Phenotypic models of evolution and development: geometry as destiny

A Framework for Investigating Rules of Life by Establishing Zones of Influence

The incredible complexity of biological processes across temporal and spatial scales hampers defining common underlying mechanisms driving the patterns of life. However, recent advances in sequencing, big data analysis, machine learning, and molecular dynamics simulation have renewed the hope and urgency of finding potential hidden rules of life. There currently exists no framework to develop such synoptic investigations. Some efforts aim to identify unifying rules of life across hierarchical levels of time, space, and biological organization, but not all phenomena occur across all the levels of these hierarchies. Instead of identifying the same parameters and rules across levels, we posit that each level of a temporal and spatial scale and each level of biological organization has unique parameters and rules that may or may not predict outcomes in neighboring levels. We define this neighborhood, or the set of levels, across which a rule functions as the zone of influence. Here, we introduce the zone of influence framework and explain using three examples: (Smocovitis, 1992) randomness in biology, where we use a Poisson process to describe processes from protein dynamics to DNA mutations to gene expressions, (Leroi, 2014) island biogeography, and (Gropp, 2016) animal coloration. The zone of influence framework may enable researchers to identify which levels are worth investigating for a particular phenomenon and reframe the narrative of searching for a unifying rule of life to the investigation of how, when, and where various rules of life operate.

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.

Geometric models for robust encoding of dynamical information into embryonic patterns

During development, cells gradually assume specialized fates via changes of transcriptional dynamics, sometimes even within the same developmental stage. For anterior-posterior (AP) patterning in metazoans, it has been suggested that the gradual transition from a dynamic genetic regime to a static one is encoded by different transcriptional modules. In that case, the static regime has an essential role in pattern formation in addition to its maintenance function. In this work, we introduce a geometric approach to study such transition. We exhibit two types of genetic regime transitions arising through local or global bifurcations, respectively. We find that the global bifurcation type is more generic, more robust, and better preserves dynamical information. This could parsimoniously explain common features of metazoan segmentation, such as changes of periods leading to waves of gene expressions, ‘speed/frequency-gradient’ dynamics, and changes of wave patterns. Geometric approaches appear as possible alternatives to gene regulatory networks to understand development.

Self-Organization in Pattern Formation

Self-organization is pervasive in development, from symmetry breaking in the early embryo to tissue patterning and morphogenesis. For a few model systems, the underlying molecular and cellular processes are now sufficiently characterized that mathematical models can be confronted with experiments, to explore the dynamics of pattern formation. Here, we review selected systems, ranging from cyanobacteria to mammals, where different forms of cell-cell communication, acting alone or together with positional cues, drive the patterning of cell fates, highlighting the insights that even very simple models can provide as well as the challenges on the path to a predictive understanding of development.

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.

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.

Critical Timing without a Timer for Embryonic Development

Timing of embryonic development is precisely controlled, but the mechanisms underlying biological timers are still unclear. Here, a validated model for timing under control of Sonic Hedgehog is revisited and generalized to an arbitrary number of genes. The developmental dynamics where a temporal sequence of gene expression recapitulates a steady-state spatial pattern can be realized through a simple network close to criticality, controlled by the duration of exposure to a morphogen. Criticality simultaneously accounts for many observed biological properties, such as timing, multistability, and canalization of genetic expression. This process can be parsimoniously generalized in many dimensions with a minimum number of genes, all repressing each other with asymmetrical strengths, which also explains sequential activation of different fates. Separation of timescales allows for a simple analytical interpretation. Finally, it is shown that even in the presence of noise, coupling between cells preserves criticality and robust patterning. The model offers a simple theoretical framework for the study of emergent developmental timers.

Positional information from oscillatory phase shifts : insights from in silico evolution

Complex cellular decisions are based on temporal dynamics of pathways, including genetic oscillators. In development, recent works on vertebrae formation have suggested that relative phase of genetic oscillators encode positional information, including differentiation front defining vertebrae positions. Precise mechanisms for this are still unknown. Here, we use computational evolution to find gene network topologies that can compute the phase difference between oscillators and convert it into a decoder morphogen concentration. Two types of networks are discovered, based on symmetry properties of the decoder gene. So called asymmetric networks are studied, and two submodules are identified converting phase information into an amplitude variable. Those networks naturally display a 'shock' for a well defined phase difference, that can be used to define a wavefront of differentiation. We show how implementation of these ideas reproduce experimental features of vertebrate segmentation.

