Noise, delays, robustness, canalization and all that

Cellular compartmentalisation and receptor promiscuity as a strategy for accurate and robust inference of position during morphogenesis

Precise spatial patterning of cell fate during morphogenesis requires accurate inference of cellular position. In making such inferences from morphogen profiles, cells must contend with inherent stochasticity in morphogen production, transport, sensing and signalling. Motivated by the multitude of signalling mechanisms in various developmental contexts, we show how cells may utilise multiple tiers of processing (compartmentalisation) and parallel branches (multiple receptor types), together with feedback control, to bring about fidelity in morphogenetic decoding of their positions within a developing tissue. By simultaneously deploying specific and nonspecific receptors, cells achieve a more accurate and robust inference. We explore these ideas in the patterning of Drosophila melanogaster wing imaginal disc by Wingless morphogen signalling, where multiple endocytic pathways participate in decoding the morphogen gradient. The geometry of the inference landscape in the high dimensional space of parameters provides a measure for robustness and delineates stiff and sloppy directions. This distributed information processing at the scale of the cell highlights how local cell autonomous control facilitates global tissue scale design.

A Dilp8-dependent time window ensures tissue size adjustment in Drosophila

The control of organ size mainly relies on precise autonomous growth programs. However, organ development is subject to random variations, called developmental noise, best revealed by the fluctuating asymmetry observed between bilateral organs. The developmental mechanisms ensuring bilateral symmetry in organ size are mostly unknown. In Drosophila, null mutations for the relaxin-like hormone Dilp8 increase wing fluctuating asymmetry, suggesting that Dilp8 plays a role in buffering developmental noise. Here we show that size adjustment of the wing primordia involves a peak of dilp8 expression that takes place sharply at the end of juvenile growth. Wing size adjustment relies on a cross-organ communication involving the epidermis as the source of Dilp8. We identify ecdysone signaling as both the trigger for epidermal dilp8 expression and its downstream target in the wing primordia, thereby establishing reciprocal hormonal feedback as a systemic mechanism, which controls organ size and bilateral symmetry in a narrow developmental time window.

Mechanisms ensuring developmental precision are poorly understood. Here Blanco-Obregon et al. report reciprocal feedback between Dilp8 and Ecdysone, two hormones required during a precise time window of Drosophila development for organ size adjustment.

A model of gene expression based on random dynamical systems reveals modularity properties of gene regulatory networks

Here we propose a new approach to modeling gene expression based on the theory of random dynamical systems (RDS) that provides a general coupling prescription between the nodes of any given regulatory network given the dynamics of each node is modeled by a RDS. The main virtues of this approach are the following: (i) it provides a natural way to obtain arbitrarily large networks by coupling together simple basic pieces, thus revealing the modularity of regulatory networks; (ii) the assumptions about the stochastic processes used in the modeling are fairly general, in the sense that the only requirement is stationarity; (iii) there is a well developed mathematical theory, which is a blend of smooth dynamical systems theory, ergodic theory and stochastic analysis that allows one to extract relevant dynamical and statistical information without solving the system; (iv) one may obtain the classical rate equations form the corresponding stochastic version by averaging the dynamic random variables (small noise limit). It is important to emphasize that unlike the deterministic case, where coupling two equations is a trivial matter, coupling two RDS is non-trivial, specially in our case, where the coupling is performed between a state variable of one gene and the switching stochastic process of another gene and, hence, it is not a priori true that the resulting coupled system will satisfy the definition of a random dynamical system. We shall provide the necessary arguments that ensure that our coupling prescription does indeed furnish a coupled regulatory network of random dynamical systems. Finally, the fact that classical rate equations are the small noise limit of our stochastic model ensures that any validation or prediction made on the basis of the classical theory is also a validation or prediction of our model. We illustrate our framework with some simple examples of single-gene system and network motifs.

