Turing's model for biological pattern formation and the robustness problem

Optimal network sizes for most robust Turing patterns

Many cellular patterns exhibit a reaction-diffusion component, suggesting that Turing instability may contribute to pattern formation. However, biological gene-regulatory pathways are more complex than simple Turing activator-inhibitor models and generally do not require fine-tuning of parameters as dictated by the Turing conditions. To address these issues, we employ random matrix theory to analyze the Jacobian matrices of larger networks with robust statistical properties. Our analysis reveals that Turing patterns are more likely to occur by chance than previously thought and that the most robust Turing networks have an optimal size, consisting of only a handful of molecular species, thus significantly increasing their identifiability in biological systems. Broadly speaking, this optimal size emerges from a trade-off between the highest stability in small networks and the greatest instability with diffusion in large networks. Furthermore, we find that with multiple immobile nodes, differential diffusion ceases to be important for Turing patterns. Our findings may inform future synthetic biology approaches and provide insights into bridging the gap to complex developmental pathways.

Trafficking in cancer: from gene deregulation to altered organelles and emerging biophysical properties

Intracellular trafficking supports all cell functions maintaining the exchange of material between membrane-bound organelles and the plasma membrane during endocytosis, cargo sorting, and exocytosis/secretion. Several proteins of the intracellular trafficking machinery are deregulated in diseases, particularly cancer. This complex and deadly disease stays a heavy burden for society, despite years of intense research activity. Here, we give an overview about trafficking proteins and highlight that in addition to their molecular functions, they contribute to the emergence of intracellular organelle landscapes. We review recent evidence of organelle landscape alterations in cancer. We argue that focusing on organelles, which represent the higher-order, cumulative behavior of trafficking regulators, could help to better understand, describe and fight cancer. In particular, we propose adopting a physical framework to describe the organelle landscape, with the goal of identifying the key parameters that are crucial for a stable and non-random organelle organization characteristic of healthy cells. By understanding these parameters, we may gain insights into the mechanisms that lead to a pathological organelle spatial organization, which could help explain the plasticity of cancer cells.

Linear coupling of patterning systems can have nonlinear effects

Isolated patterning systems have been repeatedly investigated. However, biological systems rarely work on their own. This paper presents a theoretical and quantitative analysis of a two-domain interconnected geometry, or bilayer, coupling two two-species reaction-diffusion systems mimicking interlayer communication, such as in mammary organoids. Each layer has identical kinetics and parameters, but differing diffusion coefficients. Critically, we show that despite a linear coupling between the layers, the model demonstrates nonlinear behavior; the coupling can lead to pattern suppression or pattern enhancement. Using the Routh-Hurwitz stability criterion multiple times, we investigate the pattern-forming capabilities of the uncoupled system, the weakly coupled system, and the strongly coupled system, using numerical simulations to back up the analysis. We show that although the dispersion relation of the entire system is a nontrivial octic polynomial, the patterning wave modes in the strongly coupled case can be approximated by a quartic polynomial, whose features are easier to understand.

Structure of random Turing-like patterns in discrete-time systems is determined by the initial conditions

Patterns, spatiotemporal ordered structures, are prevalent in diverse systems, arising from the emergence of complexity. Turing proposed a mechanism that involves a short-range activator and a long-range inhibitor to explain the formation of patterns, and patterns that satisfy this mechanism are called Turing patterns. Patterns with similar structures but not caused by the Turing mechanism are referred to as Turing-like patterns. In the absence of external influences, the structure of Turing patterns is generally determined by control parameters. In this study, we revealed that the structure of Turing-like patterns in discrete-time systems is only determined by the ratio of states in the initial conditions. As the ratio changes, the structure of patterns transitions from spots to labyrinth and eventually to inverse spots. We proposed the structure parameter for the quantitative description of the structure of the patterns. And the structure parameter is directly proportional to the ratio in the initial conditions. The mechanism underlying this structure control is attributed to the traversability of multiperiodic states in discrete-time systems, where each local point will go through all states in the periodic orbit. Our findings shed light on the pattern formation for Turing-like patterns in discrete-time systems.

A three-node Turing gene circuit forms periodic spatial patterns in bacteria

Turing patterns are self-organizing systems that can form spots, stripes, or labyrinths. Proposed examples in tissue organization include zebrafish pigmentation, digit spacing, and many others. The theory of Turing patterns in biology has been debated because of their stringent fine-tuning requirements, where patterns only occur within a small subset of parameters. This has complicated the engineering of synthetic Turing gene circuits from first principles, although natural genetic Turing networks have been identified. Here, we engineered a synthetic genetic reaction-diffusion system where three nodes interact according to a non-classical Turing network with improved parametric robustness. The system reproducibly generated stationary, periodic, concentric stripe patterns in growing E. coli colonies. A partial differential equation model reproduced the patterns, with a Turing parameter regime obtained by fitting to experimental data. Our synthetic Turing system can contribute to nanotechnologies, such as patterned biomaterial deposition, and provide insights into developmental patterning programs. A record of this paper's transparent peer review process is included in the supplemental information.

Versatile system cores as a conceptual basis for generality in cell and developmental biology

The discovery of general principles underlying the complexity and diversity of cellular and developmental systems is a central and long-standing aim of biology. While new technologies collect data at an ever-accelerating rate, there is growing concern that conceptual progress is not keeping pace. We contend that this is due to a paucity of conceptual frameworks that support meaningful generalizations. This led us to develop the core and periphery (C&P) hypothesis, which posits that many biological systems can be decomposed into a highly versatile core with a large behavioral repertoire and a specific periphery that configures said core to perform one particular function. Versatile cores tend to be widely reused across biology, which confers generality to theories describing them. Here, we introduce this concept and describe examples at multiple scales, including Turing patterning, actomyosin dynamics, multi-cellular morphogenesis, and vertebrate gastrulation. We also sketch its evolutionary basis and discuss key implications and open questions. We propose that the C&P hypothesis could unlock new avenues of conceptual progress in mesoscale biology.

