Circuit topology and the evolution of robustness in two-gene circadian oscillators

Body plan evolvability: The role of variability in gene regulatory networks

Recent computational modeling of early fruit fly (Drosophila) development has characterized the degree to which gene regulation networks can be robust to natural variability. In the first few hours of development, broad spatial gradients of maternally derived transcription factors activate embryonic gap genes. These gap patterns determine the subsequent segmented insect body plan through pair-rule gene expression. Gap genes are expressed with greater spatial precision than the maternal patterns. Computational modeling of the gap-gap regulatory interactions provides a mechanistic understanding for this robustness to maternal variability in wild-type (WT) patterning. A long-standing question in evolutionary biology has been how a system which is robust, such as the developmental program creating any particular species' body plan, is also evolvable, i.e. how can a system evolve or speciate, if the WT form is strongly buffered and protected? In the present work, we use the WT model to explore the breakdown of such Waddington-type 'canalization'. What levels of variability will push the system out of the WT form; are there particular pathways in the gene regulatory mechanism which are more susceptible to losing the WT form; and when robustness is lost, what types of forms are most likely to occur (i.e. what forms lie near the WT)? Manipulating maternal effects in several different pathways, we find a common gap 'peak-to-step' pattern transition in the loss of WT. We discuss these results in terms of the evolvability of insect segmentation, and in terms of experimental perturbations and mutations which could test the model predictions. We conclude by discussing the prospects for using continuum models of pattern dynamics to investigate a wider range of evo-devo problems.

Deduction of signaling mechanisms from cellular responses to multiple cues

Cell signaling networks are complex and often incompletely characterized, making it difficult to obtain a comprehensive picture of the mechanisms they encode. Mathematical modeling of these networks provides important clues, but the models themselves are often complex, and it is not always clear how to extract falsifiable predictions. Here we take an inverse approach, using experimental data at the cell level to deduce the minimal signaling network. We focus on cells' response to multiple cues, specifically on the surprising case in which the response is antagonistic: the response to multiple cues is weaker than the response to the individual cues. We systematically build candidate signaling networks one node at a time, using the ubiquitous ingredients of (i) up- or down-regulation, (ii) molecular conversion, or (iii) reversible binding. In each case, our method reveals a minimal, interpretable signaling mechanism that explains the antagonistic response. Our work provides a systematic way to deduce molecular mechanisms from cell-level data.

Rethinking the Architecture of Attachment: New Insights into the Role for Oxytocin Signaling

Social attachments, the enduring bonds between individuals and groups, are essential to health and well-being. The appropriate formation and maintenance of social relationships depend upon a number of affective processes, including stress regulation, motivation, reward, as well as reciprocal interactions necessary for evaluating the affective state of others. A genetic, molecular, and neural circuit level understanding of social attachments therefore provides a powerful substrate for probing the affective processes associated with social behaviors. Socially monogamous species form long-term pair bonds, allowing us to investigate the mechanisms underlying attachment. Now, molecular genetic tools permit manipulations in monogamous species. Studies using these tools reveal new insights into the genetic and neuroendocrine factors that design and control the neural architecture underlying attachment behavior. We focus this discussion on the prairie vole and oxytocinergic signaling in this and related species as a model of attachment behavior that has been studied in the context of genetic and pharmacological manipulations. We consider developmental processes that impact the demonstration of bonding behavior across genetic backgrounds, the modularity of mechanisms underlying bonding behaviors, and the distributed circuitry supporting these behaviors. Incorporating such theoretical considerations when interpreting reverse genetic studies in the context of the rich ethological and pharmacological data collected in monogamous species provides an important framework for studies of attachment behavior in both animal models and studies of human relationships.

Model-Based Design of a Synthetic Oscillator Based on an Epigenetic Methylation Memory System

Oscillations are an important component in biological systems; grasping their mechanisms and regulation, however, is difficult. Here, we use the theory of dynamical systems to support the design of oscillatory systems based on epigenetic control elements. Specifically, we use results that extend the Poincaré-Bendixson theorem for monotone control systems that are coupled to a negative feedback circuit. The methodology is applied to a synthetic epigenetic memory system based on DNA methylation that serves as a monotone control system, which is coupled to a negative feedback. This system is generally able to show sustained oscillations according to its structure; however, a first experimental implementation showed that fine-tuning of several parameters is required. We provide design support by exploring the experimental design space using systems-theoretic analysis of a computational model.

A counter-enzyme complex regulates glutamate metabolism in Bacillus subtilis

Multi-enzyme assemblies composed of metabolic enzymes catalyzing sequential reactions are being increasingly studied. Here, we report the discovery of a 1.6 megadalton multi-enzyme complex from Bacillus subtilis composed of two enzymes catalyzing opposite ('counter-enzymes') rather than sequential reactions: glutamate synthase (GltAB) and glutamate dehydrogenase (GudB), which make and break glutamate, respectively. In vivo and in vitro studies show that the primary role of complex formation is to inhibit the activity of GudB. Using cryo-electron microscopy, we elucidated the structure of the complex and the molecular basis of inhibition of GudB by GltAB. The complex exhibits unusual oscillatory progress curves and is necessary for both planktonic growth, in glutamate-limiting conditions, and for biofilm growth, in glutamate-rich media. The regulation of a key metabolic enzyme by complexing with its counter enzyme may thus enable cell growth under fluctuating glutamate concentrations.

Building predictive signaling models by perturbing yeast cells with time-varying stimulations resulting in distinct signaling responses

This protocol provides a step-by-step approach to perturb single cells with time-varying stimulation profiles, collect distinct signaling responses, and use these to infer a system of ordinary differential equations to capture and predict dynamics of protein-protein regulation in signal transduction pathways. The models are validated by predicting the signaling activation upon new cell stimulation conditions. In comparison to using standard step-like stimulations, application of diverse time-varying cell stimulations results in better inference of model parameters and substantially improves model predictions.

For complete details on the use and results of this protocol, please refer to Jashnsaz et al. (2020).

