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1.  Criticality Is an Emergent Property of Genetic Networks that Exhibit Evolvability 
PLoS Computational Biology  2012;8(9):e1002669.
Accumulating experimental evidence suggests that the gene regulatory networks of living organisms operate in the critical phase, namely, at the transition between ordered and chaotic dynamics. Such critical dynamics of the network permits the coexistence of robustness and flexibility which are necessary to ensure homeostatic stability (of a given phenotype) while allowing for switching between multiple phenotypes (network states) as occurs in development and in response to environmental change. However, the mechanisms through which genetic networks evolve such critical behavior have remained elusive. Here we present an evolutionary model in which criticality naturally emerges from the need to balance between the two essential components of evolvability: phenotype conservation and phenotype innovation under mutations. We simulated the Darwinian evolution of random Boolean networks that mutate gene regulatory interactions and grow by gene duplication. The mutating networks were subjected to selection for networks that both (i) preserve all the already acquired phenotypes (dynamical attractor states) and (ii) generate new ones. Our results show that this interplay between extending the phenotypic landscape (innovation) while conserving the existing phenotypes (conservation) suffices to cause the evolution of all the networks in a population towards criticality. Furthermore, the networks produced by this evolutionary process exhibit structures with hubs (global regulators) similar to the observed topology of real gene regulatory networks. Thus, dynamical criticality and certain elementary topological properties of gene regulatory networks can emerge as a byproduct of the evolvability of the phenotypic landscape.
Author Summary
Dynamically critical systems are those which operate at the border of a phase transition between two behavioral regimes often present in complex systems: order and disorder. Critical systems exhibit remarkable properties such as fast information processing, collective response to perturbations or the ability to integrate a wide range of external stimuli without saturation. Recent evidence indicates that the genetic networks of living cells are dynamically critical. This has far reaching consequences, for it is at criticality that living organisms can tolerate a wide range of external fluctuations without changing the functionality of their phenotypes. Therefore, it is necessary to know how genetic criticality emerged through evolution. Here we show that dynamical criticality naturally emerges from the delicate balance between two fundamental forces of natural selection that make organisms evolve: (i) the existing phenotypes must be resilient to random mutations, and (ii) new phenotypes must emerge for the organisms to adapt to new environmental challenges. The joint effect of these two forces, which are essential for evolvability, is sufficient in our computational models to generate populations of genetic networks operating at criticality. Thus, natural selection acting as a tinkerer of evolvable systems naturally generates critical dynamics.
doi:10.1371/journal.pcbi.1002669
PMCID: PMC3435273  PMID: 22969419
2.  Additive Functions in Boolean Models of Gene Regulatory Network Modules 
PLoS ONE  2011;6(11):e25110.
Gene-on-gene regulations are key components of every living organism. Dynamical abstract models of genetic regulatory networks help explain the genome's evolvability and robustness. These properties can be attributed to the structural topology of the graph formed by genes, as vertices, and regulatory interactions, as edges. Moreover, the actual gene interaction of each gene is believed to play a key role in the stability of the structure. With advances in biology, some effort was deployed to develop update functions in Boolean models that include recent knowledge. We combine real-life gene interaction networks with novel update functions in a Boolean model. We use two sub-networks of biological organisms, the yeast cell-cycle and the mouse embryonic stem cell, as topological support for our system. On these structures, we substitute the original random update functions by a novel threshold-based dynamic function in which the promoting and repressing effect of each interaction is considered. We use a third real-life regulatory network, along with its inferred Boolean update functions to validate the proposed update function. Results of this validation hint to increased biological plausibility of the threshold-based function. To investigate the dynamical behavior of this new model, we visualized the phase transition between order and chaos into the critical regime using Derrida plots. We complement the qualitative nature of Derrida plots with an alternative measure, the criticality distance, that also allows to discriminate between regimes in a quantitative way. Simulation on both real-life genetic regulatory networks show that there exists a set of parameters that allows the systems to operate in the critical region. This new model includes experimentally derived biological information and recent discoveries, which makes it potentially useful to guide experimental research. The update function confers additional realism to the model, while reducing the complexity and solution space, thus making it easier to investigate.
doi:10.1371/journal.pone.0025110
PMCID: PMC3221653  PMID: 22132067
3.  Quantitative analysis of regulatory flexibility under changing environmental conditions 
Day length changes with the seasons in temperate latitudes, affecting the many biological rhythms that entrain to the day/night cycle: we measure these effects on the expression of Arabidopsis clock genes, using RNA and reporter gene readouts, with a new method of phase analysis.Dusk sensitivity is proposed as a simple, natural and general mathematical measure to analyse and manipulate the changing phase of a clock output relative to the change in the day/night cycle.Dusk sensitivity shows how increasing the numbers of feedback loops in the Arabidopsis clock models allows more flexible regulation, consistent with a previously-proposed, general operating principle of biological networks.The Arabidopsis clock genes show flexibility of regulation that is characteristic of a three-loop clock model, validating aspects of the model and the operating principle, but some clock output genes show greater flexibility arising from direct light regulation.
The analysis of dynamic, non-linear regulation with the aid of mechanistic models is central to Systems Biology. This study compares the predictions of mechanistic, mathematical models of the circadian clock with molecular time-series data on rhythmic gene expression in the higher plant Arabidopsis thaliana. Analysis of the models helps us to understand (explain and predict) how the clock gene circuit balances regulation by external and endogenous factors to achieve particular behaviours. Such multi-factorial regulation is ubiquitous in, and characteristic of, living systems.
The Earth's rotation causes predictable changes in the environment, notably in the availability of sunlight for photosynthesis. Many biological processes are driven by the environmental input via sensory pathways, for example, from photoreceptors. Circadian clocks provide an alternative strategy. These endogenous, 24-h rhythms can drive biological processes that anticipate the regular environmental changes, rather than merely responding. Many rhythmic processes have both light and clock control. Indeed, the clock components themselves must balance internal timing with external inputs, because circadian clocks are reset daily through light regulation of one or more clock components. This process of entrainment is complicated by the change in day length. When the times of dawn and dusk move apart in summer, and closer together in winter, does the clock track dawn, track dusk or interpolate between them?
In plants, the clock controls leaf and petal movements, the opening and closing of stomatal pores, the discharge of floral fragrances, and many metabolic activities, especially those associated with photosynthesis. Centuries of physiological studies have shown that these rhythms can behave differently. Flowering in Ipomoea nil (Pharbitis nil, Japanese morning glory) is controlled by a rhythm that tracks the time of dusk, to give a classic example. We showed that two other rhythms associated with vegetative growth track dawn in this species (Figure 5A), so the clock system allows flexible regulation.
The relatively small number of components involved in the circadian clockwork makes it an ideal candidate for mathematical modelling. Molecular genetic studies in a variety of model eukaryotes have shown that the circadian rhythm is generated by a network of 6–20 genes. These genes form feedback loops generating a rhythm in mRNA production. A single negative feedback loop in which a gene encodes a protein that, after several hours, turns off transcription is capable of generating a circadian rhythm, in principle. A single light input can entrain the clock to ‘local time', synchronised with a light–dark cycle. However, real circadian clocks have proven to be more complicated than this, with multiple light inputs and interlocked feedback loops.
We have previously argued from mathematical analysis that multi-loop networks increase the flexibility of regulation (Rand et al, 2004) and have shown that appropriately deployed flexibility can confer functional robustness (Akman et al, 2010). Here we test whether that flexibility can be demonstrated in vivo, in the model plant, A. thaliana. The Arabidopsis clock mechanism comprises a feedback loop in which two partially redundant, myb transcription factors, LATE ELONGATED HYPOCOTYL (LHY) and CIRCADIAN CLOCK ASSOCIATED 1 (CCA1), repress the expression of their activator, TIMING OF CAB EXPRESSION 1 (TOC1). We previously modelled this single-loop circuit and showed that it was not capable of recreating important data (Locke et al, 2005a). An extended, two-loop model was developed to match observed behaviours, incorporating a hypothetical gene Y, for which the best identified candidate was the GIGANTEA gene (GI) (Locke et al, 2005b). Two further models incorporated the TOC1 homologues PSEUDO-RESPONSE REGULATOR (PRR) 9 and PRR7 (Locke et al, 2006; Zeilinger et al, 2006). In these circuits, a morning oscillator (LHY/CCA1–PRR9/7) is coupled to an evening oscillator (Y/GI–TOC1) via the original LHY/CCA1–TOC1 loop.
These clock models, like those for all other organisms, were developed using data from simple conditions of constant light, darkness or 12-h light–12-h dark cycles. We therefore tested how the clock genes in Arabidopsis responded to light–dark cycles with different photoperiods, from 3 h light to 18 h light per 24-h cycle (Edinburgh, 56° North latitude, has 17.5 h light in midsummer). The time-series assays of mRNA and in vivo reporter gene images showed a range of peak times for different genes, depending on the photoperiod (Figure 5C). A new data analysis method, mFourfit, was introduced to measure the peak times, in the Biological Rhythms Analysis Software Suite (BRASS v3.0). None of the genes showed the dusk-tracking behaviour characteristic of the Ipomoea flowering rhythm. The one-, two- and three-loop models were analysed to understand the observed patterns. A new mathematical measure, dusk sensitivity, was introduced to measure the change in timing of a model component versus a change in the time of dusk. The one- and two-loop models tracked dawn and dusk, respectively, under all conditions. Only the three-loop model (Figure 5B) had the flexibility required to match the photoperiod-dependent changes that we found in vivo, and in particular the unexpected, V-shaped pattern in the peak time of TOC1 expression. This pattern of regulation depends on the structure and light inputs to the model's evening oscillator, so the in vivo data supported this aspect of the model. LHY and CCA1 gene expression under short photoperiods showed greater dusk sensitivity, in the interval 2–6 h before dawn, than the three-loop model predicted, so these data will help to constrain future models.
The approach described here could act as a template for experimental biologists seeking to understand biological regulation using dynamic, experimental perturbations and time-series data. Simulation of mathematical models (despite known imperfections) can provide contrasting hypotheses that guide understanding. The system's detailed behaviour is complex, so a natural and general measure such as dusk sensitivity is helpful to focus on one property of the system. We used the measure to compare models, and to predict how this property could be manipulated. To enable additional analysis of this system, we provide the time-series data and experimental metadata online.
