Turing Reaction-diffusion Research Articles

A diffusible small-RNA-based Turing system dynamically coordinates organ polarity.

The formation of a flat and thin leaf presents a developmentally challenging problem, requiring intricate regulation of adaxial-abaxial (top-bottom) polarity. The patterning principles controlling the spatial arrangement of these domains during organ growth have remained unclear. Here we show that this regulation in Arabidopsis thaliana is achieved by an organ-autonomous Turing reaction-diffusion system centred on mobile small RNAs. The data illustrate how Turing dynamics transiently instructed by prepatterned information is sufficient to self-sustain properly oriented polarity in a dynamic, growing organ, presenting intriguing parallels to left-right patterning in the vertebrate embryo. Computational modelling demonstrates that this self-organizing system continuously adapts to coordinate the robust planar polarity of a flat leaf while affording flexibility to generate the tissue patterns of evolutionarily diverse organ shapes. Our findings identify a small-RNA-based Turing network as a dynamic regulator of organ polarity that accounts for leaf shape diversity at the level of the individual organ, plant or species.

Minimal reaction schemes for pattern formation.

We link continuum models of reaction-diffusion systems that exhibit diffusion-driven instability to constraints on the particle-scale interactions underpinning this instability. While innumerable biological, chemical and physical patterns have been studied through the lens of Alan Turing's reaction-diffusion pattern-forming mechanism, the connections between models of pattern formation and the nature of the particle interactions generating them have been relatively understudied in comparison with the substantial efforts that have been focused on understanding proposed continuum systems. To derive the necessary reactant combinations for the most parsimonious reaction schemes, we analyse the emergent continuum models in terms of possible generating elementary reaction schemes. This analysis results in the complete list of such schemes containing the fewest reactions; these are the simplest possible hypothetical mass-action models for a pattern-forming system of two interacting species.

Turing Instabilities are Not Enough to Ensure Pattern Formation.

Symmetry-breaking instabilities play an important role in understanding the mechanisms underlying the diversity of patterns observed in nature, such as in Turing's reaction-diffusion theory, which connects cellular signalling and transport with the development of growth and form. Extensive literature focuses on the linear stability analysis of homogeneous equilibria in these systems, culminating in a set of conditions for transport-driven instabilities that are commonly presumed to initiate self-organisation. We demonstrate that a selection of simple, canonical transport models with only mild multistable non-linearities can satisfy the Turing instability conditions while also robustly exhibiting only transient patterns. Hence, a Turing-like instability is insufficient for the existence of a patterned state. While it is known that linear theory can fail to predict the formation of patterns, we demonstrate that such failures can appear robustly in systems with multiple stable homogeneous equilibria. Given that biological systems such as gene regulatory networks and spatially distributed ecosystems often exhibit a high degree of multistability and nonlinearity, this raises important questions of how to analyse prospective mechanisms for self-organisation.

Novel Aspects in Pattern Formation Arise from Coupling Turing Reaction–Diffusion and Chemotaxis

Recent experimental studies on primary hair follicle formation and feather bud morphogenesis indicate a coupling between Turing-type diffusion driven instability and chemotactic patterning. Inspired by these findings we develop and analyse a mathematical model that couples chemotaxis to a reaction–diffusion system exhibiting diffusion–driven (Turing) instability. While both systems, reaction–diffusion systems and chemotaxis, can independently generate spatial patterns, we were interested in how the coupling impacts the stability of the system, parameter region for patterning, pattern geometry, as well as the dynamics of pattern formation. We conduct a classical linear stability analysis for different model structures, and confirm our results by numerical analysis of the system. Our results show that the coupling generally increases the robustness of the patterning process by enlarging the pattern region in the parameter space. Concerning time scale and pattern regularity, we find that an increase in the chemosensitivity can speed up the patterning process for parameters inside and outside of the Turing space, but generally reduces spatial regularity of the pattern. Interestingly, our analysis indicates that pattern formation can also occur when neither the Turing nor the chemotaxis system can independently generate pattern. On the other hand, for some parameter settings, the coupling of the two processes can extinguish the pattern formation, rather than reinforce it. These theoretical findings can be used to corroborate the biological findings on morphogenesis and guide future experimental studies. From a mathematical point of view, this work sheds a light on coupling classical pattern formation systems from the parameter space perspective.

A simple Turing reaction-diffusion model explains how PLK4 breaks symmetry during centriole duplication and assembly.

