MBI Publications

MBI Publications for Chuan Xue (20)

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

    Abstract

  • R. Erban, H. hwang, K. Painter and C. Xue
    Traveling waves in bacteria chemotaxis
    (In Preparation)

    Abstract

  • S. Dai, D. Li, C. Xue and K. Zhao
    On the mechanical origin of scratch wound healing: A particle based model
    (In Preparation)

    Abstract

  • S. Dai, A. Brown, P. Jung and C. Xue
    Model for aggregation of interacting neurofilaments in a slow axonal transport
    (In Preparation)

    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.
  • 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
    Mathematical Modelling of Natural PhenomenaVol. 4 No. 4 (2009) pp. 3-82

    Abstract

  • C. Xue, A. Friedman and C. Sen
    A mathematical model of ischemic cutaneous wounds
    PNASVol. 106 No. 39 (2009) pp. 16782-16787

    Abstract

    Chronic wounds represent a major public health problem affecting 6.5 million people in the United States. Ischemia, primarily caused by peripheral artery diseases, represents a major complicating factor in cutaneous wound healing. In this work, we sought to develop a mathematical model of ischemic dermal wounds. The model consists of a coupled system of partial differential equations in the partially healed region, with the wound boundary as a free boundary. The extracellular matrix (ECM) is assumed to be viscoelastic, and the free boundary moves with the velocity of the ECM at the boundary. The model equations involve the concentration of oxygen, PDGF and VEGF, the densities of macrophages, fibroblasts, capillary tips and sprouts, and the density and velocity of the ECM. Simulations of the model demonstrate how ischemic conditions may limit macrophage recruitment to the wound-site and impair wound closure. The results are in general agreement with experimental findings.
  • 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

  • A. Friedman, B. Hu and C. Xue
    Analysis of a mathematical model of ischemic cutaneous wounds
    SIAM J. Math. Anal.Vol. 42 No. 5 (2010) pp. 2013-2040

    Abstract

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

    Abstract

  • C. Xue, H. hwang, K. Painter and R. Erban
    Traveling waves in bacterial chemotaxis
    Bul. Math. Biol (2011)

    Abstract

    Mathematical models of bacterial populations are often written as systems of partial differential equations for the densities of bacteria and concentrations of extracellular (signal) chemicals. This approach has been employed since the seminal work of Keller and Segel in the 1970s (Keller and Segel, J. Theor. Biol. 30:235‚€“248, 1971). The system has been shown to permit travelling wave solutions which correspond to travelling band formation in bacterial colonies, yet only under specific criteria, such as a singularity in the chemotactic sensitivity function as the signal approaches zero. Such a singularity generates infinite macroscopic velocities which are biologically unrealistic. In this paper, we formulate a model that takes into consideration relevant details of the intracellular processes while avoiding the singularity in the chemotactic sensitivity. We prove the global existence of solutions and then show the existence of travelling wave solutions both numerically and analytically.
  • A. Friedman and C. Xue
    A mathematical model of chronic wounds
    Mathematical Biosciences and EngineeringVol. 8 No. 2 (2011) pp. 253-261

    Abstract

  • A. Friedman, B. Hu and C. Xue
    A three dimensional model of chronic wound healing: analysis and computation
    DCDS-B (2012) (In Preparation)

    Abstract

  • C. Xue, C. Chou, C. Kao, C. Sen and A. Friedman
    Propagation of Cutaneous Thermal Injury: A Mathematical Model
    Wound Repair and RegenVol. 20 No. 1 (2012) pp. 114-122

    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

  • C. Xue, H. hwang, K. Painter and R. Erban
    Traveling waves in hyperbolic chemotaxis equations
    Bul. Math. BiolVol. 73 No. 8 (2012) pp. 1695-1733

    Abstract

  • C. Xue, C. Chou, C. Kao, C. Sen and A. Friedman
    Propagation of Cutaneous Thermal Injury: A Mathematical Model
    Wound Repair and RegenVol. 20 No. 1 (2012) pp. 114-122

    Abstract

  • C. Xue, A. Friedman and B. Hu
    A three dimensional model of chronic wound healing: analysis and computation
    DCDS-B (2012) (Submitted)

    Abstract

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