Perspectives on the mathematics of biological patterning and morphogenesis

Garikipati, Krishna

doi:10.1016/j.jmps.2016.11.013

Cited by 26 publications

(28 citation statements)

References 85 publications

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“…The finite element formulation has been implemented in C++ by extending a code for general problems of patterning and morphology [29]. This framework uses the deal.II open source finite element library [30,31].…”

Section: Softwarementioning

confidence: 99%

A computational study of the mechanisms of growth-driven folding patterns on shells, with application to the developing brain

Verner

Garikipati

2018

Extreme Mechanics Letters

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We consider the mechanisms by which folds, or sulci (troughs) and gyri (crests), develop in the brain. This feature, common to many gyrencephalic species including humans, has attracted recent attention from soft matter physicists. It occurs due to inhomogeneous, and predominantly tangential, growth of the cortex, which causes circumferential compression, leading to a bifurcation of the solution path to a folded configuration. The problem can be framed as one of buckling in the regime of linearized elasticity. However, the brain is a very soft solid, which is subject to large strains due to inhomogeneous growth. As a consequence, the morphomechanics of the developing brain demonstrates an extensive post-bifurcation regime. Nonlinear elasticity studies of growth-driven brain folding have established the conditions necessary for the onset of folding, and for its progression to configurations broadly resembling gyrencephalic brains. The reference, unfolded, configurations in these treatments have a high degree of symmetry-typically, ellipsoidal. Depending on the boundary conditions, the folded configurations have symmetric or anti-symmetric patterns. However, these configurations do not approximate the actual morphology of, e.g., human brains, which display unsymmetric folding. More importantly, from a neurodevelopmental standpoint, many of the unsymmetric sulci and gyri are notably robust in their locations. Here, we initiate studies on the physical mechanisms and geometry that control the development of primary sulci and gyri. In this preliminary communication we carry out computations with idealized geometries,

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Section: Softwarementioning

confidence: 99%

A computational study of the mechanisms of growth-driven folding patterns on shells, with application to the developing brain

Verner

Garikipati

2018

Extreme Mechanics Letters

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“…Closed form solutions for PDEs exist only in the most simplest of cases and numerical solutions need to be employed. Packaged solvers for PDEs do exist [31] and some like deal.II [32] have been used in systems biology applications [33–35]. However, due to the overhead of generalizability and computational tractability in structuring models, we wrote our own solver.…”

Section: Methodsmentioning

confidence: 99%

Bayesian model selection for the Drosophila gap gene network

Zubair¹,

Rosen

Nuzhdin³

et al. 2019

BMC Bioinformatics

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Background The gap gene system controls the early cascade of the segmentation pathway in Drosophila melanogaster as well as other insects. Owing to its tractability and key role in embryo patterning, this system has been the focus for both computational modelers and experimentalists. The gap gene expression dynamics can be considered strictly as a one-dimensional process and modeled as a system of reaction-diffusion equations. While substantial progress has been made in modeling this phenomenon, there still remains a deficit of approaches to evaluate competing hypotheses. Most of the model development has happened in isolation and there has been little attempt to compare candidate models. Results The Bayesian framework offers a means of doing formal model evaluation. Here, we demonstrate how this framework can be used to compare different models of gene expression. We focus on the Papatsenko-Levine formalism, which exploits a fractional occupancy based approach to incorporate activation of the gap genes by the maternal genes and cross-regulation by the gap genes themselves. The Bayesian approach provides insight about relationship between system parameters. In the regulatory pathway of segmentation, the parameters for number of binding sites and binding affinity have a negative correlation. The model selection analysis supports a stronger binding affinity for Bicoid compared to other regulatory edges, as shown by a larger posterior mean. The procedure doesn’t show support for activation of Kruppel by Bicoid. Conclusions We provide an efficient solver for the general representation of the Papatsenko-Levine model. We also demonstrate the utility of Bayes factor for evaluating candidate models for spatial pattering models. In addition, by using the parallel tempering sampler, the convergence of Markov chains can be remarkably improved and robust estimates of Bayes factors obtained. Electronic supplementary material The online version of this article (10.1186/s12859-019-2888-0) contains supplementary material, which is available to authorized users.

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“…Note that the target vector in (27) and matrix Ξ in (28) are currently structured to contain all DOFs. However we can always exclude some DOFs by deleting their corresponding rows.…”

Section: Identification Of Basis Operators Via Stepwise Regressionmentioning

confidence: 99%

“…An extreme manifestation of this still-developing field is seen in "model-free" approaches that do not rely on 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 [20,21,22,23,24,25,26,27,28]. The Cahn-Hilliard phase field equation [29] has been applied to model other biological processes with evolving fronts, such as tumor growth and angiogenesis [30,31,32,33,34,35,36,37].…”

Section: Introductionmentioning

confidence: 99%

Variational system identification of the partial differential equations governing the physics of pattern-formation: Inference under varying fidelity and noise

Wang

Huan

Garikipati

2019

Computer Methods in Applied Mechanics and Engineering

102

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We present a contribution to the field of system identification of partial differential equations (PDEs), with emphasis on discerning between competing mathematical models of patternforming physics. The motivation comes from developmental biology, where pattern formation is central to the development of any multicellular organism, and from materials physics, where phase transitions similarly lead to microstructure. In both these fields there is a collection of nonlinear, parabolic PDEs that, over suitable parameter intervals and regimes of physics, can resolve the patterns or microstructures with comparable fidelity. This observation frames the question of which PDE best describes the data at hand. This question is particularly compelling because identification of the closest representation to the true PDE, while constrained by the functional spaces considered relative to the data at hand, immediately delivers insights to the physics underlying the systems. While building on recent work that uses stepwise regression, we present advances that leverage the variational framework and statistical tests. We also address the influences of variable fidelity and noise in the data. physics-based knowledge. Such a path to modelling holds the promise of very high computational efficiency by circumventing solutions to potentially complex physics. However, it draws criticism for its "black box" nature, and often for lack of interpretability. More seriously, these approaches offer scant openings for analysis when the model fails. Conversely, the availability of abundant data also presents opportunities to discover mathematical frameworks, with well-understood physical meaning, which govern the underlying behavior. This has fueled the field of system identification, which is particularly compelling for determining the governing partial differential equations (PDEs). This is so, because knowledge of the PDE directly translates to deep insights to the physics, guided by differential and integral calculus.Approaches for data-assisted and data-driven discovery of PDEs have proceeded along several fronts. (a) Early work in parameter identification can be traced to nonlinear regression approaches [1,2]. (b) If the governing equation is unknown, an approximate model-whether physical, empirical, or mixed-could be learned from data. For example, the approximate model could be a neural network [3], a reduced-order model [4,5], a phenomenological effective model formed by a linear combinations of basis functions in finite dimensions [6,7], or a linear mapping from a current to a future state where the dynamic characteristics of the mapping could be learned by Dynamic Mode Decomposition [8,9]. (c) Lastly, the complete underlying governing equations could be extracted from data by combining symbolic regression and genetic programming to infer algebraic expressions along with their coefficients [10,11]. In this context, while genetic programming balances model accuracy and complexity, it proves very expensive when searching for a few relevant...

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Perspectives on the mathematics of biological patterning and morphogenesis

Cited by 26 publications

References 85 publications

A computational study of the mechanisms of growth-driven folding patterns on shells, with application to the developing brain

A computational study of the mechanisms of growth-driven folding patterns on shells, with application to the developing brain

Bayesian model selection for the Drosophila gap gene network

Variational system identification of the partial differential equations governing the physics of pattern-formation: Inference under varying fidelity and noise

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