Classification of transient behaviours in a time-dependent toggle switch model

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.

Phenotypic noise and plasticity in cancer evolution

Non-genetic alterations can produce changes in a cell's phenotype. In cancer, these phenomena can influence a cell's fitness by conferring access to heritable, beneficial phenotypes. Herein, we argue that current discussions of 'phenotypic plasticity' in cancer evolution ignore a salient feature of the original definition: namely, that it occurs in response to an environmental change. We suggest 'phenotypic noise' be used to distinguish non-genetic changes in phenotype that occur independently from the environment. We discuss the conceptual and methodological techniques used to identify these phenomena during cancer evolution. We propose that the distinction will guide efforts to define mechanisms of phenotype change, accelerate translational work to manipulate phenotypes through treatment, and, ultimately, improve patient outcomes.

Unveiling the Dynamics behind Glioblastoma Multiforme Single-Cell Data Heterogeneity

Glioblastoma Multiforme is a brain tumor distinguished by its aggressiveness. We suggested that this aggressiveness leads single-cell RNA-sequence data (scRNA-seq) to span a representative portion of the cancer attractors domain. This conjecture allowed us to interpret the scRNA-seq heterogeneity as reflecting a representative trajectory within the attractor's domain. We considered factors such as genomic instability to characterize the cancer dynamics through stochastic fixed points. The fixed points were derived from centroids obtained through various clustering methods to verify our method sensitivity. This methodological foundation is based upon sample and time average equivalence, assigning an interpretative value to the data cluster centroids and supporting parameters estimation. We used stochastic simulations to reproduce the dynamics, and our results showed an alignment between experimental and simulated dataset centroids. We also computed the Waddington landscape, which provided a visual framework for validating the centroids and standard deviations as characterizations of cancer attractors. Additionally, we examined the stability and transitions between attractors and revealed a potential interplay between subtypes. These transitions might be related to cancer recurrence and progression, connecting the molecular mechanisms of cancer heterogeneity with statistical properties of gene expression dynamics. Our work advances the modeling of gene expression dynamics and paves the way for personalized therapeutic interventions.

Tipping points in epithelial-mesenchymal lineages from single-cell transcriptomics data

Understanding cell fate decision-making during complex biological processes is an open challenge that is now aided by high-resolution single-cell sequencing technologies. Specifically, it remains challenging to identify and characterize transition states corresponding to "tipping points" whereby cells commit to new cell states. Here, we present a computational method that takes advantage of single-cell transcriptomics data to infer the stability and gene regulatory networks (GRNs) along cell lineages. Our method uses the unspliced and spliced counts from single-cell RNA sequencing data and cell ordering along lineage trajectories to train an RNA splicing multivariate model, from which cell-state stability along the lineage is inferred based on spectral analysis of the model's Jacobian matrix. Moreover, the model infers the RNA cross-species interactions resulting in GRNs and their variation along the cell lineage. When applied to epithelial-mesenchymal transition in ovarian and lung cancer-derived cell lines, our model predicts a saddle-node transition between the epithelial and mesenchymal states passing through an unstable, intermediate cell state. Furthermore, we show that the underlying GRN controlling epithelial-mesenchymal transition rearranges during the transition, resulting in denser and less modular networks in the intermediate state. Overall, our method represents a flexible tool to study cell lineages with a combination of theory-driven modeling and single-cell transcriptomics data.

Rethinking cancer initiation: The role of large-scale mutational events

Cancer initiation is revisited in light of recent discoveries in cancer pathogenesis. Of note is the detection of mutated cancer genes in benign conditions. More significantly, somatic clones, which harbor mutations in cancer genes, arise in normal tissues from early development through adulthood, but seldom do they transform into cancer. Further, clustered mutational events-kataegis, chromothripsis and chromoplexy-are widespread in cancer, generating point mutations and chromosomal rearrangements in a single cellular catastrophe. These observations are contrary to the prevailing somatic mutation theory, which states that a cancer is caused by the gradual accumulation of mutations over time. A different perspective is proposed within the framework of Waddington's epigenetic landscape wherein tumorigenesis is viewed primarily as a disruption of cell development. Cell types are defined by their specific gene-expression profiles, determined by the gene regulatory network, and can be regarded as attractor states of the network dynamics: they represent specific, self-stabilizing patterns of gene activities across the genome. However, large-scale mutational events reshape the landscape topology, creating abnormal "unphysiological" attractors. This is the crux of the process of initiation. Initiation primes the cell for conversion into a tumor phenotype by oncogenes and tumor suppressor genes, which drive cell proliferation and clonal diversification. This view of tumorigenesis calls for a different approach to therapy.

Automated design of gene circuits with optimal mushroom-bifurcation behavior

Recent advances in synthetic biology are enabling exciting technologies, including the next generation of biosensors, the rational design of cell memory, modulated synthetic cell differentiation, and generic multifunctional biocircuits. These novel applications require the design of gene circuits leading to sophisticated behaviors and functionalities. At the same time, designs need to be kept minimal to avoid compromising cell viability. Bifurcation theory addresses such challenges by associating circuit dynamical properties with molecular details of its design. Nevertheless, incorporating bifurcation analysis into automated design processes has not been accomplished yet. This work presents an optimization-based method for the automated design of synthetic gene circuits with specified bifurcation diagrams that employ minimal network topologies. Using this approach, we designed circuits exhibiting the mushroom bifurcation, distilled the most robust topologies, and explored its multifunctional behavior. We then outline potential applications in biosensors, memory devices, and synthetic cell differentiation.

