Beyond Turing: mechanochemical pattern formation in biological tissues

Geometry-induced patterns through mechanochemical coupling

Intracellular protein patterns regulate a variety of vital cellular processes such as cell division and motility, which often involve dynamic cell-shape changes. These changes in cell shape may in turn affect the dynamics of pattern-forming proteins, hence leading to an intricate feedback loop between cell shape and chemical dynamics. While several computational studies have examined the rich resulting dynamics, the underlying mechanisms are not yet fully understood. To elucidate some of these mechanisms, we explore a conceptual model for cell polarity on a dynamic one-dimensional manifold. Using concepts from differential geometry, we derive the equations governing mass-conserving reaction-diffusion systems on time-evolving manifolds. Analyzing these equations mathematically, we show that dynamic shape changes of the membrane can induce pattern-forming instabilities in parts of the membrane, which we refer to as regional instabilities. Deformations of the local membrane geometry can also (regionally) suppress pattern formation and spatially shift already existing patterns. We explain our findings by applying and generalizing the local equilibria theory of mass-conserving reaction-diffusion systems. This allows us to determine a simple onset criterion for geometry-induced pattern-forming instabilities, which is linked to the phase-space structure of the reaction-diffusion system. The feedback loop between membrane shape deformations and reaction-diffusion dynamics then leads to a surprisingly rich phenomenology of patterns, including oscillations, traveling waves, and standing waves, even if these patterns do not occur in systems with a fixed membrane shape. Our paper reveals that the local conformation of the membrane geometry acts as an important dynamical control parameter for pattern formation in mass-conserving reaction-diffusion systems.

Cell shape anisotropy contributes to self-organized feather pattern fidelity in birds

Developing tissues can self-organize into a variety of patterned structures through the stabilization of stochastic fluctuations in their molecular and cellular properties. While molecular factors and cell dynamics contributing to self-organization have been identified in vivo, events channeling self-organized systems such that they achieve stable pattern outcomes remain unknown. Here, we described natural variation in the fidelity of self-organized arrays formed by feather follicle precursors in bird embryos. By surveying skin cells prior to and during tissue self-organization and performing species-specific ex vivo drug treatments and mechanical stress tests, we demonstrated that pattern fidelity depends on the initial amplitude of cell anisotropy in regions of the developing dermis competent to produce a pattern. Using live imaging, we showed that cell shape anisotropy is associated with a limited increase in cell motility for sharp and precisely located primordia formation, and thus, proper pattern geometry. These results evidence a mechanism through which initial tissue properties ensure stability in self-organization and thus, reproducible pattern production.

A study of natural variation in feather pattern geometry and a combination of pharmacological and mechanical manipulations ex vivo provides evidence for a mechanism by which initial tissue properties ensure stability in self-organization and species-specific pattern production.

Pattern formation features might explain homoplasy: fertile surfaces in higher fungi as an example

Fungi show a high degree of morphological convergence. Regarded for a long time as an obstacle for phylogenetic studies, homoplasy has also been proposed as a source of information about underlying morphogenetic patterning mechanisms. The "local-activation and long-range inhibition principle" (LALIP), underlying the famous reaction-diffusion model proposed by Alan Turing in 1952, appears to be one of the universal phenomena that can explain the ontogenetic origin of seriate patterns in living organisms. Reproductive structures of fungi in the class Agaricomycetes show a highly periodic structure resulting in, for example, poroid, odontoid, lamellate or labyrinthic hymenophores. In this paper, we claim that self-organized patterns might underlie the basic ontogenetic processes of these structures. Simulations based on LALIP-driven models and covering a wide range of parameters show an absolute mutual correspondence with the morphospace explored by extant agaricomycetes. This could not only explain geometric particularities but could also account for the limited possibilities displayed by hymenial configurations, thus making homoplasy a direct consequence of the limited morphospace resulting from the proposed patterning dynamics.

Mechanical Regulation of Limb Bud Formation

Early limb bud development has been of considerable interest for the study of embryological development and especially morphogenesis. The focus has long been on biochemical signalling and less on cell biomechanics and mechanobiology. However, their importance cannot be understated since tissue shape changes are ultimately controlled by active forces and bulk tissue rheological properties that in turn depend on cell-cell interactions as well as extracellular matrix composition. Moreover, the feedback between gene regulation and the biomechanical environment is still poorly understood. In recent years, novel experimental techniques and computational models have reinvigorated research on this biomechanical and mechanobiological side of embryological development. In this review, we consider three stages of early limb development, namely: outgrowth, elongation, and condensation. For each of these stages, we summarize basic biological regulation and examine the role of cellular and tissue mechanics in the morphogenetic process.