Unraveling biochemical spatial patterns: Machine learning approaches to the inverse problem of stationary Turing patterns

The diffusion-driven Turing instability is a potential mechanism for spatial pattern formation in numerous biological and chemical systems. However, engineering these patterns and demonstrating that they are produced by this mechanism is challenging. To address this, we aim to solve the inverse problem in artificial and experimental Turing patterns. This task is challenging since patterns are often corrupted by noise and slight changes in initial conditions can lead to different patterns. We used both least squares to explore the problem and physics-informed neural networks to build a noise-robust method. We elucidate the functionality of our network in scenarios mimicking biological noise levels and showcase its application using an experimentally obtained chemical pattern. The findings reveal the significant promise of machine learning in steering the creation of synthetic patterns in bioengineering, thereby advancing our grasp of morphological intricacies within biological systems while acknowledging existing limitations.

Parsing patterns: Emerging roles of tissue self-organization in health and disease

Patterned morphologies, such as segments, spirals, stripes, and spots, frequently emerge during embryogenesis through self-organized coordination between cells. Yet, complex patterns also emerge in adults, suggesting that the capacity for spontaneous self-organization is a ubiquitous property of biological tissues. We review current knowledge on the principles and mechanisms of self-organized patterning in embryonic tissues and explore how these principles and mechanisms apply to adult tissues that exhibit features of patterning. We discuss how and why spontaneous pattern generation is integral to homeostasis and healing of tissues, illustrating it with examples from regenerative biology. We examine how aberrant self-organization underlies diverse pathological states, including inflammatory skin disorders and tumors. Lastly, we posit that based on such blueprints, targeted engineering of pattern-driving molecular circuits can be leveraged for synthetic biology and the generation of organoids with intricate patterns.

Robustness of Turing models and gene regulatory networks with a sweet spot

Traditional linear stability analysis based on matrix diagonalization is a computationally intensive process for high-dimensional systems of differential equations, posing substantial limitations for the exploration of Turing systems of pattern formation where an additional wave-number parameter needs to be investigated. In this paper, we introduce an efficient and intuitive technique that leverages Gershgorin's theorem to determine upper limits on regions of parameter space and the wave number beyond which Turing instabilities cannot occur. This method offers a streamlined avenue for exploring the phase diagrams of other complex multi-parametric models, such as those found in gene regulatory networks in systems biology. Due to its suitability for the asymptotic limit of infinitely large systems, it predicts the existence of a sweet spot in network size for maximal Jacobian stability.

Gap junctions in Turing-type periodic feather pattern formation

Periodic patterning requires coordinated cell-cell interactions at the tissue level. Turing showed, using mathematical modeling, how spatial patterns could arise from the reactions of a diffusive activator-inhibitor pair in an initially homogeneous 2D field. Most activators and inhibitors studied in biological systems are proteins, and the roles of cell-cell interaction, ions, bioelectricity, etc. are only now being identified. Gap junctions (GJs) mediate direct exchanges of ions or small molecules between cells, enabling rapid long-distance communications in a cell collective. They are therefore good candidates for propagating nonprotein-based patterning signals that may act according to the Turing principles. Here, we explore the possible roles of GJs in Turing-type patterning using feather pattern formation as a model. We found 7 of the 12 investigated GJ isoforms are highly dynamically expressed in the developing chicken skin. In ovo functional perturbations of the GJ isoform, connexin 30, by siRNA and the dominant-negative mutant applied before placode development led to disrupted primary feather bud formation. Interestingly, inhibition of gap junctional intercellular communication (GJIC) in the ex vivo skin explant culture allowed the sequential emergence of new feather buds at specific spatial locations relative to the existing primary buds. The results suggest that GJIC may facilitate the propagation of long-distance inhibitory signals. Thus, inhibition of GJs may stimulate Turing-type periodic feather pattern formation during chick skin development, and the removal of GJ activity would enable the emergence of new feather buds if the local environment were competent and the threshold to form buds was reached. We further propose Turing-based computational simulations that can predict the sequential appearance of these ectopic buds. Our models demonstrate how a Turing activator-inhibitor system can continue to generate patterns in the competent morphogenetic field when the level of intercellular communication at the tissue scale is modulated.

Developmental plasticity and variability in the formation of egg-spots, a pigmentation ornament in the cichlid Astatotilapia calliptera

Vertebrate pigmentation patterns are highly diverse, yet we have a limited understanding of how evolutionary changes to genetic, cellular, and developmental mechanisms generate variation. To address this, we examine the formation of a sexually-selected male ornament exhibiting inter- and intraspecific variation, the egg-spot pattern, consisting of circular yellow-orange markings on the male anal fins of haplochromine cichlid fishes. We focus on Astatotilapia calliptera, the ancestor-type species of the Malawi cichlid adaptive radiation of over 850 species. We identify a key role for iridophores in initializing egg-spot aggregations composed of iridophore-xanthophore associations. Despite adult sexual dimorphism, aggregations initially form in both males and females, with development only diverging between the sexes at later stages. Unexpectedly, we found that the timing of egg-spot initialization is plastic. The earlier individuals are socially isolated, the earlier the aggregations form, with iridophores being the cell type that responds to changes to the social environment. Furthermore, we observe apparent competitive interactions between adjacent egg-spot aggregations, which strongly suggests that egg-spot patterning results mostly from cell-autonomous cellular interactions. Together, these results demonstrate that A. calliptera egg-spot development is an exciting model for investigating pigment pattern formation at the cellular level in a system with developmental plasticity, sexual dimorphism, and intraspecific variation. As A. calliptera represents the ancestral bauplan for egg-spots, these findings provide a baseline for informed comparisons across the incredibly diverse Malawi cichlid radiation.

Mechanical guidance to self-organization and pattern formation of stem cells

The self-organization of stem cells (SCs) constitutes the fundamental basis of the development of biological organs and structures. SC-driven patterns are essential for tissue engineering, yet unguided SCs tend to form chaotic patterns, impeding progress in biomedical engineering. Here, we show that simple geometric constraints can be used as an effective mechanical modulation approach that promotes the development of controlled self-organization and pattern formation of SCs. Using the applied SC guidance with geometric constraints, we experimentally uncover a remarkable deviation in cell aggregate orientation from a random direction to a specific orientation. Subsequently, we propose a dynamic mechanical framework, including cells, the extracellular matrix (ECM), and the culture environment, to characterize the specific orientation deflection of guided cell aggregates relative to initial geometric constraints, which agrees well with experimental observation. Based on this framework, we further devise various theoretical strategies to realize complex biological patterns, such as radial and concentric structures. Our study highlights the key role of mechanical factors and geometric constraints in governing SCs' self-organization. These findings yield critical insights into the regulation of SC-driven pattern formation and hold great promise for advancements in tissue engineering and bioactive material design for regenerative application.

