University of Minnesota
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## 1998-1999 Research Accomplishments in Mathematics in Biology

Workshop Organizers     Visitors      Postdocs

1. ANNUAL PROGRAM ORGANIZERS

2. WORKSHOP ORGANIZERS

Lisa Fauci of Tulane University (Mathematics) is one of the annual program organizers for the 1998-99 year in "Mathematics in Biology" She and Shay Gueron of Technion - I.I.T. (Mathematics) express the following:

As the organizers of the workshop on Computational Modeling in Biological Fluid Dynamics that was held during January 25-29, we feel that it is our pleasant duty to thank you and the IMA staff for this wonderful workshop.

The workshop was a success in all respects: friendly atmosphere, high quality talks, fruitful scientific interactions, and extraordinary logistics. Many of the participants addressed us during the workshop and after it was over, and expressed their satisfaction. All of them ranked the workshop as one of the best scientific meetings in their entire academic career. We feel the same. Although we would happily take the credit for this success, we feel that it is your perfect organization that is actually entitled to it.

We were extremely impressed by the friendliness, efficiency, willingness, and resourcefulness of the entire IMA staff: meticulous and professional organization contributed immensely to the smooth operation during the week of the workshop, and the very cordial staff generated the friendly and casual atmosphere which was also an essential ingredient.

We would like to take advantage of this opportunity to particularly acknowledge the important contribution of Dr. Fred Dulles to the success of the workshop.

As the workshop's organizers and on behalf of the participants and speakers - thank you again for organizing a wonderful event which benefited our scientific community.

Philip Maini of University of Oxford (Mathematics Institute) is one of the organizers of the following sessions:

- IMA Tutorial: Mathematical and Computational Issues in Pattern Formation, September 3-4, 1998

- IMA Workshop: Pattern Formation and Morphogenesis: The Basic Process, September 8-12, 1998

- IMA Workshop: Pattern Formation and Morphogenesis: Model Systems

He writes:

I was at the IMA during the period September-December, 1998. During that time I ran, together with Hans Othmer, a 2-day tutorial and two one-week long workshops on pattern formation. We are presently nearing the completion of a proceedings volume that came out of the workshops. I also ran a four-day long workshop on mathematical modelling in cancer. This was an area I was not involved in but subsequent to this workshop I have become involved in it and, together with one of the participants, Dr Byrne, we are now the UK partner in a European Network that recently secured funding of approx 1.5 m dollars to work in this area.

The very good computing and library facilities at Minneapolis allowed me to make good progress in my own research and grant writing activities. The workshops enabled me to interact with co-workers and finish papers or begin new projects, and to meet many new people in the field. Having a base in the US allowed me to visit a number of collaborators within the continent.

I also had some interaction with the postdocs but I found that being a visitor for only a term did not really give enough time to develop proper collaborative projects.

The IMA, with its excellent support facilities, provided a very good place to spend a sabbatical.

Lee Segel of Weizmann Institute of Science (Applied Mathematics and Computer Science) is one of the organizers of the IMA Period of Concentration "Forging an Appropriate Immune Response as a Problem in Distributed Artificial Intelligence" held on October 19-23, 1998. He also delivered a lecture as part of the School of Mathematics Ordway Lecture on November 10, 1999. He writes:

One feature of the workshop that I organized was participation of a relatively large number of U Minn faculty: N Papanikolopolous (Electrical Engineering), M Mescher and Marc Jenkins (Immunology) and Deborah Richards (Political Science). All have interests in distributed autonomous systems. My own interaction was particularly close with Jenkins. We had several conversations, which certainly helped me with deeper understanding of various relevant aspects of immunology. Jenkins asserted that he might do some experiments based on some of my ideas. These are connected with "adjuvants" that the immunologists use to boost otherwise weak immunological reactions. Someone called adjuvants "the dirty little secret of immunology" but in my view adjuvants can be viewed as the vehicles by which the immune system adjusts its actions in accord with feedbacks from sensing progress toward a variety of "goals". If this view is correct, then there may be a long range contribution here to vaccine design; this is what interested Jenkins. I talked with several of the post docs, and offered some advice that was nontrivial I believe, but not in any way decisive: to Tracey, Patrick and Kathleen. Patrick mediated an interaction with Ron Siegel, another U Minn faculty member, which resulted in some genuine collaborative work, and a paper I believe, by Siegel and another of the post docs.

Aside from general and informative discussions with IMA visitors, I enjoyed getting up to date on the work of some of my distinguished friends from U Minn, Dan Joseph (Mechanical Engineering), and Chemical Engineers Skip Scriven, Gus Aris, and Bob Tranquillo. I gave a lecture in Mechanical Engineering which seemed well received. And of course I gave those Ordway lectures in mathematics.

Both in the Ordway lectures and in personal conversations (with Naresh Jain, Wei Min Ni, Don Aronson, and Hans Weinberg) I pushed my views on the importance of subject-matter oriented applied mathematics. I don't know what influence this had.

Annual Program Organizers      Workshop Organizers     Postdocs

3. VISITORS/SPEAKERS

Graham A. Dunn of MRC Muscle and Cell Motility Unit, The Randall Institute King's College London visits the IMA from January 1-30, 1999. His report follows:

Analysing the Marginal Activity of Moving Cells

My stay at the IMA began with the Workshop on "Cell Adhesion and Motility" (Jan 4-8) during which I presented a talk on "Using Microinferometry to Study the Function of the Cytoskeleton in Cell Motility". This described a project to investigate the dynamics of protrusion and retraction of the cell margin during crawling cell motility. The DRIMAPS system of microinferometry that we have developed enables the margin of living, thinly spread, vertebrate cells to be located with unprecedented accuracy. The crawling form of cell locomotion is ubiquitous among vertebrate cells and plays a central role in embryogenesis and tissue repair. The main protein involved in crawling locomotion is actin which interacts with a host of other proteins and provides the basis of the cell's cytoskeleton. Understanding the dynamics of disassembly, relocation and assembly of actin-based structures is therefore the key to understanding how vertebrate cells move. Earlier work in our laboratory and in that of Wolfgang Alt (one of the organizers of the Workshop) had demonstrated that temporal and spatial correlations in the activity of the cell margin could reveal interesting dynamical properties of the actin cytoskeleton of the cell. In the case of temporal correlations, there is often a strong positive correlation between the protrusion activity of the cell margin and its retraction activity a short time later after a lag of about one minute.

In collaboration with Michelle Peckham of the University of Leeds, we have studied how these correlations are affected by transfecting mouse myoblast cells with the human beta-actin gene. Beta-actin and gamma-actin are the two forms of actin commonly present in non-muscle cells and beta-actin is preferentially located near the cell margin which suggests that it may be more directly involved in protrusion and retraction of the cell margin. Transfection with beta-actin increases its preponderance within the cell and probably also leads to a down regulation of gamma-actin. The effects of this transfection on temporal correlations of marginal activity were analysed during my stay at the IMA. Cross correlations between protrusion and retraction were obtained after pre-whitening each time series to eliminate autocorrelations which can give rise to spurious cross correlations. One effect of the transfection is to reduce dramatically the positive correlation between the protrusion and the retraction of one minute later. Another effect is to double the speed of cell locomotion. It is too early to understand the implications of these observations, and more experiments will be needed, but it does suggest that the absolute and/or relative sizes of pools of beta- and gamma-actins within the cell may be critical parameters in the regulation of cell motility. Total cell actin is known to be increased in most cells when they are activated to perform motile tasks and these observations indicate that the extent of this increase, and/or the type of actin increased, may determine the type of task which the cell is able to perform.

Frithjof Lutscher of Universitat Tubingen (Lehrstuhl Biomathematik) is a graduate student who participates in the following sessions:

- Workshop: Local Interaction and Global Phenomena in Vegetation and Other Systems, April 19-23, 1999

- Tutorial: Introduction to Epidemiology and Immunology, May 13-14, 1999

- Workshop: From Individual to Aggregation: Modeling Animal Grouping, June 7-11, 1999.

He writes:

I am in the middle of my Ph.D. thesis work in biomathematics on the formation of fish schools through alignment. I found out about the IMA's special program "Mathematics in Biology" via the internet. It was clear to me that all of spring quarter's activities would be very interesting for me, in particular the last workshop on "Animal Aggregation." All the leading experts in my field (Schooling and Alignment) were going to be there.

When I first asked about the possibilities to participate I got an immediate and very encouraging answer. And from then on the IMA staff was always very helpful during my preparation of the stay with everything from finding housing to visa information.

I was lucky enough to get a fellowship from the DAAD (German Academic Exchange Program) and arrived in Minneapolis in the beginning of April. I was welcomed very warmly and everything was already set up (office space, computer account,...) so that I could get to work immediately.

