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1.  Robustness Can Evolve Gradually in Complex Regulatory Gene Networks with Varying Topology 
PLoS Computational Biology  2007;3(2):e15.
The topology of cellular circuits (the who-interacts-with-whom) is key to understand their robustness to both mutations and noise. The reason is that many biochemical parameters driving circuit behavior vary extensively and are thus not fine-tuned. Existing work in this area asks to what extent the function of any one given circuit is robust. But is high robustness truly remarkable, or would it be expected for many circuits of similar topology? And how can high robustness come about through gradual Darwinian evolution that changes circuit topology gradually, one interaction at a time? We here ask these questions for a model of transcriptional regulation networks, in which we explore millions of different network topologies. Robustness to mutations and noise are correlated in these networks. They show a skewed distribution, with a very small number of networks being vastly more robust than the rest. All networks that attain a given gene expression state can be organized into a graph whose nodes are networks that differ in their topology. Remarkably, this graph is connected and can be easily traversed by gradual changes of network topologies. Thus, robustness is an evolvable property. This connectedness and evolvability of robust networks may be a general organizational principle of biological networks. In addition, it exists also for RNA and protein structures, and may thus be a general organizational principle of all biological systems.
Author Summary
Living things are astonishingly complex, yet unlike houses of cards they are also highly robust. That is, they have persisted for billions of years, despite being exposed to an endless stream of environmental stressors and random mutations. Is this robustness an evolvable property? Do different biological systems vary in their robustness? Has natural selection shaped this robustness? These questions are very difficult to answer experimentally for most systems, be they proteins or large gene networks. Here we address these questions with a model of the transcription regulation networks that regulate both cellular functions and embryonic development in many organisms. We examine millions of such networks that differ in the topology or architecture of their regulatory interactions, that is, in the “who interacts with whom” of a network. We find that radically different network architectures can show the same gene expression pattern. The networks' robustness to both mutations and gene expression noise shows a broad distribution: some network architectures are highly robust, whereas others are quite fragile. Importantly, the entire space of network architectures can be traversed through small changes of individual regulatory interactions, without changing a network's gene expression pattern. This means that high robustness in gene expression can evolve through gradual and neutral evolution in the space of network architectures. Our results show that the robustness of transcriptional regulation networks is an evolvable trait that natural selection can change like any other trait.
PMCID: PMC1794322  PMID: 17274682
2.  Synthesizing a novel genetic sequential logic circuit: a push-on push-off switch 
We designed and constructed a genetic sequential logic circuit that can function as a push-on push-off switch. The circuit consists of a bistable switch module and a NOR gate module.The bistable switch module and NOR gate module were rationally designed and constructed.The two above modules were coupled by two interconnecting parts, cIind- and lacI. When optimizing the defined function, we fine-tuned the expression of the two interconnecting parts by directed evolution.Three control circuits were constructed to show the interconnecting parts are essential for achieving the defined function.
Design and synthesis of basic functional circuits are the fundamental tasks of synthetic biologists. Before it is possible to engineer higher-order genetic networks that can perform complex functions, a toolkit of basic devices must be developed. Among those devices, sequential logic circuits are expected to be the foundation of genetic information-processing systems.
As in electronics, combinational and sequential logic circuits are two kinds of fundamental processors in cells. In a combinational logic circuit, the output depends only on the present inputs, whereas in a sequential logic circuit, the output also depends on the history of the input due to its own memory. If we can successfully construct the two kinds of basic logic circuits in a cell, they can serve as building blocks to be assembled into high-order genetic circuits and implement more sophisticated computation.
Construction of genetic combinational logic circuits (GSLCs), such as AND, OR, and NOR gates, has been frequently reported in the last decade (Guet et al, 2002; Dueber et al, 2003; Anderson et al, 2007; Win and Smolke, 2008). Meanwhile toggle switches, which can function as memory modules, have been implemented in prokaryotic and eukaryotic cells (Becskei et al, 2001; Kramer et al, 2004; Ajo-Franklin et al, 2007).
Here, we constructed a novel GSLC that functions as a push-on push-off switch by coupling a combinational logic module with a bistable switch module (Figure 1A). When the internal state of the memory is in the ‘ON' state, the external UV input makes the circuit's output promoter PNOR generate an ‘OFF' pulse signal and register the ‘OFF' state into the memory; when the internal state is in the ‘OFF' state, the same external UV input induces the circuit's output promoter PNOR to generate an ‘ON' pulse signal and register the ‘ON' state into the memory.
In our design, the combinatorial logic gate is a NOR gate and the switch module is a clearable bistable switch (Figure 1C). Two interconnecting parts are designed to connect the NOR gate and the bistable switch (Figure 1D). UV irradiation was used as both an external input signal and a reset signal for the clearable bistable switch (Figure 1B).
Before implementing the experimental construction, we used a set of ordinary differential equations to simulate the dynamic process. With a set of reasonable parameters, the simulation results showed that the circuit could function as a push-on push-off switch (Figure 1E). Then the bistable switch module and NOR gate module were rationally designed and constructed. Our experimental results showed that the corresponding functions were implemented very well.
After the construction of the memory and the NOR gate module, we coupled the two modules together by fine-tuning the expression of two interconnecting parts lacI and cIind−. The two libraries for the ribosome-binding sites (RBSs) of lacI and cIind− were simultaneously transformed into Escherichia coli cells harboring the memory module plasmid. After growth on agar plates with appropriate antibiotics, colonies containing all three plasmids were selected.
With efficient mutation libraries, we developed a new screening method to select the functional circuits. The experimental process is described in Figure 4A. It consists of two rounds of selection. In the first round of selection, approximatelybout 300 mutants out of 1000 were chosen. In the second round, only three mutants were selected. As shown in Figure 4B, if the initial state was ‘OFF' with green color, the fraction of green cells in the population was near 100% before UV stimulus, whereas less than 10% of cells remained in the green ‘OFF' state after UV stimulus (Figure 4B). This result indicates that the switch from ‘OFF' to ‘ON' is quite complete. Unfortunately, the switch from ‘ON' to ‘OFF' was not as efficient: only about one-third of the population switched to the ‘OFF' state after UV triggering (Figure 4C). Nonetheless, the switch is still significant compared with that of the population not exposed to UV irradiation (Figure 4B and C). These results show that the fine-tuned GSLC can generate different output signals under the same input on the basis of the internal state of its memory, and register the output signal into its memory as the new internal state.
To show that decoupled circuits cannot achieve the sequential logic function, we also constructed three control circuits. The bistable switch module and the NOR gate module were decoupled by removing either or both of the interconnecting parts. In the first control circuit, LacI was removed; without LacI, LexA becomes the only effective input for the NOR gate. As a consequence, upon UV stimulus, promoter PNOR always generates a high output signal, and the ‘ON' state (high CI and low CI434) is latched in the memory with the help of CIind−. Correspondingly, the color of the cells will change to red. In the other two control circuits, CIind− or both LacI and CIind− were removed. Owing to the lack of the feedback part CIind−, when the output of the promoter PNOR is ‘ON', no output signal can be registered into the memory. In this case, the memory module will spontaneously enter into the low CI/high CI434 state after UV stimulus. All experimental results are consistent with the above expectation.
Finally, to show the property of the push-on push-off switch of the circuit, we sequentially stimulated a homogeneous population of cells with the same dose of UV signal multiple times. The first UV stimulus caused the fraction of green cells in the population to decrease from 99.3% to 8.4%, so that more than 90% of the population switched from the ‘OFF' to the ‘ON' state. The second UV stimulus resulted in the fraction of green cells increasing from 8.4% to 34.5%. Therefore, only 26.1% of the population switched back to the ‘OFF' state. These results are comparable to the results of switching efficiency measurement shown in Figure 4B and C. With repeated exposure to UV irradiation, the population increasingly appeared like a mixture of the two states, the ratio of which gradually reached a steady state. The push-on–push-off function of the circuit was thus lost at the population level.
In summary, we successfully assembled a bistable switch module and a combinatorial NOR gate module into a functional sequential logic circuit. We combined rational design with directed evolution to generate the desired system behavior. In this work, we showed that simultaneous mutation of multiple RBS targets, followed by directed evolution, is a powerful tool to search the in vivo parameter space to generate functional circuits from multiple rationally designed synthetic device modules. We anticipate that this approach will lend itself well to the next step in synthetic biology, combining multiple circuits, each composed of several device modules, to create useful synthetic systems that perform sophisticated computation.
Design and synthesis of basic functional circuits are the fundamental tasks of synthetic biologists. Before it is possible to engineer higher-order genetic networks that can perform complex functions, a toolkit of basic devices must be developed. Among those devices, sequential logic circuits are expected to be the foundation of the genetic information-processing systems. In this study, we report the design and construction of a genetic sequential logic circuit in Escherichia coli. It can generate different outputs in response to the same input signal on the basis of its internal state, and ‘memorize' the output. The circuit is composed of two parts: (1) a bistable switch memory module and (2) a double-repressed promoter NOR gate module. The two modules were individually rationally designed, and they were coupled together by fine-tuning the interconnecting parts through directed evolution. After fine-tuning, the circuit could be repeatedly, alternatively triggered by the same input signal; it functions as a push-on push-off switch.
PMCID: PMC2858441  PMID: 20212522
bistable switch; coupling modules; genetic sequential logic circuit; NOR gate; push-on push-off switch
3.  Dynamic interaction networks in a hierarchically organized tissue 
We have integrated gene expression profiling with database and literature mining, mechanistic modeling, and cell culture experiments to identify intercellular and intracellular networks regulating blood stem cell self-renewal.Blood stem cell fate in vitro is regulated non-autonomously by a coupled positive–negative intercellular feedback circuit, composed of megakaryocyte-derived stimulatory growth factors (VEGF, PDGF, EGF, and serotonin) versus monocyte-derived inhibitory factors (CCL3, CCL4, CXCL10, TGFB2, and TNFSF9).The antagonistic signals converge in a core intracellular network focused around PI3K, Raf, PLC, and Akt.Model simulations enable functional classification of the novel endogenous ligands and signaling molecules.
Intercellular (between cell) communication networks are required to maintain homeostasis and coordinate regenerative and developmental cues in multicellular organisms. Despite the recognized importance of intercellular networks in regulating adult stem and progenitor cell fate, the specific cell populations involved, and the underlying molecular mechanisms are largely undefined. Although a limited number of studies have applied novel bioinformatic approaches to unravel intercellular signaling in other cell systems (Frankenstein et al, 2006), a comprehensive analysis of intercellular communication in a stem cell-derived, hierarchical tissue network has yet to be reported.
