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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.  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
3.  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
4.  Proportionality between variances in gene expression induced by noise and mutation: consequence of evolutionary robustness 
Background
Characterization of robustness and plasticity of phenotypes is a basic issue in evolutionary and developmental biology. The robustness and plasticity are concerned with changeability of a biological system against external perturbations. The perturbations are either genetic, i.e., due to mutations in genes in the population, or epigenetic, i.e., due to noise during development or environmental variations. Thus, the variances of phenotypes due to genetic and epigenetic perturbations provide quantitative measures for such changeability during evolution and development, respectively.
Results
Using numerical models simulating the evolutionary changes in the gene regulation network required to achieve a particular expression pattern, we first confirmed that gene expression dynamics robust to mutation evolved in the presence of a sufficient level of transcriptional noise. Under such conditions, the two types of variances in the gene expression levels, i.e. those due to mutations to the gene regulation network and those due to noise in gene expression dynamics were found to be proportional over a number of genes. The fraction of such genes with a common proportionality coefficient increased with an increase in the robustness of the evolved network. This proportionality was generally confirmed, also under the presence of environmental fluctuations and sexual recombination in diploids, and was explained from an evolutionary robustness hypothesis, in which an evolved robust system suppresses the so-called error catastrophe - the destabilization of the single-peaked distribution in gene expression levels. Experimental evidences for the proportionality of the variances over genes are also discussed.
Conclusions
The proportionality between the genetic and epigenetic variances of phenotypes implies the correlation between the robustness (or plasticity) against genetic changes and against noise in development, and also suggests that phenotypic traits that are more variable epigenetically have a higher evolutionary potential.
doi:10.1186/1471-2148-11-27
PMCID: PMC3045907  PMID: 21269459
5.  Dynamic Properties of Network Motifs Contribute to Biological Network Organization 
PLoS Biology  2005;3(11):e343.
Biological networks, such as those describing gene regulation, signal transduction, and neural synapses, are representations of large-scale dynamic systems. Discovery of organizing principles of biological networks can be enhanced by embracing the notion that there is a deep interplay between network structure and system dynamics. Recently, many structural characteristics of these non-random networks have been identified, but dynamical implications of the features have not been explored comprehensively. We demonstrate by exhaustive computational analysis that a dynamical property—stability or robustness to small perturbations—is highly correlated with the relative abundance of small subnetworks (network motifs) in several previously determined biological networks. We propose that robust dynamical stability is an influential property that can determine the non-random structure of biological networks.
The authors model how network motifs respond to small-scale perturbations and find a strong correlation between motif stability and abundance in a network, suggesting that dynamic properties of network motifs may play a role in overall network structure.
doi:10.1371/journal.pbio.0030343
PMCID: PMC1239925  PMID: 16187794
6.  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
7.  Ground State Robustness as an Evolutionary Design Principle in Signaling Networks 
PLoS ONE  2009;4(12):e8001.
The ability of an organism to survive depends on its capability to adapt to external conditions. In addition to metabolic versatility and efficient replication, reliable signal transduction is essential. As signaling systems are under permanent evolutionary pressure one may assume that their structure reflects certain functional properties. However, despite promising theoretical studies in recent years, the selective forces which shape signaling network topologies in general remain unclear. Here, we propose prevention of autoactivation as one possible evolutionary design principle. A generic framework for continuous kinetic models is used to derive topological implications of demanding a dynamically stable ground state in signaling systems. To this end graph theoretical methods are applied. The index of the underlying digraph is shown to be a key topological property which determines the so-called kinetic ground state (or off-state) robustness. The kinetic robustness depends solely on the composition of the subdigraph with the strongly connected components, which comprise all positive feedbacks in the network. The component with the highest index in the feedback family is shown to dominate the kinetic robustness of the whole network, whereas relative size and girth of these motifs are emphasized as important determinants of the component index. Moreover, depending on topological features, the maintenance of robustness differs when networks are faced with structural perturbations. This structural off-state robustness, defined as the average kinetic robustness of a network's neighborhood, turns out to be useful since some structural features are neutral towards kinetic robustness, but show up to be supporting against structural perturbations. Among these are a low connectivity, a high divergence and a low path sum. All results are tested against real signaling networks obtained from databases. The analysis suggests that ground state robustness may serve as a rationale for some structural peculiarities found in intracellular signaling networks.