I did participate in all the three workshops and found them very inspiring. Most of the talks were very good. And at least equally important was the fact that there was enough time in between the talks for discussion with other participants. And not only the time but also the space and an atmosphere to do so was set up with white boards and tea and coffee. For me this is much more efficient than on big conferences where twice as many scheduled talks per day don't permit time to talk individually. (The only time I found it difficult to follow was when there were many short talks scheduled for some afternoons.)

In the time between the workshops I was able to get quite some of my own work done. It was great to have so many experts around, visiting professors at the IMA as well as faculty of the mathematics department, and to be able to talk to them. The big open space at the institute and the coffee breaks there support and facilitate communication as well as the (joint) seminars. I certainly found very good technical support (computer system).

In short: it was a very valuable time, inspiring for my own work. I met many researchers and had fruitful discussions. And I learned about many new ideas in mathematical modeling.

Annual Program Organizers      Workshop Organizers     Visitors

4. POSTDOCS

Kevin Anderson reports:

Workshops and Seminars

I spent the past year as a postdoctoral member of the I.M.A. participating in the special year of emphasis on Mathematical Biology. A large portion of my time has been spent participating in the fifteen workshops which comprised the majority of the year's programming. In these workshops I gained an outstanding overview of current research in mathematical biology. This was both through attending the workshops and tutorials, and through interacting with workshop participants and long term visitors. In particular, I enjoyed the workshops in the fall quarter, which discussed developmental biology and immunity, and the spring quarter which covered ecosystems and epidemics. Some of the visitors with whom I had the most useful interactions have were Carla Wofsy of the University of New Mexico, Phillip Mani of Oxford University, Lee Segal of the Weizmann Institute of Science, Yoh Iwasa of Kyushu University, and Mark Lewis of the University of Utah.

I also participated in the I.M.A. postdoctoral seminar series on Mathematical Biology. I also organized this seminar during the winter quarter.

Talks

During the year I also gave three talks. On November 14, I spoke on the topic Estimating Mean Time to Extinction: An Overview of Some Popular Methods'' at the Western Regional Meeting of the American Mathematical Society in Tucson Arizona. On April 2, I spoke on the topic "Measure Valued Markov Processes and Bacteria" at the Probability Seminar of the University of Minnesota's School of Mathematics, and on May 25 I spoke on "Spatial Patterns in Gynodioecious Populations" at the I.M.A. postdoctoral seminar series. Research

I spent a considerable amount of time during the beginning of the year finishing up projects which I had started while a graduate student in the Program in Applied Mathematics at the University of Arizona. The work culminated in the submission of two papers. The first, "A New Mathematical Approach Predicts Individual Cell Growth Behavior Using Bacterial Population Information," which was written along with Dr Joseph Watkins and Dr. Neil Mendelson, was submitted to Journal of Theoretical Biology. This paper described work which I had done as a part of my dissertation. The second, "A Test of Popular Methods for Estimating Mean Time to Extinction" was written with Dr Wade Leitner of the Department of Ecology and Evolutionary Biology at the University of Arizona. This was submitted to the journal Conservation Biology.

I also began some new research projects. With Claudia Neuhauser of the University of Minnesota's School of Mathematics, I began a theoretical study of bacteriocin producing bacteria. Bacteriocins are poisonous substances secreted by some bacterial strains. This is a purely spiteful behavior, as the production is lethal to the bacteria who produce it. Systems in which bacteriocin producing strains compete with bacteriocin susceptible strains make very interesting models for studying competition between species. This competition has been the focus of our study. Dr Neuhauser was my faculty mentor, and we met weekly to discuss our research. In addition, I participated in a graduate ecology course on Spatial Processes in Ecology which Dr Neuhauser taught.

Another project upon which I worked, was one with Dr Neuhauser and Dr Yoh Iwasa of Kyushu University's Department of Biology. Dr Iwasa was a visitor to the I.M.A. during the spring quarter. This work was initiated during his visit, and continued after he left. In this project we studied the spatial patterning of gynodioecious plant populations. Gynodioecious populations are one which are comprised of both female and hermaphroditic plants. When observed in nature, the female plants are often clustered together. We asked what conditions of pollen and seed dispersal lead to this clustering behavior. To answer this we studied a system of differential integro equations which models the population. We have prepared a manuscript on this work entitled "Spatial Patterns in Gynodioecious Plant Populations" which we plan to submit to Journal of Mathematical Biology.

A third new project upon which I worked this year was an analysis of Gott's Formula. Gott's formula gives one confidence intervals for the time remaining in an epoch, using only the time since the beginning of the epoch as data. It is a completely non-parametric method, assuming only that the observer occupies no privileged place in time. I showed that one makes implicit assumptions in using the method, and that it is therefore it is not as non-parametric or as powerful as previously thought. This work resulted in my submitting the paper "Implicit Assumptions in the Application of Gott's Formula" to the journal Nature.

Conclusion

I feel that by participating in the special year on mathematical biology at the I.M.A., I have gained a broad overview of the field. I am now familiar with many more areas of current research than I was upon finishing my graduate studies. I have also learned new techniques for studying such systems, and met many of the people who are actively researchers in the field of biomathematics. I feel all of this will provide an invaluable foundation for my career.

Bruce P. Ayati, formerly of the University of Chicago, is completing the first of two years as an IMA Postdoctoral Member.

Workshops and Seminars

As a Postdoctoral Member of the IMA during the special year Mathematics in Biology, my main responsibility was participating in twelve weeks of workshops, seminars and tutorials. These were meant to give a broad view of the diverse field of mathematical biology, a goal which I feel was accomplished.

My main area of interest during the special year was population biology, the theme for the spring quarter. The workshops on vegetation, epidemiology, and animal aggregation each brought different problems that had common themes. In particular, the use of structured population models was prevalent in all three workshops. Since my current research is in numerical methods for structured population models, these workshops gave motivation for future research as well as provided areas of application for the current numerical methods.

A particular project I am currently working on is the derivation and numerical solution of size and space structured forest models. This is in conjunction with Claudia Neuhauser, who also co-organized the vegetation workshop. As a result, the vegetation workshop proved invaluable to my current research.

After the completion of the special year at the IMA, I attended a NATO Advanced Study Institute on Mathematics Arising from Biology at the Fields Institute in Toronto. This two week workshop was also co-organized by Claudia Neuhauser. It consisted of one week of ecology and another week of population genetics.

Presentations

The IMA has a Postdoctoral Seminar Series, which is organized by the IMA postdocs but which gathers speakers and attendees from the University of Minnesota as well as longer term visitors to the IMA. I co-organized the spring seminar series with Ralf Wittenberg.

I spoke during the winter postdoc seminar on Numerical Analysis and Population Dynamics. In addition, I spoke at University of Iowa Applied Mathematical and Computational Sciences Colloquium on Galerkin Methods for PDE Models of Age and Space Structured Populations and the University of Iowa Numerical Analysis Seminar on BuGS for Parabolic Problems (some software I had written). At the Fields Institute, I presented a poster on The Influence of Discontinuities in the Population Density and Migration Rate on Neutral Models of Geographical Variation.

Research

While at the IMA, postdocs are expected to pursue their own research programs with the aid of University of Minnesota Department of Mathematics mentors. My mentors are Mitch Luskin, a numerical analyst, and Claudia Neuhauser, a probabilist who also holds an appointment in the ecology department.

During my first year at the IMA, I completed and submitted three papers. The first paper was a generalization of my thesis work to higher order finite element spaces in age, Galerkin Methods in Age and Space for a Population Model with Nonlinear Diffusion. The second paper was the completion of work begun at the University of Chicago with my thesis advisor, Todd F. Dupont, and Thomas Nagylaki of the Department of Evolution and Ecology at the University of Chicago. It was a population genetics paper entitled The Influence of Spatial Inhomogeneities on Neutral Models of Geographical Variation IV: Discontinuities in the Population Density and Migration Rate, which is in press in Theoretical Population Biology. The third paper was a work in numerical analysis on an adaptive time stepping method for parabolic PDE's entitled Convergence of a Step-doubling Galerkin Method for Parabolic Problems.

I am currently working on simulations of Proteus mirabilis swarm colony development, a bacteria that forms interesting spatial and temporal patters on petri dishes. My main project is the derivation and numerical solution of size and space structured forest models. This work is being done with Claudia Neuhauser.

From September 1998 till August 31 1999, I was an industrial mathematics postdoctoral member working with the Computer Graphics group at IBM, T.J.Watson Research Center, Hawthorne in New York. For the first six months I stayed in residence at T.J.Watson working with Gabriel Taubin on the problem he assigned to me. More specific the problem was to find efficient compression algorithms for 3D model geometry. The first method we tried was based on an eigenfunction decomposition of the discretized Laplace operator on a fictive uniform mesh. We came up with an iterative eigenproblem solver adapted to the topological surgery that is taking place on the mesh. Unfortunately, the initial intuition was not correct and the algorithm turned out not to compress the geometry, but rather to expand it. This aspect, on the other hand, shed more light on some spectral inter-relations between coarser and finer meshes in the level of detail sequence. Then, at around the time I moved in residence to Minneapolis, we changed the approach and we looked for linear filtering techniques on the sequence of meshes. It turned out as a fairly robust and good compression method with possible applications to some other problems as well (I would mention the fairing - or smoothing - problem).