As a model system to explore intercellular communication networks in a hierarchically organized tissue, we cultured human umbilical cord blood (UCB)-derived stem and progenitor cells in defined, minimal cytokine-supplemented liquid culture (Madlambayan et al, 2006). To systematically explore the molecular and cellular dynamics underlying primitive progenitor growth and differentiation, gene expression profiles of primitive (lineage negative; Lin−) and mature (lineage positive; Lin+) populations were generated during phases of stem cell expansion versus depletion. Parallel phenotypic and subproteomic experiments validated that mRNA expression correlated with complex measures of proteome activity (protein secretion and cell surface expression). Using a curated list of secreted ligand–receptor interactions and published expression profiles of purified mature blood populations, we implemented a novel algorithm to reconstruct the intercellular signaling networks established between stem cells and multi-lineage progeny in vitro. By correlating differential expression patterns with stem cell growth, we predict cell populations, pathways, and secreted ligands associated with stem cell self-renewal and differentiation (Figure 3A).
We then tested the correlative predictions in a series of cell culture experiments. UCB progenitor cell cultures were supplemented with saturating amounts of 18 putative regulatory ligands, or cocultured with purified mature blood lineages (megakaryocytes, monocytes, and erythrocytes), and analyzed for effects on total cell, progenitor, and primitive progenitor growth. At the primitive progenitor level, 3/5 novel predicted stimulatory ligands (EGF, PDGFB, and VEGF) displayed significant positive effects, 5/7 predicted inhibitory factors (CCL3, CCL4, CXCL10, TNFSF9, and TGFB2) displayed negative effects, whereas only 1/5 non-correlated ligand (CXCL7) displayed an effect. Also consistent with predictions from gene expression data, megakaryocytes and monocytes were found to stimulate and inhibit primitive progenitor growth, respectively, and these effects were attributable to differential secretome profiles of stimulatory versus inhibitory ligands.
Cellular responses to external stimuli, particularly in heterogeneous and dynamic cell populations, represent complex functions of multiple cell fate decisions acting both directly and indirectly on the target (stem cell) populations. Experimentally distinguishing the mode of action of cytokines is thus a difficult task. To address this we used our previously published interactive model of hematopoiesis (Kirouac et al, 2009) to classify experimentally identified regulatory ligands into one of four distinct functional categories based on their differential effects on cell population growth. TGFB2 was classified as a proliferation inhibitor, CCL4, CXCL10, SPARC, and TNFSF9 as self-renewal inhibitors, CCL3 a proliferation stimulator, and EGF, VEGF, and PDGFB as self-renewal stimulators.
Stem and progenitor cells exposed to combinatorial extracellular signals must propagate this information through intracellular molecular networks, and respond appropriately by modifying cell fate decisions. To explore how our experimentally identified positive and negative regulatory signals are integrated at the intracellular level, we constructed a blood stem cell self-renewal signaling network through extensive literature curation and protein–protein interaction (PPI) network mapping. We find that signal transduction pathways activated by the various stimulatory and inhibitory ligands converge on a limited set of molecular control nodes, forming a core subnetwork enriched for known regulators of self-renewal (Figure 6A). To experimentally test the intracellular signaling molecules computationally predicted as regulators of stem cell self-renewal, we obtained five small molecule antagonists against the kinases Phosphatidylinositol 3-kinase (PI3K), Raf, Akt, Phospholipase C (PLC), and MEK1. Liquid cultures were supplemented with the five molecules individually, and resultant cell population outputs compared against model simulations to deconvolute the functional effects on proliferation (and survival) versus self-renewal. This analysis classifies inhibition of PI3K and Raf activity as selectively targeting self-renewal, PLC as selectively targeting survival, and Akt as selectively targeting proliferation; MEK inhibition appears non-specific for these processes.
This represents the first systematic characterization of how cell fate decisions are regulated non-autonomously through lineage-specific interactions with differentiated progeny. The complex intercellular communication networks can be approximated as an antagonistic positive–negative feedback circuit, wherein progenitor expansion is modulated by a balance of megakaryocyte-derived stimulatory factors (EGF, PDGF, VEGF, and possibly serotonin) versus monocyte-derived inhibitory factors (CCL3, CCL4, CXCL10, TGFB2, and TNFSF9). This complex milieu of endogenous regulatory signals is integrated and processed within a core intracellular signaling network, resulting in modulation of cell-level kinetic parameters (proliferation, survival, and self-renewal). We reconstruct a stem cell associated intracellular network, and identify PI3K, Raf, Akt, and PLC as functionally distinct signal integration nodes, linking extracellular and intracellular signaling. These findings lay the groundwork for novel strategies to control blood stem cell self-renewal in vitro and in vivo.
Intercellular (between cell) communication networks maintain homeostasis and coordinate regenerative and developmental cues in multicellular organisms. Despite the importance of intercellular networks in stem cell biology, their rules, structure and molecular components are poorly understood. Herein, we describe the structure and dynamics of intercellular and intracellular networks in a stem cell derived, hierarchically organized tissue using experimental and theoretical analyses of cultured human umbilical cord blood progenitors. By integrating high-throughput molecular profiling, database and literature mining, mechanistic modeling, and cell culture experiments, we show that secreted factor-mediated intercellular communication networks regulate blood stem cell fate decisions. In particular, self-renewal is modulated by a coupled positive–negative intercellular feedback circuit composed of megakaryocyte-derived stimulatory growth factors (VEGF, PDGF, EGF, and serotonin) versus monocyte-derived inhibitory factors (CCL3, CCL4, CXCL10, TGFB2, and TNFSF9). We reconstruct a stem cell intracellular network, and identify PI3K, Raf, Akt, and PLC as functionally distinct signal integration nodes, linking extracellular, and intracellular signaling. This represents the first systematic characterization of how stem cell fate decisions are regulated non-autonomously through lineage-specific interactions with differentiated progeny.
PMCID: PMC2990637  PMID: 20924352
cellular networks; hematopoiesis; intercellular signaling; self-renewal; stem cells
4.  Noise Attenuation in the ON and OFF States of Biological Switches 
ACS Synthetic Biology  2013;2(10):587-593.
Biological switches must sense changes in signal concentration and at the same time buffer against signal noise. While many studies have focused on the response of switching systems to noise in the ON state, how systems buffer noise at both ON and OFF states is poorly understood. Through analytical and computational approaches, we find that switching systems require different dynamics at the OFF state than at the ON state in order to have good noise buffering capability. Specifically, we introduce a quantity called the input-associated Signed Activation Time (iSAT) that concisely captures an intrinsic temporal property at either the ON or OFF state. We discover a trade-off between achieving good noise buffering in the ON versus the OFF states: a large iSAT corresponds to noise amplification in the OFF state in contrast to noise buffering in the ON state. To search for biological circuits that can buffer noise in both ON and OFF states, we systematically analyze all three-node circuits and identify mutual activation as a central motif. We also study connections among signal sensitivity, iSAT, and noise amplification. We find that a large iSAT at the ON state maintains signaling sensitivity while minimizing noise propagation. Taken together, the analysis of iSATs helps reveal the noise properties of biological networks and should aid in the design of robust switches that can both repress noise at the OFF state and maintain a reliable ON state.
PMCID: PMC3805451  PMID: 23768065
noise; signaling switch; SAT; feedbacks; sensitivity; network motif
5.  The Fidelity of Dynamic Signaling by Noisy Biomolecular Networks 
PLoS Computational Biology  2013;9(3):e1002965.
Cells live in changing, dynamic environments. To understand cellular decision-making, we must therefore understand how fluctuating inputs are processed by noisy biomolecular networks. Here we present a general methodology for analyzing the fidelity with which different statistics of a fluctuating input are represented, or encoded, in the output of a signaling system over time. We identify two orthogonal sources of error that corrupt perfect representation of the signal: dynamical error, which occurs when the network responds on average to other features of the input trajectory as well as to the signal of interest, and mechanistic error, which occurs because biochemical reactions comprising the signaling mechanism are stochastic. Trade-offs between these two errors can determine the system's fidelity. By developing mathematical approaches to derive dynamics conditional on input trajectories we can show, for example, that increased biochemical noise (mechanistic error) can improve fidelity and that both negative and positive feedback degrade fidelity, for standard models of genetic autoregulation. For a group of cells, the fidelity of the collective output exceeds that of an individual cell and negative feedback then typically becomes beneficial. We can also predict the dynamic signal for which a given system has highest fidelity and, conversely, how to modify the network design to maximize fidelity for a given dynamic signal. Our approach is general, has applications to both systems and synthetic biology, and will help underpin studies of cellular behavior in natural, dynamic environments.
Author Summary
Cells do not live in constant conditions, but in environments that change over time. To adapt to their surroundings, cells must therefore sense fluctuating concentrations and ‘interpret’ the state of their environment to see whether, for example, a change in the pattern of gene expression is needed. This task is achieved via the noisy computations of biomolecular networks. But what levels of signaling fidelity can be achieved and how are dynamic signals encoded in the network's outputs? Here we present a general technique for analyzing such questions. We identify two sources of signaling error: dynamic error, which occurs when the network responds to features of the input other than the signal of interest; and mechanistic error, which arises because of the inevitable stochasticity of biochemical reactions. We show analytically that increased biochemical noise can sometimes improve fidelity and that, for genetic autoregulation, feedback can be deleterious. Our approach also allows us to predict the dynamic signal for which a given signaling network has highest fidelity and to design networks to maximize fidelity for a given signal. We thus propose a new way to analyze the flow of information in signaling networks, particularly for the dynamic environments expected in nature.
PMCID: PMC3610653  PMID: 23555208
6.  An integrated machine learning approach for predicting DosR-regulated genes in Mycobacterium tuberculosis 
BMC Systems Biology  2010;4:37.
DosR is an important regulator of the response to stress such as limited oxygen availability in Mycobacterium tuberculosis. Time course gene expression data enable us to dissect this response on the gene regulatory level. The mRNA expression profile of a regulator, however, is not necessarily a direct reflection of its activity. Knowing the transcription factor activity (TFA) can be exploited to predict novel target genes regulated by the same transcription factor. Various approaches have been proposed to reconstruct TFAs from gene expression data. Most of them capture only a first-order approximation to the complex transcriptional processes by assuming linear gene responses and linear dynamics in TFA, or ignore the temporal information in data from such systems.