Predicting Ancestral Segmentation Phenotypes from Drosophila to Anopheles Using In Silico Evolution

Molecular evolution is an established technique for inferring gene homology but regulatory DNA turns over so rapidly that inference of ancestral networks is often impossible. In silico evolution is used to compute the most parsimonious path in regulatory space for anterior-posterior patterning linking two Dipterian species. The expression pattern of gap genes has evolved between Drosophila (fly) and Anopheles (mosquito), yet one of their targets, eve, has remained invariant. Our model predicts that stripe 5 in fly disappears and a new posterior stripe is created in mosquito, thus eve stripe modules 3+7 and 4+6 in fly are homologous to 3+6 and 4+5 in mosquito. We can place Clogmia on this evolutionary pathway and it shares the mosquito homologies. To account for the evolution of the other pair-rule genes in the posterior we have to assume that the ancestral Dipterian utilized a dynamic method to phase those genes in relation to eve.

The last common ancestor of the fruit fly (Drosophila) and mosquito (Anopheles) lived more than 200 Million years ago. Can we use available data on insects alive today to infer what their ancestor looked like? In this manuscript, we focus on early embryonic development, when stripes of genetic expression appear and define the location of insect segments (“segmentation”). We use an evolutionary algorithm to reconstruct and predict dynamics of genes controlling stripes in the last common ancestor of fly and mosquito. We predict a new and different combinatorial logic of stripe formation in mosquito compared to fly, which is fully consistent with development of intermediate species such as moth-fly (Clogmia). Our simulations further suggest that the dynamics of gene expression in this last common ancestor were similar to other insects, such as wasps (Nasonia). Our method illustrates how computational methods inspired by machine learning and non-linear physics can be used to infer gene dynamics in species that disappeared millions of years ago.

Bridging Mechanistic and Phenomenological Models of Complex Biological Systems

The inherent complexity of biological systems gives rise to complicated mechanistic models with a large number of parameters. On the other hand, the collective behavior of these systems can often be characterized by a relatively small number of phenomenological parameters. We use the Manifold Boundary Approximation Method (MBAM) as a tool for deriving simple phenomenological models from complicated mechanistic models. The resulting models are not black boxes, but remain expressed in terms of the microscopic parameters. In this way, we explicitly connect the macroscopic and microscopic descriptions, characterize the equivalence class of distinct systems exhibiting the same range of collective behavior, and identify the combinations of components that function as tunable control knobs for the behavior. We demonstrate the procedure for adaptation behavior exhibited by the EGFR pathway. From a 48 parameter mechanistic model, the system can be effectively described by a single adaptation parameter τ characterizing the ratio of time scales for the initial response and recovery time of the system which can in turn be expressed as a combination of microscopic reaction rates, Michaelis-Menten constants, and biochemical concentrations. The situation is not unlike modeling in physics in which microscopically complex processes can often be renormalized into simple phenomenological models with only a few effective parameters. The proposed method additionally provides a mechanistic explanation for non-universal features of the behavior.

Gap Gene Regulatory Dynamics Evolve along a Genotype Network

Developmental gene networks implement the dynamic regulatory mechanisms that pattern and shape the organism. Over evolutionary time, the wiring of these networks changes, yet the patterning outcome is often preserved, a phenomenon known as “system drift.” System drift is illustrated by the gap gene network—involved in segmental patterning—in dipteran insects. In the classic model organism Drosophila melanogaster and the nonmodel scuttle fly Megaselia abdita, early activation and placement of gap gene expression domains show significant quantitative differences, yet the final patterning output of the system is essentially identical in both species. In this detailed modeling analysis of system drift, we use gene circuits which are fit to quantitative gap gene expression data in M. abdita and compare them with an equivalent set of models from D. melanogaster. The results of this comparative analysis show precisely how compensatory regulatory mechanisms achieve equivalent final patterns in both species. We discuss the larger implications of the work in terms of “genotype networks” and the ways in which the structure of regulatory networks can influence patterns of evolutionary change (evolvability).