The circadian clock controls 24-h rhythms in many biological processes, allowing appropriate timing of biological rhythms relative to dawn and dusk. Known clock circuits include multiple, interlocked feedback loops. Theory suggested that multiple loops contribute the flexibility for molecular rhythms to track multiple phases of the external cycle. Clear dawn- and dusk-tracking rhythms illustrate the flexibility of timing in Ipomoea nil. Molecular clock components in Arabidopsis thaliana showed complex, photoperiod-dependent regulation, which was analysed by comparison with three contrasting models. A simple, quantitative measure, Dusk Sensitivity, was introduced to compare the behaviour of clock models with varying loop complexity. Evening-expressed clock genes showed photoperiod-dependent dusk sensitivity, as predicted by the three-loop model, whereas the one- and two-loop models tracked dawn and dusk, respectively. Output genes for starch degradation achieved dusk-tracking expression through light regulation, rather than a dusk-tracking rhythm. Model analysis predicted which biochemical processes could be manipulated to extend dusk tracking. Our results reveal how an operating principle of biological regulators applies specifically to the plant circadian clock.
doi:10.1038/msb.2010.81
PMCID: PMC3010117  PMID: 21045818
Arabidopsis thaliana; biological clocks; dynamical systems; gene regulatory networks; mathematical models; photoperiodism
4.  The regulatory network of E. coli metabolism as a Boolean dynamical system exhibits both homeostasis and flexibility of response 
BMC Systems Biology  2008;2:21.
Background
Elucidating the architecture and dynamics of large scale genetic regulatory networks of cells is an important goal in systems biology. We study the system level dynamical properties of the genetic network of Escherichia coli that regulates its metabolism, and show how its design leads to biologically useful cellular properties. Our study uses the database (Covert et al., Nature 2004) containing 583 genes and 96 external metabolites which describes not only the network connections but also the Boolean rule at each gene node that controls the switching on or off of the gene as a function of its inputs.
Results
We have studied how the attractors of the Boolean dynamical system constructed from this database depend on the initial condition of the genes and on various environmental conditions corresponding to buffered minimal media. We find that the system exhibits homeostasis in that its attractors, that turn out to be fixed points or low period cycles, are highly insensitive to initial conditions or perturbations of gene configurations for any given fixed environment. At the same time the attractors show a wide variation when external media are varied implying that the system mounts a highly flexible response to changed environmental conditions. The regulatory dynamics acts to enhance the cellular growth rate under changed media.
Conclusion
Our study shows that the reconstructed genetic network regulating metabolism in E. coli is hierarchical, modular, and largely acyclic, with environmental variables controlling the root of the hierarchy. This architecture makes the cell highly robust to perturbations of gene configurations as well as highly responsive to environmental changes. The twin properties of homeostasis and response flexibility are achieved by this dynamical system even though it is not close to the edge of chaos.
doi:10.1186/1752-0509-2-21
PMCID: PMC2322946  PMID: 18312613
5.  Phase Transitions in the Multi-cellular Regulatory Behavior of Pancreatic Islet Excitability 
PLoS Computational Biology  2014;10(9):e1003819.
The pancreatic islets of Langerhans are multicellular micro-organs integral to maintaining glucose homeostasis through secretion of the hormone insulin. β-cells within the islet exist as a highly coupled electrical network which coordinates electrical activity and insulin release at high glucose, but leads to global suppression at basal glucose. Despite its importance, how network dynamics generate this emergent binary on/off behavior remains to be elucidated. Previous work has suggested that a small threshold of quiescent cells is able to suppress the entire network. By modeling the islet as a Boolean network, we predicted a phase-transition between globally active and inactive states would emerge near this threshold number of cells, indicative of critical behavior. This was tested using islets with an inducible-expression mutation which renders defined numbers of cells electrically inactive, together with pharmacological modulation of electrical activity. This was combined with real-time imaging of intracellular free-calcium activity [Ca2+]i and measurement of physiological parameters in mice. As the number of inexcitable cells was increased beyond ∼15%, a phase-transition in islet activity occurred, switching from globally active wild-type behavior to global quiescence. This phase-transition was also seen in insulin secretion and blood glucose, indicating physiological impact. This behavior was reproduced in a multicellular dynamical model suggesting critical behavior in the islet may obey general properties of coupled heterogeneous networks. This study represents the first detailed explanation for how the islet facilitates inhibitory activity in spite of a heterogeneous cell population, as well as the role this plays in diabetes and its reversal. We further explain how islets utilize this critical behavior to leverage cellular heterogeneity and coordinate a robust insulin response with high dynamic range. These findings also give new insight into emergent multicellular dynamics in general which are applicable to many coupled physiological systems, specifically where inhibitory dynamics result from coupled networks.
Author Summary
As science has successfully broken down the elements of many biological systems, the network dynamics of large-scale cellular interactions has emerged as a new frontier. One way to understand how dynamical elements within large networks behave collectively is via mathematical modeling. Diabetes, which is of increasing international concern, is commonly caused by a deterioration of these complex dynamics in a highly coupled micro-organ called the islet of Langerhans. Therefore, if we are to understand diabetes and how to treat it, we must understand how coupling affects ensemble dynamics. While the role of network connectivity in islet excitation under stimulatory conditions has been well studied, how connectivity also suppresses activity under fasting conditions remains to be elucidated. Here we use two network models of islet connectivity to investigate this process. Using genetically altered islets and pharmacological treatments, we show how suppression of islet activity is solely dependent on a threshold number of inactive cells. We found that the islet exhibits critical behavior in the threshold region, rapidly transitioning from global activity to inactivity. We therefore propose how the islet and multicellular systems in general can generate a robust stimulated response from a heterogeneous cell population.
doi:10.1371/journal.pcbi.1003819
PMCID: PMC4154652  PMID: 25188228
6.  Dynamical Modeling of the Cell Cycle and Cell Fate Emergence in Caulobacter crescentus 
PLoS ONE  2014;9(11):e111116.
The division of Caulobacter crescentus, a model organism for studying cell cycle and differentiation in bacteria, generates two cell types: swarmer and stalked. To complete its cycle, C. crescentus must first differentiate from the swarmer to the stalked phenotype. An important regulator involved in this process is CtrA, which operates in a gene regulatory network and coordinates many of the interactions associated to the generation of cellular asymmetry. Gaining insight into how such a differentiation phenomenon arises and how network components interact to bring about cellular behavior and function demands mathematical models and simulations. In this work, we present a dynamical model based on a generalization of the Boolean abstraction of gene expression for a minimal network controlling the cell cycle and asymmetric cell division in C. crescentus. This network was constructed from data obtained from an exhaustive search in the literature. The results of the simulations based on our model show a cyclic attractor whose configurations can be made to correspond with the current knowledge of the activity of the regulators participating in the gene network during the cell cycle. Additionally, we found two point attractors that can be interpreted in terms of the network configurations directing the two cell types. The entire network is shown to be operating close to the critical regime, which means that it is robust enough to perturbations on dynamics of the network, but adaptable to environmental changes.
doi:10.1371/journal.pone.0111116
PMCID: PMC4219702  PMID: 25369202
7.  Identification of a Topological Characteristic Responsible for the Biological Robustness of Regulatory Networks 
PLoS Computational Biology  2009;5(7):e1000442.
Attribution of biological robustness to the specific structural properties of a regulatory network is an important yet unsolved problem in systems biology. It is widely believed that the topological characteristics of a biological control network largely determine its dynamic behavior, yet the actual mechanism is still poorly understood. Here, we define a novel structural feature of biological networks, termed ‘regulation entropy’, to quantitatively assess the influence of network topology on the robustness of the systems. Using the cell-cycle control networks of the budding yeast (Saccharomyces cerevisiae) and the fission yeast (Schizosaccharomyces pombe) as examples, we first demonstrate the correlation of this quantity with the dynamic stability of biological control networks, and then we establish a significant association between this quantity and the structural stability of the networks. And we further substantiate the generality of this approach with a broad spectrum of biological and random networks. We conclude that the regulation entropy is an effective order parameter in evaluating the robustness of biological control networks. Our work suggests a novel connection between the topological feature and the dynamic property of biological regulatory networks.
Author Summary
Living organisms exert very complicated control on the functionality of their components. Such control systems can often operate in a surprisingly robust manner, in spite of constant perturbations from fluctuating internal conditions and a volatile external environment. What feature makes such control mechanisms robust? Is there a general way to achieve robustness? Here, we address these questions by investigating the wiring of interaction networks, which contains the most condensed information about the control mechanisms of biological systems. We suggest that one of the most important factors in the realization of biological robustness rests in the global coherency of the control strategy, i.e., the consistency of commands flowing through different routes in the network to the same destination. To implement this idea, we propose an order parameter termed ‘regulation entropy’ to quantitatively describe this control consistency of networks. We find that this order parameter correlates with the resistance of the system to external perturbations as well as internal fluctuations. Our results suggest that the self-consistency of the control strategy is important for the vitality and robustness of living organisms.
doi:10.1371/journal.pcbi.1000442
PMCID: PMC2704863  PMID: 19629157
8.  Buffered Qualitative Stability explains the robustness and evolvability of transcriptional networks 
eLife  2014;3:e02863.
The gene regulatory network (GRN) is the central decision‐making module of the cell. We have developed a theory called Buffered Qualitative Stability (BQS) based on the hypothesis that GRNs are organised so that they remain robust in the face of unpredictable environmental and evolutionary changes. BQS makes strong and diverse predictions about the network features that allow stable responses under arbitrary perturbations, including the random addition of new connections. We show that the GRNs of E. coli, M. tuberculosis, P. aeruginosa, yeast, mouse, and human all verify the predictions of BQS. BQS explains many of the small- and large‐scale properties of GRNs, provides conditions for evolvable robustness, and highlights general features of transcriptional response. BQS is severely compromised in a human cancer cell line, suggesting that loss of BQS might underlie the phenotypic plasticity of cancer cells, and highlighting a possible sequence of GRN alterations concomitant with cancer initiation.
DOI: http://dx.doi.org/10.7554/eLife.02863.001
eLife digest
The genomes of living organisms consist of thousands of genes, which produce proteins that perform many essential functions. Cells receive signals from both their internal and external environments, and respond by changing how they express their genes. This allows a cell to make the right amount of different proteins when needed. The proteins that a cell produces can then, in turn, influence how the cell's genes are expressed. This set of interactions between genes and proteins is called a gene regulatory network, and is akin to a computer program that the cell runs to define its behaviour. At present, we understand very little about why these networks take on the forms seen in living cells.
A remarkable feature of living organisms is their ability to withstand an extremely wide variety of predicaments, such as DNA damage, physical trauma or exposure to toxins. This ability, generally called robustness, requires a cell to rapidly activate different gene sets and maintain their activity for as long as necessary. However, very little is known about how cells are programmed to respond appropriately, whatever happens, and keep themselves in a stable state.