Single-cell multi-omics and lineage tracing to dissect cell fate decision-making

The concept of cell fate relates to the future identity of a cell, and its daughters, which is obtained via cell differentiation and division. Understanding, predicting, and manipulating cell fate has been a long-sought goal of developmental and regenerative biology. Recent insights obtained from single-cell genomic and integrative lineage-tracing approaches have further aided to identify molecular features predictive of cell fate. In this perspective, we discuss these approaches with a focus on theoretical concepts and future directions of the field to dissect molecular mechanisms underlying cell fate.

spliceJAC: transition genes and state-specific gene regulation from single-cell transcriptome data

Extracting dynamical information from single-cell transcriptomics is a novel task with the promise to advance our understanding of cell state transition and interactions between genes. Yet, theory-oriented, bottom-up approaches that consider differences among cell states are largely lacking. Here, we present spliceJAC, a method to quantify the multivariate mRNA splicing from single-cell RNA sequencing (scRNA-seq). spliceJAC utilizes the unspliced and spliced mRNA count matrices to constructs cell state-specific gene-gene regulatory interactions and applies stability analysis to predict putative driver genes critical to the transitions between cell states. By applying spliceJAC to biological systems including pancreas endothelium development and epithelial-mesenchymal transition (EMT) in A549 lung cancer cells, we predict genes that serve specific signaling roles in different cell states, recover important differentially expressed genes in agreement with pre-existing analysis, and predict new transition genes that are either exclusive or shared between different cell state transitions.

Context-Dependent Stability and Robustness of Genetic Toggle Switches with Leaky Promoters

Multistable switches are ubiquitous building blocks in both systems and synthetic biology. Given their central role, it is thus imperative to understand how their fundamental properties depend not only on the tunable biophysical properties of the switches themselves, but also on their genetic context. To this end, we reveal in this article how these factors shape the essential characteristics of toggle switches implemented using leaky promoters such as their stability and robustness to noise, both at single-cell and population levels. In particular, our results expose the roles that competition for scarce transcriptional and translational resources, promoter leakiness, and cell-to-cell heterogeneity collectively play. For instance, the interplay between protein expression from leaky promoters and the associated cost of relying on shared cellular resources can give rise to tristable dynamics even in the absence of positive feedback. Similarly, we demonstrate that while promoter leakiness always acts against multistability, resource competition can be leveraged to counteract this undesirable phenomenon. Underpinned by a mechanistic model, our results thus enable the context-aware rational design of multistable genetic switches that are directly translatable to experimental considerations, and can be further leveraged during the synthesis of large-scale genetic systems using computer-aided biodesign automation platforms.

Homology of process: developmental dynamics in comparative biology

Comparative biology builds up systematic knowledge of the diversity of life, across evolutionary lineages and levels of organization, starting with evidence from a sparse sample of model organisms. In developmental biology, a key obstacle to the growth of comparative approaches is that the concept of homology is not very well defined for levels of organization that are intermediate between individual genes and morphological characters. In this paper, we investigate what it means for ontogenetic processes to be homologous, focusing specifically on the examples of insect segmentation and vertebrate somitogenesis. These processes can be homologous without homology of the underlying genes or gene networks, since the latter can diverge over evolutionary time, while the dynamics of the process remain the same. Ontogenetic processes like these therefore constitute a dissociable level and distinctive unit of comparison requiring their own specific criteria of homology. In addition, such processes are typically complex and nonlinear, such that their rigorous description and comparison requires not only observation and experimentation, but also dynamical modelling. We propose six criteria of process homology, combining recognized indicators (sameness of parts, morphological outcome and topological position) with novel ones derived from dynamical systems modelling (sameness of dynamical properties, dynamical complexity and evidence for transitional forms). We show how these criteria apply to animal segmentation and other ontogenetic processes. We conclude by situating our proposed dynamical framework for homology of process in relation to similar research programmes, such as process structuralism and developmental approaches to morphological homology.

Time and space in segmentation

Arthropod segmentation and vertebrate somitogenesis are leading fields in the experimental and theoretical interrogation of developmental patterning. However, despite the sophistication of current research, basic conceptual issues remain unresolved. These include: (i) the mechanistic origins of spatial organization within the segment addition zone (SAZ); (ii) the mechanistic origins of segment polarization; (iii) the mechanistic origins of axial variation; and (iv) the evolutionary origins of simultaneous patterning. Here, I explore these problems using coarse-grained models of cross-regulating dynamical processes. In the morphogenetic framework of a row of cells undergoing axial elongation, I simulate interactions between an 'oscillator', a 'switch' and up to three 'timers', successfully reproducing essential patterning behaviours of segmenting systems. By comparing the output of these largely cell-autonomous models to variants that incorporate positional information, I find that scaling relationships, wave patterns and patterning dynamics all depend on whether the SAZ is regulated by temporal or spatial information. I also identify three mechanisms for polarizing oscillator output, all of which functionally implicate the oscillator frequency profile. Finally, I demonstrate significant dynamical and regulatory continuity between sequential and simultaneous modes of segmentation. I discuss these results in the context of the experimental literature.