Centrioles duplicate when a mother centriole gives birth to a daughter that grows from its side. Polo-like-kinase 4 (PLK4), the master regulator of centriole duplication, is recruited symmetrically around the mother centriole, but it then concentrates at a single focus that defines the daughter centriole assembly site. How PLK4 breaks symmetry is unclear. Here, we propose that phosphorylated and unphosphorylated species of PLK4 form the 2 components of a classical Turing reaction-diffusion system. These 2 components bind to/unbind from the surface of the mother centriole at different rates, allowing a slow-diffusing activator species of PLK4 to accumulate at a single site on the mother, while a fast-diffusing inhibitor species of PLK4 suppresses activator accumulation around the rest of the centriole. This "short-range activation/long-range inhibition," inherent to Turing systems, can drive PLK4 symmetry breaking on a either a continuous or compartmentalised Plk4-binding surface, with PLK4 overexpression producing multiple PLK4 foci and PLK4 kinase inhibition leading to a lack of symmetry-breaking and PLK4 accumulation-as observed experimentally.

Patterning of nonlocal transport models in biology: The impact of spatial dimension

Throughout developmental biology and ecology, transport can be driven by nonlocal interactions. Examples include cells that migrate based on contact with pseudopodia extended from other cells, and animals that move based on their awareness of other animals. Nonlocal integro-PDE models have been used to investigate contact attraction and repulsion in cell populations in 1D. In this paper, we generalise the analysis of pattern formation in such a model from 1D to higher spatial dimensions. Numerical simulations in 2D demonstrate complex behaviour in the model, including spatio-temporal patterns, multi-stability, and patterns with wavelength and shape that differ significantly depending on whether interactions are attractive or repulsive. Through linear stability analysis in N dimensions, we demonstrate how, unlike in local Turing reaction–diffusion models, the capacity for pattern formation fundamentally changes with dimensionality for this nonlocal model. Most notably, pattern formation is possible only in higher than one spatial dimension for both the single species system with repulsive interactions, and the two species system with ‘run-and-chase’ interactions. The latter case may be relevant to zebrafish stripe formation, which has been shown to be driven by run-and-chase dynamics between melanophore and xanthophore pigment cells.

The developmental basis of fingerprint pattern formation and variation.

Fingerprints are complex and individually unique patterns in the skin. Established prenatally, the molecular and cellular mechanisms that guide fingerprint ridge formation and their intricate arrangements are unknown. Here we show that fingerprint ridges are epithelial structures that undergo a truncated hair follicle developmental program and fail to recruit a mesenchymal condensate. Their spatial pattern is established by a Turing reaction-diffusion system, based on signaling between EDAR, WNT, and antagonistic BMP pathways. These signals resolve epithelial growth into bands of focalized proliferation under a precociously differentiated suprabasal layer. Ridge formation occurs as a set of waves spreading from variable initiation sites defined by the local signaling environments and anatomical intricacies of the digit, with the propagation and meeting of these waves determining the type of pattern that forms. Relying on a dynamic patterning system triggered at spatially distinct sites generates the characteristic types and unending variation of human fingerprint patterns.

Secondary cell wall patterning-connecting the dots, pits and helices.

All plant cells are encased in primary cell walls that determine plant morphology, but also protect the cells against the environment. Certain cells also produce a secondary wall that supports mechanically demanding processes, such as maintaining plant body stature and water transport inside plants. Both these walls are primarily composed of polysaccharides that are arranged in certain patterns to support cell functions. A key requisite for patterned cell walls is the arrangement of cortical microtubules that may direct the delivery of wall polymers and/or cell wall producing enzymes to certain plasma membrane locations. Microtubules also steer the synthesis of cellulose-the load-bearing structure in cell walls-at the plasma membrane. The organization and behaviour of the microtubule array are thus of fundamental importance to cell wall patterns. These aspects are controlled by the coordinated effort of small GTPases that probably coordinate a Turing's reaction-diffusion mechanism to drive microtubule patterns. Here, we give an overview on how wall patterns form in the water-transporting xylem vessels of plants. We discuss systems that have been used to dissect mechanisms that underpin the xylem wall patterns, emphasizing the VND6 and VND7 inducible systems, and outline challenges that lay ahead in this field.

The feather pattern autosomal barring in chicken is strongly associated with segregation at the MC1R locus.

Color patterns within individual feathers are common in birds but little is known about the genetic mechanisms causing such patterns. Here, we investigate the genetic basis for autosomal barring in chicken, a horizontal striping pattern on individual feathers. Using an informative backcross, we demonstrate that the MC1R locus is strongly associated with this phenotype. A deletion at SOX10, underlying the dark brown phenotype on its own, affects the manifestation of the barring pattern. The coding variant L133Q in MC1R is the most likely causal mutation for autosomal barring in this pedigree. Furthermore, a genetic screen across six different breeds showing different patterning phenotypes revealed that the most striking shared characteristics among these breeds were that they all carried the MC1R alleles Birchen or brown. Our data suggest that the presence of activating MC1R mutations enhancing pigment synthesis is an important mechanism underlying pigmentation patterns on individual feathers in chicken. We propose that MC1R and its antagonist ASIP play a critical role for determining within-feather pigmentation patterns in birds by acting as activator and inhibitor possibly in a Turing reaction-diffusion model.