Diffusiophoresis-enhanced Turing patterns

Turing patterns are fundamental in biophysics, emerging from short-range activation and long-range inhibition processes. However, their paradigm is based on diffusive transport processes that yield patterns with shallower gradients than those observed in nature. A complete physical description of this discrepancy remains unknown. We propose a solution to this phenomenon by investigating the role of diffusiophoresis, which is the propulsion of colloids by a chemical gradient, in Turing patterns. Diffusiophoresis enables robust patterning of colloidal particles with substantially finer length scales than the accompanying chemical Turing patterns. A scaling analysis and a comparison to recent experiments indicate that chromatophores, ubiquitous in biological pattern formation, are likely diffusiophoretic and the colloidal Péclet number controls the pattern enhancement. This discovery suggests that important features of biological pattern formation can be explained with a universal mechanism that is quantified straightforwardly from the fundamental physics of colloids.

Development shapes the evolutionary diversification of rodent stripe patterns

Vertebrate groups have evolved strikingly diverse color patterns. However, it remains unknown to what extent the diversification of such patterns has been shaped by the proximate, developmental mechanisms that regulate their formation. While these developmental mechanisms have long been inaccessible empirically, here we take advantage of recent insights into rodent pattern formation to investigate the role of development in shaping pattern diversification across rodents. Based on a broad survey of museum specimens, we first establish that various rodents have independently evolved diverse patterns consisting of longitudinal stripes, varying across species in number, color, and relative positioning. We then interrogate this diversity using a simple model that incorporates recent molecular and developmental insights into stripe formation in African striped mice. Our results suggest that, on the one hand, development has facilitated pattern diversification: The diversity of patterns seen across species can be generated by a single developmental process, and small changes in this process suffice to recapitulate observed evolutionary changes in pattern organization. On the other hand, development has constrained diversification: Constraints on stripe positioning limit the scope of evolvable patterns, and although pattern organization appears at first glance phylogenetically unconstrained, development turns out to impose a cryptic constraint. Altogether, this work reveals that pattern diversification in rodents can in part be explained by the underlying development and illustrates how pattern formation models can be leveraged to interpret pattern evolution.

Stochastic Turing patterns of trichomes in Arabidopsis leaves

Biological patterns that emerge during the morphogenesis of multicellular organisms can display high precision at large scales, while at cellular scales, cells exhibit large fluctuations stemming from cell-cell differences in molecular copy numbers also called demographic noise. We study the conflicting interplay between high precision and demographic noise in trichome patterns on the epidermis of wild-type Arabidopsis thaliana leaves, as a two-dimensional model system. We carry out a statistical characterization of these patterns and show that their power spectra display fat tails-a signature compatible with noise-driven stochastic Turing patterns-which are absent in power spectra of patterns driven by deterministic instabilities. We then present a theoretical model that includes demographic noise stemming from birth-death processes of genetic regulators which we study analytically and by stochastic simulations. The model captures the observed experimental features of trichome patterns.

Principles and challenges of modeling temporal and spatial omics data

Studies with temporal or spatial resolution are crucial to understand the molecular dynamics and spatial dependencies underlying a biological process or system. With advances in high-throughput omic technologies, time- and space-resolved molecular measurements at scale are increasingly accessible, providing new opportunities to study the role of timing or structure in a wide range of biological questions. At the same time, analyses of the data being generated in the context of spatiotemporal studies entail new challenges that need to be considered, including the need to account for temporal and spatial dependencies and compare them across different scales, biological samples or conditions. In this Review, we provide an overview of common principles and challenges in the analysis of temporal and spatial omics data. We discuss statistical concepts to model temporal and spatial dependencies and highlight opportunities for adapting existing analysis methods to data with temporal and spatial dimensions.

Mechano-induced homotypic patterned domain formation by monocytes

Matrix stiffness and corresponding mechano-signaling play indispensable roles in cellular phenotypes and functions. How tissue stiffness influences the behavior of monocytes, a major circulating leukocyte of the innate system, and how it may promote the emergence of collective cell behavior is less understood. Here, using tunable collagen-coated hydrogels of physiological stiffness, we show that human primary monocytes undergo a dynamic local phase separation to form highly regular, reversible, multicellular, multi-layered domains on soft matrix. Local activation of the β2 integrin initiates inter-cellular adhesion, while global soluble inhibitory factors maintain the steady state domain pattern over days. Patterned domain formation generated by monocytes is unique among other key immune cells, including macrophages, B cells, T cells, and NK cells. While inhibiting their phagocytic capability, domain formation promotes monocytes' survival. We develop a computational model based on the Cahn-Hilliard equation of phase separation, combined with a Turing mechanism of local activation and global inhibition suggested by our experiments, and provides experimentally validated predictions of the role of seeding density and both chemotactic and random cell migration on domain pattern formation. This work reveals that, unlike active matters, cells can generate complex cell phases by exploiting their mechanosensing abilities and combined short-range interactions and long-range signals to enhance their survival.

Mechano-induced homotypic patterned domain formation by monocytes

Matrix stiffness and corresponding mechano-signaling play indispensable roles in cellular phenotypes and functions. How tissue stiffness influences the behavior of monocytes, a major circulating leukocyte of the innate system, and how it may promote the emergence of collective cell behavior is less understood. Here, using tunable collagen-coated hydrogels of physiological stiffness, we show that human primary monocytes undergo a dynamic local phase separation to form highly patterned multicellular multi-layered domains on soft matrix. Local activation of the β2 integrin initiates inter-cellular adhesion, while global soluble inhibitory factors maintain the steady-state domain pattern over days. Patterned domain formation generated by monocytes is unique among other key immune cells, including macrophages, B cells, T cells, and NK cells. While inhibiting their phagocytic capability, domain formation promotes monocytes' survival. We develop a computational model based on the Cahn-Hilliard equation, which includes combined local activation and global inhibition mechanisms of intercellular adhesion suggested by our experiments, and provides experimentally validated predictions of the role of seeding density and both chemotactic and random cell migration on pattern formation.