doi:10.1371/journal.pone.0008001
PMCID: PMC2779451  PMID: 19956601
8.  GA-based Design Algorithms for the Robust Synthetic Genetic Oscillators with Prescribed Amplitude, Period and Phase 
In the past decade, the development of synthetic gene networks has attracted much attention from many researchers. In particular, the genetic oscillator known as the repressilator has become a paradigm for how to design a gene network with a desired dynamic behaviour. Even though the repressilator can show oscillatory properties in its protein concentrations, their amplitudes, frequencies and phases are perturbed by the kinetic parametric fluctuations (intrinsic molecular perturbations) and external disturbances (extrinsic molecular noises) of the environment. Therefore, how to design a robust genetic oscillator with desired amplitude, frequency and phase under stochastic intrinsic and extrinsic molecular noises is an important topic for synthetic biology.
In this study, based on periodic reference signals with arbitrary amplitudes, frequencies and phases, a robust synthetic gene oscillator is designed by tuning the kinetic parameters of repressilator via a genetic algorithm (GA) so that the protein concentrations can track the desired periodic reference signals under intrinsic and extrinsic molecular noises. GA is a stochastic optimization algorithm which was inspired by the mechanisms of natural selection and evolution genetics. By the proposed GA-based design algorithm, the repressilator can track the desired amplitude, frequency and phase of oscillation under intrinsic and extrinsic noises through the optimization of fitness function.
The proposed GA-based design algorithm can mimic the natural selection in evolutionary process to select adequate kinetic parameters for robust genetic oscillators. The design method can be easily extended to any synthetic gene network design with prescribed behaviours.
PMCID: PMC2881514  PMID: 20535234
synthetic gene network; robust synthetic gene oscillator; genetic algorithm (GA); nature selection; evolutionary genetic repressilator
9.  Robustness from flexibility in the fungal circadian clock 
BMC Systems Biology  2010;4:88.
Background
Robustness is a central property of living systems, enabling function to be maintained against environmental perturbations. A key challenge is to identify the structures in biological circuits that confer system-level properties such as robustness. Circadian clocks allow organisms to adapt to the predictable changes of the 24-hour day/night cycle by generating endogenous rhythms that can be entrained to the external cycle. In all organisms, the clock circuits typically comprise multiple interlocked feedback loops controlling the rhythmic expression of key genes. Previously, we showed that such architectures increase the flexibility of the clock's rhythmic behaviour. We now test the relationship between flexibility and robustness, using a mathematical model of the circuit controlling conidiation in the fungus Neurospora crassa.
Results
The circuit modelled in this work consists of a central negative feedback loop, in which the frequency (frq) gene inhibits its transcriptional activator white collar-1 (wc-1), interlocked with a positive feedback loop in which FRQ protein upregulates WC-1 production. Importantly, our model reproduces the observed entrainment of this circuit under light/dark cycles with varying photoperiod and cycle duration. Our simulations show that whilst the level of frq mRNA is driven directly by the light input, the falling phase of FRQ protein, a molecular correlate of conidiation, maintains a constant phase that is uncoupled from the times of dawn and dusk. The model predicts the behaviour of mutants that uncouple WC-1 production from FRQ's positive feedback, and shows that the positive loop enhances the buffering of conidiation phase against seasonal photoperiod changes. This property is quantified using Kitano's measure for the overall robustness of a regulated system output. Further analysis demonstrates that this functional robustness is a consequence of the greater evolutionary flexibility conferred on the circuit by the interlocking loop structure.
Conclusions
Our model shows that the behaviour of the fungal clock in light-dark cycles can be accounted for by a transcription-translation feedback model of the central FRQ-WC oscillator. More generally, we provide an example of a biological circuit in which greater flexibility yields improved robustness, while also introducing novel sensitivity analysis techniques applicable to a broader range of cellular oscillators.
doi:10.1186/1752-0509-4-88
PMCID: PMC2913929  PMID: 20576110
10.  Exact and Heuristic Methods for Network Completion for Time-Varying Genetic Networks 
BioMed Research International  2014;2014:684014.