Effects of elongation delay in transcription dynamics

In the transcription process, elongation delay is induced by the movement of RNA polymerases (RNAP) along the DNA sequence, and can result in changes in the transcription dynamics. This paper studies the transcription dynamics that involved the elongation delay and effects of cell division and DNA replication. The stochastic process of gene expression is modeled with delay chemical master equation with periodic coefficients, and is studied numerically through the stochastic simulation algorithm with delay. We show that the average transcription level approaches to a periodic dynamics over cell cycles at homeostasis, and the elongation delay can reduce the transcription level and increase the transcription noise. Moreover, the transcription elongation can induce bimodal distribution of mRNA levels that can be measured by the techniques of flow cytometry.

Asymmetric flies: the control of developmental noise in Drosophila

What are the sources of phenotypic variation and which factors shape this variation are fundamental questions of developmental and evolutionary biology. Despite this simple formulation and intense research, controversy remains. Three points are particularly discussed: (1) whether adaptive developmental mechanisms buffering variation exist at all; (2) if yes, do they involve specific genes and processes, i.e., different from those involved in the development of the traits that are buffered?; and (3) whether different mechanisms specifically buffer the various sources of variation, i.e., genetic, environmental and stochastic, or whether a generalist process buffers them all at once. We advocate that experimental work integrating different levels of analysis will improve our understanding of the origin of phenotypic variation and thus help answering these contentious questions. In this paper, we first survey the current views on these issues, highlighting potential sources of controversy. We then focus on the stochastic part of phenotypic variation, as measured by fluctuating asymmetry, and on current knowledge about the genetic basis of developmental stability. We report our recent discovery that an individual gene, Cyclin G, plays a central role-adaptive or not-in developmental stability in Drosophila. ( 1) We discuss the implications of this discovery on the regulation of organ size and shape, and finally point out open questions.

What is complex about complex disorders?

Rather than being polygenic, complex disorders probably represent umbrella terms for collections of conditions caused by rare, recent mutations in any of a large number of different genes.

Key role of time-delay and connection topology in shaping the dynamics of noisy genetic regulatory networks

This paper focuses on a paced genetic regulatory small-world network with time-delayed coupling. How the dynamical behaviors including temporal resonance and spatial synchronization evolve under the influence of time-delay and connection topology is explored through numerical simulations. We reveal the phenomenon of delay-induced resonance when the network topology is fixed. For a fixed time-delay, temporal resonance is shown to be degraded by increasing the rewiring probability of the network. On the other hand, for small rewiring probability, temporal resonance can be enhanced by an appropriately tuned small delay but degraded by a large delay, while conversely, temporal resonance is always reduced by time-delay for large rewiring probability. Finally, an optimal spatial synchrony is detected by a proper combination of time-delay and connection topology.

ON SIMULATING THE GENERATION OF MOSAICISM DURING MAMMALIAN CEREBRAL CORTICAL DEVELOPMENT

Renewed calls for a systems biology reflect the hope hat enduring biological questions at single-cell and cell-population scales will be resolved as modern molecular biology, with its reductionist program, approaches a nearly-complete characterization of the molecular mechanisms of specific cellular processes. Due to the confounding complexity of biological organization across these scales, computational science is sought to complement the intuition of experimentalists. However, with respect to the molecular basis of cellular processes during development and disease, a gulf between feasible simulations and realistic biology persists. Formidable are the mathematical and computational challenges to conducting and validating cell population-scale simulations, drawn from single-cell level and molecular level details. Nonetheless, in some biological contexts, a focus on core processes crafted by evolution can yield coarse-grained mathematical models that retain explanatory potential despite drastic simplification of known biochemical kinetics.

In this article, we bring this modeling philosophy to bear on the nature of neural progenitor cell decision making during mammalian cerebral cortical development. Specifically, we present the computational component to a research program addressing developmental links between (i) the cellular response to endogenous DNA damage, (ii) primary mechanisms of neuronal genetic heterogeneity, or mosaicism, and (iii) the cell fate decision making that defines the population kinetics of neurogenesis.

An atlas of gene regulatory networks reveals multiple three-gene mechanisms for interpreting morphogen gradients

Although >450 different topologies can achieve the same multicellular patterning function, they can be grouped into six main classes, which operate using different underlying dynamics.