Implications of diffusion and time-varying morphogen gradients for the dynamic positioning and precision of bistable gene expression boundaries

The earliest models for how morphogen gradients guide embryonic patterning failed to account for experimental observations of temporal refinement in gene expression domains. Following theoretical and experimental work in this area, dynamic positional information has emerged as a conceptual framework to discuss how cells process spatiotemporal inputs into downstream patterns. Here, we show that diffusion determines the mathematical means by which bistable gene expression boundaries shift over time, and therefore how cells interpret positional information conferred from morphogen concentration. First, we introduce a metric for assessing reproducibility in boundary placement or precision in systems where gene products do not diffuse, but where morphogen concentrations are permitted to change in time. We show that the dynamics of the gradient affect the sensitivity of the final pattern to variation in initial conditions, with slower gradients reducing the sensitivity. Second, we allow gene products to diffuse and consider gene expression boundaries as propagating wavefronts with velocity modulated by local morphogen concentration. We harness this perspective to approximate a PDE model as an ODE that captures the position of the boundary in time, and demonstrate the approach with a preexisting model for Hunchback patterning in fruit fly embryos. We then propose a design that employs antiparallel morphogen gradients to achieve accurate boundary placement that is robust to scaling. Throughout our work we draw attention to tradeoffs among initial conditions, boundary positioning, and the relative timescales of network and gradient evolution. We conclude by suggesting that mathematical theory should serve to clarify not just our quantitative, but also our intuitive understanding of patterning processes.

Quantifying cell transitions in C. elegans with data-fitted landscape models

Increasing interest has emerged in new mathematical approaches that simplify the study of complex differentiation processes by formalizing Waddington's landscape metaphor. However, a rational method to build these landscape models remains an open problem. Here we study vulval development in C. elegans by developing a framework based on Catastrophe Theory (CT) and approximate Bayesian computation (ABC) to build data-fitted landscape models. We first identify the candidate qualitative landscapes, and then use CT to build the simplest model consistent with the data, which we quantitatively fit using ABC. The resulting model suggests that the underlying mechanism is a quantifiable two-step decision controlled by EGF and Notch-Delta signals, where a non-vulval/vulval decision is followed by a bistable transition to the two vulval states. This new model fits a broad set of data and makes several novel predictions.

An ABA-GA bistable switch can account for natural variation in the variability of Arabidopsis seed germination time

Genetically identical plants growing in the same conditions can display heterogeneous phenotypes. Here we use Arabidopsis seed germination time as a model system to examine phenotypic variability and its underlying mechanisms. We show extensive variation in seed germination time variability between Arabidopsis accessions and use a multiparent recombinant inbred population to identify two genetic loci involved in this trait. Both loci include genes implicated in modulating abscisic acid (ABA) sensitivity. Mutually antagonistic regulation between ABA, which represses germination, and gibberellic acid (GA), which promotes germination, underlies the decision to germinate and can act as a bistable switch. A simple stochastic model of the ABA-GA network shows that modulating ABA sensitivity can generate the range of germination time distributions we observe experimentally. We validate the model by testing its predictions on the effects of exogenous hormone addition. Our work provides a foundation for understanding the mechanism and functional role of phenotypic variability in germination time.

Stability and Robustness of Unbalanced Genetic Toggle Switches in the Presence of Scarce Resources

While the vision of synthetic biology is to create complex genetic systems in a rational fashion, system-level behaviors are often perplexing due to the context-dependent dynamics of modules. One major source of context-dependence emerges due to the limited availability of shared resources, coupling the behavior of disconnected components. Motivated by the ubiquitous role of toggle switches in genetic circuits ranging from controlling cell fate differentiation to optimizing cellular performance, here we reveal how their fundamental dynamic properties are affected by competition for scarce resources. Combining a mechanistic model with nullcline-based stability analysis and potential landscape-based robustness analysis, we uncover not only the detrimental impacts of resource competition, but also how the unbalancedness of the switch further exacerbates them. While in general both of these factors undermine the performance of the switch (by pushing the dynamics toward monostability and increased sensitivity to noise), we also demonstrate that some of the unwanted effects can be alleviated by strategically optimized resource competition. Our results provide explicit guidelines for the context-aware rational design of toggle switches to mitigate our reliance on lengthy and expensive trial-and-error processes, and can be seamlessly integrated into the computer-aided synthesis of complex genetic systems.

Controlling spatiotemporal pattern formation in a concentration gradient with a synthetic toggle switch

The formation of spatiotemporal patterns of gene expression is frequently guided by gradients of diffusible signaling molecules. The toggle switch subnetwork, composed of two cross-repressing transcription factors, is a common component of gene regulatory networks in charge of patterning, converting the continuous information provided by the gradient into discrete abutting stripes of gene expression. We present a synthetic biology framework to understand and characterize the spatiotemporal patterning properties of the toggle switch. To this end, we built a synthetic toggle switch controllable by diffusible molecules in Escherichia coli. We analyzed the patterning capabilities of the circuit by combining quantitative measurements with a mathematical reconstruction of the underlying dynamical system. The toggle switch can produce robust patterns with sharp boundaries, governed by bistability and hysteresis. We further demonstrate how the hysteresis, position, timing, and precision of the boundary can be controlled, highlighting the dynamical flexibility of the circuit.