In this paper, we approach the problem of inferring dynamic hidden TFAs using Gaussian processes (GP). We are able to model dynamic TFAs and to account for both linear and nonlinear gene responses. To test the validity of the proposed approach, we reconstruct the hidden TFA of p53, a tumour suppressor activated by DNA damage, using published time course gene expression data. Our reconstructed TFA is closer to the experimentally determined profile of p53 concentration than that from the original study. We then apply the model to time course gene expression data obtained from chemostat cultures of M. tuberculosis under reduced oxygen availability. After estimation of the TFA of DosR based on a number of known target genes using the GP model, we predict novel DosR-regulated genes: the parameters of the model are interpreted as relevance parameters indicating an existing functional relationship between TFA and gene expression. We further improve the prediction by integrating promoter sequence information in a logistic regression model. Apart from the documented DosR-regulated genes, our prediction yields ten novel genes under direct control of DosR.
Chemostat cultures are an ideal experimental system for controlling noise and variability when monitoring the response of bacterial organisms such as M. tuberculosis to finely controlled changes in culture conditions and available metabolites. Nonlinear hidden TFA dynamics of regulators can be reconstructed remarkably well with Gaussian processes from such data. Moreover, estimated parameters of the GP can be used to assess whether a gene is controlled by the reconstructed TFA or not. It is straightforward to combine these parameters with further information, such as the presence of binding motifs, to increase prediction accuracy.
PMCID: PMC2867773  PMID: 20356371
7.  Combined Model of Intrinsic and Extrinsic Variability for Computational Network Design with Application to Synthetic Biology 
PLoS Computational Biology  2013;9(3):e1002960.
Biological systems are inherently variable, with their dynamics influenced by intrinsic and extrinsic sources. These systems are often only partially characterized, with large uncertainties about specific sources of extrinsic variability and biochemical properties. Moreover, it is not yet well understood how different sources of variability combine and affect biological systems in concert. To successfully design biomedical therapies or synthetic circuits with robust performance, it is crucial to account for uncertainty and effects of variability. Here we introduce an efficient modeling and simulation framework to study systems that are simultaneously subject to multiple sources of variability, and apply it to make design decisions on small genetic networks that play a role of basic design elements of synthetic circuits. Specifically, the framework was used to explore the effect of transcriptional and post-transcriptional autoregulation on fluctuations in protein expression in simple genetic networks. We found that autoregulation could either suppress or increase the output variability, depending on specific noise sources and network parameters. We showed that transcriptional autoregulation was more successful than post-transcriptional in suppressing variability across a wide range of intrinsic and extrinsic magnitudes and sources. We derived the following design principles to guide the design of circuits that best suppress variability: (i) high protein cooperativity and low miRNA cooperativity, (ii) imperfect complementarity between miRNA and mRNA was preferred to perfect complementarity, and (iii) correlated expression of mRNA and miRNA – for example, on the same transcript – was best for suppression of protein variability. Results further showed that correlations in kinetic parameters between cells affected the ability to suppress variability, and that variability in transient states did not necessarily follow the same principles as variability in the steady state. Our model and findings provide a general framework to guide design principles in synthetic biology.
Author Summary
Variability is inherent in biological systems, and in order to understand them, we need to be able to model different sources of variability. Systems have evolved to harness and control the variability, and more recently, synthetic biologists are trying to learn how to control variability in engineered biological systems. Several sources of variability exist; they arise due to stochastic expression of genes, which is most pronounced when numbers of mRNA and protein molecules are low, as well as due to differences between individual cells. Here we propose a modeling framework that combines different sources of biological variability. Furthermore, current research seeks to control biological variability though robust design of synthetic biological circuits, for example for use in therapies and other biomedical or biotechnological applications. Here we apply our framework to guide design of synthetic circuits that use transcriptional and post-transcriptional regulation to suppress variability in the output protein of interest. We find that certain properties and network designs are better than others in their ability to control variability, and here we report on the design guidelines to aid synthetic circuit design to suppress variability, in spite of our uncertain knowledge of parameters.
PMCID: PMC3610654  PMID: 23555205
8.  Optimal regulatory strategies for metabolic pathways in Escherichia coli depending on protein costs 
Pathways in Escherichia coli show large differences in the extent to which enzymes from the same pathway are expressed in a coordinated manner.Using dynamic optimization, we show that regulation of the initial and terminal reactions of a pathway is the minimum requirement for a precise control of flux.We find that in E. coli a regulation of initial and terminal reactions is predominantly used to control pathways with low costs of enzymes while a regulation of all enzymes occurs if protein costs are high.A trade-off between minimization of protein investment and minimization of response time can explain the preference for transcriptional regulation at key positions (leading to high protein costs, but low response time) or coordinated transcriptional regulation of all enzymes (leading to low protein costs, but high response time).
The increasing availability and decreasing prices of experimental techniques have led to an explosion in the number of available experimental data sets (Ishii et al, 2007; Lu et al, 2007; Bennett et al, 2009; Lewis et al, 2010). However, approaches to integrate these diverse data sets into a coherent model of cellular mechanisms have lagged behind (Palsson and Zengler, 2010). In this study, we want to contribute to this effort through the analysis of a large number of data sets in order to identify global principles in the regulation of metabolism in Escherichia coli. While previous studies have shed light onto the link between the transcriptional regulation of metabolism and its structure (Ihmels et al, 2004; Reed and Palsson, 2004; Schwartz et al, 2007; Seshasayee et al, 2009), the extent to which transcriptional regulation controls metabolism has remained elusive.
To address this problem, we investigated the coexpression of enzymes within the same pathway in all biochemically annotated subsystems of E. coli metabolism. As a reference for metabolic pathways, we used elementary flux patterns, a recently introduced concept for pathway analysis in genome-scale metabolic networks (Kaleta et al, 2009). Through this analysis, we found that while pathways in many subsystems of metabolism show a high degree of coexpression, pathways in the subsystems cofactor and prosthetic group biosynthesis, glycerophospholipid metabolism, murein recycling, nucleotide salvage pathway and pentose phosphate pathway show only weak coexpression. We refer to these subsystems with a low coordination of transcriptional regulation as transcriptionally sparsely regulated subsystems.
In order to understand these different patterns of regulation, we constructed a simplified model of a linear metabolic pathway that converts a substrate s via four intermediates into a product p. We then used dynamic optimization to identify a regulatory program (i.e. a time course for the enzyme concentrations), which allows the cell to maintain the concentration of the product p in a changing environment while obeying a set of physiological constraints. As an objective function we used the minimization of the level of transcriptional regulation, specified through absolute deviations of enzyme concentrations from their initial values, and the minimization of protein costs. Protein costs are measured as the sum of the initial enzyme concentrations.
The optimization results revealed that for a full control of the flux through a pathway, transcriptional regulation of initial and terminal reactions of the pathway is sufficient (sparse transcriptional regulation). Regulation of the first reaction is required to control the flux into the pathway, and hence, the intermediate concentrations. In contrast, regulation at the terminal position is required to tightly control the rate of synthesis of the product. By performing the same optimization for randomly chosen kinetic parameters, we found that this pattern is also optimal in most cases with differences in the catalytic properties of enzymes. Moreover, we found that with increasing enzyme costs (i.e. increasing enzyme concentrations), there is a shift from sparse transcriptional regulation to coordinated transcriptional regulation of all enzymes within a pathway (pervasive transcriptional regulation).
To verify these predictions, we analyzed the position-specific frequency of regulatory events in the pathways of the transcriptionally sparsely regulated subsystems. We could confirm that there is a significant increase in the frequency of transcriptional regulation at the end and a less pronounced increase at the beginning of pathways. Performing the same analysis for post-translational regulation, we found that there is a statistically significant increase at the beginning of pathways. Thus, the control at the beginning of pathways is achieved through a combination of transcriptional and post-translational regulation. In other subsystems that were not identified as transcriptionally sparsely regulated, we did not find this pattern of transcriptional regulation while the same pattern of post-translational regulation could be observed. By analyzing protein abundance data, we confirmed that particularly pathways within subsystems, for which enzyme costs are low, are transcriptionally sparsely regulated.
Having confirmed the predictions made by the optimization, we found that there appears to be a mechanism favoring sparse transcriptional regulation in pathways with low-cost enzymes. We suggest an evolutionary trade-off between the cellular objectives of protein cost minimization and response time minimization as a cause of this mechanism. The optimal strategy to reduce average protein costs is to transcriptionally control enzymes within a pathway. However, responses on a transcriptional level are usually very slow. In contrast, short response times can be achieved through a constitutive expression of enzymes with a focused regulation of key steps within a pathway. The interplay between the two cellular objectives leads to the observation that particularly pathways with highly abundant and thus costly enzymes are transcriptionally pervasively regulated (Figure 7A). In contrast, pathways with low abundance enzymes are transcriptionally sparsely regulated (Figure 7B). In agreement with these results, we found that pathways such as the pentose phosphate pathway, for which rapid response times are required, are sparsely regulated even if they contain costly enzymes (Figure 7C). Finally, if the fitness advantage achieved through following either of the cellular objectives is low, sparse transcriptional regulation is the minimum requirement to control flux through a pathway (Figure 7D).
In summary, our results demonstrate that, in contrast to the classical picture, regulation of key positions of metabolic pathways is sufficient for full control of flux and is implemented in vivo. This pattern of sparse regulation is particularly useful if a higher fitness advantage can be achieved through rapid response times compared to the fitness advantage achieved through the reduced protein cost of pervasive transcriptional regulation.
Analysis of optimal strategies for the control of metabolic pathways in Escherichia coli reveals that the extent of transcriptional regulation reflects an evolutionary trade-off between the minimization of response time and protein costs.
While previous studies have shed light on the link between the structure of metabolism and its transcriptional regulation, the extent to which transcriptional regulation controls metabolism has not yet been fully explored. In this work, we address this problem by integrating a large number of experimental data sets with a model of the metabolism of Escherichia coli. Using a combination of computational tools including the concept of elementary flux patterns, methods from network inference and dynamic optimization, we find that transcriptional regulation of pathways reflects the protein investment into these pathways. While pathways that are associated to a high protein cost are controlled by fine-tuned transcriptional programs, pathways that only require a small protein cost are transcriptionally controlled in a few key reactions. As a reason for the occurrence of these different regulatory strategies, we identify an evolutionary trade-off between the conflicting requirements to reduce protein investment and the requirement to be able to respond rapidly to changes in environmental conditions.
PMCID: PMC3159982  PMID: 21772263
cost-optimal regulatory strategies; evolutionary optimization; genome-scale metabolic networks; proteomics; transcriptomics
9.  Robust Network Topologies for Generating Switch-Like Cellular Responses 
PLoS Computational Biology  2011;7(6):e1002085.
