigate the specific experiments in the laboratory. This work involves many exciting concepts concerning the biology of photosynthesis (the original solar energy technology), and the mathematics of random walks (including certain limits under which these converge to diffusion equations) and systems of ordinary differential equations. It also involves a lot of work with computers, learning something about the Linux operating system, and programming in Fortran and Matlab.
| Faculty Mentor(s): | Mark Schumaker, Mathematics, and David Kramer, Institute of Biological Chemistry. |
| Math Topics: | Random walks, ordinary differential equations (ODEs). |
| Biology Topics: | Molecular mechanism of photosynthesis reactions. |
| Math Work: | Conduct simulations of random walk models to study electron transport (coding in FORTRAN and MATLAB required). |
| Biology Work: | Do experiments to study photosynthesis, measure fluorescence on spinach leaves, also use data from previous studies. |
Modeling fitness benefits of extra-pair reproductive strategies
In many populations males and females form monogamous social pairs during the breeding season, but recent genetic studies have clearly shown that reproduction occurs outside of these pair bonds. For example, male birds often copulate with females other than their social mates, and females often lay their eggs in the nests of other females. The frequency of these "extra-pair" reproductive tactics varies dramatically across populations and across species, yet we have little understanding of the selective costs and benefits creating this variation. One hypothesis is that males who copulate with extra-pair females and females who lay their eggs in other nests are benefiting from "risk spreading" -- for example by spreading offspring across a number of nests they are more likely to have some offspring survive predators. In this project we will examine the rationale for this hypothesis and develop quantitative predictions by developing and analyzing mathematical models of the benefits of extra-pair reproduction under a number of different predation scenarios. We will compare these predictions with data from long-term studies of two populations of birds.
| Faculty Mentor(s): | Michael Webster, Biology and Richard Gomulkiewicz, Biology and Mathematics. |
| Math Topics: | Predator-prey models, differential equations, statistics, simulation. |
| Biology Topics: | Population biology. |
| Math Work: | Modeling, solving systems of differential equations, simple calculations in MATLAB or Mathematica. |
| Biology Work: | Literature review, use data from previous studies. |