Modeling of reaction-diffusion processes of synthesis of materials with regular (periodic) microstructure

The paper represents results of modeling of reaction-diffusion processes that ensure the formation of materials with a regular (periodic) interconnected microstructure. A numerical method is proposed for solving the reaction-diffusion system of Turing equations. Using the Gray-Scott model as an example, it is shown that under certain conditions a system is formed with the geometry of triply periodic minimal surface (TPMS). It is shown that Turing reaction-diffusion processes can become the basis of a new technology of materials with an adjustable (periodic) microstructure.

New Chemical Technologies Based on Turing Reaction–Diffusion Processes

Diamond–SiC composite materials were obtained by impregnation with gaseous and liquid Si. The mechanisms of the impregnation and consolidation of the diamond–silicon carbide composites were studied. The effect of the Turing reaction–diffusion mechanism on the sintering of the diamond–SiC materials, the formation of the microstructure, and the mechanical properties was shown for the first time.

Coupling Turing stripes to active flows.

We numerically solve the active nematohydrodynamic equations of motion, coupled to a Turing reaction-diffusion model, to study the effect of active nematic flow on the stripe patterns resulting from a Turing instability. If the activity is uniform across the system, the Turing patterns dissociate when the flux from active advection balances that from the reaction-diffusion process. If the activity is coupled to the concentration of Turing morphogens, and neighbouring stripes have equal and opposite activity, the system self organises into a pattern of shearing flows, with stripes tending to fracture and slip sideways to join their neighbours. We discuss the role of active instabilities in controlling the crossover between these limits. Our results are of relevance to mechanochemical coupling in biological systems.

Neuroepithelial Organoid Patterning is Mediated by Wnt-Driven Turing Mechanism

Cell patterning in epithelia is critical for the establishment of tissue function during development. The organization of patterns in these tissues is mediated by the interpretation of signals operating across multiple length scales. How epithelial tissues coordinate changes in cell identity across these length scales to orchestrate cellular rearrangements and fate specification remains poorly understood. Here, we use human neural tube organoids as model systems to interrogate epithelial patterning principles that guide domain specification. In silico modeling of the patterning process by cellular automata, validated by in vitro experiments, reveal that the initial positions of floor plate cells, coupled with activator-inhibitor signaling interactions, deterministically dictate the patterning outcome according to a discretized Turing reaction-diffusion mechanism. This model predicts an enhancement of organoid patterning by modulating inhibitor levels. Receptor-ligand interaction analysis of scRNAseq data from multiple organoid domains reveals WNT-pathway ligands as the specific inhibitory agents, thereby allowing for the experimental validation of model predictions. These results demonstrate that neuroepithelia employ reaction-diffusion-based mechanisms during early embryonic human development to organize cellular identities and morphogen sources to achieve patterning. The wider implementation of such in vitro organoid models in combination with in-silico agent-based modeling coupled to receptor-ligand analysis of scRNAseq data opens avenues for a broader understanding of dynamic tissue patterning processes.

French flag gradients and Turing reaction-diffusion versus differentiation waves as models of morphogenesis

The Turing reaction-diffusion model and the French Flag Model are widely accepted in the field of development as the best models for explaining embryogenesis. Virtually all current attempts to understand cell differentiation in embryos begin and end with the assumption that some combination of these two models works. The result may become a bias in embryogenesis in assuming the problem has been solved by these two-chemical substance-based models. Neither model is applied consistently. We review the differences between the French Flag, Turing reaction-diffusion model, and a mechanochemical model called the differentiation wave/cell state splitter model. The cytoskeletal cell state splitter and the embryonic differentiation waves was first proposed in 1987 as a combined physics and chemistry model for cell differentiation in embryos, based on empirical observations on urodele amphibian embryos. We hope that the development of theory can be advanced and observations relevant to distinguishing the embryonic differentiation wave model from the French Flag model and reaction-diffusion equations will be taken up by experimentalists. Experimentalists rely on mathematical biologists for theory, and therefore depend on them for what parameters they choose to measure and ignore. Therefore, mathematical biologists need to fully understand the distinctions between these three models.

Temporal Modulations of NODAL, BMP, and WNT Signals Guide the Spatial Patterning in Self-Organized Human Ectoderm Tissues

Summary Developmental biology studies using model organisms suggest that the emergence of spatial patterning in the ectoderm is mediated by the morphogen gradients. However, it is still unclear whether the morphogen gradient is necessary and dominates the cell spatial patterning, particularly in human genetic background. Here, we demonstrate that human pluripotent stem cells can self-organize to concentric rings of all major cell types in ectoderm when cultured on micropatterned surfaces in a chemically defined condition. We reveal that modulating the dynamics of NODAL, BMP, and WNT signals is sufficient to control the spatial order of different cell types. Our mathematical model suggests that changes in wavelength and phase of signaling patterns formed via reaction-diffusion may be the mechanism by which temporal information is translated into spatial information. Together, our work demonstrates that in vitro human ectoderm microtissues have great potential in understanding the mechanisms of early-stage human development.

