Perspectives on the mathematics of biological patterning and morphogenesis

doi:10.1016/j.jmps.2016.11.013

Journal of the Mechanics and Physics of Solids

Volume 99, February 2017, Pages 192-210

https://doi.org/10.1016/j.jmps.2016.11.013 Get rights and content

Highlights

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I have identified the dual roles of patterning and morphogenesis, and their coupling in determining the form of organisms.
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The manuscript draws a rigorous connection between organization of multicellular tissues and phase segregation phenomena.
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It goes on to highlight several aspects of phase field models and their bearing on diverse features of biological patterning.
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Throughout the manuscript the central questions of developmental biology are placed within context of the considered models: How are size and position controlled by the physics that the mathematical models describe?

Abstract

A central question in developmental biology is how size and position are determined. The genetic code carries instructions on how to control these properties in order to regulate the pattern and morphology of structures in the developing organism. Transcription and protein translation mechanisms implement these instructions. However, this cannot happen without some manner of sampling of epigenetic information on the current patterns and morphological forms of structures in the organism. Any rigorous description of space- and time-varying patterns and morphological forms reduces to one among various classes of spatio-temporal partial differential equations. Reaction-transport equations represent one such class. Starting from simple Fickian diffusion, the incorporation of reaction, phase segregation and advection terms can represent many of the patterns seen in the animal and plant kingdoms. Morphological form, requiring the development of three-dimensional structure, also can be represented by these equations of mass transport, albeit to a limited degree. The recognition that physical forces play controlling roles in shaping tissues leads to the conclusion that (nonlinear) elasticity governs the development of morphological form. In this setting, inhomogeneous growth drives the elasticity problem. The combination of reaction-transport equations with those of elasto-growth makes accessible a potentially unlimited spectrum of patterning and morphogenetic phenomena in developmental biology. This perspective communication is a survey of the partial differential equations of mathematical physics that have been proposed to govern patterning and morphogenesis in developmental biology. Several numerical examples are included to illustrate these equations and the corresponding physics, with the intention of providing physical insight wherever possible.

Section snippets

Introduction and background

Developmental biology is concerned with the development of patterns and morphological form (morphogenesis) in organisms. It is useful to make these terms precise at the outset in order to enable a mathematical physics-centered discussion: The term pattern will be applied here to a scalar field in one-, two- or three-dimensional manifolds. Morphological form will be taken to refer to the vector placement of material points of the developing organism, in the spirit of D'Arcy Thompson. (Thompson,

Reaction-diffusion equations and patterning; Turing instabilities

Before reviewing the role of reaction-diffusion equations in biological pattern generation, it helps to consider the simpler case of Fickian diffusion. For a scalar field whose concentration is c, the canonical diffusion problem can be posed over a domain $Ω \in R^{3}$ that represents an organ or developing system with a combination of concentration boundary conditions, and zero flux (vanishing concentration gradient for homogeneous, isotropic diffusion) boundary conditions being appropriate: $\frac{\partial c}{\partial t} = D \nabla^{2} c in Ω$

Patterning by cell segregation in tissues

While reaction-diffusion models do give rise to patterning, stable or steady state patterns only arise for special ranges of coefficients predicted by (4) in the linear case, and examples such as Schnakenberg kinetics in the nonlinear case. The thesis that Turing-like biological patterns can be generated by reaction-diffusion equations relies on the identification of morphogens subject to Turing patterns that then trigger cell differentiation. Morphogen distributions controlled by Fickian

Morphogenesis by elastic buckling, and the post-bifurcated shape

Many morphogenetic phenomena in three dimensions can be explained by inhomogeneous growth. For the case of sulcification and gyrification of the brain, a competing theory held that axonal tension played a role (Essen, 1997). However, that hypothesis has largely been ruled out in favor of the idea that inhomogeneous growth causes an initial buckling of elastic layers in the brain, followed by folding, wrinkling or creasing, after the bifurcation (Xu et al., 2010, Bayly et al., 2013, Tallinen et

Summary and outlook

The primacy of partial differential equations for the representation of position and size, and therefore to control patterning and morphogenesis has been amply laid out in the literature, and summarized in the Introduction. While the dominant patterning models have been based on reaction-diffusion equations, it is worth noting that for tissue patterning at least, cell segregation and the resulting system of fourth-order transport equations present a compelling alternative rooted in a derivation

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Perspectives on the mathematics of biological patterning and morphogenesis

Highlights

Abstract

Section snippets

Introduction and background

Reaction-diffusion equations and patterning; Turing instabilities

Patterning by cell segregation in tissues

Morphogenesis by elastic buckling, and the post-bifurcated shape

Summary and outlook

J. Mech. Phys. Solids

Bull. Math. Biol.

Appl. Math. Lett.

Cell

J. Mech. Phys. Solids

J. Mech. Phys. Solids

Math. Biosci.

J. Theor. Biol.

Phys. Rep.

J. Biomech.

Cell

Model for the regulation of size in the wing imaginal disc of Drosophila

Mech. Dev.

Digit patterning during limb development as a result of the BMP-receptor interaction

Sci. Rep.

The deal.II library, version 8.4

J. Num. Math.

deal.II - a general purpose object oriented finite element library

ACM Trans. Math. Softw.

A cortical folding model incorporating stress-dependent growth explains gyral wavelengths and stress patterns in the developing brain

Phys. Biol.

Swelling instability of surface-attached gels as a model of soft tissue growth under geometric constraints

J. Mech. Phys. Solids

Anisotropic growth shapes intestinal tissues during embryogenesis

Proc. Natl. Acad. Sci.

Petal shapes of sympetalous flowersthe interplay between growth, geometry and elasticity

New J. Phys.

Scaling of morphogen gradients

Curr. Opin. Gen. Dev.

The roleofmechanicsduringbraindevelopment

J. Mech. Phys. Solids

Modelling solid tumour growth using the theory of mixtures

Math. Med. Biol.

Free energy of a nonuniform system. i interfacial energy

J. Chem. Phys.

Osmotic collapse of a void in an elastomerbreathing, buckling and creasing

Soft Matter

Emergence of microstructural patterns in skin cancera phase separation analysis in a binary mixture

New J. Phys.

Mechanical basis of morphogenesis and convergent evolution of spiny seashells

Proc. Natl. Acad. Sci.

On global minimizers for a variational problem with long-range interactions

Quart. Appl. Math. LXX

Reaction and diffusion on growing domainsscenarios for robust pattern formation

Bull. Math. Biol.

Nonlinear simulations of solid tumor growth using a mixture modelinvasion and branching

J. Math. Biol.

Multiscale Modeling of Cancer

Geometric evolution of bilayers under the functionalized cahn-hilliard equation

Proc. R. Soc. Ser. A

Measuring dimensionsthe regulation of size and shape

Development

Morphogenesis of growing soft tissues

Phys. Rev. Lett.

Pattern formation in generalized turing systems isteady-state patterns in systems with mixed boundary conditions

J. Math. Biol.

A tension-based theory of morphogenesis and compact wiring in the central nervous system

Nature

Coordination of signaling and tissue mechanics during morphogenesis of murine intestinal villi: a role for mitotic cell rounding

Integ. Biol.

A theory of biological pattern formation

Kybernetik

Evolution of patterns on Conus shells

Proc. Natl. Acad. Sci.

Mechanics of the brainperspectives, challenges, and opportunities

Biomech. Model. Mechanobiol.

Simulation of biological cell sorting using a two dimensional extended Potts model

Phys. Rev. Lett.

Instabilities of monolayered epitheliashape and structure of villi and crypts

Phys. Rev. Lett.