The segment polarity network is a robust developmental module

Exploring gene regulation and biological processes in insects: Insights from omics data using gene regulatory network models

Gene regulatory network (GRN) comprises complicated yet intertwined gene-regulator relationships. Understanding the GRN dynamics will unravel the complexity behind the observed gene expressions. Insect gene regulation is often complicated due to their complex life cycles and diverse ecological adaptations. The main interest of this review is to have an update on the current mathematical modelling methods of GRNs to explain insect science. Several popular GRN architecture models are discussed, together with examples of applications in insect science. In the last part of this review, each model is compared from different aspects, including network scalability, computation complexity, robustness to noise and biological relevancy.

A mathematical framework for measuring and tuning tempo in developmental gene regulatory networks

Embryo development is a dynamic process governed by the regulation of timing and sequences of gene expression, which control the proper growth of the organism. Although many genetic programmes coordinating these sequences are common across species, the timescales of gene expression can vary significantly among different organisms. Currently, substantial experimental efforts are focused on identifying molecular mechanisms that control these temporal aspects. In contrast, the capacity of established mathematical models to incorporate tempo control while maintaining the same dynamical landscape remains less understood. Here, we address this gap by developing a mathematical framework that links the functionality of developmental programmes to the corresponding gene expression orbits (or landscapes). This unlocks the ability to find tempo differences as perturbations in the dynamical system that preserve its orbits. We demonstrate that this framework allows for the prediction of molecular mechanisms governing tempo, through both numerical and analytical methods. Our exploration includes two case studies: a generic network featuring coupled production and degradation, with a particular application to neural progenitor differentiation; and the repressilator. In the latter, we illustrate how altering the dimerisation rates of transcription factors can decouple the tempo from the shape of the resulting orbits. We conclude by highlighting how the identification of orthogonal molecular mechanisms for tempo control can inform the design of circuits with specific orbits and tempos.

Establishing a conceptual framework for holistic cell states and state transitions

Cell states were traditionally defined by how they looked, where they were located, and what functions they performed. In this post-genomic era, the field is largely focused on a molecular view of cell state. Moving forward, we anticipate that the observables used to define cell states will evolve again as single-cell imaging and analytics are advancing at a breakneck pace via the collection of large-scale, systematic cell image datasets and the application of quantitative image-based data science methods. This is, therefore, a key moment in the arc of cell biological research to develop approaches that integrate the spatiotemporal observables of the physical structure and organization of the cell with molecular observables toward the concept of a holistic cell state. In this perspective, we propose a conceptual framework for holistic cell states and state transitions that is data-driven, practical, and useful to enable integrative analyses and modeling across many data types.

Deciphering the origin of developmental stability: The role of intracellular expression variability in evolutionary conservation

Progress in evolutionary developmental biology (evo-devo) has deepened our understanding of how intrinsic properties of embryogenesis, along with natural selection and population genetics, shape phenotypic diversity. A focal point of recent empirical and theoretical research is the idea that highly developmentally stable phenotypes are more conserved in evolution. Previously, we demonstrated that in Japanese medaka (Oryzias latipes), embryonic stages and genes with high stability, estimated through whole-embryo RNA-seq, are highly conserved in subsequent generations. However, the precise origin of the stability of gene expression levels evaluated at the whole-embryo level remained unclear. Such stability could be attributed to two distinct sources: stable intracellular expression levels or spatially stable expression patterns. Here we demonstrate that stability observed in whole-embryo RNA-seq can be attributed to stability at the cellular level (low variability in gene expression at the cellular levels). We quantified the intercellular variations in expression levels and spatial gene expression patterns for seven key genes involved in patterning dorsoventral and rostrocaudal regions during early development in medaka. We evaluated intracellular variability by counting transcripts and found its significant correlation with variation observed in whole-embryo RNA-seq data. Conversely, variation in spatial gene expression patterns, assessed through intraindividual left-right asymmetry, showed no correlation. Given the previously reported correlation between stability and conservation of expression levels throughout embryogenesis, our findings suggest a potential general trend: the stability or instability of developmental systems-and the consequent evolutionary diversity-may be primarily anchored in intrinsic fundamental elements such as the variability of intracellular states.

Promises and limits of an agency perspective in evolutionary developmental biology

An agent-based perspective in the study of complex systems is well established in diverse disciplines, yet is only beginning to be applied to evolutionary developmental biology. In this essay, we begin by defining agency and associated terminology formally. We then explore the assumptions and predictions of an agency perspective, apply these to select processes and key concept areas relevant to practitioners of evolutionary developmental biology, and consider the potential epistemic roles that an agency perspective might play in evo devo. Throughout, we discuss evidence supportive of agential dynamics in biological systems relevant to evo devo and explore where agency thinking may enrich the explanatory reach of research efforts in evolutionary developmental biology.

Integrated Systems Biology Pipeline to Compare Co-Expression Networks in Plants and Elucidate Differential Regulators

To identify sets of genes that exhibit similar expression characteristics, co-expression networks were constructed from transcriptome datasets that were obtained from plant samples at various stages of growth and development or treated with diverse biotic, abiotic, and other environmental stresses. In addition, co-expression network analysis can provide deeper insights into gene regulation when combined with transcriptomics. The coordination and integration of all these complex networks to deduce gene regulation are major challenges for plant biologists. Python and R have emerged as major tools for managing complex scientific data over the past decade. In this study, we describe a reproducible protocol POTFUL (pant co-expression transcription factor regulators), implemented in Python 3, for integrating co-expression and transcription factor target protein networks to infer gene regulation.

Efficient parameter generation for constrained models using MCMC

Mathematical models of complex systems rely on parameter values to produce a desired behavior. As mathematical and computational models increase in complexity, it becomes correspondingly difficult to find parameter values that satisfy system constraints. We propose a Markov Chain Monte Carlo (MCMC) approach for the problem of constrained model parameter generation by designing a Markov chain that efficiently explores a model's parameter space. We demonstrate the use of our proposed methodology to analyze responses of a newly constructed bistability-constrained model of protein phosphorylation to perturbations in the underlying protein network. Our results suggest that parameter generation for constrained models using MCMC provides powerful tools for modeling-aided analysis of complex natural processes.

A bistable autoregulatory module in the developing embryo commits cells to binary expression fates

Bistable autoactivation has been proposed as a mechanism for cells to adopt binary fates during embryonic development. However, it is unclear whether the autoactivating modules found within developmental gene regulatory networks are bistable, unless their parameters are quantitatively determined. Here, we combine in vivo live imaging with mathematical modeling to dissect the binary cell fate dynamics of the fruit fly pair-rule gene fushi tarazu (ftz), which is regulated by two known enhancers: the early (non-autoregulating) element and the autoregulatory element. Live imaging of transcription and protein concentration in the blastoderm revealed that binary Ftz fates are achieved as Ftz expression rapidly transitions from being dictated by the early element to the autoregulatory element. Moreover, we discovered that Ftz concentration alone is insufficient to activate the autoregulatory element, and that this element only becomes responsive to Ftz at a prescribed developmental time. Based on these observations, we developed a dynamical systems model and quantitated its kinetic parameters directly from experimental measurements. Our model demonstrated that the ftz autoregulatory module is indeed bistable and that the early element transiently establishes the content of the binary cell fate decision to which the autoregulatory module then commits. Further in silico analysis revealed that the autoregulatory element locks the Ftz fate quickly, within 35 min of exposure to the transient signal of the early element. Overall, our work confirms the widely held hypothesis that autoregulation can establish developmental fates through bistability and, most importantly, provides a framework for the quantitative dissection of cellular decision-making.

