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Binary encoding of the phase of the oscillation by bistability. Creation of high (A) and low (B) values of the segmentation marker E in two cells by the coupled effects of oscillatory and bistable dynamics. The network corresponds to Figure 4D and the colors and scalings are identical. While the morphogen G is high, E is high and oscillates in response to the clock variable R. As time passes, G decreases and at a given moment (black arrow), it can no longer significantly activate gene E. The cell fate is determined by the concentration of E at this particular moment relative to a threshold E0 (shown by a dashed line). E0 is the (unstable) fixed point (for G=R=0) that separates protein concentrations E>E0 converging to the high state of E expression, from smaller values that end in the low state of E expression. In (A), E is high enough at the arrowed time so that G and R can disappear while leaving E>E0. In (B), the concentration of E at the arrowed time is under the threshold E0.
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Segmentation is a common feature of disparate clades of metazoans, and its evolution is a central problem of evolutionary developmental biology. We evolved in silico regulatory networks by a mutation/selection process that just rewards the number of segment boundaries. For segmentation controlled by a static gradient, as in long-germ band insects,...
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Context 1
... these conditions are satisfied, R drives E to zero well ahead of the morphogen front, while in the morphogen- decaying tail, an island of autoactivated E remains ( Figure 4C and Supplementary Figure S5). Subsequent evolution creates, however, a more spectacular improvement ( Figure 4D) and a large fitness increase. ...Context 2
... negative feedback loop produces temporal oscillations in R so long as G is large enough to induce R. The oscillations in R produce islands of E for the same reasons as before. The oscillation phase is encoded in a binary way and rendered permanent by the bistable dynamics of E as G decreases to 0. Figure 5 illustrates this mechanism at the level of individual cells. As a consequence of this process (and with the exception of the very first segments), all segments are of the same size as can be seen in Figure 4E: segment size is simply given by the product of the velocity of the front times the period of the clock. ...Similar publications
A fundamental quest for evolutionary biologists has been to uncover the molecular mechanisms underlying animal diversity. In recent decades, researchers have turned their attention to a set of genes conserved across phyla that control identity and pattern of body parts during embryonic development. The finding of this developmental-genetic toolkit...
Citations
... A less abstract set of alternatives would trade off reproducibility against a term that minimizes the divergence of the model-generated patterns from a desired (usually measured) pattern (60). This allows exploring trade-offs induced by biophysical constraints and maximizing reproducible patterning outcomes that are similar to the observed pattern (61)(62)(63). Another productive approach is to start with the PI itself as the utility (63, 64) to be maximized. ...
A key feature of many developmental systems is their ability to self-organize spatial patterns of functionally distinct cell fates. To ensure proper biological function, such patterns must be established reproducibly, by controlling and even harnessing intrinsic and extrinsic fluctuations. While the relevant molecular processes are increasingly well understood, we lack a principled framework to quantify the performance of such stochastic self-organizing systems. To that end, we introduce an information-theoretic measure for self-organized fate specification during embryonic development. We show that the proposed measure assesses the total information content of fate patterns and decomposes it into interpretable contributions corresponding to the positional and correlational information. By optimizing the proposed measure, our framework provides a normative theory for developmental circuits, which we demonstrate on lateral inhibition, cell type proportioning, and reaction–diffusion models of self-organization. This paves a way toward a classification of developmental systems based on a common information-theoretic language, thereby organizing the zoo of implicated chemical and mechanical signaling processes.
... Hypotheses about the possible relationships between development and evolution can then be tested. Theoretical hypotheses have been proposed using this approach regarding the structure of gene regulatory networks that induce segmentation [27][28][29], testing plausible evolutionary scenarios of segmentation mechanisms [30,31], the relationship between developmental and mutational robustness [32], and congruence in the developmental and evolutionary processes in both RNA [33] and gene expression patterns [34]. ...
... In this study, we conducted an evolutionary simulation to investigate the mechanism behind the establishment of a developmental hourglass. For this purpose, we adopted a dynamical-system scheme to generate spatial patterns, which is commonly adopted in theoretical studies [27,[29][30][31]34]. The model for the simulation can represent cellular states that are determined by gene expression patterns governed by gene regulatory networks and spatial cell-cell interactions, leading to the differentiation of cell types and pattern formation. ...
Author summary
Understanding the intriguing relationship between development and evolution in multicellular organisms has long been a challenge in biology. A recent hypothesis called the developmental hourglass proposes that there is a conserved middle stage during development across species of the same animal group. Despite growing evidence supporting this hypothesis, the underlying mechanisms and reasons for its emergence have remained elusive due to limited experimental data.