Self-organization of primitive metabolic cycles due to non-reciprocal interactions

One of the greatest mysteries concerning the origin of life is how it has emerged so quickly after the formation of the earth. In particular, it is not understood how metabolic cycles, which power the non-equilibrium activity of cells, have come into existence in the first instances. While it is generally expected that non-equilibrium conditions would have been necessary for the formation of primitive metabolic structures, the focus has so far been on externally imposed non-equilibrium conditions, such as temperature or proton gradients. Here, we propose an alternative paradigm in which naturally occurring non-reciprocal interactions between catalysts that can partner together in a cyclic reaction lead to their recruitment into self-organized functional structures. We uncover different classes of self-organized cycles that form through exponentially rapid coarsening processes, depending on the parity of the cycle and the nature of the interaction motifs, which are all generic but have readily tuneable features.

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.

Physical interactions in non-ideal fluids promote Turing patterns

Turing's mechanism is often invoked to explain periodic patterns in nature, although direct experimental support is scarce. Turing patterns form in reaction-diffusion systems when the activating species diffuse much slower than the inhibiting species, and the involved reactions are highly nonlinear. Such reactions can originate from cooperativity, whose physical interactions should also affect diffusion. We here take direct interactions into account and show that they strongly affect Turing patterns. We find that weak repulsion between the activator and inhibitor can substantially lower the required differential diffusivity and reaction nonlinearity. By contrast, strong interactions can induce phase separation, but the resulting length scale is still typically governed by the fundamental reaction-diffusion length scale. Taken together, our theory connects traditional Turing patterns with chemically active phase separation, thus describing a wider range of systems. Moreover, we demonstrate that even weak interactions affect patterns substantially, so they should be incorporated when modelling realistic systems.

Self-organized collective cell behaviors as design principles for synthetic developmental biology

Over the past two decades, molecular cell biology has graduated from a mostly analytic science to one with substantial synthetic capability. This success is built on a deep understanding of the structure and function of biomolecules and molecular mechanisms. For synthetic biology to achieve similar success at the scale of tissues and organs, an equally deep understanding of the principles of development is required. Here, we review some of the central concepts and recent progress in tissue patterning, morphogenesis and collective cell migration and discuss their value for synthetic developmental biology, emphasizing in particular the power of (guided) self-organization and the role of theoretical advances in making developmental insights applicable in synthesis.

Gap junctions in Turing-type periodic feather pattern formation

Periodic patterning requires coordinated cell-cell interactions at the tissue level. Turing showed, using mathematical modeling, how spatial patterns could arise from the reactions of a diffusive activator-inhibitor pair in an initially homogenous two-dimensional field. Most activators and inhibitors studied in biological systems are proteins, and the roles of cell-cell interaction, ions, bioelectricity, etc. are only now being identified. Gap junctions (GJs) mediate direct exchanges of ions or small molecules between cells, enabling rapid long-distance communications in a cell collective. They are therefore good candidates for propagating non-protein-based patterning signals that may act according to the Turing principles. Here, we explore the possible roles of GJs in Turing-type patterning using feather pattern formation as a model. We found seven of the twelve investigated GJ isoforms are highly dynamically expressed in the developing chicken skin. In ovo functional perturbations of the GJ isoform, connexin 30, by siRNA and the dominant-negative mutant applied before placode development led to disrupted primary feather bud formation, including patches of smooth skin and buds of irregular sizes. Later, after the primary feather arrays were laid out, inhibition of gap junctional intercellular communication in the ex vivo skin explant culture allowed the emergence of new feather buds in temporal waves at specific spatial locations relative to the existing primary buds. The results suggest that gap junctional communication may facilitate the propagation of long-distance inhibitory signals. Thus, the removal of GJ activity would enable the emergence of new feather buds if the local environment is competent and the threshold to form buds is reached. We propose Turing-based computational simulations that can predict the appearance of these ectopic bud waves. Our models demonstrate how a Turing activator-inhibitor system can continue to generate patterns in the competent morphogenetic field when the level of intercellular communication at the tissue scale is modulated.

Self-Regulatory Micro- and Macroscale Patterning of ATP-Mediated Nanobioconjugate

Directional interactions and the assembly of a nanobioconjugate in clusters at a specific location are important for patterning and microarrays in biomedical research. Herein, we report that self-assembly and spatial control in surface patterning of the surfactant-functionalized nanoparticles can be governed in micro- and macroscale environments by two factors, synergistic enzyme-substrate-nanoparticle affinity and the phoretic effect. First, we show that aggregation of cationic gold nanoparticles (GNP) can be modulated by multivalent anionic nanoparticle binding of an adenosine-based nucleotide and enzyme, alkaline phosphatase. We further demonstrate two different types of their autonomous aggregation pattern: (i) by introducing an enzyme gradient that modulates the synergistic nonequilibrium interactivity of the nanoparticle, nucleotide, and enzyme both in microfluidic conditions and at the macroscale; and (ii) the surface deposition pattern from evaporating droplets via the coffee ring effect. Here, temporal control over the width and site of the patterning area inside the microfluidic channel under catalytic and noncatalytic conditions has also been demonstrated. Finally, we show a change in capillary phoresis parameters responsible for the coffee ring due to introduction of ATP-loaded GNP in the blood serum, showing applicability in low-cost disease diagnostics. Overall, an enzyme-actuated surface nanobiopatterning method has been demonstrated that has potential application in controlled micro- and macroscale area patterning with a diverse cascade catalytic surface and spatiotemporal multisensory-based application.

