# Research Interests and Activities

My research area is interdisciplinary: it is a combination of mathematics, biology, bioinformatics, and ecology. I am principally interested in applied mathematics, partial differential equations, and inverse problems. I am using mathematical and quantitative techniques to address biological and ecological problems. My main goal is to contribute to the improvement of mathematical methods and models for advancements of mathematical analysis in parallel to the advancements in biology and to address the mismatch between current limited quantitative methods and the applied ecological needs.

Biology looks to be a driving force for the scientific improvement of this century, as physics was for the previous couple of centuries. In the era of nano-technology, biological knowledge is getting more complicated and more advanced every day, with probably a doubling time of five to ten years. Advancements in biology are so rapid that quantitative analysis of biological phenomena does not keep pace with them.

Environmental issues have taken a central place in human concerns with new and growing understandings from science that unlimited population growth, associated resource demands, and continuing technological advances are not indefinitely sustainable on a finite planet. On the other hand, traditional ecology has an applied nature and still in the empirical stage of development; a first principles-based formal theory has yet to emerge in its mainstream framework. This narrows the field's scope of applicability and compromises its ability to deal with complex organism-environment relationships. To that extent, ecology and environmental science are limited in their applied reach by a general inability to realistically model and analyze the complex systems of man and nature. Mathematical theories and modeling have significant potential to lead the way to a more formalistic and theoretical ecological science devoted to the discovery of scientific laws on which to base understanding and prediction, and out of this, more exact, precise, and incisive environmental applications can be expected to materialize.

### Environmental Ecology

Although the steady-state analysis of ecological networks is well established, dynamic and nonlinear analytical methods have remained a long-standing, open problem. One such static theory, an environmental system theory known as environ theory, has been developed over recent decades for linear, static models. The mathematical formulation of it is based on the principle of conservation of mass or energy.

I recently formulated a dynamic method for analysis of ecological networks--an age-old, open problem, successfully. The dynamic method is based on the novel dynamic system partitioning methodology that allows partitioning the whole system, including storages and flows, analytically and explicitly into mutually exclusive and exhaustive subsystems, each of which is driven by a single input (see Fig below). Consequently, this methodology refines the system analysis from the static linear compartmental to the dynamic nonlinear subcompartmental level to explore full complexity of the systems. It forms a platform for definitions of more delicate dynamic system analysis tools. This methodology, in effect, brings a novel formal, deterministic, complex system theory to the service of urgent applied ecological problems of the day. The dynamic method is compared with earlier approaches from literature. Our dynamic results at steady state are shown to be consistent with those of the prior static and linear methods. In that sense, the proposed dynamic methodology can also be considered as a natural extension of the established static network analysis methods, currently applicable only to continuous linear models at steady-state, to general dynamic analysis. The dynamic methodology also generates a mathematical framework in which all separate concepts and individual quantities of static theory are combined in a coherently unified mathematical context.

__Related publications__:

- Huseyin Coskun, Analysis of Nonlinear Dynamical Compartmental Systems, under review

### Cell Motility

I developed models for cell movements and introduced novel model-based inverse problem formulations for extracting useful information from single cell motion. Cell motility is a vital phenomenon in almost all living organisms. Abnormalities related to cell motility are often times the manifestation of a variety of diseases. An increase in the motility of tumor cells, for example, may be associated with cancer aggressiveness and metastasis. Behavioral or conformal changes of a cell naturally bear information about the underlying mechanisms that generate such changes. Reading cell motion, that is, understanding these biophysical and mechano-chemical processes is, therefore, of paramount importance.

The mathematical models we developed determine some physical features and material properties of the cells locally through analysis of live cell image sequences and uses this information to make further inferences about the molecular structures, dynamics, and processes within the cells. We propose that such analysis can identify cell pathologies, as analogous to the physical examination of patients to diagnose possible etiologic factors causing the diseases. Thus, the importance of the research program we initiated lies in its ability to translate the abstract theoretical understanding of cell motion into practical data that is useable for diagnosis and prognosis at the cellular level. Analysis of cell's behavioral changes under different chemical and physical conditions is also possible. We therefore propose that systematic applications of our novel approach can lead to cell profiling, characterization, and motility--based quantitative classification. Similarly, it may have impact on designing effective treatment regimens, early detection of diseases such as cancer, novel drug development, and on personalized and cellular medicine. The generality of the principals used in formation of the models ensure their wide applicability to different phenomena as for diagnostic and prognostic tools. As an immediate application of this technique, the morphological changes and motility of single cancer cells are being investigated and their comparison to normal cells are being studied for characterization and classification of cancer cells in an ongoing work.

