A novel method for analysis of the periodicity of chondrogenic patterns in limb bud cell culture: correlation of in vitro pattern formation with theoretical models

Contractility-Induced Phase Separation in Active Solids

Experiments over many decades are suggestive that the combination of cellular contractility and active phase separation in cell-matrix composites can enable spatiotemporal patterning in multicellular tissues across scales. To characterize these phenomena, we provide a general theory that incorporates active cellular contractility into the classical Cahn-Hilliard-Larché model for phase separation in passive viscoelastic solids. Within this framework, we show how a homogeneous cell-matrix mixture can be destabilized by activity via either a pitchfork or Hopf bifurcation, resulting in stable phase separation and/or traveling waves. Numerical simulations of the full equations allow us to track the evolution of the resulting self-organized patterns in periodic and mechanically constrained domains, and in different geometries. Altogether, our study underscores the importance of integrating both cellular activity and mechanical phase separation in understanding patterning in soft, active biosolids in both in vivo and in vitro settings.

Geometric analysis of chondrogenic self-organisation of embryonic limb bud cells in micromass culture

Spatial and temporal control of chondrogenesis generates precise, species-specific patterns of skeletal structures in the developing vertebrate limb. The pattern-template is laid down when mesenchymal cells at the core of the early limb bud condense and undergo chondrogenic differentiation. Although the mechanisms involved in organising such complex patterns are not fully understood, the interplay between BMP and Wnt signalling pathways is fundamental. Primary embryonic limb bud cells grown under high-density micromass culture conditions spontaneously create a simple cartilage nodule pattern, presenting a model to investigate pattern generation. We describe a novel analytical approach to quantify geometric properties and spatial relationships between chondrogenic condensations, utilizing the micromass model. We follow the emergence of pattern in live cultures with nodules forming at regular distances, growing and changing shape over time. Gene expression profiling supports rapid chondrogenesis and transition to hypertrophy, mimicking the process of endochondral ossification within the limb bud. Manipulating the signalling environment through addition of BMP or Wnt ligands, as well as the BMP pathway antagonist Noggin, altered the differentiation profile and nodule pattern. BMP2 addition increased chondrogenesis while WNT3A or Noggin had the opposite effect, but with distinct pattern outcomes. Titrating these pro- and anti-chondrogenic factors and examining the resulting patterns support the hypothesis that regularly spaced cartilage nodules formed by primary limb bud cells in micromass culture are influenced by the balance of Wnt and BMP signalling under a Turing-like mechanism. This study demonstrates an approach for investigating the mechanisms governing chondrogenic spatial organization using simple micromass culture.

Capturing and analyzing pattern diversity: an example using the melanistic spotted patterns of leopard geckos

Animal color patterns are widely studied in ecology, evolution, and through mathematical modeling. Patterns may vary among distinct body parts such as the head, trunk or tail. As large amounts of photographic data is becoming more easily available, there is a growing need for general quantitative methods for capturing and analyzing the full complexity and details of pattern variation. Detailed information on variation in color pattern elements is necessary to understand how patterns are produced and established during development, and which evolutionary forces may constrain such a variation. Here, we develop an approach to capture and analyze variation in melanistic color pattern elements in leopard geckos. We use this data to study the variation among different body parts of leopard geckos and to draw inferences about their development. We compare patterns using 14 different indices such as the ratio of melanistic versus total area, the ellipticity of spots, and the size of spots and use these to define a composite distance between two patterns. Pattern presence/absence among the different body parts indicates a clear pathway of pattern establishment from the head to the back legs. Together with weak within-individual correlation between leg patterns and main body patterns, this suggests that pattern establishment in the head and tail may be independent from the rest of the body. We found that patterns vary greatest in size and density of the spots among body parts and individuals, but little in their average shapes. We also found a correlation between the melanistic patterns of the two front legs, as well as the two back legs, and also between the head, tail and trunk, especially for the density and size of the spots, but not their shape or inter-spot distance. Our data collection and analysis approach can be applied to other organisms to study variation in color patterns between body parts and to address questions on pattern formation and establishment in animals.

Statistical approach for parameter identification by Turing patterns

Prevailing theories in biological pattern formation, such as in morphogenesis or multicellular structures development, have been based on purely chemical processes, with the Turing models as the prime example. Recent studies have challenged the approach, by underlining the role of mechanical forces. A quantitative discrimination of competing theories is difficult, however, due to the elusive character of the processes: different mechanisms may result in similar patterns, while patterns obtained with a fixed model and fixed parameter values, but with small random perturbations of initial values, will significantly differ in shape, while being of the "same" type. In this sense each model parameter value corresponds to a family of patterns, rather than a fixed solution. For this situation we create a likelihood that allows a statistically sound way to distinguish the model parameters that correspond to given patterns. The method allows us to identify model parameters of reaction-diffusion systems by using Turing patterns only, i.e., the steady-state solutions of the respective equations without the use of transient data or initial values. The method is tested with three classical models of pattern formation: the FitzHugh-Nagumo model, Gierer-Meinhardt system and Brusselator reaction-diffusion system. We quantify the accuracy achieved by different amounts of training data by Bayesian sampling methods. We demonstrate how a large enough ensemble of patterns leads to detection of very small but systematic structural changes, practically impossible to distinguish with the naked eye.

