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"Mathematics is biology’s next microscope,
only better; biology is mathematics’ next
physics, only better"  Joel E. Cohen
My research area is mathematical biology. I develop new mathematical models and methods to address fundamental questions in cell and developmental biology. In particular, we are interested in understanding how transport and interaction of intracellular molecules lead to robust wholecell functioning, how cells communicate with each other and selforganize into multicellular structures, and how different cells coordinate with each other to maintain health and prevent diseases.
My research is datadriven and deeply rooted in realworld applications. I extract interesting applied math problems and solve them using a wide variety of tools including stochastic processes, stochastic differential equations, ordinary and partial differential equations, asymptotic analysis, machine learning and scientific computing. My research has been supported by NSF DMS 1312966 (20132017), NSF CAREER Award 1553637 (20162021). Descriptions of some projects are listed below.
Axonal transport and intracellular traffic jams
Neurons are highly polarized cells that typically contain several dendrites and a single long axon. The development and maintenance of neuron morphology is crucial for the proper propagation of electric signals in neuronal networks. The axonal cytoskeleton is a dynamic polymer system providing structural support for the axon, and it also acts as highways and local roads for the active transport of proteins and organelles inside the axon.
Focal axonal and/or dendritic swellings occur in traumatic brain injuries and most neurodegenerative diseases including ALS, Alzheimer's, Parkinson's, Multiple Sclerosis and CharcotMarieTooth. These phenomena are associated with disruptions of axonal transport, causing intracellular traffic jams in a similar, but more complicated manner as everyday traffic. A frequently observed early event is aberrant redistribution of the axonal cytoskeleton, such as segregation of different types of cytoskeletal polymers, before axonal swelling becomes observable.
In close collaboration with experimentalists, we have been developing stochastic and PDE models to help understand axonal cytoskeleton segregation, axonal traffic jams and axonal swelling. This project is supported by my NSF CAREER Award and we have made significant progress recently.
Collaborators: Related publications:
Intracellular transport in pollen tube
Plant sperms and vegetative nucleus form a complex and move along the pollen tube towards the egg cells in the ovule. We use machine learning and stochastic models to help understand the underlying molecular mechanism of this crucial process for plant fertilization. Please contact me for more details.
Collaborator:
Cell encapsulation and phasefield modeling
Coming soon...
Chemotaxis of cell populations  a multiscale approach
Chemotaxis is the directed movement of cells or organisms in response to external chemical signals. Modeling the chemotaxis of cell populations is essential for accurate and quantitative descriptions of many medical and environmental processes such as biofilm formation, bioremediation, wound healing, and cancer metastasis. Continuum PDE models of chemotaxis have been developed in the past and the most popular approach is to use the PatlakKellerSegel (PKS) equation. However, these models are phenomenological and their connections with the molecular basis of chemotaxis are not well established. On the other hand, individualbased stochastic models that faithfully replicate mechanisms of cell signaling and movement can be used to simulate processes that involve chemotaxis of cell populations. However, for applications that involve large numbers of cells, this is computationally prohibitive. To develop mechanistic, quantitative, and efficient models of chemotaxis, systematic methods to connect continuum PDE models from individualbased models must be developed.
We have made significant progress in these aspects for bacterial chemotaxis. We have shown that the PKS equation is only accurate for chemotaxis in small signal gradients, and for this situation we derived explicit formulas to map parameters at the singlecell level to parameters of the PKS equation. We found that the observed logarithmic chemotactic sensitivity is rooted in the structure of the intracellular signaling network. Moreover, we found that if the signal gradient becomes large, the PKS model cannot accurately approximate the cell population dynamics, and for this situation we derived new momentflux models which highlight the necessity to include information on the distribution of intracellular states in continuum models.
Related publications:
 C. Xue and X. Yang. Momentflux models for bacterial chemotaxis in large signal gradients, Journal of Mathematical Biology, 2016.