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Diverse time-varying cell stimulations result in distinct signaling activation dynamics

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Signaling models fit step stimuli responses well but result in poor predictions

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Distinct responses upon diverse time-varying stimulations improve model predictions

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Temporal stimulation of pathways result in novel signaling dynamics and mechanisms

Model-based robustness and bistability analysis for methylation-based, epigenetic memory systems

In recent years, epigenetic memory systems have been developed based on DNA methylation and positive feedback systems. Achieving a robust design for these systems is generally a challenging and multifactorial task. We developed and validated a novel mathematical model to describe methylation-based epigenetic memory systems that capture switching dynamics of methylation levels and methyltransferase amounts induced by different inputs. A bifurcation analysis shows that the system operates in the bistable range, but in its current setup is not robust to changes in parameters. An expansion of the model captures heterogeneity of cell populations by accounting for distributed cell division rates. Simulations predict that the system is highly sensitive to variations in temperature, which affects cell division and the efficiency of the zinc finger repressor. A moderate decrease in temperature leads to a highly heterogeneous response to input signals and bistability on a single-cell level. The predictions of our model were confirmed by flow cytometry experiments conducted in this study. Overall, the results of our study give insights into the functional mechanisms of methylation-based memory systems and demonstrate that the switching dynamics can be highly sensitive to experimental conditions.

Identifying a stochastic clock network with light entrainment for single cells of Neurospora crassa

Stochastic networks for the clock were identified by ensemble methods using genetic algorithms that captured the amplitude and period variation in single cell oscillators of Neurospora crassa. The genetic algorithms were at least an order of magnitude faster than ensemble methods using parallel tempering and appeared to provide a globally optimum solution from a random start in the initial guess of model parameters (i.e., rate constants and initial counts of molecules in a cell). The resulting goodness of fit [Formula: see text] was roughly halved versus solutions produced by ensemble methods using parallel tempering, and the resulting [Formula: see text] per data point was only [Formula: see text] = 2,708.05/953 = 2.84. The fitted model ensemble was robust to variation in proxies for "cell size". The fitted neutral models without cellular communication between single cells isolated by microfluidics provided evidence for only one Stochastic Resonance at one common level of stochastic intracellular noise across days from 6 to 36 h of light/dark (L/D) or in a D/D experiment. When the light-driven phase synchronization was strong as measured by the Kuramoto (K), there was degradation in the single cell oscillations away from the stochastic resonance. The rate constants for the stochastic clock network are consistent with those determined on a macroscopic scale of 10⁷ cells.

Asymmetry between Activators and Deactivators in Functional Protein Networks

Are “turn-on” and “turn-off” functions in protein-protein interaction networks exact opposites of each other? To answer this question, we implement a minimal model for the evolution of functional protein-interaction networks using a sequence-based mutational algorithm, and apply the model to study neutral drift in networks that yield oscillatory dynamics. We study the roles of activators and deactivators, two core components of oscillatory protein interaction networks, and find a striking asymmetry in the roles of activating and deactivating proteins, where activating proteins tend to be synergistic and deactivating proteins tend to be competitive.

Can policy maintain habitat connectivity under landscape fragmentation? A case study of Shenzhen, China

The acceleration of urbanization has aggravated the fragmentation of ecological patches and increased the uncertainty risk of habitat connectivity. In the context of landscape fragmentation, government need to establish sound policies that effectively protect the stepping-stones of habitat connection and realize urban ecological integration. In this study, a circuit theory model was used to identify the potential ecological corridors in the city and corresponding stepping-stone groups, with two important stepping-stone groups selected as our areas of focus. By establishing the potential linkages between stepping-stone nodes, we constructed stepping-stone networks within the potential ecological corridors and formulated four scenarios to analyze the robustness of the stepping-stone networks under different policies. The results show that there are 46 important habitats and 22 potential ecological corridors in Shenzhen, including 22 stepping-stone networks. The most important stepping-stone network in the central Shenzhen area connects 7 important habitats and contains 110 stepping-stone nodes. The most important stepping-stone network in the northeast area of Shenzhen connects 5 important habitats and contains 130 stepping-stone nodes. Comparing the robustness and form of the stepping-stone networks under the four policy scenarios, we found that the centrality of stepping-stones is of great significance for maintaining the connectivity of important habitats. For instance, there are five stepping-stone networks in Shenzhen that have not been successfully protected by existing ecological protection policies because their highly central nodes are vulnerable to threats. This study analyzed the maintenance of habitat connectivity under different ecological protection policies in the process of urbanization and discussed the significance of highly central stepping-stone nodes in maintaining habitat connectivity. With this effort, a new perspective on habitat connectivity protection under landscape fragmentation is provided, producing a reference for the formulation of urban ecological protection policy.

Incoherent Inputs Enhance the Robustness of Biological Oscillators

Robust biological oscillators retain the critical ability to function in the presence of environmental perturbations. Although central architectures that support robust oscillations have been extensively studied, networks containing the same core vary drastically in their potential to oscillate, and it remains elusive what peripheral modifications to the core contribute to this functional variation. Here, we have generated a complete atlas of two- and three-node oscillators computationally, then systematically analyzed the association between network structure and robustness. We found that, while certain core topologies are essential for producing a robust oscillator, local structures can substantially modulate the robustness of oscillations. Notably, local nodes receiving incoherent or coherent inputs respectively promote or attenuate the overall network robustness in an additive manner. We validated these relationships in larger-scale networks reflective of real biological oscillators. Our findings provide an explanation for why auxiliary structures not required for oscillation are evolutionarily conserved and suggest simple ways to evolve or design robust oscillators.

Constructing network topologies for multiple signal-encoding functions

Background

Cells use signaling protein networks to sense their environment and mediate specific responses. Information about environmental stress is usually encoded in the dynamics of the signaling molecules, and qualitatively distinct dynamics of the same signaling molecule can lead to dramatically different cell fates. Exploring the design principles of networks with multiple signal-encoding functions is important for understanding how distinct dynamic patterns are shaped and integrated by real cellular networks, and for building cells with targeted sensing–response functions via synthetic biology.

Results

In this paper, we investigate multi-node enzymatic regulatory networks with three signal-encoding functions, i.e., dynamic responses of oscillation, transient activation, and sustained activation upon step stimulation by three different inducers, respectively. Taking into account competition effects of the substrates for the same enzyme in the enzymatic reactions, we searched for robust subnetworks for each signal-encoding function by three-node-network enumeration and then integrated the three subnetworks together via node-merging. The obtained tri-functional networks consisted of four to six nodes, and the core structures of these networks were hybrids of the motifs for the subfunctions.