Beyond Turing: far-from-equilibrium patterns and mechano-chemical feedback

Turing patterns are commonly understood as specific instabilities of a spatially homogeneous steady state, resulting from activator-inhibitor interaction destabilized by diffusion. We argue that this view is restrictive and its agreement with biological observations is problematic. We present two alternatives to the classical Turing analysis of patterns. First, we employ the abstract framework of evolution equations to enable the study of far-from-equilibrium patterns. Second, we introduce a mechano-chemical model, with the surface on which the pattern forms being dynamic and playing an active role in the pattern formation, effectively replacing the inhibitor. We highlight the advantages of these two alternatives vis-à-vis the classical Turing analysis, and give an overview of recent results and future challenges for both approaches. This article is part of the theme issue 'Recent progress and open frontiers in Turing's theory of morphogenesis'.

Mathematical modeling of the eyespots in butterfly wings

Butterfly wing color patterns are a representative model system for studying biological pattern formation, due to their two-dimensional simple structural and high inter- and intra-specific variabilities. Moreover, butterfly color patterns have demonstrated roles in mate choice, thermoregulation, and predator avoidance via disruptive coloration, attack deflection, aposematism, mimicry, and masquerade. Because of the importance of color patterns to many aspects of butterfly biology and their apparent tractability for study, color patterns have been the subjects of many attempts to model their development. Early attempts focused on generalized mechanisms of pattern formation such as reaction-diffusion, diffusion gradient, lateral inhibition, and threshold responses, without reference to any specific gene products. As candidate genes with expression patterns that resembled incipient color patterns were identified, genetic regulatory networks were proposed for color pattern formation based on gene functions inferred from other insects with wings, such as Drosophila. Particularly detailed networks incorporating the gene products, Distal-less (Dll), Engrailed (En), Hedgehog (Hh), Cubitus interruptus (Ci), Transforming growth factor-β (TGF-β), and Wingless (Wg), have been proposed for butterfly border ocelli (eyespots) which helps the investigation of the formation of these patterns. Thus, in this work, we develop a mathematical model including the gene products En, Hh, Ci, TGF-β, and Wg to mimic and investigate the eyespot formation in butterflies. Our simulations show that the level of En has peaks in the inner and outer rings and the level of Ci has peaks in the inner and middle rings. The interactions among these peaks activate cells to produce white, black, and yellow pigments in the inner, middle, and outer rings, respectively, which captures the eyespot pattern of wild type Bicyclus anynana butterflies. Additionally, our simulations suggest that lack of En generates a single black spot and lack of Hh or Ci generates a single white spot, and a deficiency of TGF-β or Wg will cause the loss of the outer yellow ring. These deficient patterns are similar to those observed in the eyespots of Vanessa atalanta, Vanessa altissima, and Chlosyne nycteis. Thus, our model also provides a hypothesis to explain the mechanism of generating the deficient patterns in these species.

The design principles of discrete turing patterning systems

The formation of spatial structures lies at the heart of developmental processes. However, many of the underlying gene regulatory and biochemical processes remain poorly understood. Turing patterns constitute a main candidate to explain such processes, but they appear sensitive to fluctuations and variations in kinetic parameters, raising the question of how they may be adopted and realised in naturally evolved systems. The vast majority of mathematical studies of Turing patterns have used continuous models specified in terms of partial differential equations. Here, we complement this work by studying Turing patterns using discrete cellular automata models. We perform a large-scale study on all possible two-species networks and find the same Turing pattern producing networks as in the continuous framework. In contrast to continuous models, however, we find these Turing pattern topologies to be substantially more robust to changes in the parameters of the model. We also find that diffusion-driven instabilities are substantially weaker predictors for Turing patterns in our discrete modelling framework in comparison to the continuous case, in the sense that the presence of an instability does not guarantee a pattern emerging in simulations. We show that a more refined criterion constitutes a stronger predictor. The similarity of the results for the two modelling frameworks suggests a deeper underlying principle of Turing mechanisms in nature. Together with the larger robustness in the discrete case this suggests that Turing patterns may be more robust than previously thought.

