Mathematical modeling of vertebrate limb development

Cellular self-organization: An overdrive in Cambrian diversity?

During the early Cambrian period metazoan life forms diverged at an accelerated rate to occupy multiple ecological niches on earth. A variety of explanations have been proposed to address this major evolutionary phenomenon termed the "Cambrian explosion." While most hypotheses address environmental, developmental, and ecological factors that facilitated evolutionary innovations, the biological basis for accelerated emergence of species diversity in the Cambrian period remains largely conjectural. Herein, we posit that morphogenesis by self-organization enables the uncoupling of genomic mutational landscape from phenotypic diversification. Evidence is provided for a two-tiered interpretation of genomic changes in metazoan animals wherein mutations not only impact upon function of individual cells, but also alter the self-organization outcome during developmental morphogenesis. We provide evidence that the morphological impacts of mutations on self-organization could remain repressed if associated with an unmet negative energetic cost. We posit that accelerated morphological diversification in transition to the Cambrian period has occurred by emergence of dormant (i.e., reserved) morphological novelties whose molecular underpinnings were seeded in the Precambrian period.

Combined computational modeling and experimental analysis integrating chemical and mechanical signals suggests possible mechanism of shoot meristem maintenance

Stem cell maintenance in multilayered shoot apical meristems (SAMs) of plants requires strict regulation of cell growth and division. Exactly how the complex milieu of chemical and mechanical signals interact in the central region of the SAM to regulate cell division plane orientation is not well understood. In this paper, simulations using a newly developed multiscale computational model are combined with experimental studies to suggest and test three hypothesized mechanisms for the regulation of cell division plane orientation and the direction of anisotropic cell expansion in the corpus. Simulations predict that in the Apical corpus, WUSCHEL and cytokinin regulate the direction of anisotropic cell expansion, and cells divide according to tensile stress on the cell wall. In the Basal corpus, model simulations suggest dual roles for WUSCHEL and cytokinin in regulating both the direction of anisotropic cell expansion and cell division plane orientation. Simulation results are followed by a detailed analysis of changes in cell characteristics upon manipulation of WUSCHEL and cytokinin in experiments that support model predictions. Moreover, simulations predict that this layer-specific mechanism maintains both the experimentally observed shape and structure of the SAM as well as the distribution of WUSCHEL in the tissue. This provides an additional link between the roles of WUSCHEL, cytokinin, and mechanical stress in regulating SAM growth and proper stem cell maintenance in the SAM.

THREE-DIMENSIONAL COMPUTATIONAL MODEL OF EARLY UPPER LIMB DEVELOPMENT

Limb development begins during embryogenesis when a series of biochemical interactions are triggered between a particular region of the mesoderm and the ectoderm. These processes affect the morphogenesis and growth of bones, joints, and all the other constituent elements of limbs; nevertheless, how the biochemical regulation affects mesenchymal condensation is not entirely clear. In this study, a three-dimensional computational model is designed to predict the appearance and location of the mesenchymal condensation in the stylopod and zeugopod; the biochemical events were described with reaction–diffusion equations that were solved using the finite elements method. The result of the gene expression in our model was consistent with the one reported in literature; the obtained patterns of Fgf8, Fgf10, and Wnt3a can predict the shape of the mesenchymal condensation of early upper limb development; the simple diffusive patterns of molecules were suitable to explain the areas where sox9 is expressed. Furthermore, our results suggest that the expression of Tgf-[Formula: see text] in the upper limb could be due to the inhibition of retinoic acid. These results suggest the importance of building computational scenarios where pathologies may be comprehensively examined.

Multiscale modeling of vertebrate limb development

We review the current state of mathematical modeling of cartilage pattern formation in vertebrate limbs. We place emphasis on several reaction-diffusion type models that have been proposed in the last few years. These models are grounded in more detailed knowledge of the relevant regulatory processes than previous ones but generally refer to different molecular aspects of these processes. Considering these models in light of comparative phylogenomics permits framing of hypotheses on the evolutionary order of appearance of the respective mechanisms and their roles in the fin-to-limb transition. This article is categorized under: Analytical and Computational Methods > Computational Methods Models of Systems Properties and Processes > Mechanistic Models Developmental Biology > Developmental Processes in Health and Disease Analytical and Computational Methods > Analytical Methods.