Albergante et al. propose that a fully robust gene regulatory network should be able to stabilize itself. This means that the robustness of a gene regulatory network should only depend on how it is wired up, and not on quantitative changes to any features that may change unpredictably—for example the concentration of a protein.
By analysing data that is already available about gene regulatory networks in a wide selection of organisms ranging from bacteria to humans, Albergante et al. show that all known gene regulatory networks are wired up in a way that any quantitative change to the network will not cause the state of network to change. In addition, gene regulatory networks tend to remain stable even if new regulatory links are randomly added. Albergante et al. call this property Buffered Qualitative Stability (BQS): the network is qualitatively stable because its state does not change when the activity of particular regulatory links in the network changes, and it is buffered against its stability being compromised by the random addition of new links.
Albergante et al. also found that the gene regulatory network of a cancer cell does not match up with the predictions of BQS, suggesting that the robustness of the network is compromised in these cells. This could explain why cancer cells are able to easily change their characteristics in response to changes in the environment. In addition, using BQS to analyse the gene regulatory network of bacteria such as E. coli reveals points in the network that, if disrupted, would make the network unstable, potentially harming the cell. Therefore, in the future, an understanding of BQS could help efforts to design new drugs to treat a range of infections and diseases.
DOI: http://dx.doi.org/10.7554/eLife.02863.002
doi:10.7554/eLife.02863
PMCID: PMC4151086  PMID: 25182846
robustness; gene regulatory network; systems biology; cancer; antibiotics resistance; evolvability; E. coli; human; mouse; S. cerevisiae
9.  Floral Morphogenesis: Stochastic Explorations of a Gene Network Epigenetic Landscape 
PLoS ONE  2008;3(11):e3626.
In contrast to the classical view of development as a preprogrammed and deterministic process, recent studies have demonstrated that stochastic perturbations of highly non-linear systems may underlie the emergence and stability of biological patterns. Herein, we address the question of whether noise contributes to the generation of the stereotypical temporal pattern in gene expression during flower development. We modeled the regulatory network of organ identity genes in the Arabidopsis thaliana flower as a stochastic system. This network has previously been shown to converge to ten fixed-point attractors, each with gene expression arrays that characterize inflorescence cells and primordial cells of sepals, petals, stamens, and carpels. The network used is binary, and the logical rules that govern its dynamics are grounded in experimental evidence. We introduced different levels of uncertainty in the updating rules of the network. Interestingly, for a level of noise of around 0.5–10%, the system exhibited a sequence of transitions among attractors that mimics the sequence of gene activation configurations observed in real flowers. We also implemented the gene regulatory network as a continuous system using the Glass model of differential equations, that can be considered as a first approximation of kinetic-reaction equations, but which are not necessarily equivalent to the Boolean model. Interestingly, the Glass dynamics recover a temporal sequence of attractors, that is qualitatively similar, although not identical, to that obtained using the Boolean model. Thus, time ordering in the emergence of cell-fate patterns is not an artifact of synchronous updating in the Boolean model. Therefore, our model provides a novel explanation for the emergence and robustness of the ubiquitous temporal pattern of floral organ specification. It also constitutes a new approach to understanding morphogenesis, providing predictions on the population dynamics of cells with different genetic configurations during development.
doi:10.1371/journal.pone.0003626
PMCID: PMC2572848  PMID: 18978941
10.  Robust synchronization control scheme of a population of nonlinear stochastic synthetic genetic oscillators under intrinsic and extrinsic molecular noise via quorum sensing 
BMC Systems Biology  2012;6:136.
Background
Collective rhythms of gene regulatory networks have been a subject of considerable interest for biologists and theoreticians, in particular the synchronization of dynamic cells mediated by intercellular communication. Synchronization of a population of synthetic genetic oscillators is an important design in practical applications, because such a population distributed over different host cells needs to exploit molecular phenomena simultaneously in order to emerge a biological phenomenon. However, this synchronization may be corrupted by intrinsic kinetic parameter fluctuations and extrinsic environmental molecular noise. Therefore, robust synchronization is an important design topic in nonlinear stochastic coupled synthetic genetic oscillators with intrinsic kinetic parameter fluctuations and extrinsic molecular noise.
Results
Initially, the condition for robust synchronization of synthetic genetic oscillators was derived based on Hamilton Jacobi inequality (HJI). We found that if the synchronization robustness can confer enough intrinsic robustness to tolerate intrinsic parameter fluctuation and extrinsic robustness to filter the environmental noise, then robust synchronization of coupled synthetic genetic oscillators is guaranteed. If the synchronization robustness of a population of nonlinear stochastic coupled synthetic genetic oscillators distributed over different host cells could not be maintained, then robust synchronization could be enhanced by external control input through quorum sensing molecules. In order to simplify the analysis and design of robust synchronization of nonlinear stochastic synthetic genetic oscillators, the fuzzy interpolation method was employed to interpolate several local linear stochastic coupled systems to approximate the nonlinear stochastic coupled system so that the HJI-based synchronization design problem could be replaced by a simple linear matrix inequality (LMI)-based design problem, which could be solved with the help of LMI toolbox in MATLAB easily.
Conclusion
If the synchronization robustness criterion, i.e. the synchronization robustness ≥ intrinsic robustness + extrinsic robustness, then the stochastic coupled synthetic oscillators can be robustly synchronized in spite of intrinsic parameter fluctuation and extrinsic noise. If the synchronization robustness criterion is violated, external control scheme by adding inducer can be designed to improve synchronization robustness of coupled synthetic genetic oscillators. The investigated robust synchronization criteria and proposed external control method are useful for a population of coupled synthetic networks with emergent synchronization behavior, especially for multi-cellular, engineered networks.
doi:10.1186/1752-0509-6-136
PMCID: PMC3554485  PMID: 23101662
11.  Guiding the self-organization of random Boolean networks 
Theory in Biosciences  2011;131(3):181-191.
Random Boolean networks (RBNs) are models of genetic regulatory networks. It is useful to describe RBNs as self-organizing systems to study how changes in the nodes and connections affect the global network dynamics. This article reviews eight different methods for guiding the self-organization of RBNs. In particular, the article is focused on guiding RBNs toward the critical dynamical regime, which is near the phase transition between the ordered and dynamical phases. The properties and advantages of the critical regime for life, computation, adaptability, evolvability, and robustness are reviewed. The guidance methods of RBNs can be used for engineering systems with the features of the critical regime, as well as for studying how natural selection evolved living systems, which are also critical.
doi:10.1007/s12064-011-0144-x
PMCID: PMC3414703  PMID: 22127955
Guided self-organization; Random Boolean networks; Phase transitions; Criticality; Adaptability; Evolvability; Robustness
12.  Genome-wide transcriptional plasticity underlies cellular adaptation to novel challenge 
By recruiting the essential HIS3 gene to the GAL regulatory system and switching to a repressing glucose medium, we confronted yeast cells with a novel challenge they had not encountered before along their history in evolution.Adaptation to this challenge involved a global transcriptional response of a sizeable fraction of the genome, which relaxed on the time scale of the population adaptation, of order of 10 generations.For a large fraction of the responding genes there is no simple biological interpretation, connecting them to the specific cellular demands imposed by the novel challenge.Strikingly, repeating the experiment did not reproduce similar transcription patterns neither in the transient phase nor in the adapted state in glucose.These results suggest that physiological selection operates on the new metabolic configurations generated by the non-specific large scale transcriptional response to eventually stabilize an adaptive state.
Cells adjust their transcriptional state to accommodate environmental and genetic perturbations. Some common perturbations, such as changes in nutrient composition, elicit well-characterized transcriptional responses that can be understood by simple engineering-like design principles as satisfying specific demands imposed by the perturbation. However, cells also have the ability to adapt to novel and unforeseen challenges. This ability is central in realizing the evolvability potential of cells as they respond to dramatic genetic or environmental changes along evolution. Little is known about the mechanisms underlying such adaptations to novel challenges; in particular, the role of the transcriptional regulatory network in such adaptations has not been characterized. Genome-wide measurements have revealed that, in many cases, perturbations lead to a global transcriptional response involving a sizeable fraction of the genome (Gasch et al, 2000; Jelinsky et al, 2000; Causton et al, 2001; Ideker et al, 2001; Lai et al, 2005). Such global behavior suggests that general collective properties of the genetic network, rather than specific pre-designed pathways, determine an important part of the transcriptional response. It is not known however what fraction of genes within such massive transcriptional responses is essential to the specific cellular demands. It is also unknown whether the non-pre-designed part of the response can have a functional role in adaptation to novel challenges.
To study these questions, we confronted yeast cells with a novel challenge they had not encountered before along their history in evolution. A strain of the yeast Saccharomyces cerevisiae was engineered to recruit the gene HIS3, an essential enzyme from the histidine biosynthesis pathway (Hinnebusch, 1992), to the GAL regulatory system, responsible for galactose utilization (Stolovicki et al, 2006). The GAL system is known to be strongly repressed when the cells are exposed to glucose. Therefore, upon switching to a medium containing glucose and lacking histidine, the GAL system and with it HIS3 are highly repressed immediately following the switch and the cells encounter a severe challenge. We have recently shown that a cell population carrying this rewired genome can adapt to grow competitively in a chemostat in a medium containing pure glucose (Stolovicki et al, 2006). This adaptation occurred on a timescale of ∼10 generations; applying a stronger environmental pressure in the form of a competitive inhibitor to HIS3 (3AT) resulted in a similar adaptation albeit with a longer timescale. Figure 1 shows the dynamics of the population's cell density (blue lines, measured by OD) following a medium switch from galactose to glucose in the chemostat without (A) and with (B) 3AT. The experiments revealed that adaptation occurs on physiological timescales (much shorter than required by spontaneous random mutations), but the mechanisms underlying this adaptation have remained unclear (Stolovicki et al, 2006).
Yeast cells had not encountered recruitment of HIS3 to the GAL system along their evolutionary history, and their genome could not possibly have been selected to specifically address glucose repression of HIS3. This experiment, therefore, provides a unique opportunity to characterize the spontaneous transcriptional response during adaptation to a novel challenge and to assess the functional role of the regulatory system in this adaptation. We used DNA microarrays to measure the genome-wide expression levels at time points along the adaptation process, with and without 3AT. These measurements revealed that a sizeable fraction of the genome responded by induction or repression to the switch into glucose. Superimposed on the OD traces, Figure 1 shows the results of a clustering analysis of the expression of genes as measured by the arrays along time in the experiments. This analysis revealed two dominant clusters, each containing hundreds of genes in each experiment, which responded to the medium switch to glucose by a strong transient induction or repression followed by relaxation to steady state on the timescale of the adaptation process, ∼ 10 generations. The two clusters in each experiment show similar but opposite dynamics.