Chemically controlled pattern formation in self-oscillating elastic shells

Patterns and morphology develop in living systems such as embryos in response to chemical signals. To understand and exploit the interplay of chemical reactions with mechanical transformations, chemomechanical polymer systems have been synthesized by attaching chemicals into hydrogels. In this work, we design autonomous responsive elastic shells that undergo morphological changes induced by chemical reactions. We couple the local mechanical response of the gel with the chemical processes on the shell. This causes swelling and deswelling of the gel, generating diverse morphological changes, including periodic oscillations. We further introduce a mechanical instability and observe buckling-unbuckling dynamics with a response time delay. Moreover, we investigate the mechanical feedback on the chemical reaction and demonstrate the dynamic patterns triggered by an initial deformation. We show the chemical characteristics that account for the shell morphology and discuss the future designs for autonomous responsive materials.

A general principle for spontaneous genetic symmetry breaking and pattern formation within cell populations

Elements within biological systems interact and frequently self-organize from initially disordered states into highly structured patterns. The local self-activation and lateral inhibition mechanism, derived from the coupling between two reacting and diffusing chemicals, has been believed to be one of the main causes for biological pattern formation. Graded positional information can be produced by the limited diffusion of one single signaling molecule through cell populations with no pre-patterns being required. We demonstrate, using multiscale computations, that spontaneous symmetry breaking can be driven within expanding and non-expanding cell populations, without local self-enhancement of activators and long-range inhibition. Instead, cells can self-organize into structured gene patterns via a combination of timing gene expression in cells and the graded positional information which has been coupled to the gene expression. We show that the genetic symmetry breaking in expanding E. coli populations occurs at a critical colony size, which is independent of the cell doubling time but scales with the diffusion speed of the signaling molecule. We also show the quasi-3D structure of gene patterns, and observe that the wave length of periodic genetic stripes is in proportion to the genetic oscillation cycle time and in inverse proportion to cell doubling time. Our results provide insights into relevant biological development processes.

Trends and variation in vertebrate patterns as outcomes of self-organization

In extant vertebrates, natural motifs such as coat markings, spongy bone structures, neuronal arborescence or collective swarms correspond to diverse pattern types, from fractals to periodic stripes or tessellations, and so on. In this subphylum, evolution produced an apparent paradox: a given pattern may vary tremendously in its extent, periodicity or regularity, but follows general geometrical trends and is produced with meticulous precision. In this review, we discuss the role of self-organization, a patterning strategy in which spontaneous order arises through local interactions without gradient formation, in shaping both natural pattern differences and common themes. Mathematical models evidenced a wide high adaptability of self-organizing dynamics, long-advocating for their contribution to natural pattern diversity. Recent empirical and theoretical approaches taking into account network topologies and natural variation also replaced outcomes of self-organization in more constrained biological contexts, shedding light on mechanisms ensuring pattern fidelity.

From embryos to embryoids: How external signals and self-organization drive embryonic development

Embryonic development has been traditionally seen as an inductive process directed by exogenous maternal inputs and extra-embryonic signals. Increasing evidence, however, is showing that, in addition to exogenous signals, the development of the embryo involves endogenous self-organization. Recently, this self-organizing potential has been highlighted by a number of stem cell models known as embryoids that can recapitulate different aspects of embryogenesis in vitro. Here, we review the self-organizing behaviors observed in different embryoid models and seek to reconcile this new evidence with classical knowledge of developmental biology. This analysis leads to reexamine embryonic development as a guided self-organizing process, where patterning and morphogenesis are controlled by a combination of exogenous signals and endogenous self-organization. Finally, we discuss the multidisciplinary approach required to investigate the genetic and cellular basis of self-organization.

A "Numerical Evo-Devo" Synthesis for the Identification of Pattern-Forming Factors

Animals display extensive diversity in motifs adorning their coat, yet these patterns have reproducible orientation and periodicity within species or groups. Morphological variation has been traditionally used to dissect the genetic basis of evolutionary change, while pattern conservation and stability in both mathematical and organismal models has served to identify core developmental events. Two patterning theories, namely instruction and self-organisation, emerged from this work. Combined, they provide an appealing explanation for how natural patterns form and evolve, but in vivo factors underlying these mechanisms remain elusive. By bridging developmental biology and mathematics, novel frameworks recently allowed breakthroughs in our understanding of pattern establishment, unveiling how patterning strategies combine in space and time, or the importance of tissue morphogenesis in generating positional information. Adding results from surveys of natural variation to these empirical-modelling dialogues improves model inference, analysis, and in vivo testing. In this evo-devo-numerical synthesis, mathematical models have to reproduce not only given stable patterns but also the dynamics of their emergence, and the extent of inter-species variation in these dynamics through minimal parameter change. This integrative approach can help in disentangling molecular, cellular and mechanical interaction during pattern establishment.