Conclusions

The simplest but relatively robust tri-functional networks demonstrated that the three functions were compatible within a simple negative feedback loop. Depending on the network structure, the competition effects of the substrates for the same enzyme within the networks could promote or hamper the target functions, and can create implicit functional motifs. Overall, the networks we obtained could in principle be synthesized to construct dynamic control circuits with multiple target functions.

Electronic supplementary material

The online version of this article (10.1186/s12918-018-0676-5) contains supplementary material, which is available to authorized users.

The role of stochastic sequestration dynamics for intrinsic noise filtering in signaling network motifs

The relation between design principles of signaling network motifs and their robustness against intrinsic noise still remains illusive. In this work we investigate the role of cascading for coping with intrinsic noise due to stochasticity in molecular reactions. We use stochastic approaches to quantify fluctuations in the terminal kinase of phosphorylation-dephosphorylation cascade motifs and demonstrate that cascading highly affects these fluctuations. We show that this purely stochastic effect can be explained by time-varying sequestration of upstream kinase molecules. In particular, we discuss conditions on time scales and parameter regimes which lead to a reduction of output fluctuations. Our results are put into biological context by adapting rate parameters of our modeling approach to biologically feasible ranges for general binding-unbinding and phosphorylation-dephosphorylation mechanisms. Overall, this study reveals a novel role of stochastic sequestration for dynamic noise filtering in signaling cascade motifs.

Synthetic circuits reveal how mechanisms of gene regulatory networks constrain evolution

Phenotypic variation is the raw material of adaptive Darwinian evolution. The phenotypic variation found in organismal development is biased towards certain phenotypes, but the molecular mechanisms behind such biases are still poorly understood. Gene regulatory networks have been proposed as one cause of constrained phenotypic variation. However, most pertinent evidence is theoretical rather than experimental. Here, we study evolutionary biases in two synthetic gene regulatory circuits expressed in Escherichia coli that produce a gene expression stripe-a pivotal pattern in embryonic development. The two parental circuits produce the same phenotype, but create it through different regulatory mechanisms. We show that mutations cause distinct novel phenotypes in the two networks and use a combination of experimental measurements, mathematical modelling and DNA sequencing to understand why mutations bring forth only some but not other novel gene expression phenotypes. Our results reveal that the regulatory mechanisms of networks restrict the possible phenotypic variation upon mutation. Consequently, seemingly equivalent networks can indeed be distinct in how they constrain the outcome of further evolution.

Modeling Reveals a Key Mechanism for Light-Dependent Phase Shifts of Neurospora Circadian Rhythms

Light shifts and synchronizes the phase of the circadian clock to daily environments, which is critical for maintaining the daily activities of an organism. It has been proposed that such light-dependent phase shifts are triggered by light-induced upregulation of a negative element of the core circadian clock (i.e., frq, Per1/2) in many organisms, including fungi. However, we find, using systematic mathematical modeling of the Neurospora crassa circadian clock, that the upregulation of the frq gene expression alone is unable to reproduce the observed light-dependent phase responses. Indeed, we find that the depression of the transcriptional activator white-collar-1, previously shown to be promoted by FRQ and VVD, is a key molecular mechanism for accurately simulating light-induced phase response curves for wild-type and mutant strains of Neurospora. Our findings elucidate specific molecular pathways that can be utilized to control phase resetting of circadian rhythms.

It's not magic - Hsp90 and its effects on genetic and epigenetic variation

Canalization, or phenotypic robustness in the face of environmental and genetic perturbation, is an emergent property of living systems. Although this phenomenon has long been recognized, its molecular underpinnings have remained enigmatic until recently. Here, we review the contributions of the molecular chaperone Hsp90, a protein that facilitates the folding of many key regulators of growth and development, to canalization of phenotype - and de-canalization in times of stress - drawing on studies in eukaryotes as diverse as baker's yeast, mouse ear cress, and blind Mexican cavefish. Hsp90 is a hub of hubs that interacts with many so-called 'client proteins,' which affect virtually every aspect of cell signaling and physiology. As Hsp90 facilitates client folding and stability, it can epistatically suppress or enable the expression of genetic variants in its clients and other proteins that acquire client status through mutation. Hsp90's vast interaction network explains the breadth of its phenotypic reach, including Hsp90-dependent de novo mutations and epigenetic effects on gene regulation. Intrinsic links between environmental stress and Hsp90 function thus endow living systems with phenotypic plasticity in fluctuating environments. As environmental perturbations alter Hsp90 function, they also alter Hsp90's interaction with its client proteins, thereby re-wiring networks that determine the genotype-to-phenotype map. Ensuing de-canalization of phenotype creates phenotypic diversity that is not simply stochastic, but often has an underlying genetic basis. Thus, extreme phenotypes can be selected, and assimilated so that they no longer require environmental stress to manifest. In addition to acting on standing genetic variation, Hsp90 perturbation has also been linked to increased frequency of de novo variation and several epigenetic phenomena, all with the potential to generate heritable phenotypic change. Here, we aim to clarify and discuss the multiple means by which Hsp90 can affect phenotype and possibly evolutionary change, and identify their underlying common feature: at its core, Hsp90 interacts epistatically through its chaperone function with many other genes and their gene products. Its influence on phenotypic diversification is thus not magic but rather a fundamental property of genetics.

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.

Systems and synthetic biology approaches in understanding biological oscillators

Background

Self-sustained oscillations are a ubiquitous and vital phenomenon in living systems. From primitive single-cellular bacteria to the most sophisticated organisms, periodicities have been observed in a broad spectrum of biological processes such as neuron firing, heart beats, cell cycles, circadian rhythms, etc. Defects in these oscillators can cause diseases from insomnia to cancer. Elucidating their fundamental mechanisms is of great significance to diseases, and yet challenging, due to the complexity and diversity of these oscillators.

Results

Approaches in quantitative systems biology and synthetic biology have been most effective by simplifying the systems to contain only the most essential regulators. Here, we will review major progress that has been made in understanding biological oscillators using these approaches. The quantitative systems biology approach allows for identification of the essential components of an oscillator in an endogenous system. The synthetic biology approach makes use of the knowledge to design the simplest, de novo oscillators in both live cells and cell-free systems. These synthetic oscillators are tractable to further detailed analysis and manipulations.