A detailed analysis of the gene content in the two clusters revealed that only a small portion of the response was induced by a change in carbon source (15% overlap between the corresponding clusters in the two experiments, with and without 3AT). Moreover, it revealed a very low overlap with the universal stress response observed for a wide range of environmental stresses (Gasch et al, 2000; Causton et al, 2001) and with the typical response to amino-acid starvation (Natarajan et al, 2001). Additionally, all known specific responses to stress in the literature are characterized by transient induction or repression with relaxation to steady state within a generation time (Gasch et al, 2000; Koerkamp et al, 2002; Wu et al, 2004), whereas in our experiments relaxation of the transcriptional response occurs over many generations. Taken together, these results show that the transcriptional response observed here is neither a metabolic response to the change in carbon source nor is it a standard response to stress or amino-acid starvation. This raises the possibility that it is a spontaneous collective response that is largely composed of genes that do not have a specific function. This possibility was tested directly by repeating the experiment with different populations and comparing their responses. This procedure revealed reproducible adaptation dynamics and steady states in terms of population density, but showed significantly different transcriptional transient responses and steady states for the two repeated experiments. Thus, a significant portion of the genes that changed their expression during the adaptation process do not have a well-defined and reproducible function in the challenging environment.
The application of a stronger environmental pressure in the form of 3AT had a dramatic effect on the global characteristics of the transcriptional response: it induced a markedly higher correlation among the hundreds of responding genes. Figure 3A compares the array data in color code for the two experiments. It is seen that the emergent pattern of transcription exhibits a higher degree of order by the introduction of high external pressure. Observation of the transcriptional patterns for specific metabolic pathways illustrates the different contributions to the correlated dynamics (Figure 3B–D). A general energetic module such as glycolysis exhibited similar patterns of induction and relaxation in experiments with and without 3AT (Figure 3B). However, in general, we found that more than one-third of the known metabolic modules (30 out of 88 modules described in KEGG) exhibited high expression correlation among their genes when the environmental pressure was high but not when it was low. As an example, Figure 3C shows the histidine biosynthesis pathway and Figure 3D the purine pathway. Note the highly ordered trajectories in the lower panels (with 3AT) compared to the disordered ones in the upper panels (no 3AT). This order extends also between genes belonging to different and even distant metabolic modules. It indicates that a global transcriptional regulatory mechanism is in operation, rather than a local specific one. Surprisingly, genes belonging to the same metabolic pathway exhibited simultaneous positively and negatively correlated dynamics. Thus, an important conclusion of this work is that the global transcriptional response to a novel challenge cannot be explained by a simple cellular or metabolic logic. This is to be expected if the response had not been specifically selected in evolution and was not pre-designed for the challenge.
Our data clearly reveal that the massive transcriptional response underlies the adaptation process to a novel challenge. The novelty of the challenge presented to the cells excludes the possibility that this response has been specifically selected toward this challenge. Thus, transcriptional regulation has dynamic properties resulting in a general massive nonspecific response to a novel perturbation. Such a response in turn allows for metabolic rearrangements, which by feeding back on transcription lead to adaptation of the cells to the unforeseen situation. The drastic change in the expression state of the cell opens multiple new metabolic pathways. Physiological selection works then on these multiple metabolic pathways to stabilize an adaptive state that causes relaxation of the perturbed expression pattern. This scenario, involving the creation of a library of possibilities and physiological selection over this library, is compatible with our understanding of a broad class of biological systems, placing the cellular metabolic/regulatory networks on the same footing as the neural or the immune systems (Gerhart and Kirschner, 1997).
Cells adjust their transcriptional state to accommodate environmental and genetic perturbations. An open question is to what extent transcriptional response to perturbations has been specifically selected along evolution. To test the possibility that transcriptional reprogramming does not need to be ‘pre-designed' to lead to an adaptive metabolic state on physiological timescales, we confronted yeast cells with a novel challenge they had not previously encountered. We rewired the genome by recruiting an essential gene, HIS3, from the histidine biosynthesis pathway to a foreign regulatory system, the GAL network responsible for galactose utilization. Switching medium to glucose in a chemostat caused repression of the essential gene and presented the cells with a severe challenge to which they adapted over approximately 10 generations. Using genome-wide expression arrays, we show here that a global transcriptional reprogramming (>1200 genes) underlies the adaptation. A large fraction of the responding genes is nonreproducible in repeated experiments. These results show that a nonspecific transcriptional response reflecting the natural plasticity of the regulatory network supports adaptation of cells to novel challenges.
doi:10.1038/msb4100147
PMCID: PMC1865588  PMID: 17453047
adaptation; cellular metabolism; expression arrays; plasticity; transcriptional response
13.  The Stochastic Evolutionary Game for a Population of Biological Networks Under Natural Selection 
In this study, a population of evolutionary biological networks is described by a stochastic dynamic system with intrinsic random parameter fluctuations due to genetic variations and external disturbances caused by environmental changes in the evolutionary process. Since information on environmental changes is unavailable and their occurrence is unpredictable, they can be considered as a game player with the potential to destroy phenotypic stability. The biological network needs to develop an evolutionary strategy to improve phenotypic stability as much as possible, so it can be considered as another game player in the evolutionary process, ie, a stochastic Nash game of minimizing the maximum network evolution level caused by the worst environmental disturbances. Based on the nonlinear stochastic evolutionary game strategy, we find that some genetic variations can be used in natural selection to construct negative feedback loops, efficiently improving network robustness. This provides larger genetic robustness as a buffer against neutral genetic variations, as well as larger environmental robustness to resist environmental disturbances and maintain a network phenotypic traits in the evolutionary process. In this situation, the robust phenotypic traits of stochastic biological networks can be more frequently selected by natural selection in evolution. However, if the harbored neutral genetic variations are accumulated to a sufficiently large degree, and environmental disturbances are strong enough that the network robustness can no longer confer enough genetic robustness and environmental robustness, then the phenotype robustness might break down. In this case, a network phenotypic trait may be pushed from one equilibrium point to another, changing the phenotypic trait and starting a new phase of network evolution through the hidden neutral genetic variations harbored in network robustness by adaptive evolution. Further, the proposed evolutionary game is extended to an n-tuple evolutionary game of stochastic biological networks with m players (competitive populations) and k environmental dynamics.
doi:10.4137/EBO.S13227
PMCID: PMC3928070  PMID: 24558296
network robustness; phenotype robustness; evolutionary biological network; evolvability; evolutionary biology; poisson process; stochastic nash game; evolutionary game
14.  The auxin signalling network translates dynamic input into robust patterning at the shoot apex 
We provide a comprehensive expression map of the different genes (TIR1/AFBs, ARFs and Aux/IAAs) involved in the signalling pathway regulating gene transcription in response to auxin in the shoot apical meristem (SAM).We demonstrate a relatively simple structure of this pathway using a high-throughput yeast two-hybrid approach to obtain the Aux/IAA-ARF full interactome.The topology of the signalling network was used to construct a model for auxin signalling and to predict a role for the spatial regulation of auxin signalling in patterning of the SAM.We used a new sensor to monitor the input in the auxin signalling pathway and to confirm the model prediction, thus demonstrating that auxin signalling is essential to create robust patterns at the SAM.
The plant hormone auxin is a key morphogenetic signal involved in the control of cell identity throughout development. A striking example of auxin action is at the shoot apical meristem (SAM), a population of stem cells generating the aerial parts of the plant. Organ positioning and patterning depends on local accumulations of auxin in the SAM, generated by polar transport of auxin (Vernoux et al, 2010). However, it is still unclear how auxin is distributed at cell resolution in tissues and how the hormone is sensed in space and time during development. A complex ensemble of 29 Aux/IAAs and 23 ARFs is central to the regulation of gene transcription in response to auxin (for review, see Leyser, 2006; Guilfoyle and Hagen, 2007; Chapman and Estelle, 2009). Protein–protein interactions govern the properties of this transduction pathway (Del Bianco and Kepinski, 2011). Limited interaction studies suggest that, in the absence of auxin, the Aux/IAA repressors form heterodimers with the ARF transcription factors, preventing them from regulating target genes. In the presence of auxin, the Aux/IAA proteins are targeted to the proteasome by an SCF E3 ubiquitin ligase complex (Chapman and Estelle, 2009; Leyser, 2006). In this process, auxin promotes the interaction between Aux/IAA proteins and the TIR1 F-box of the SCF complex (or its AFB homologues) that acts as an auxin co-receptor (Dharmasiri et al, 2005a, 2005b; Kepinski and Leyser, 2005; Tan et al, 2007). The auxin-induced degradation of Aux/IAAs would then release ARFs to regulate transcription of their target genes. This includes activation of most of the Aux/IAA genes themselves, thus establishing a negative feedback loop (Guilfoyle and Hagen, 2007). Although this general scenario provides a framework for understanding gene regulation by auxin, the underlying protein–protein network remains to be fully characterized.
In this paper, we combined experimental and theoretical analyses to understand how this pathway contributes to sensing auxin in space and time (Figure 1). We first analysed the expression patterns of the ARFs, Aux/IAAs and TIR1/AFBs genes in the SAM. Our results demonstrate a general tendency for most of the 25 ARFs and Aux/IAAs detected in the SAM: a differential expression with low levels at the centre of the meristem (where the stem cells are located) and high levels at the periphery of the meristem (where organ initiation takes place). We also observed a similar differential expression for TIR1/AFB co-receptors. To understand the functional significance of the distribution of ARFs and Aux/IAAs in the SAM, we next investigated the global structure of the Aux/IAA-ARF network using a high-throughput yeast two-hybrid approach and uncover a rather simple topology that relies on three basic generic features: (i) Aux/IAA proteins interact with themselves, (ii) Aux/IAA proteins interact with ARF activators and (iii) ARF repressors have no or very limited interactions with other proteins in the network.