Robustness in biological networks can be regarded as an important feature of living systems. A system maintains its functions against internal and external perturbations, leading to topological changes in the network with varying delays. To understand the flexibility of biological networks, we propose a novel approach to analyze time-dependent networks, based on the framework of network completion, which aims to make the minimum amount of modifications to a given network so that the resulting network is most consistent with the observed data. We have developed a novel network completion method for time-varying networks by extending our previous method for the completion of stationary networks. In particular, we introduce a double dynamic programming technique to identify change time points and required modifications. Although this extended method allows us to guarantee the optimality of the solution, this method has relatively low computational efficiency. In order to resolve this difficulty, we developed a heuristic method for speeding up the calculation of minimum least squares errors. We demonstrate the effectiveness of our proposed methods through computational experiments using synthetic data and real microarray gene expression data. The results indicate that our methods exhibit good performance in terms of completing and inferring gene association networks with time-varying structures.
doi:10.1155/2014/684014
PMCID: PMC3966496  PMID: 24738067
11.  Studying the Relationship between Robustness against Mutations in Metabolic Networks and Lifestyle of Organisms 
The Scientific World Journal  2013;2013:615697.
Robustness is the key feature of biological networks that enables living organisms to keep their homeostatic state and to survive against external and internal perturbations. Variations in environmental conditions or nutrients and intracellular changes such as genetic mutations have the potential to change stability and efficiency of an organism. Structural robustness helps biological systems to choose alternative routes of adaptation to varying conditions. In this study, in order to estimate the structural robustness in metabolic networks we presented a novel flux balance-based approach inspired by bond percolation theory. Fourteen in silico metabolic models were studied in this work in order to examine the possible relationship between the lifestyle of organisms and their metabolic robustness. The results of this study confirm that in organisms which are highly adapted to their environment robustness to mutations may decrease compared to other organisms.
doi:10.1155/2013/615697
PMCID: PMC3848384  PMID: 24348175
12.  Differential Dynamic Properties of Scleroderma Fibroblasts in Response to Perturbation of Environmental Stimuli 
PLoS ONE  2008;3(2):e1693.
Diseases are believed to arise from dysregulation of biological systems (pathways) perturbed by environmental triggers. Biological systems as a whole are not just the sum of their components, rather ever-changing, complex and dynamic systems over time in response to internal and external perturbation. In the past, biologists have mainly focused on studying either functions of isolated genes or steady-states of small biological pathways. However, it is systems dynamics that play an essential role in giving rise to cellular function/dysfunction which cause diseases, such as growth, differentiation, division and apoptosis. Biological phenomena of the entire organism are not only determined by steady-state characteristics of the biological systems, but also by intrinsic dynamic properties of biological systems, including stability, transient-response, and controllability, which determine how the systems maintain their functions and performance under a broad range of random internal and external perturbations. As a proof of principle, we examine signal transduction pathways and genetic regulatory pathways as biological systems. We employ widely used state-space equations in systems science to model biological systems, and use expectation-maximization (EM) algorithms and Kalman filter to estimate the parameters in the models. We apply the developed state-space models to human fibroblasts obtained from the autoimmune fibrosing disease, scleroderma, and then perform dynamic analysis of partial TGF-β pathway in both normal and scleroderma fibroblasts stimulated by silica. We find that TGF-β pathway under perturbation of silica shows significant differences in dynamic properties between normal and scleroderma fibroblasts. Our findings may open a new avenue in exploring the functions of cells and mechanism operative in disease development.
doi:10.1371/journal.pone.0001693
PMCID: PMC2246014  PMID: 18301770
13.  A systematic design method for robust synthetic biology to satisfy design specifications 
BMC Systems Biology  2009;3:66.
Background
Synthetic biology is foreseen to have important applications in biotechnology and medicine, and is expected to contribute significantly to a better understanding of the functioning of complex biological systems. However, the development of synthetic gene networks is still difficult and most newly created gene networks are non-functioning due to intrinsic parameter uncertainties, external disturbances and functional variations of intra- and extra-cellular environments. The design method for a robust synthetic gene network that works properly in a host cell under these intrinsic parameter uncertainties and external disturbances is the most important topic in synthetic biology.