Mathematical modeling of chondrogenic pattern formation during limb development: Recent advances in continuous models

The phenomenon of chondrogenic pattern formation in the vertebrate limb is one of the best studied examples of organogenesis. Many different models, mathematical as well as conceptual, have been proposed for it in the last fifty years or so. In this review, we give a brief overview of the fundamental biological background, then describe in detail several models which aim to describe qualitatively and quantitatively the corresponding biological phenomena. We concentrate on several new models that have been proposed in recent years, taking into account recent experimental progress. The major mathematical tools in these approaches are ordinary and partial differential equations. Moreover, we discuss models with non-local flux terms used to account for cell-cell adhesion forces and a structured population model with diffusion. We also include a detailed list of gene products and potential morphogens which have been identified to play a role in the process of limb formation and its growth.

From the perspective of embryonic tendon development: various cells applied to tendon tissue engineering

There is a high risk of injury from damage to the force-bearing tissue of the tendon. Due to its poor self-healing ability, clinical interventions for tendon injuries are limited and yield unsatisfying results. Tissue engineering might supply an alternative to this obstacle. As one of the key elements of tissue engineering, various cell sources have been used for tendon engineering, but there is no consensue concerning a single optimal source. In this review, we summarized the development of tendon tissue from the embryonic stage and categorized the used cell sources in tendon engineering. By comparing various cell sources as the candidates for tendon regeneration, each cell type was found to have its advantages and limitations; therefore, it is difficult to define the best cell source for tendon engineering. The microenvironment cells located is also crucial for cell growth and differentiation; so, the optimal cells are unlikely to be the same for each patient. In the future, the clinical application of tendon engineering might be more precise and customized in contrast to the current use of a standardized/generic one-size-fits-all procedure. The best cell source for tendon engineering will require a case-based assessment.

The developmental-genetics of canalization

Canalization, or robustness to genetic or environmental perturbations, is fundamental to complex organisms. While there is strong evidence for canalization as an evolved property that varies among genotypes, the developmental and genetic mechanisms that produce this phenomenon are very poorly understood. For evolutionary biology, understanding how canalization arises is important because, by modulating the phenotypic variation that arises in response to genetic differences, canalization is a determinant of evolvability. For genetics of disease in humans and for economically important traits in agriculture, this subject is important because canalization is a potentially significant cause of missing heritability that confounds genomic prediction of phenotypes. We review the major lines of thought on the developmental-genetic basis for canalization. These fall into two groups. One proposes specific evolved molecular mechanisms while the other deals with robustness or canalization as a more general feature of development. These explanations for canalization are not mutually exclusive and they overlap in several ways. General explanations for canalization are more likely to involve emergent features of development than specific molecular mechanisms. Disentangling these explanations is also complicated by differences in perspectives between genetics and developmental biology. Understanding canalization at a mechanistic level will require conceptual and methodological approaches that integrate quantitative genetics and developmental biology.

The vertebrate limb: An evolving complex of self-organizing systems

The paired appendages (fins or limbs) of jawed vertebrates contain an endoskeleton consisting of nodules, bars and, in some groups, plates of cartilage, or bone arising from replacement of cartilaginous templates. The generation of the endoskeletal elements occurs by processes involving production and diffusion of morphogens, with, variously, positive and negative feedback circuits, adhesion, and receptor dynamics with similarities to the mechanism for chemical pattern formation proposed by Alan Turing. This review presents a unified interpretation of the evolution and functioning of these mechanisms. Studies are described indicating that protocondensations, compacted mesenchymal cell aggregates that prefigure the appendicular skeleton, arise through the adhesive activity of galectin-1, a matricellular protein with skeletogenic homologs in all jawed vertebrates. In the cartilaginous and lobe-finned fishes (and to a variable extent in ray-finned fishes) it additionally cooperates with an isoform of galectin-8 to constitute a self-organizing network capable of generating arrays of preskeletal nodules, bars and plates. Further, in the tetrapods, a putative galectin-8 control module was acquired that may have enabled proximodistal increase in the number of protocondensations. In parallel to this, other self-organizing networks emerged that acted, via Bmp, Wnt, Sox9 and Runx2, as well as transforming factor-β and fibronectin, to convert protocondensations into skeletal tissues. The progressive appearance and integration of these skeletogenic networks over evolution occurred in the context of an independently evolved system of Hox protein and Shh gradients that interfaced with them to tune the spatial wavelengths and refine the identities of the resulting arrays of elements.