The results of our interaction analysis suggest a model for the Aux/IAA-ARF signalling pathway in the SAM, where transcriptional activation by ARF activators would be negatively regulated by two independent systems, one involving the ARF repressors, the other the Aux/IAAs. The presence of auxin would remove the inhibitory action of Aux/IAAs, but leave the ARF repressors to compete with ARF activators for promoter-binding sites. To explore the regulatory properties of this signalling network, we developed a mathematical model to describe the transcriptional output as a function of the signalling input that is the combinatorial effect of auxin concentration and of its perception. We then used the model and a simplified view of the meristem (where the same population of Aux/IAAs and ARFs exhibit a low expression at the centre and a high expression in the peripheral zone) for investigating the role of auxin signalling in SAM function. We show that in the model, for a given ARF activator-to-repressor ratio, the gene induction capacity increases with the absolute levels of ARF proteins. We thus predict that the differential expression of the ARFs generates differences in auxin sensitivities between the centre (low sensitivity) and the periphery (high sensitivity), and that the expression of TIR1/AFB participates to this regulation (prediction 1). We also use the model to analyse the transcriptional response to rapidly changing auxin concentrations. By simulating situations equivalent either to the centre or the periphery of our simplified representation of the SAM, we predict that the signalling pathway buffers its response to the auxin input via the balance between ARF activators and repressors, in turn generated by their differential spatial distributions (prediction 2).
To test the predictions from the model experimentally, we needed to assess both the input (auxin level and/or perception) and the output (target gene induction) of the signalling cascade. For measuring the transcriptional output, the widely used DR5 reporter is perfectly adapted (Figure 5) (Ulmasov et al, 1997; Sabatini et al, 1999; Benkova et al, 2003; Heisler et al, 2005). For assaying pathway input, we designed DII-VENUS, a novel auxin signalling sensor that comprises a constitutively expressed fusion of the auxin-binding domain (termed domain II or DII) (Dreher et al, 2006; Tan et al, 2007) of an IAA to a fast-maturating variant of YFP, VENUS (Figure 5). The degradation patterns from DII-VENUS indicate a high auxin signalling input both in flower primordia and at the centre of the SAM. This is in contrast to the organ-specific expression pattern of DR5::VENUS (Figure 5). These results indicate that the signalling pathway limits gene activation in response to auxin at the meristem centre and confirm the differential sensitivity to auxin between the centre and the periphery (prediction 1). We further confirmed the buffering capacities of the signalling pathway (prediction 2) by carrying out live imaging experiments to monitor DII-VENUS and DR5::VENUS expression in real time (Figure 5). This analysis reveals the presence of important temporal variations of DII-VENUS fluorescence, while DR5::VENUS does not show such global variations. Our approach thus provides evidence that the Aux/IAA-ARF pathway has a key role in patterning in the SAM, alongside the auxin transport system. Our results illustrate how the tight spatio-temporal regulation of both the distribution of a morphogenetic signal and the activity of the downstream signalling pathway provides robustness to a dynamic developmental process.
A comprehensive expression and interaction map of auxin signalling factors in the Arabidopsis shoot apical meristem is constructed and used to derive a mathematical model of auxin signalling, from which key predictions are experimentally confirmed.
The plant hormone auxin is thought to provide positional information for patterning during development. It is still unclear, however, precisely how auxin is distributed across tissues and how the hormone is sensed in space and time. The control of gene expression in response to auxin involves a complex network of over 50 potentially interacting transcriptional activators and repressors, the auxin response factors (ARFs) and Aux/IAAs. Here, we perform a large-scale analysis of the Aux/IAA-ARF pathway in the shoot apex of Arabidopsis, where dynamic auxin-based patterning controls organogenesis. A comprehensive expression map and full interactome uncovered an unexpectedly simple distribution and structure of this pathway in the shoot apex. A mathematical model of the Aux/IAA-ARF network predicted a strong buffering capacity along with spatial differences in auxin sensitivity. We then tested and confirmed these predictions using a novel auxin signalling sensor that reports input into the signalling pathway, in conjunction with the published DR5 transcriptional output reporter. Our results provide evidence that the auxin signalling network is essential to create robust patterns at the shoot apex.
doi:10.1038/msb.2011.39
PMCID: PMC3167386  PMID: 21734647
auxin; biosensor; live imaging; ODE; signalling
15.  A model of yeast cell-cycle regulation based on multisite phosphorylation 
Multisite phosphorylation of CDK target proteins provides the requisite nonlinearity for cell cycle modeling using elementary reaction mechanisms.Stochastic simulations, based on Gillespie's algorithm and using realistic numbers of protein and mRNA molecules, compare favorably with single-cell measurements in budding yeast.The role of transcription–translation coupling is critical in the robust operation of protein regulatory networks in yeast cells.
Progression through the eukaryotic cell cycle is governed by the activation and inactivation of a family of cyclin-dependent kinases (CDKs) and auxiliary proteins that regulate CDK activities (Morgan, 2007). The many components of this protein regulatory network are interconnected by positive and negative feedback loops that create bistable switches and transient pulses (Tyson and Novak, 2008). The network must ensure that cell-cycle events proceed in the correct order, that cell division is balanced with respect to cell growth, and that any problems encountered (in replicating the genome or partitioning chromosomes to daughter cells) are corrected before the cell proceeds to the next phase of the cycle. The network must operate robustly in the context of unavoidable molecular fluctuations in a yeast-sized cell. With a volume of only 5×10−14 l, a yeast cell contains one copy of the gene for each component of the network, a handful of mRNA transcripts of each gene, and a few hundreds to thousands of protein molecules carrying out each gene's function. How large are the molecular fluctuations implied by these numbers, and what effects do they have on the functioning of the cell-cycle control system?
To answer these questions, we have built a new model (Figure 1) of the CDK regulatory network in budding yeast, based on the fact that the targets of CDK activity are typically phosphorylated on multiple sites. The activity of each target protein depends on how many sites are phosphorylated. The target proteins feedback on CDK activity by controlling cyclin synthesis (SBF's role) and degradation (Cdh1's role) and by releasing a CDK-counteracting phosphatase (Cdc14). Every reaction in Figure 1 can be described by a mass-action rate law, with an accompanying rate constant that must be estimated from experimental data. As the transcription and translation of mRNA molecules have major effects on fluctuating numbers of protein molecules (Pedraza and Paulsson, 2008), we have included mRNA transcripts for each protein in the model.
To create a deterministic model, the rate laws are combined, according to standard principles of chemical kinetics, into a set of 60 differential equations that govern the temporal dynamics of the control system. In the stochastic version of the model, the rate law for each reaction determines the probability per unit time that a particular reaction occurs, and we use Gillespie's stochastic simulation algorithm (Gillespie, 1976) to compute possible temporal sequences of reaction events. Accurate stochastic simulations require knowledge of the expected numbers of mRNA and protein molecules in a single yeast cell. Fortunately, these numbers are available from several sources (Ghaemmaghami et al, 2003; Zenklusen et al, 2008). Although the experimental estimates are not always in good agreement with each other, they are sufficiently reliable to populate a stochastic model with realistic numbers of molecules.
By simulating thousands of cells (as in Figure 5), we can build up representative samples for computing the mean and s.d. of any measurable cell-cycle property (e.g. interdivision time, size at division, duration of G1 phase). The excellent fit of simulated statistics to observations of cell-cycle variability is documented in the main text and Supplementary Information.
Of particular interest to us are observations of Di Talia et al (2007) of the timing of a crucial G1 event (export of Whi5 protein from the nucleus) in a population of budding yeast cells growing at a specific growth rate α=ln2/(mass-doubling time). Whi5 export is a consequence of Whi5 phosphorylation, and it occurs simultaneously with the release (activation) of SBF (see Figure 1). Using fluorescently labeled Whi5, Di Talia et al could easily measure (in individual yeast cells) the time, T1, from cell birth to the abrupt loss of Whi5 from the nucleus. Correlating T1 to the size of the cell at birth, Vbirth, they found that, for a sample of daughter cells, αT1 versus ln(Vbirth) could be fit with two straight lines of slope −0.7 and −0.3. Our simulation of this experiment (Figure 7 of the main text) compares favorably with Figure 3d and e in Di Talia et al (2007).
The major sources of noise in our model (and in protein regulatory networks in yeast cells, in general) are related to gene transcription and the small number of unique mRNA transcripts. As each mRNA molecule may instruct the synthesis of dozens of protein molecules, the coefficient of variation of molecular fluctuations at the protein level (CVP) may be dominated by fluctuations at the mRNA level, as expressed in the formula (Pedraza and Paulsson, 2008) where NM, NP denote the number of mRNA and protein molecules, respectively, and ρ=τM/τP is the ratio of half-lives of mRNA and protein molecules. For a yeast cell, typical values of NM and NP are 8 and 800, respectively (Ghaemmaghami et al, 2003; Zenklusen et al, 2008). If ρ=1, then CVP≈25%. Such large fluctuations in protein levels are inconsistent with the observed variability of size and age at division in yeast cells, as shown in the simplified cell-cycle model of Kar et al (2009) and as we have confirmed with our more realistic model. The size of these fluctuations can be reduced to a more acceptable level by assuming a shorter half-life for mRNA (say, ρ=0.1).
There must be some mechanisms whereby yeast cells lessen the protein fluctuations implied by transcription–translation coupling. Following Pedraza and Paulsson (2008), we suggest that mRNA gestation and senescence may resolve this problem. Equation (3) is based on a simple, one-stage, birth–death model of mRNA turnover. In Supplementary Appendix 1, we show that a model of mRNA processing, with 10 stages each of mRNA gestation and senescence, gives reasonable fluctuations at the protein level (CVP≈5%), even if the effective half-life of mRNA is 10 min. A one-stage model with τM=1 min gives comparable fluctuations (CVP≈5%). In the main text, we use a simple birth–death model of mRNA turnover with an ‘effective' half-life of 1 min, in order to limit the computational complexity of the full cell-cycle model.
In order for the cell's genome to be passed intact from one generation to the next, the events of the cell cycle (DNA replication, mitosis, cell division) must be executed in the correct order, despite the considerable molecular noise inherent in any protein-based regulatory system residing in the small confines of a eukaryotic cell. To assess the effects of molecular fluctuations on cell-cycle progression in budding yeast cells, we have constructed a new model of the regulation of Cln- and Clb-dependent kinases, based on multisite phosphorylation of their target proteins and on positive and negative feedback loops involving the kinases themselves. To account for the significant role of noise in the transcription and translation steps of gene expression, the model includes mRNAs as well as proteins. The model equations are simulated deterministically and stochastically to reveal the bistable switching behavior on which proper cell-cycle progression depends and to show that this behavior is robust to the level of molecular noise expected in yeast-sized cells (∼50 fL volume). The model gives a quantitatively accurate account of the variability observed in the G1-S transition in budding yeast, which is governed by an underlying sizer+timer control system.
doi:10.1038/msb.2010.55
PMCID: PMC2947364  PMID: 20739927
bistability; cell-cycle variability; size control; stochastic model; transcription–translation coupling
16.  Niche adaptation by expansion and reprogramming of general transcription factors 
Experimental analysis of TFB family proteins in a halophilic archaeon reveals complex environment-dependent fitness contributions. Gene conversion events among these proteins can generate novel niche adaptation capabilities, a process that may have contributed to archaeal adaptation to extreme environments.
