MBI Publications

MBI Publications for Hans Othmer (16)

  • C. Xue and H. Othmer
    Macroscopic equations from cell-based models in bacterial pattern formation
    SIAM J. Appl. Math. (Under Review)

    Abstract

  • J. Hu, H. Kang and H. Othmer
    Stochastic analysis of reaction-diffusion processes
    Bulletin of Mathematical Biology (Accepted)

    Abstract

    Reaction and diffusion processes are used to model chemical and biological processes over a wide range of spatial and temporal scales. Several routes to the diffusion process at various levels of description in time and space are discussed and the master equation for spatially-discretized systems involving reaction and diffusion is developed. We discuss an estimator for the appropriate compartment size for simulating reaction-diffusion systems and introduce a measure of fluctuations in a discretized system. We then describe a new computational algorithm for implementing a modified Gillespie method for compartmental systems in which reactions are aggregated into equivalence classes and computational cells are searched via an optimized tree structure. Finally, we discuss several examples that illustrate the issues that have to be addressed in general systems.
  • Y. Kim and H. Othmer
    A hybrid model for tumor spheroid grown in vitro I: Theoretical development and early results
    Math. Models Methods in Appl ScisVol. 17 (2007) pp. 1773-1798

    Abstract

    Tumor spheroids grown in vitro have been widely used as models of in vivo tumor growth because they display many of the characteristics of in vivo growth, including the effects of nutrient limitations and perhaps the effect of stress on growth. In either case there are numerous biochemical and biophysical processes involved whose interactions can only be understood via a detailed mathematical model. Previous models have focused on either a continuum description or a cell-based description, but both have limitations. In this paper we propose a new mathematical model of tumor spheroid growth that incorporates both continuum and cell-level descriptions, and thereby retains the advantages of each while circumventing some of their disadvantages. In this model the cell-based description is used in the region where the majority of growth and cell division occurs, at the periphery of a tumor, while a continuum description is used for the quiescent and necrotic zones of the tumor and for the extracellular matrix. Reaction-diffusion equations describe the transport and consumption of two important nutrients, oxygen and glucose, throughout the entire domain. The cell-based component of this hybrid model allows us to examine the effects of cell‚€“cell adhesion and variable growth rates at the cellular level rather than at the continuum level. We show that the model can predict a number of cellular behaviors that have been observed experimentally.
  • M. Stolarska, Y. Kim and H. Othmer
    Multiscale models of cell and tissue dynamics
    Phil. Trans. Roy. Soc.Vol. 367 (2009) pp. 3525-3553

    Abstract

  • C. Xue and H. Othmer
    Multiscale models of taxis-driven patterning in bacterial populations
    SIAM Journal of Applied MathematicsVol. 70 No. 1 (2009) pp. 133-167

    Abstract

    Spatially-distributed populations of various types of bacteria often display intricate spatial patterns that are thought to result from the cellular response to gradients of nutrients or other attractants. In the past decade a great deal has been learned about signal transduction, metabolism and movement in E. coli and other bacteria, but translating the individual-level behavior into population-level dynamics is still a challenging problem. However, this is a necessary step because it is computationally impractical to use a strictly cell-based model to understand patterning in growing populations, since the total number of cells may reach 1012 - 1014 in some experiments. In the past phenomenological equations such as the Patlak-Keller-Segel equations have been used in modeling the cell movement that is involved in the formation of such patterns, but the question remains as to how the microscopic behavior can be correctly described by a macroscopic equation. Significant progress has been made for bacterial species that employ a ‚??run-and-tumble‚?? strategy of movement, in that macroscopic equations based on simplified schemes for signal transduction and turning behavior have been derived [14, 15]. Here we extend previous work in a number of directions: (i) we allow for time-dependent signals, which extends the applicability of the equations to natural environments, (ii) we use a more general turning rate function that better describes the biological behavior, and (iii) we incorporate the effect of hydrodynamic forces that arise when cells swim in close proximity to a surface. We also develop a new approach to solving the moment equations derived from the transport equation that does not involve closure assumptions. Numerical examples show that the solution of the lowest-order macroscopic equation agrees well with the solution obtained from a Monte Carlo simulation of cell movement under a variety of temporal protocols for the signal. We also apply the method to derive equations of chemotactic movement that are governed by multiple chemotactic signals.
  • H. Othmer, K. Painter, D. Umulis and C. Xue
    The intersection of theory and application in elucidating pattern formation in developmental biology
    Mathematical Modelling of Natural PhenomenaVol. 4 No. 4 (2009) pp. 3-82

    Abstract

  • C. Xue, H. Othmer and R. Erban
    From Individual to Collective Behavior of Unicellular Organisms: Recent Results and Open Problems, Multiscale Phenomena in Biology
    Proceedings of the 2nd Okinawa Conference on Mathematics and Biology, AIPVol. 1167 (2009) pp. 3-14

    Abstract

  • M. Stolarska, Y. Kim and H. Othmer
    Multiscale models of cell and tissue dynamics
    Phil. Trans. Toy. Soc.Vol. 367 (2009) pp. 3525-3553