Signaling networks that convert graded stimuli into binary, all-or-none cellular responses are critical in processes ranging from cell-cycle control to lineage commitment. To exhaustively enumerate topologies that exhibit this switch-like behavior, we simulated all possible two- and three-component networks on random parameter sets, and assessed the resulting response profiles for both steepness (ultrasensitivity) and extent of memory (bistability). Simulations were used to study purely enzymatic networks, purely transcriptional networks, and hybrid enzymatic/transcriptional networks, and the topologies in each class were rank ordered by parametric robustness (i.e., the percentage of applied parameter sets exhibiting ultrasensitivity or bistability). Results reveal that the distribution of network robustness is highly skewed, with the most robust topologies clustering into a small number of motifs. Hybrid networks are the most robust in generating ultrasensitivity (up to 28%) and bistability (up to 18%); strikingly, a purely transcriptional framework is the most fragile in generating either ultrasensitive (up to 3%) or bistable (up to 1%) responses. The disparity in robustness among the network classes is due in part to zero-order ultrasensitivity, an enzyme-specific phenomenon, which repeatedly emerges as a particularly robust mechanism for generating nonlinearity and can act as a building block for switch-like responses. We also highlight experimentally studied examples of topologies enabling switching behavior, in both native and synthetic systems, that rank highly in our simulations. This unbiased approach for identifying topologies capable of a given response may be useful in discovering new natural motifs and in designing robust synthetic gene networks.
Author Summary
Biomolecular signaling networks enable cells to mediate responses to extracellular and intracellular stimuli and are hence crucial for the functioning of all organisms. Such networks do not merely forward information, but perform signal processing: specific modules have evolved to produce complex, dynamic behaviors from input cues. Switching, or the conversion of a graded stimulus into a binary, all-or-none response, is a ubiquitous behavior that regulates critical processes ranging from cell division to stem cell differentiation. While a number of switch-generating networks have been identified, a comprehensive understanding of network architectures that can yield switch-like behavior remains elusive. In this work, we assessed the entire space of minimal networks to identify architectures that can not only exhibit switching behavior but can do so robustly in the dynamic and noisy cellular environment. Our results reveal that these robust networks fit into a small number of topological motifs. Furthermore, network composition (i.e., whether a signaling component functions as an enzyme or a transcription factor) can dramatically impact robustness in generating switching behavior. Topologies presented in this work can be used to identify additional circuits in nature that may exhibit switching behavior and suggest design strategies for engineering switching behavior in synthetic circuits.
PMCID: PMC3121696  PMID: 21731481
10.  Dynamics and Design Principles of a Basic Regulatory Architecture Controlling Metabolic Pathways 
PLoS Biology  2008;6(6):e146.
The dynamic features of a genetic network's response to environmental fluctuations represent essential functional specifications and thus may constrain the possible choices of network architecture and kinetic parameters. To explore the connection between dynamics and network design, we have analyzed a general regulatory architecture that is commonly found in many metabolic pathways. Such architecture is characterized by a dual control mechanism, with end product feedback inhibition and transcriptional regulation mediated by an intermediate metabolite. As a case study, we measured with high temporal resolution the induction profiles of the enzymes in the leucine biosynthetic pathway in response to leucine depletion, using an automated system for monitoring protein expression levels in single cells. All the genes in the pathway are known to be coregulated by the same transcription factors, but we observed drastically different dynamic responses for enzymes upstream and immediately downstream of the key control point—the intermediate metabolite α-isopropylmalate (αIPM), which couples metabolic activity to transcriptional regulation. Analysis based on genetic perturbations suggests that the observed dynamics are due to differential regulation by the leucine branch-specific transcription factor Leu3, and that the downstream enzymes are strictly controlled and highly expressed only when αIPM is available. These observations allow us to build a simplified mathematical model that accounts for the observed dynamics and can correctly predict the pathway's response to new perturbations. Our model also suggests that transient dynamics and steady state can be separately tuned and that the high induction levels of the downstream enzymes are necessary for fast leucine recovery. It is likely that principles emerging from this work can reveal how gene regulation has evolved to optimize performance in other metabolic pathways with similar architecture.
Author Summary
Single-cell organisms must constantly adjust their gene expression programs to survive in a changing environment. Interactions between different molecules form a regulatory network to mediate these changes. While the network connections are often known, figuring out how the network responds dynamically by looking at a static picture of its structure presents a significant challenge. Measuring the response at a finer time scales could reveal the link between the network's function and its structure. The architecture of the system we studied in this work—the leucine biosynthesis pathway in yeast—is shared by other metabolic pathways: a metabolic intermediate binds to a transcription factor to activate the pathway genes, creating an intricate feedback structure that links metabolism with gene expression. We measured protein abundance at high temporal resolution for genes in this pathway in response to leucine depletion and studied the effects of various genetic perturbations on gene expression dynamics. Our measurements and theoretical modeling show that only the genes immediately downstream from the intermediate are highly regulated by the metabolite, a feature that is essential to fast recovery from leucine depletion. Since the architecture we studied is common, we believe that our work may lead to general principles governing the dynamics of gene expression in other metabolic pathways.
A quantitative, high-temporal resolution study of gene induction in a metabolic pathway reveals an intricate connection between the regulatory architecture and the dynamic response of the system, pointing to possible principles underlying the design of these pathways.
PMCID: PMC2429954  PMID: 18563967
11.  Orientation Selectivity in Inhibition-Dominated Networks of Spiking Neurons: Effect of Single Neuron Properties and Network Dynamics 
PLoS Computational Biology  2015;11(1):e1004045.
The neuronal mechanisms underlying the emergence of orientation selectivity in the primary visual cortex of mammals are still elusive. In rodents, visual neurons show highly selective responses to oriented stimuli, but neighboring neurons do not necessarily have similar preferences. Instead of a smooth map, one observes a salt-and-pepper organization of orientation selectivity. Modeling studies have recently confirmed that balanced random networks are indeed capable of amplifying weakly tuned inputs and generating highly selective output responses, even in absence of feature-selective recurrent connectivity. Here we seek to elucidate the neuronal mechanisms underlying this phenomenon by resorting to networks of integrate-and-fire neurons, which are amenable to analytic treatment. Specifically, in networks of perfect integrate-and-fire neurons, we observe that highly selective and contrast invariant output responses emerge, very similar to networks of leaky integrate-and-fire neurons. We then demonstrate that a theory based on mean firing rates and the detailed network topology predicts the output responses, and explains the mechanisms underlying the suppression of the common-mode, amplification of modulation, and contrast invariance. Increasing inhibition dominance in our networks makes the rectifying nonlinearity more prominent, which in turn adds some distortions to the otherwise essentially linear prediction. An extension of the linear theory can account for all the distortions, enabling us to compute the exact shape of every individual tuning curve in our networks. We show that this simple form of nonlinearity adds two important properties to orientation selectivity in the network, namely sharpening of tuning curves and extra suppression of the modulation. The theory can be further extended to account for the nonlinearity of the leaky model by replacing the rectifier by the appropriate smooth input-output transfer function. These results are robust and do not depend on the state of network dynamics, and hold equally well for mean-driven and fluctuation-driven regimes of activity.
Author Summary
It is not yet fully clear how sensory information is being processed when it arrives in primary cortical areas. We studied this general question in the context of rodent vision. We focused on the example of orientation selectivity, namely the selectivity of cortical neurons for specific orientations of an elongated stimulus. Our results show that a large body of experimental findings regarding the basic computations performed in early sensory processing can already be explained by linear processing of firing rates in neuronal networks with realistic parameters. The distribution of selectivity in our networks, as well as the exact shape of output tuning curves, including all details and inhomogeneities of structure and function, can be computed from the known connectivity matrix of the network and the known gain function of single neurons. A simple but essential form of nonlinearity, namely rectification of firing rates, accounts for sharpening of tuning curves and leads to some normalization of output modulation, even in networks without feature-specific connectivity. Our results and analyses demonstrate that none of these functional properties depend crucially on a fluctuation-driven or a mean-driven regime of activity, and that synchronous and asynchronous states of network dynamics can equally well serve these functions.
PMCID: PMC4287576  PMID: 25569445
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.
PMCID: PMC1865588  PMID: 17453047
adaptation; cellular metabolism; expression arrays; plasticity; transcriptional response
13.  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.
PMCID: PMC3010117  PMID: 21045818
Arabidopsis thaliana; biological clocks; dynamical systems; gene regulatory networks; mathematical models; photoperiodism
14.  Noise Management by Molecular Networks 
PLoS Computational Biology  2009;5(9):e1000506.
Fluctuations in the copy number of key regulatory macromolecules (“noise”) may cause physiological heterogeneity in populations of (isogenic) cells. The kinetics of processes and their wiring in molecular networks can modulate this molecular noise. Here we present a theoretical framework to study the principles of noise management by the molecular networks in living cells. The theory makes use of the natural, hierarchical organization of those networks and makes their noise management more understandable in terms of network structure. Principles governing noise management by ultrasensitive systems, signaling cascades, gene networks and feedback circuitry are discovered using this approach. For a few frequently occurring network motifs we show how they manage noise. We derive simple and intuitive equations for noise in molecule copy numbers as a determinant of physiological heterogeneity. We show how noise levels and signal sensitivity can be set independently in molecular networks, but often changes in signal sensitivity affect noise propagation. Using theory and simulations, we show that negative feedback can both enhance and reduce noise. We identify a trade-off; noise reduction in one molecular intermediate by negative feedback is at the expense of increased noise in the levels of other molecules along the feedback loop. The reactants of the processes that are strongly (cooperatively) regulated, so as to allow for negative feedback with a high strength, will display enhanced noise.
Author Summary
Within cells, fluctuations in molecule numbers are inevitable, since the synthesis and degradation of molecules are not synchronised. Such molecular noise can be transferred to other molecules through regulatory interactions. Noise in molecular networks, and especially in gene expression, has been studied extensively over the past years, both experimentally and through mathematical modelling. In this work, we present a theoretical framework that merges concepts derived from metabolic control analysis (which was originally developed to describe the control in metabolic pathways) with linear noise approximation (a concept from statistical physics). This framework is useful to analyse how noise propagates through molecular networks, how noise can be managed within the networks and how different network designs reduce or enhance noise. The present theory makes use of the natural, hierarchical organization of regulatory networks and makes their noise management more understandable in terms of network structure. Within this paper, we apply the framework to signaling and regulatory cascades, and analyse how feedback and time scale separation influence noise propagation in molecular networks.
PMCID: PMC2731877  PMID: 19763166
15.  Starling Flock Networks Manage Uncertainty in Consensus at Low Cost 
PLoS Computational Biology  2013;9(1):e1002894.