Assessing biological network dynamics: comparing numerical simulations with analytical decomposition of parameter space

Mathematical modeling of the emergent dynamics of gene regulatory networks (GRN) faces a double challenge of (a) dependence of model dynamics on parameters, and (b) lack of reliable experimentally determined parameters. In this paper we compare two complementary approaches for describing GRN dynamics across unknown parameters: (1) parameter sampling and resulting ensemble statistics used by RACIPE (RAndom CIrcuit PErturbation), and (2) use of rigorous analysis of combinatorial approximation of the ODE models by DSGRN (Dynamic Signatures Generated by Regulatory Networks). We find a very good agreement between RACIPE simulation and DSGRN predictions for four different 2- and 3-node networks typically observed in cellular decision making. This observation is remarkable since the DSGRN approach assumes that the Hill coefficients of the models are very high while RACIPE assumes the values in the range 1-6. Thus DSGRN parameter domains, explicitly defined by inequalities between systems parameters, are highly predictive of ODE model dynamics within a biologically reasonable range of parameters.

Reproducibility and FAIR principles: the case of a segment polarity network model

The issue of reproducibility of computational models and the related FAIR principles (findable, accessible, interoperable, and reusable) are examined in a specific test case. I analyze a computational model of the segment polarity network in Drosophila embryos published in 2000. Despite the high number of citations to this publication, 23 years later the model is barely accessible, and consequently not interoperable. Following the text of the original publication allowed successfully encoding the model for the open source software COPASI. Subsequently saving the model in the SBML format allowed it to be reused in other open source software packages. Submission of this SBML encoding of the model to the BioModels database enables its findability and accessibility. This demonstrates how the FAIR principles can be successfully enabled by using open source software, widely adopted standards, and public repositories, facilitating reproducibility and reuse of computational cell biology models that will outlive the specific software used.

Using sensitivity analyses to understand bistable system behavior

BACKGROUND

Bistable systems, i.e., systems that exhibit two stable steady states, are of particular interest in biology. They can implement binary cellular decision making, e.g., in pathways for cellular differentiation and cell cycle regulation. The onset of cancer, prion diseases, and neurodegenerative diseases are known to be associated with malfunctioning bistable systems. Exploring and characterizing parameter spaces in bistable systems, so that they retain or lose bistability, is part of a lot of therapeutic research such as cancer pharmacology.

RESULTS

We use eigenvalue sensitivity analysis and stable state separation sensitivity analysis to understand bistable system behaviors, and to characterize the most sensitive parameters of a bistable system. While eigenvalue sensitivity analysis is an established technique in engineering disciplines, it has not been frequently used to study biological systems. We demonstrate the utility of these approaches on a published bistable system. We also illustrate scalability and generalizability of these methods to larger bistable systems.

CONCLUSIONS

Eigenvalue sensitivity analysis and separation sensitivity analysis prove to be promising tools to define parameter design rules to make switching decisions between either stable steady state of a bistable system and a corresponding monostable state after bifurcation. These rules were applied to the smallest two-component bistable system and results were validated analytically. We showed that with multiple parameter settings of the same bistable system, we can design switching to a desirable state to retain or lose bistability when the most sensitive parameter is varied according to our parameter perturbation recommendations. We propose eigenvalue and stable state separation sensitivity analyses as a framework to evaluate large and complex bistable systems.

Stability in gene expression and body-plan development leads to evolutionary conservation

BACKGROUND

Phenotypic evolution is mainly explained by selection for phenotypic variation arising from factors including mutation and environmental noise. Recent theoretical and experimental studies have suggested that phenotypes with greater developmental stability tend to have a constant phenotype and gene expression level within a particular genetic and environmental condition, and this positively correlates with stronger evolutionary conservation, even after the accumulation of genetic changes. This could reflect a novel mechanism that contributes to evolutionary conservation; however, it remains unclear whether developmental stability is the cause, or whether at least it contributes to their evolutionary conservation. Here, using Japanese medaka lines, we tested experimentally whether developmental stages and gene expression levels with greater stability led to their evolutionary conservation.

RESULTS

We first measured the stability of each gene expression level and developmental stage (defined here as the whole embryonic transcriptome) in the inbred F0 medaka population. We then measured their evolutionary conservation in the F3 generation by crossing the F0 line with the distantly related Japanese medaka line (Teradomori), followed by two rounds of intra-generational crossings. The results indicated that the genes and developmental stages that had smaller variations in the F0 generation showed lower diversity in the hybrid F3 generation, which implies a causal relationship between stability and evolutionary conservation.

CONCLUSIONS

These findings suggest that the stability in phenotypes, including the developmental stages and gene expression levels, leads to their evolutionary conservation; this most likely occurs due to their low potential to generate phenotypic variation. In addition, since the highly stable developmental stages match with the body-plan-establishment stage, it also implies that the developmental stability potentially contributed to the strict conservation of animal body plan.

Localization of Noise in Biochemical Networks

Noise, or uncertainty in biochemical networks, has become an important aspect of many biological problems. Noise can arise and propagate from external factors and probabilistic chemical reactions occurring in small cellular compartments. For species survival, it is important to regulate such uncertainties in executing vital cell functions. Regulated noise can improve adaptability, whereas uncontrolled noise can cause diseases. Simulation can provide a detailed analysis of uncertainties, but parameters such as rate constants and initial conditions are usually unknown. A general understanding of noise dynamics from the perspective of network structure is highly desirable. In this study, we extended the previously developed law of localization for characterizing noise in terms of (co)variances and developed noise localization theory. With linear noise approximation, we can expand a biochemical network into an extended set of differential equations representing a fictitious network for pseudo-components consisting of variances and covariances, together with chemical species. Through localization analysis, perturbation responses at the steady state of pseudo-components can be summarized into a sensitivity matrix that only requires knowledge of network topology. Our work allows identification of buffering structures at the level of species, variances, and covariances and can provide insights into noise flow under non-steady-state conditions in the form of a pseudo-chemical reaction. We tested noise localization in various systems, and here we discuss its implications and potential applications. Results show that this theory is potentially applicable in discriminating models, scanning network topologies with interesting noise behavior, and designing and perturbing networks with the desired response.

Early effects of gene duplication on the robustness and phenotypic variability of gene regulatory networks

Background

Research on gene duplication is abundant and comes from a wide range of approaches, from high-throughput analyses and experimental evolution to bioinformatics and theoretical models. Notwithstanding, a consensus is still lacking regarding evolutionary mechanisms involved in evolution through gene duplication as well as the conditions that affect them. We argue that a better understanding of evolution through gene duplication requires considering explicitly that genes do not act in isolation. It demands studying how the perturbation that gene duplication implies percolates through the web of gene interactions. Due to evolution’s contingent nature, the paths that lead to the final fate of duplicates must depend strongly on the early stages of gene duplication, before gene copies have accumulated distinctive changes.

