Department of Applied and Computational Mathematics and Statistics, University of Notre Dame, Notre Dame, IN 46556, USA.
Math Biosci. 2013 May;243(1):1-17. doi: 10.1016/j.mbs.2012.11.003. Epub 2012 Dec 3.
In this paper, we review the major mathematical and computational models of vertebrate limb development and their roles in accounting for different aspects of this process. The main aspects of limb development that have been modeled include outgrowth and shaping of the limb bud, establishment of molecular gradients within the bud, and formation of the skeleton. These processes occur interdependently during development, although (as described in this review), there are various interpretations of the biological relationships among them. A wide range of mathematical and computational methods have been used to study these processes, including ordinary and partial differential equation systems, cellular automata and discrete, stochastic models, finite difference methods, finite element methods, the immersed boundary method, and various combinations of the above. Multiscale mathematical modeling and associated computational simulation have become integrated into the study of limb morphogenesis and pattern formation to an extent with few parallels in the field of developmental biology. These methods have contributed to the design and analysis of experiments employing microsurgical and genetic manipulations, evaluation of hypotheses for limb bud outgrowth, interpretation of the effects of natural mutations, and the formulation of scenarios for the origination and evolution of the limb skeleton.
在本文中，我们回顾了脊椎动物肢体发育的主要数学和计算模型，以及它们在解释该过程不同方面的作用。已建模的肢体发育的主要方面包括肢体芽的生长和塑造、芽内分子梯度的建立以及骨骼的形成。这些过程在发育过程中相互依存，尽管（如本综述所述），它们之间的生物学关系有各种解释。广泛使用了各种数学和计算方法来研究这些过程，包括常微分和偏微分方程组、元胞自动机和离散、随机模型、有限差分方法、有限元方法、浸入边界方法以及上述方法的各种组合。多尺度数学建模和相关的计算模拟已经成为肢体形态发生和模式形成研究的一部分，在发育生物学领域很少有类似的情况。这些方法有助于设计和分析采用显微手术和遗传操作的实验，评估肢体芽生长的假设，解释自然突变的影响，以及制定肢体骨骼起源和进化的方案。