During this period, I regularly visited T.J.Watson to keep in contact with Gabriel Taubin. I also had fruitful discussion with several members if IMA or University of Minnesota. Firstly, I regularly consulted with Fadil Santosa who gave me the idea of using the simplex algorithm for solving an optimal linear l1-problem. Unrelated to the IBM project, but rather to a paper that I finished writing this period, I had a very interesting discussion with Scot Adams who suggested a topologic invariant (some homology group) as the main obstruction to what is called the Balian-Low nonlocalization theorem. I also had some interactions with the analysis group on the Monday's Rivire-Fabes Seminars on Real Analysis. A special mention is for Max Jodeit, the organizer of these seminars. Lastly, but not least, I had interesting discussions with Radu Marculescu, who is assistant professor within the Electrical Engineering department. Our discussions evolved several stochastic models of digital integrated circuits (more specific, how the heat dissipation can be modeled on a Markov chain).

During this year I gave a number of talks and I took part in several workshops and conferences. First I gave some lectures on wavelets at IBM T.J.Watson. These ran for about 3 months during which I had the opportunity to find out of some other research over there. At IMA I gave a talk at the Vertical Integration Applied Mathematics Seminar (the Monday one) on the status of the 3D geometry compression problem. Then I presented to the analysis group, in a Riviere-Fabes seminar, a density result concerning super Weyl-Heisenberg frames. I presented the same result at the AMS-MAA joint meeting in San Antonio, as an invited speaker to the special session on harmonic analysis of wavelets and frames. Toward the end of my stay at IMA, I gave a talk in the Postdoc Seminar on a signal processing problem, namely the identification of singular multivariate AR processes with application to the blind source separation problem. This constituted my research for Siemens Research Corporation, my next employer.

Finally, I should mention an extraordinary opportunity that one of the workshops gave it to me. It all started in the first days I moved to Minneapolis. An workshop on audition was just about to start and, because of my Siemens project, I was curios and particularly interested in the subject. However, in one of the talk breaks I had the chance to meet Zeph Landau, then a postdoc at UCSF. He happens to be one of the authors of a particularly important paper on Weyl-Heisenberg frames, and also an expert on von Neumann algebras. After a couple of talks we realized we can do something in this field (Weyl-Heisenberg sets) and we decided to write down our ideas. So now we are about to write a lengthy paper on von Neumann algebra methods applied to Weyl-Heisenberg sets (also known as windowed Fourier transform). Recently we have been invited to a concentration week at Texas A&M to present some results. There, at College Station, we had very interesting discussions with the people in the same field.

Nicholas Coult, IMA Industrial Postdoc Industrial Partner: Fast Mathematical Algorithms and Hardware Corp. He writes:

Following is a brief report on my activities for my first year at the IMA, from September 1, 1998 to August 31, 1999. I have broken the report into four parts:

- Industrial Research. Overview of research relevant to industrial problem.
- Local collaborations. Description of collaborations with researchers at the University of Minnesota.
- Papers. Brief descriptions of papers published, submitted, or in preparation.
- Presentations. Brief descriptions of presentations given during the year.

Industrial Research:

I pursued research on the primary aspect of my industrial problem, which is compression of three-dimensional travel-time data for seismic applications. The basic difference between this compression problem and others such as image compression is that one would like to keep the data in an appropriately compressed form so that one may use it as a sort of compressed interpolation table. I wrote prototype code to perform this compression using piecewise-polynomial wavelets. During the course of developing and testing this code it became clear that a number of improvements were necessary both in the mathematics and the implementation. Specifically, the relationship between the error level and compression ratio was not as good as one would like. I would like to emphasize that the bi-weekly industrial postdoc meetings played an essential role in the process of my understanding these aspects of the problem. I spent the remainder of my research time for this problem on improving these parts of the algorithm, and recently I have made some significant developments towards achieving a dramatically improved compression-ratio to error-level relationship.

Related to this problem is that of travel-time computation for seismic problems. On this topic I was mostly a novice, so I spent a good amount of time reading books and literature (and occasionally discussing it with my industrial colleagues) to better understand the state of the art. The goal is to eventually design an algorithm which can compute travel-times directly in a compressed format. I did not expect to make much headway on this problem during this year, but I am planning to pursue this more vigorously next year.

Local Collaborations:

As part of the IMA's industrial postdoc program I was assigned Professor Bernardo Cockburn as my faculty mentor. With some exceptions, we met regularly and discussed the state of my research and also his research in the area of discontinuous Galerkin methods for PDE's. The basis that I used for compression of travel-time data is the same basis that he uses for DGM. Thus, we saw potential for some collaboration. We continued meeting regularly and discussing this matter, and I gave a talk at a conference that he organized (the talk and associated paper will be described later). We have committed to writing a paper together on this topic in the coming months.

Papers:

During the first part of this year at the IMA, I submitted a paper on computational quantum physics together with my Ph.D. advisor Gregory Beylkin and Martin Mohlenkamp, a postdoc of his. We submitted revisions in January, and the paper was published this spring. I am continuing to collaborate with them remotely on this research area. Following is the citation:

G. Beylkin, N. Coult, and M. Mohlenkamp. "Fast Spectral Projection Algorithms for Density Matrix Computations." J. of Comp. Phys. (152) 1999, no. 1, 32-54.

This May I submitted a paper for the refereed proceedings of the International Symposium on Discontinuous Galerkin Methods, at which I also gave a presentation. The paper grew out of discussions with Bernardo Cockburn, and gives a basic description of discontinuous wavelets:

N. Coult. "Introduction to Discontinuous Wavelets." Accepted for publication in Proc. Int. Symp. Discontinuous Galerkin Methods.

Finally, I am working on a manuscript which extends the discontinuous wavelet construction to arbitrary triangles and tetrahedra. This construction will allow the use of fast wavelet algorithms for problems on non-trivial domains in two and three dimensions, something which to my knowledge has not yet been feasible:

N. Coult "Triangular Multiwavelets for the Solution of PDE's and Integral Equations." In preparation.

Professor Cockburn and I are planning on further collaboration to tie together his work on discontinuous Galerkin methods with the automatic adaptivity that wavelets provide. He and I, together with his student Paul Castillo, will work together in the coming year on this area.

Presentations:

During the fall, I was invited to give a presentation on wavelets for the Math Department's Real Analysis seminar. The title of my talk was "Wavelet-based Homogenization of PDE's and Eigenvalue Problems."

I was also invited to give a talk at Carleton College's Math/CS colloquium. This took place in January. The title of this talk was "Data Compression and Fast Algorithms Using Wavelets."

Finally, I was a plenary speaker at the International Symposium on Discontinuous Galerkin Methods. The title of this talk was "Wavelet-based Discontinuous Galerkin Methods."

Trachette L. Jackson reports:

I was a postdoctoral member of the Institute for Mathematics and Its Applications during the 1998-1999 year devoted to highlighting mathematical challenges emerging from the consideration of issues in the biological sciences. This general topic has been of great interest to me for some time and is in line with the subject of my doctoral dissertation, Mathematical Models in Two Step Cancer Chemotherapy. After being focused on such a specific aspect of mathematical biology during my graduate training, I accepted this postdoctoral position in order to gain a broader appreciation of the significant applications of mathematics to medicine.

The IMA brought in world leaders in several areas of mathematical biology to organize and participant in a selected series of workshops and tutorials. The first three months of my tenure were centered around developing new collaborations and studying theoretical problems in developmental biology and immunology. During this time I attended tutorials on mathematical and computational issues in pattern formation, the physiology of the immune system, and mathematical models of AIDS. These tutorials were an invaluable introduction to the topics, terminology, and biology of up coming workshops. Also during this time, I came to know several leaders in the fields of pattern formation and mathematical immunology.

Applying mathematical models to the field of cancer biology and treatment has been my main focus. Recognizing the high degree of interest, the IMA arranged a mini-symposium on this subject. They invited a small group of clinicians, experimentalists, and theoreticians to present their latest discoveries in the area in hopes of stimulating cross-disciplinary collaboration. I was invited to speak during this workshop to represent the contributions of mathematical models to new chemotherapeutic approaches. This mini-symposium was extremely important to me and resulted in several collaborations which are discussed in detail below.