On the problem of biological form

Embryonic development, which inspired the first theories of biological form, was eventually excluded from the conceptual framework of the Modern Synthesis as irrelevant. A major question during the last decades has centred on understanding whether new advances in developmental biology are compatible with the standard view or whether they compel a new theory. Here, I argue that the answer to this question depends on which concept of morphogenesis is held. Morphogenesis can be conceived as (1) a chemically driven or (2) a mechanically driven process. According to the first option, genetic regulatory networks drive morphogenesis. According to the second, morphogenesis results from an invariant tendency of embryonic tissues to restore changes in mechanical stress. While chemically driven morphogenesis allows an extension of the standard view, mechanically driven morphogenesis would deeply transform it. Which of these hypotheses has wider explanatory power is unknown. At present, the problem of biological form remains unsolved.

Tissue rheology in embryonic organization

Tissue morphogenesis in multicellular organisms is brought about by spatiotemporal coordination of mechanical and chemical signals. Extensive work on how mechanical forces together with the well-established morphogen signalling pathways can actively shape living tissues has revealed evolutionary conserved mechanochemical features of embryonic development. More recently, attention has been drawn to the description of tissue material properties and how they can influence certain morphogenetic processes. Interestingly, besides the role of tissue material properties in determining how much tissues deform in response to force application, there is increasing theoretical and experimental evidence, suggesting that tissue material properties can abruptly and drastically change in development. These changes resemble phase transitions, pointing at the intriguing possibility that important morphogenetic processes in development, such as symmetry breaking and self-organization, might be mediated by tissue phase transitions. In this review, we summarize recent findings on the regulation and role of tissue material properties in the context of the developing embryo. We posit that abrupt changes of tissue rheological properties may have important implications in maintaining the balance between robustness and adaptability during embryonic development.

Emergence of form in embryogenesis

The development of form in an embryo is the result of a series of topological and informational symmetry breakings. We introduce the vector-reaction-diffusion-drift (VRDD) system where the limit cycle of spatial dynamics is morphogen concentrations with Dirac delta-type distributions. This is fundamentally different from the Turing reaction-diffusion system, as VRDD generates system-wide broken symmetry. We developed 'fundamental forms' from spherical blastula with a single organizing axis (rotational symmetry), double axis (mirror symmetry) and triple axis (no symmetry operator in three dimensions). We then introduced dynamics for cell differentiation, where genetic regulatory states are modelled as a finite-state machine (FSM). The state switching of an FSM is based on local morphogen concentrations as epigenetic information that changes dynamically. We grow complicated forms hierarchically in spatial subdomains using the FSM model coupled with the VRDD system. Using our integrated simulation model with four layers (topological, physical, chemical and regulatory), we generated life-like forms such as hydra. Genotype-phenotype mapping was investigated with continuous and jump mutations. Our study can have applications in morphogenetic engineering, soft robotics and biomimetic design.

Post-Turing tissue pattern formation: Advent of mechanochemistry

Chemical and mechanical pattern formation is fundamental during embryogenesis and tissue development. Yet, the underlying molecular and cellular mechanisms are still elusive in many cases. Most current theories assume that tissue development is driven by chemical processes: either as a sequence of chemical patterns each depending on the previous one, or by patterns spontaneously arising from specific chemical interactions (such as “Turing-patterns”). Within both theories, mechanical patterns are usually regarded as passive by-products of chemical pre-patters. However, several experiments question these theories, and an increasing number of studies shows that tissue mechanics can actively influence chemical patterns during development. In this study, we thus focus on the interplay between chemical and mechanical processes during tissue development. On one hand, based on recent experimental data, we develop new mechanochemical simulation models of evolving tissues, in which the full 3D representation of the tissue appears to be critical for obtaining a realistic mechanochemical behaviour. The presented modelling approach is flexible and numerically studied using state of the art finite element methods. Thus, it may serve as a basis to combine simulations with new experimental methods in tissue development. On the other hand, we apply the developed approach and demonstrate that even simple interactions between tissue mechanics and chemistry spontaneously lead to robust and complex mechanochemical patterns. Especially, we demonstrate that the main contradictions arising in the framework of purely chemical theories are naturally and automatically resolved using the mechanochemical patterning theory.