To address this gap, we employed numerical evolution of gene regulation networks controlling pattern formation. Remarkably, our simulations revealed that species that diverged relatively recently in phylogeny displayed the highest similarity during the middle stage of development, which gradually diminished as they diverged further phylogenetically. Our findings satisfied not only the criteria of the developmental hourglass but also confirmed several essential characteristics of the developmental hourglass reported in recent experiments. Through theoretical analysis, we further demonstrated that the emergence of the developmental hourglass could be attributed to the acquisition of genes that change slowly and govern developmental processes, which also foster the robustness of development.
By integrating computational simulations, theoretical insights, and previous experimental evidence, our study thus provides a comprehensive understanding of the developmental hourglass, which will unravel the intricate relationship between development and evolution.
... Hypotheses about the possible relationships between development and evolution can then be tested. Theoretical hypotheses have been proposed using this approach regarding the structure of gene regulatory networks that induce segmentation [27,28,29], testing plausible evolutionary scenarios of segmentation mechanisms [30,31], the relationship between developmental and mutational robustness [32], and congruence in the developmental and evolutionary processes in both RNA [33] and gene expression patterns [34]. ...
... In this study, we conducted an evolutionary simulation to investigate the mechanism behind the establishment of a developmental hourglass. For this purpose, we adopted a dynamical-system scheme to generate spatial patterns, which is commonly adopted in theoretical studies [27,29,30,31,34]. The model for the simulation can represent cellular states that are determined by gene expression patterns governed by gene regulatory networks and spatial cell-cell interactions, leading to the differentiation of cell types and pattern formation. ...
Determining the general laws between evolution and development is a fundamental biological challenge. Developmental hourglasses have attracted increased attention as candidates for such laws, but the necessity of their emergence remains elusive. We conducted evolutionary simulations of developmental processes to confirm the emergence of the developmental hourglass and unveiled its establishment. We considered organisms consisting of cells containing identical gene networks that control morphogenesis and evolved them under selection pressure to induce more cell types. By computing the similarity between the spatial patterns of gene expression of two species that evolved from a common ancestor, a developmental hourglass was observed, that is, there was a correlation peak in the intermediate stage of development. The fraction of pleiotropic genes increased, whereas the variance in individuals decreased, consistent with previous experimental reports. Reduction of the unavoidable variance by initial or developmental noise, essential for survival, was achieved up to the hourglass bottleneck stage, followed by diversification in developmental processes, whose timing is controlled by the slow expression dynamics conserved among organisms sharing the hourglass. This study suggests why developmental hourglasses are observed within a certain phylogenetic range of species.
Author Summary
Understanding the intriguing relationship between development and evolution in multicellular organisms has long been a challenge in biology. A recent hypothesis called the developmental hourglass proposes that there is a conserved middle stage during development across species of the same animal group. Despite growing evidence supporting this hypothesis, the underlying mechanisms and reasons for its emergence have remained elusive due to limited experimental data.
To address this gap, we employed numerical evolution of gene regulation networks controlling pattern formation. Remarkably, our simulations revealed that species that diverged relatively recently in phylogeny displayed the highest similarity during the middle stage of development, which gradually diminished as they diverged further phylogenetically. Our findings satisfied not only the criteria of the developmental hourglass but also confirmed several essential characteristics of the developmental hourglass reported in recent experiments. Through theoretical analysis, we further demonstrated that the emergence of the developmental hourglass could be attributed to the acquisition of genes that change slowly and govern developmental processes, which also foster the robustness of development.
By integrating computational simulations, theoretical insights, and previous experimental evidence, our study thus provides a comprehensive understanding of the developmental hourglass, which will unravel the intricate relationship between development and evolution.
... We focused on the Drosophila gap gene system, one of the paradigms for developmental biology and for physical precision measurements in living systems (62). Our work extends previous mathematical models of this system (26)(27)(28)(29)(30)(31)(32)(33)(34)(63)(64)(65)(66), as well as attempts to predict it ab initio (67)(68)(69). In contrast to previously studied models, we systematically incorporate the unavoidable physical sources of noise, highlighting how patterning precision can emerge from noisy signals by a synergistic combination of multiple mechanisms. ...
Many biological systems approach physical limits to their performance, motivating the idea that their behavior and underlying mechanisms could be determined by such optimality. Nevertheless, optimization as a predictive principle has only been applied in very simplified setups. Here, in contrast, we explore a mechanistically-detailed class of models for the gap gene network of the Drosophila embryo, and determine its 50+ parameters by optimizing the information that gene expression levels convey about nuclear positions, subject to physical constraints on the number of available molecules. Optimal networks recapitulate the architecture and spatial gene expression profiles of the real organism. Our framework makes precise the many tradeoffs involved in maximizing functional performance, and allows us to explore alternative networks to address the questions of necessity vs contingency. Multiple solutions to the optimization problem may be realized in closely related organisms.