Results
In this study, we propose a stochastic model that includes parameter fluctuations and external disturbances to mimic the dynamic behaviors of a synthetic gene network in the host cell. Then, based on this stochastic model, four design specifications are introduced to guarantee that a synthetic gene network can achieve its desired steady state behavior in spite of parameter fluctuations, external disturbances and functional variations in the host cell. We propose a systematic method to select a set of appropriate design parameters for a synthetic gene network that will satisfy these design specifications so that the intrinsic parameter fluctuations can be tolerated, the external disturbances can be efficiently filtered, and most importantly, the desired steady states can be achieved. Thus the synthetic gene network can work properly in a host cell under intrinsic parameter uncertainties, external disturbances and functional variations. Finally, a design procedure for the robust synthetic gene network is developed and a design example is given in silico to confirm the performance of the proposed method.
Conclusion
Based on four design specifications, a systematic design procedure is developed for designers to engineer a robust synthetic biology network that can achieve its desired steady state behavior under parameter fluctuations, external disturbances and functional variations in the host cell. Therefore, the proposed systematic design method has good potential for the robust synthetic gene network design.
doi:10.1186/1752-0509-3-66
PMCID: PMC2732592  PMID: 19566953
14.  A Unifying Mathematical Framework for Genetic Robustness, Environmental Robustness, Network Robustness and their Trade-off on Phenotype Robustness in Biological Networks Part I: Gene Regulatory Networks in Systems and Evolutionary Biology 
Robust stabilization and environmental disturbance attenuation are ubiquitous systematic properties observed in biological systems at 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 enough to confer intrinsic robustness in order to tolerate intrinsic parameter fluctuations, genetic robustness for buffering genetic variations, and environmental robustness for resisting environmental disturbances. With this, the phenotypic stability of biological network can be maintained, thus guaranteeing phenotype robustness. This paper presents a survey on biological systems and then develops a unifying mathematical framework for investigating the principles of both robust stabilization and environmental disturbance attenuation in systems and evolutionary biology. Further, from the unifying mathematical framework, it was discovered that the phenotype robustness criterion for biological networks at different levels relies upon intrinsic robustness + genetic robustness + environmental robustness ≦ network robustness. When this is true, 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 systems and evolutionary biology can also be investigated through their corresponding phenotype robustness criterion from the systematic point of view.
doi:10.4137/EBO.S10080
PMCID: PMC3596976  PMID: 23515240
phenotype robustness; network robustness; network sensitivity; evolvability; systems biology; evolutionary biology
15.  Dynamics of Drosophila embryonic patterning network perturbed in space and time using microfluidics 
Nature  2005;434(7037):1134-1138.
Biochemical networks are perturbed both by fluctuations in environmental conditions and genetic variation. These perturbations must be compensated for, especially when they occur during embryonic pattern formation. Complex chemical reaction networks displaying spatiotemporal dynamics have been controlled and understood by perturbing their environment in space and time1–3. Here, we apply this approach using microfluidics to investigate the robust network in Drosophila melanogaster that compensates for variation in the Bicoid morphogen gradient. We show that the compensation system can counteract the effects of extremely unnatural environmental conditions—a temperature step—in which the anterior and posterior halves of the embryo are developing at different temperatures and thus at different rates. Embryonic patterning was normal under this condition, suggesting that a simple reciprocal gradient system is not the mechanism of compensation. Time-specific reversals of the temperature step narrowed down the critical period for compensation to between 65 and 100 min after onset of embryonic development. The microfluidic technology used here may prove useful to future studies, as it allows spatial and temporal regulation of embryonic development.
doi:10.1038/nature03509
PMCID: PMC2656922  PMID: 15858575
16.  Robustness: mechanisms and consequences 
Trends in genetics : TIG  2009;25(9):395-403.