Conclusion

With the recent development of biological and computational tools, both approaches have made significant achievements.

Robust network topologies for generating oscillations with temperature-independent periods

Nearly all living systems feature a temperature-independent oscillation period in circadian clocks. This ubiquitous property occurs at the system level and is rooted in the network architecture of the clock machinery. To investigate the mechanism of this prominent property of the circadian clock and provide general guidance for generating robust genetic oscillators with temperature-compensated oscillations, we theoretically explored the design principle and core network topologies preferred by oscillations with a temperature-independent period. By enumerating all topologies of genetic regulatory circuits with three genes, we obtained four network motifs, namely, a delayed negative feedback oscillator, repressilator, activator-inhibitor oscillator and substrate-depletion oscillator; hybrids of these motifs constitute the vast majority of target network topologies. These motifs are biased in their capacities for achieving oscillations and the temperature sensitivity of the period. The delayed negative feedback oscillator and repressilator are more robust for oscillations, whereas the activator-inhibitor and substrate-depletion oscillators are superior for maintaining a temperature-independent oscillation period. These results suggest that thermally robust oscillation can be more plausibly achieved by hybridizing these two categories of network motifs. Antagonistic balance and temperature insulation mechanisms for achieving temperature compensation are typically found in these topologies with temperature robustness. In the temperature insulation approach, the oscillation period relies on very few parameters, and these parameters are influenced only slightly by temperature. This approach prevents the temperature from affecting the oscillation period and generates circadian rhythms that are robust against environmental perturbations.

Sampling-based Bayesian approaches reveal the importance of quasi-bistable behavior in cellular decision processes on the example of the MAPK signaling pathway in PC-12 cell lines

BACKGROUND

Positive and negative feedback loops are ubiquitous motifs in biochemical signaling pathways. The mitogen-activated protein kinase (MAPK) pathway module is part of many distinct signaling networks and comprises several of these motifs, whose functioning depends on the cell line at hand and on the particular context. The maintainance of specificity of the response of the MAPK module to distinct stimuli has become a key paradigm especially in PC-12 cells, where the same module leads to different cell fates, depending on the stimulating growth factor. This cell fate is regulated by differences in the ERK (MAPK) activation profile, which shows a transient response upon stimulation with EGF, while the response is sustained in case of NGF. This behavior was explained by different effective network topologies. It is widely believed that this sustained response requires a bistable system.

RESULTS

In this study we present a sampling-based Bayesian model analysis on a dataset, in which PC-12 cells have been stimulated with different growth factors. This is combined with novel analysis methods to investigate the role of feedback interconnections to shape ERK response. Results strongly suggest that, besides bistability, an additional effect called quasi-bistability can contribute to explain the observed responses of the system to different stimuli. Quasi-bistability is the ability of a monostable system to maintain two distinct states over a long time period upon a transient signal, which is also related to positive feedback, but cannot be detected by standard steady state analysis methods.

CONCLUSIONS

Although applied on a specific example, our framework is generic enough to be also relevant for other regulatory network modeling studies that comprise positive feedback to explain cellular decision making processes. Overall, this study advices to focus not only on steady states, but also to take transient behavior into account in the analysis.

Feedback, Mass Conservation and Reaction Kinetics Impact the Robustness of Cellular Oscillations

Oscillations occur in a wide variety of cellular processes, for example in calcium and p53 signaling responses, in metabolic pathways or within gene-regulatory networks, e.g. the circadian system. Since it is of central importance to understand the influence of perturbations on the dynamics of these systems a number of experimental and theoretical studies have examined their robustness. The period of circadian oscillations has been found to be very robust and to provide reliable timing. For intracellular calcium oscillations the period has been shown to be very sensitive and to allow for frequency-encoded signaling. We here apply a comprehensive computational approach to study the robustness of period and amplitude of oscillatory systems. We employ different prototype oscillator models and a large number of parameter sets obtained by random sampling. This framework is used to examine the effect of three design principles on the sensitivities towards perturbations of the kinetic parameters. We find that a prototype oscillator with negative feedback has lower period sensitivities than a prototype oscillator relying on positive feedback, but on average higher amplitude sensitivities. For both oscillator types, the use of Michaelis-Menten instead of mass action kinetics in all degradation and conversion reactions leads to an increase in period as well as amplitude sensitivities. We observe moderate changes in sensitivities if replacing mass conversion reactions by purely regulatory reactions. These insights are validated for a set of established models of various cellular rhythms. Overall, our work highlights the importance of reaction kinetics and feedback type for the variability of period and amplitude and therefore for the establishment of predictive models.

Rhythmic behavior is omnipresent in biology and has many crucial functions. In cells the activation levels and abundances of signaling molecules such as NF-κB, p53, EGFR or calcium repeatedly increase and decrease in response to stimuli. Such a dynamic behavior can also be observed monitoring the concentrations of mRNAs and proteins in the circadian clock and the cell cycle. Period and amplitude which are the time span between peaks and the peak height, respectively, as well as their variabilities are important features of oscillations. The circadian period is very stable allowing for proper time keeping, whereas in calcium signaling the period is very variable encoding different stimulation strengths. Our goal is to examine the origin of differences in sensitivities of periods and amplitudes using a computational approach. We use prototype oscillators and demonstrate that they can be used to derive general principles that explain the degree of robustness in period and amplitude for a set of commonly used models of cellular oscillators. Our findings imply that the robustness of oscillating systems can be influenced by feedback type and kinetic properties to which special attention should be paid when designing mathematical models of cellular rhythms.

The circuit-breaking algorithm for monotone systems

In earlier work, we have introduced the circuit-breaking algorithm (CBA) for the analysis of intracellular regulation networks. This algorithm uses the network topology to construct a one-dimensional circuit-characteristic whose zeros correspond to the fixed points of the system. In this study, we apply the CBA to monotone systems whose flow preserves a partial order with respect to some orthant cone. We consider relations between stability of fixed points and the derivative of the corresponding zeros of the circuit-characteristic. In particular, we derive sufficient conditions for instability in case of global asymptotic stability of the open-loop system. Furthermore, we fully characterize stability of the fixed points if in addition the system is monotone. Combined with the theory of monotone systems, our results are used to characterize the long-term behavior of two models for different intracellular regulation processes.