Evolution-development congruence in pattern formation dynamics: Bifurcations in gene expression and regulation of networks structures

Search for possible relationships between phylogeny and ontogeny is important in evolutionary-developmental biology. Here we uncover such relationships by numerical evolution and unveil their origin in terms of dynamical systems theory. By representing developmental dynamics of spatially located cells with gene expression dynamics with cell-to-cell interaction under external morphogen gradient, gene regulation networks are evolved under mutation and selection with the fitness to approach a prescribed spatial pattern of expressed genes. For most numerical evolution experiments, evolution of pattern over generations and development of pattern by an evolved network exhibit remarkable congruence. Both in the evolution and development pattern changes consist of several epochs where stripes are formed in a short time, while for other temporal regimes, pattern hardly changes. In evolution, these quasi-stationary regimes are generations needed to hit relevant mutations, while in development, they are due to some gene expression that varies slowly and controls the pattern change. The morphogenesis is regulated by combinations of feedback or feedforward regulations, where the upstream feedforward network reads the external morphogen gradient, and generates a pattern used as a boundary condition for the later patterns. The ordering from up to downstream is common in evolution and development, while the successive epochal changes in development and evolution are represented as common bifurcations in dynamical-systems theory, which lead to the evolution-development congruence. Mechanism of exceptional violation of the congruence is also unveiled. Our results provide a new look on developmental stages, punctuated equilibrium, developmental bottlenecks, and evolutionary acquisition of novelty in morphogenesis.

Dynamics of the slowing segmentation clock reveal alternating two-segment periodicity

The formation of reiterated somites along the vertebrate body axis is controlled by the segmentation clock, a molecular oscillator expressed within presomitic mesoderm (PSM) cells. Although PSM cells oscillate autonomously, they coordinate with neighboring cells to generate a sweeping wave of cyclic gene expression through the PSM that has a periodicity equal to that of somite formation. The velocity of each wave slows as it moves anteriorly through the PSM, although the dynamics of clock slowing have not been well characterized. Here, we investigate segmentation clock dynamics in the anterior PSM in developing zebrafish embryos using an in vivo clock reporter, her1:her1-venus. The her1:her1-venus reporter has single-cell resolution, allowing us to follow segmentation clock oscillations in individual cells in real-time. By retrospectively tracking oscillations of future somite boundary cells, we find that clock reporter signal increases in anterior PSM cells and that the periodicity of reporter oscillations slows to about ∼1.5 times the periodicity in posterior PSM cells. This gradual slowing of the clock in the anterior PSM creates peaks of clock expression that are separated at a two-segment periodicity both spatially and temporally, a phenomenon we observe in single cells and in tissue-wide analyses. These results differ from previous predictions that clock oscillations stop or are stabilized in the anterior PSM. Instead, PSM cells oscillate until they incorporate into somites. Our findings suggest that the segmentation clock may signal somite formation using a phase gradient with a two-somite periodicity.

Transcription factor competition allows embryonic stem cells to distinguish authentic signals from noise

Stem cells occupy variable environments where they must distinguish stochastic fluctuations from developmental cues. Here, we use optogenetics to investigate how the pluripotency network in embryonic stem (ES) cells achieves a robust response to differentiation cues but not to gene expression fluctuations. We engineered ES cells in which we could quantitatively ontrol the endogenous mechanism of neural differentiation through a light-inducible Brn2 transgene and monitor differentiation status through a genome-integrated Nanog-GFP reporter. By exposing cells to pulses of Brn2, we find that the pluripotency network rejects Brn2 inputs that are below specific magnitude or duration thresholds, but allows rapid differentiation when both thresholds are satisfied. The filtering properties of the network arise through its positive feedback architecture and the intrinsic half-life of Nanog, which determines the duration threshold in the network. Together our results suggest that the dynamic properties of positive-feedback networks might determine how inputs are classified as signal or noise by stem cells.

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.

Evolving phenotypic networks in silico

Evolved gene networks are constrained by natural selection. Their structures and functions are consequently far from being random, as exemplified by the multiple instances of parallel/convergent evolution. One can thus ask if features of actual gene networks can be recovered from evolutionary first principles. I review a method for in silico evolution of small models of gene networks aiming at performing predefined biological functions. I summarize the current implementation of the algorithm, insisting on the construction of a proper "fitness" function. I illustrate the approach on three examples: biochemical adaptation, ligand discrimination and vertebrate segmentation (somitogenesis). While the structure of the evolved networks is variable, dynamics of our evolved networks are usually constrained and present many similar features to actual gene networks, including properties that were not explicitly selected for. In silico evolution can thus be used to predict biological behaviours without a detailed knowledge of the mapping between genotype and phenotype.