 C. Xue. Macroscopic equations for bacterial chemotaxis: integration of detailed biochemistry of cell signaling, Journal of Mathematical Biology, 2015. PDF
 B. Franz, C. Xue, K. Painter, and R. Erban. Travelling waves in hybrid chemotaxis models, Bulletin of Mathematical Biology, 2014. PDF
 H. G. 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 (edited by M. Lewis, P. Maini and S. Petrovskii), 2013. PDF
 C. Xue and H. G. Othmer, Multiscale models of taxisdriven
patterning in bacterial populations, SIAM J. Appl. Math., Vol. 70, No. 1, pp. 133167, 2009.
PDF
 C. Xue, H. G. 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, AIP, Vol. 1167, pp. 314, 2009. PDF
Biological experiments show that swimming bacteria can selforganize into complex population patterns when grown in Petri plates. These patterns are primitive forms of biofilms, which cause big industrial and medical problems. Thus it is important to uncover the underlying mechanisms that lead to these patterns. We addressed such problems for patterns formed by bacteria P. mirabilis and E. coli.
Related publications:
 C. Xue , E. BudreneKac, and H. G. Othmer, Radial and
spiral streams in Proteus mirabilis colonies, PLOS Computational Biology, 2011. PDF
 C. Xue, H. Hwang, K. Painter, and R. Erban, Traveling waves in hyperbolic chemotaxis equations,
Bul. Math. Biol, 73(8):1695733, 2011. PDF
 B. Franz, C. Xue, K. Painter, and R. Erban. Travelling waves in hybrid chemotaxis models, Bulletin of Mathematical Biology, 2014. PDF
 H. Othmer, X. Xin, and C. Xue. Excitation and adaptation in bacteria  a model signal
transduction system that controls taxis and spatial pattern formation, International Journal of Molecular Sciences, 2013, 14(5), 92059248. PDF
 H. G. 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, pp. 382, 2009. PDF
Chronic wound healing as a moving boundary problem
Wound healing is a complex and fragile process in which injured tissue repairs itself. It involves interactions among a large array of soluble chemical mediators, cell types, and the extracellular matrix (ECM). Interruption of certain subprocesses leads to nonhealing chronic wounds such as diabetic foot ulcers and pressure ulcers. Major complicating factors of wound healing include insufficient oxygen supply, biofilmassociated infections, etc. To obtain a quantitative understanding on how wound heals and how to improve healing, it is important to develop mathematical models that integrate the complex biology and justify these models using experimental data.
The ultimate goal is to use these models as tools to test new hypotheses of treatments and guide clinical practice. To use these models in clinical settings, the following two steps need to be carried out in sequence: 1) Personalize the parameters in these models using individual patient data collected during the first few days post wounding. 2) Take the personalized parameters, solve the model forward with treatments and therapies, and predict how wound heals in the coming days.
Related publications:
 C. Xue, A. Friedman, and C. K. Sen, A mathematical model of ischemic cutaneous wounds, PNAS, Vol. 106, No. 39, pp. 1678216787, 2009.
(with PNAS highlight)
PDF
 A. Friedman, B. Hu, and C. Xue, Analysis of a mathematical model of ischemic cutaneous wounds,
SIAM J. Math. Anal., Vol 42, Issue 5, pp. 20132040, 2010. PDF
 A. Friedman and C. Xue, A mathematical model of chronic wounds,
Mathematical Biosciences and Engineering, 8(2):25361, 2011. PDF
 C. Xue, C. S. Chou, C. Y. Kao, C. Sen, and A. Friedman, Propagation of Cutaneous Thermal Injury:
A Mathematical Model, Wound Repair and Regen., 20(1):11422, 2012. PDF

PNAS highlights: Math takes stab at stubborn wounds
 SIAM nuggets: A Mathematical Model for Ischemic Wound Healing
 MAA news: Math Model May Speed Healing of Chronic Wounds
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