Evolution of robust circadian clocks in Drosophila melanogaster populations reared in constant dark for over 330 generations

Robustness is considered to be an important feature of biological systems which may evolve when the functionality of a trait is associated with higher fitness across multiple environmental conditions. Thus, the ability to maintain stable biological phenotypes across environments is thought to be of adaptive value. Previously, we have reported higher intrinsic activity levels (activity levels of free-running rhythm in constant darkness) and power of rhythm (as assessed by amplitude of the periodogram) in Drosophila melanogaster populations (stocks) reared in constant darkness (DD stocks) as compared to those reared in constant light (LL stocks) and 12:12-h light-dark cycles (LD stocks) for over 19 years (∼330 generations). In the current study, we intended to examine whether the enhanced levels of activity observed in DD stocks persist under various environments such as photoperiods, ambient temperatures, non-24-h light-dark (LD) cycles, and semi-natural conditions (SN). We found that DD stocks largely retain their phenotype of enhanced activity levels across most of the above-mentioned environments suggesting the evolution of robust circadian clocks in DD stocks. Furthermore, we compared the peak activity levels of the three stocks across different environmental conditions relative to their peaks in constant darkness and found that the change in peak activity levels upon entrainment was not significantly different across the three stocks for any of the examined environmental conditions. This suggests that the enhancement of activity levels in DD stocks is not due to differential sensitivity to environment. Thus, these results suggest that rearing in constant darkness (DD) leads to evolution of robust circadian clocks suggesting a possible adaptive value of possessing such rhythms under constant dark environments.

Adaptation through proportion

Adaptation is a ubiquitous feature in biological sensory and signaling networks. It has been suggested that adaptive systems may follow certain simple design principles across diverse organisms, cells and pathways. One class of networks that can achieve adaptation utilizes an incoherent feedforward control, in which two parallel signaling branches exert opposite but proportional effects on the output at steady state. In this paper, we generalize this adaptation mechanism by establishing a steady-state proportionality relationship among a subset of nodes in a network. Adaptation can be achieved by using any two nodes in the sub-network to respectively regulate the output node positively and negatively. We focus on enzyme networks and first identify basic regulation motifs consisting of two and three nodes that can be used to build small networks with proportional relationships. Larger proportional networks can then be constructed modularly similar to LEGOs. Our method provides a general framework to construct and analyze a class of proportional and/or adaptation networks with arbitrary size, flexibility and versatile functional features.

A Statistical Approach Reveals Designs for the Most Robust Stochastic Gene Oscillators

The engineering of transcriptional networks presents many challenges due to the inherent uncertainty in the system structure, changing cellular context, and stochasticity in the governing dynamics. One approach to address these problems is to design and build systems that can function across a range of conditions; that is they are robust to uncertainty in their constituent components. Here we examine the parametric robustness landscape of transcriptional oscillators, which underlie many important processes such as circadian rhythms and the cell cycle, plus also serve as a model for the engineering of complex and emergent phenomena. The central questions that we address are: Can we build genetic oscillators that are more robust than those already constructed? Can we make genetic oscillators arbitrarily robust? These questions are technically challenging due to the large model and parameter spaces that must be efficiently explored. Here we use a measure of robustness that coincides with the Bayesian model evidence, combined with an efficient Monte Carlo method to traverse model space and concentrate on regions of high robustness, which enables the accurate evaluation of the relative robustness of gene network models governed by stochastic dynamics. We report the most robust two and three gene oscillator systems, plus examine how the number of interactions, the presence of autoregulation, and degradation of mRNA and protein affects the frequency, amplitude, and robustness of transcriptional oscillators. We also find that there is a limit to parametric robustness, beyond which there is nothing to be gained by adding additional feedback. Importantly, we provide predictions on new oscillator systems that can be constructed to verify the theory and advance design and modeling approaches to systems and synthetic biology.

Synthetic biology approaches in cancer immunotherapy, genetic network engineering, and genome editing

Investigations into cells and their contents have provided evolving insight into the emergence of complex biological behaviors. Capitalizing on this knowledge, synthetic biology seeks to manipulate the cellular machinery towards novel purposes, extending discoveries from basic science to new applications. While these developments have demonstrated the potential of building with biological parts, the complexity of cells can pose numerous challenges. In this review, we will highlight the broad and vital role that the synthetic biology approach has played in applying fundamental biological discoveries in receptors, genetic circuits, and genome-editing systems towards translation in the fields of immunotherapy, biosensors, disease models and gene therapy. These examples are evidence of the strength of synthetic approaches, while also illustrating considerations that must be addressed when developing systems around living cells.

Dynamic information routing in complex networks

Flexible information routing fundamentally underlies the function of many biological and artificial networks. Yet, how such systems may specifically communicate and dynamically route information is not well understood. Here we identify a generic mechanism to route information on top of collective dynamical reference states in complex networks. Switching between collective dynamics induces flexible reorganization of information sharing and routing patterns, as quantified by delayed mutual information and transfer entropy measures between activities of a network's units. We demonstrate the power of this mechanism specifically for oscillatory dynamics and analyse how individual unit properties, the network topology and external inputs co-act to systematically organize information routing. For multi-scale, modular architectures, we resolve routing patterns at all levels. Interestingly, local interventions within one sub-network may remotely determine nonlocal network-wide communication. These results help understanding and designing information routing patterns across systems where collective dynamics co-occurs with a communication function.

Flexible information routing underlies the function of many biological and artificial networks. Here, the authors present a theoretical framework that shows how information can be flexibly routed across networks using collective reference dynamics and how local changes may induce remote rerouting.

Design principles for robust oscillatory behavior

Oscillatory responses are ubiquitous in regulatory networks of living organisms, a fact that has led to extensive efforts to study and replicate the circuits involved. However, to date, design principles that underlie the robustness of natural oscillators are not completely known. Here we study a three-component enzymatic network model in order to determine the topological requirements for robust oscillation. First, by simulating every possible topological arrangement and varying their parameter values, we demonstrate that robust oscillators can be obtained by augmenting the number of both negative feedback loops and positive autoregulations while maintaining an appropriate balance of positive and negative interactions. We then identify network motifs, whose presence in more complex topologies is a necessary condition for obtaining oscillatory responses. Finally, we pinpoint a series of simple architectural patterns that progressively render more robust oscillators. Together, these findings can help in the design of more reliable synthetic biomolecular networks and may also have implications in the understanding of other oscillatory systems.