Collaborations and Research Projects

In the field of mathematical biology, communication and collaboration with experimental biologists and clinical researchers is crucial to the development of realistic mathematical models which can be validated by experimental data. I wanted to use my time at the IMA to develop collaborations with experimentalist as well as continue my training with senior mathematicians. To this end I began work on four distinct projects.

Investigating the Timescales Involved in the Angiogenic Switch:

With Carla Wofsy (Longterm Visitor) and Sundaram Ramakrishnan (Dept of Pharmacology, UM)

Carla Wofsy of the department of mathematics at the University of New Mexico was a long term visitor in residence at the IMA for three months. We both became interested in tumor-induced angiogenesis after attending a UM Medical School sponsored lecture given by Judah Folkman, a world leader in the field. After a bit of research into the subject, we sought out local researchers who were studying vessel formation and anti-angiogenic treatment strategies. Sundaram Ramakrishnan of the University of Minnesota's department of pharmacology, is associated with the University of Minnesota Cancer Center and is designing novel therapeutics to specifically kill endothelial cells in the tumor-associated neovasculature. Carla and I began attending the weekly lab meetings of Professor Ramakrishnan and his group. They showed us experimental data which suggest that there are \textit{in vivo} differences in the time it takes for cell lines which grow at the same rate in culture to make the angiogenic switch. This led us to the research project described below.

There are two distinct stages of growth during solid tumorigenesis - the avascular stage and the vascular stage. During the avascular stage, the tumor cells obtain nutrients from surrounding tissues and can grow to a few millimeters in diameter, containing about one million cells. As nutrient levels are depleted, the tumor forms a necrotic core consisting of oxygen starved cells and dead cellular debris; viable, proliferating cells are usually only found in a thin layer near the periphery of the tumor. There is some experimental evidence that it is these hypoxic cells in the necrotic core that begin to secrete certain chemicals which promote the formation of new blood vessels from the existing vasculature in the surrounding tissue. When the generation of new blood vessels is not regulated, this increased blood supply can sustain the progression of many neoplastic diseases. We are developing mathematical models that describe the mechanisms by which an avascular tumor can make the transition to a vascularized state. Particular attention paid to the timescales involved.

From the models and the analysis we hope provide an explanation for the in vivo differences observed in the time it takes for cell lines with similar growth dynamics in culture to make the angiogenic switch. We also wish determine critical parameters that lead to ways of controlling the angiogenic switch, thereby forcing tumors to remain in the avascular state.

Mathematical Modeling of Benign Tumor Encapsulation: With Sharon Lubkin (North Carolina State University)

As I was finishing up my doctoral dissertation at the University of Washington, I began looking for new problem in tumor biology which I could investigate mathematically. I began reading papers on benign tumor encapsulation and found that there is very little experimental research on the underlying mechanisms which cause the capsule to form. I later brought this modeling idea to Sharon Lubkin of North Carolina State University's department of mathematics during one of her many visits to the IMA; our work is discussed briefly below.

The phenomenon of tumor encapsulation is well documented. In fact, the presence of a capsule surrounding a tumor mass is a significant morphological determinant of the clinical outcome. Encapsulated tumors are generally benign and have a favorable prognosis. Tumors which do not form this containing capsule are generally malignant and the cells can invade the surrounding normal tissue and even enter the blood stream to relocate to other sites in the body. We are working to develop a mathematical model which describes the process of tumor encapsulation. From the model we hope to determine ways of eliciting the encapsulation of otherwise invasive cancer cells.

The Response of Vascular Tumors to Chemotherapeutic Treatment: With Helen Byrne (University of Nottingham)

Helen Byrne of the Mathematics Department at the University of Nottingham, UK, was one of the invited speakers for the mini-symposium on cancer. After hearing each other speak we set up a meeting to discuss a possible collaboration. Our mutual interests include tumor growth and treatment; a description of our first joint effort follows.

There are several features inherent in the physiology of vascular tumors which make them difficult treat successfully via blood borne strategies. These barriers to treatment include the spatial heterogeneity of the vascular network which nourishes the tumor. This leads to high density regions of blood vessels to which the chemotherapeutic agent is accessible, and low density regions of blood vessels where the agent can reach in only small quantities.

Another barrier to treatment with a single agent is the frequent mutation of cancer cells to phenotypes which are resistant to the anti-cancer agent being administered. We are developing mathematical models to study how vascular tumors respond to specific chemotherapeutic treatment strategies. Particular attention is paid to the dynamics that result from the incorporation of a semi-resistant tumor cell type. The models consist of a system of partial differential equations governing intratumoral drug concentrations and cell densities. In the model the tumor is treated as a continuum of cells which differ in their proliferation rates and their responses the the chemotherapeutic agent. The balance between cell proliferation and death with the tumor generates a velocity field which drives expansion or regression of the spheriod. Insight into the tumor's temporal response to therapy is gained by applying a combination of analytical and numerical techniques to the model equations.

Spatio-temporal Studies of the Mitotic Clock in Avascular Tumor Growth and Treatment: With Helen Byrne (University of Nottingham)

After completing our paper on the response of vascular tumors to chemotherapy, Helen Byrne and I decided to continue our collaboration and we are now studying the effects of nutrient mediated cell-cycle progression on the growth and treatment of avascular tumors. A brief summary is provided below.

It has been suggested that a single biochemical mechanism involving the activation and inactivation of the maturation promoting factor (MPF) by the protein cyclin underlies the progression of all eukaryotic cells through the cell cycle. We are developing and analyzing a mathematical model that incorporates the details of the intra-cellular biochemistry of MPF and cyclin into a model for avascular tumor growth. We hope to determine the spatial variation in the cell cycle as we move toward the center of the tumor where the nutrient concentration is lower. We then wish to include the effects of a cell-cycle specific drug and a drug resistant cell type into the model.

Scientific Contributions

• Jackson, T.L., Lubkin, S.R, and Murray, J.D. Theoretical Analysis of Conjugate Localization in Two-Step Cancer Chemotherapy. (Accepted: To appear in J. of Math. Bio., 1999)

Revised and accepted for publication while at the IMA.

• Jackson, T.L., Senter, P.D., and Murray, J.D. Development and Validation of a Mathematical Model to Describe Anti-cancer Prodrug Activation by Antibody-Enzyme Conjugates. (Accepted: To appear in J. of Theo. Med., 1999)

Revised and accepted for publication while at the IMA.

• Jackson, T.L., and Byrne H. A Mathematical Model to Study the Effects of Drug resistance and Vasculature on the Response of Solid Tumors to Chemotherapy. (submitted)

Started, completed, and submitted for publication while at the IMA.

• Jackson, T.L., and Byrne H. A Spatio-temporal Model of the Mototic Clock in Avascular Tumor Growth and Treatment. (In preparation)

Started while at the IMA.

• Lubkin, S.R., and Jackson, T.L. Mechanics of Capsule Formulation in Tumors. (submitted)

Started and submitted for publication while at the IMA.

Invited Presentations

• Association for Women in Mathematics, Mini-symposium on Mathematical Biology. In conjunction with the SIAM annual meeting, Atlanta, GA. (May)
• Duke University Mathematical Biology Fest, Center for Mathematics and Computation in the Life Sciences and Medicine, Department of Mathematics, Duke University, Durham, NC. (May)
• Minority Access to Research Careers, Departments of Mathematics and Zoology, Arizona State University, Tempe, Az. (April)
• Cancer Mini-symposium, Institute for Mathematics and Its Applications, University of Minnesota, Minneapolis, MN. (November)

Accepting a postdoctoral appointment at the IMA for the special year in Mathematical Biology was the best way I could have chosen to begin my post-graduate career in the Mathematical Sciences. During my one year stay, I developed several fruitful collaborations with researchers here and abroad. As a result, I have co-authored two papers which were submitted for publication during my appointment and I have two more in preparation. Due to the diversity of long term visitors and invited speakers, I have also come to know the most prominent and influential scientists in the field of Mathematical Biology and have broadened my own mathematical and biomedical knowledge.

Bingtuan Li is one of the IMA Postdocs. His report follows:

The year I spent at the IMA was really exciting, rewarding, and productive. I spent part of my time attending workshops and seminars, interacting with people, and part of my time doing research. I learned a great deal about several areas, such as pattern information, dynamics of the immune response, hormone secretion and control, animal grouping, cell adhesion and mobility, etc. I benefited very much from interacting with people from different fields. The research I conducted at the IMA included chemostat modeling, drug deliver modeling, and the spread of competing species in the habitat, involving ordinary differential equations, integral differential equations, difference equations, and partial differential equations.