Universal dynamics of biological pattern formation in spatio-temporal morphogen variations

In biological systems, chemical signals termed morphogens self-organize into patterns that are vital for many physiological processes. As observed by Turing in 1952, these patterns are in a state of continual development, and are usually transitioning from one pattern into another. How do cells robustly decode these spatio-temporal patterns into signals in the presence of confounding effects caused by unpredictable or heterogeneous environments? Here, we answer this question by developing a general theory of pattern formation in spatio-temporal variations of ‘pre-pattern’ morphogens, which determine gene-regulatory network parameters. Through mathematical analysis, we identify universal dynamical regimes that apply to wide classes of biological systems. We apply our theory to two paradigmatic pattern-forming systems, and predict that they are robust with respect to non-physiological morphogen variations. More broadly, our theoretical framework provides a general approach to classify the emergent dynamics of pattern-forming systems based on how the bifurcations in their governing equations are traversed.

Concentration-Dependent Domain Evolution in Reaction-Diffusion Systems

Pattern formation has been extensively studied in the context of evolving (time-dependent) domains in recent years, with domain growth implicated in ameliorating problems of pattern robustness and selection, in addition to more realistic modelling in developmental biology. Most work to date has considered prescribed domains evolving as given functions of time, but not the scenario of concentration-dependent dynamics, which is also highly relevant in a developmental setting. Here, we study such concentration-dependent domain evolution for reaction-diffusion systems to elucidate fundamental aspects of these more complex models. We pose a general form of one-dimensional domain evolution and extend this to N-dimensional manifolds under mild constitutive assumptions in lieu of developing a full tissue-mechanical model. In the 1D case, we are able to extend linear stability analysis around homogeneous equilibria, though this is of limited utility in understanding complex pattern dynamics in fast growth regimes. We numerically demonstrate a variety of dynamical behaviours in 1D and 2D planar geometries, giving rise to several new phenomena, especially near regimes of critical bifurcation boundaries such as peak-splitting instabilities. For sufficiently fast growth and contraction, concentration-dependence can have an enormous impact on the nonlinear dynamics of the system both qualitatively and quantitatively. We highlight crucial differences between 1D evolution and higher-dimensional models, explaining obstructions for linear analysis and underscoring the importance of careful constitutive choices in defining domain evolution in higher dimensions. We raise important questions in the modelling and analysis of biological systems, in addition to numerous mathematical questions that appear tractable in the one-dimensional setting, but are vastly more difficult for higher-dimensional models.

A computational modeling approach for predicting multicell spheroid patterns based on signaling-induced differential adhesion

Physiological and pathological processes including embryogenesis and tumorigenesis rely on the ability of individual cells to work collectively to form multicell patterns. In these heterogeneous multicell systems, cell-cell signaling induces differential adhesion between cells that leads to tissue-level patterning. However, the sensitivity of pattern formation to changes in the strengths of signaling or cell adhesion processes is not well understood. Prior work has explored these issues using synthetically engineered heterogeneous multicell spheroid systems, in which cell subpopulations engage in bidirectional intercellular signaling to regulate the expression of different cadherins. While engineered cell systems provide excellent experimental tools to observe pattern formation in cell populations, computational models of these systems may be leveraged to explore more systematically how specific combinations of signaling and adhesion parameters can drive the emergence of unique patterns. We developed and validated two- and three-dimensional agent-based models (ABMs) of spheroid patterning for previously described cells engineered with a bidirectional signaling circuit that regulates N- and P-cadherin expression. Systematic exploration of model predictions, some of which were experimentally validated, revealed how cell seeding parameters, the order of signaling events, probabilities of induced cadherin expression, and homotypic adhesion strengths affect pattern formation. Unsupervised clustering was also used to map combinations of signaling and adhesion parameters to these unique spheroid patterns predicted by the ABM. Finally, we demonstrated how the model may be deployed to design new synthetic cell signaling circuits based on a desired final multicell pattern.

Distributed information encoding and decoding using self-organized spatial patterns

Dynamical systems often generate distinct outputs according to different initial conditions, and one can infer the corresponding input configuration given an output. This property captures the essence of information encoding and decoding. Here, we demonstrate the use of self-organized patterns that generate high-dimensional outputs, combined with machine learning, to achieve distributed information encoding and decoding. Our approach exploits a critical property of many natural pattern-formation systems: in repeated realizations, each initial configuration generates similar but not identical output patterns due to randomness in the patterning process. However, for sufficiently small randomness, different groups of patterns that arise from different initial configurations can be distinguished from one another. Modulating the pattern-generation and machine learning model training can tune the tradeoff between encoding capacity and security. We further show that this strategy is scalable by implementing the encoding and decoding of all characters of the standard English keyboard.

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.

Fixed and Distributed Gene Expression Time Delays in Reaction-Diffusion Systems

Time delays, modelling the process of intracellular gene expression, have been shown to have important impacts on the dynamics of pattern formation in reaction-diffusion systems. In particular, past work has shown that such time delays can shrink the Turing space, thereby inhibiting patterns from forming across large ranges of parameters. Such delays can also increase the time taken for pattern formation even when Turing instabilities occur. Here, we consider reaction-diffusion models incorporating fixed or distributed time delays, modelling the underlying stochastic nature of gene expression dynamics, and analyse these through a systematic linear instability analysis and numerical simulations for several sets of different reaction kinetics. We find that even complicated distribution kernels (skewed Gaussian probability density functions) have little impact on the reaction-diffusion dynamics compared to fixed delays with the same mean delay. We show that the location of the delay terms in the model can lead to changes in the size of the Turing space (increasing or decreasing) as the mean time delay, [Formula: see text], is increased. We show that the time to pattern formation from a perturbation of the homogeneous steady state scales linearly with [Formula: see text], and conjecture that this is a general impact of time delay on reaction-diffusion dynamics, independent of the form of the kinetics or location of the delayed terms. Finally, we show that while initial and boundary conditions can influence these dynamics, particularly the time-to-pattern, the effects of delay appear robust under variations of initial and boundary data. Overall, our results help clarify the role of gene expression time delays in reaction-diffusion patterning, and suggest clear directions for further work in studying more realistic models of pattern formation.