Methods

Here we use a widely-known model of gene regulatory networks to study how gene duplication affects network behavior in early stages. Such networks comprise sets of genes that cross-regulate. They organize gene activity creating the gene expression patterns that give cells their phenotypic properties. We focus on how duplication affects two evolutionarily relevant properties of gene regulatory networks: mitigation of the effect of new mutations and access to new phenotypic variants through mutation.

Results

Among other observations, we find that those networks that are better at maintaining the original phenotype after duplication are usually also better at buffering the effect of single interaction mutations and that duplication tends to enhance further this ability. Moreover, the effect of mutations after duplication depends on both the kind of mutation and genes involved in it. We also found that those phenotypes that had easier access through mutation before duplication had higher chances of remaining accessible through new mutations after duplication.

Conclusion

Our results support that gene duplication often mitigates the impact of new mutations and that this effect is not merely due to changes in the number of genes. The work that we put forward helps to identify conditions under which gene duplication may enhance evolvability and robustness to mutations.

Robustness in phenotypic plasticity and heterogeneity patterns enabled by EMT networks

Epithelial-mesenchymal plasticity (EMP) is a key arm of cancer metastasis and is observed across many contexts. Cells undergoing EMP can reversibly switch between three classes of phenotypes: epithelial (E), mesenchymal (M), and hybrid E/M. While a large number of multistable regulatory networks have been identified to be driving EMP in various contexts, the exact mechanisms and design principles that enable robustness in driving EMP across contexts are not yet fully understood. Here, we investigated dynamic and structural robustness in EMP networks with regard to phenotypic heterogeneity and plasticity. We use two different approaches to simulate these networks: a computationally inexpensive, parameter-independent continuous state space Boolean model, and an ODE-based parameter-agnostic framework (RACIPE), both of which yielded similar phenotypic distributions. While the latter approach is useful for measurements of plasticity, the former model enabled us to extensively investigate robustness in phenotypic heterogeneity. Using perturbations to network topology and by varying network parameters, we show that multistable EMP networks are structurally and dynamically more robust compared with their randomized counterparts, thereby highlighting their topological hallmarks. These features of robustness are governed by a balance of positive and negative feedback loops embedded in these networks. Using a combination of the number of negative and positive feedback loops weighted by their lengths, we identified a metric that can explain the structural and dynamical robustness of these networks. This metric enabled us to compare networks across multiple sizes, and the network principles thus obtained can be used to identify fragilities in large networks without simulating their dynamics. Our analysis highlights a network topology-based approach to quantify robustness in the phenotypic heterogeneity and plasticity emergent from EMP networks.

Leveraging network structure in nonlinear control

Over the last twenty years, dynamic modeling of biomolecular networks has exploded in popularity. Many of the classical tools for understanding dynamical systems are unwieldy in the highly nonlinear, poorly constrained, high-dimensional systems that often arise from these modeling efforts. Understanding complex biological systems is greatly facilitated by purpose-built methods that leverage common features of such models, such as local monotonicity, interaction graph sparsity, and sigmoidal kinetics. Here, we review methods for controlling the systems of ordinary differential equations used to model biomolecular networks. We focus on methods that make use of the structure of the network of interactions to help inform, which variables to target for control, and highlight the computational and experimental advantages of such approaches. We also discuss the importance of nonperturbative methods in biomedical and experimental molecular biology applications, where finely tuned interventions can be difficult to implement. It is well known that feedback loops, and positive feedback loops in particular, play a major determining role in the dynamics of biomolecular networks. In many of the methods we cover here, control over system trajectories is realized by overriding the behavior of key feedback loops.

Genetic networks underlying salinity tolerance in wheat uncovered with genome-wide analyses and selective sweeps

A genetic framework underpinning salinity tolerance at reproductive stage was revealed by genome-wide SNP markers and major adaptability genes in synthetic-derived wheats, and trait-associated loci were used to predict phenotypes. Using wild relatives of crops to identify genes related to improved productivity and resilience to climate extremes is a prioritized area of crop genetic improvement. High salinity is a widespread crop production constraint, and development of salt-tolerant cultivars is a sustainable solution. We evaluated a panel of 294 wheat accessions comprising synthetic-derived wheat lines (SYN-DERs) and modern bread wheat advanced lines under control and high salinity conditions at two locations. The GWAS analysis revealed a quantitative genetic framework of more than 200 loci with minor effect underlying salinity tolerance at reproductive stage. The significant trait-associated SNPs were used to predict phenotypes using a GBLUP model, and the prediction accuracy (r²) ranged between 0.57 and 0.74. The r² values for flag leaf weight, days to flowering, biomass, and number of spikes per plant were all above 0.70, validating the phenotypic effects of the loci discovered in this study. Furthermore, the germplasm sets were compared to identify selection sweeps associated with salt tolerance loci in SYN-DERs. Six loci associated with salinity tolerance were found to be differentially selected in the SYN-DERs (12.4 Mb on chromosome (chr)1B, 7.1 Mb on chr2A, 11.2 Mb on chr2D, 200 Mb on chr3D, 600 Mb on chr6B, and 700.9 Mb on chr7B). A total of 228 reported markers and genes, including 17 well-characterized genes, were uncovered using GWAS and EigenGWAS. A linkage disequilibrium (LD) block on chr5A, including the Vrn-A1 gene at 575 Mb and its homeologs on chr5D, were strongly associated with multiple yield-related traits and flowering time under salinity stress conditions. The diversity panel was screened with more than 68 kompetitive allele-specific PCR (KASP) markers of functional genes in wheat, and the pleiotropic effects of superior alleles of Rht-1, TaGASR-A1, and TaCwi-A1 were revealed under salinity stress. To effectively utilize the extensive genetic information obtained from the GWAS analysis, a genetic interaction network was constructed to reveal correlations among the investigated traits. The genetic network data combined with GWAS, selective sweeps, and the functional gene survey provided a quantitative genetic framework for identifying differentially retained loci associated with salinity tolerance in wheat.

Thermodynamic Modelling of Transcriptional Control: A Sensitivity Analysis

Modelling is a tool used to decipher the biochemical mechanisms involved in transcriptional control. Experimental evidence in genetics is usually supported by theoretical models in order to evaluate the effects of all the possible interactions that can occur in these complicated processes. Models derived from the thermodynamic method are critical in this labour because they are able to take into account multiple mechanisms operating simultaneously at the molecular micro-scale and relate them to transcriptional initiation at the tissular macro-scale. This work is devoted to adapting computational techniques to this context in order to theoretically evaluate the role played by several biochemical mechanisms. The interest of this theoretical analysis relies on the fact that it can be contrasted against those biological experiments where the response to perturbations in the transcriptional machinery environment is evaluated in terms of genetically activated/repressed regions. The theoretical reproduction of these experiments leads to a sensitivity analysis whose results are expressed in terms of the elasticity of a threshold function determining those activated/repressed regions. The study of this elasticity function in thermodynamic models already proposed in the literature reveals that certain modelling approaches can alter the balance between the biochemical mechanisms considered, and this can cause false/misleading outcomes. The reevaluation of classical thermodynamic models gives us a more accurate and complete picture of the interactions involved in gene regulation and transcriptional control, which enables more specific predictions. This sensitivity approach provides a definite advantage in the interpretation of a wide range of genetic experimental results.