Developmental nonlinearity drives phenotypic robustness

Robustness to perturbation is a fundamental feature of complex organisms. Mutations are the raw material for evolution, yet robustness to their effects is required for species survival. The mechanisms that produce robustness are poorly understood. Nonlinearities are a ubiquitous feature of development that may link variation in development to phenotypic robustness. Here, we manipulate the gene dosage of a signaling molecule, Fgf8, a critical regulator of vertebrate development. We demonstrate that variation in Fgf8 expression has a nonlinear relationship to phenotypic variation, predicting levels of robustness among genotypes. Differences in robustness are not due to gene expression variance or dysregulation, but emerge from the nonlinearity of the genotype–phenotype curve. In this instance, embedded features of development explain robustness differences. How such features vary in natural populations and relate to genetic variation are key questions for unraveling the origin and evolvability of this feature of organismal development.

Developmental processes often involve nonlinearities, but the consequences for translating genotype to phenotype are not well characterized. Here, Green et al. vary Fgf8 signaling across allelic series of mice and show that phenotypic robustness in craniofacial shape is explained by a nonlinear effect of Fgf8 expression.

A minimally sufficient model for rib proximal-distal patterning based on genetic analysis and agent-based simulations

For decades, the mechanism of skeletal patterning along a proximal-distal axis has been an area of intense inquiry. Here, we examine the development of the ribs, simple structures that in most terrestrial vertebrates consist of two skeletal elements-a proximal bone and a distal cartilage portion. While the ribs have been shown to arise from the somites, little is known about how the two segments are specified. During our examination of genetically modified mice, we discovered a series of progressively worsening phenotypes that could not be easily explained. Here, we combine genetic analysis of rib development with agent-based simulations to conclude that proximal-distal patterning and outgrowth could occur based on simple rules. In our model, specification occurs during somite stages due to varying Hedgehog protein levels, while later expansion refines the pattern. This framework is broadly applicable for understanding the mechanisms of skeletal patterning along a proximal-distal axis.

Systems morphodynamics: understanding the development of tissue hardware

Systems morphodynamics describes a multi-level analysis of mechanical morphogenesis that draws on new microscopy and computational technologies and embraces a systems biology-informed scope. We present a selection of articles that illustrate and explain this rapidly progressing field.This article is part of the themed issue 'Systems morphodynamics: understanding the development of tissue hardware'.

'Biogeneric' developmental processes: drivers of major transitions in animal evolution

Using three examples drawn from animal systems, I advance the hypothesis that major transitions in multicellular evolution often involved the constitution of new cell-based materials with unprecedented morphogenetic capabilities. I term the materials and formative processes that arise when highly evolved cells are incorporated into mesoscale matter 'biogeneric', to reflect their commonality with, and distinctiveness from, the organizational properties of non-living materials. The first transition arose by the innovation of classical cell-adhesive cadherins with transmembrane linkage to the cytoskeleton and the appearance of the morphogen Wnt, transforming some ancestral unicellular holozoans into 'liquid tissues', and thereby originating the metazoans. The second transition involved the new capabilities, within a basal metazoan population, of producing a mechanically stable basal lamina, and of planar cell polarization. This gave rise to the eumetazoans, initially diploblastic (two-layered) forms, and then with the addition of extracellular matrices promoting epithelial-mesenchymal transformation, three-layered triploblasts. The last example is the fin-to-limb transition. Here, the components of a molecular network that promoted the development of species-idiosyncratic endoskeletal elements in gnathostome ancestors are proposed to have evolved to a dynamical regime in which they constituted a Turing-type reaction-diffusion system capable of organizing the stereotypical arrays of elements of lobe-finned fish and tetrapods. The contrasting implications of the biogeneric materials-based and neo-Darwinian perspectives for understanding major evolutionary transitions are discussed.This article is part of the themed issue 'The major synthetic evolutionary transitions'.