Flocks of starlings exhibit a remarkable ability to maintain cohesion as a group in highly uncertain environments and with limited, noisy information. Recent work demonstrated that individual starlings within large flocks respond to a fixed number of nearest neighbors, but until now it was not understood why this number is seven. We analyze robustness to uncertainty of consensus in empirical data from multiple starling flocks and show that the flock interaction networks with six or seven neighbors optimize the trade-off between group cohesion and individual effort. We can distinguish these numbers of neighbors from fewer or greater numbers using our systems-theoretic approach to measuring robustness of interaction networks as a function of the network structure, i.e., who is sensing whom. The metric quantifies the disagreement within the network due to disturbances and noise during consensus behavior and can be evaluated over a parameterized family of hypothesized sensing strategies (here the parameter is number of neighbors). We use this approach to further show that for the range of flocks studied the optimal number of neighbors does not depend on the number of birds within a flock; rather, it depends on the shape, notably the thickness, of the flock. The results suggest that robustness to uncertainty may have been a factor in the evolution of flocking for starlings. More generally, our results elucidate the role of the interaction network on uncertainty management in collective behavior, and motivate the application of our approach to other biological networks.
Author Summary
Starling flocks move in beautiful ways that both captivate and intrigue the observer. Previous work has shown that starlings pay attention to their seven closest neighbors, but until now it was not understood why this number is seven. Our paper explains the mystery: when uncertainty in sensing is present, interacting with six or seven neighbors optimizes the balance between group cohesiveness and individual effort. To prove this result we develop a new systems-theoretic approach for understanding noisy consensus dynamics. The approach allows the evaluation of robustness over a family of hypothesized sensing strategies using observations of the spatial positions of birds within the flock. We apply this approach to experimental data from wild starling flocks, and find that six or seven neighbors yield maximal robustness. The implication that robustness of cohesion may have been a factor in the evolution of flocking has significant consequences for evolutionary biology. In addition, the results and the versatility of the approach have implications for uncertainty management in social and technological networks.
PMCID: PMC3561045  PMID: 23382667
16.  Balance between Noise and Information Flow Maximizes Set Complexity of Network Dynamics 
PLoS ONE  2013;8(3):e56523.
Boolean networks have been used as a discrete model for several biological systems, including metabolic and genetic regulatory networks. Due to their simplicity they offer a firm foundation for generic studies of physical systems. In this work we show, using a measure of context-dependent information, set complexity, that prior to reaching an attractor, random Boolean networks pass through a transient state characterized by high complexity. We justify this finding with a use of another measure of complexity, namely, the statistical complexity. We show that the networks can be tuned to the regime of maximal complexity by adding a suitable amount of noise to the deterministic Boolean dynamics. In fact, we show that for networks with Poisson degree distributions, all networks ranging from subcritical to slightly supercritical can be tuned with noise to reach maximal set complexity in their dynamics. For networks with a fixed number of inputs this is true for near-to-critical networks. This increase in complexity is obtained at the expense of disruption in information flow. For a large ensemble of networks showing maximal complexity, there exists a balance between noise and contracting dynamics in the state space. In networks that are close to critical the intrinsic noise required for the tuning is smaller and thus also has the smallest effect in terms of the information processing in the system. Our results suggest that the maximization of complexity near to the state transition might be a more general phenomenon in physical systems, and that noise present in a system may in fact be useful in retaining the system in a state with high information content.
PMCID: PMC3596377  PMID: 23516395
17.  An atlas of gene regulatory networks reveals multiple three-gene mechanisms for interpreting morphogen gradients 
Although >450 different topologies can achieve the same multicellular patterning function, they can be grouped into six main classes, which operate using different underlying dynamics.Alternative designs for the same functions can therefore split into two types: (a) topology alterations that retain the same underlying dynamics and (b) alterations that utilize a completely different underlying dynamical mechanism.This segregation of networks into distinct dynamical mechanisms can be revealed by the shape of the topology atlas itself.Cell–cell communication is not usually part of the causal mechanism underlying a band-pass response during morphogen interpretation, but it can tune the result or increase robustness.
Understanding how gene regulatory networks (GRNs) achieve particular biological functions is a central question in systems biology. Systems biology promises to go beyond a case-by-case understanding of individual networks to map out the complete design space of mechanistic possibilities that underlie biological functions. Can such maps serve as useful theoretical frameworks in which to explore the general design principles for these functions? Towards addressing these questions, we created the first design space for a morphogen interpretation function.
In order to generate a design space for such a function, we enumerated all possible wiring designs of GRNs consisting of three genes and tested their ability to perform one particular morphogen interpretation function; stripe formation, as it represents a simplified form of the much studied French flag problem and is a commonly found gene expression pattern (Figure 1A). We found that only 5% of GRNs had the ability to generate a single stripe of gene expression when simulated with a fixed morphogen input in a one-dimensional model.
We hypothesized that the core mechanisms for producing the stripe of gene expression should be represented by topologies that contain only the necessary and sufficient gene–gene interactions for that function. Hence, we utilized the notions of complexity and neighborhood to generate a complexity atlas. GRNs of such an atlas (represented by nodes) are considered neighbors if they differ by a single gene–gene interaction (neighboring GRN nodes are connected by edges). Such a metagraph (graph of graphs) can then be reorganized using complexity (number of gene–gene interactions) to determine a GRNs position in the y axis, whereas GRNs are spaced in the x axis with the aim of reducing edge crossing (Figure 5A). This reorganization reveals a striking structure, where ‘stalactites' of complexity can be seen protruding from the bottom of the atlas. Each of these stalactites converges on a single ‘core' topology that by extensive analysis we find represents a distinct mechanism.
The mechanisms employ a diverse range of distinct space–time behaviors, and the underlying core topologies display design features such as modularity and feed-forward. We mapped the mechanisms to the complexity atlas by analyzing how each particular GRN of the atlas was working. The GRNs functioning via the different mechanisms are highlighted by the different colors in Figure 5A. Mechanisms thus occupy large regions of separated topology space, suggesting them to be discrete. Analyzing transitions between mechanisms through parameter space confirms this to be the case.
We find that three of the mechanisms are employed in real patterning systems, including both blastoderm patterning in Drosophila and mesoderm specification in Xenopus (Figure 5B). The remaining three mechanisms are thus candidates for employment in other patterning systems. We explored the performance features of these mechanisms, which suggest that some have features such as robustness to parameter variation that make them highly likely to be employed in particular patterning contexts.
Only one of the six-core mechanisms absolutely requires cell–cell communication for functionality, prompting us to predict that cell–cell communication will rarely be responsible for the basic dose response of morphogen interpretation networks. However, we show how cell–cell communication has an important role in robust stripe generation in the face of a noisy morphogen input and in fine tuning the quantitative details of stripe patterning.
In summary, the complexity atlas approach is an amendable approach to any system with a clear genotype–function relationship. We demonstrate how certain functions such as morphogen interpretation may have a range of potential solutions in contrast to previous studies that analyzed more constrained functions. Furthermore, we demonstrate how such an approach can be utilized to define a ‘design space' for a given biological function that describes the different mechanistic possibilities and how they relate to one another (Figure 5). Such a design space can be used practically as a guide to discern which patterning mechanisms are likely be at work in a particular context throwing up less intuitive possibilities with powerful performance features.
The interpretation of morphogen gradients is a pivotal concept in developmental biology, and several mechanisms have been proposed to explain how gene regulatory networks (GRNs) achieve concentration-dependent responses. However, the number of different mechanisms that may exist for cells to interpret morphogens, and the importance of design features such as feedback or local cell–cell communication, is unclear. A complete understanding of such systems will require going beyond a case-by-case analysis of real morphogen interpretation mechanisms and mapping out a complete GRN ‘design space.' Here, we generate a first atlas of design space for GRNs capable of patterning a homogeneous field of cells into discrete gene expression domains by interpreting a fixed morphogen gradient. We uncover multiple very distinct mechanisms distributed discretely across the atlas, thereby expanding the repertoire of morphogen interpretation network motifs. Analyzing this diverse collection of mechanisms also allows us to predict that local cell–cell communication will rarely be responsible for the basic dose-dependent response of morphogen interpretation networks.
PMCID: PMC3010108  PMID: 21045819
design space; gene network; morphogen; patterning; systems biology
18.  A modular gradient-sensing network for chemotaxis in Escherichia coli revealed by responses to time-varying stimuli 
Combining in vivo FRET with time-varying stimuli, such as steps, ramps, and sinusoids allowed deduction of the molecular mechanisms underlying cellular signal processing.The bacterial chemotaxis pathway can be described as a two-module feedback circuit, the transfer functions of which we have characterized quantitatively by experiment. Model-driven experimental design allowed the use of a single FRET pair for measurements of both transfer functions of the pathway.The adaptation module's transfer function revealed that feedback near steady state is weak, consistent with high sensitivity to shallow gradients, but also strong steady-state fluctuations in pathway output.The measured response to oscillatory stimuli defines the frequency band over which the chemotaxis system can compute time derivatives.
In searching for better environments, bacteria sample their surroundings by random motility, and make temporal comparisons of experienced sensory cues to bias their movement toward favorable directions (Berg and Brown, 1972). Thus, the problem of sensing spatial gradients is reduced to time-derivative computations, carried out by a signaling pathway that is well characterized at the molecular level in Escherichia coli. Here, we study the physiology of this signal processing system in vivo by fluorescence resonance energy transfer (FRET) experiments in which live cells are stimulated by time-varying chemoeffector signals. By measuring FRET between the active response regulator of the pathway CheY-P and its phosphatase CheZ, each labeled with GFP variants, we obtain a readout that is directly proportional to pathway activity (Sourjik et al, 2007). We analyze the measured response functions in terms of mechanistic models of signaling, and discuss functional consequences of the observed quantitative characteristics.
Experiments are guided by a coarse-grained modular model (Tu et al, 2008) of the sensory network (Figure 1), in which we identify two important ‘transfer functions': one corresponding to the receptor–kinase complex, which responds to changes in input ligand concentration on a fast time scale, and another corresponding to the adaptation system, which provides negative feedback, opposing the effect of ligand on a slower time scale. For the receptor module, we calibrate an allosteric MWC-type model of the receptor–kinase complex by FRET measurements of the ‘open-loop' transfer function G([L],m) using step stimuli. This calibration provides a basis for using the same FRET readout (between CheY-P and CheZ) to further study properties of the adaptation module.