The origins of mutational robustness

Biological systems are resistant to genetic changes; a property known as mutational robustness, the origin of which remains an open question. In recent years, researchers have explored emergent properties of biological systems and mechanisms of genetic redundancy to reveal how mutational robustness emerges and persists. Several mechanisms have been proposed to explain the origin of mutational robustness, including molecular chaperones and gene duplication. The latter has received much attention, but its role in robustness remains controversial. Here, I examine recent findings linking genetic redundancy through gene duplication and mutational robustness. Experimental evolution and genome resequencing have made it possible to test the role of gene duplication in tolerating mutations at both the coding and regulatory levels. This evidence as well as previous findings on regulatory reprogramming of duplicates support the role of gene duplication in the origin of robustness.

Phenotype accessibility and noise in random threshold gene regulatory networks

Evolution requires phenotypic variation in a population of organisms for selection to function. Gene regulatory processes involved in organismal development affect the phenotypic diversity of organisms. Since only a fraction of all possible phenotypes are predicted to be accessed by the end of development, organisms may evolve strategies to use environmental cues and noise-like fluctuations to produce additional phenotypic diversity, and hence to enhance the speed of adaptation. We used a generic model of organismal development --gene regulatory networks-- to investigate how different levels of noise on gene expression states (i.e. phenotypes) may affect access to new, unique phenotypes, thereby affecting phenotypic diversity. We studied additional strategies that organisms might adopt to attain larger phenotypic diversity: either by augmenting their genome or the number of gene expression states. This was done for different types of gene regulatory networks that allow for distinct levels of regulatory influence on gene expression or are more likely to give rise to stable phenotypes. We found that if gene expression is binary, increasing noise levels generally decreases phenotype accessibility for all network types studied. If more gene expression states are considered, noise can moderately enhance the speed of discovery if three or four gene expression states are allowed, and if there are enough distinct regulatory networks in the population. These results were independent of the network types analyzed, and were robust to different implementations of noise. Hence, for noise to increase the number of accessible phenotypes in gene regulatory networks, very specific conditions need to be satisfied. If the number of distinct regulatory networks involved in organismal development is large enough, and the acquisition of more genes or fine tuning of their expression states proves costly to the organism, noise can be useful in allowing access to more unique phenotypes.

Causal drift, robust signaling, and complex disease

The phenotype of many regulatory circuits in which mutations can cause complex, polygenic diseases is to some extent robust to DNA mutations that affect circuit components. Here I demonstrate how such mutational robustness can prevent the discovery of genetic disease determinants. To make my case, I use a mathematical model of the insulin signaling pathway implicated in type 2 diabetes, whose signaling output is governed by 15 genetically determined parameters. Using multiple complementary measures of a parameter's importance for this phenotype, I show that any one disease determinant that is crucial in one genetic background will be virtually irrelevant in other backgrounds. In an evolving population that drifts through the parameter space of this or other robust circuits through DNA mutations, the genetic changes that can cause disease will vary randomly over time. I call this phenomenon causal drift. It means that mutations causing disease in one (human or non-human) population may have no effect in another population, and vice versa. Causal drift casts doubt on our ability to infer the molecular mechanisms of complex diseases from non-human model organisms.

Evolving robust gene regulatory networks

Design and implementation of robust network modules is essential for construction of complex biological systems through hierarchical assembly of 'parts' and 'devices'. The robustness of gene regulatory networks (GRNs) is ascribed chiefly to the underlying topology. The automatic designing capability of GRN topology that can exhibit robust behavior can dramatically change the current practice in synthetic biology. A recent study shows that Darwinian evolution can gradually develop higher topological robustness. Subsequently, this work presents an evolutionary algorithm that simulates natural evolution in silico, for identifying network topologies that are robust to perturbations. We present a Monte Carlo based method for quantifying topological robustness and designed a fitness approximation approach for efficient calculation of topological robustness which is computationally very intensive. The proposed framework was verified using two classic GRN behaviors: oscillation and bistability, although the framework is generalized for evolving other types of responses. The algorithm identified robust GRN architectures which were verified using different analysis and comparison. Analysis of the results also shed light on the relationship among robustness, cooperativity and complexity. This study also shows that nature has already evolved very robust architectures for its crucial systems; hence simulation of this natural process can be very valuable for designing robust biological systems.

Analytical study of robustness of a negative feedback oscillator by multiparameter sensitivity

Background

One of the distinctive features of biological oscillators such as circadian clocks and cell cycles is robustness which is the ability to resume reliable operation in the face of different types of perturbations. In the previous study, we proposed multiparameter sensitivity (MPS) as an intelligible measure for robustness to fluctuations in kinetic parameters. Analytical solutions directly connect the mechanisms and kinetic parameters to dynamic properties such as period, amplitude and their associated MPSs. Although negative feedback loops are known as common structures to biological oscillators, the analytical solutions have not been presented for a general model of negative feedback oscillators.

Results

We present the analytical expressions for the period, amplitude and their associated MPSs for a general model of negative feedback oscillators. The analytical solutions are validated by comparing them with numerical solutions. The analytical solutions explicitly show how the dynamic properties depend on the kinetic parameters. The ratio of a threshold to the amplitude has a strong impact on the period MPS. As the ratio approaches to one, the MPS increases, indicating that the period becomes more sensitive to changes in kinetic parameters. We present the first mathematical proof that the distributed time-delay mechanism contributes to making the oscillation period robust to parameter fluctuations. The MPS decreases with an increase in the feedback loop length (i.e., the number of molecular species constituting the feedback loop).

Conclusions

Since a general model of negative feedback oscillators was employed, the results shown in this paper are expected to be true for many of biological oscillators. This study strongly supports that the hypothesis that phosphorylations of clock proteins contribute to the robustness of circadian rhythms. The analytical solutions give synthetic biologists some clues to design gene oscillators with robust and desired period.

