Contemporary Mathematics Developmental Biology and Mathematics: the rules of an embryo Berta Alsina, Adri´ an L. Garc´ ıa de Lomana, Jordi Vill`a-Freixa, and Fernando Gir´aldez Introduction The use of mathematical concepts and theories to understand our world has a long journey. Although the interrelations between Mathematics and Physics have been the most prevalent, some biological phenomena such as genetic evolution and epidemiology embraced mathematical modelling long ago, using it to get a better understanding of such typically non-linear underlying mechanisms. Developmental Biology deals with how specialized cells emerge in a spatial and temporal pattern during embryogenesis to properly organize an adult organism. Is the development of an embryo just genes and proteins with no inherent formulations? The intersection of Mathematics and Developmental Biology was pioneered in the 1950s but it is just recently that this new area is developing rapidly. In this chapter we will review some concepts and biological processes of Developmental Biology to then pinpoint the emerging areas of Applied Mathematics and computer simulations in Developmental Biology. Developmental Biology has to do with the instructions that are required to build up an organism. Those are coded in the genome of each animal species. The unfolding of this genetic information is done from the fertilized egg to the embryo, in a sequence of gene expression patterns and cellular interactions that is invariant for any given animal species, and that generates the different stages of the embryo along its development. The state of this area of biology is the result of more than one hundred years of observation and experimental work. During the last thirty years it has plunged into the molecular description. But it is still at the level of the description of the elements and their gross interactions where Developmental Biology resides. Most of the literature on Developmental Biology is devoted to identify genes that are relevant to development, and to characterise their basic effects on cells and on other developmental genes. More recently, genome-wide techniques permitted this work to be done on more massive basis, but the approach is, still, to identify the actors and their basic interactions. The understanding of The work was supported by grants BFU 2005-03045 Ministerio de Educaci´ on y Ciencia, CTQ2008-0075/BQU Ministerio de Ciencia e Innovaci´ on and the EC funded QosCosGrid project (IST-033883)and VPH NOE (ICT-223920). c XXXX American Mathematical Society 1 Volume 492, 2009 c
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