Mathematical modelling of digit specification by a sonic hedgehog gradient

Boundary Conditions Cause Different Generic Bifurcation Structures in Turing Systems

Turing’s theory of morphogenesis is a generic mechanism to produce spatial patterning from near homogeneity. Although widely studied, we are still able to generate new results by returning to common dogmas. One such widely reported belief is that the Turing bifurcation occurs through a pitchfork bifurcation, which is true under zero-flux boundary conditions. However, under fixed boundary conditions, the Turing bifurcation becomes generically transcritical. We derive these algebraic results through weakly nonlinear analysis and apply them to the Schnakenberg kinetics. We observe that the combination of kinetics and boundary conditions produce their own uncommon boundary complexities that we explore numerically. Overall, this work demonstrates that it is not enough to only consider parameter perturbations in a sensitivity analysis of a specific application. Variations in boundary conditions should also be considered.

Bespoke Turing Systems

Reaction-diffusion systems are an intensively studied form of partial differential equation, frequently used to produce spatially heterogeneous patterned states from homogeneous symmetry breaking via the Turing instability. Although there are many prototypical "Turing systems" available, determining their parameters, functional forms, and general appropriateness for a given application is often difficult. Here, we consider the reverse problem. Namely, suppose we know the parameter region associated with the reaction kinetics in which patterning is required-we present a constructive framework for identifying systems that will exhibit the Turing instability within this region, whilst in addition often allowing selection of desired patterning features, such as spots, or stripes. In particular, we show how to build a system of two populations governed by polynomial morphogen kinetics such that the: patterning parameter domain (in any spatial dimension), morphogen phases (in any spatial dimension), and even type of resulting pattern (in up to two spatial dimensions) can all be determined. Finally, by employing spatial and temporal heterogeneity, we demonstrate that mixed mode patterns (spots, stripes, and complex prepatterns) are also possible, allowing one to build arbitrarily complicated patterning landscapes. Such a framework can be employed pedagogically, or in a variety of contemporary applications in designing synthetic chemical and biological patterning systems. We also discuss the implications that this freedom of design has on using reaction-diffusion systems in biological modelling and suggest that stronger constraints are needed when linking theory and experiment, as many simple patterns can be easily generated given freedom to choose reaction kinetics.

A mathematical model of the role of aggregation in sonic hedgehog signalling

Effective regulation of the sonic hedgehog (Shh) signalling pathway is essential for normal development in a wide variety of species. Correct Shh signalling requires the formation of Shh aggregates on the surface of producing cells. Shh aggregates subsequently diffuse away and are recognised in receiving cells located elsewhere in the developing embryo. Various mechanisms have been postulated regarding how these aggregates form and what their precise role is in the overall signalling process. To understand the role of these mechanisms in the overall signalling process, we formulate and analyse a mathematical model of Shh aggregation using nonlinear ordinary differential equations. We consider Shh aggregate formation to comprise of multimerisation, association with heparan sulfate proteoglycans (HSPG) and binding with lipoproteins. We show that the size distribution of the Shh aggregates formed on the producing cell surface resembles an exponential distribution, a result in agreement with experimental data. A detailed sensitivity analysis of our model reveals that this exponential distribution is robust to parameter changes, and subsequently, also to variations in the processes by which Shh is recruited by HSPGs and lipoproteins. The work demonstrates the time taken for different sized Shh aggregates to form and the important role this likely plays in Shh diffusion.

The sonic hedgehog (Shh) pathway is vital for normal development in a wide variety of species and its activity is strictly regulated to ensure correct spatiotemporal patterning of numerous developing tissues. Shh signalling requires the formation of Shh aggregates, formed on producing cells via a range of different mechanisms, that then diffuse to receiving cells. We formulate and analyse a mathematical model of the most well described mechanisms, namely monomer multimerisation, and recruitment of Shh by heparan sulfate proteoglycans and lipoproteins. Our results illustrate a distribution of the size and quantities of aggregates formed by these mechanisms. We found that as a consequence of competition between the mechanisms for Shh monomers the shape distribution of Shh aggregates resembles an exponential distribution. We also found the distribution to be robust to both parameter changes and variations to the processes by which mechanisms recruit Shh. We report that our approach and subsequent results demonstrate that these mechanisms act in synergy allowing Shh to aggregate in various quantities with diverse diffusive abilities. We postulate that this regulation contributes significantly to aid precision in signalling for Shh in areas of development.

Of mitogens and morphogens: modelling Sonic Hedgehog mechanisms in vertebrate development

Sonic Hedgehog (Shh) Is a critical protein in vertebrate development, orchestrating patterning and growth in many developing systems. First described as a classic morphogen that patterns tissues through a spatial concentration gradient, subsequent studies have revealed a more complex mechanism, in which Shh can also regulate proliferation and differentiation. While the mechanism of action of Shh as a morphogen is well understood, it remains less clear how Shh might integrate patterning, proliferation and differentiation in a given tissue, to ultimately direct its morphogenesis. In tandem with experimental studies, mathematical modelling can help gain mechanistic insights into these processes and bridge the gap between Shh-regulated patterning and growth, by integrating these processes into a common theoretical framework. Here, we briefly review the roles of Shh in vertebrate development, focusing on its functions as a morphogen, mitogen and regulator of differentiation. We then discuss the contributions that modelling has made to our understanding of the action of Shh and highlight current challenges in using mathematical models in a quantitative and predictive way. This article is part of a discussion meeting issue 'Contemporary morphogenesis'.