Inferring the Effects of Honokiol on the Notch Signaling Pathway in SW480 Colon Cancer Cells

In a tumor cell, the development of acquired therapeutic resistance and the ability to survive in extracellular environments that differ from the primary site are the result of molecular adaptations in potentially highly plastic molecular networks. The accurate prediction of intracellular networks in a tumor remains a difficult problem in cancer informatics. In order to make truly rational patient-driven therapeutic decisions, it will be critical to develop methodologies that can accurately infer the molecular circuitry in the cells of a specific tumor. Despite enormous heterogeneity, cellular networks elicit deterministic digital-like responses. We discuss the use and limitations of methodologies that model molecular networks in cancer cells as a digital circuit. We also develop a network model of Notch signaling in colon cancer using a novel reverse engineering logic-based method and published western blot data to elucidate the interactions likely present in the circuits of the SW480 colon cancer cell line. Within this framework, we make predictions related to the role that honokiol may be playing as an anti-cancer drug.

The dynamics of p53 in single cells: physiologically based ODE and reaction-diffusion PDE models

The intracellular signalling network of the p53 protein plays important roles in genome protection and the control of cell cycle phase transitions. Recently observed oscillatory behaviour in single cells under stress conditions has inspired several research groups to simulate and study the dynamics of the protein with the aim of gaining a proper understanding of the physiological meanings of the oscillations. We propose compartmental ODE and PDE models of p53 activation and regulation in single cells following DNA damage and we show that the p53 oscillations can be retrieved by plainly involving p53-Mdm2 and ATM-p53-Wip1 negative feedbacks, which are sufficient for oscillations experimentally, with no further need to introduce any delays into the protein responses and without considering additional positive feedback.

Network representations and methods for the analysis of chemical and biochemical pathways

Systems biologists increasingly use network representations to investigate biochemical pathways and their dynamic behaviours. In this critical review, we discuss four commonly used network representations of chemical and biochemical pathways. We illustrate how some of these representations reduce network complexity but result in the ambiguous representation of biochemical pathways. We also examine the current theoretical approaches available to investigate the dynamic behaviour of chemical and biochemical networks. Finally, we describe how the critical chemical and biochemical pathways responsible for emergent dynamic behaviour can be identified using network mining and functional mapping approaches.

A model for the epigenetic switch linking inflammation to cell transformation: deterministic and stochastic approaches

Recently, a molecular pathway linking inflammation to cell transformation has been discovered. This molecular pathway rests on a positive inflammatory feedback loop between NF-κB, Lin28, Let-7 microRNA and IL6, which leads to an epigenetic switch allowing cell transformation. A transient activation of an inflammatory signal, mediated by the oncoprotein Src, activates NF-κB, which elicits the expression of Lin28. Lin28 decreases the expression of Let-7 microRNA, which results in higher level of IL6 than achieved directly by NF-κB. In turn, IL6 can promote NF-κB activation. Finally, IL6 also elicits the synthesis of STAT3, which is a crucial activator for cell transformation. Here, we propose a computational model to account for the dynamical behavior of this positive inflammatory feedback loop. By means of a deterministic model, we show that an irreversible bistable switch between a transformed and a non-transformed state of the cell is at the core of the dynamical behavior of the positive feedback loop linking inflammation to cell transformation. The model indicates that inhibitors (tumor suppressors) or activators (oncogenes) of this positive feedback loop regulate the occurrence of the epigenetic switch by modulating the threshold of inflammatory signal (Src) needed to promote cell transformation. Both stochastic simulations and deterministic simulations of a heterogeneous cell population suggest that random fluctuations (due to molecular noise or cell-to-cell variability) are able to trigger cell transformation. Moreover, the model predicts that oncogenes/tumor suppressors respectively decrease/increase the robustness of the non-transformed state of the cell towards random fluctuations. Finally, the model accounts for the potential effect of competing endogenous RNAs, ceRNAs, on the dynamics of the epigenetic switch. Depending on their microRNA targets, the model predicts that ceRNAs could act as oncogenes or tumor suppressors by regulating the occurrence of cell transformation.

An increasing amount of evidence demonstrates a close relation between inflammation and cancer development, which reveals the importance of the tumor microenvironment for the development of cancers. Recently, a molecular pathway linking inflammation to cell transformation, which is a prerequisite to cancer development, has been discovered. This molecular pathway is based on a positive inflammatory feedback loop between NF-κB, Lin28, Let-7 microRNA and IL6, allowing the occurrence of an epigenetic switch leading to cell transformation. Here, we propose a computational model to account for the dynamics of this epigenetic switch. We show that an irreversible bistable switch is at the core of the dynamics of the system. The model further indicates that oncogenes (activators of the switch) and tumor suppressors (inhibitors of the switch) regulate the occurrence of cell transformation by modulating the threshold of inflammatory signal needed to induce the switch. Stochastic simulations of the model suggest that molecular fluctuations are able to trigger cell transformation, highlighting possible links between stochasticity and cancer development. Finally, the model predicts a crucial role of competing endogenous RNAs (ceRNAs) for the dynamics of the epigenetic switch and the occurrence of cell transformation.

The p53 miRNA interactome and its potential role in the cancer clinic

p53 is one of the most frequently mutated tumor suppressors. It regulates protein-coding genes and noncoding RNAs involved in many cellular processes, functioning predominantly at the transcriptional level but also through nontranscriptional processes. miRNAs have recently been identified as key mediators of the p53 stress-response pathway. p53 regulates miRNA transcription and processing, and miRNAs regulate p53 activity and expression and, accordingly, various feedback/feed-forward loops have been identified. Many chemotherapeutic agents induce cancer cell death or senescence via DNA damage and the subsequent activation of p53. Resistance to chemotherapy can occur due to the mutation of components in p53 signaling networks. A better understanding of the role of the various components within these pathways and their interactions with each other may allow the modification and improvement of current treatments, and the design of novel therapies. Improving our knowledge of the role of miRNAs in such p53 signaling networks may be crucial to achieving this.

Stochastic effects as a force to increase the complexity of signaling networks

Cellular signaling networks are complex and appear to include many nonfunctional elements. Recently, it was suggested that nonfunctional interactions of proteins cause signaling noise, which, perhaps, shapes the signal transduction mechanism. However, the conditions under which molecular noise influences cellular information processing remain unclear. Here, we explore a large number of simple biological models of varying network sizes to understand the architectural conditions under which the interactions of signaling proteins can exhibit specific stochastic effects—called deviant effects—in which the average behavior of a biological system is substantially altered in the presence of molecular noise. We find that a small fraction of these networks does exhibit deviant effects and shares a common architectural feature whereas most of the networks show only insignificant levels of deviations. Interestingly, addition of seemingly unimportant interactions into protein networks gives rise to deviant effects.