Tissue landscape alters adjacent cell fates during Drosophila egg development

Extracellular signalling molecules control many biological processes, but the influence of tissue architecture on the local concentrations of these factors is unclear. Here we examine this issue in the Drosophila egg chamber, where two anterior cells secrete Unpaired (Upd) to activate Signal transducer and activator of transcription (STAT) signalling in the epithelium. High STAT signalling promotes cell motility. Genetic analysis shows that all cells near the Upd source can respond. However, using upright imaging, we show surprising asymmetries in STAT activation patterns, suggesting that some cells experience different Upd levels than predicted by their location. We develop a three-dimensional mathematical model to characterize the spatio-temporal distribution of the activator. Simulations show that irregular tissue domains can produce asymmetric distributions of Upd, consistent with results in vivo. Mutant analysis substantiates this idea. We conclude that cellular landscape can heavily influence the effect of diffusible activators and should be more widely considered.

Limb, tooth, beak: three modes of development and evolutionary innovation of form

The standard model of evolutionary change of form, deriving from Darwin's theory via the Modern Synthesis, assumes a gradualistic reshaping of anatomical structures, with major changes only occurring by many cycles of natural selection for marginal adaptive advantage. This model, with its assertion that a single mechanism underlies both micro- and macroevolutionary change, contains an implicit notion of development which is only applicable in some cases. Here we compare the embryological processes that shape the vertebrate limb bud, the mammalian tooth and the avian beak. The implied notion of development in the standard evolutionary picture is met only in the case of the vertebrate limb, a single-primordium organ with morphostatic shaping, in which cells rearrange in response to signalling centres which are essentially unchanged by cell movement. In the case of the tooth, a single-primordium organ with morphodynamic shaping in which the strengths and relationships between signalling centres is influenced by the cell and tissue movements they induce, and the beak, in which the final form is influenced by the collision and rearrangement of multiple tissue primordia, abrupt appearance of qualitatively different forms (i.e. morphological novelties) can occur with small changes in system parameters induced by a genetic change, or by an environmental factor whose effects can be subsequently canalized genetically. Bringing developmental mechanisms and, specifically, the material properties of tissues as excitable media into the evolutionary picture, demonstrates that gradualistic change for incremental adaptive advantage is only one of the possible modes of morphological evolution.

Modeling the morphodynamic galectin patterning network of the developing avian limb skeleton

We present a mathematical model for the morphogenesis and patterning of the mesenchymal condensations that serve as primordia of the avian limb skeleton. The model is based on the experimentally established dynamics of a multiscale regulatory network consisting of two glycan-binding proteins expressed early in limb development: CG (chicken galectin)-1A, CG-8 and their counterreceptors that determine the formation, size, number and spacing of the "protocondensations" that give rise to the condensations and subsequently the cartilaginous elements that serve as the templates of the bones. The model, a system of partial differential and integro-differential equations containing a flux term to represent local adhesion gradients, is simulated in a "full" and a "reduced" form to confirm that the system has pattern-forming capabilities and to explore the nature of the patterning instability. The full model recapitulates qualitatively and quantitatively the experimental results of network perturbation and leads to new predictions, which are verified by further experimentation. The reduced model is used to demonstrate that the patterning process is inherently morphodynamic, with cell motility being intrinsic to it. Furthermore, subtle relationships between cell movement and the positive and negative interactions between the morphogens produce regular patterns without the requirement for activators and inhibitors with widely separated diffusion coefficients. The described mechanism thus represents an extension of the category of activator-inhibitor processes capable of generating biological patterns with repetitive elements beyond the morphostatic mechanisms of the Turing/Gierer-Meinhardt type.