In the fall of 1998, I prepared a paper about a chemostat model with two perfectly complementary resources and n species. This was a continuation of my Ph.D thesis. The difference is that my Ph.D thesis deals with models with one limiting resource. I established predicted biological conditions for the survive of one species and coexistence of two species for the model. The results generalize those of Hsu, Cheng, and Hubbell (SIAM J. Appl. Math. 41 (1981), 422-444) and Butler and Wolkowicz (Math. Biosci. 83 (1987), 1-48) where only two species are considered. The difficulty was that the reduced system is no longer a two dimensional system and therefore I had to examine a high dimensional system. The key idea was to divide the relevant region into disjoint sets. The global stability of steady states was then captured on each set. While preparing this paper, I began collaborating with Ronald Siegel (department of Pharmaceutics and Biomedical Engineering, this university). The problem is about a model for a drug delivery system. It involves negative feedback action, with hysteresis, of an enzyme on a membrane through which substrate diffuses to reach the enzyme. The model was proposed by Zou and Siegel (J. Chem. Phys. 110 (1999), 2267-2279) and some primary analysis and extensive simulations were given there. In the joint work with Ronald Siegel, we gave rigorous mathematical proofs regarding the global stability of steady states as well as the existence and global stability of the limit cycle solution. The globally asymptotic behavior of this model is fully understood. The final version of a paper on the drug delivery model is in preparation.

In the spring of 1999, I started studying a chemostat model with two perfectly complementary resources, two species and delay. The model incorporates distributed time delay in the form of integral differential equations in order to describe the time delay involved in converting nutrient to biomass. This study was motivated by Ellermeyer (SIAM J. Appl. Math. 54 (1994), 456-465) and Wolkowicz, Xia, and Ruan (SIAM J. Appl. Math. 57 (1997), 1019-1043) where only one limiting resource is involved. I worked with Gail Wolkowicz (McMaster University) and Yang Kuang (Arizona State University), and we prepared a paper. We gave sufficient conditions to predict competitive exclusion for certain parameters ranges and coexistence for others. We demonstrated that when delays are large, ignoring them may result in incorrect predictions. The main technique we used was the linear chain trick which allowed us to transform the original system into a huge system of nonlinear ordinary differential equations.

Mark Lewis (University of Utah) visited the IMA in the spring of 1999, and we had many discussions on the spread of two competing species in the habitat. This led us to begin working with Hans Weinberger (School of Mathematics, this university). We had regular discussions every week. I learned a lot from our discussions and from Hans Weinberger's early publications. Currently, we are working on the spreading speed of two-species diffusion-competition (continuous and discrete) models. In many applications, the asymptotic speed of propagation of a nonlinear system is the same as that of its linearization. This principle has been referred to as the linear conjecture. Hosono (Bull. Math. Bio. 60 (1998), 435-448) showed that for the two-species diffusion-competition model, the linear conjecture is not true for certain parameters by using extensive numerical simulations. We first gave a general result for the discrete model, and applied that to the continuous model (a nice feature of this analysis is that the continuous model can be put in the form of discrete one). We obtained sufficient conditions for various cases which guarantee the linear conjecture is true. We then obtained the speed of propagation for each case. We are now preparing a paper about these results.

I had several discussions with Carlos Castillo-Chavez (Cornell University) about some models in epidemiology, and with John Mittler (Los Alamos National Laboratory) about predatory-prey models and chemostat models. I also had many interesting discussions with George Sell and Wei-Ming Ni (School of Mathematics, this university). All the discussions are very invaluable, and will certainly influence my future studies.

I visited Arizona State University in April of 1999, and gave a talk on the chemastat modeling in the mathematical biology seminar there. I gave a talk about my Ph.D thesis in the IMA postdoc seminar. I was invited to give a lecture in two international conferences ( in Canada and Bulgaria). But I was not able to make to the conferences.

During my stay at the IMA, I participated in other activities. I became a reviewer for Mathematical Reviews and for Zentralblatt fur Mathematick, and reviewed a number of papers. I refereed papers for several journals, including SIAM Journal on Mathematical Analysis, Proceedings of the American Mathematical Society, Applied Mathematics Letters, and Journal of Theoretical Biology.

Xianfeng (Dave) Meng submits the following report:

As a postdoctoral member for the year 1998-1999 on Mathematics in Biology, I spent a large part of my time attending workshops, seminars, and tutorials. Coming to IMA, my major goal was to learn more about mathematical biology, to see how other people use mathematics to explain biological phenomena. I feel I mostly achieved that goal. Through attending the tutorials, workshops and through interacting with the other workshop participants, I became acquainted with various biological area that mathematics plays an important role in. My knowledge was definitely broadened. I saw many applications of mathematics in biology. For example, reaction-diffusion systems are used in pattern formation to describe how patterns are generated reliably in the face of biological variation, systems of nonlinear ordinary differential equations, partial differential equations, cellular automata and stochastic processes are involved in the modeling of immune systems and cell signaling, traditionally well studied Navier-Stokes equations are used to model the swimming of some living organisms and deterministic and stochastic mathematical models are used to describe spectral patterns of plant communities, etc. The opportunities of meeting leading researchers in their respective field and of seeing how and what they are doing in their research fields are priceless. It was an exciting year for me. I particularly like the idea of tutorials. It gives us a chance to learn some basic concepts which are not familiar to us and prepares us for the workshop talks.

I started to collaborate with my mentor Dr. John Lowengrub in the fall of 1998 on an interesting computational fluid dynamics problem. We are considering a case of fluid flowing through a channel with the fluid outside the channel flowing in the opposite direction as opposed to the fluid inside the channel. It is a two dimensional problem. The channel walls are assumed immersed in the fluid. At first, We don't consider any tensions that hold the the channel walls together against the moving fluid. With this assumption, the channel walls will eventually roll up. Later, we will add tensions to the walls to make them more rigid to move. We are interested in comparing how different numerical computational methods handle the roll-up phenomenon and how accurately the methods can simulate it. On my part, I will use Immersed Boundary method to simulate the roll-up phenomenon. The Immersed Boundary method was proposed by Dr. Charles Peskin of Courant Institute to model blood flow in the heart. I have done the case without the tension in the channel walls. Currently, I am still working on the case with tensions added to the channel walls. The main difficulty I encountered is the accuracy problem of my program. No paper has come out from this work yet. We plan to continue our collaboration after my job at IMA. I enjoyed working with Dr. Lowengrub a lot. As my mentor, he has helped me both in my research and in my making career choices. I like IMA's mentor system very much. As a newly graduate just starting my career, I feel this system helped me tremendously. My mentor's help definitely has a very positive impact on my career development.

I also spent part of my time writing papers based on my dissertation research. One is about proposing a computational model of porous media at the pore level. It is going to be submitted to the Conference Proceedings of the IMA Workshop on Computational Modeling in Biological Fluid Dynamics. Another one is about the comparison between the computational method I used and the theoretical analysis. It is still under writing.

I participated the postdoc seminars and gave a talk in February. I am always interested in mathematical applications in industry. I attended all the talks of Seminar on Industrial Problems. As an applied mathematician, I think it is always important and also helpful to understand how mathematical researches are done and what kind of mathematics is needed in industrial environment. Personally, I think pursuing a career in industry is also a very good option. It is as challenging as that in academia.

At the end of June, I had a job interview with Louisiana Tech University. It is a regular tenure track position and I got the job.

Looking back, I feel this is a pretty fruitful year for me. I learned a lot in mathematical biology. I also established some contacts with people which I hope will last. My one-year stay at IMA is definitely an important point in my career development. It certainly will help to shape my future academic career.

Patrick Nelson writes:

The past year I have participated, as a postdoctoral fellow, in the Institute for Mathematics and Its Applications (IMA) year long program on Mathematical Biology. My year can be sectioned into three areas.

Area I

I participated in numerous week long workshops dealing with a wide range of topics in the biological sciences. My participation included attending lectures, panel discussions, poster sessions and social events.

Area II

I spent time developing a new collaboration with some of the long term visitors. In the fall quarter, I worked with Dr. Byron Goldstein, of the Los Alamos National Laboratory. This collaboration is continuing and we are currently developing models which explain the cellular response to allergens. I spent a week in March visiting Byron at the Los Alamos National Laboratory to continue this research. During the winter quarter, I started working with Dr. Harold Layton, of Duke University. He introduced me to models of renal flow in the kidneys. The modeling included partial differential equation with delays, an area I was currently working on with another project. This collaboration is continuing as I am starting a position at Duke University as a postdoctoral fellow in August and one of my projects will be the continued work with Dr. Layton. Finally, in the spring quarter, I started working with Dr. Jorge Velasco-Hernandez, of the Universidad Autonoma Metropolitan. During a discussion we determined that we were each working on models which could be combined to examine Chagas Infection. This work will be continued after I arrive at Duke University.

Area III

The third area and most important was the continued work on my research. Most of this time I continued my research in the area of HIV modeling with Dr. Alan Perelson of the Los Alamos National Laboratory. But there were a few other projects which were completed.