Enhanced entrainability of genetic oscillators by period mismatch

Biological oscillators coordinate individual cellular components so that they function coherently and collectively. They are typically composed of multiple feedback loops, and period mismatch is unavoidable in biological implementations. We investigated the advantageous effect of this period mismatch in terms of a synchronization response to external stimuli. Specifically, we considered two fundamental models of genetic circuits: smooth and relaxation oscillators. Using phase reduction and Floquet multipliers, we numerically analysed their entrainability under different coupling strengths and period ratios. We found that a period mismatch induces better entrainment in both types of oscillator; the enhancement occurs in the vicinity of the bifurcation on their limit cycles. In the smooth oscillator, the optimal period ratio for the enhancement coincides with the experimentally observed ratio, which suggests biological exploitation of the period mismatch. Although the origin of multiple feedback loops is often explained as a passive mechanism to ensure robustness against perturbation, we study the active benefits of the period mismatch, which include increasing the efficiency of the genetic oscillators. Our findings show a qualitatively different perspective for both the inherent advantages of multiple loops and their essentiality.

Design principles of regulatory networks: searching for the molecular algorithms of the cell

A challenge in biology is to understand how complex molecular networks in the cell execute sophisticated regulatory functions. Here we explore the idea that there are common and general principles that link network structures to biological functions, principles that constrain the design solutions that evolution can converge upon for accomplishing a given cellular task. We describe approaches for classifying networks based on abstract architectures and functions, rather than on the specific molecular components of the networks. For any common regulatory task, can we define the space of all possible molecular solutions? Such inverse approaches might ultimately allow the assembly of a design table of core molecular algorithms that could serve as a guide for building synthetic networks and modulating disease networks.

Analyzing fixed points of intracellular regulation networks with interrelated feedback topology

Background

Modeling the dynamics of intracellular regulation networks by systems of ordinary differential equations has become a standard method in systems biology, and it has been shown that the behavior of these networks is often tightly connected to the network topology. We have recently introduced the circuit-breaking algorithm, a method that uses the network topology to construct a one-dimensional circuit-characteristic of the system. It was shown that this characteristic can be used for an efficient calculation of the system’s fixed points.

Results

Here we extend previous work and show several connections between the circuit-characteristic and the stability of fixed points. In particular, we derive a sufficient condition on the characteristic for a fixed point to be unstable for certain graph structures and demonstrate that the characteristic does not contain the information to decide whether a fixed point is asymptotically stable. All statements are illustrated on biological network models.

Conclusions

Single feedback circuits and their role for complex dynamic behavior of biological networks have extensively been investigated, but a transfer of most of these concepts to more complex topologies is difficult. In this context, our algorithm is a powerful new approach for the analysis of regulation networks that goes beyond single isolated feedback circuits.

Dose-response aligned circuits in signaling systems

Cells use biological signal transduction pathways to respond to environmental stimuli and the behavior of many cell types depends on precise sensing and transmission of external information. A notable property of signal transduction that was characterized in the Saccharomyces cerevisiae yeast cell and many mammalian cells is the alignment of dose-response curves. It was found that the dose response of the receptor matches closely the dose responses of the downstream. This dose-response alignment (DoRA) renders equal sensitivities and concordant responses in different parts of signaling system and guarantees a faithful information transmission. The experimental observations raise interesting questions about the nature of the information transmission through DoRA signaling networks and design principles of signaling systems with this function. Here, we performed an exhaustive computational analysis on network architectures that underlie the DoRA function in simple regulatory networks composed of two and three enzymes. The minimal circuits capable of DoRA were examined with Michaelis-Menten kinetics. Several motifs that are essential for the dynamical function of DoRA were identified. Systematic analysis of the topology space of robust DoRA circuits revealed that, rather than fine-tuning the network's parameters, the function is primarily realized by enzymatic regulations on the controlled node that are constrained in limiting regions of saturation or linearity.

Validation and invalidation of systems biology models using robustness analysis

Robustness, the ability of a system to function correctly in the presence of both internal and external uncertainty, has emerged as a key organising principle in many biological systems. Biological robustness has thus become a major focus of research in Systems Biology, particularly on the engineering-biology interface, since the concept of robustness was first rigorously defined in the context of engineering control systems. This review focuses on one particularly important aspect of robustness in Systems Biology, that is, the use of robustness analysis methods for the validation or invalidation of models of biological systems. With the explosive growth in quantitative modelling brought about by Systems Biology, the problem of validating, invalidating and discriminating between competing models of a biological system has become an increasingly important one. In this review, the authors provide a comprehensive overview of the tools and methods that are available for this task, and illustrate the wide range of biological systems to which this approach has been successfully applied.

Efficient characterization of high-dimensional parameter spaces for systems biology

Background

A biological system's robustness to mutations and its evolution are influenced by the structure of its viable space, the region of its space of biochemical parameters where it can exert its function. In systems with a large number of biochemical parameters, viable regions with potentially complex geometries fill a tiny fraction of the whole parameter space. This hampers explorations of the viable space based on "brute force" or Gaussian sampling.

Results

We here propose a novel algorithm to characterize viable spaces efficiently. The algorithm combines global and local explorations of a parameter space. The global exploration involves an out-of-equilibrium adaptive Metropolis Monte Carlo method aimed at identifying poorly connected viable regions. The local exploration then samples these regions in detail by a method we call multiple ellipsoid-based sampling. Our algorithm explores efficiently nonconvex and poorly connected viable regions of different test-problems. Most importantly, its computational effort scales linearly with the number of dimensions, in contrast to "brute force" sampling that shows an exponential dependence on the number of dimensions. We also apply this algorithm to a simplified model of a biochemical oscillator with positive and negative feedback loops. A detailed characterization of the model's viable space captures well known structural properties of circadian oscillators. Concretely, we find that model topologies with an essential negative feedback loop and a nonessential positive feedback loop provide the most robust fixed period oscillations. Moreover, the connectedness of the model's viable space suggests that biochemical oscillators with varying topologies can evolve from one another.

Conclusions

Our algorithm permits an efficient analysis of high-dimensional, nonconvex, and poorly connected viable spaces characteristic of complex biological circuitry. It allows a systematic use of robustness as a tool for model discrimination.