Morphogen-directed cell fate boundaries: slow passage through bifurcation and the role of folded saddles

One of the fundamental mechanisms in embryogenesis is the process by which cells differentiate and create tissues and structures important for functioning as a multicellular organism. Morphogenesis involves diffusive process of chemical signalling involving morphogens that pre-pattern the tissue. These morphogens influence cell fate through a highly nonlinear process of transcriptional signalling. In this paper, we consider this multiscale process in an idealised model for a growing domain. We focus on intracellular processes that lead to robust differentiation into two cell lineages through interaction of a single morphogen species with a cell fate variable that undergoes a bifurcation from monostability to bistability. In particular, we investigate conditions that result in successful and robust pattern formation into two well-separated domains, as well as conditions where this fails and produces a pinned boundary wave where only one part of the domain grows. We show that successful and unsuccessful patterning scenarios can be characterised in terms of presence or absence of a folded saddle singularity for a system with two slow variables and one fast variable; this models the interaction of slow morphogen diffusion, slow parameter drift through bifurcation and fast transcription dynamics. We illustrate how this approach can successfully model acquisition of three cell fates to produce three-domain "French flag" patterning, as well as for a more realistic model of the cell fate dynamics in terms of two mutually inhibiting transcription factors.

Enhancing Mechanical Metamodels with a Generative Model-Based Augmented Training Dataset

Modeling biological soft tissue is complex in part due to material heterogeneity. Microstructural patterns, which play a major role in defining the mechanical behavior of these tissues, are both challenging to characterize, and difficult to simulate. Recently, machine learning (ML)-based methods to predict the mechanical behavior of heterogeneous materials have made it possible to more thoroughly explore the massive input parameter space associated with heterogeneous blocks of material. Specifically, we can train ML models to closely approximate computationally expensive heterogeneous material simulations where the ML model is trained on datasets of simulations with relevant spatial heterogeneity. However, when it comes to applying these techniques to tissue, there is a major limitation: the number of useful examples available to characterize the input domain under study is often limited. In this work, we investigate the efficacy of both ML-based generative models and procedural methods as tools for augmenting limited input pattern datasets. We find that a Style-based Generative Adversarial Network with an adaptive discriminator augmentation mechanism is able to successfully leverage just 1,000 example patterns to create authentic generated patterns. And, we find that diverse generated patterns with adequate resemblance to real patterns can be used as inputs to finite element simulations to meaningfully augment the training dataset. To enable this methodological contribution, we have created an open access Finite Element Analysis simulation dataset based on Cahn-Hilliard patterns. We anticipate that future researchers will be able to leverage this dataset and build on the work presented here.

Synthetic spatial patterning in bacteria: advances based on novel diffusible signals

Engineering multicellular patterning may help in the understanding of some fundamental laws of pattern formation and thus may contribute to the field of developmental biology. Furthermore, advanced spatial control over gene expression may revolutionize fields such as medicine, through organoid or tissue engineering. To date, foundational advances in spatial synthetic biology have often been made in prokaryotes, using artificial gene circuits. In this review, engineered patterns are classified into four levels of increasing complexity, ranging from spatial systems with no diffusible signals to systems with complex multi-diffusor interactions. This classification highlights how the field was held back by a lack of diffusible components. Consequently, we provide a summary of both previously characterized and some new potential candidate small-molecule signals that can regulate gene expression in Escherichia coli. These diffusive signals will help synthetic biologists to successfully engineer increasingly intricate, robust and tuneable spatial structures.

Patterning, From Conifers to Consciousness: Turing's Theory and Order From Fluctuations

This is a brief account of Turing's ideas on biological pattern and the events that led to their wider acceptance by biologists as a valid way to investigate developmental pattern, and of the value of theory more generally in biology. Periodic patterns have played a key role in this process, especially 2D arrays of oriented stripes, which proved a disappointment in theoretical terms in the case of Drosophila segmentation, but a boost to theory as applied to skin patterns in fish and model chemical reactions. The concept of "order from fluctuations" is a key component of Turing's theory, wherein pattern arises by selective amplification of spatial components concealed in the random disorder of molecular and/or cellular processes. For biological examples, a crucial point from an analytical standpoint is knowing the nature of the fluctuations, where the amplifier resides, and the timescale over which selective amplification occurs. The answer clarifies the difference between "inelegant" examples such as Drosophila segmentation, which is perhaps better understood as a programmatic assembly process, and "elegant" ones expressible in equations like Turing's: that the fluctuations and selection process occur predominantly in evolutionary time for the former, but in real time for the latter, and likewise for error suppression, which for Drosophila is historical, in being lodged firmly in past evolutionary events. The prospects for a further extension of Turing's ideas to the complexities of brain development and consciousness is discussed, where a case can be made that it could well be in neuroscience that his ideas find their most important application.

Turing pattern design principles and their robustness

Turing patterns have morphed from mathematical curiosities into highly desirable targets for synthetic biology. For a long time, their biological significance was sometimes disputed but there is now ample evidence for their involvement in processes ranging from skin pigmentation to digit and limb formation. While their role in developmental biology is now firmly established, their synthetic design has so far proved challenging. Here, we review recent large-scale mathematical analyses that have attempted to narrow down potential design principles. We consider different aspects of robustness of these models and outline why this perspective will be helpful in the search for synthetic Turing-patterning systems. We conclude by considering robustness in the context of developmental modelling more generally. This article is part of the theme issue 'Recent progress and open frontiers in Turing's theory of morphogenesis'.

Modern perspectives on near-equilibrium analysis of Turing systems

In the nearly seven decades since the publication of Alan Turing's work on morphogenesis, enormous progress has been made in understanding both the mathematical and biological aspects of his proposed reaction-diffusion theory. Some of these developments were nascent in Turing's paper, and others have been due to new insights from modern mathematical techniques, advances in numerical simulations and extensive biological experiments. Despite such progress, there are still important gaps between theory and experiment, with many examples of biological patterning where the underlying mechanisms are still unclear. Here, we review modern developments in the mathematical theory pioneered by Turing, showing how his approach has been generalized to a range of settings beyond the classical two-species reaction-diffusion framework, including evolving and complex manifolds, systems heterogeneous in space and time, and more general reaction-transport equations. While substantial progress has been made in understanding these more complicated models, there are many remaining challenges that we highlight throughout. We focus on the mathematical theory, and in particular linear stability analysis of 'trivial' base states. We emphasize important open questions in developing this theory further, and discuss obstacles in using these techniques to understand biological reality. This article is part of the theme issue 'Recent progress and open frontiers in Turing's theory of morphogenesis'.