Evolution of archaeal lineages correlate with duplication events in the TFB family.Each TFB is required for adaptation to multiple environments.The relative fitness contributions of TFBs change with environmental context.Changes in the regulation of duplicated TFBs can generate new adaptation capabilities.
The evolutionary success of an organism depends on its ability to continually adapt to changes in the patterns of constant, periodic, and transient challenges within its environment. This process of ‘niche adaptation' requires reprogramming of the organism's environmental response networks by reorganizing interactions among diverse parts including environmental sensors, signal transducers, and transcriptional and post-transcriptional regulators. Gene duplications have been discovered to be one of the principal strategies in this process, especially for reprogramming of gene regulatory networks (GRNs). Whereas eukaryotes require dozens of factors for recruitment of RNA polymerase, archaea require just two general transcription factors (GTFs) that are orthologous to eukaryotic TFIIB (TFB in archaea) and TATA-binding protein (TBP) (Bell et al, 1998). Both of these GTFs have expanded extensively in nearly 50% of all archaea whose genomes have been fully sequenced. The phylogenetic analysis presented in this study reveal lineage-specific expansions of TFBs, suggesting that they might encode functionally specialized gene regulatory programs for the unique environments to which these organisms have adapted. This hypothesis is particularly appealing when we consider that the greatest expansion is observed within the group of halophilic archaea whose habitats are associated with routine and dynamic changes in a number of environmental factors including light, temperature, oxygen, salinity, and ionic composition (Rodriguez-Valera, 1993; Litchfield, 1998).
We have previously demonstrated that variations in the expanded set of TFBs (a through e) in Halobacterium salinarum NRC-1 manifests at the level of physical interactions within and across the two families, their DNA-binding specificity, their differential regulation in varying environments, and, ultimately, on the large-scale segregation of transcription of all genes into overlapping yet distinct sets of functionally related groups (Facciotti et al, 2007). We have extended findings from this earlier study with a systematic survey of the fitness consequences of perturbing the TFB network of H. salinarum NRC-1 across 17 environments. Notably, each TFB conferred fitness in two or more environmental conditions tested, and the relative fitness contributions (see Table I) of the five TFBs varied significantly by environment. From an evolutionary perspective, the relationships among these fitness landscapes reveal that two classes of TFBs (c/g- and f-type) appear to have played an important role in the evolution of halophilic archaea by overseeing regulation of core physiological capabilities in these organisms. TFBs of the other clades (b/d and a/e) seem to have emerged much more recently through gene duplications or horizontal gene transfers (HGTs) and are being utilized for adaptation to specialized environmental conditions.
We also investigated higher-order functional interactions and relationships among the duplicated TFBs by performing competition experiments and by mapping genetic interactions in different environments. This demonstrated that depending on environmental context, the TFBs have strikingly different functional hierarchies and genetic interactions with one another. This is remarkable as it makes each TFB essential albeit at different times in a dynamically changing environment.
In order to understand the process by which such gene family expansions shape architecture and functioning of a GRN, we performed integrated analysis of phylogeny, physical interactions, regulation, and fitness landscapes of the seven TFBs in H. salinarum NRC-1. This revealed that evolution of both their protein-coding sequence and their promoter has been instrumental in the encoding of environment-specific regulatory programs. Importantly, the convergent and divergent evolution of regulation and binding properties of TFBs suggested that, aside from HGT and random mutations, a third plausible (and perhaps most interesting) mechanism for acquiring a novel TFB variant is through gene conversion. To test this hypothesis, we synthesized a novel TFBx by transferring TFBa/e clade-specific residues to a TFBd backbone, transformed this variant under the control of either the TFBd or the TFBe promoter (PtfbD or PtfbE) into three different host genetic backgrounds (Δura3 (parent), ΔtfbD, and ΔtfbE), and analyzed fitness and gene expression patterns during growth at 25 and 37°C. This showed that gene conversion events spanning the coding sequence and the promoter, environmental context, and genetic background of the host are all extremely influential in the functional integration of a TFB into the GRN. Importantly, this analysis suggested that altering the regulation of an existing set of expanded TFBs might be an efficient mechanism to reprogram the GRN to rapidly generate novel niche adaptation capability. We have confirmed this experimentally by increasing fitness merely by moving tfbE to PtfbD control, and by generating a completely novel phenotype (biofilm-like appearance) by overexpression of tfbE.
Altogether this study clearly demonstrates that archaea can rapidly generate novel niche adaptation programs by simply altering regulation of duplicated TFBs. This is significant because expansions in the TFB family is widespread in archaea, a class of organisms that not only represent 20% of biomass on earth but are also known to have colonized some of the most extreme environments (DeLong and Pace, 2001). This strategy for niche adaptation is further expanded through interactions of the multiple TFBs with members of other expanded TF families such as TBPs (Facciotti et al, 2007) and sequence-specific regulators (e.g. Lrp family (Peeters and Charlier, 2010)). This is analogous to combinatorial solutions for other complex biological problems such as recognition of pathogens by Toll-like receptors (Roach et al, 2005), generation of antibody diversity by V(D)J recombination (Early et al, 1980), and recognition and processing of odors (Malnic et al, 1999).
Numerous lineage-specific expansions of the transcription factor B (TFB) family in archaea suggests an important role for expanded TFBs in encoding environment-specific gene regulatory programs. Given the characteristics of hypersaline lakes, the unusually large numbers of TFBs in halophilic archaea further suggests that they might be especially important in rapid adaptation to the challenges of a dynamically changing environment. Motivated by these observations, we have investigated the implications of TFB expansions by correlating sequence variations, regulation, and physical interactions of all seven TFBs in Halobacterium salinarum NRC-1 to their fitness landscapes, functional hierarchies, and genetic interactions across 2488 experiments covering combinatorial variations in salt, pH, temperature, and Cu stress. This systems analysis has revealed an elegant scheme in which completely novel fitness landscapes are generated by gene conversion events that introduce subtle changes to the regulation or physical interactions of duplicated TFBs. Based on these insights, we have introduced a synthetically redesigned TFB and altered the regulation of existing TFBs to illustrate how archaea can rapidly generate novel phenotypes by simply reprogramming their TFB regulatory network.
doi:10.1038/msb.2011.87
PMCID: PMC3261711  PMID: 22108796
evolution by gene family expansion; fitness; niche adaptation; reprogramming of gene regulatory network; transcription factor B
17.  Robustness under Functional Constraint: The Genetic Network for Temporal Expression in Drosophila Neurogenesis 
PLoS Computational Biology  2010;6(4):e1000760.
Precise temporal coordination of gene expression is crucial for many developmental processes. One central question in developmental biology is how such coordinated expression patterns are robustly controlled. During embryonic development of the Drosophila central nervous system, neural stem cells called neuroblasts express a group of genes in a definite order, which leads to the diversity of cell types. We produced all possible regulatory networks of these genes and examined their expression dynamics numerically. From the analysis, we identified requisite regulations and predicted an unknown factor to reproduce known expression profiles caused by loss-of-function or overexpression of the genes in vivo, as well as in the wild type. Following this, we evaluated the stability of the actual Drosophila network for sequential expression. This network shows the highest robustness against parameter variations and gene expression fluctuations among the possible networks that reproduce the expression profiles. We propose a regulatory module composed of three types of regulations that is responsible for precise sequential expression. This study suggests that the Drosophila network for sequential expression has evolved to generate the robust temporal expression for neuronal specification.
Author Summary
Cell fate specification is of key importance in the development of multicellular organisms. To specify various cell fates correctly, genetic networks precisely coordinate spatial and temporal gene expression patterns during various developmental stages. One central question in developmental biology is to elucidate the relationship between the pattern formation and the network architecture. During embryonic development of the Drosophila central nervous system, the neural stem cells express a group of genes in a definite order, which is responsible for the diversity of neural cells. To elucidate the underlying mechanism of the process, we analyzed the structure and dynamics of the genetic network for the temporal changes occurring in the Drosophila neural stem cells. Searching all the possible regulatory networks of these genes using a computer program, we detected the requisite regulations that reproduce observed gene expression profiles. By comparing the stability of the dynamics among the functional networks, we uncovered the robust nature of the actual Drosophila network against environmental and intrinsic fluctuations. These results indicate that the genetic network for sequential expression has evolved to be robust under functional constraints. Our study proposes regulatory modules that are responsible for the precise sequential expressions, which might exist in genetic networks for other temporal patterning processes.
doi:10.1371/journal.pcbi.1000760
PMCID: PMC2861627  PMID: 20454677
18.  An algebra-based method for inferring gene regulatory networks 
BMC Systems Biology  2014;8:37.
Background
The inference of gene regulatory networks (GRNs) from experimental observations is at the heart of systems biology. This includes the inference of both the network topology and its dynamics. While there are many algorithms available to infer the network topology from experimental data, less emphasis has been placed on methods that infer network dynamics. Furthermore, since the network inference problem is typically underdetermined, it is essential to have the option of incorporating into the inference process, prior knowledge about the network, along with an effective description of the search space of dynamic models. Finally, it is also important to have an understanding of how a given inference method is affected by experimental and other noise in the data used.
Results
This paper contains a novel inference algorithm using the algebraic framework of Boolean polynomial dynamical systems (BPDS), meeting all these requirements. The algorithm takes as input time series data, including those from network perturbations, such as knock-out mutant strains and RNAi experiments. It allows for the incorporation of prior biological knowledge while being robust to significant levels of noise in the data used for inference. It uses an evolutionary algorithm for local optimization with an encoding of the mathematical models as BPDS. The BPDS framework allows an effective representation of the search space for algebraic dynamic models that improves computational performance. The algorithm is validated with both simulated and experimental microarray expression profile data. Robustness to noise is tested using a published mathematical model of the segment polarity gene network in Drosophila melanogaster. Benchmarking of the algorithm is done by comparison with a spectrum of state-of-the-art network inference methods on data from the synthetic IRMA network to demonstrate that our method has good precision and recall for the network reconstruction task, while also predicting several of the dynamic patterns present in the network.