Alternative designs for the same functions can therefore split into two types: (a) topology alterations that retain the same underlying dynamics and (b) alterations that utilize a completely different underlying dynamical mechanism.

This segregation of networks into distinct dynamical mechanisms can be revealed by the shape of the topology atlas itself.

Cell–cell communication is not usually part of the causal mechanism underlying a band-pass response during morphogen interpretation, but it can tune the result or increase robustness.

Understanding how gene regulatory networks (GRNs) achieve particular biological functions is a central question in systems biology. Systems biology promises to go beyond a case-by-case understanding of individual networks to map out the complete design space of mechanistic possibilities that underlie biological functions. Can such maps serve as useful theoretical frameworks in which to explore the general design principles for these functions? Towards addressing these questions, we created the first design space for a morphogen interpretation function.

In order to generate a design space for such a function, we enumerated all possible wiring designs of GRNs consisting of three genes and tested their ability to perform one particular morphogen interpretation function; stripe formation, as it represents a simplified form of the much studied French flag problem and is a commonly found gene expression pattern (Figure 1A). We found that only 5% of GRNs had the ability to generate a single stripe of gene expression when simulated with a fixed morphogen input in a one-dimensional model.

We hypothesized that the core mechanisms for producing the stripe of gene expression should be represented by topologies that contain only the necessary and sufficient gene–gene interactions for that function. Hence, we utilized the notions of complexity and neighborhood to generate a complexity atlas. GRNs of such an atlas (represented by nodes) are considered neighbors if they differ by a single gene–gene interaction (neighboring GRN nodes are connected by edges). Such a metagraph (graph of graphs) can then be reorganized using complexity (number of gene–gene interactions) to determine a GRNs position in the y axis, whereas GRNs are spaced in the x axis with the aim of reducing edge crossing (Figure 5A). This reorganization reveals a striking structure, where ‘stalactites' of complexity can be seen protruding from the bottom of the atlas. Each of these stalactites converges on a single ‘core' topology that by extensive analysis we find represents a distinct mechanism.

The mechanisms employ a diverse range of distinct space–time behaviors, and the underlying core topologies display design features such as modularity and feed-forward. We mapped the mechanisms to the complexity atlas by analyzing how each particular GRN of the atlas was working. The GRNs functioning via the different mechanisms are highlighted by the different colors in Figure 5A. Mechanisms thus occupy large regions of separated topology space, suggesting them to be discrete. Analyzing transitions between mechanisms through parameter space confirms this to be the case.

We find that three of the mechanisms are employed in real patterning systems, including both blastoderm patterning in Drosophila and mesoderm specification in Xenopus (Figure 5B). The remaining three mechanisms are thus candidates for employment in other patterning systems. We explored the performance features of these mechanisms, which suggest that some have features such as robustness to parameter variation that make them highly likely to be employed in particular patterning contexts.

Only one of the six-core mechanisms absolutely requires cell–cell communication for functionality, prompting us to predict that cell–cell communication will rarely be responsible for the basic dose response of morphogen interpretation networks. However, we show how cell–cell communication has an important role in robust stripe generation in the face of a noisy morphogen input and in fine tuning the quantitative details of stripe patterning.

In summary, the complexity atlas approach is an amendable approach to any system with a clear genotype–function relationship. We demonstrate how certain functions such as morphogen interpretation may have a range of potential solutions in contrast to previous studies that analyzed more constrained functions. Furthermore, we demonstrate how such an approach can be utilized to define a ‘design space' for a given biological function that describes the different mechanistic possibilities and how they relate to one another (Figure 5). Such a design space can be used practically as a guide to discern which patterning mechanisms are likely be at work in a particular context throwing up less intuitive possibilities with powerful performance features.