Biological systems are robust to perturbation by mutations and environmental fluctuations. New data are shedding light on the biochemical and network-level mechanisms responsible for robustness. Robustness to mutation might have evolved as an adaptation to reduce the effect of mutations, as a congruent byproduct of adaptive robustness to environmental variation, or as an intrinsic property of biological systems selected for their primary functions. Whatever its mechanism or origin, robustness to mutation results in the accumulation of phenotypically cryptic genetic variation. Partial robustness can lead to pre-adaptation, and thereby might contribute to evolvability. The identification and characterization of phenotypic capacitors — which act as switches of the degree of robustness — are critical to understanding the mechanisms and consequences of robustness.
doi:10.1016/j.tig.2009.07.005
PMCID: PMC2770586  PMID: 19717203
17.  Estimating the Stochastic Bifurcation Structure of Cellular Networks 
PLoS Computational Biology  2010;6(3):e1000699.
High throughput measurement of gene expression at single-cell resolution, combined with systematic perturbation of environmental or cellular variables, provides information that can be used to generate novel insight into the properties of gene regulatory networks by linking cellular responses to external parameters. In dynamical systems theory, this information is the subject of bifurcation analysis, which establishes how system-level behaviour changes as a function of parameter values within a given deterministic mathematical model. Since cellular networks are inherently noisy, we generalize the traditional bifurcation diagram of deterministic systems theory to stochastic dynamical systems. We demonstrate how statistical methods for density estimation, in particular, mixture density and conditional mixture density estimators, can be employed to establish empirical bifurcation diagrams describing the bistable genetic switch network controlling galactose utilization in yeast Saccharomyces cerevisiae. These approaches allow us to make novel qualitative and quantitative observations about the switching behavior of the galactose network, and provide a framework that might be useful to extract information needed for the development of quantitative network models.
Author Summary
Decades ago, Waddington, and later Kauffman, likened the dynamics of a differentiating cell to a marble rolling downhill on bumpy terrain—the epigenetic landscape. In this metaphor, the valleys of the landscape represent the paths that cells can follow towards a stable cell type, and the fate of the cell is determined by the constant modulation of the epigenetic landscape by internal and external signals. With new technologies for measuring single-cell gene expression, it is increasingly feasible to map out these valleys and how external variables influence cellular responses. Moreover, it is possible to quantify population level effects, such as what fraction of a population of cells arrives at one valley or another, and variability at the cellular level, such as how individual cells bounce around within, and possibly between, valleys due to the stochasticity of cellular biochemistry. In this paper, we discuss which characteristics of the epigenetic landscape can readily be extracted from single-cell gene expression data, and describe computational methods for doing so.
doi:10.1371/journal.pcbi.1000699
PMCID: PMC2832680  PMID: 20221261
18.  Attraction Basins as Gauges of Robustness against Boundary Conditions in Biological Complex Systems 
PLoS ONE  2010;5(8):e11793.
One fundamental concept in the context of biological systems on which researches have flourished in the past decade is that of the apparent robustness of these systems, i.e., their ability to resist to perturbations or constraints induced by external or boundary elements such as electromagnetic fields acting on neural networks, micro-RNAs acting on genetic networks and even hormone flows acting both on neural and genetic networks. Recent studies have shown the importance of addressing the question of the environmental robustness of biological networks such as neural and genetic networks. In some cases, external regulatory elements can be given a relevant formal representation by assimilating them to or modeling them by boundary conditions. This article presents a generic mathematical approach to understand the influence of boundary elements on the dynamics of regulation networks, considering their attraction basins as gauges of their robustness. The application of this method on a real genetic regulation network will point out a mathematical explanation of a biological phenomenon which has only been observed experimentally until now, namely the necessity of the presence of gibberellin for the flower of the plant Arabidopsis thaliana to develop normally.
doi:10.1371/journal.pone.0011793
PMCID: PMC2916819  PMID: 20700525
19.  Extended Kalman Filter for Estimation of Parameters in Nonlinear State-Space Models of Biochemical Networks 
PLoS ONE  2008;3(11):e3758.