It is well known that adaptation in E. coli's chemotaxis system uses integral feedback, which guarantees exact restoration of the baseline activity after transient responses to step stimuli (Barkai and Leibler, 1997; Yi et al, 2000). However, the output of time-derivative computations during smoothly varying stimuli depends not only on the presence of integral feedback, but also on what is being integrated. As this integrand can in general be any function of the output, we represent it by a black-box function F(a) in our model, and set out to determine its shape by experiments with time-varying stimuli.
We first apply exponential ramp stimuli—waveforms in which the logarithm of the stimulus level varies linearly with time, at a fixed rate r. It was shown many years ago that during such a stimulus, the kinase output of the pathway changes to a new constant value, ac that is dependent on the applied ramp rate, r (Block et al, 1983). A plot of ac versus r (Figure 5A) can thus be considered as an output of time-derivative computations by the network, and could also be used to study the ‘gradient sensitivity' of bacteria traveling at constant speeds.
To obtain the feedback transfer function, F(a), we apply a simple coordinate transformation, identified using our model, to the same ramp-response data (Figure 5B). This function reveals how the temporal rate of change of the feedback signal m depends on the current output signal a. The shape of this function is analyzed using a biochemical reaction scheme, from which in vivo kinetic parameters of the feedback enzymes, CheR and CheB, are extracted. The fitted Michaelis constants for these enzymatic reactions are small compared with the steady-state abundance of their substrates, thus indicating that these enzymes operate close to saturation in vivo. The slope of the function near steady state can be used to assess the strength of feedback, and to compute the relaxation time of the system, τm. Relaxation is found to be slow (i.e. large τm), consistent with large fluctuations about the steady-state activity caused by the near-saturation kinetics of the feedback enzymes (Emonet and Cluzel, 2008).
Finally, exponential sine-wave stimuli are used to map out the system's frequency response (Figure 5C). The measured data points for both the amplitude and phase of the response are found to be in excellent agreement with model predictions based on parameters from the independently measured step and ramp responses. No curve fitting was required to obtain this agreement. Although the amplitude response as a function of frequency resembles a first-order high-pass filter with a well-defined cutoff frequency, νm, we point out that the chemotaxis pathway is actually a low-pass filter if the time derivative of the input is viewed as the input signal. In this latter perspective, νm defines an upper bound for the frequency band over which time-derivative computations can be carried out.
The two types of measurements yield complementary information regarding time-derivative computations by E. coli. The ramp-responses characterize the asymptotically constant output when a temporal gradient is held fixed over extended periods. Interestingly, the ramp responses do not depend on receptor cooperativity, but only on properties of the adaptation system, and thus can be used to reveal the in vivo adaptation kinetics, even outside the linear regime of the kinase response. The frequency response is highly relevant in considering spatial searches in the real world, in which experienced gradients are not held fixed in time. The characteristic cutoff frequency νm is found by working within the linear regime of the kinase response, and depends on parameters from both modules (it increases with both cooperativity in the receptor module, and the strength of feedback in the adaptation module).
Both ramp responses and sine-wave responses were measured at two different temperatures (22 and 32°C), and found to differ significantly. Both the slope of F(a) near steady state, from ramp experiments, and the characteristic cutoff frequency, from sine-wave experiments, were higher by a factor of ∼3 at 32°C. Fits of the enzymatic model to F(a) suggest that temperature affects the maximal velocity (Vmax) more strongly than the Michaelis constants (Km) for CheR and CheB.
Successful application of inter-molecular FRET in live cells using GFP variants always requires some degree of serendipity. Genetic fusions to these bulky fluorophores can impair the function of the original proteins, and even when fusions are functional, efficient FRET still requires the fused fluorophores to come within the small (<10 nm) Förster radius on interactions between the labeled proteins. Thus, when a successful FRET pair is identified, it is desirable to make the most of it. We have shown here that combined with careful temporal control of input stimuli, and appropriately calibrated models, a single FRET pair can be used to study the structure of multiple transfer functions within a signaling network.
The Escherichia coli chemotaxis-signaling pathway computes time derivatives of chemoeffector concentrations. This network features modules for signal reception/amplification and robust adaptation, with sensing of chemoeffector gradients determined by the way in which these modules are coupled in vivo. We characterized these modules and their coupling by using fluorescence resonance energy transfer to measure intracellular responses to time-varying stimuli. Receptor sensitivity was characterized by step stimuli, the gradient sensitivity by exponential ramp stimuli, and the frequency response by exponential sine-wave stimuli. Analysis of these data revealed the structure of the feedback transfer function linking the amplification and adaptation modules. Feedback near steady state was found to be weak, consistent with strong fluctuations and slow recovery from small perturbations. Gradient sensitivity and frequency response both depended strongly on temperature. We found that time derivatives can be computed by the chemotaxis system for input frequencies below 0.006 Hz at 22°C and below 0.018 Hz at 32°C. Our results show how dynamic input–output measurements, time honored in physiology, can serve as powerful tools in deciphering cell-signaling mechanisms.
PMCID: PMC2913400  PMID: 20571531
adaptation; feedback; fluorescence resonance energy transfer (FRET); frequency response; Monod–Wyman–Changeux (MWC) model
19.  Synthetic incoherent feedforward circuits show adaptation to the amount of their genetic template 
Variable gene dosage is a major source of fluctuations in gene expression in both endogenous and synthetic circuits. Synthetic incoherent feedforward regulatory motifs using RNA interference are shown to robustly adapt to changes in DNA template amounts in mammalian cells.
Variable gene dosage is a major source of fluctuations in gene product levels in both endogenous and synthetic circuits.To mitigate gene expression variability, we designed, simulated, constructed, and tested regulatory circuits, implementing an incoherent feedforward motif.A number of control mechanisms including transcription and post-transcriptional regulation were tested in mammalian cells.Feedforward regulation displayed better adaptation than negative feedback, and circuits based on RNA interference were the most robust to variation in DNA template amounts.
Natural and synthetic biological networks must function reliably in the face of fluctuating stoichiometry of their molecular components. These fluctuations are caused in part by changes in relative expression efficiency and the DNA template amount of the network-coding genes. Indeed, changes in gene dosage are clearly a major source of variation in cells, and yet those changes are very common in both normal processes (sex determination, ploidy change) and disease (gene amplification in cancer). In synthetic networks, the problem is exacerbated due to commonly used transient delivery methods that result in very large cell-to-cell variability in gene dosage. The basic question on gene dosage compensation in nature (Veitia et al, 2008; Acar et al, 2010) and a practical challenge of overcoming sensitivity to DNA copy number in synthetic circuits prompted us to investigate mechanisms to reduce this variability using sophisticated internal regulatory mechanisms. Indeed, the baseline expression unit in many synthetic circuits is an open-loop promoter-ORF combination. We hypothesized that some sort of internal regulation will result in ‘expression units' whose gene product (i.e. protein) output will depend only mildly on the intracellular concentration of its DNA template. In other words, we searched for architecture that would lead to ‘adaptation' of the gene product to the amount of DNA template.
By examining large body of published work, we found frequent reference to a so-called ‘incoherent feedforward' network motif (Mangan and Alon, 2003). The canonical three-node incoherent loop contains input, auxiliary regulator, and output nodes. The output is controlled directly by the input and the auxiliary regulator. The latter is also controlled by the input, introducing an additional indirect effect of the input on the output. In incoherent loops, the input controls the auxiliary regulator node in such a way that input's overall indirect action on the output via this node counteracts its direct effect. In a motif named ‘type I incoherent feedforward loop' (I1-FFL), the input's direct effect is activating, as is its control of the auxiliary node, while the auxiliary node controls the output through repression. One of the most studied properties of such motifs is their transient response to persistent stimulus, that is visually characterized as a ‘bump' or ‘pulse' (hence the name ‘pulse generator') that then goes back to the original steady state of the system (Basu et al, 2004). We hypothesized that changing DNA amount could serve as an input to an incoherent circuit if the auxiliary regulator and the output nodes are coexpressed from this DNA; in other words DNA can be viewed as an ‘activator' of both the regulator and the output. We conjectured that this might lead to adaptation to changes in DNA template.
We designed and simulated in silico a number of network architectures that all exhibit incoherent feedforward connectivity. We also compared them with the well-studied feedback loop circuit that in theory weakens but does not eliminate gene product dependency on the DNA template amount. The schematics of the circuits are shown in Figure 1.
Experimental measurement of input–output response of these circuits, or their transfer function, indeed uncovered adaptation of the output to DNA template abundance. Such adaptation has not been observed with feedback loop, as expected. Among various architectures, the post-transcriptional circuits showed faster adaptation, higher absolute expression levels and lower ‘noise' (Figure 4).
We also simulated and measured stochastic variability in the circuits by collecting all the cells with similar input values and statistically analyzing output values in those cells. We found that substantial noise component could not be accounted for by known noise sources, and concluded that the very step of negative regulation, both by a repressor LacI and by a microRNA, significantly increases cell-to-cell variability. This needs to be addressed in further studies. We also found that the negative feedback loop did not result in reduced noise as we expected, yet it did not result in noise increase as in the incoherent motif. This means that there may be effective noise reduction but it is not sufficient to produce narrow distributions of outputs for a given input.
We conclude that expression units that incorporate incoherent feedforward control of the gene product provide adaptation to the amount of DNA template and can be superior to simple combinations of constitutive promoters with ORFs. We also emphasize the relevance of our findings to the long-standing question of gene dosage compensation in cells, and note that similar incoherent architectures with microRNA negative regulators have been found in cells, suggesting that their physiological role is to curb variability in gene dosage and/or promoter strength.
Natural and synthetic biological networks must function reliably in the face of fluctuating stoichiometry of their molecular components. These fluctuations are caused in part by changes in relative expression efficiency and the DNA template amount of the network-coding genes. Gene product levels could potentially be decoupled from these changes via built-in adaptation mechanisms, thereby boosting network reliability. Here, we show that a mechanism based on an incoherent feedforward motif enables adaptive gene expression in mammalian cells. We modeled, synthesized, and tested transcriptional and post-transcriptional incoherent loops and found that in all cases the gene product adapts to changes in DNA template abundance. We also observed that the post-transcriptional form results in superior adaptation behavior, higher absolute expression levels, and lower intrinsic fluctuations. Our results support a previously hypothesized endogenous role in gene dosage compensation for such motifs and suggest that their incorporation in synthetic networks will improve their robustness and reliability.