Insights from chemical systems into Turing-type morphogenesis

In 1952, Alan Turing proposed a theory showing how morphogenesis could occur from a simple two morphogen reaction-diffusion system [Turing, A. M. (1952) Phil. Trans. R. Soc. Lond. A 237, 37-72. (doi:10.1098/rstb.1952.0012)]. While the model is simple, it has found diverse applications in fields such as biology, ecology, behavioural science, mathematics and chemistry. Chemistry in particular has made significant contributions to the study of Turing-type morphogenesis, providing multiple reproducible experimental methods to both predict and study new behaviours and dynamics generated in reaction-diffusion systems. In this review, we highlight the historical role chemistry has played in the study of the Turing mechanism, summarize the numerous insights chemical systems have yielded into both the dynamics and the morphological behaviour of Turing patterns, and suggest future directions for chemical studies into Turing-type morphogenesis. This article is part of the theme issue 'Recent progress and open frontiers in Turing's theory of morphogenesis'.

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.

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.

On the bandwidth of stable nonlinear stripe patterns in finite size systems

Nonlinear stripe patterns occur in many different systems, from the small scales of biological cells to geological scales as cloud patterns. They all share the universal property of being stable at different wavenumbers q, i.e., they are multistable. The stable wavenumber range of the stripe patterns, which is limited by the Eckhaus- and zigzag instabilities even in finite systems for several boundary conditions, increases with decreasing system size. This enlargement comes about because suppressing degrees of freedom from the two instabilities goes along with the system reduction, and the enlargement depends on the boundary conditions, as we show analytically and numerically with the generic Swift-Hohenberg (SH) model and the universal Newell-Whitehead-Segel equation. We also describe how, in very small system sizes, any periodic pattern that emerges from the basic state is simultaneously stable in certain parameter ranges, which is especially important for the Turing pattern in cells. In addition, we explain why below a certain system width, stripe patterns behave quasi-one-dimensional in two-dimensional systems. Furthermore, we show with numerical simulations of the SH model in medium-sized rectangular domains how unstable stripe patterns evolve via the zigzag instability differently into stable patterns for different combinations of boundary conditions.

Spatial localisation meets biomolecular networks

Spatial organisation through localisation/compartmentalisation of species is a ubiquitous but poorly understood feature of cellular biomolecular networks. Current technologies in systems and synthetic biology (spatial proteomics, imaging, synthetic compartmentalisation) necessitate a systematic approach to elucidating the interplay of networks and spatial organisation. We develop a systems framework towards this end and focus on the effect of spatial localisation of network components revealing its multiple facets: (i) As a key distinct regulator of network behaviour, and an enabler of new network capabilities (ii) As a potent new regulator of pattern formation and self-organisation (iii) As an often hidden factor impacting inference of temporal networks from data (iv) As an engineering tool for rewiring networks and network/circuit design. These insights, transparently arising from the most basic considerations of networks and spatial organisation, have broad relevance in natural and engineered biology and in related areas such as cell-free systems, systems chemistry and bionanotechnology.

Complex biomolecular networks are fundamental to the functioning of living systems, both at the cellular level and beyond. In this paper, the authors develop a systems framework to elucidate the interplay of networks and the spatial localisation of network components.

Quantitative developmental biology in vitro using micropatterning

Micropatterning encompasses a set of methods aimed at precisely controlling the spatial distribution of molecules onto the surface of materials. Biologists have borrowed the idea and adapted these methods, originally developed for electronics, to impose physical constraints on biological systems with the aim of addressing fundamental questions across biological scales from molecules to multicellular systems. Here, I approach this topic from a developmental biologist's perspective focusing specifically on how and why micropatterning has gained in popularity within the developmental biology community in recent years. Overall, this Primer provides a concise overview of how micropatterns are used to study developmental processes and emphasises how micropatterns are a useful addition to the developmental biologist's toolbox.

Active flows and deformable surfaces in development

We review progress in active hydrodynamic descriptions of flowing media on curved and deformable manifolds: the state-of-the-art in continuum descriptions of single-layers of epithelial and/or other tissues during development. First, after a brief overview of activity, flows and hydrodynamic descriptions, we highlight the generic challenge of identifying the dependence on dynamical variables of so-called active kinetic coefficients- active counterparts to dissipative Onsager coefficients. We go on to describe some of the subtleties concerning how curvature and active flows interact, and the issues that arise when surfaces are deformable. We finish with a broad discussion around the utility of such theories in developmental biology. This includes limitations to analytical techniques, challenges associated with numerical integration, fitting-to-data and inference, and potential tools for the future, such as discrete differential geometry.

Isolating Patterns in Open Reaction-Diffusion Systems

Realistic examples of reaction-diffusion phenomena governing spatial and spatiotemporal pattern formation are rarely isolated systems, either chemically or thermodynamically. However, even formulations of 'open' reaction-diffusion systems often neglect the role of domain boundaries. Most idealizations of closed reaction-diffusion systems employ no-flux boundary conditions, and often patterns will form up to, or along, these boundaries. Motivated by boundaries of patterning fields related to the emergence of spatial form in embryonic development, we propose a set of mixed boundary conditions for a two-species reaction-diffusion system which forms inhomogeneous solutions away from the boundary of the domain for a variety of different reaction kinetics, with a prescribed uniform state near the boundary. We show that these boundary conditions can be derived from a larger heterogeneous field, indicating that these conditions can arise naturally if cell signalling or other properties of the medium vary in space. We explain the basic mechanisms behind this pattern localization and demonstrate that it can capture a large range of localized patterning in one, two, and three dimensions and that this framework can be applied to systems involving more than two species. Furthermore, the boundary conditions proposed lead to more symmetrical patterns on the interior of the domain and plausibly capture more realistic boundaries in developmental systems. Finally, we show that these isolated patterns are more robust to fluctuations in initial conditions and that they allow intriguing possibilities of pattern selection via geometry, distinct from known selection mechanisms.