Organization at criticality enables processing of time-varying signals by receptor networks

How cells utilize surface receptors for chemoreception is a recurrent question spanning between physics and biology over the past few decades. However, the dynamical mechanism for processing time-varying signals is still unclear. Using dynamical systems formalism to describe criticality in non-equilibrium systems, we propose generic principle for temporal information processing through phase space trajectories using dynamic transient memory. In contrast to short-term memory, dynamic memory generated via "ghost" attractor enables signal integration depending on stimulus history and thereby uniquely promotes integrating and interpreting complex temporal growth factor signals. We argue that this is a generic feature of receptor networks, the first layer of the cell that senses the changing environment. Using the experimentally established epidermal growth factor sensing system, we propose how recycling could provide self-organized maintenance of the critical receptor concentration at the plasma membrane through a simple, fluctuation-sensing mechanism. Processing of non-stationary signals, a feature previously attributed only to neural networks, thus uniquely emerges for receptor networks organized at criticality.

Intracellular Energy Variability Modulates Cellular Decision-Making Capacity

Cells generate phenotypic diversity both during development and in response to stressful and changing environments, aiding survival. Functionally vital cell fate decisions from a range of phenotypic choices are made by regulatory networks, the dynamics of which rely on gene expression and hence depend on the cellular energy budget (and particularly ATP levels). However, despite pronounced cell-to-cell ATP differences observed across biological systems, the influence of energy availability on regulatory network dynamics is often overlooked as a cellular decision-making modulator, limiting our knowledge of how energy budgets affect cell behaviour. Here, we consider a mathematical model of a highly generalisable, ATP-dependent, decision-making regulatory network, and show that cell-to-cell ATP variability changes the sets of decisions a cell can make. Our model shows that increasing intracellular energy levels can increase the number of supported stable phenotypes, corresponding to increased decision-making capacity. Model cells with sub-threshold intracellular energy are limited to a singular phenotype, forcing the adoption of a specific cell fate. We suggest that energetic differences between cells may be an important consideration to help explain observed variability in cellular decision-making across biological systems.

Dynamic positional information: Patterning mechanism versus precision in gradient-driven systems

There is much talk about information in biology. In developmental biology, this takes the form of "positional information," especially in the context of morphogen-based pattern formation. Unfortunately, the concept of "information" is rarely defined in any precise manner. Here, we provide two alternative interpretations of "positional information," and examine the complementary meanings and uses of each concept. Positional information defined as Shannon information helps us understand decoding and error propagation in patterning systems. General relativistic positional information, in contrast, provides a metric to assess the output of pattern-forming mechanisms. Both interpretations provide powerful conceptual tools that do not compete, but are best used in combination to gain a proper mechanistic understanding of robust patterning.

Discrete and continuous models of probability flux of switching dynamics: Uncovering stochastic oscillations in a toggle-switch system

The probability flux and velocity in stochastic reaction networks can help in characterizing dynamic changes in probability landscapes of these networks. Here, we study the behavior of three different models of probability flux, namely, the discrete flux model, the Fokker-Planck model, and a new continuum model of the Liouville flux. We compare these fluxes that are formulated based on, respectively, the chemical master equation, the stochastic differential equation, and the ordinary differential equation. We examine similarities and differences among these models at the nonequilibrium steady state for the toggle switch network under different binding and unbinding conditions. Our results show that at a strong stochastic condition of weak promoter binding, continuum models of Fokker-Planck and Liouville fluxes deviate significantly from the discrete flux model. Furthermore, we report the discovery of stochastic oscillation in the toggle-switch system occurring at weak binding conditions, a phenomenon captured only by the discrete flux model.

Arthropod segmentation

There is now compelling evidence that many arthropods pattern their segments using a clock-and-wavefront mechanism, analogous to that operating during vertebrate somitogenesis. In this Review, we discuss how the arthropod segmentation clock generates a repeating sequence of pair-rule gene expression, and how this is converted into a segment-polarity pattern by ‘timing factor’ wavefronts associated with axial extension. We argue that the gene regulatory network that patterns segments may be relatively conserved, although the timing of segmentation varies widely, and double-segment periodicity appears to have evolved at least twice. Finally, we describe how the repeated evolution of a simultaneous (Drosophila-like) mode of segmentation within holometabolan insects can be explained by heterochronic shifts in timing factor expression plus extensive pre-patterning of the pair-rule genes.

Modularity, criticality, and evolvability of a developmental gene regulatory network

The existence of discrete phenotypic traits suggests that the complex regulatory processes which produce them are functionally modular. These processes are usually represented by networks. Only modular networks can be partitioned into intelligible subcircuits able to evolve relatively independently. Traditionally, functional modularity is approximated by detection of modularity in network structure. However, the correlation between structure and function is loose. Many regulatory networks exhibit modular behaviour without structural modularity. Here we partition an experimentally tractable regulatory network-the gap gene system of dipteran insects-using an alternative approach. We show that this system, although not structurally modular, is composed of dynamical modules driving different aspects of whole-network behaviour. All these subcircuits share the same regulatory structure, but differ in components and sensitivity to regulatory interactions. Some subcircuits are in a state of criticality, while others are not, which explains the observed differential evolvability of the various expression features in the system.

Approximating the energy landscape of a two-dimensional bistable gene autoregulation model by separating slow and fast dynamics

The energy landscape is widely used to quantify the stability of multistable nonlinear systems, such as bistable gene regulation networks. In physics, the potential can be obtained through integration only for gradient systems. However, multidimensional nonlinear systems are often nongradient, for which the potential is calculated by decomposing the dynamics to gradient and nongradient parts. This potential is then called a quasipotential. Given that one-dimensional (1D) systems can be regarded as gradient systems, we attempt to separate the two-dimensional (2D) system into two 1D systems working on distinct timescales, and the potential can be easily calculated for the two 1D systems separately. This method is used in this study to estimate the energy landscape of a two-variable gene autoregulation model. This elegant and comprehensive method is accessible for 2D nonlinear systems in which the dynamics can be divided into slow and fast parts.