NETISCE: a network-based tool for cell fate reprogramming

The search for effective therapeutic targets in fields like regenerative medicine and cancer research has generated interest in cell fate reprogramming. This cellular reprogramming paradigm can drive cells to a desired target state from any initial state. However, methods for identifying reprogramming targets remain limited for biological systems that lack large sets of experimental data or a dynamical characterization. We present NETISCE, a novel computational tool for identifying cell fate reprogramming targets in static networks. In combination with machine learning algorithms, NETISCE estimates the attractor landscape and predicts reprogramming targets using signal flow analysis and feedback vertex set control, respectively. Through validations in studies of cell fate reprogramming from developmental, stem cell, and cancer biology, we show that NETISCE can predict previously identified cell fate reprogramming targets and identify potentially novel combinations of targets. NETISCE extends cell fate reprogramming studies to larger-scale biological networks without the need for full model parameterization and can be implemented by experimental and computational biologists to identify parts of a biological system relevant to the desired reprogramming task.

Serial Homology and Segment Identity in the Arthropod Head

The anterior-most unit of the crown-group arthropod body plan includes three segments, the pre-gnathal segments, that contain three neuromeres that together comprise the brain. Recent work on the development of this anterior region has shown that its three units exhibit many developmental differences to the more posterior segments, to the extent that they should not be considered serial homologs. Building on this revised understanding of the development of the pre-gnathal segments, we suggest a novel scenario for arthropod head evolution. We posit an expansion of an ancestral single-segmented head at the transition from Radiodonta to Deuteropoda in the arthropod stem group. The expanded head subdivided into three segmental units, each maintaining some of the structures of the ancestral head. This scenario is consistent with what we know of head evolution from the fossil record and helps reconcile some of the debates about early arthropod evolution.

Modeling the C. elegans germline stem cell genetic network using automated reasoning

Computational methods and tools are a powerful complementary approach to experimental work for studying regulatory interactions in living cells and systems. We demonstrate the use of formal reasoning methods as applied to the Caenorhabditis elegans germ line, which is an accessible system for stem cell research. The dynamics of the underlying genetic networks and their potential regulatory interactions are key for understanding mechanisms that control cellular decision-making between stem cells and differentiation. We model the “stem cell fate” versus entry into the “meiotic development” pathway decision circuit in the young adult germ line based on an extensive study of published experimental data and known/hypothesized genetic interactions. We apply a formal reasoning framework to derive predictive networks for control of differentiation. Using this approach we simultaneously specify many possible scenarios and experiments together with potential genetic interactions, and synthesize genetic networks consistent with all encoded experimental observations. In silico analysis of knock-down and overexpression experiments within our model recapitulate published phenotypes of mutant animals and can be applied to make predictions on cellular decision-making. A methodological contribution of this work is demonstrating how to effectively model within a formal reasoning framework a complex genetic network with a wealth of known experimental data and constraints. We provide a summary of the steps we have found useful for the development and analysis of this model and can potentially be applicable to other genetic networks. This work also lays a foundation for developing realistic whole tissue models of the C. elegans germ line where each cell in the model will execute a synthesized genetic network.

Elucidating multi-input processing 3-node gene regulatory network topologies capable of generating striped gene expression patterns

A central problem in developmental and synthetic biology is understanding the mechanisms by which cells in a tissue or a Petri dish process external cues and transform such information into a coherent response, e.g., a terminal differentiation state. It was long believed that this type of positional information could be entirely attributed to a gradient of concentration of a specific signaling molecule (i.e., a morphogen). However, advances in experimental methodologies and computer modeling have demonstrated the crucial role of the dynamics of a cell’s gene regulatory network (GRN) in decoding the information carried by the morphogen, which is eventually translated into a spatial pattern. This morphogen interpretation mechanism has gained much attention in systems biology as a tractable system to investigate the emergent properties of complex genotype-phenotype maps. In this study, we apply a Markov chain Monte Carlo (MCMC)-like algorithm to probe the design space of three-node GRNs with the ability to generate a band-like expression pattern (target phenotype) in the middle of an arrangement of 30 cells, which resemble a simple (1-D) morphogenetic field in a developing embryo. Unlike most modeling studies published so far, here we explore the space of GRN topologies with nodes having the potential to perceive the same input signal differently. This allows for a lot more flexibility during the search space process, and thus enables us to identify a larger set of potentially interesting and realizable morphogen interpretation mechanisms. Out of 2061 GRNs selected using the search space algorithm, we found 714 classes of network topologies that could correctly interpret the morphogen. Notably, the main network motif that generated the target phenotype in response to the input signal was the type 3 Incoherent Feed-Forward Loop (I3-FFL), which agrees with previous theoretical expectations and experimental observations. Particularly, compared to a previously reported pattern forming GRN topologies, we have uncovered a great variety of novel network designs, some of which might be worth inquiring through synthetic biology methodologies to test for the ability of network design with minimal regulatory complexity to interpret a developmental cue robustly.

Systems biology is a fast growing field largely powered by advances in high-performance computing and sophisticated mathematical modeling of biological systems. Based on these advances, we are now in a position to mechanistically understand and accurately predict the behavior of complex biological processes, including cell differentiation and spatial pattern formation during embryogenesis. In this article, we use an in silico approach to probe the design space of multi-input, three-node Gene Regulatory Networks (GRNs) capable of generating a striped gene expression pattern in the context of a simplified 1-D morphogenetic field.

System drift and speciation

Even if a species' phenotype does not change over evolutionary time, the underlying mechanism may change, as distinct molecular pathways can realize identical phenotypes. Here we use linear system theory to explore the consequences of this idea, describing how a gene network underlying a conserved phenotype evolves, as the genetic drift of small changes to these molecular pathways causes a population to explore the set of mechanisms with identical phenotypes. To do this, we model an organism's internal state as a linear system of differential equations for which the environment provides input and the phenotype is the output, in which context there exists an exact characterization of the set of all mechanisms that give the same input-output relationship. This characterization implies that selectively neutral directions in genotype space should be common and that the evolutionary exploration of these distinct but equivalent mechanisms can lead to the reproductive incompatibility of independently evolving populations. This evolutionary exploration, or system drift, is expected to proceed at a rate proportional to the amount of intrapopulation genetic variation divided by the effective population size ( Ne$N_e$ ). At biologically reasonable parameter values this could lead to substantial interpopulation incompatibility, and thus speciation, on a time scale of Ne$N_e$ generations. This model also naturally predicts Haldane's rule, thus providing a concrete explanation of why heterogametic hybrids tend to be disrupted more often than homogametes during the early stages of speciation.