IRX3/5 regulate mitotic chromatid segregation and limb bud shape

Pattern formation is influenced by transcriptional regulation as well as by morphogenetic mechanisms that shape organ primordia, although factors that link these processes remain under-appreciated. Here we show that, apart from their established transcriptional roles in pattern formation, IRX3/5 help to shape the limb bud primordium by promoting the separation and intercalation of dividing mesodermal cells. Surprisingly, IRX3/5 are required for appropriate cell cycle progression and chromatid segregation during mitosis, possibly in a nontranscriptional manner. IRX3/5 associate with, promote the abundance of, and share overlapping functions with co-regulators of cell division such as the cohesin subunits SMC1, SMC3, NIPBL and CUX1. The findings imply that IRX3/5 coordinate early limb bud morphogenesis with skeletal pattern formation.

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.

Pattern production through a chiral chasing mechanism

Recent experiments on zebrafish pigmentation suggests that their typical black and white striped skin pattern is made up of a number of interacting chromatophore families. Specifically, two of these cell families have been shown to interact through a nonlocal chasing mechanism, which has previously been modeled using integro-differential equations. We extend this framework to include the experimentally observed fact that the cells often exhibit chiral movement, in that the cells chase, and run away, at angles different to the line connecting their centers. This framework is simplified through the use of multiple small limits leading to a coupled set of partial differential equations which are amenable to Fourier analysis. This analysis results in the production of dispersion relations and necessary conditions for a patterning instability to occur. Beyond the theoretical development and the production of new pattern planiforms we are able to corroborate the experimental hypothesis that the global pigmentation patterns can be dependent on the chirality of the chromatophores.

Sonic Hedgehog Signaling in Limb Development

The gene encoding the secreted protein Sonic hedgehog (Shh) is expressed in the polarizing region (or zone of polarizing activity), a small group of mesenchyme cells at the posterior margin of the vertebrate limb bud. Detailed analyses have revealed that Shh has the properties of the long sought after polarizing region morphogen that specifies positional values across the antero-posterior axis (e.g., thumb to little finger axis) of the limb. Shh has also been shown to control the width of the limb bud by stimulating mesenchyme cell proliferation and by regulating the antero-posterior length of the apical ectodermal ridge, the signaling region required for limb bud outgrowth and the laying down of structures along the proximo-distal axis (e.g., shoulder to digits axis) of the limb. It has been shown that Shh signaling can specify antero-posterior positional values in limb buds in both a concentration- (paracrine) and time-dependent (autocrine) fashion. Currently there are several models for how Shh specifies positional values over time in the limb buds of chick and mouse embryos and how this is integrated with growth. Extensive work has elucidated downstream transcriptional targets of Shh signaling. Nevertheless, it remains unclear how antero-posterior positional values are encoded and then interpreted to give the particular structure appropriate to that position, for example, the type of digit. A distant cis-regulatory enhancer controls limb-bud-specific expression of Shh and the discovery of increasing numbers of interacting transcription factors indicate complex spatiotemporal regulation. Altered Shh signaling is implicated in clinical conditions with congenital limb defects and in the evolution of the morphological diversity of vertebrate limbs.

How domain growth is implemented determines the long-term behavior of a cell population through its effect on spatial correlations

Domain growth plays an important role in many biological systems, and so the inclusion of domain growth in models of these biological systems is important to understanding how these systems function. In this work we present methods to include the effects of domain growth on the evolution of spatial correlations in a continuum approximation of a lattice-based model of cell motility and proliferation. We show that, depending on the way in which domain growth is implemented, different steady-state densities are predicted for an agent population. Furthermore, we demonstrate that the way in which domain growth is implemented can result in the evolution of the agent density depending on the size of the domain. Continuum approximations that ignore spatial correlations cannot capture these behaviors, while those that account for spatial correlations do. These results will be of interest to researchers in developmental biology, as they suggest that the nature of domain growth can determine the characteristics of cell populations.

Spatiotemporal analysis of different mechanisms for interpreting morphogen gradients

During development, multicellular organisms must accurately control both temporal and spatial aspects of tissue patterning. This is often achieved using morphogens, signaling molecules that form spatially varying concentrations and so encode positional information. Typical analysis of morphogens assumes that spatial information is decoded in steady state by measuring the value of the morphogen concentration. However, recent experimental work suggests that both pre-steady-state readout and measurement of spatial and temporal derivatives of the morphogen concentration can play important roles in defining boundaries. Here, we undertake a detailed theoretical and numerical study of the accuracy of patterning-both in space and time-in models where readout is provided not by the morphogen concentration but by its spatial and temporal derivatives. In both cases we find that accurate patterning can be achieved, with sometimes even smaller errors than directly reading the morphogen concentration. We further demonstrate that such models provide other potential benefits to the system, such as the ability to switch on and off gene response with a high degree of spatiotemporal accuracy. Finally, we discuss how such derivatives might be calculated biologically and examine these models in relation to Sonic Hedgehog signaling in the vertebrate central nervous system. We show that, when coupled to a downstream transcriptional network, pre-steady-state measurement of the temporal change in the Shh morphogen is a plausible mechanism for determining precise gene boundaries in both space and time.