The interpretation of morphogen gradients is a pivotal concept in developmental biology, and several mechanisms have been proposed to explain how gene regulatory networks (GRNs) achieve concentration-dependent responses. However, the number of different mechanisms that may exist for cells to interpret morphogens, and the importance of design features such as feedback or local cell–cell communication, is unclear. A complete understanding of such systems will require going beyond a case-by-case analysis of real morphogen interpretation mechanisms and mapping out a complete GRN ‘design space.' Here, we generate a first atlas of design space for GRNs capable of patterning a homogeneous field of cells into discrete gene expression domains by interpreting a fixed morphogen gradient. We uncover multiple very distinct mechanisms distributed discretely across the atlas, thereby expanding the repertoire of morphogen interpretation network motifs. Analyzing this diverse collection of mechanisms also allows us to predict that local cell–cell communication will rarely be responsible for the basic dose-dependent response of morphogen interpretation networks.

Rethinking the genetic architecture of schizophrenia

BACKGROUND

For many years, the prevailing paradigm has stated that in each individual with schizophrenia (SZ) the genetic risk is due to a combination of many genetic variants, individually of small effect. Recent empirical data are prompting a re-evaluation of this polygenic, common disease-common variant (CDCV) model. Evidence includes a lack of the expected strong positive findings from genome-wide association studies and the concurrent discovery of many different mutations that individually strongly predispose to SZ and other psychiatric disorders. This has led some to adopt a mixed model wherein some cases are caused by polygenic mechanisms and some by single mutations. This model runs counter to a substantial body of theoretical literature that had supposedly conclusively rejected Mendelian inheritance with genetic heterogeneity. Here we ask how this discrepancy between theory and data arose and propose a rationalization of the recent evidence base.

METHOD

In light of recent empirical findings, we reconsider the methods and conclusions of early theoretical analyses and the explicit assumptions underlying them.

RESULTS

We show that many of these assumptions can now be seen to be false and that the model of genetic heterogeneity is consistent with observed familial recurrence risks, endophenotype studies and other population-wide parameters.

CONCLUSIONS

We argue for a more biologically consilient mixed model that involves interactions between disease-causing and disease-modifying variants in each individual. We consider the implications of this model for moving SZ research beyond statistical associations to pathogenic mechanisms.

A delay model for noise-induced bi-directional switching

Many biological systems can switch between two distinct states. Once switched, the system remains stable for a period of time and may switch back to its original state. A gene network with bistability is usually required for the switching and stochastic effect in the gene expression may induce such switching. A typical bistable system allows one-directional switching, in which the switch from the low state to the high state or from the high state to the low state occurs under different conditions. It is usually difficult to enable bi-directional switching such that the two switches can occur under the same condition. Here, we present a model consisting of standard positive feedback loops and an extra negative feedback loop with a time delay to study its capability to produce bi-directional switching induced by noise. We find that the time delay in the negative feedback is critical for robust bi-directional switching and the length of delay affects its switching frequency.

Long delay times in reaction rates increase intrinsic fluctuations

In spatially distributed cellular systems, it is often convenient to represent complicated auxiliary pathways and spatial transport by time-delayed reaction rates. Furthermore, many of the reactants appear in low numbers necessitating a probabilistic description. The coupling of delayed rates with stochastic dynamics leads to a probability conservation equation characterizing a non-Markovian process. A systematic approximation is derived that incorporates the effect of delayed rates on the characterization of molecular noise valid in the limit of long delay time. By way of a simple example, we show that delayed reaction dynamics can only increase intrinsic fluctuations about the steady state. The method is general enough to accommodate nonlinear transition rates allowing characterization of fluctuations around a delay-induced limit cycle.

Limb development takes a measured step toward systems analysis

Growth and patterning of the embryonic vertebrate limb is regulated by feedback loops signaling between the epithelium and mesenchyme of the limb bud. Fibroblast growth factor (FGF) signaling from the epithelium regulates Sonic hedgehog (Shh) expression in the mesenchyme. In turn, SHH activity maintains the expression of Gremlin1, which encodes a bone morphogenetic protein (BMP) antagonist; the interaction between BMP and Gremlin1 regulates Fgf gene expression. A computational model using data from complex genetic analysis and quantitative measurements of gene induction kinetics demonstrates that limb development is robust and thus buffered against certain mutational alterations and epigenetic changes because of these feedback loops.