Project I

I completed and submitted the following work in June to The Bulletin of Mathematical Biology on Models of macrophage activation in response to pathogens.

The immune response to infection can be classified into two compartments; innate and cell-mediated. Macrophages, part of the innate system, recognize and digest foreign particles. This leads to a cascade of events, one of which is the signalling of the cell-mediate system. In the past decade, mathematical models have become an integral part in the study of infection and the immune response. Models have been developed which examine the interactions of the innate and cell-mediated system to infection and have provided much insight into the disease dynamics. Unfortunatly, there have only been a few mathematical works which focus on the innate system's response. We study the changes in the dynamics of macrophages in response to a pathogen and extend the previous works by including two compartments for macrophages, resident and activated. The model is then applied to experimental data and estimates for certain, previously unknown, kinetic parameters are obtained.

Project II

I completed and submitted the following work in June to The Journal of Emerging Infectious Diseases on Predicting the effect of judicious antibiotic use on drug-resistant Streptococcus pneumoniae colonization among children in day-care. This work is in collaboration with Tao Sheng Kwan-Gett, M.D. and Dr. James P. Hughes, both at The University of Washington. The objective was to predict the effects of reducing antibiotic use on the prevalence of drug-resistant S. pneumoniae colonization in a child day-care population.

Work on HIV modeling included the following:

Publication

Myself and Dr. Alan Perelson had a paper published in March, 1999 in Siam Review title Mathematical models of HIV dynamics in vivo.

Abstract: Mathematical models have proven valuable in understanding the dynamics of HIV-1 infection in vivo. By comparing these models to data obtained from patients undergoing antiretroviral drug therapy, it has been possible to determine many quantitative features of the interaction between HIV-1, the virus that causes AIDS, and the cells that are infected by the virus. The most dramatic finding has been that even though AIDS is a disease that occurs on a time scale of about 10 years, there are very rapid dynamical processes that occur on time scales of hours to days, as well as slower processes that occur on time scales of weeks to months. We show how dynamical modeling and parameter estimation techniques have uncovered these important features of HIV pathogenesis and impacted the way in which AIDS patients are treated with potent antiretroviral drugs.

Project III A

Extensions of this work included examining a model which included intracellular delays and recently we submitted, to Mathematical Biosciences, "A model of intracellular delay used to study HIV pathogenesis."

Abstract: Mathematical modeling combined with experimental measurements have provided profound results in the study of HIV-1 pathogenesis. Experiments in which HIV-infected patients are given potent antiretroviral drugs that perturb the infection process have provided data necessary for mathematical models to predict kinetic parameters such as the productively infected T cell loss and viral decay rates. Many of the models used to analyze data have assumed drug treatments to be completely efficacious and that upon infection a cell instantly begins producing virus. We consider a model which allows for less then perfect drug effects and which includes a delay process. We present detailed analysis of this delay differential equation model and compare results between a model with instantaneous behavior to a model with a constant delay between infection and viral production. Our analysis shows that when drug efficacy is not 100%, as may be the case in vivo, the predicted rate of decline in plasma virus concentration depends on three factors: the death rate of virus producing cells, the efficacy of therapy, and the length of the delay. Thus, previous estimates of infected cell loss rates can be improved upon by considering more realistic models of viral infection.

Project III B

Concurrently with the above work we submitted to The Journal of Virology a paper titled "Effect of the eclipse phase of the viral life cycle on estimation of HIV viral dynamic parameters." This work focuses on using potent antiretroviral therapy to perturb the steady state viral load in HIV-1 infected patients has yielded estimates of the lifespan of virally infected cells. Here we show that including a delay that accounts for the eclipse phase of the viral life-cycle in HIV dynamics models decreases the estimate of the productively infected cell lifespan. Thus, productively infected cells may have a half-life that is shorter than the estimate of 1.6 days published by Perelson et al.

Project III C

I am currently working on extensions of both models above and am planning to submit, when completed, a detailed paper on the analysis of the delay models to Siam journal of Applied Mathematics. This work will be completed while I am at Duke University.

Professional Activities

Besides research, the past year I co-organized a tutorial on HIV modeling which was given in November of 1998. I was invited to give a seminar at Arizona State University on my delay models in HIV. I visited Dr's Goldstein and Perelson, at Los Alamos National Laboratory in March. I was also invited to participate in a week long workshop on Mathematical Biology given at Duke University in May where I mentored a group of students on a modeling project and gave a lecture on my current research. I also spent time this year reviewing research articles for The Journal of Theoretical Biology and Mathematical Biosciences and I am currently reviewing a book, titled "Investigating Biological Systems Using Modeling for the Society of Mathematical Biology.

To conclude, I found this year to be very productive and stimulating and the most important aspect of this year and what I am most proud of was the birth of our baby boy, Joshua in January.

Kathleen A. (Rogers) Hoffman reports:

Workshops and Seminars

As a Postdoctoral Member of the IMA during the theme year in Mathematical Biology, my main responsibilities included attending workshops, seminars and tutorials. This provided a unique opportunity for me to gain a broad perspective of the latest developments in the area of mathematical biology. This broad perspective allowed me to sample the very different research areas that exist in this broad field. Identifying interesting research areas in mathematical biology is of particular interest to me since I have previously worked in the field and I hope to continue to expand my research program.

My biggest professional accomplishment this year culminated in accepting a tenure track position at UMBC. After months of submitting applications and traveling thousands of miles to interviews, I was thrilled to accept the offer from UMBC. Not only was it a tenure track position in an applied math department, it was also in a location in which my husband found a job as well. So although the seemingly endless hours dedicated to the job search didn't produce any quantifiable mathematics (theorems or papers), I believe that it was worth it since it produced a (non-unique) solution to one of the biggest problems in mathematics--the two body problem!

Talks

In addition to participating in the workshops and seminars associated with the theme year, postdoctoral members are expected to organize, attend and speak at the weekly Postdoc Seminar'. As with all the postdocs, I gave a talk in the postdoc seminar. In addition, I gave talks at the following series of talks:

• SIAM Annual Meeting, Atlanta GA, May 1999
• Mathematics Colloquium, Drexel University, Philadelphia, PA, February 1999
• Non-Linear Science Seminar, Naval Research Lab, Washington DC, February 1999
• Mathematics Colloquium, George Mason University, Fairfax VA, February 1999
• Research Colloquium, Southern Methodist University, Dallas TX, February 1999
• Mathematics Colloquium, University of Florida, Gainesville FL, February 1999
• Dynamics Seminar, Boston University, Boston MA, February, 1999
• Mathematics Colloquium, UMBC, Baltimore MD, January, 1999
• Mathematics Colloquium, Case Western Reserve University, Cleveland, OH, January 1999
• Postdoc Seminar, Institute for Mathematics and its Applications, University of Minnesota, Minneapolis MN, January, 1999

I was also invited to give the keynote address at the annual Sonia Kovalevsky Day at the University of Minnesota in October 1998. I found this particularly challenging since the audience consisted of high school students, some of whom hadn't even had algebra!

Research

My research accomplishments for this year encompassed three separate projects. The first project involved research on the stability of twisted elastic rods as a model for supercoiling in DNA minicircles. The second project was an industrial problem presented to the IMA by General Motors involving welding and clamping of beams. The third project investigated a system of four ordinary differential equations that serve as an idealized model of two reciprocally inhibitory neurons.

DNA: A twisted elastic rod is widely accepted to be a qualitative model of supercoiled DNA. Mathematically, a twisted elastic rod is represented by an isoperimetrically constrained calculus of variations problem. That is, the equilibria of the rod exactly correspond to critical points of a certain functional subject to integral constraints. Similarly, critical points which correspond to constrained minima are said to be stable equilibria. My thesis comprises a series of practical tests which determine which critical points correspond to constrained minima, or equivalently, which equilibria are stable. My research goals in this particular area were to complete papers that were based on my thesis research. During this year, one of the papers

- R. S. Manning, K. A. Rogers, & J. H. Maddocks, Isoperimetric Conjugate Points with Application to the Stability of DNA Minicircles
appeared in the Proceedings of the Royal Society of London: Mathematical, Physical and Engineering Sciences Vol 454, No. 1980, p. 3047-3074, Dec. 1998. Additionally, a paper with Leon Greenberg and John Maddocks

- L. Greenberg, J.H. Maddocks, & K.A. Rogers, The Bordered Operator and the Index of a Constrained Critical Point.
was accepted to Mathematische Nachrichten.

During a trip to visit my advisor, the results for the final paper from my dissertation were strengthened and generalized to include stability exchange results at non-simple folds as well as simple folds. These generalizations required a significant rewrite of the paper

- K.A. Rogers & J.H. Maddocks, Distinguished Bifurcation Diagrams for Isoperimetric Calculus of Variations Problems and the Stability of a Twisted Elastic Loop.