Modulation of dynamic modes by interplay between positive and negative feedback loops in gene regulatory networks

A positive and a negative feedback loop can induce bistability and oscillation, respectively, in biological networks. Nevertheless, they are frequently interlinked to perform more elaborate functions in many gene regulatory networks. Coupled positive and negative feedback loops may exhibit either oscillation or bistability depending on the intensity of the stimulus in some particular networks. It is less understood how the transition between the two dynamic modes is modulated by the positive and negative feedback loops. We developed an abstract model of such systems, largely based on the core p53 pathway, to explore the mechanism for the transformation of dynamic behaviors. Our results show that enhancing the positive feedback may promote or suppress oscillations depending on the strength of both feedback loops. We found that the system oscillates with low amplitudes in response to a moderate stimulus and switches to the on state upon a strong stimulus. When the positive feedback is activated much later than the negative one in response to a strong stimulus, the system exhibits long-term oscillations before switching to the on state. We explain this intriguing phenomenon using quasistatic approximation. Moreover, early switching to the on state may occur when the system starts from a steady state in the absence of stimuli. The interplay between the positive and negative feedback plays a key role in the transitions between oscillation and bistability. Of note, our conclusions should be applicable only to some specific gene regulatory networks, especially the p53 network, in which both oscillation and bistability exist in response to a certain type of stimulus. Our work also underscores the significance of transient dynamics in determining cellular outcome.

Computing with biological switches and clocks

The complex dynamics of biological systems is primarily driven by molecular interactions that underpin the regulatory networks of cells. These networks typically contain positive and negative feedback loops, which are responsible for switch-like and oscillatory dynamics, respectively. Many computing systems rely on switches and clocks as computational modules. While the combination of such modules in biological systems leads to a variety of dynamical behaviours, it is also driving development of new computing algorithms. Here we present a historical perspective on computation by biological systems, with a focus on switches and clocks, and discuss parallels between biology and computing. We also outline our vision for the future of biological computing.

Minimum Action Path Theory Reveals the Details of Stochastic Transitions Out of Oscillatory States

Cell state determination is the outcome of intrinsically stochastic biochemical reactions. Transitions between such states are studied as noise-driven escape problems in the chemical species space. Escape can occur via multiple possible multidimensional paths, with probabilities depending nonlocally on the noise. Here we characterize the escape from an oscillatory biochemical state by minimizing the Freidlin-Wentzell action, deriving from it the stochastic spiral exit path from the limit cycle. We also use the minimized action to infer the escape time probability density function.

Combining a Toggle Switch and a Repressilator within the AC-DC Circuit Generates Distinct Dynamical Behaviors

Although the structure of a genetically encoded regulatory circuit is an important determinant of its function, the relationship between circuit topology and the dynamical behaviors it can exhibit is not well understood. Here, we explore the range of behaviors available to the AC-DC circuit. This circuit consists of three genes connected as a combination of a toggle switch and a repressilator. Using dynamical systems theory, we show that the AC-DC circuit exhibits both oscillations and bistability within the same region of parameter space; this generates emergent behaviors not available to either the toggle switch or the repressilator alone. The AC-DC circuit can switch on oscillations via two distinct mechanisms, one of which induces coherence into ensembles of oscillators. In addition, we show that in the presence of noise, the AC-DC circuit can behave as an excitable system capable of spatial signal propagation or coherence resonance. Together, these results demonstrate how combinations of simple motifs can exhibit multiple complex behaviors.

A damped oscillator imposes temporal order on posterior gap gene expression in Drosophila

Insects determine their body segments in two different ways. Short-germband insects, such as the flour beetle Tribolium castaneum, use a molecular clock to establish segments sequentially. In contrast, long-germband insects, such as the vinegar fly Drosophila melanogaster, determine all segments simultaneously through a hierarchical cascade of gene regulation. Gap genes constitute the first layer of the Drosophila segmentation gene hierarchy, downstream of maternal gradients such as that of Caudal (Cad). We use data-driven mathematical modelling and phase space analysis to show that shifting gap domains in the posterior half of the Drosophila embryo are an emergent property of a robust damped oscillator mechanism, suggesting that the regulatory dynamics underlying long- and short-germband segmentation are much more similar than previously thought. In Tribolium, Cad has been proposed to modulate the frequency of the segmentation oscillator. Surprisingly, our simulations and experiments show that the shift rate of posterior gap domains is independent of maternal Cad levels in Drosophila. Our results suggest a novel evolutionary scenario for the short- to long-germband transition and help explain why this transition occurred convergently multiple times during the radiation of the holometabolan insects.

Different insect species exhibit one of two distinct modes of determining their body segments (known as segmentation) during development: they either use a molecular oscillator to position segments sequentially, or they generate segments simultaneously through a hierarchical gene-regulatory cascade. The sequential mode is ancestral, while the simultaneous mode has been derived from it independently several times during evolution. In this paper, we present evidence suggesting that simultaneous segmentation also involves an oscillator in the posterior end of the embryo of the vinegar fly, Drosophila melanogaster. This surprising result indicates that both modes of segment determination are much more similar than previously thought. Such similarity provides an important step towards our understanding of the frequent evolutionary transitions observed between sequential and simultaneous segmentation.