Perturbation analysis of a multi-morphogen Turing reaction-diffusion stripe patterning system reveals key regulatory interactions.

ABSTRACTPeriodic patterning is widespread in development and can be modelled by reaction-diffusion (RD) processes. However, minimal two-component RD descriptions are vastly simpler than the multi-molecular events that actually occur and are often hard to relate to real interactions measured experimentally. Addressing these issues, we investigated the periodic striped patterning of the rugae (transverse ridges) in the mammalian oral palate, focusing on multiple previously implicated pathways: FGF, Hh, Wnt and BMP. For each, we experimentally identified spatial patterns of activity and distinct responses of the system to inhibition. Through numerical and analytical approaches, we were able to constrain substantially the number of network structures consistent with the data. Determination of the dynamics of pattern appearance further revealed its initiation by ‘activators’ FGF and Wnt, and ‘inhibitor’ Hh, whereas BMP and mesenchyme-specific-FGF signalling were incorporated once stripes were formed. This further limited the number of possible networks. Experimental constraint thus limited the number of possible minimal networks to 154, just 0.004% of the number of possible diffusion-driven instability networks. Together, these studies articulate the principles of multi-morphogen RD patterning and demonstrate the utility of perturbation analysis for constraining RD systems.This article has an associated ‘The people behind the papers’ interview.

Exploring Mechanisms of Cranial Vault Development using a Coupled Turing‐Biomechanical Model

Bones of the mouse cranial vault are formed by differentiation of mesenchymal cells into osteoblasts, a process called intramembranous ossification that is primarily thought to be controlled by a cascade of reactions between extracellular molecules and cells that may be significantly guided by mechanical forces. In this study, we use a computational approach and an extensive experimental database to elucidate the role of biomechanics in morphogenesis of the cranial vault. We assume that the biochemical process can be modeled using Turing's reaction-diffusion model, a mathematical model describing the pattern formation controlled by interacting molecules (activators and inhibitors). In addition, we hypothesize that mechanical stimuli of the cells due to growth of the underlying brain contributes significantly to the process of cell differentiation. We have developed a coupled framework that incorporates these modeling constructs. The mathematical formulation was solved using the finite volume method. The computational domain and model parameters are determined using a large collection of experimental data that provides precise three-dimensional measures of mouse cranial geometry and cranial vault bone formation for specific embryonic time points. The results of this study suggest that mechanical strain contributes information to specific aspects of bone formation. Our mechanobiological model predicts some key features of cranial vault bone formation that were verified by experimental observations in normal case, including the relative location of ossification centers of the individual vault bones, the pattern of cranial vault bone growth over time, and the position of cranial vault sutures. Support or Funding Information Computations for this research were performed on the Pennsylvania State University's Institute for Cyber Science Advanced Cyber Infrastructure (ICS-ACI). This work was supported in part through instrumentation funded by a National Science Foundation grant OCI0821527, a Burroughs-Wellcome Fund 2013 Collaborative Research Travel Grant, Pennsylvania Department of Health using Tobacco Cure Funds, and by the National Institutes of Health grants R01DE022988 and P01HD078233. The model predicts the pattern of cranial vault bone growth. a) Computational prediction of the distribution of differentiating osteoblasts and cranial vault bone formation by embryonic day. b) Observed cranial vault bone formation and growth in embryonic mice. This abstract is from the Experimental Biology 2019 Meeting. There is no full text article associated with this abstract published in The FASEB Journal.

An ancient Turing-like patterning mechanism regulates skin denticle development in sharks.

Vertebrates have a vast array of epithelial appendages, including scales, feathers, and hair. The developmental patterning of these diverse structures can be theoretically explained by Alan Turing's reaction-diffusion system. However, the role of this system in epithelial appendage patterning of early diverging lineages (compared to tetrapods), such as the cartilaginous fishes, is poorly understood. We investigate patterning of the unique tooth-like skin denticles of sharks, which closely relates to their hydrodynamic and protective functions. We demonstrate through simulation models that a Turing-like mechanism can explain shark denticle patterning and verify this system using gene expression analysis and gene pathway inhibition experiments. This mechanism bears remarkable similarity to avian feather patterning, suggesting deep homology of the system. We propose that a diverse range of vertebrate appendages, from shark denticles to avian feathers and mammalian hair, use this ancient and conserved system, with slight genetic modulation accounting for broad variations in patterning.

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.

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.