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

BACKGROUND

Waddington's epigenetic landscape is an intuitive metaphor for the developmental and evolutionary potential of biological regulatory processes. It emphasises time-dependence and transient behaviour. Nowadays, we can derive this landscape by modelling a specific regulatory network as a dynamical system and calculating its so-called potential surface. In this sense, potential surfaces are the mathematical equivalent of the Waddingtonian landscape metaphor. In order to fully capture the time-dependent (non-autonomous) transient behaviour of biological processes, we must be able to characterise potential landscapes and how they change over time. However, currently available mathematical tools focus on the asymptotic (steady-state) behaviour of autonomous dynamical systems, which restricts how biological systems are studied.

RESULTS

We present a pragmatic first step towards a methodology for dealing with transient behaviours in non-autonomous systems. We propose a classification scheme for different kinds of such dynamics based on the simulation of a simple genetic toggle-switch model with time-variable parameters. For this low-dimensional system, we can calculate and explicitly visualise numerical approximations to the potential landscape. Focussing on transient dynamics in non-autonomous systems reveals a range of interesting and biologically relevant behaviours that would be missed in steady-state analyses of autonomous systems. Our simulation-based approach allows us to identify four qualitatively different kinds of dynamics: transitions, pursuits, and two kinds of captures. We describe these in detail, and illustrate the usefulness of our classification scheme by providing a number of examples that demonstrate how it can be employed to gain specific mechanistic insights into the dynamics of gene regulation.

CONCLUSIONS

The practical aim of our proposed classification scheme is to make the analysis of explicitly time-dependent transient behaviour tractable, and to encourage the wider use of non-autonomous models in systems biology. Our method is applicable to a large class of biological processes.

Mechanisms and constraints shaping the evolution of body plan segmentation

Segmentation of the major body axis into repeating units is arguably one of the major inventions in the evolution of animal body plan pattering. It is found in current day vertebrates, annelids and arthropods. Most segmented animals seem to use a clock-and-wavefront type mechanism in which oscillations emanating from a posterior growth zone become transformed into an anterior posterior sequence of segments. In contrast, few animals such as Drosophila use a complex gene regulatory hierarchy to simultaneously subdivide their entire body axis into segments. Here I discuss how in silico models simulating the evolution of developmental patterning can be used to investigate the forces and constraints that helped shape these two developmental modes. I perform an analysis of a series of previous simulation studies, exploiting the similarities and differences in their outcomes in relation to model characteristics to elucidate the circumstances and constraints likely to have been important for the evolution of sequential and simultaneous segmentation modes. The analysis suggests that constraints arising from the involved growth process and spatial patterning signal--posterior elongation producing a propagating wavefront versus a tissue wide morphogen gradient--and the evolutionary history--ancestral versus derived segmentation mode--strongly shaped both segmentation mechanisms. Furthermore, this implies that these patterning types are to be expected rather than random evolutionary outcomes and supports the likelihood of multiple parallel evolutionary origins.

Evolution of transcription networks in response to temporal fluctuations

Organisms respond to changes in their environment over a wide range of biological and temporal scales. Such phenotypic plasticity can involve developmental, behavioral, physiological, and genetic shifts. The adaptive value of a plastic response is known to depend on the nature of the information that is available to the organism as well as the direct and indirect costs of the plastic response. We modeled the dynamic process of simple gene regulatory networks as they responded to temporal fluctuations in environmental conditions. We simulated the evolution of networks to determine when genes that function solely as transcription factors, with no direct function of their own, are beneficial to the function of the network. When there is perfect information about the environment and there is no timing information to be extracted then there is no advantage to adding pure transcription factor genes to the network. In contrast, when there is either timing information that can be extracted or only indirect information about the current state of the environment then additional transcription factor genes improve the evolved network fitness.

Evolution in developmental phenotype space

Developmental systems can produce a variety of patterns and morphologies when the molecular and cellular activities within them are varied. With the advent of quantitative modeling, the range of phenotypic output of a developmental system can be assessed by exploring model parameter space. Here I review recent examples where developmental evolution is studied using quantitative models, which increasingly rely on empirically determined molecular signaling pathways and their crosstalk. Quantitative pathway evolution may result in dramatic morphological changes. Alternatively, in many developmental systems, the phenotypic output is robust to a range of parameter variation, and cryptic developmental evolution may occur without morphological change. Formalization and measurements of the relationship between genetic variation and parameter variation in developmental models remain in their infancy.