It is system dynamics that determines the function of cells, tissues and organisms. To develop mathematical models and estimate their parameters are an essential issue for studying dynamic behaviors of biological systems which include metabolic networks, genetic regulatory networks and signal transduction pathways, under perturbation of external stimuli. In general, biological dynamic systems are partially observed. Therefore, a natural way to model dynamic biological systems is to employ nonlinear state-space equations. Although statistical methods for parameter estimation of linear models in biological dynamic systems have been developed intensively in the recent years, the estimation of both states and parameters of nonlinear dynamic systems remains a challenging task. In this report, we apply extended Kalman Filter (EKF) to the estimation of both states and parameters of nonlinear state-space models. To evaluate the performance of the EKF for parameter estimation, we apply the EKF to a simulation dataset and two real datasets: JAK-STAT signal transduction pathway and Ras/Raf/MEK/ERK signaling transduction pathways datasets. The preliminary results show that EKF can accurately estimate the parameters and predict states in nonlinear state-space equations for modeling dynamic biochemical networks.
doi:10.1371/journal.pone.0003758
PMCID: PMC2582954  PMID: 19018286
20.  Archaeal Genome Guardians Give Insights into Eukaryotic DNA Replication and Damage Response Proteins 
Archaea  2014;2014:206735.
As the third domain of life, archaea, like the eukarya and bacteria, must have robust DNA replication and repair complexes to ensure genome fidelity. Archaea moreover display a breadth of unique habitats and characteristics, and structural biologists increasingly appreciate these features. As archaea include extremophiles that can withstand diverse environmental stresses, they provide fundamental systems for understanding enzymes and pathways critical to genome integrity and stress responses. Such archaeal extremophiles provide critical data on the periodic table for life as well as on the biochemical, geochemical, and physical limitations to adaptive strategies allowing organisms to thrive under environmental stress relevant to determining the boundaries for life as we know it. Specifically, archaeal enzyme structures have informed the architecture and mechanisms of key DNA repair proteins and complexes. With added abilities to temperature-trap flexible complexes and reveal core domains of transient and dynamic complexes, these structures provide insights into mechanisms of maintaining genome integrity despite extreme environmental stress. The DNA damage response protein structures noted in this review therefore inform the basis for genome integrity in the face of environmental stress, with implications for all domains of life as well as for biomanufacturing, astrobiology, and medicine.
doi:10.1155/2014/206735
PMCID: PMC3950489  PMID: 24701133
21.  A Unifying Mathematical Framework for Genetic Robustness, Environmental Robustness, Network Robustness and their Tradeoff on Phenotype Robustness in Biological Networks Part II: Ecological Networks 
In ecological networks, network robustness should be large enough to confer intrinsic robustness for tolerating intrinsic parameter fluctuations, as well as environmental robustness for resisting environmental disturbances, so that the phenotype stability of ecological networks can be maintained, thus guaranteeing phenotype robustness. However, it is difficult to analyze the network robustness of ecological systems because they are complex nonlinear partial differential stochastic systems. This paper develops a unifying mathematical framework for investigating the principles of both robust stabilization and environmental disturbance sensitivity in ecological networks. We found that the phenotype robustness criterion for ecological networks is that if intrinsic robustness + environmental robustness ≦ network robustness, then the phenotype robustness can be maintained in spite of intrinsic parameter fluctuations and environmental disturbances. These results in robust ecological networks are similar to that in robust gene regulatory networks and evolutionary networks even they have different spatial-time scales.
doi:10.4137/EBO.S10685
PMCID: PMC3596974  PMID: 23515112
phenotype robustness; network robustness; network sensitivity; ecological networks; spatial-temporal domain; PDE
22.  Inferring cell cycle feedback regulation from gene expression data 
Journal of biomedical informatics  2011;44(4):565-575.
Feedback control is an important regulatory process in biological systems, which confers robustness against external and internal disturbances. Genes involved in feedback structures are therefore likely to have a major role in regulating cellular processes.
Here we rely on a dynamic Bayesian network approach to identify feedback loops in cell cycle regulation. We analyzed the transcriptional profile of the cell cycle in HeLa cancer cells and identified a feedback loop structure composed of 10 genes. In silico analyses showed that these genes hold important roles in system's dynamics. The results of published experimental assays confirmed the central role of 8 of the identified feedback loop genes in cell cycle regulation. In conclusion, we provide a novel approach to identify critical genes for the dynamics of biological processes. This may lead to the identification of therapeutic targets in diseases that involve perturbations of these dynamics.