This last paper is still in preparation, but should be submitted shortly.

- Welding and Clamping of Beams: Experiments on shells have demonstrated that the sequence in which two shells are clamped and welded affects the final shape of the shells. Such a situation arises in assembling automobiles. In that setting, the consequences of different final shapes can be costly if, for instance, the final shape of the two shells (or automobile parts) causes the larger structure not to meet required specifications. In order to understand why this sequence dependence arises Dr. Danny Baker and Dr. Samuel Marin of General Motors Research and Development Center, Fadil Santosa, Associate Director for Industrial Programs at the IMA, and I proposed models of clamping and welding of beams which demonstrate this sequence dependence.

In the model that we propose, a series of rigid links connected by torsional springs represents a simplified model of the beam. Using the simplified model, we are able to show that the horizontal sliding that occurs during the clamping process gives rise to the sequence dependence. Although the model that does not allow horizontal sliding, and hence does not produce sequence dependence, can be solved analytically the model that does allow horizontal sliding is solved numerically using a constrained optimization routine from Matlab. Additionally, we are able to perform variational simulations that statistically demonstrate that clamping and welding the beams from the inside out produces the least amount of variation in the final assembly.

Currently, we are in the process of writing an article

- F. Santosa & K.A. Rogers, A Simple Model of Sheet Metal Assembly
that we intend to submit to the Education segment of SIAM Review. In summary, we are proposing that this problem could serve as one topics in an applied math topics class. The problem provides a great mix of modeling, computation and simulation while solving a real world problem.

- Reciprocal Inhibitory Neurons: Many of the behaviors observed in the solutions of the Hodgkin and Huxley equations can also be seen in simpler, yet still biologically reasonable, models. In particular, simple models of the action potential of neurons connected by reciprocally inhibited synapses have been studied to further understand such biological phenomena as heartbeat, swimming, and feeding. Two identical oscillatory neurons connected by reciprocally inhibitory synapses will oscillate exactly out of phase of each other, that is, while one neuron is active the other is quiescent. John Guckenheimer, Warren Weckesser and I studied an idealized model of a pair of reciprocally inhibited neurons in the gastric mill circuit of a lobster. Our goal is to understand solutions of a set of four differential equations which model two asymmetric oscillators in terms of geometric singular perturbation theory, an effective tool for understanding equations with multiple time scales. Essentially, singular perturbation theory pieces together solutions from the fast system and solutions from the slow system to get a solution of the singularly perturbed system. Behavior of a singularly perturbed system consists of motion on the slow manifold (the set of equilibria of the fast system) and fast jumps between different parts of the slow manifold. These fast transitions occur at folds in the slow manifold. A periodic solution to the system of equations which describes a pair of identical reciprocally inhibitory neurons can be described in terms of singular perturbation theory as consisting of two fast transitions. These fast transitions correspond to one neuron jumping from an active to a quiescent state and the other jumping from a quiescent state to an active state.

As we investigated the solution space of the asymmetric problem, we found many solutions that were qualitatively similar to the solutions of the symmetric system. We also found other very different types of behavior. For instance, in a small parameter range, there exists (at least) two stable periodic orbits of the full system. Both of these periodic solutions correspond to more complicated behavior than the typical reciprocally inhibitory behavior described above. Instead of the orbit consisting of two fast transitions, the periodic orbits consist of nine and eleven fast transitions, respectively, and the behavior of the two neurons can no longer be classified simply as active or quiescent. The possible implications of the existence of two stable periodic solutions as well as the structure of these solutions are a source of continued research.

In addition to bistability in the system, we also found canard solutions, that is, solutions in which part of the orbit occurs on an unstable portion of the slow manifold. We identified two different types of canard solutions. One type of canard solution consists of a fast transition to an unstable part of the slow manifold. In the other type of canard solution, the orbit continues past a fold in the slow manifold onto the unstable part of the manifold. For a small parameter range, the family of canard solutions is stable. Continuation of the two stable periodic solutions reveals that the canard solutions persist for a much larger parameter regime but are unstable.

We have observed that existing integration algorithms have difficulty computing accurate numerical representations of canard solutions. The portion of the canard solution on the unstable part of the slow manifold is simply too sensitive to small changes to be computed using the most sophisticated initial value problem sovlers. Instead, we have had limited success in calculating canard solutions using AUTO, a package for solving boundary value problems that uses continuation methods to track solutions. However, even this approach is not without its computational difficulties. For instance, the Floquet multiplier routine in AUTO seems to have trouble with this problem, thus making stability information difficult to obtain.

Marina Osipchuk, Industrial Postdoctoral Member shares the following:

My second year as an Industrial postdoc affiliated with the Honeywell Technology Center (HTC) I worked on disturbance rejection control in decentralized systems. It was a joint project with Dr. Michael Elgersma and Dr. Blaise Morton, HTC. We developed an algorithm and implemented it in a software package that finds a decentralized control achieving the maximum attenuation of a disturbance signal.

I presented the results of our project on a research seminar at the Honeywell Technology Center.

In addition I continued my collaboration with Edriss Titi and Yannis Kevrekidis on finite-dimensional control of reaction-diffusion systems. I also gave an invited presentation on the results at the Dynamical Systems and Control Seminar at the Aerospace Engineering and Mechanics department, UMN. The further results on this project were presented at the SIAM Conference on Applications of Dynamical Systems.

Research Disturbance Rejection Control in Systems with Decentralized Control Architectures

Many industrial control problems are associated with the control of complex interconnected systems such as those for electric power distribution, chemical process control, and expandingly constructed systems. Such systems are often characterized by practical restrictions on information flow, rendering conventional centralized control impractical. This system architecture calls for decentralized control, wherein a given controller observes only local subsystem outputs and controls only local inputs and all controllers function in concert to regulate the composite system. Compared to centralized control, this architecture provides for implementation simplicity and tolerance of many types of failures.

In practice, the decentralized control should demonstrate adequate performance in the presence of disturbances as well as internal stability. We designed a decentralized control that achieves the global minimum of the system response to a disturbance output. The developed efficient algorithm reduces the optimal control problem to a sequence of polynomial systems and subsequently formulate them as matrix equations. The solutions of the matrix equations corresponding to all local minima were found using eigensystem solvers. Simulations performed using this algorithm for a variety of systems and disturbance types indicate that it has significant promise for practical application to decentralized control.

Overall, the two years spent at the IMA were productive and stimulating for me. The environment of the IMA had a broadening influence on my research. I have established several contacts that I believe will have a strong impact on my career.

Anthony Varghese writes:

I was an industrial postdoctoral fellow working on projects from Medtronic Inc. from Oct. 1, 1998 to Sept. 30, 1999. The year was very fruitful for me as I have learned a great deal about industrial collaborations. The industrial project I was working on was the general area of atrial fibrillation. Atrial fibrillation is a condition that afflicts a large number of individuals and although not immediately life-threatening as ventricular fibrillation is, it can cause chronic fatigue and can greatly increase the risk of stroke. Atrial fibrillation takes place in the atria, the upper chambers of the heart and is characterized by travelling waves of electrical activity that appear unorganized compared to the regular pattern of activation in the normal heart. Although there are drugs as well as implantable devices made by companies like Medtronic to control atrial fibrillation, both approaches have serious side-effects. My task was to set up models of cellular excitability to understand how atrial fibrillation is initiated with a view towards coming up with ways of terminating fibrillation. Given that there were two recent cellular models of human atrial cell electrical activity published last year, my project was to code these models and study its properties. This approach has been taken since the 1960s when some prominent cardiologists together with Werner Rheinboldt set up a coupled cellular automata lattice to study spiral waves. The difference in my case is that much has been learned in the intervening years about the properties of heart cells and so instead of a discrete-state cellular automaton model for each cell, I used systems of coupled nonlinear ordinary differential equations for each cell. Propagation of electrical activity can be modeled by combining the above mentioned cell models with a parabolic equation. A chronic condition like chronic atrial fibrillation can be modeled by changing a number of parameters based on experiments on cells from human hearts. The interesting feature of atrial fibrillation is that it starts as short periods of fibrillation that occur spontaneously but the more that these short events occur, the more likely it is for the fibrillation to persist. It is not clear why this is so.