During embryogenesis, biological tissues gradually increase their complexity by self-organised creation of diverse chemical and mechanical patterns. Detailed mechanisms driving and controlling these patterns are not well understood. Previous theories mostly assume that these patterns are driven by chemical processes. Based on these theories, mechanical patterns are usually considered being mainly determined by chemical pre-patterns. However, experimental evidence for these theories is sparse, and several inconsistencies have been discovered. Furthermore, an increasing amount of data shows that tissue mechanics plays an important role in pattern formation. In this study, we present 3D computer simulations of evolving tissues to investigate the capacity of mechanochemical interactions for pattern formation. We show that even simple interactions between tissue mechanics and tissue chemistry spontaneously lead to robust chemical and mechanical pattern formation. We additionally demonstrate that main contradictions arising in the framework of purely chemical theories are naturally and automatically resolved using the mechanochemical patterning theory. The presented modelling approach can be used to combine simulations with recent experimental developments, to help unravel one of the big mysteries in biology: The mechanisms of self-organised pattern formation during embryogenesis.

Role of the boundary in feather bud formation on one-dimensional bioengineered skin

The role of a boundary in pattern formation from a homogenous state in Turing's reaction–diffusion equations is important, particularly when the domain size is comparable to the pattern scale. Such experimental conditions may be achieved for in vitro regeneration of ectodermal appendages such as feathers, via reconstruction of embryonic single cells. This procedure can eliminate a predefined genetic map, such as the midline of chick feather bud formation, leaving uniformly distributed identical cells as a bioengineered skin. Here, the self-organizing nature of multiple feather bud formation was examined in bioengineered 1D-skin samples. Primal formation of feather buds occurred at a fixed length from the skin edge. This formation was numerically recapitulated by a standard two-component reaction-diffusion model, suggesting that the boundary effect caused this observation. The proper boundary conditions were nonstandard, either mixed Dirichlet–Neumann or partial-flux. In addition, the model implies imperfect or hindered bud formation as well as nearly equal distances between buds. In contrast, experimental observations indicated that the skin curvature, which was not included in our model, also strongly affected bud formation. Thus, bioengineered skin may provide an ideal template for modeling a self-organized process from a homogenous state. This study will examine the possible diffusion activities of activator or inhibitor molecular candidates and mechanical activities during cell aggregation, which will advance our understanding of skin appendage regeneration from pluripotent or embryonic stem cells.

Combining Turing and 3D vertex models reproduces autonomous multicellular morphogenesis with undulation, tubulation, and branching

This study demonstrates computational simulations of multicellular deformation coupled with chemical patterning in the three-dimensional (3D) space. To address these aspects, we proposes a novel mathematical model, where a reaction–diffusion system is discretely expressed at a single cell level and combined with a 3D vertex model. To investigate complex phenomena emerging from the coupling of patterning and deformation, as an example, we employed an activator–inhibitor system and converted the activator concentration of individual cells into their growth rate. Despite the simplicity of the model, by growing a monolayer cell vesicle, the coupling system provided rich morphological dynamics such as undulation, tubulation, and branching. Interestingly, the morphological variety depends on the difference in time scales between patterning and deformation, and can be partially understood by the intrinsic hysteresis in the activator-inhibitor system with domain growth. Importantly, the model can be applied to 3D multicellular dynamics that couple the reaction–diffusion patterning with various cell behaviors, such as deformation, rearrangement, division, apoptosis, differentiation, and proliferation. Thus, the results demonstrate the significant advantage of the proposed model as well as the biophysical importance of exploring spatiotemporal dynamics of the coupling phenomena of patterning and deformation in 3D space.

Inceptions of biomathematics from Lotka to Thom

Mathematical biology occupies a special place at the interface between the physical, mathematical and life sciences. Is this interface merely a meeting point for dabblers venturing out of their own proper domains to work on problems of mutual interest? Or is it an incipient science in its own right, with its own particular character, principles, and practices? The past century has seen vast advances in the application of mathematical and physical ideas and techniques to biological problems, in the process transforming many of them almost beyond recognition. Nonetheless, the question of a biomathematics as a new kind of science remains open, despite several fascinating, if sometimes problematic, attempts.