PMCID: PMC3202791  PMID: 21811230
feedforward motifs; gene dosage and noise; mammalian cells; microRNAs; negative autoregulation
20.  Systematic evaluation of objective functions for predicting intracellular fluxes in Escherichia coli 
The in vivo distribution of metabolic fluxes in Escherichia coli can be predicted from optimality principles At least two different sets of optimality principles govern the operation of the metabolic network under different environmental conditionsMetabolism during unlimited growth on glucose in batch culture is best described by the nonlinear maximization of ATP yield per unit of flux
Based on a long history of biochemical and lately genomic research, metabolic networks, in particular microbial ones, are among the best characterized cellular networks. Most components (genes, proteins and metabolites) and their interactions are known. This topological knowledge of the reaction stoichiometry allows to construct metabolic models up to the level of genome scale (Price et al, 2004). Experimentally, sophisticated 13C-tracer-based methodologies were developed that enable tracking of the intracellular flux traffic through the reaction network (Sauer, 2006). With the accumulation of such experimental flux data, the question arises why a particular distribution of flux within the network is realized and not one of many alternatives?
Here, we address the question whether the intracellular flux state can be predicted from optimality principles, with the underlying rational that evolution might have optimized metabolic operation toward particular objectives or combinations of multiple objectives. For this purpose, we performed a systematic and rigorous comparison between computational flux predictions and available experimental flux data (Emmerling et al, 2002; Perrenoud and Sauer, 2005; Nanchen et al, 2006) under six different environmental conditions for the model bacterium E. coli. For computational flux predictions, we used a constraint-based modeling approach that requires a stoichiometric model of metabolism (Stelling, 2004). More specifically, we employed flux balance analysis (FBA) where objective functions are defined that represent optimality principles of network operation (Price et al, 2004). This approach has been applied successfully to predict gene deletion lethality (Edwards and Palsson, 2000a, bEdwards and Palsson, 2000a, b; Forster et al, 2003; Kuepfer et al, 2005), network capacities and feasible network states (Edwards 2001, Ibarra 2002), but in only few cases to predict the intracellular flux state (Beard et al, 2002; Holzhütter, 2004).
While different objective functions were proposed for different biological systems (Holzhütter, 2004; Price et al, 2004; Knorr et al, 2006), by far the most common assumption is that microbial cells maximize their growth. To address this issue more generally, we evaluated the accuracy of FBA-based flux predictions for 11 linear and nonlinear objective functions that were combined with eight adjustable constraints. For this purpose, we constructed a highly interconnected stoichiometric network model with 98 reactions and 60 metabolites of E. coli central carbon metabolism. Based on mathematical analyses, the overall model could be reduced to a set of 10 reactions that summarize the actual systemic degree of freedom.
As a quantitative measure of how accurate the experimental data are predicted, we defined predictive fidelity as a single value to quantify the overall deviation between in silico and in vivo fluxes. By comparing all in silico predictions to 13C-based in vivo fluxes, we show that prediction of intracellular steady-state fluxes from network stoichiometry alone is, within limits, possible. An unexpected key result is that no further assumptions on network operation in the form of additional and potentially artificial constraints are necessary, provided the appropriate objective function is chosen for a given condition.
While no single objective was able to describe the flux states under all six conditions, we identified two sets of objectives for biologically meaningful predictions without the need for further constraints. For unlimited growth on glucose in aerobic or nitrate-respiring batch cultures, we find that the most accurate and robust results are obtained with the nonlinear maximization of ATP yield per flux unit (Figure 1). Under nutrient scarcity in glucose- or ammonium-limited continuous cultures, in contrast, linear maximization of the overall ATP or biomass yields achieved the highest predictive accuracy.
Since these identified optimality principles describe the system behavior without preconditioning of the network through further constraints, they reflect, to some extent, the evolutionary selection of metabolic network regulation that realizes the various flux states. For conditions of nutrient scarcity, the maximization of energy or biomass yield objective is consistent with the generally observed physiology (Russell and Cook, 1995). The meaning of the maximization of ATP yield per flux unit objective for unlimited growth, however, is less obvious. Generally, it selects for small networks with yet high, albeit suboptimal ATP formation, which has three biological consequences. Firstly, resources are economically allocated since expenditures for enzyme synthesis are, on average, greater for longer pathways. Secondly, suboptimal ATP yields dissipate more energy and thus enable higher catabolic rates. Thirdly, at a constant catabolic rate, a small network results in shorter residence times of substrate molecules until they generate ATP. The relative contribution of these consequences to the evolution of network regulation is unclear, but simultaneous optimization for ATP yield and catabolic rate under this optimality principle identifies a trade-off between the contradicting objectives of maximum overall ATP yield and maximum rate of ATP formation (Pfeiffer et al, 2001).
To which extent can optimality principles describe the operation of metabolic networks? By explicitly considering experimental errors and in silico alternate optima in flux balance analysis, we systematically evaluate the capacity of 11 objective functions combined with eight adjustable constraints to predict 13C-determined in vivo fluxes in Escherichia coli under six environmental conditions. While no single objective describes the flux states under all conditions, we identified two sets of objectives for biologically meaningful predictions without the need for further, potentially artificial constraints. Unlimited growth on glucose in oxygen or nitrate respiring batch cultures is best described by nonlinear maximization of the ATP yield per flux unit. Under nutrient scarcity in continuous cultures, in contrast, linear maximization of the overall ATP or biomass yields achieved the highest predictive accuracy. Since these particular objectives predict the system behavior without preconditioning of the network structure, the identified optimality principles reflect, to some extent, the evolutionary selection of metabolic network regulation that realizes the various flux states.
PMCID: PMC1949037  PMID: 17625511
13C-flux; evolution; flux balance analysis; metabolic network; network optimality
21.  Dose-Response Aligned Circuits in Signaling Systems 
PLoS ONE  2012;7(4):e34727.
Cells use biological signal transduction pathways to respond to environmental stimuli and the behavior of many cell types depends on precise sensing and transmission of external information. A notable property of signal transduction that was characterized in the Saccharomyces cerevisiae yeast cell and many mammalian cells is the alignment of dose-response curves. It was found that the dose response of the receptor matches closely the dose responses of the downstream. This dose-response alignment (DoRA) renders equal sensitivities and concordant responses in different parts of signaling system and guarantees a faithful information transmission. The experimental observations raise interesting questions about the nature of the information transmission through DoRA signaling networks and design principles of signaling systems with this function. Here, we performed an exhaustive computational analysis on network architectures that underlie the DoRA function in simple regulatory networks composed of two and three enzymes. The minimal circuits capable of DoRA were examined with Michaelis-Menten kinetics. Several motifs that are essential for the dynamical function of DoRA were identified. Systematic analysis of the topology space of robust DoRA circuits revealed that, rather than fine-tuning the network's parameters, the function is primarily realized by enzymatic regulations on the controlled node that are constrained in limiting regions of saturation or linearity.
PMCID: PMC3320644  PMID: 22496849
22.  Intrinsic Neuronal Properties Switch the Mode of Information Transmission in Networks 
PLoS Computational Biology  2014;10(12):e1003962.
Diverse ion channels and their dynamics endow single neurons with complex biophysical properties. These properties determine the heterogeneity of cell types that make up the brain, as constituents of neural circuits tuned to perform highly specific computations. How do biophysical properties of single neurons impact network function? We study a set of biophysical properties that emerge in cortical neurons during the first week of development, eventually allowing these neurons to adaptively scale the gain of their response to the amplitude of the fluctuations they encounter. During the same time period, these same neurons participate in large-scale waves of spontaneously generated electrical activity. We investigate the potential role of experimentally observed changes in intrinsic neuronal properties in determining the ability of cortical networks to propagate waves of activity. We show that such changes can strongly affect the ability of multi-layered feedforward networks to represent and transmit information on multiple timescales. With properties modeled on those observed at early stages of development, neurons are relatively insensitive to rapid fluctuations and tend to fire synchronously in response to wave-like events of large amplitude. Following developmental changes in voltage-dependent conductances, these same neurons become efficient encoders of fast input fluctuations over few layers, but lose the ability to transmit slower, population-wide input variations across many layers. Depending on the neurons' intrinsic properties, noise plays different roles in modulating neuronal input-output curves, which can dramatically impact network transmission. The developmental change in intrinsic properties supports a transformation of a networks function from the propagation of network-wide information to one in which computations are scaled to local activity. This work underscores the significance of simple changes in conductance parameters in governing how neurons represent and propagate information, and suggests a role for background synaptic noise in switching the mode of information transmission.
Author Summary
Differences in ion channel composition endow different neuronal types with distinct computational properties. Understanding how these biophysical differences affect network-level computation is an important frontier. We focus on a set of biophysical properties, experimentally observed in developing cortical neurons, that allow these neurons to efficiently encode their inputs despite time-varying changes in the statistical context. Large-scale propagating waves are autonomously generated by the developing brain even before the onset of sensory experience. Using multi-layered feedforward networks, we examine how changes in intrinsic properties can lead to changes in the network's ability to represent and transmit information on multiple timescales. We demonstrate that measured changes in the computational properties of immature single neurons enable the propagation of slow-varying wave-like inputs. In contrast, neurons with more mature properties are more sensitive to fast fluctuations, which modulate the slow-varying information. While slow events are transmitted with high fidelity in initial network layers, noise degrades transmission in downstream network layers. Our results show how short-term adaptation and modulation of the neurons' input-output firing curves by background synaptic noise determine the ability of neural networks to transmit information on multiple timescales.
PMCID: PMC4256072  PMID: 25474701
23.  Modeling Integrated Cellular Machinery Using Hybrid Petri-Boolean Networks 
PLoS Computational Biology  2013;9(11):e1003306.
The behavior and phenotypic changes of cells are governed by a cellular circuitry that represents a set of biochemical reactions. Based on biological functions, this circuitry is divided into three types of networks, each encoding for a major biological process: signal transduction, transcription regulation, and metabolism. This division has generally enabled taming computational complexity dealing with the entire system, allowed for using modeling techniques that are specific to each of the components, and achieved separation of the different time scales at which reactions in each of the three networks occur. Nonetheless, with this division comes loss of information and power needed to elucidate certain cellular phenomena. Within the cell, these three types of networks work in tandem, and each produces signals and/or substances that are used by the others to process information and operate normally. Therefore, computational techniques for modeling integrated cellular machinery are needed. In this work, we propose an integrated hybrid model (IHM) that combines Petri nets and Boolean networks to model integrated cellular networks. Coupled with a stochastic simulation mechanism, the model simulates the dynamics of the integrated network, and can be perturbed to generate testable hypotheses. Our model is qualitative and is mostly built upon knowledge from the literature and requires fine-tuning of very few parameters. We validated our model on two systems: the transcriptional regulation of glucose metabolism in human cells, and cellular osmoregulation in S. cerevisiae. The model produced results that are in very good agreement with experimental data, and produces valid hypotheses. The abstract nature of our model and the ease of its construction makes it a very good candidate for modeling integrated networks from qualitative data. The results it produces can guide the practitioner to zoom into components and interconnections and investigate them using such more detailed mathematical models.