A self-regulatory system of interlinked signaling feedback loops controls mouse limb patterning

Embryogenesis depends on self-regulatory interactions between spatially separated signaling centers, but few of these are well understood. Limb development is regulated by epithelial-mesenchymal (e-m) feedback loops between sonic hedgehog (SHH) and fibroblast growth factor (FGF) signaling involving the bone morphogenetic protein (BMP) antagonist Gremlin1 (GREM1). By combining mouse molecular genetics with mathematical modeling, we showed that BMP4 first initiates and SHH then propagates e-m feedback signaling through differential transcriptional regulation of Grem1 to control digit specification. This switch occurs by linking a fast BMP4/GREM1 module to the slower SHH/GREM1/FGF e-m feedback loop. This self-regulatory signaling network results in robust regulation of distal limb development that is able to compensate for variations by interconnectivity among the three signaling pathways.

Robustness: confronting lessons from physics and biology

The term robustness is encountered in very different scientific fields, from engineering and control theory to dynamical systems to biology. The main question addressed herein is whether the notion of robustness and its correlates (stability, resilience, self-organisation) developed in physics are relevant to biology, or whether specific extensions and novel frameworks are required to account for the robustness properties of living systems. To clarify this issue, the different meanings covered by this unique term are discussed; it is argued that they crucially depend on the kind of perturbations that a robust system should by definition withstand. Possible mechanisms underlying robust behaviours are examined, either encountered in all natural systems (symmetries, conservation laws, dynamic stability) or specific to biological systems (feedbacks and regulatory networks). Special attention is devoted to the (sometimes counterintuitive) interrelations between robustness and noise. A distinction between dynamic selection and natural selection in the establishment of a robust behaviour is underlined. It is finally argued that nested notions of robustness, relevant to different time scales and different levels of organisation, allow one to reconcile the seemingly contradictory requirements for robustness and adaptability in living systems.

Noise-driven stem cell and progenitor population dynamics

BACKGROUND

The balance between maintenance of the stem cell state and terminal differentiation is influenced by the cellular environment. The switching between these states has long been understood as a transition between attractor states of a molecular network. Herein, stochastic fluctuations are either suppressed or can trigger the transition, but they do not actually determine the attractor states.

METHODOLOGY/PRINCIPAL FINDINGS

We present a novel mathematical concept in which stem cell and progenitor population dynamics are described as a probabilistic process that arises from cell proliferation and small fluctuations in the state of differentiation. These state fluctuations reflect random transitions between different activation patterns of the underlying regulatory network. Importantly, the associated noise amplitudes are state-dependent and set by the environment. Their variability determines the attractor states, and thus actually governs population dynamics. This model quantitatively reproduces the observed dynamics of differentiation and dedifferentiation in promyelocytic precursor cells.

CONCLUSIONS/SIGNIFICANCE

Consequently, state-specific noise modulation by external signals can be instrumental in controlling stem cell and progenitor population dynamics. We propose follow-up experiments for quantifying the imprinting influence of the environment on cellular noise regulation.

Raw mediates antagonism of AP-1 activity in Drosophila

High baselines of transcription factor activities represent fundamental obstacles to regulated signaling. Here we show that in Drosophila, quenching of basal activator protein 1 (AP-1) transcription factor activity serves as a prerequisite to its tight spatial and temporal control by the JNK (Jun N-terminal kinase) signaling cascade. Our studies indicate that the novel raw gene product is required to limit AP-1 activity to leading edge epidermal cells during embryonic dorsal closure. In addition, we provide the first evidence that the epidermis has a Basket JNK-independent capacity to activate AP-1 targets and that raw function is required broadly throughout the epidermis to antagonize this activity. Finally, our mechanistic studies of the three dorsal-open group genes [raw, ribbon (rib), and puckered (puc)] indicate that these gene products provide at least two tiers of JNK/AP-1 regulation. In addition to Puckered phosphatase function in leading edge epidermal cells as a negative-feedback regulator of JNK signaling, the three dorsal-open group gene products (Raw, Ribbon, and Puckered) are required more broadly in the dorsolateral epidermis to quench a basal, signaling-independent activity of the AP-1 transcription factor.