Achilles and the tortoise: Some caveats to mathematical modeling in biology

Mathematical modeling has recently become a much-lauded enterprise, and many funding agencies seek to prioritize this endeavor. However, there are certain dangers associated with mathematical modeling, and knowledge of these pitfalls should also be part of a biologist's training in this set of techniques. (1) Mathematical models are limited by known science; (2) Mathematical models can tell what can happen, but not what did happen; (3) A model does not have to conform to reality, even if it is logically consistent; (4) Models abstract from reality, and sometimes what they eliminate is critically important; (5) Mathematics can present a Platonic ideal to which biologically organized matter strives, rather than a trial-and-error bumbling through evolutionary processes. This "Unity of Science" approach, which sees biology as the lowest physical science and mathematics as the highest science, is part of a Western belief system, often called the Great Chain of Being (or Scala Natura), that sees knowledge emerge as one passes from biology to chemistry to physics to mathematics, in an ascending progression of reason being purification from matter. This is also an informal model for the emergence of new life. There are now other informal models for integrating development and evolution, but each has its limitations.

A spectrum of modularity in multi-functional gene circuits

A major challenge in systems biology is to understand the relationship between a circuit's structure and its function, but how is this relationship affected if the circuit must perform multiple distinct functions within the same organism? In particular, to what extent do multi‐functional circuits contain modules which reflect the different functions? Here, we computationally survey a range of bi‐functional circuits which show no simple structural modularity: They can switch between two qualitatively distinct functions, while both functions depend on all genes of the circuit. Our analysis reveals two distinct classes: hybrid circuits which overlay two simpler mono‐functional sub‐circuits within their circuitry, and emergent circuits, which do not. In this second class, the bi‐functionality emerges from more complex designs which are not fully decomposable into distinct modules and are consequently less intuitive to predict or understand. These non‐intuitive emergent circuits are just as robust as their hybrid counterparts, and we therefore suggest that the common bias toward studying modular systems may hinder our understanding of real biological circuits.

Dynamic Maternal Gradients Control Timing and Shift-Rates for Drosophila Gap Gene Expression

Pattern formation during development is a highly dynamic process. In spite of this, few experimental and modelling approaches take into account the explicit time-dependence of the rules governing regulatory systems. We address this problem by studying dynamic morphogen interpretation by the gap gene network in Drosophila melanogaster. Gap genes are involved in segment determination during early embryogenesis. They are activated by maternal morphogen gradients encoded by bicoid (bcd) and caudal (cad). These gradients decay at the same time-scale as the establishment of the antero-posterior gap gene pattern. We use a reverse-engineering approach, based on data-driven regulatory models called gene circuits, to isolate and characterise the explicitly time-dependent effects of changing morphogen concentrations on gap gene regulation. To achieve this, we simulate the system in the presence and absence of dynamic gradient decay. Comparison between these simulations reveals that maternal morphogen decay controls the timing and limits the rate of gap gene expression. In the anterior of the embyro, it affects peak expression and leads to the establishment of smooth spatial boundaries between gap domains. In the posterior of the embryo, it causes a progressive slow-down in the rate of gap domain shifts, which is necessary to correctly position domain boundaries and to stabilise the spatial gap gene expression pattern. We use a newly developed method for the analysis of transient dynamics in non-autonomous (time-variable) systems to understand the regulatory causes of these effects. By providing a rigorous mechanistic explanation for the role of maternal gradient decay in gap gene regulation, our study demonstrates that such analyses are feasible and reveal important aspects of dynamic gene regulation which would have been missed by a traditional steady-state approach. More generally, it highlights the importance of transient dynamics for understanding complex regulatory processes in development.

Animal development is a highly dynamic process. Biochemical or environmental signals can cause the rules that shape it to change over time. We know little about the effects of such changes. For the sake of simplicity, we usually leave them out of our models and experimental assays. Here, we do exactly the opposite. We characterise precisely those aspects of pattern formation caused by changing signalling inputs to a gene regulatory network, the gap gene system of Drosophila melanogaster. Gap genes are involved in determining the body segments of flies and other insects during early development. Gradients of maternal morphogens activate the expression of the gap genes. These gradients are highly dynamic themselves, as they decay while being read out. We show that this decay controls the peak concentration of gap gene products, produces smooth boundaries of gene expression, and slows down the observed positional shifts of gap domains in the posterior of the embryo, thereby stabilising the spatial pattern. Our analysis demonstrates that the dynamics of gene regulation not only affect the timing, but also the positioning of gene expression. This suggests that we must pay closer attention to transient dynamic aspects of development than is currently the case.

Intrinsic Noise Profoundly Alters the Dynamics and Steady State of Morphogen-Controlled Bistable Genetic Switches

During tissue development, patterns of gene expression determine the spatial arrangement of cell types. In many cases, gradients of secreted signalling molecules—morphogens—guide this process by controlling downstream transcriptional networks. A mechanism commonly used in these networks to convert the continuous information provided by the gradient into discrete transitions between adjacent cell types is the genetic toggle switch, composed of cross-repressing transcriptional determinants. Previous analyses have emphasised the steady state output of these mechanisms. Here, we explore the dynamics of the toggle switch and use exact numerical simulations of the kinetic reactions, the corresponding Chemical Langevin Equation, and Minimum Action Path theory to establish a framework for studying the effect of gene expression noise on patterning time and boundary position. This provides insight into the time scale, gene expression trajectories and directionality of stochastic switching events between cell states. Taking gene expression noise into account predicts that the final boundary position of a morphogen-induced toggle switch, although robust to changes in the details of the noise, is distinct from that of the deterministic system. Moreover, the dramatic increase in patterning time close to the boundary predicted from the deterministic case is substantially reduced. The resulting stochastic switching introduces differences in patterning time along the morphogen gradient that result in a patterning wave propagating away from the morphogen source with a velocity determined by the intrinsic noise. The wave sharpens and slows as it advances and may never reach steady state in a biologically relevant time. This could explain experimentally observed dynamics of pattern formation. Together the analysis reveals the importance of dynamical transients for understanding morphogen-driven transcriptional networks and indicates that gene expression noise can qualitatively alter developmental patterning.