Author Summary
Within the cell of an organism, three networks—signaling, transcriptional, and metabolic—are always at work to determine the response of the cell to signals from its environment, and consequently, its fate. Evidence from experimental studies is painting a picture of complex crosstalk among these networks. Thus, while a wide array of computational techniques exist for analyzing each of these network types, there is clear need for new modeling techniques that allow for simultaneously analyzing integrated networks, which combine elements from all three networks. Here, we provide a step towards achieving this task by combining two population modeling techniques—Petri nets and Boolean networks—to produce an integrated hybrid model. We demonstrate the accuracy and utility of this model on two biological systems: transcriptional regulation of glucose metabolism in human cells, and cellular osmoregulation in yeast.
PMCID: PMC3820535  PMID: 24244124
24.  Dynamic Analysis of Integrated Signaling, Metabolic, and Regulatory Networks 
PLoS Computational Biology  2008;4(5):e1000086.
Extracellular cues affect signaling, metabolic, and regulatory processes to elicit cellular responses. Although intracellular signaling, metabolic, and regulatory networks are highly integrated, previous analyses have largely focused on independent processes (e.g., metabolism) without considering the interplay that exists among them. However, there is evidence that many diseases arise from multifunctional components with roles throughout signaling, metabolic, and regulatory networks. Therefore, in this study, we propose a flux balance analysis (FBA)–based strategy, referred to as integrated dynamic FBA (idFBA), that dynamically simulates cellular phenotypes arising from integrated networks. The idFBA framework requires an integrated stoichiometric reconstruction of signaling, metabolic, and regulatory processes. It assumes quasi-steady-state conditions for “fast” reactions and incorporates “slow” reactions into the stoichiometric formalism in a time-delayed manner. To assess the efficacy of idFBA, we developed a prototypic integrated system comprising signaling, metabolic, and regulatory processes with network features characteristic of actual systems and incorporating kinetic parameters based on typical time scales observed in literature. idFBA was applied to the prototypic system, which was evaluated for different environments and gene regulatory rules. In addition, we applied the idFBA framework in a similar manner to a representative module of the single-cell eukaryotic organism Saccharomyces cerevisiae. Ultimately, idFBA facilitated quantitative, dynamic analysis of systemic effects of extracellular cues on cellular phenotypes and generated comparable time-course predictions when contrasted with an equivalent kinetic model. Since idFBA solves a linear programming problem and does not require an exhaustive list of detailed kinetic parameters, it may be efficiently scaled to integrated intracellular systems that incorporate signaling, metabolic, and regulatory processes at the genome scale, such as the S. cerevisiae system presented here.
Author Summary
Cellular systems comprise many diverse components and component interactions spanning signal transduction, transcriptional regulation, and metabolism. Although signaling, metabolic, and regulatory activities are often investigated independently of one another, there is growing evidence that considerable interplay occurs among them, and that the malfunctioning of this interplay is associated with disease. The computational analysis of integrated networks has been challenging because of the varying time scales involved as well as the sheer magnitude of such systems (e.g., the numbers of rate constants involved). To this end, we developed a novel computational framework called integrated dynamic flux balance analysis (idFBA) that generates quantitative, dynamic predictions of species concentrations spanning signaling, regulatory, and metabolic processes. idFBA extends an existing approach called flux balance analysis (FBA) in that it couples “fast” and “slow” reactions, thereby facilitating the study of whole-cell phenotypes and not just sub-cellular network properties. We applied this framework to a prototypic integrated system derived from literature as well as a representative integrated yeast module (the high-osmolarity glycerol [HOG] pathway) and generated time-course predictions that matched with available experimental data. By extending this framework to larger-scale systems, phenotypic profiles of whole-cell systems could be attained expeditiously.
PMCID: PMC2377155  PMID: 18483615
25.  Single-gene tuning of Caulobacter cell cycle period and noise, swarming motility, and surface adhesion 
We established that the sensor histidine kinase DivJ has an important role in the regulation of C. crescentus cell cycle period and noise. This was accomplished by designing and conducting single-cell experiments to probe the dependence of cell cycle noise on divJ expression and constructing a simplified cell cycle model that captures the dependence of cell cycle noise on DivJ with molecular details.In addition to its role in regulating the cell cycle, DivJ also affects polar cell development in C. crescentus, regulating swarming motility and surface adhesion. We propose that pleiotropic control of polar cell development by the DivJ–DivK–PleC signaling pathway underlies divJ-dependent tuning of cell swarming and adhesion behaviors.We have integrated the study of single-cell fluorescence dynamics with a kinetic model simulation to provide direct quantitative evidence that the DivJ histidine kinase is localized to the cell pole through a dynamic diffusion-and-capture mechanism during the C. crescentus cell cycle.
Temporally-coordinated localization of various structural and signaling proteins is critical for proper cell cycle regulation and polar cell development in the bacterium, Caulobacter crescentus. Included among these dynamically-localized regulatory proteins is the sensor histidine kinase, DivJ (Wheeler and Shapiro, 1999). Co-localized with DivJ in the early stalked phase is the phosphorylated response regulator DivK∼P (Jacobs et al, 2001), and the protease ClpXP (McGrath et al, 2006), which degrades the master cell cycle regulator, CtrA (Jenal and Fuchs, 1998). Recent single-cell measurements of surface attached C. crescentus cells have revealed an intriguing role for DivJ in the control of noise in cell division period (Siegal-Gaskins and Crosson, 2008). The noise of the cell cycle increases significantly upon disruption of the divJ gene, with a relatively small accompanying increase in the mean cell cycle time. The deterministic nature of the existing cell cycle models (Li et al, 2008, 2009; Shen et al, 2008) cannot explain the measured increase in cell cycle period and noise in a divJ null strain. Moreover, mechanistic descriptions of how DivJ and its signaling partners are localized and how these proteins underlie the control of polar cell development and cell adhesion in C. crescentus remain immature.
The single-cell experiments and analysis presented herein reveal that C. crescentus cell cycle period and noise can be tuned by DivJ (Figure 2). Specifically, in the case of low (or no) divJ expression the cell cycle is perturbed, and this is quantified by way of the (measured) noise in the cell cycle period. The level of noise is readily controlled through regulated expression of the divJ gene (Figure 2B). A simplified protein interaction network of stalked C. crescentus cell cycle regulation involving minimal components (CtrA, CtrA∼P, DivK, DivK∼P, and DivJ) was constructed to explore such tunability at the molecular level. The agreement of our model with our (and other) experiments suggests this simplified protein regulatory network is sufficient to explain the major features of the C. crescentus cell cycle. Indeed, stochastic simulations of this model using the Gillespie method (Gillespie, 1976) establish the importance of robust DivJ-mediated phosphorylation of its cognate receiver protein, DivK, in regulating the variance of cell cycle oscillations. Increased variability in the concentration of DivK∼P at the single cell level under divJ depletion subsequently leads to increased noise in the regulation of CtrA phosphorylation and degradation. Our experiments and simulations provide evidence that the steady state level of DivK∼P at the single-cell level (as maintained by DivJ) is essential in maintaining regular timing of the cell division period in C. crescentus.
In addition to its role in regulating cell cycle, divJ expression also affects polar cell development in C. crescentus. Specifically, the capacity of swarmer cells to adhere to a glass surface is suppressed at high levels of divJ expression. The effect of elevated divJ expression on the adhesive capacity of the cell is reflected in a reduced rate of two-dimensional biofilm formation. This effect is quantitatively captured by our mathematical model that relates single-cell surface adhesion physiology and biofilm formation dynamics. This result, and our observation that divJ expression tunes swarming motility in semi-solid growth medium, suggests a model in which increased DivJ concentration in the swarmer compartment (due to constitutive overexpression) ultimately results in improper development of polar organelles that are required for adhesion of swarming motility.
Despite the appreciated significance of protein localization for bacterial physiological functions, the molecular mechanism of how polar protein localization is achieved has only been tested in a few cases (Shapiro et al, 2002; Thanbichler and Shapiro, 2008). Mechanisms such as the polar insertion model and diffusion-and-capture have been proposed but the community's knowledge is limited to very few examples (Charles et al, 2001; Rudner et al, 2002). We provide direct evidence from experiments and simulations that the DivJ histidine kinase becomes localized to the cell pole through a dynamic diffusion-and-capture mechanism during the C. crescentus cell cycle (Figure 7). We show that a kinetic model based on a Langmuir adsorption/desorption relationship (Figure 7D) is sufficient to explain the time evolution of the single cell fluorescence time traces (Figure 7C and E) and allows establishing quantitative correspondences between the simulated dynamics and experimentally determined DivJ–EGFP dynamics. This localization mechanism is consistent with a diffusion-and-capture model. In short, the model posits that proteins are randomly distributed and are freely diffusing until they are captured at the site where they ultimately reside (Rudner et al, 2002; Shapiro et al, 2002; Bardy and Maddock, 2007). With a diffusion-and-capture pathway, it has been argued that proteins can be adsorbed either dynamically or statically (Shapiro et al, 2009). Our analysis of DivJ–EGFP in single cells supports a dynamic diffuse-and-capture mechanism for DivJ localization.
Sensor histidine kinases underlie the regulation of a range of physiological processes in bacterial cells, from chemotaxis to cell division. In the gram-negative bacterium Caulobacter crescentus, the membrane-bound histidine kinase, DivJ, is a polar-localized regulator of cell cycle progression and development. We show that DivJ localizes to the cell pole through a dynamic diffusion and capture mechanism rather than by active localization. Analysis of single C. crescentus cells in microfluidic culture demonstrates that controlled expression of divJ permits facile tuning of both the mean and noise of the cell division period. Simulations of the cell cycle that use a simplified protein interaction network capture previously measured oscillatory protein profiles, and recapitulate the experimental observation that deletion of divJ increases the cell cycle period and noise. We further demonstrate that surface adhesion and swarming motility of C. crescentus in semi-solid media can also be tuned by divJ expression. We propose a model in which pleiotropic control of polar cell development by the DivJ–DivK–PleC signaling pathway underlies divJ-dependent tuning of cell swarming and adhesion behaviors.
PMCID: PMC3018171  PMID: 21179017
cell cycle; histidine kinase; protein interaction network; protein localization; single cell

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