Perspectives on the mathematics of biological patterning and morphogenesis
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 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:
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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2024, Journal of the Mechanics and Physics of SolidsBio-chemo-mechanical coupling models of soft biological materials: A review
2022, Advances in Applied MechanicsCitation Excerpt :In this framework, the deformation gradient tensor F can be multiplicatively decomposed into the growth portion Fg and the elastic deformation portion Fe (Rodriguez, Hoger, & McCulloch, 1994), as shown in Fig. 2. This method can well describe large deformations in soft tissues, estimate the stress field during the organism development, and decipher the driving forces of growth at multiple scales (Garikipati, 2017; Göktepe, Abilez, & Kuhl, 2010; Göktepe, Abilez, Parker, & Kuhl, 2010; Kuhl et al., 2007). In the case of small deformation, the additive decomposition of the strain tensor, which is calculated from F, is also a feasible method to simulate the deformation and stress fields in a growing tissue (Liang & Mahadevan, 2009).
Bio-chemo-mechanical theory of active shells
2021, Journal of the Mechanics and Physics of SolidsCitation Excerpt :Biological morphogenesis is a fundamental bio-chemo-mechanical process that relies on the genetic template and subsequent interplay of biochemical signals and mechanical cues (Gross et al., 2017). Many biochemical patterns, such as fish and butterfly markings (Garikipati, 2017), calcium waves (Carroll et al., 2003), and axon action potential waves (Deneke and Di Talia, 2018), have been widely observed and studied in the framework of reaction–diffusion system. The reaction–diffusion theory, which dates back to Turing’s seminal paper in 1952, is a theoretical hypothesis for biological morphogenesis (Turing, 1952).
Variational system identification of the partial differential equations governing microstructure evolution in materials: Inference over sparse and spatially unrelated data
2021, Computer Methods in Applied Mechanics and EngineeringCitation Excerpt :Pattern formation during phase transformations in materials physics can happen as the result of instability-induced bifurcations from a uniform composition [1–3], which was the original setting of the Cahn–Hilliard treatment [4]. Following Alan Turing’s seminal work on reaction–diffusion systems [5], a robust literature has developed on the application of nonlinear versions of this class of partial differential equations (PDEs) to model pattern formation in developmental biology [6–14]. The Cahn–Hilliard phase field equation has also been applied to model other biological processes with evolving fronts, such as tumor growth and angiogenesis [15–22].
A perspective on regression and Bayesian approaches for system identification of pattern formation dynamics
2020, Theoretical and Applied Mechanics LettersVariational system identification of the partial differential equations governing the physics of pattern-formation: Inference under varying fidelity and noise
2019, Computer Methods in Applied Mechanics and EngineeringCitation Excerpt :For context, we briefly discuss the role of pattern forming systems of equations in developmental biology and materials physics. Following Alan Turing’s seminal work on reaction–diffusion systems [19], a robust literature has developed on the application of nonlinear versions of this class of PDEs to model pattern formation in developmental biology [20–28]. The Cahn–Hilliard phase field equation [29] has been applied to model other biological processes with evolving fronts, such as tumor growth and angiogenesis [30–37].