Annals of Mathematical Sciences and Applications

Volume 1 (2016)

Number 2

Mathematical models of morphogen dynamics and growth control

Pages: 427 – 471

DOI: http://dx.doi.org/10.4310/AMSA.2016.v1.n2.a6

Authors

Jinzhi Lei (Zhou Pei-Yuan Center for Applied Mathematics, MOE Key Laboratory of Bioinformatics, Tsinghua University, Beijing, China)

Wing-Cheong Lo (Department of Mathematics, City University of Hong Kong, Kowloon, Hong Kong)

Qing Nie (Dept. of Mathematics, Dept. of Biomedical Engineering, Center for Complex Biological Systems, and Center for Mathematical and Computational Biology, University of California, Irvine, Calif., U.S.A.)

Abstract

Morphogens are diffusive molecules produced by cells, sending signals to neighboring cells in tissues for communication. As a result, tissues develop cellular patterns that depend on the concentration levels of the morphogens. The formation of morphogen gradients is among the most fundamental biological processes during development, regeneration, and disease. During the past two decades, sophisticated mathematical models have been utilized to decipher the complex biological mechanisms that regulate the spatial and temporal dynamics of morphogens. Here, we review the model formulations for morphogen systems and present the mathematical questions and challenges that arise from the model analysis, with an emphasis on Drosophila. We discuss several important aspects of modeling frameworks: robustness, stochastic dynamics, growth control, and mechanics of morphogen-mediated patterning.

Keywords

pattern formation, morphogen gradients, robustness, boundary value problem, reaction-diffusion equations

2010 Mathematics Subject Classification

Primary 35K57, 92C15. Secondary 34B08, 92B05.

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