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

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

Fitting thermodynamic-based models: Incorporating parameter sensitivity improves the performance of an evolutionary algorithm

A detailed comprehension of transcriptional regulation is critical to understanding the genetic control of development and disease across many different organisms. To more fully investigate the complex molecular interactions controlling the precise expression of genes, many groups have constructed mathematical models to complement their experimental approaches. A critical step in such studies is choosing the most appropriate parameter estimation algorithm to enable detailed analysis of the parameters that contribute to the models. In this study, we develop a novel set of evolutionary algorithms that use a pseudo-random Sobol Set to construct the initial population and incorporate parameter sensitivities into the adaptation of mutation rates, using local, global, and hybrid strategies. Comparison of the performance of these new algorithms to a number of current state-of-the-art global parameter estimation algorithms on a range of continuous test functions, as well as synthetic biological data representing models of gene regulatory systems, reveals improved performance of the new algorithms in terms of runtime, error and reproducibility. In addition, by analyzing the ability of these algorithms to fit datasets of varying quality, we provide the experimentalist with a guide to how the algorithms perform across a range of noisy data. These results demonstrate the improved performance of the new set of parameter estimation algorithms and facilitate meaningful integration of model parameters and predictions in our understanding of the molecular mechanisms of gene regulation.

Computational analysis of GAL pathway pinpoints mechanisms underlying natural variation

Quantitative traits are measurable phenotypes that show continuous variation over a wide phenotypic range. Enormous effort has recently been put into determining the genetic influences on a variety of quantitative traits with mixed success. We identified a quantitative trait in a tractable model system, the GAL pathway in yeast, which controls the uptake and metabolism of the sugar galactose. GAL pathway activation depends both on galactose concentration and on the concentrations of competing, preferred sugars such as glucose. Natural yeast isolates show substantial variation in the behavior of the pathway. All studied yeast strains exhibit bimodal responses relative to external galactose concentration, i.e. a set of galactose concentrations existed at which both GAL-induced and GAL-repressed subpopulations were observed. However, these concentrations differed in different strains. We built a mechanistic model of the GAL pathway and identified parameters that are plausible candidates for capturing the phenotypic features of a set of strains including standard lab strains, natural variants, and mutants. In silico perturbation of these parameters identified variation in the intracellular galactose sensor, Gal3p, the negative feedback node within the GAL regulatory network, Gal80p, and the hexose transporters, HXT, as the main sources of the bimodal range variation. We were able to switch the phenotype of individual yeast strains in silico by tuning parameters related to these three elements. Determining the basis for these behavioral differences may give insight into how the GAL pathway processes information, and into the evolution of nutrient metabolism preferences in different strains. More generally, our method of identifying the key parameters that explain phenotypic variation in this system should be generally applicable to other quantitative traits.

Microbes adopt elaborate strategies for the preferred uptake and use of nutrients to cope with complex and fluctuating environments. As a result, yeast strains originating from different ecological niches show significant variation in the way they induce genes in the galactose metabolism (GAL) pathway in response to nutrient signals. To identify the mechanistic sources of this variation, we built a mathematical model to simulate the dynamics of the galactose metabolic regulation network, and studied how parameters with different biological implications contributed to the natural variation. We found that variations in the behavior of the galactose sensor Gal3p, the negative feedback node Gal80p, and the hexose transporters HXT were critical elements in the GAL pathway response. Tuning single parameters in silico was sufficient to achieve phenotype switching between different yeast strains. Our computational approach should be generally useful to help pinpoint the genetic and molecular bases of natural variation in other systems.

The developmental hourglass model is applicable to the spinal cord based on single-cell transcriptomes and non-conserved cis-regulatory elements

The developmental hourglass model predicts that embryonic morphology is most conserved at the mid‐embryonic stage and diverges at the early and late stages. To date, this model has been verified by examining the anatomical features or gene expression profiles at the whole embryonic level. Here, by data mining approach utilizing multiple genomic and transcriptomic datasets from different species in combination, and by experimental validation, we demonstrate that the hourglass model is also applicable to a reduced element, the spinal cord. In the middle of spinal cord development, dorsoventrally arrayed neuronal progenitor domains are established, which are conserved among vertebrates. By comparing the publicly available single‐cell transcriptome datasets of mice and zebrafish, we found that ventral subpopulations of post‐mitotic spinal neurons display divergent molecular profiles. We also detected the non‐conservation of cis‐regulatory elements located around the progenitor fate determinants, indicating that the cis‐regulatory elements contributing to the progenitor specification are evolvable. These results demonstrate that, despite the conservation of the progenitor domains, the processes before and after the progenitor domain specification diverged. This study will be helpful to understand the molecular basis of the developmental hourglass model.

The evolution of the gene regulatory networks patterning the Drosophila Blastoderm

The Drosophila blastoderm gene regulatory network is one of the best studied networks in biology. It is composed of a series of tiered sub-networks that act sequentially to generate a primary segmental pattern. Many of these sub-networks have been studied in other arthropods, allowing us to reconstruct how each of them evolved over the transition from the arthropod ancestor to the situation seen in Drosophila today. I trace the evolution of each of these networks, showing how some of them have been modified significantly in Drosophila relative to the ancestral state while others are largely conserved across evolutionary timescales. I compare the putative ancestral arthropod segmentation network with that found in Drosophila and discuss how and why it has been modified throughout evolution, and to what extent this modification is unusual.

Development of the Pre-gnathal Segments in the Milkweed Bug Oncopeltus fasciatus Suggests They Are Not Serial Homologs of Trunk Segments

The three anterior-most segments in arthropods contain the ganglia that make up the arthropod brain. These segments, the pre-gnathal segments (PGS), are known to exhibit many developmental differences to other segments, believed to reflect their divergent morphology. We have analyzed the expression and function of the genes involved in the conserved segment-polarity network, including genes from the Wnt and Hedgehog pathways, in the PGS, compared with the trunk segments, in the hemimetabolous insect Oncopeltus fasciatus. Gene function was tested by manipulating expression through RNA interference against components of the two pathways. We show that there are fundamental differences in the expression patterns of the segment polarity genes, in the timing of their expression and in the interactions among them in the process of pre-gnathal segment generation, relative to all other segments. We argue that given these differences, the PGS should not be considered serially homologous to trunk segments. This realization raises important questions about the differing evolutionary ancestry of different regions of the arthropod head.

Serotonin signaling modulates aging-associated metabolic network integrity in response to nutrient choice in Drosophila melanogaster

Aging arises from complex interactions among multiple biochemical products. Systems-level analyses of biological networks may provide insights into the causes and consequences of aging that evade single-gene studies. We have previously found that dietary choice is sufficient to modulate aging in the vinegar fly, Drosophila melanogaster. Here we show that nutrient choice influenced several measures of metabolic network integrity, including connectivity, community structure, and robustness. Importantly, these effects are mediated by serotonin signaling, as a mutation in serotonin receptor 2A (5-HT2A) eliminated the effects of nutrient choice. Changes in network structure were associated with organism resilience and increased susceptibility to genetic perturbation. Our data suggest that the behavioral or perceptual consequences of exposure to individual macronutrients, involving serotonin signaling through 5-HT2A, qualitatively change the state of metabolic networks throughout the organism from one that is highly connected and robust to one that is fragmented, fragile, and vulnerable to perturbations.