3D Interrelationship between Osteocyte Network and Forming Mineral during Human Bone Remodeling

During bone remodeling, osteoblasts are known to deposit unmineralized collagenous tissue (osteoid), which mineralizes after some time lag. Some of the osteoblasts differentiate into osteocytes, forming a cell network within the lacunocanalicular network (LCN) of bone. To get more insight into the potential role of osteocytes in the mineralization process of osteoid, sites of bone formation are three-dimensionally imaged in nine forming human osteons using focused ion beam-scanning electron microscopy (FIB-SEM). In agreement with previous observations, the mineral concentration is found to gradually increase from the central Haversian canal toward pre-existing mineralized bone. Most interestingly, a similar feature is discovered on a length scale more than 100-times smaller, whereby mineral concentration increases from the LCN, leaving around the canaliculi a zone virtually free of mineral, the size of which decreases with progressing mineralization. This suggests that the LCN controls mineral formation but not just by diffusion of mineralization precursors, which would lead to a continuous decrease of mineral concentration from the LCN. The observation is, however, compatible with the codiffusion and reaction of precursors and inhibitors from the LCN into the bone matrix.

Mechanical Models of Pattern and Form in Biological Tissues: The Role of Stress-Strain Constitutive Equations

Mechanical and mechanochemical models of pattern formation in biological tissues have been used to study a variety of biomedical systems, particularly in developmental biology, and describe the physical interactions between cells and their local surroundings. These models in their original form consist of a balance equation for the cell density, a balance equation for the density of the extracellular matrix (ECM), and a force-balance equation describing the mechanical equilibrium of the cell-ECM system. Under the assumption that the cell-ECM system can be regarded as an isotropic linear viscoelastic material, the force-balance equation is often defined using the Kelvin-Voigt model of linear viscoelasticity to represent the stress-strain relation of the ECM. However, due to the multifaceted bio-physical nature of the ECM constituents, there are rheological aspects that cannot be effectively captured by this model and, therefore, depending on the pattern formation process and the type of biological tissue considered, other constitutive models of linear viscoelasticity may be better suited. In this paper, we systematically assess the pattern formation potential of different stress-strain constitutive equations for the ECM within a mechanical model of pattern formation in biological tissues. The results obtained through linear stability analysis and the dispersion relations derived therefrom support the idea that fluid-like constitutive models, such as the Maxwell model and the Jeffrey model, have a pattern formation potential much higher than solid-like models, such as the Kelvin-Voigt model and the standard linear solid model. This is confirmed by the results of numerical simulations, which demonstrate that, all else being equal, spatial patterns emerge in the case where the Maxwell model is used to represent the stress-strain relation of the ECM, while no patterns are observed when the Kelvin-Voigt model is employed. Our findings suggest that further empirical work is required to acquire detailed quantitative information on the mechanical properties of components of the ECM in different biological tissues in order to furnish mechanical and mechanochemical models of pattern formation with stress-strain constitutive equations for the ECM that provide a more faithful representation of the underlying tissue rheology.

Symmetry breaking meets multisite modification

Multisite modification is a basic way of conferring functionality to proteins and a key component of post-translational modification networks. Additional interest in multisite modification stems from its capability of acting as complex information processors. In this paper, we connect two seemingly disparate themes: symmetry and multisite modification. We examine different classes of random modification networks of substrates involving separate or common enzymes. We demonstrate that under different instances of symmetry of the modification network (invoked explicitly or implicitly and discussed in the literature), the biochemistry of multisite modification can lead to the symmetry being broken. This is shown computationally and consolidated analytically, revealing parameter regions where this can (and in fact does) happen, and characteristics of the symmetry-broken state. We discuss the relevance of these results in situations where exact symmetry is not present. Overall, through our study we show how symmetry breaking (i) can confer new capabilities to protein networks, including concentration robustness of different combinations of species (in conjunction with multiple steady states); (ii) could have been the basis for ordering of multisite modification, which is widely observed in cells; (iii) can significantly impact information processing in multisite modification and in cell signalling networks/pathways where multisite modification is present; and (iv) can be a fruitful new angle for engineering in synthetic biology and chemistry. All in all, the emerging conceptual synthesis provides a new vantage point for the elucidation and the engineering of molecular systems at the junction of chemical and biological systems.

Proteins help our cells perform the chemical reactions necessary for life. Once proteins are made, they can also be modified in different ways. This can simply change their activity, or otherwise make them better suited for their specific jobs within the cell. Biological ‘catalysts’ called enzymes carry out protein modifications by reversibly adding (or removing) chemical groups, such as phosphate groups.

‘Multisite modifications’ occur when a protein has two or more modifications in different areas, which can be added randomly or in a specific sequence. The combination of all the modifications attached to a protein acts like a chemical barcode and confers a specific function to the protein.

Modification networks add levels of complexity above individual proteins. These encompass not only the proteins in a cell or tissue, but also the different enzymes that can modify them, and how they all interact with each other. Although our knowledge of these networks is substantial, basic aspects, such as how the ordering of multisite modification systems emerges, is still not well understood.

Using a simple set of multisite modifications, Ramesh and Krishnan set out to study the potential mechanisms allowing the creation of order in this context. Symmetry is a pervasive theme across the sciences. In biology, symmetry and how it may be broken, is important to understand, for example, how organism develop. Ramesh and Krishnan used the perspective of symmetry in protein networks to uncover the origins of ordering.

First, mathematical models of simple modification networks were created based on their basic descriptions. This system centred on proteins that could have phosphate modifications at two possible sites. The network was ‘symmetric’, meaning that the rate of different sets of chemical reactions was identical, as were the amounts of all the enzymes involved.

Dissecting the simulated network using a variety of mathematical approaches showed that its initial symmetry could break, giving rise to sets of ordered multisite modifications. Breaking symmetry did not require any additional features or factors; the basic chemical ‘ingredients’ of protein modification were all that was needed. The prism of symmetry also revealed other aspects of these multisite modification networks, such as robustness and oscillations.

This study sheds new light on the mechanism behind ordering of protein modifications. In the future, Ramesh and Krishnan hope that this approach can be applied to the study of not just proteins but also a wider range of biochemical networks.