Systems for intricate patterning of the vertebrate anatomy

Periodic patterns form intricate arrays in the vertebrate anatomy, notably the hair and feather follicles of the skin, but also internally the villi of the gut and the many branches of the lung, kidney, mammary and salivary glands. These tissues are composite structures, being composed of adjoined epithelium and mesenchyme, and the patterns that arise within them require interaction between these two tissue layers. In embryonic development, cells change both their distribution and state in a periodic manner, defining the size and relative positions of these specialized structures. Their placement is determined by simple spacing mechanisms, with substantial evidence pointing to a variety of local enhancement/lateral inhibition systems underlying the breaking of symmetry. The nature of the cellular processes involved, however, has been less clear. While much attention has focused on intercellular soluble signals, such as protein growth factors, experimental evidence has grown for contributions of cell movement or mechanical forces to symmetry breaking. In the mesenchyme, unlike the epithelium, cells may move freely and can self-organize into aggregates by chemotaxis, or through generation and response to mechanical strain on their surrounding matrix. Different modes of self-organization may coexist, either coordinated into a single system or with hierarchical relationships. Consideration of a broad range of distinct biological processes is required to advance understanding of biological pattern formation. This article is part of the theme issue 'Recent progress and open frontiers in Turing's theory of morphogenesis'.

Cost-precision trade-off relation determines the optimal morphogen gradient for accurate biological pattern formation

Spatial boundaries formed during animal development originate from the pre-patterning of tissues by signaling molecules, called morphogens. The accuracy of boundary location is limited by the fluctuations of morphogen concentration that thresholds the expression level of target gene. Producing more morphogen molecules, which gives rise to smaller relative fluctuations, would better serve to shape more precise target boundaries; however, it incurs more thermodynamic cost. In the classical diffusion-depletion model of morphogen profile formation, the morphogen molecules synthesized from a local source display an exponentially decaying concentration profile with a characteristic length λ. Our theory suggests that in order to attain a precise profile with the minimal cost, λ should be roughly half the distance to the target boundary position from the source. Remarkably, we find that the profiles of morphogens that pattern the Drosophila embryo and wing imaginal disk are formed with nearly optimal λ. Our finding underscores the thermodynamic cost as a key physical constraint in the morphogen profile formation in Drosophila development.

Non-linear Analysis of Models for Biological Pattern Formation: Application to Ocular Dominance Stripes

We present a technique for the analysis of pattern formation by a class of models for the formation of ocular dominance stripes in the striate cortex of some mammals. The method, which employs the adiabatic approximation to derive a set of ordinary differential equations for patterning modes, has been successfully applied to reaction-diffusion models for striped patterns. Models of ocular dominance stripes have been studied by computation, or by linearization of the model equations. These techniques do not provide a rationale for the origin of the stripes. We show here that stripe formation is a non-linear property of the models. Our analysis indicates that stripe selection is closely linked to a property in the dynamics of the models which arises from a symmetry between ipsilateral and contralateral synapses to the visual cortex of a given hemisphere.

Mechanical Patterning in Animal Morphogenesis

Morphogenesis is one of the most remarkable examples of biological pattern formation. Despite substantial progress in the field, we still do not understand the organizational principles responsible for the robust convergence of the morphogenesis process across scales to form viable organisms under variable conditions. Achieving large-scale coordination requires feedback between mechanical and biochemical processes, spanning all levels of organization and relating the emerging patterns with the mechanisms driving their formation. In this review, we highlight the role of mechanics in the patterning process, emphasizing the active and synergistic manner in which mechanical processes participate in developmental patterning rather than merely following a program set by biochemical signals. We discuss the value of applying a coarse-grained approach toward understanding this complex interplay, which considers the large-scale dynamics and feedback as well as complementing the reductionist approach focused on molecular detail. A central challenge in this approach is identifying relevant coarse-grained variables and developing effective theories that can serve as a basis for an integrated framework for understanding this remarkable pattern-formation process. Expected final online publication date for the Annual Review of Cell and Developmental Biology, Volume 37 is October 2021. Please see http://www.annualreviews.org/page/journal/pubdates for revised estimates.

Spatio-temporal Brusselator Model and Biological Pattern Formation

This paper explores a two-species non-homogeneous reaction-diffusion model for the study of pattern formation with the Brusselator model. We scrutinize the pattern formation with initial conditions and Neumann boundary conditions in a spatially heterogeneous environment. In the whole investigation, we assume the case for random diffusion strategy. The dynamics of model behaviors show that the nature of pattern formation with varying parameters and initial conditions thoroughly. The model also studies in the absence of diffusion terms. The theoretical and numerical observations explain pattern formation using the reaction-diffusion model in both one and two dimensions.