... However, they did not provide a mechanistic explanation for the emergence and the arrest of oscillations. Recent studies have described this patterning event by using one-dimensional dynamical systems, where the freezing of oscillations was modeled as a bifurcation from an unstable state to a bi-stable regime along the anterior posterior axis [15,19]. Our two-dimensional implementation of the Clock and Wavefront model is based on the same principle and has a phase space that resemble the ones in [19]. ...
During mouse development, presomitic mesoderm cells synchronize oscillations of Wnt and Notch signaling forming sequential waves that pattern somites. Tail explants have shown that these waves can be formed in the absence of global embryonic signals in vitro but the self-organizing mechanism that underlies this process remains unknown. Here, we present a model called Sevilletor that synchronizes two oscillatory reactants via local cell communication to generate a variety of self-organizing patterns. This model provides a unifying minimal framework to study somitogenesis showing that classical models cannot recapitulate the waves formed in explants. Nevertheless, we show that explant patterning can be recapitulated by an extended Clock and Wavefront model where Fgf and Retinoic acid signals modulate a self-organizing oscillator implemented by Wnt and Notch. This model captures the change in relative phase of Wnt and Notch observed in mouse and explains how cells can be excited to oscillate by local cell communication.
... Broadly categorised, segments can be generated simultaneously or one by one sequentially. Examples of simultaneous mechanisms that evolved are: hierarchical mechanisms (reminiscent of the mechanism in Drosophila) [38][39][40], reaction-diffusion mechanisms [38,41], and noise-amplifying mechanisms [42]. Sequential mechanisms include: stereotyped asymmetric divisions in a growth zone (reminiscent of annelid worms) [42]; timed, segment-specific determination of a band [43]; and the transformation of gene expression oscillations into a spatial pattern (like the clock-and-wavefront mechanism in vertebrates) [39,[41][42][43][44]. ...
... Examples of simultaneous mechanisms that evolved are: hierarchical mechanisms (reminiscent of the mechanism in Drosophila) [38][39][40], reaction-diffusion mechanisms [38,41], and noise-amplifying mechanisms [42]. Sequential mechanisms include: stereotyped asymmetric divisions in a growth zone (reminiscent of annelid worms) [42]; timed, segment-specific determination of a band [43]; and the transformation of gene expression oscillations into a spatial pattern (like the clock-and-wavefront mechanism in vertebrates) [39,[41][42][43][44]. ...
Evolution has been an inventive process since its inception, about 4 billion years ago. It has generated an astounding diversity of novel mechanisms and structures for adaptation to the environment, for competition and cooperation, and for organisation of the internal and external dynamics of the organism. How does this novelty come about? Evolution builds with the tools available, and on top of what it has already built – therefore, much novelty consists in repurposing old functions in a different context. In the process, the tools themselves evolve, allowing yet more novelty to arise.
Despite evolutionary novelty being the most striking observable of evolution, it is not accounted for in classical evolutionary theory. Nevertheless, mathematical and computational models that illustrate mechanisms of evolutionary innovation have been developed. In the present review, we present and compare several examples of computational evo–devo models that capture two aspects of novelty: ‘between-level novelty’ and ‘constructive novelty.’ Novelty can evolve between predefined levels of organisation to dynamically transcode biological information across these levels – as occurs during development. Constructive novelty instead generates a level of biological organisation by exploiting the lower level as an informational scaffold to open a new space of possibilities – an example being the evolution of multicellularity. We propose that the field of computational evo–devo is well-poised to reveal many more exciting mechanisms for the evolution of novelty. A broader theory of evolutionary novelty may well be attainable in the near future.
... Iterative or exhaustive approaches have been developed in the past to uncover the common features in networks that perform temporal functions, such as signal adaptation 8 or robust oscillations 9 ; spatial functions, such as embryonic patterning [10][11][12] ; or information processing functions, such as noise reduction 13 . Related approaches have derived simple networks via in-silico evolution in the service of particular fitness goals 7,14 . Many of these works have focused on the performance of a specific function in response to a single input, whereas here we are interested in deducing minimal networks that underlie a system's ability to multiplex, i.e., to respond to multiple inputs either individually or together. ...