An in vitro model of tumor heterogeneity resolves genetic, epigenetic, and stochastic sources of cell state variability

Tumor heterogeneity is a primary cause of treatment failure and acquired resistance in cancer patients. Even in cancers driven by a single mutated oncogene, variability in response to targeted therapies is well known. The existence of additional genomic alterations among tumor cells can only partially explain this variability. As such, nongenetic factors are increasingly seen as critical contributors to tumor relapse and acquired resistance in cancer. Here, we show that both genetic and nongenetic factors contribute to targeted drug response variability in an experimental model of tumor heterogeneity. We observe significant variability to epidermal growth factor receptor (EGFR) inhibition among and within multiple versions and clonal sublines of PC9, a commonly used EGFR mutant nonsmall cell lung cancer (NSCLC) cell line. We resolve genetic, epigenetic, and stochastic components of this variability using a theoretical framework in which distinct genetic states give rise to multiple epigenetic "basins of attraction," across which cells can transition driven by stochastic noise. Using mutational impact analysis, single-cell differential gene expression, and correlations among Gene Ontology (GO) terms to connect genomics to transcriptomics, we establish a baseline for genetic differences driving drug response variability among PC9 cell line versions. Applying the same approach to clonal sublines, we conclude that drug response variability in all but one of the sublines is due to epigenetic differences; in the other, it is due to genetic alterations. Finally, using a clonal drug response assay together with stochastic simulations, we attribute subclonal drug response variability within sublines to stochastic cell fate decisions and confirm that one subline likely contains genetic resistance mutations that emerged in the absence of drug treatment.

A comprehensive survey of developmental programs reveals a dearth of tree-like lineage graphs and ubiquitous regeneration

Background

Multicellular organisms are characterized by a wide diversity of forms and complexity despite a restricted set of key molecules and mechanisms at the base of organismal development. Development combines three basic processes—asymmetric cell division, signaling, and gene regulation—in a multitude of ways to create this overwhelming diversity of multicellular life forms. Here, we use a generative model to test the limits to which such processes can be combined to generate multiple differentiation paths during development, and attempt to chart the diversity of multicellular organisms generated.

Results

We sample millions of biologically feasible developmental schemes, allowing us to comment on the statistical properties of cell differentiation trajectories they produce. We characterize model-generated “organisms” using the graph topology of their cell type lineage maps. Remarkably, tree-type lineage differentiation maps are the rarest in our data. Additionally, a majority of the “organisms” generated by our model appear to be endowed with the ability to regenerate using pluripotent cells.

Conclusions

Our results indicate that, in contrast to common views, cell type lineage graphs are unlikely to be tree-like. Instead, they are more likely to be directed acyclic graphs, with multiple lineages converging on the same terminal cell type. Furthermore, the high incidence of pluripotent cells in model-generated organisms stands in line with the long-standing hypothesis that whole body regeneration is an epiphenomenon of development. We discuss experimentally testable predictions of our model and some ways to adapt the generative framework to test additional hypotheses about general features of development.

Supplementary Information

The online version contains supplementary material available at (10.1186/s12915-021-01013-4).

Dynamical systems approach to evolution-development congruence: Revisiting Haeckel's recapitulation theory

It is acknowledged that embryonic development has a tendency to proceed from common toward specific. Ernst Haeckel raised the question of why that tendency prevailed through evolution, and the question remains unsolved. Here, we revisit Haeckel's recapitulation theory, that is, the parallelism between evolution and development through numerical evolution and dynamical systems theory. By using intracellular gene expression dynamics with cell‐to‐cell interaction over spatially aligned cells to represent the developmental process, gene regulation networks (GRN) that govern these dynamics evolve under the selection pressure to achieve a prescribed spatial gene expression pattern. For most numerical evolutionary experiments, the evolutionary pattern changes over generations, as well as the developmental pattern changes governed by the evolved GRN exhibit remarkable similarity. Changes in both patterns consisted of several epochs where stripes are formed in a short time, whereas for other temporal regimes, the pattern hardly changes. In evolution, these quasi‐stationary generations are needed to achieve relevant mutations, whereas, in development, they are due to some gene expressions that vary slowly and control the pattern change. These successive epochal changes in development and evolution are represented as common bifurcations in dynamical systems theory, regulating working network structure from feedforward subnetwork to those containing feedback loops. The congruence is the correspondence between successive acquisitions of subnetworks through evolution and changes in working subnetworks in development. Consistency of the theory with the segmentation gene‐expression dynamics is discussed. Novel outlook on recapitulation and heterochrony are provided, testable experimentally by the transcriptome and network analysis.

We confirmed recapitulation, parallelism between evolution and development, by evolutionary simulation of pattern formation. Its origin is analyzed in terms of bifurcation in dynamical systems theory, slow‐mode control, and network structures.

Deciphering the Role of Wnt and Rho Signaling Pathway in iPSC-Derived ARVC Cardiomyocytes by In Silico Mathematical Modeling

Arrhythmogenic Right Ventricular cardiomyopathy (ARVC) is an inherited cardiac muscle disease linked to genetic deficiency in components of the desmosomes. The disease is characterized by progressive fibro-fatty replacement of the right ventricle, which acts as a substrate for arrhythmias and sudden cardiac death. The molecular mechanisms underpinning ARVC are largely unknown. Here we propose a mathematical model for investigating the molecular dynamics underlying heart remodeling and the loss of cardiac myocytes identity during ARVC. Our methodology is based on three computational models: firstly, in the context of the Wnt pathway, we examined two different competition mechanisms between β-catenin and Plakoglobin (PG) and their role in the expression of adipogenic program. Secondly, we investigated the role of RhoA-ROCK pathway in ARVC pathogenesis, and thirdly we analyzed the interplay between Wnt and RhoA-ROCK pathways in the context of the ARVC phenotype. We conclude with the following remark: both Wnt/β-catenin and RhoA-ROCK pathways must be inactive for a significant increase of PPARγ expression, suggesting that a crosstalk mechanism might be responsible for mediating ARVC pathogenesis.

Quantifying the roles of space and stochasticity in computer simulations for cell biology and cellular biochemistry

Most of the fascinating phenomena studied in cell biology emerge from interactions among highly organized multimolecular structures embedded into complex and frequently dynamic cellular morphologies. For the exploration of such systems, computer simulation has proved to be an invaluable tool, and many researchers in this field have developed sophisticated computational models for application to specific cell biological questions. However, it is often difficult to reconcile conflicting computational results that use different approaches to describe the same phenomenon. To address this issue systematically, we have defined a series of computational test cases ranging from very simple to moderately complex, varying key features of dimensionality, reaction type, reaction speed, crowding, and cell size. We then quantified how explicit spatial and/or stochastic implementations alter outcomes, even when all methods use the same reaction network, rates, and concentrations. For simple cases, we generally find minor differences in solutions of the same problem. However, we observe increasing discordance as the effects of localization, dimensionality reduction, and irreversible enzymatic reactions are combined. We discuss the strengths and limitations of commonly used computational approaches for exploring cell biological questions and provide a framework for decision making by researchers developing new models. As computational power and speed continue to increase at a remarkable rate, the dream of a fully comprehensive computational model of a living cell may be drawing closer to reality, but our analysis demonstrates that it will be crucial to evaluate the accuracy of such models critically and systematically.

