Mathematical Modeling of Morphogenesis in Living Materials
From a mathematical viewpoint, the study of morphogenesis focuses on the description of all geometric and structural changes which locally orchestrate the underlying biological processes directing the formation of a macroscopic shape in living matter. In
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Mathematical Modeling of Morphogenesis in Living Materials Valentina Balbi and Pasquale Ciarletta
Abstract From a mathematical viewpoint, the study of morphogenesis focuses on the description of all geometric and structural changes which locally orchestrate the underlying biological processes directing the formation of a macroscopic shape in living matter. In this chapter, we introduce a continuous chemo-mechanical approach of morphogenesis, deriving the balance principles and evolution laws for both volumetric and interfacial processes. The proposed theory is applied to the study of pattern formation for either a fluid-like or a solid-like biological system model, using both theoretical methods and simulation tools.
4.1 Introduction In developmental biology, morphogenesis can be defined as the ensemble of the underlying biological processes directing the formation of a macroscopic shape in living matter. Morphogenetic events span over a wide range of molecular and cellular mechanisms, often inter-playing for controlling events at individual or collective level. Thus, the shape of an organism emerges over time as a morphological transition coordinated by these complex multi-scale interactions, revealing the intrinsic dynamic nature of its driving forces. Indeed, from a mathematical viewpoint, the study of morphogenesis focuses on the description of all geometric and structural changes which locally orchestrate the occurrence of a global shape in living materials. Accordingly, a mathematical model must consider the two most fundamental processes during tissue development, namely growth and remodeling, intended as the biological mechanisms determining any variation of the mass and of
V. Balbi School of Mathematics, University of Manchester, Manchester M13 9PL, UK e-mail: [email protected] P. Ciarletta () Institut Jean le Rond d’Alembert, CNRS and Sorbonne Universités, Université Paris 6, UMR 7190, 4 place Jussieu case 162, 75005 Paris, France Mox, Dipartimento di Matematica, Politecnico di Milano, piazza Leonardo da Vinci 33, 20133 Milano, Italy e-mail: [email protected]; [email protected] © Springer International Publishing Switzerland 2016 L. Preziosi et al. (eds.), Mathematical Models and Methods for Living Systems, Lecture Notes in Mathematics 2167, DOI 10.1007/978-3-319-42679-2_4
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the micro-structural organization, respectively [110]. Nonetheless, morphogenesis is by far a more complex phenomenon, since these basic mechanisms often occur simultaneously, involving a cascade of inter-related developmental events from the molecular to the tissue scale, provoked by either genetic or epigenetic causes [31]. It is therefore mandatory for a mathematical theory of morphogenesis to provide a dynamic modelling of the morphology as a function of the interplay between geometry and physical forces, as well as chemical and biomolecular cues. Moreover, the suitability of a mathematical model should be evaluated not only for the accuracy of reproducing
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