The Anchor Model is applied to the neutrophil chasing two bacteria (neutrophil.mov) and keratocyte motionĀ (keratocyte.mov). The original movies are annotated based on the model outcomes.

For more delicate biochemical and biophysical analysis, the protein-protein interactions that give rise to single cell motion and force generation are analyzed separately. Yet, for a more realistic account of mechanics, a multiphase model is formulated and is currently being studied. Based on the mathematical model's outcomes, we hypothesized a novel biological model for collective biomechanical and molecular mechanism of cell motion which incorporates the actin network and adhesion site formation, microdomains, chemotaxis, and retrograde flow, and membrane ruffling. We propose that micro domain signaling dynamics organizes cytoskeleton and its interaction with substratum. As microdomains trigger and maintain active polymerization of actin filaments, their propagation and zigzagging motion on the membrane generate a highly interlinked network of curved or linear filaments oriented at a wide spectrum of angles to the cell boundary. Microdomain interaction may also mark the formation of new focal adhesion sites at the cell periphery. Myosin interaction with the actin network then generate membrane retraction/ruffling, retrograde flow, and contractile forces for forward motion. Finally, continuous application of stress on the old focal adhesion sites could result in the calcium-induced cal pain activation, and consequently the detachment of focal adhesions which completes the cycle.

Images: (left) Schematic representation of the collective biomechanical and molecular mechanism of cell motion: Actin network is represented by vertical, thin, black lines for clear visualization of the idea presented. Filaments marked with red ends represent actively pushing, polymerizing filaments following signal by the microdomains. Green curves on the membrane represent microdomains and left/right arrows represent microdomain/signal propagation speed, $\nu$. Cyan dots represent integrins and yellow dumbbells represent myosin family proteins. Blue line segment represents the region pulled forward by the actomyosin system. Up and down thick, magenta arrows represent cell contraction and myosin induced retrograde flow and/or membrane ruffling, respectively (right). Components of the deformation index for comparison of three model applications; neutrophil, keratocyte, and brain tumor (glioblastoma) cell line motion.

The proposed mathematical models have forward and inverse problem components. The forward problems formulate governing equations of cell motion where a set of material parameters, such as elasticity and viscosity, is given. Our novel model--based inverse problem approach introduces systems of equations that determine these physical parameters, and underlying molecular mechanisms involved in the motion. The material parameters in turn can be used as input for the forward problem to mimic the given cell motion. In the case of our continuum model, functional relations between molecular dynamics and continuum mechanics are set up coherently through the formulation of dynamic material properties and the description of force generation in a modular fashion. The forward problems use the theory of differential equations including free boundary value problems and differential geometric tools and the inverse problems use that of algebraic systems, least squares approximations, and Kalman filtering in formulation of the model and its solution. The models are, in general, solved numerically. In a recent work, qualitative analysis of the governing system of equations for the continuum model, which forms a free boundary value problem of mixed type, is approached using fix point arguments. Shauder type estimates and consequently existence and uniqueness results are being studied. The results and analysis will be completed and appear in a separate paper.