The bistable switch, a common regulatory sub-network, is found in many biological processes. It consists of cross-repressing components that generate a switch-like transition between two possible states. In developing tissues, bistable switches, created by cross-repressing transcriptional determinants, are often controlled by gradients of secreted signalling molecules—morphogens. These provide a mechanism to convert a morphogen gradient into stripes of gene expression that determine the arrangement of distinct cell types. Here we use mathematical models to analyse the temporal response of such a system. We find that the behaviour is highly dependent on the intrinsic fluctuations that result from the stochastic nature of gene expression. This noise has a marked effect on both patterning time and the location of the stripe boundary. One of the techniques we use, Minimum Action Path theory, identifies key features of the switch without computationally expensive calculations. The results reveal a noise driven switching wave that propels the stripe boundary away from the morphogen source to eventually settle, at steady state, further from the morphogen source than in the deterministic description. Together the analysis highlights the importance dynamics in patterning and demonstrates a set of mathematical tools for studying this problem.

Transition states and cell fate decisions in epigenetic landscapes

Waddington's epigenetic landscape is an abstract metaphor frequently used to represent the relationship between gene activity and cell fates during development. Over the past few years, it has become a useful framework for interpreting results from single-cell transcriptomics experiments. It has led to the proposal that, during fate transitions, cells experience smooth, continuous progressions of global transcriptional activity, which can be captured by (pseudo)temporal dynamics. Here, focusing strictly on the fate decision events, we suggest an alternative view: that fate transitions occur in a discontinuous, stochastic manner whereby signals modulate the probability of the transition events.

Developmental Pattern Formation in Phases

Cells in developing organs undergo a series of changes in their transcriptional state until a complete repertoire of cell types is specified. These changes in cell identity, together with the control of tissue growth, determine the pattern of gene expression in the tissue. Recent studies explore the dynamics of pattern formation during development and provide new insights into the control mechanisms. Changes in morphogen signalling and transcriptional networks control the specification of cell types. This is often followed by a distinct second phase, where pattern is elaborated by tissue growth. Here, we discuss the transitions between distinct phases in pattern formation. We consider the implications of the underlying mechanisms for understanding how reproducible patterns form during development.

Gene switching rate determines response to extrinsic perturbations in the self-activation transcriptional network motif

Gene switching dynamics is a major source of randomness in genetic networks, also in the case of large concentrations of the transcription factors. In this work, we consider a common network motif - the positive feedback of a transcription factor on its own synthesis - and assess its response to extrinsic noises perturbing gene deactivation in a variety of settings where the network might operate. These settings are representative of distinct cellular types, abundance of transcription factors and ratio between gene switching and protein synthesis rates. By investigating noise-induced transitions among the different network operative states, our results suggest that gene switching rates are key parameters to shape network response to external perturbations, and that such response depends on the particular biological setting, i.e. the characteristic time scales and protein abundance. These results might have implications on our understanding of irreversible transitions for noise-related phenomena such as cellular differentiation. In addition these evidences suggest to adopt the appropriate mathematical model of the network in order to analyze the system consistently to the reference biological setting.

Pattern selection by dynamical biochemical signals

The development of multicellular organisms involves cells to decide their fate upon the action of biochemical signals. This decision is often spatiotemporally coordinated such that a spatial pattern arises. The dynamics that drive pattern formation usually involve genetic nonlinear interactions and positive feedback loops. These complex dynamics may enable multiple stable patterns for the same conditions. Under these circumstances, pattern formation in a developing tissue involves a selection process: why is a certain pattern formed and not another stable one? Herein we computationally address this issue in the context of the Notch signaling pathway. We characterize a dynamical mechanism for developmental selection of a specific pattern through spatiotemporal changes of the control parameters of the dynamics, in contrast to commonly studied situations in which initial conditions and noise determine which pattern is selected among multiple stable ones. This mechanism can be understood as a path along the parameter space driven by a sequence of biochemical signals. We characterize the selection process for three different scenarios of this dynamical mechanism that can take place during development: the signal either 1) acts in all the cells at the same time, 2) acts only within a cluster of cells, or 3) propagates along the tissue. We found that key elements for pattern selection are the destabilization of the initial pattern, the subsequent exploration of other patterns determined by the spatiotemporal symmetry of the parameter changes, and the speeds of the path compared to the timescales of the pattern formation process itself. Each scenario enables the selection of different types of patterns and creates these elements in distinct ways, resulting in different features. Our approach extends the concept of selection involved in cellular decision-making, usually applied to cell-autonomous decisions, to systems that collectively make decisions through cell-to-cell interactions.

Reshaping the epigenetic landscape during early flower development: induction of attractor transitions by relative differences in gene decay rates

BACKGROUND

Gene regulatory network (GRN) dynamical models are standard systems biology tools for the mechanistic understanding of developmental processes and are enabling the formalization of the epigenetic landscape (EL) model.