Distinguishing serial homologs from novel traits: Experimental limitations and ideas for improvements

One of the central but yet unresolved problems in evolutionary biology concerns the origin of novel complex traits. One hypothesis is that complex traits derive from pre-existing gene regulatory networks (GRNs) reused and modified to specify a novel trait somewhere else in the body. This simple explanation encounters problems when the novel trait that emerges in a body is in a region that is known to harbor a latent or repressed trait that has been silent for millions of years. Is the novel trait merely a re-emerged de-repressed trait or a truly novel trait that emerged via a novel deployment of an old GRN? A couple of new studies sided on opposite sides of this question when investigating the origin of horns in dung beetles and helmets in treehoppers that develop in the first thoracic segment (T1) of their bodies, a segment known to harbor a pair of repressed/modified wings in close relatives. Here, I point to some key limitations of the experimental approaches used and highlight additional experiments that could be done in future to resolve the developmental origin of these and other traits.

Of mitogens and morphogens: modelling Sonic Hedgehog mechanisms in vertebrate development

Sonic Hedgehog (Shh) Is a critical protein in vertebrate development, orchestrating patterning and growth in many developing systems. First described as a classic morphogen that patterns tissues through a spatial concentration gradient, subsequent studies have revealed a more complex mechanism, in which Shh can also regulate proliferation and differentiation. While the mechanism of action of Shh as a morphogen is well understood, it remains less clear how Shh might integrate patterning, proliferation and differentiation in a given tissue, to ultimately direct its morphogenesis. In tandem with experimental studies, mathematical modelling can help gain mechanistic insights into these processes and bridge the gap between Shh-regulated patterning and growth, by integrating these processes into a common theoretical framework. Here, we briefly review the roles of Shh in vertebrate development, focusing on its functions as a morphogen, mitogen and regulator of differentiation. We then discuss the contributions that modelling has made to our understanding of the action of Shh and highlight current challenges in using mathematical models in a quantitative and predictive way. This article is part of a discussion meeting issue 'Contemporary morphogenesis'.

Genetic circuit characterization by inferring RNA polymerase movement and ribosome usage

To perform their computational function, genetic circuits change states through a symphony of genetic parts that turn regulator expression on and off. Debugging is frustrated by an inability to characterize parts in the context of the circuit and identify the origins of failures. Here, we take snapshots of a large genetic circuit in different states: RNA-seq is used to visualize circuit function as a changing pattern of RNA polymerase (RNAP) flux along the DNA. Together with ribosome profiling, all 54 genetic parts (promoters, ribozymes, RBSs, terminators) are parameterized and used to inform a mathematical model that can predict circuit performance, dynamics, and robustness. The circuit behaves as designed; however, it is riddled with genetic errors, including cryptic sense/antisense promoters and translation, attenuation, incorrect start codons, and a failed gate. While not impacting the expected Boolean logic, they reduce the prediction accuracy and could lead to failures when the parts are used in other designs. Finally, the cellular power (RNAP and ribosome usage) required to maintain a circuit state is calculated. This work demonstrates the use of a small number of measurements to fully parameterize a regulatory circuit and quantify its impact on host.

Canalization and Robustness in Human Genetics and Disease

Canalization refers to the evolution of populations such that the number of individuals who deviate from the optimum trait, or experience disease, is minimized. In the presence of rapid cultural, environmental, or genetic change, the reverse process of decanalization may contribute to observed increases in disease prevalence. This review starts by defining relevant concepts, drawing distinctions between the canalization of populations and robustness of individuals. It then considers evidence pertaining to three continuous traits and six domains of disease. In each case, existing genetic evidence for genotype-by-environment interactions is insufficient to support a strong inference of decanalization, but we argue that the advent of genome-wide polygenic risk assessment now makes an empirical evaluation of the role of canalization in preventing disease possible. Finally, the contributions of both rare and common variants to congenital abnormality and adult onset disease are considered in light of a new kerplunk model of genetic effects.

Enhancement of gene expression noise from transcription factor binding to genomic decoy sites

The genome contains several high-affinity non-functional binding sites for transcription factors (TFs) creating a hidden and unexplored layer of gene regulation. We investigate the role of such “decoy sites” in controlling noise (random fluctuations) in the level of a TF that is synthesized in stochastic bursts. Prior studies have assumed that decoy-bound TFs are protected from degradation, and in this case decoys function to buffer noise. Relaxing this assumption to consider arbitrary degradation rates for both bound/unbound TF states, we find rich noise behaviors. For low-affinity decoys, noise in the level of unbound TF always monotonically decreases to the Poisson limit with increasing decoy numbers. In contrast, for high-affinity decoys, noise levels first increase with increasing decoy numbers, before decreasing back to the Poisson limit. Interestingly, while protection of bound TFs from degradation slows the time-scale of fluctuations in the unbound TF levels, the decay of bound TFs leads to faster fluctuations and smaller noise propagation to downstream target proteins. In summary, our analysis reveals stochastic dynamics emerging from nonspecific binding of TFs and highlights the dual role of decoys as attenuators or amplifiers of gene expression noise depending on their binding affinity and stability of the bound TF.

On the problem of biological form

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

Mechanical Networks Have Robustness Built into Their Topology, Too

Network topology ensures robust outcomes for information-transferring networks in biology (e.g., signaling and gene regulatory networks). In this issue of Developmental Cell, Yevick et al. (2019) demonstrate that a similar principle holds for the mechanical networks that drive morphogenesis.

Robustness and parameter geography in post-translational modification systems

Biological systems are acknowledged to be robust to perturbations but a rigorous understanding of this has been elusive. In a mathematical model, perturbations often exert their effect through parameters, so sizes and shapes of parametric regions offer an integrated global estimate of robustness. Here, we explore this “parameter geography” for bistability in post-translational modification (PTM) systems. We use the previously developed “linear framework” for timescale separation to describe the steady-states of a two-site PTM system as the solutions of two polynomial equations in two variables, with eight non-dimensional parameters. Importantly, this approach allows us to accommodate enzyme mechanisms of arbitrary complexity beyond the conventional Michaelis-Menten scheme, which unrealistically forbids product rebinding. We further use the numerical algebraic geometry tools Bertini, Paramotopy, and alphaCertified to statistically assess the solutions to these equations at ∼10⁹ parameter points in total. Subject to sampling limitations, we find no bistability when substrate amount is below a threshold relative to enzyme amounts. As substrate increases, the bistable region acquires 8-dimensional volume which increases in an apparently monotonic and sigmoidal manner towards saturation. The region remains connected but not convex, albeit with a high visibility ratio. Surprisingly, the saturating bistable region occupies a much smaller proportion of the sampling domain under mechanistic assumptions more realistic than the Michaelis-Menten scheme. We find that bistability is compromised by product rebinding and that unrealistic assumptions on enzyme mechanisms have obscured its parametric rarity. The apparent monotonic increase in volume of the bistable region remains perplexing because the region itself does not grow monotonically: parameter points can move back and forth between monostability and bistability. We suggest mathematical conjectures and questions arising from these findings. Advances in theory and software now permit insights into parameter geography to be uncovered by high-dimensional, data-centric analysis.