__Related publications__:

- Hasan Coskun, Huseyin Coskun, Cell Physician: Reading Cell Motion. A Mathematical Diagnostic Technique Through Analysis of Single Cell Motion, Bulletin of Mathematical Biology, doi: 10.1007/s11538-010-9580-x, Epub: 9/28/2010. (PDF)
in the OSU and other scientific news portals and blogs.**Research reviewed** - Ahmet Sacan, Hakan Ferhatosmanoglu, Huseyin Coskun. CellTrack: An Open-Source Software for Cell Tracking and Motility Analysis, Bioinformatics, 24(14):1647-9, 2008. (PDF)
- Huseyin Coskun, Yi Li, Michael Mackey. Amoeboid Cell Motility: A Model, and Inverse Problem with an application to live cell imaging data, Journal of Theoretical Biology, Elsevier, 244(2):169-179, 2007. (PDF)
- Huseyin Coskun. PhD Thesis, Mathematical Models for Cell Motility and Model Based Inverse Problems, July, 2006.
- Huseyin Coskun. Molecular mechanism of single cell motion, in preparation
- Huseyin Coskun. A Continuum Model with Free Boundary Formulation and the Inverse Problem for Ameboid Cell Motility, in preparation
- Qualitative analysis of a Continuum Model with Free Boundary Formulation and the Inverse Problem for Ameboid Cell Motility (with Avner Friedman, PhD, OSU), in preparation
- Cancer Cell Characterization and Classification through Single Cell Motility Analysis (with Hakan Ferhatosmanoglu, PhD, OSU, Enver Ozer, MD, OSU), in preparation

### Bioinformatics

For analysis of single cell motion, quantification of experimental data in the form of image sequences is necessary. For this nontrivial task, an open access software, CellTrack, which is based on the extension of known quantitative image analysis techniques, is developed. CellTrack is a self-contained, extensible, and cross-platform software package for cell tracking and motility analysis. Besides the general purpose image enhancement, object segmentation and tracking algorithms, we have implemented a novel edge-based method for sensitive tracking of the cell boundaries, and constructed an ensemble of methods that achieves refined tracking results even under large displacements or deformations of the cells. The software determines other characteristic features of cell motion, including cell's speed, area, etc., too. A novel point set registration technique formulated in combination of nonlinear transformations and Kalman filter approaches and currently being studied for further refinement of data quantification.

__Software__: Open-access software can be downloaded here.

__Related publications__:

- Ahmet Sacan, Hakan Ferhatosmanoglu, Huseyin Coskun. CellTrack: An Open-Source Software for Cell Tracking and Motility Analysis, Bioinformatics, 24(14):1647-9, 2008. (PDF)
- Hasan Coskun, Huseyin Coskun, Cell Physician: Reading Cell Motion. A Mathematical Diagnostic Technique Through Analysis of Single Cell Motion, Bulletin of Mathematical Biology, doi: 10.1007/s11538-010-9580-x, Epub: 9/28/2010. (PDF)
in the OSU and other scientific news portals and blogs.**Research reviewed** - Huseyin Coskun, Fatih Olmez, Hasan Coskun. Point set registration for 2D nonrigid motion of single cells, in preparation

### Stem Cell Research

I was also able to develop models for various other biological phenomena including fat cell fate determination that is related to stem cell research. White adipose tissue is the major energy storage depot for neutral lipids and is also a key endocrine regulator of a host of homeostatic activities, including metabolism, feeding behaviors, cardiovascular functions and reproduction. Abnormal fat accretion in the setting of obesity can lead to insulin resistance and type 2 diabetes, and has been linked to some cancers and arteriosclerosis. Thus, a thorough appreciation of the intricate signaling events that must take place as quiescent adipocyte precursors are recruited into the proliferating cell population that then must `decide` to differentiate into fully functional fat cells is critical to our understanding of diseases related to excess adipogenesis. We are beginning to gain insights into the molecular regulators of adipocyte differentiation. A significant advance would be to construct mathematical modeling tools that would assist cell biologists in predicting how environmental and/or intrinsic inputs could influence preadipocyte fate decision making. We have developed a model of how preadipocytes may use the dynamic interplay of two transcription factors, nuclear factor-kappaB (NF-kappaB) and peroxisome proliferator-activated receptor-gamma (PPAR-gamma) in early proliferation and differentiation events in vitro. Critical to the model is the feedback signaling between NF-kappaB and its inhibitor, IkappaB. The model is based on differential equations that describe the interactions of stimuli for NF-kappaB activation and mitogen concentrations and allows one to make predictions about how mouse 3T3-L1 preadipocytes choose between proliferation, differentiation or quiescence.

__Related publications__:

- Huseyin Coskun, Taryn L. Summerfield, Doug A. Kniss, Avner Friedman. Mathematical Modeling of Preadipocyte Fate Determination, Journal of Theoretical Biology, 265(1), July 7, 2010. (PDF)
in the OSU and other scientific news portals.**Research reviewed**