Hierarchical patterning modes orchestrate hair follicle morphogenesis

Two theories address the origin of repeating patterns, such as hair follicles, limb digits, and intestinal villi, during development. The Turing reaction-diffusion system posits that interacting diffusible signals produced by static cells first define a prepattern that then induces cell rearrangements to produce an anatomical structure. The second theory, that of mesenchymal self-organisation, proposes that mobile cells can form periodic patterns of cell aggregates directly, without reference to any prepattern. Early hair follicle development is characterised by the rapid appearance of periodic arrangements of altered gene expression in the epidermis and prominent clustering of the adjacent dermal mesenchymal cells. We assess the contributions and interplay between reaction-diffusion and mesenchymal self-organisation processes in hair follicle patterning, identifying a network of fibroblast growth factor (FGF), wingless-related integration site (WNT), and bone morphogenetic protein (BMP) signalling interactions capable of spontaneously producing a periodic pattern. Using time-lapse imaging, we find that mesenchymal cell condensation at hair follicles is locally directed by an epidermal prepattern. However, imposing this prepattern's condition of high FGF and low BMP activity across the entire skin reveals a latent dermal capacity to undergo spatially patterned self-organisation in the absence of epithelial direction. This mesenchymal self-organisation relies on restricted transforming growth factor (TGF) β signalling, which serves to drive chemotactic mesenchymal patterning when reaction-diffusion patterning is suppressed, but, in normal conditions, facilitates cell movement to locally prepatterned sources of FGF. This work illustrates a hierarchy of periodic patterning modes operating in organogenesis.

Computational and mathematical models of chondrogenesis in vertebrate limbs

The production of cartilage (chondrogenic patterning) in the limb is one of the best-studied examples of the emergence of form in developmental biology. At the core of the theoretical study is an effort to understand the mechanism that establishes the characteristic distribution of cartilage in the embryonic limb, which defines the future sites and shapes of bones that will be present in the mature limb. This review article gives an overview of the history and current state of a rich literature of mathematical and computational models that seek to contribute to this problem. We describe models for the mechanisms of limb growth and shaping via interaction with various chemical fields, as well as models addressing the intrinsic self-organization capabilities of the embryonic mesenchymal tissue, such as reaction-diffusion and mechanochemical models. We discuss the contributions of these models to the current understanding of chondrogenesis in vertebrate limbs, as well as their relation to the varied conceptual models that have been proposed by experimentalists.

Patterns of mesenchymal condensation in a multiscale, discrete stochastic model

Cells of the embryonic vertebrate limb in high-density culture undergo chondrogenic pattern formation, which results in the production of regularly spaced “islands” of cartilage similar to the cartilage primordia of the developing limb skeleton. The first step in this process, in vitro and in vivo, is the generation of “cell condensations,” in which the precartilage cells become more tightly packed at the sites at which cartilage will form. In this paper we describe a discrete, stochastic model for the behavior of limb bud precartilage mesenchymal cells in vitro. The model uses a biologically motivated reaction–diffusion process and cell-matrix adhesion (haptotaxis) as the bases of chondrogenic pattern formation, whereby the biochemically distinct condensing cells, as well as the size, number, and arrangement of the multicellular condensations, are generated in a self-organizing fashion. Improving on an earlier lattice-gas representation of the same process, it is multiscale (i.e., cell and molecular dynamics occur on distinct scales), and the cells are represented as spatially extended objects that can change their shape. The authors calibrate the model using experimental data and study sensitivity to changes in key parameters. The simulations have disclosed two distinct dynamic regimes for pattern self-organization involving transient or stationary inductive patterns of morphogens. The authors discuss these modes of pattern formation in relation to available experimental evidence for the in vitro system, as well as their implications for understanding limb skeletal patterning during embryonic development.