METHODS

In this work we propose a modeling framework which integrates standard mathematical analyses to extend the simple GRN Boolean model in order to address questions regarding the impact of gene specific perturbations in cell-fate decisions during development.

RESULTS

We systematically tested the propensity of individual genes to produce qualitative changes to the EL induced by modification of gene characteristic decay rates reflecting the temporal dynamics of differentiation stimuli. By applying this approach to the flower specification GRN (FOS-GRN) we uncovered differences in the functional (dynamical) role of their genes. The observed dynamical behavior correlates with biological observables. We found a relationship between the propensity of undergoing attractor transitions between attraction basins in the EL and the direction of differentiation during early flower development - being less likely to induce up-stream attractor transitions as the course of development progresses. Our model also uncovered a potential mechanism at play during the transition from EL basins defining inflorescence meristem to those associated to flower organs meristem. Additionally, our analysis provided a mechanistic interpretation of the homeotic property of the ABC genes, being more likely to produce both an induced inter-attractor transition and to specify a novel attractor. Finally, we found that there is a close relationship between a gene's topological features and its propensity to produce attractor transitions.

CONCLUSIONS

The study of how the state-space associated with a dynamical model of a GRN can be restructured by modulation of genes' characteristic expression times is an important aid for understanding underlying mechanisms occurring during development. Our contribution offers a simple framework to approach such problem, as exemplified here by the case of flower development. Different GRN models and the effect of diverse inductive signals can be explored within the same framework. We speculate that the dynamical role of specific genes within a GRN, as uncovered here, might give information about which genes are more likely to link a module to other regulatory circuits and signaling transduction pathways.

Modeling the epigenetic attractors landscape: toward a post-genomic mechanistic understanding of development

Robust temporal and spatial patterns of cell types emerge in the course of normal development in multicellular organisms. The onset of degenerative diseases may result from altered cell fate decisions that give rise to pathological phenotypes. Complex networks of genetic and non-genetic components underlie such normal and altered morphogenetic patterns. Here we focus on the networks of regulatory interactions involved in cell-fate decisions. Such networks modeled as dynamical non-linear systems attain particular stable configurations on gene activity that have been interpreted as cell-fate states. The network structure also restricts the most probable transition patterns among such states. The so-called Epigenetic Landscape (EL), originally proposed by C. H. Waddington, was an early attempt to conceptually explain the emergence of developmental choices as the result of intrinsic constraints (regulatory interactions) shaped during evolution. Thanks to the wealth of molecular genetic and genomic studies, we are now able to postulate gene regulatory networks (GRN) grounded on experimental data, and to derive EL models for specific cases. This, in turn, has motivated several mathematical and computational modeling approaches inspired by the EL concept, that may be useful tools to understand and predict cell-fate decisions and emerging patterns. In order to distinguish between the classical metaphorical EL proposal of Waddington, we refer to the Epigenetic Attractors Landscape (EAL), a proposal that is formally framed in the context of GRNs and dynamical systems theory. In this review we discuss recent EAL modeling strategies, their conceptual basis and their application in studying the emergence of both normal and pathological developmental processes. In addition, we discuss how model predictions can shed light into rational strategies for cell fate regulation, and we point to challenges ahead.

The Comet Cometh: Evolving Developmental Systems

In a recent opinion piece, Denis Duboule has claimed that the increasing shift towards systems biology is driving evolutionary and developmental biology apart, and that a true reunification of these two disciplines within the framework of evolutionary developmental biology (EvoDevo) may easily take another 100 years. He identifies methodological, epistemological, and social differences as causes for this supposed separation. Our article provides a contrasting view. We argue that Duboule’s prediction is based on a one-sided understanding of systems biology as a science that is only interested in functional, not evolutionary, aspects of biological processes. Instead, we propose a research program for an evolutionary systems biology, which is based on local exploration of the configuration space in evolving developmental systems. We call this approach—which is based on reverse engineering, simulation, and mathematical analysis—the natural history of configuration space. We discuss a number of illustrative examples that demonstrate the past success of local exploration, as opposed to global mapping, in different biological contexts. We argue that this pragmatic mode of inquiry can be extended and applied to the mathematical analysis of the developmental repertoire and evolutionary potential of evolving developmental mechanisms and that evolutionary systems biology so conceived provides a pragmatic epistemological framework for the EvoDevo synthesis.

Bioattractors: dynamical systems theory and the evolution of regulatory processes

In this paper, we illustrate how dynamical systems theory can provide a unifying conceptual framework for evolution of biological regulatory systems. Our argument is that the genotype-phenotype map can be characterized by the phase portrait of the underlying regulatory process. The features of this portrait--such as attractors with associated basins and their bifurcations--define the regulatory and evolutionary potential of a system. We show how the geometric analysis of phase space connects Waddington's epigenetic landscape to recent computational approaches for the study of robustness and evolvability in network evolution. We discuss how the geometry of phase space determines the probability of possible phenotypic transitions. Finally, we demonstrate how the active, self-organizing role of the environment in phenotypic evolution can be understood in terms of dynamical systems concepts. This approach yields mechanistic explanations that go beyond insights based on the simulation of evolving regulatory networks alone. Its predictions can now be tested by studying specific, experimentally tractable regulatory systems using the tools of modern systems biology. A systematic exploration of such systems will enable us to understand better the nature and origin of the phenotypic variability, which provides the substrate for evolution by natural selection.