Biological organisms are often said to have robust properties but it is difficult to understand how such robustness arises from molecular interactions. Here, we use a mathematical model to study how the molecular mechanism of protein modification exhibits the property of multiple internal states, which has been suggested to underlie memory and decision making. The robustness of this property is revealed by the size and shape, or “geography,” of the parametric region in which the property holds. We use advances in reducing model complexity and in rapidly solving the underlying equations, to extensively sample parameter points in an 8-dimensional space. We find that under realistic molecular assumptions the size of the region is surprisingly small, suggesting that generating multiple internal states with such a mechanism is much harder than expected. While the shape of the region appears straightforward, we find surprising complexity in how the region grows with increasing amounts of the modified substrate. Our approach uses statistical analysis of data generated from a model, rather than from experiments, but leads to precise mathematical conjectures about parameter geography and biological robustness.

Regulatory feedback on receptor and non-receptor synthesis for robust signaling

Elaborate regulatory feedback processes are thought to make biological development robust, that is, resistant to changes induced by genetic or environmental perturbations. How this might be done is still not completely understood. Previous numerical simulations on reaction-diffusion models of Dpp gradients in Drosophila wing imaginal disc have showed that feedback (of the Hill function type) on (signaling) receptors and/or non-(signaling) receptors are of limited effectiveness in promoting robustness. Spatial nonuniformity of the feedback processes has also been shown theoretically to lead to serious shape distortion and a principal cause for ineffectiveness. Through mathematical modeling and analysis, the present article shows that spatially uniform nonlocal feedback mechanisms typically modify gradient shape through a shape parameter (that does not change with location). This in turn enables us to uncover new multi-feedback instrument for effective promotion of robust signaling gradients.

A computational framework for a Lyapunov-enabled analysis of biochemical reaction networks

Complex molecular biological processes such as transcription and translation, signal transduction, post-translational modification cascades, and metabolic pathways can be described in principle by biochemical reactions that explicitly take into account the sophisticated network of chemical interactions regulating cell life. The ability to deduce the possible qualitative behaviors of such networks from a set of reactions is a central objective and an ongoing challenge in the field of systems biology. Unfortunately, the construction of complete mathematical models is often hindered by a pervasive problem: despite the wealth of qualitative graphical knowledge about network interactions, the form of the governing nonlinearities and/or the values of kinetic constants are hard to uncover experimentally. The kinetics can also change with environmental variations. This work addresses the following question: given a set of reactions and without assuming a particular form for the kinetics, what can we say about the asymptotic behavior of the network? Specifically, it introduces a class of networks that are "structurally (mono) attractive" meaning that they are incapable of exhibiting multiple steady states, oscillation, or chaos by virtue of their reaction graphs. These networks are characterized by the existence of a universal energy-like function called a Robust Lyapunov function (RLF). To find such functions, a finite set of rank-one linear systems is introduced, which form the extremals of a linear convex cone. The problem is then reduced to that of finding a common Lyapunov function for this set of extremals. Based on this characterization, a computational package, Lyapunov-Enabled Analysis of Reaction Networks (LEARN), is provided that constructs such functions or rules out their existence. An extensive study of biochemical networks demonstrates that LEARN offers a new unified framework. Basic motifs, three-body binding, and genetic networks are studied first. The work then focuses on cellular signalling networks including various post-translational modification cascades, phosphotransfer and phosphorelay networks, T-cell kinetic proofreading, and ERK signalling. The Ribosome Flow Model is also studied.

Lighting up the central dogma for predictive developmental biology

Although the last 30years have witnessed the mapping of the wiring diagrams of the gene regulatory networks that dictate cell fate and animal body plans, specific understanding building on such network diagrams that shows how DNA regulatory regions control gene expression lags far behind. These networks have yet to yield the predictive power necessary to, for example, calculate how the concentration dynamics of input transcription factors and DNA regulatory sequence prescribes output patterns of gene expression that, in turn, determine body plans themselves. Here, we argue that reaching a predictive understanding of developmental decision-making calls for an interplay between theory and experiment aimed at revealing how the regulation of the processes of the central dogma dictate network connections and how network topology guides cells toward their ultimate developmental fate. To make this possible, it is crucial to break free from the snapshot-based understanding of embryonic development facilitated by fixed-tissue approaches and embrace new technologies that capture the dynamics of developmental decision-making at the single cell level, in living embryos.

An in silico analysis of robust but fragile gene regulation links enhancer length to robustness

Organisms must ensure that expression of genes is directed to the appropriate tissues at the correct times, while simultaneously ensuring that these gene regulatory systems are robust to perturbation. This idea is captured by a mathematical concept called r-robustness, which says that a system is robust to a perturbation in up to r - 1 randomly chosen parameters. r-robustness implies that the biological system has a small number of sensitive parameters and that this number can be used as a robustness measure. In this work we use this idea to investigate the robustness of gene regulation using a sequence level model of the Drosophila melanogaster gene even-skipped. We consider robustness with respect to mutations of the enhancer sequence and with respect to changes of the transcription factor concentrations. We find that gene regulation is r-robust with respect to mutations in the enhancer sequence and identify a number of sensitive nucleotides. In both natural and in silico predicted enhancers, the number of nucleotides that are sensitive to mutation correlates negatively with the length of the sequence, meaning that longer sequences are more robust. The exact degree of robustness obtained is dependent not only on DNA sequence, but also on the local concentration of regulatory factors. We find that gene regulation can be remarkably sensitive to changes in transcription factor concentrations at the boundaries of expression features, while it is robust to perturbation elsewhere.

The Design Principles of Biochemical Timers: Circuits that Discriminate between Transient and Sustained Stimulation

Many cellular responses for which timing is critical display temporal filtering-the ability to suppress response until stimulated for longer than a given minimal time. To identify biochemical circuits capable of kinetic filtering, we comprehensively searched the space of three-node enzymatic networks. We define a metric of "temporal ultrasensitivity," the steepness of activation as a function of stimulus duration. We identified five classes of core network motifs capable of temporal filtering, each with distinct functional properties such as rejecting high-frequency noise, committing to response (bistability), and distinguishing between long stimuli. Combinations of the two most robust motifs, double inhibition (DI) and positive feedback with AND logic (PF_AND), underlie several natural timer circuits involved in processes such as cell cycle transitions, T cell activation, and departure from the pluripotent state. The biochemical network motifs described in this study form a basis for understanding common ways cells make dynamic decisions.