    Abstract

    Cell and tissue movement are essential processes at various stages in the life cycle of most organisms. The early development of multi-cellular organisms involves individual and collective cell movement; leukocytes must migrate towards sites of infection as part of the immune response; and in cancer, directed movement is involved in invasion and metastasis. The forces needed to drive movement arise from actin polymerization, molecular motors and other processes, but understanding the cell- or tissue-level organization of these processes that is needed to produce the forces necessary for directed movement at the appropriate point in the cell or tissue is a major challenge. In this paper, we present three models that deal with the mechanics of cells and tissues: a model of an arbitrarily deformable single cell, a discrete model of the onset of tumour growth in which each cell is treated individually, and a hybrid continuum-discrete model of the later stages of tumour growth. While the models are different in scope, their underlying mechanical and mathematical principles are similar and can be applied to a variety of biological systems.
  • C. Xue, H. Othmer and R. Erban
    From Individual to Collective Behavior of Unicellular Organisms: Recent Results and Open Problems
    Multiscale Phenomena in Biology: Proceedings of the 2nd Okinawa Conference on Mathematics and Biology, AIPVol. 1167 (2009) pp. 3-14

    Abstract

  • H. Othmer, K. Painter, D. Umulis and C. Xue
    The intersection of theory and application in elucidating pattern formation in developmental biology
    Math. Model. Nat. Phenom.Vol. 4 No. 4 (2009) pp. 3-82

    Abstract

  • C. Xue, E. Budrene and H. Othmer
    Radial and spiral streams in Proteus mirabilis colonies
    PLOS Computational BiologyVol. 7 No. 12 (2011)

    Abstract

  • Y. Kim, M. Stolarska and H. Othmer
    The Role of the Microenvironment in Tumor Growth and Invasion
    Progress in Biophysics and Molecular BiologyVol. 106 (2011) pp. 353-379 (Submitted)

    Abstract

    Mathematical modeling and computational analysis are essential for understanding the dynamics of the complex gene networks that control normal development and homeostasis, and can help to under- stand how circumvention of that control leads to abnormal outcomes such as cancer. Our objectives here are to discuss the different mechanisms by which the local biochemical and mechanical microenvironment, which is comprised of various signaling molecules, cell types and the extracellular matrix (ECM), affects the progression of potentially-cancerous cells, and to present new results on two aspects of these effects. We first deal with the major processes involved in the progression from a normal cell to a cancerous cell at a level accessible to a general scientific readership, and we then outline a number of mathematical and computational issues that arise in cancer modeling. In Section 2 we present results from a model that deals with the effects of the mechanical properties of the environment on tumor growth, and in Section 3 we report results from a model of the signaling pathways and the tumor microenvironment (TME), and how their interactions affect the development of breast cancer. The results emphasize anew the complexities of the interactions within the TME and their effect on tumor growth, and show that tumor progression is not solely determined by the presence of a clone of mutated immortal cells, but rather that it can be √Ę‚?¨ň?community-controlled√Ę‚?¨‚?Ę.
  • Y. Kim and H. Othmer
    A hybrid model of tumor-stromal interactions in breast cancer
    Bull. Math. Biol. (2012) (Submitted)

    Abstract

  • H. Othmer and C. Xue
    The mathematical analysis of biological aggregation and dispersal: progress, problems and perspectives
    Dispersal, individual movement and spatial ecology: A mathematical perspective (2012) (In Preparation)

    Abstract

  • L. Zheng, H. Othmer and H. Kang
    The effect of the signaling scheme on the robustness of pattern formation in development
    Interface FocusVol. 2 No. 4 (2012) pp. 465-486 (To Appear)

    Abstract

  • H. Kang, L. Zheng and H. Othmer
    A new method for choosing the computational cell in stochastic reaction-diffusion systems
    Journal of Mathematical BiologyVol. 65 No. Series 6-7 (2012) pp. 1017-1099

    Abstract

    How to choose the computational compartment or cell size for the stochastic simulation of a reaction‚€“diffusion system is still an open problem, and a number of criteria have been suggested. A generalized measure of the noise for finite-dimensional systems based on the largest eigenvalue of the covariance matrix of the number of molecules of all species has been suggested as a measure of the overall fluctuations in a multivariate system, and we apply it here to a discretized reaction‚€“diffusion system. We show that for a broad class of first-order reaction networks this measure converges to the square root of the reciprocal of the smallest mean species number in a compartment at the steady state. We show that a suitably re-normalized measure stabilizes as the volume of a cell approaches zero, which leads to a criterion for the maximum volume of the compartments in a computational grid. We then derive a new criterion based on the sensitivity of the entire network, not just of the fastest step, that predicts a grid size that assures that the concentrations of all species converge to a spatially-uniform solution. This criterion applies for all orders of reactions and for reaction rate functions derived from singular perturbation or other reduction methods, and encompasses both diffusing and non-diffusing species. We show that this predicts the maximal allowable volume found in a linear problem, and we illustrate our results with an example motivated by anterior-posterior pattern formation in Drosophila, and with several other examples.

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