Specifying positional information in the embryo: looking beyond morphogens

Concentration gradients of small diffusible molecules called morphogens are key regulators of development, specifying position during pattern formation in the embryo. It is now becoming clear that additional or alternative mechanisms involving interactions among cells are also crucial for positional specification.

Stochastic Transcriptional Regulatory Systems with Time Delays: A Mean-Field Approximation

Modeling transcriptional regulation with time delays is an important problem of computational cell biology. In this paper, we propose a computational tool for studying transcriptional regulation in single cells based on a mean-field approximation method. The main idea is to replace the occurrence probabilities of the underlying transcriptional events by their mean values and use appropriately chosen additive noise terms to model statistical variations not accounted by this approximation. The proposed methodology allows us to characterize the transient and steady-state behavior of transcriptional regulation. Moreover, it provides a rather simple and computationally attractive tool for rapid statistical characterization of the dynamic behavior of a nonlinear transcriptional regulatory system with time delays.

Using a state-space model with hidden variables to infer transcription factor activities

MOTIVATION

In a gene regulatory network, genes are typically regulated by transcription factors (TFs). Transcription factor activity (TFA) is more difficult to measure than gene expression levels are. Other models have extracted information about TFA from gene expression data, but without explicitly modeling feedback from the genes. We present a state-space model (SSM) with hidden variables. The hidden variables include regulatory motifs in the gene network, such as feedback loops and auto-regulation, making SSM a useful complement to existing models.

RESULTS

A gene regulatory network incorporating, for example, feed-forward loops, auto-regulation and multiple-inputs was constructed with an SSM model. First, the gene expression data were simulated by SSM and used to infer the TFAs. The ability of SSM to infer TFAs was evaluated by comparing the profiles of the inferred and simulated TFAs. Second, SSM was applied to gene expression data obtained from Escherichia coli K12 undergoing a carbon source transition and from the Saccharomyces cerevisiae cell cycle. The inferred activity profile for each TF was validated either by measurement or by activity information from the literature. The SSM model provides a probabilistic framework to simulate gene regulatory networks and to infer activity profiles of hidden variables.

AVAILABILITY

Supplementary data and Matlab code will be made available at the URL below.

SUPPLEMENTARY INFORMATION

http://www.chems.msu.edu/groups/chan/ssm.zip.

Stochasticity in gene expression: from theories to phenotypes

Genetically identical cells exposed to the same environmental conditions can show significant variation in molecular content and marked differences in phenotypic characteristics. This variability is linked to stochasticity in gene expression, which is generally viewed as having detrimental effects on cellular function with potential implications for disease. However, stochasticity in gene expression can also be advantageous. It can provide the flexibility needed by cells to adapt to fluctuating environments or respond to sudden stresses, and a mechanism by which population heterogeneity can be established during cellular differentiation and development.

The ups and downs of biological timers

BACKGROUND

The need to execute a sequence of events in an orderly and timely manner is central to many biological processes, including cell cycle progression and cell differentiation. For self-perpetuating systems, such as the cell cycle oscillator, delay times between events are defined by the network of interacting proteins that propagates the system. However, protein levels inside cells are subject to genetic and environmental fluctuations, raising the question of how reliable timing is maintained.

RESULTS

We compared the robustness of different mechanisms for encoding delay times to fluctuations in protein expression levels. Gradual accumulation and gradual decay of a regulatory protein have an equivalent capacity for defining delay times. Yet, we find that the former is highly sensitive to fluctuations in gene dosage, while the latter can buffer such perturbations. In particular, a positive feedback where the degrading protein auto-enhances its own degradation may render delay times practically insensitive to gene dosage.

CONCLUSION

While our understanding of biological timing mechanisms is still rudimentary, it is clear that there is an ample use of degradation as well as self-enhanced degradation in processes such as cell cycle and circadian clocks. We propose that degradation processes, and specifically self-enhanced degradation, will be preferred in processes where maintaining the robustness of timing is important.