In addition to modeling atrial fibrillation, I was also involved in modeling the kinetics of ion channels with Linda Boland who was in the Dept. of Physiology at first and who switched to the newly formed Dept. of Neuroscience in June. I set up a Markov state model based using a published model and used a robust numerical scheme to handle the resulting stiff equations. Dr. Boland acquires data from experiments and we were able to compare the data with the results of the model. During the summer I helped supervise an undergraduate who set up a scheme to search in parameter space for model parameters that gave the best fit in a least-squares sense. Some of these results will be presented in abstract form at a meeting on the Biology of Potassium Channels in Sept. 99 at Colorado and a paper is in preparation. I also worked with a pain researcher named George Wilcox on setting up partial differential equation models of conduction of electrical activity in primary afferent neurons responsible for conducting pain signals. In addition the action of estrogen on certain potassium channels were investigated along with a cardiologist, Scott Sakaguchi, in the Medical School and these results were presented as an abstract. I was able to collaborate very fruitfully with a researcher in Bristol, UK, on the effect of genetic mutations on a particular potassium channel that is responsible for deadly arrhythmias of the heart. These results were published last December. I also worked with experimenters in Oxford, UK, on modeling certain arrhythmias related to excess nervous input and this work has been submitted for publication.

Other activities:

I was asked to recruit speakers for the Mathematical Physiology seminars and managed to get speakers from Chemical Engineering as well as Biochemistry and Physiology.

Conferences:

I was invited to a conference on Modeling and Defibrillation organized by a Swiss cardiologist and Medtronic in Lausanne, Switzerland in December 1998. I was able to present results that indicated that simple square domains were insufficient as far as being able to reconstruct atrial fibrillation. This was an extremely useful and interesting meeting since it brought together a small group of mathematical modelers with experimenters and clinicians to discuss cardiac arrhythmias and the contribution of modeling in particular. In June I was invited to a workshop organized by Craig Henriquez at the Dept. of Biomedical Engineering at Duke University on numerical methods for models of heart electrical activity. This meeting brought together mathematicians, computer scientists and engineers to examine numerical schemes. I was surprised to find that a number of schemes relied at least in part on the simple forward-difference scheme. In July I visited Linda Petzold to discuss the use of her numerical codes for the problems I am investigating. Since I had been using her codes since January, I had a number of very specific problems to discuss and the meeting was very fruitful.

Publications: (Work during the year)

1. Varghese, A., In Press for 2000,Membrane Models', In: Biomedical Engineering Handbook}, Second Edition, CRC Press, Boca Raton.

2. Hancox, J.C., H.J. Wichtel, and A. Varghese, 1998, Alteration of HERG current profile during cardiac ventricular action potential, following a pore mutation" Biochemical and Biophysical Research Communications, vol. 253, pp. 719-724.

Submitted:

1. Nash, M.P., J.M. Thornton, C.E. Sears, A. Varghese, M. O'Neill, and D.J. Paterson, "Epicardial Activation Sequence During a Norepinephrine-Induced Ventricular Arrhythmia and its Computational Reconstruction."

Abstracts:

1. Nash, M.P., J.M. Thornton, A. Varghese, and D.J. Paterson, 1999, "Electromechanical Characterization and Computer Simulation of a Noradrenaline Induced Ventricular Arrhythmia." FASEB J. 13 (5) A 1075, Mar. 1999

2. Varghese, A., G.L. Wilcox, S. Sakaguchi, 1999, "Modulation of I_Ks by Estradiol." European Working Group on Cellular Cardiac Electrophysiology, September, 1999.

3. Hancox, J.C., H.J. Witchel, J.S. Mitcheson, and A. Varghese, 1999, "Insights into the Rapid Delayed Rectifier K Current, IKr, from Action Potential Clamp Experiments." European Working Group on Cellular Cardiac Electrophysiology, September, 1999.

4. Boland, L.M. and A. Varghese, 1999, "Immobilization of Shaker Potassium Channel Gating Currents by the Beta Subunits Kvbeta1.1 and Kvbeta1.3" Biology of Potassium Channels: From Molecules to Disease, September, 1999.

Warren Weckesser is another IMA postdoc. He writes:

My activities this year focused on attending the many exciting workshops in mathematical biology, continuing my collaboration with Kathleen Rogers and John Guckenheimer on a study of a model of two coupled neurons, studying some interesting properties of an inverted pendulum with a rapidly vibrating suspension point, and continuing my research on the stability of whirling modes in rotating mechanical systems.

This year's theme of Mathematical Biology provided a fascinating variety of mathematical applications in many fields of biology. The workshops on pattern formation, immunology (including AIDS and cancer), cell motility, renal physiology, ecosystems, epidemics, and more have been an invaluable educational experience. My next academic position is at the University of Michigan, and while there I will maintain an active research program in mathematical biology.

My primary research effort this year has been a collaboration with John Guckenheimer and Kathleen Rogers on an intensive study of the rich dynamical behavior found in a system of two coupled relaxation oscillators. More specifically, we are considering two non-identical Van der Pol-like oscillators. The coupling is based on reciprocal inhibition, as occurs in membrane models of neurons. Our model of two coupled neurons results in a singularly perturbed system of differential equations, with two fast variables and two slow variables. Our observations so far include several families of complicated periodic orbits, a range of parameters for which there are two stable periodic orbits, families of orbits that exhibit a variety of canards (solutions that track an invariant unstable slow manifold for long times), a possible homoclinic explosion associated with a homoclinic bifurcation from a periodic orbit, and several mechanism for the formation of canards. One goal of this research is to classify the types of bifurcations that occur in singularly perturbed systems with more than two dimensions. We have gained great insight into the importance of canards in the bifurcation of periodic orbits in singularly perturbed systems. Especially important for this work are the methods of geometric singular perturbation theory. Another important component of the work so far has been the numerical continuation of periodic orbits with the software package AUTO. We are currently preparing a paper that focuses on the numerical aspects of this problem; subsequent papers will discuss bifurcation theory for singularly perturbed systems. Kathleen and I presented a poster on this research at a conference in mathematical biology at the University of Pittsburgh.

This year I began working on a new project with Mark Levi (my thesis advisor and a visitor last year). We are considering a generalization of the much studied pendulum with a rapidly vibrating suspension point. In the classical problem, the suspension point vibrates in a straight line. We are extending the analysis to the case where the vibration is periodic, but the path of the suspension point is an arbitrary closed curve. We find that the averaged equations contain terms that have interesting geometric interpretations. One is purely geometric, depending on the area of the closed curve followed by the suspension point. Another term can be interpreted as the effective force exerted by a nonholonomic constraint, even though the full system is holonomic.

I have also continued my research on mechanical systems composed of symmetric rigid bodies coupled with constant velocity joints. This is a study that I began last year, during a visit by Mark Levi. Unlike a universal joint, a constant velocity joint creates a kinematic constraint that directly couples the angular velocities of the rigid bodies. I am investigating the bifurcation and stability of whirling configurations of chains of coupled rigid bodies. This work may shed new light on certain gyroscopic phenomena in spinning beams and related mechanical systems.

Aleksandar Zatezalo is one of the industrial postdoctoral associates. He reports:

My postdoctoral appointment started on June 15th of 1998. During 1998-99 academic year I was working closely on problems and development of passive surveillance system (PSS) as part of the research and development group at Lockheed Martin Tactical Defense Systems, Eagan. We derived and stated mathematical models for the bistatic Doppler radar system in order to localize in three dimensional space positions of flying objects in Twin Cities area using several transmitters and determining direction of arrival (DOA) of electromagnetic waves which scatter from them by using standard beamforming techniques. We developed algorithms and simulations for several scenarios of localizations and tracking on the real trajectories, the so called geo-tracker which for example localizes and tracks flying objects using the data from six transmitters, four transmitters, or the direction of arrival in conjunction with two transmitters. We also developed algorithms for only tracking using simultaneously three transmitters or one transmitter together with the direction of arrival. Straight forward analysis of these algorithms is performed. We proceeded with analysis of the real data collected in December of 1998 and January of 1999 since when we have been working on mathematical models of the signals and determining statistical parameters important for their extraction from the noisy environment. We developed and implemented Bayesian update tracker using these mathematical models and calculating statistical prediction by applying the Alternating Direction Implicit methods where we were dealing with problematic boundary condition by using Markov process approximations. Since we needed the track initializer and because of the computational complexity of the straight forward Bayesian approach which partially comes from the uniform noise in the phase which appears in the mathematical model we developed simplified line tracker for tracking Dopper lines which were appearing on the time-frequency grams. We are still developing the simplified line tracker for the best possible performance in order to compare our measurements with the data which were collected from the radars located in Twin Cities area and to associate signals which are coming from the different transmitters but from the same scatterer.

The goal of our research is to demonstrate localization and tracking of flying objects in Twin Cities area using several transmitters and directions of arrivals by the end of the current year.

By the end of the last year (1998) Professor Nicolai Vladimirovic Krylov and I submitted paper under title A direct approach to deriving filtering equations for diffusion processes to Applied Mathematics & Optimization as natural continuation of the research from my Ph.D. thesis. Several publications connected with my work on industrial problems are in preparation.

I benefited from discussions with Professor Walter Littman, Professor Fernando Reitich, Professor John Baxter, Dr. Marina Osipchuk, Dr. Nicholas Coult, and Dr. Marco Fontelos.

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