doi:10.1016/j.jbi.2011.02.002
PMCID: PMC3143236  PMID: 21310265
gene expression; cell cycle; dynamic Bayesian network; feedback
23.  Survival of the sparsest: robust gene networks are parsimonious 
Biological gene networks appear to be dynamically robust to mutation, stochasticity, and changes in the environment and also appear to be sparsely connected. Studies with computational models, however, have suggested that denser gene networks evolve to be more dynamically robust than sparser networks. We resolve this discrepancy by showing that misassumptions about how to measure robustness in artificial networks have inadvertently discounted the costs of network complexity. We show that when the costs of complexity are taken into account, that robustness implies a parsimonious network structure that is sparsely connected and not unnecessarily complex; and that selection will favor sparse networks when network topology is free to evolve. Because a robust system of heredity is necessary for the adaptive evolution of complex phenotypes, the maintenance of frugal network complexity is likely a crucial design constraint that underlies biological organization.
doi:10.1038/msb.2008.52
PMCID: PMC2538912  PMID: 18682703
complexity; evolvability; gene network; robustness
24.  Evolution of Robustness to Noise and Mutation in Gene Expression Dynamics 
PLoS ONE  2007;2(5):e434.
Phenotype of biological systems needs to be robust against mutation in order to sustain themselves between generations. On the other hand, phenotype of an individual also needs to be robust against fluctuations of both internal and external origins that are encountered during growth and development. Is there a relationship between these two types of robustness, one during a single generation and the other during evolution? Could stochasticity in gene expression have any relevance to the evolution of these types of robustness? Robustness can be defined by the sharpness of the distribution of phenotype; the variance of phenotype distribution due to genetic variation gives a measure of ‘genetic robustness’, while that of isogenic individuals gives a measure of ‘developmental robustness’. Through simulations of a simple stochastic gene expression network that undergoes mutation and selection, we show that in order for the network to acquire both types of robustness, the phenotypic variance induced by mutations must be smaller than that observed in an isogenic population. As the latter originates from noise in gene expression, this signifies that the genetic robustness evolves only when the noise strength in gene expression is larger than some threshold. In such a case, the two variances decrease throughout the evolutionary time course, indicating increase in robustness. The results reveal how noise that cells encounter during growth and development shapes networks' robustness to stochasticity in gene expression, which in turn shapes networks' robustness to mutation. The necessary condition for evolution of robustness, as well as the relationship between genetic and developmental robustness, is derived quantitatively through the variance of phenotypic fluctuations, which are directly measurable experimentally.
doi:10.1371/journal.pone.0000434
PMCID: PMC1855988  PMID: 17502916
25.  Metabolic gene regulation in a dynamically changing environment 
Nature  2008;454(7208):1119-1122.
Natural selection dictates that cells constantly adapt to dynamically changing environments in a context-dependent manner. Gene-regulatory networks often mediate the cellular response to perturbation 1-3, and an understanding of cellular adaptation will require experimental approaches aimed at subjecting cells to a dynamic environment that mimics their natural habitat 4-9. Here, we monitor the response of S. cerevisiae metabolic gene regulation to periodic changes in the external carbon source by utilizing a microfluidic platform that allows precise, dynamic control over environmental conditions. We find that the metabolic system acts as a low-pass filter that reliably responds to a slowly changing environment, while effectively ignoring fluctuations that are too fast for the cell to mount an efficient response. We use computational modeling calibrated with experimental data to determine how frequency selection in the system is controlled by the interaction of coupled regulatory networks governing the signal transduction of alternative carbon sources. Experimental verification of model predictions leads to the discovery of two novel properties of the regulatory network. First, we reveal a previously unknown mechanism for post-transcriptional control, by demonstrating that two key transcripts are degraded at a rate that depends on the carbon source. Second, we compare two S. cerevisiae strains and find that they exhibit the same frequency response despite having markedly different induction characteristics. Our results suggest that while certain characteristics of the complex networks may differ when probed in a static environment, the system has been optimized for a robust response to a dynamically changing environment. Importantly, the integration of a novel experimental platform with numerical simulations revealed previously masked network properties, and the approach establishes a framework for dynamically probing organisms in order to reveal mechanisms that have evolved to mediate cellular responses to unpredictable environments.
doi:10.1038/nature07211
PMCID: PMC2654342  PMID: 18668041

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