Conclusions
Boolean polynomial dynamical systems provide a powerful modeling framework for the reverse engineering of gene regulatory networks, that enables a rich mathematical structure on the model search space. A C++ implementation of the method, distributed under LPGL license, is available, together with the source code, at http://www.paola-vera-licona.net/Software/EARevEng/REACT.html.
doi:10.1186/1752-0509-8-37
PMCID: PMC4022379  PMID: 24669835
Reverse-engineering; network inference; Boolean networks; molecular networks; gene regulatory networks; polynomial dynamical systems; algebraic dynamic models; evolutionary computation; DNA microarray data; time series data; data noise
19.  A Unifying Mathematical Framework for Genetic Robustness, Environmental Robustness, Network Robustness and their Trade-offs on Phenotype Robustness in Biological Networks. Part III: Synthetic Gene Networks in Synthetic Biology 
Robust stabilization and environmental disturbance attenuation are ubiquitous systematic properties that are observed in biological systems at many different levels. The underlying principles for robust stabilization and environmental disturbance attenuation are universal to both complex biological systems and sophisticated engineering systems. In many biological networks, network robustness should be large enough to confer: intrinsic robustness for tolerating intrinsic parameter fluctuations; genetic robustness for buffering genetic variations; and environmental robustness for resisting environmental disturbances. Network robustness is needed so phenotype stability of biological network can be maintained, guaranteeing phenotype robustness. Synthetic biology is foreseen to have important applications in biotechnology and medicine; it is expected to contribute significantly to a better understanding of functioning of complex biological systems. This paper presents a unifying mathematical framework for investigating the principles of both robust stabilization and environmental disturbance attenuation for synthetic gene networks in synthetic biology. Further, from the unifying mathematical framework, we found that the phenotype robustness criterion for synthetic gene networks is the following: if intrinsic robustness + genetic robustness + environmental robustness ≦ network robustness, then the phenotype robustness can be maintained in spite of intrinsic parameter fluctuations, genetic variations, and environmental disturbances. Therefore, the trade-offs between intrinsic robustness, genetic robustness, environmental robustness, and network robustness in synthetic biology can also be investigated through corresponding phenotype robustness criteria from the systematic point of view. Finally, a robust synthetic design that involves network evolution algorithms with desired behavior under intrinsic parameter fluctuations, genetic variations, and environmental disturbances, is also proposed, together with a simulation example.
doi:10.4137/EBO.S10686
PMCID: PMC3596975  PMID: 23515190
phenotype robustness; network robustness; network sensitivity; gene network; synthetic biology
20.  Cooperative Adaptive Responses in Gene Regulatory Networks with Many Degrees of Freedom 
PLoS Computational Biology  2013;9(4):e1003001.
Cells generally adapt to environmental changes by first exhibiting an immediate response and then gradually returning to their original state to achieve homeostasis. Although simple network motifs consisting of a few genes have been shown to exhibit such adaptive dynamics, they do not reflect the complexity of real cells, where the expression of a large number of genes activates or represses other genes, permitting adaptive behaviors. Here, we investigated the responses of gene regulatory networks containing many genes that have undergone numerical evolution to achieve high fitness due to the adaptive response of only a single target gene; this single target gene responds to changes in external inputs and later returns to basal levels. Despite setting a single target, most genes showed adaptive responses after evolution. Such adaptive dynamics were not due to common motifs within a few genes; even without such motifs, almost all genes showed adaptation, albeit sometimes partial adaptation, in the sense that expression levels did not always return to original levels. The genes split into two groups: genes in the first group exhibited an initial increase in expression and then returned to basal levels, while genes in the second group exhibited the opposite changes in expression. From this model, genes in the first group received positive input from other genes within the first group, but negative input from genes in the second group, and vice versa. Thus, the adaptation dynamics of genes from both groups were consolidated. This cooperative adaptive behavior was commonly observed if the number of genes involved was larger than the order of ten. These results have implications in the collective responses of gene expression networks in microarray measurements of yeast Saccharomyces cerevisiae and the significance to the biological homeostasis of systems with many components.
Author Summary
Homeostasis is an inherent property of biological systems, which have a general tendency to adapt, i.e., to recover their original state following environmental changes. In cells, this adaptation is mediated by changes in protein expression. Initially, cells respond to environmental changes by altered gene/protein expression; subsequently, the expression of most genes returns to basal levels, albeit not completely, as shown by recent experimental analyses of yeast. Although simple mechanisms for adaptation through network motifs, composed of just a few genes, are well understood, how regulatory networks involving many genes that activate or repress each other can generate adaptive behaviors is unclear. Here, by numerically evolving gene regulatory networks, we obtained a class of genes whose expression dynamics showed adaptation over almost all genes, from which we revealed the general logic underlying such adaptive dynamics with many degrees of freedom, which was not reducible to motifs with a few genes. This adaptation was cooperative, i.e., adaptation of one gene mutually relied upon others' adaptive expressions. Moreover, this collective behavior was robust to noise and mutations. The present study sheds a light on the nature of collective gene expression dynamics allowing for biological homeostasis.
doi:10.1371/journal.pcbi.1003001
PMCID: PMC3616990  PMID: 23592959
21.  Evolution under Fluctuating Environments Explains Observed Robustness in Metabolic Networks 
PLoS Computational Biology  2010;6(8):e1000907.
A high level of robustness against gene deletion is observed in many organisms. However, it is still not clear which biochemical features underline this robustness and how these are acquired during evolution. One hypothesis, specific to metabolic networks, is that robustness emerges as a byproduct of selection for biomass production in different environments. To test this hypothesis we performed evolutionary simulations of metabolic networks under stable and fluctuating environments. We find that networks evolved under the latter scenario can better tolerate single gene deletion in specific environments. Such robustness is underlined by an increased number of independent fluxes and multifunctional enzymes in the evolved networks. Observed robustness in networks evolved under fluctuating environments was “apparent,” in the sense that it decreased significantly as we tested effects of gene deletions under all environments experienced during evolution. Furthermore, when we continued evolution of these networks under a stable environment, we found that any robustness they had acquired was completely lost. These findings provide evidence that evolution under fluctuating environments can account for the observed robustness in metabolic networks. Further, they suggest that organisms living under stable environments should display lower robustness in their metabolic networks, and that robustness should decrease upon switching to more stable environments.
Author Summary
One of the most surprising recent biological findings is the high level of tolerance organisms show towards loss of single genes. This observation suggests that there are certain features of biological systems that give them a high tolerance (i.e. robustness) towards gene loss. We still lack an exact understanding of what these features might be and how they could have been acquired during evolution. Here, we offer a possible answer for these questions in the context of metabolic networks. Using mathematical models capturing the structure and dynamics of metabolic networks, we simulate their evolution under stable and fluctuating environments (i.e., available metabolites). We find that the latter scenario leads to evolution of metabolic networks that display high robustness against gene loss. This robustness of in silico evolved networks is underlined by an increased number of multifunctional enzymes and independent paths leading from initial metabolites to biomass. These findings provide evidence that fluctuating environments can be a major evolutionary force leading to the emergence of robustness as a side effect. A direct prediction resulting from this study is that organisms living in stable and fluctuating environments should display differing levels of robustness against gene loss.
doi:10.1371/journal.pcbi.1000907
PMCID: PMC2928748  PMID: 20865149
22.  Effects of Transcriptional Pausing on Gene Expression Dynamics 
PLoS Computational Biology  2010;6(3):e1000704.
Stochasticity in gene expression affects many cellular processes and is a source of phenotypic diversity between genetically identical individuals. Events in elongation, particularly RNA polymerase pausing, are a source of this noise. Since the rate and duration of pausing are sequence-dependent, this regulatory mechanism of transcriptional dynamics is evolvable. The dependency of pause propensity on regulatory molecules makes pausing a response mechanism to external stress. Using a delayed stochastic model of bacterial transcription at the single nucleotide level that includes the promoter open complex formation, pausing, arrest, misincorporation and editing, pyrophosphorolysis, and premature termination, we investigate how RNA polymerase pausing affects a gene's transcriptional dynamics and gene networks. We show that pauses' duration and rate of occurrence affect the bursting in RNA production, transcriptional and translational noise, and the transient to reach mean RNA and protein levels. In a genetic repressilator, increasing the pausing rate and the duration of pausing events increases the period length but does not affect the robustness of the periodicity. We conclude that RNA polymerase pausing might be an important evolvable feature of genetic networks.
Author Summary
Investigation on how phenotypic diversity of genetically identical organisms is generated and regulated has focused on noise in gene expression. It is unknown to what extent noise in gene expression and genetic networks is evolvable, and by which mechanisms it evolves. The noise has several sources, e.g., noise in transcription initiation and during elongation. We focus on RNA polymerase (RNAP) pausing and show that it can regulate, to some extent, noise in gene expression. RNAP frequently pauses during elongation. The pausing frequency and average duration are sequence-specific, thus evolvable. The dependency of pause propensity on regulatory molecules makes pausing a mechanism adaptable to rapidly changing environments. We study, in a stochastic model of bacterial transcription at the single nucleotide level that includes the promoter open complex formation, pausing, arrest, misincorporation and editing, pyrophosphorolysis, and premature termination, how pausing affects the dynamics of gene expression and gene networks. In a model of a genetic clock, with periodic dynamics, pauses affect the period length but do not disrupt the periodicity. We conclude that RNAP pausing is an important evolvable feature of gene regulatory networks, that can be used by organisms to adapt to changing environments and regulate phenotypic diversity.
doi:10.1371/journal.pcbi.1000704
PMCID: PMC2837387  PMID: 20300642
23.  Modularity and hormone sensitivity of the Drosophila melanogaster insulin receptor/target of rapamycin interaction proteome 
First systematic analysis of the evolutionary conserved InR/TOR pathway interaction proteome in Drosophila.Quantitative mass spectrometry revealed that 22% of identified protein interactions are regulated by the growth hormone insulin affecting membrane proximal as well as intracellular signaling complexes.Systematic RNA interference linked a significant fraction of network components to the control of dTOR kinase activity.Combined biochemical and genetic data suggest dTTT, a dTOR-containing complex required for cell growth control by dTORC1 and dTORC2 in vivo.