Cost-precision trade-off relation determines the optimal morphogen gradient for accurate biological pattern formation

Spatial boundaries growing into macroscopic structures through animal development originate from the pre-patterning of tissues by signaling molecules, called morphogens. To establish accurate boundaries, the morphogen concentration which thresholds the expression of target gene at the boundary should be precise enough, exhibiting large gradient and small fluctuations. Producing more morphogens would better serve to shape more precise target boundaries; however, it incurs more thermodynamic cost. In the classical diffusion-degradation model of morphogen profile formation, the morphogens synthesized from a local source display an exponentially decaying concentration profile with a characteristic length λ. Our theory suggests that in order to attain a precise morphogen profile with the minimal cost, λ should be roughly half the distance to the target boundary position from the source, so that the boundary is formed at the position where the morphogen concentration is ~10% of the value at the source. Remarkably, we find that the well characterized morphogens that pattern the fruit fly embryo and wing imaginal disk form profiles with nearly optimal λ, which underscores the thermodynamic cost as a key physical constraint in the morphogen profile formation.

Convective instability in a diffusive predator–prey system

It is well known that biological pattern formation is the Turing mechanism, in which a homogeneous steady state is destabilized by the addition of diffusion, though it is stable in the kinetic ODEs. However, steady states that are unstable in the kinetic ODEs are rarely mentioned. This paper concerns a reaction diffusion advection system under Neumann boundary conditions, where steady states that are unstable in the kinetic ODEs. Our results provide a stabilization strategy for the same steady state, the combination of large advection rate and small diffusion rate can stabilize the homogeneous equilibrium. Moreover, we investigate the existence and stability of nonconstant positive steady states to the system through rigorous bifurcation analysis.

Evolving Consciousness: Insights From Turing, and the Shaping of Experience

A number of conceptual difficulties arise when considering the evolutionary origin of consciousness from the pre-conscious condition. There are parallels here with biological pattern formation, where, according to Alan Turing’s original formulation of the problem, the statistical properties of molecular-level processes serve as a source of incipient pattern. By analogy, the evolution of consciousness can be thought of as depending in part on a competition between alternative variants in the microstructure of synaptic networks and/or the activity patterns they generate, some of which then serve as neural correlates of consciousness (NCCs). Assuming that NCCs perform this function only if reliably ordered in a particular and precise way, Turing’s formulation provides a useful conceptual framework for thinking about how this is achieved developmentally, and how changes in neural structure might correlate with change at the level of conscious experience. The analysis is largely silent concerning the nature and ultimate source of conscious experience, but shows that achieving sentience is sufficient to begin the process by which evolution elaborates and shapes that first experience. By implication, much of what evolved consciousness achieves in adaptive terms can in principle be investigated irrespective of whether or not the ultimate source of real-time experience is known or understood. This includes the important issue of how precisely NCCs must be structured to ensure that each evokes a particular experience as opposed to any other. Some terminological issues are clarified, including that of “noise,” which here refers to the statistical variations in neural structure that arise during development, not to sensory noise as experienced in real time.

Synthetic Turing patterns in engineered microbial consortia

Multicellular entities are characterized by exquisite spatial patterns, intimately related to the functions they perform. Oftentimes these patterns emerge as periodic structures with a well-defined characteristic scale. A candidate mechanism to explain their origins was early introduced by Alan Turing through the interaction and diffusion of two so called morphogens. Unfortunately, most available evidence for Turing patterns in biology is usually obscured by the tangled nature of regulatory phenomena, making difficult to validate Turing’s proposal in developmental processes. Here we follow a different approach, by designing synthetic genetic circuits in engineered E. coli strains that implement the essential activator-inhibitor motif (AIM) using a two-cell consortium. The two diffusible compartments are one cell type (activator, small-diffusion component) and a small signal molecule (a homoserine lactone, acting as the fast-diffusing inhibitor). Using both experimental results, we show that the AIM is capable of generating diffusion-induced instabilities leading to regular spatial patterns. The artificial construction taken here can help validate developmental theories and identify universal properties underpinning biological pattern formation. The implications of the work for the area of synthetic developmental biology are outlined.