The development of an organism from embryo to adult includes processes of pattern formation that involve the interactions over space and time of independent cells to form multicellular structures. Computational models permit exploration of possible alternative mechanisms that reproduce biological patterns and thereby provide hypotheses for empirical testing. In this article, we describe a biologically motivated discrete stochastic model that shows that the patterns of spots and stripes of tightly packed cells observed in cultures derived from the embryonic vertebrate limb can occur by a mechanism that uses only cell–cell signaling via diffusible molecules (morphogens) and cell substratum adhesion (haptotaxis). Moreover, similar-looking patterns can arise both from stable stationary dynamics and unstable transient dynamics of the same underlying core molecular–genetic mechanism. Simulations also show that spot and stripe patterns (which also correspond to the nodules and bars of the developing limb skeleton in vivo) are close in parameter space and can be generated in multiple ways with single-parameter variations. An important implication is that some developmental processes do not require a strict progression from one stable dynamic regime to another, but can occur by a succession of transient dynamic regimes tuned (e.g., by natural selection) to achieve a particular morphological outcome.

On multiscale approaches to three-dimensional modelling of morphogenesis

In this paper we present the foundation of a unified, object-oriented, three-dimensional biomodelling environment, which allows us to integrate multiple submodels at scales from subcellular to those of tissues and organs. Our current implementation combines a modified discrete model from statistical mechanics, the Cellular Potts Model, with a continuum reaction-diffusion model and a state automaton with well-defined conditions for cell differentiation transitions to model genetic regulation. This environment allows us to rapidly and compactly create computational models of a class of complex-developmental phenomena. To illustrate model development, we simulate a simplified version of the formation of the skeletal pattern in a growing embryonic vertebrate limb.

Periodic pattern formation in reaction-diffusion systems: an introduction for numerical simulation

The aim of the present review is to provide a comprehensive explanation of Turing reaction-diffusion systems in sufficient detail to allow readers to perform numerical calculations themselves. The reaction-diffusion model is widely studied in the field of mathematical biology, serves as a powerful paradigm model for self-organization and is beginning to be applied to actual experimental systems in developmental biology. Despite the increase in current interest, the model is not well understood among experimental biologists, partly because appropriate introductory texts are lacking. In the present review, we provide a detailed description of the definition of the Turing reaction-diffusion model that is comprehensible without a special mathematical background, then illustrate a method for reproducing numerical calculations with Microsoft Excel. We then show some examples of the patterns generated by the model. Finally, we discuss future prospects for the interdisciplinary field of research involving mathematical approaches in developmental biology.

Dynamical mechanisms for skeletal pattern formation in the vertebrate limb

We describe a 'reactor-diffusion' mechanism for precartilage condensation based on recent experiments on chondrogenesis in the early vertebrate limb and additional hypotheses. Cellular differentiation of mesenchymal cells into subtypes with different fibroblast growth factor (FGF) receptors occurs in the presence of spatio-temporal variations of FGFs and transforming growth factor-betas (TGF-betas). One class of differentiated cells produces elevated quantities of the extracellular matrix protein fibronectin, which initiates adhesion-mediated preskeletal mesenchymal condensation. The same class of cells also produces an FGF-dependent laterally acting inhibitor that keeps condensations from expanding beyond a critical size. We show that this 'reactor-diffusion' mechanism leads naturally to patterning consistent with skeletal form, and describe simulations of spatio-temporal distribution of these differentiated cell types and the TGF-beta and inhibitor concentrations in the developing limb bud.

Interplay between activator-inhibitor coupling and cell-matrix adhesion in a cellular automaton model for chondrogenic patterning

We present a stochastic cellular automaton model for the behavior of limb bud precartilage mesenchymal cells undergoing chondrogenic patterning. This "agent-oriented" model represents cells by points on a lattice that obey rules motivated by experimental findings. The "cells" follow these rules as autonomous agents, interacting with other cells and with the microenvironments cell activities produce. The rules include random cell motion, production and lateral deposition of a substrate adhesion molecule (SAM, corresponding to fibronectin), production and release of a diffusible growth factor ("activator," corresponding to TGF-beta) that stimulates production of the SAM, and another diffusible factor ("inhibitor") that suppresses the activity of the activator. We implemented the cellular automaton on a two-dimensional (2D) square lattice to emulate the quasi-2D micromass culture extensively used to study patterning in avian limb bud precartilage cells. We identified parameters that produce nodular patterns that resemble, in size and distribution, cell condensations in leg-cell cultures, thus establishing a correspondence between in vitro and in silico results. We then studied the in vitro and in silico micromass cultures experimentally. We altered the standard in vitro micromass culture by diluting the initial cell density, transiently exposing it to exogenous activator, suppressing the inhibitor, and constitutively activating fibronectin production. We altered the standard in silico micromass culture in each case by changing the corresponding parameter. In vitro and in silico experiments agreed well. We also used the model to test hypotheses for differences in the in vitro patterns of cells derived from chick embryo forelimb and hindlimb. We discuss the applicability of this model to limb development in vivo and to other organ development.