Cellular growth is a fundamental process that requires constant adaptations to changing environmental conditions, like growth factor and nutrient availability, energy levels and more. Over the years, the insulin receptor/target of rapamycin pathway (InR/TOR) emerged as a key signaling system for the control of metazoan cell growth. Genetic screens carried out in the fruit fly Drosophila melanogaster identified key InR/TOR pathway components and their relationships. Phenotypes such as altered cell growth are likely to emerge from perturbed dynamic networks containing InR/TOR pathway components, which stably or transiently interact with other cellular proteins to form complexes and networks thereof. Systematic studies on the topology and dynamics of protein interaction networks become therefore highly relevant to gain systems level understanding of deregulated cell growth. Despite much progress in genetic analysis only few systematic protein interaction studies have been reported for Drosophila, which in most cases lack quantitative information representing the dynamic nature of such networks. Here, we present the first quantitative affinity purification mass spectrometry (AP–MS/MS) analysis on the evolutionary conserved InR/TOR signaling network in Drosophila. Systematic RNAi-based functional analysis of identified network components revealed key components linked to the regulation of the central effector kinase dTOR. This includes also dTTT, a novel dTOR-containing complex required for the control of dTORC1 and dTORC2 in vivo.
For systematic AP–MS analysis, we generated Drosophila Kc167 cell lines inducibly expressing affinity-tagged bait proteins previously linked to InR/TOR signaling. Bait expressing Kc167 cell lines were harvested before and after insulin stimulation for subsequent affinity purification. Following LC–MS/MS analysis and probabilistic data filtering using SAINT (Choi et al, 2010), we generated a quantitative network model from 97 high confidence protein–protein interactions and 58 network components (Figure 2). The presented network displayed a high degree of orthologous interactions conserved also in human cells and identified a number of novel molecular interactions with InR/TOR signaling components for future hypothesis driven analysis.
To measure insulin-induced changes within the InR/TOR interaction proteome, we applied a recently introduced label-free quantitative MS approach (Rinner et al, 2007). The obtained quantitative data suggest that 22% of all interactions in the network are regulated by insulin. Major changes could be observed within the membrane proximal InR/chico/PI3K signaling complexes, and also in 14-3-3 protein containing signaling complexes and dTORC1, a complex that contains besides dTOR all major orthologous proteins found also in human mTORC1 including the two dTORC1 substrates d4E-BP (Thor) and S6 Kinase (S6K). Insulin triggered both, dissociation and association of dTORC1 proteins. Among the proteins that showed enhanced binding to dTORC1 upon insulin stimulation we found Unkempt, a RING-finger protein with a proposed role in ubiquitin-mediated protein degradation (Lores et al, 2010). Besides dTORC1 our systematic AP–MS analysis also revealed the presence of dTORC2, the second major TOR complex in Drosophila. dTORC2 contains the Drosophila orthologous of human mTORC2 proteins, but in contrast to dTORC1 was not affected upon insulin stimulation. Interestingly, we also found a specific set of proteins that were not linked to the canonical TOR complexes TORC1 and TORC2 in dTOR purifications. These include LqfR (liquid facets related), Pontin, Reptin, Spaghetti and the gene product of CG16908. We found the same set of proteins when we used CG16908 as a bait, suggesting complex formation among the identified proteins. None of the dTORC1/2 components besides dTOR was identified in CG16908 purifications, indicating that these proteins form dTOR complexes distinct from dTORC1 and dTORC2. Based on known interaction information from other species and data obtained from this study we refer to this complex as dTTT (Drosophila TOR, TELO2, TTI1) (Horejsi et al, 2010; [18]Hurov et al, 2010; [20]Kaizuka et al, 2010). A directed quantitative MS analysis of dTOR complex components suggests that dTORC1 is the most abundant dTOR complex we identified in Kc167 cells.
We next studied the potential roles of the identified network components for controlling the activity of the dInR/TOR pathway using systematic RNAi depletion and quantitative western blotting to measure the changes in abundance of phosphorylated substrates of dTORC1 (Thor/d4E-BP, dS6K) and dTORC2 (dPKB) in RNAi-treated cells (Figure 5). Overall, we could identify 16 proteins (out of 58) whose depletion caused an at least 50% increase or decrease in the levels of phosphorylated d4E-BP, S6K and/or PKB compared with control GFP RNAi. Besides established pathway components, we found several novel regulators within the dInR/TOR interaction network. For example, RNAi against the novel insulin-regulated dTORC1 component Unkempt resulted in enhanced phosphorylation of the dTORC1 substrate d4E-BP, which suggests a negative role for Unkempt on dTORC1 activity. In contrast, depletion of CG16908 and LqfR caused hypo-phosphorylation of all dTOR substrates similar to dTOR itself, suggesting a positive role for the dTTT complex on dTOR activity. Subsequently, we tested whether dTTT components also plays a role in dTOR-mediated cell growth in vivo. Depletion of both dTTT components, CG16908 and LqfR, in the Drosophila eye resulted in a substantial decrease in eye size. Likewise, FLP-FRT-mediated mitotic recombination resulted in CG16908 and LqfR mutant clones with a similar reduced growth phenotype as observed in dTOR mutant clones. Hence, the combined biochemical and genetic analysis revealed dTTT as a dTOR-containing complex required for the activity of both dTORC1 and dTORC2 and thus plays a critical role in controlling cell growth.
Taken together, these results illustrate how a systematic quantitative AP–MS approach when combined with systematic functional analysis in Drosophila can reveal novel insights into the dynamic organization of regulatory networks for cell growth control in metazoans.
Using quantitative mass spectrometry, this study reports how insulin affects the modularity of the interaction proteome of the Drosophila InR/TOR pathway, an evolutionary conserved signaling system for the control of metazoan cell growth. Systematic functional analysis linked a significant number of identified network components to the control of dTOR activity and revealed dTTT, a dTOR complex required for in vivo cell growth control by dTORC1 and dTORC2.
Genetic analysis in Drosophila melanogaster has been widely used to identify a system of genes that control cell growth in response to insulin and nutrients. Many of these genes encode components of the insulin receptor/target of rapamycin (InR/TOR) pathway. However, the biochemical context of this regulatory system is still poorly characterized in Drosophila. Here, we present the first quantitative study that systematically characterizes the modularity and hormone sensitivity of the interaction proteome underlying growth control by the dInR/TOR pathway. Applying quantitative affinity purification and mass spectrometry, we identified 97 high confidence protein interactions among 58 network components. In all, 22% of the detected interactions were regulated by insulin affecting membrane proximal as well as intracellular signaling complexes. Systematic functional analysis linked a subset of network components to the control of dTORC1 and dTORC2 activity. Furthermore, our data suggest the presence of three distinct dTOR kinase complexes, including the evolutionary conserved dTTT complex (Drosophila TOR, TELO2, TTI1). Subsequent genetic studies in flies suggest a role for dTTT in controlling cell growth via a dTORC1- and dTORC2-dependent mechanism.
doi:10.1038/msb.2011.79
PMCID: PMC3261712  PMID: 22068330
cell growth; InR/TOR pathway; interaction proteome; quantitative mass spectrometry; signaling
24.  Broadband Criticality of Human Brain Network Synchronization 
PLoS Computational Biology  2009;5(3):e1000314.
Self-organized criticality is an attractive model for human brain dynamics, but there has been little direct evidence for its existence in large-scale systems measured by neuroimaging. In general, critical systems are associated with fractal or power law scaling, long-range correlations in space and time, and rapid reconfiguration in response to external inputs. Here, we consider two measures of phase synchronization: the phase-lock interval, or duration of coupling between a pair of (neurophysiological) processes, and the lability of global synchronization of a (brain functional) network. Using computational simulations of two mechanistically distinct systems displaying complex dynamics, the Ising model and the Kuramoto model, we show that both synchronization metrics have power law probability distributions specifically when these systems are in a critical state. We then demonstrate power law scaling of both pairwise and global synchronization metrics in functional MRI and magnetoencephalographic data recorded from normal volunteers under resting conditions. These results strongly suggest that human brain functional systems exist in an endogenous state of dynamical criticality, characterized by a greater than random probability of both prolonged periods of phase-locking and occurrence of large rapid changes in the state of global synchronization, analogous to the neuronal “avalanches” previously described in cellular systems. Moreover, evidence for critical dynamics was identified consistently in neurophysiological systems operating at frequency intervals ranging from 0.05–0.11 to 62.5–125 Hz, confirming that criticality is a property of human brain functional network organization at all frequency intervals in the brain's physiological bandwidth.
Author Summary
Systems in a critical state are poised on the cusp of a transition between ordered and random behavior. At this point, they demonstrate complex patterning of fluctuations at all scales of space and time. Criticality is an attractive model for brain dynamics because it optimizes information transfer, storage capacity, and sensitivity to external stimuli in computational models. However, to date there has been little direct experimental evidence for critical dynamics of human brain networks. Here, we considered two measures of functional coupling or phase synchronization between components of a dynamic system: the phase lock interval or duration of synchronization between a specific pair of time series or processes in the system and the lability of global synchronization among all pairs of processes. We confirmed that both synchronization metrics demonstrated scale invariant behaviors in two computational models of critical dynamics as well as in human brain functional systems oscillating at low frequencies (<0.5 Hz, measured using functional MRI) and at higher frequencies (1–125 Hz, measured using magnetoencephalography). We conclude that human brain functional networks demonstrate critical dynamics in all frequency intervals, a phenomenon we have described as broadband criticality.
doi:10.1371/journal.pcbi.1000314
PMCID: PMC2647739  PMID: 19300473
25.  New Measurement Methods of Network Robustness and Response Ability via Microarray Data 
PLoS ONE  2013;8(1):e55230.
“Robustness”, the network ability to maintain systematic performance in the face of intrinsic perturbations, and “response ability”, the network ability to respond to external stimuli or transduce them to downstream regulators, are two important complementary system characteristics that must be considered when discussing biological system performance. However, at present, these features cannot be measured directly for all network components in an experimental procedure. Therefore, we present two novel systematic measurement methods – Network Robustness Measurement (NRM) and Response Ability Measurement (RAM) – to estimate the network robustness and response ability of a gene regulatory network (GRN) or protein-protein interaction network (PPIN) based on the dynamic network model constructed by the corresponding microarray data. We demonstrate the efficiency of NRM and RAM in analyzing GRNs and PPINs, respectively, by considering aging- and cancer-related datasets. When applied to an aging-related GRN, our results indicate that such a network is more robust to intrinsic perturbations in the elderly than in the young, and is therefore less responsive to external stimuli. When applied to a PPIN of fibroblast and HeLa cells, we observe that the network of cancer cells possesses better robustness than that of normal cells. Moreover, the response ability of the PPIN calculated from the cancer cells is lower than that from healthy cells. Accordingly, we propose that generalized NRM and RAM methods represent effective tools for exploring and analyzing different systems-level dynamical properties via microarray data. Making use of such properties can facilitate prediction and application, providing useful information on clinical strategy, drug target selection, and design specifications of synthetic biology from a systems biology perspective.
doi:10.1371/journal.pone.0055230
PMCID: PMC3557243  PMID: 23383119

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