Cell signaling networks are complex and often incompletely characterized, making it difficult to obtain a comprehensive picture of the mechanisms they encode. Mathematical modeling of these networks provides important clues, but the models themselves are often complex, and it is not always clear how to extract falsifiable predictions. Here we take an inverse approach, using experimental data at the cell level to deduce the minimal signaling network. We focus on cells' response to multiple cues, specifically on the surprising case in which the response is antagonistic: the response to multiple cues is weaker than the response to the individual cues. We systematically build candidate signaling networks one node at a time, using the ubiquitous ingredients of (i) up- or down-regulation, (ii) molecular conversion, or (iii) reversible binding. In each case, our method reveals a minimal, interpretable signaling mechanism that explains the antagonistic response. Our work provides a systematic way to deduce molecular mechanisms from cell-level data.
... Here we introduce an inverse approach to understanding complex signaling networks. Instead of modeling all known features of a signaling network to predict a cell response, we use observed cell responses to infer a minimally sufficient signaling network [7]. Our approach is systematic: given a particular class of biochemical interactions, we build candidate networks one piece at a time, continually evaluating whether the observed set of * andrew.mugler@pitt.edu ...
Cell signaling networks are complex and often incompletely characterized, making it difficult to obtain a comprehensive picture of the mechanisms they encode. Mathematical modeling of these networks provides important clues, but the models themselves are often complex, and it is not always clear how to extract falsifiable predictions. Here we take an inverse approach, using experimental data at the cell level to infer the minimal signaling network. We focus on cells' response to multiple cues, specifically on the surprising case in which the response is antagonistic: the response to multiple cues is weaker than the response to the individual cues. We systematically build candidate signaling networks one node at a time, using the ubiquitous ingredients of (i) up- or down-regulation, (ii) molecular conversion, or (iii) reversible binding. In each case, our method reveals a minimal, interpretable signaling mechanism that explains the antagonistic response. Our work provides a systematic way to infer molecular mechanisms from cell-level data.
... Popular data-driven methods include, but are not limited to, symbolic regression [29,30], recurrent neural networks [32], evolved regulatory networks [13], and physics-informed neural networks [25]. These methods have demonstrated the ability to accurately replicate HFM after sufficient training and testing. ...
Determining the proper level of details to develop and solve physical models is usually difficult when one encounters new engineering problems. Such difficulty comes from how to balance the time (simulation cost) and accuracy for the physical model simulation afterwards. We propose a framework for automatic development of a family of surrogate models of physical systems that provide flexible cost-accuracy tradeoffs to assist making such determinations. We present both a model-based and a data-driven strategy to generate surrogate models. The former starts from a high-fidelity model generated from first principles and applies a bottom-up model order reduction (MOR) that preserves stability and convergence while providing a priori error bounds, although the resulting reduced-order model may lose its interpretability. The latter generates interpretable surrogate models by fitting artificial constitutive relations to a presupposed topological structure using experimental or simulation data. For the latter, we use Tonti diagrams to systematically produce differential equations from the assumed topological structure using algebraic topological semantics that are common to various lumped-parameter models (LPM). The parameter for the constitutive relations are estimated using standard system identification algorithms. Our framework is compatible with various spatial discretization schemes for distributed parameter models (DPM), and can supports solving engineering problems in different domains of physics.
... These waves allow Dictyostelium to self-organize and develop into a stalk-and-spore structure in complex and ill-defined environments over group sizes that can vary by many orders of magnitude [4]. Identifying how this level of control is achieved is still an open question as most studies on design principles that coordinate multicellular phenomena have focused on developmental networks in metazoans that use well-defined morphogen gradients to pattern their development [9][10][11][12][13]. ...
Complex multicellular behaviors are coordinated at the level of biochemical signaling networks, yet how this decentralized control mechanism enables robust coordination in variable environments and over many orders of magnitude of spatiotemporal scales remains an open question. A stunning example of these behaviors is found in the microbe Dictyostelium discoideum , which uses the small molecule cyclic AMP (cAMP) to drive the propagation of collective signaling oscillations and spatial waves leading to multicellular development. The critical design features of the Dictyostelium signaling network remain unclear despite decades of mathematical modeling and experimental research because the mathematical models make different assumptions about the network architecture. To resolve this discrepancy, we use recent experimental data to normalize the time and response scales of five major signal relay network models to one another and assess their ability to recapitulate experimentally-observed population and single-cell dynamics. We find that to successfully reproduce the full range of observed behaviors, single cells must be excitable and respond to the relative fold-change of environmental signals, suggesting that these features represent robust principles for controlling cellular populations and that single-cell excitable dynamics are a generalizable route for controlling population behaviors.
