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1.  Pre-dispositions and epigenetic inheritance in the Escherichia coli lactose operon bistable switch 
Under conditions of bistable induction of Escherichia coli lac operon, epigenetic patterns of sublineages of ‘on' and ‘off' cells originate from distinguishable ancestors up to two generations before induction.We found two switching pre-disposing factors, namely low repressor levels and slow growth, demonstrating that stochasticity in gene expression and global physiology synergistically determine the single-cell responses.A quantitative model where growth rate acts through simple dilution of intracellular content and repressor level controls the basal activity of the operon demonstrates that both growth rate and repressor concentration influence the cell switching ability.
The bacterium Escherichia coli, like many other microorganisms can use different sugars as a carbon source and uses some of these sugars in preference to others. For example, when grown in the presence of both lactose and glucose, the bacteria first consume glucose and use lactose only when glucose is exhausted. To this end, the enzymes necessary for lactose uptake and metabolism, grouped in one transcriptional unit called the lac operon (lacZ, Y, A encoding for the lactose degrading enzyme (β-galactosidase), permease and transacetylase, respectively) are produced only in the absence of glucose and in the presence of lactose or its analogs, such as the non-metabolizable analog thiomethyl-β-galactoside (TMG). In the absence of such inducers, the transcription of the operon is inhibited by the repressor molecule LacI. This inhibition is relieved by the inducers, which bind and inactivate LacI, initiating an amplifying feedback loop through the expression of the permease that ensures a high influx of inducer to maintain the operon's expression in the ‘on' state. This phenomenon of adaptive enzyme production has been widely studied since its discovery by Jacques Monod and François Jacob and is one of the most famous and best characterized examples of transcriptional gene regulation. E. coli lactose operon is also a paradigm of cellular differentiation. Indeed, in the presence of an intermediate concentration of TMG, an isogenic bacterial population is divided in two subpopulations of cells with the operon's genes either turned on or remaining off. The differentiation step is generally hypothesized to depend on fluctuations in expression of the operon's proteins. Nevertheless, it is still poorly characterized. On the basis of experimental and theoretical approaches, we explored the determinants of cell fate in this system.
We designed a microfluidic device allowing the observation of single cells growing within a microcolony under conditions that can be changed at will. We used this setup to study phenotypic variability in the lactose operon induction under conditions leading to a transient bimodality of lac expression in the population. We used an E. coli strain modified to express the yellow fluorescent protein (YFP) and the cyan fluorescent protein (CFP), both under the control of a promoter regulated by LacI (PLlacO1). Therefore, yellow and cyan fluorescence intensities both represent the concentration of active repressor molecules and indirectly, the expression state of the lactose operon. Microcolonies originating from a single cell were grown in the microfluidic device and followed by time-lapse microscopy. During the first generations of growth, cells were grown in the absence (or with a very low concentration) of inducer and after several generations, TMG was introduced at intermediate concenteration into the medium and maintained thereafter. In the absence of TMG, cells exhibit an overall weak fluorescence yet with significant variations between cells that were shown to correspond in part to the variability in the intracellular concentration of active LacI molecules. Upon induction, transient bimodality is observed, as the cells are divided between two subpopulations of bright and dim fluorescence.
We found a strong clustering of induced cells within their genealogical trees, indicating a substantial epigenetic inheritance. This epigenetic inheritance can be traced back up to two generations prior induction, suggesting that some molecular determinants of cell fate are epigenetically inherited with a short-range memory lasting around two divisions.
The promoter used to control fluorescence proteins expression is sensitive to small variations in active LacI concentration. Thus, in the absence of inducer, these variations result into correlated variations of YFP and CFP levels. We used the arithmetical mean of yellow and cyan fluorescence intensities to estimate the concentration of active LacI in the cells. We found that the cells exhibiting a low LacI concentration before induction are more likely to be induced upon TMG introduction. Likewise, the slowly growing cells were found to have a higher switching probability than the fast-growing ones. We used a multivariate analysis based on a generalized linear model to estimate the correlations of pre-induction LacI concentration and growth rate with the switching probability (Figure 5C). This analysis confirms that both LacI concentration and growth rate are correlated with the switching probability and demonstrates that even though LacI concentration and growth rate can be linked, their correlations with the switching probability represent independent effects. Together, these effects can account for 90% of the observed switching events.
To gain a better understanding of the possible influence of LacI expression fluctuations and growth rate on the switching probability of a cell, we used a model consisting in a system of differential equations and describing the dynamics of the lactose utilization network. In this model, LacI concentration controls the basal level of expression of the operon and the growth rate acts through the dilution of intracellular components. According to this model, depending on both LacI concentration and growth rate, a cell can be in a monostable or bistable regime. Therefore, monostable and bistable cells can coexist in the population due to parameters' variability. In addition, for cells in the bistable regime, the size of the minimal LacY burst necessary to trigger induction increases with LacI concentration and growth rate. Thus, in agreement with our experimental results, these two variables control the sensitivity of the cell to permease bursts and therefore influence its switching probability.
We thus found pre-disposing factors governing the lactose operon switching in a regime of transient bimodality. Some factors, such as LacI and LacY concentrations result from stochasticity at the local level of the network. On the contrary, growth rate variability represents variations in the cell global physiology. Therefore, the effects of local stochasticity are coupled with the influence of the global physiology, demonstrating the importance of considering the embedding of a particular genetic network in the whole cellular physiology to understand fully its dynamics.
The lactose operon regulation in Escherichia coli is a primary model of phenotypic switching, reminiscent of cell fate determination in higher organisms. Under conditions of bistability, an isogenic cell population partitions into two subpopulations, with the operon's genes turned on or remaining off. It is generally hypothesized that the final state of a cell depends solely on stochastic fluctuations of the network's protein concentrations, particularly on bursts of lactose permease expression. Nevertheless, the mechanisms underlying the cell switching decision are not fully understood. We designed a microfluidic system to follow the formation of a transiently bimodal population within growing microcolonies. The analysis of genealogy and cell history revealed the existence of pre-disposing factors for switching that are epigenetically inherited. Both the pre-induction expression stochasticity of the lactose operon repressor LacI and the cellular growth rate are predictive factors of the cell's response upon induction, with low LacI concentration and slow growth correlating with higher switching probability. Thus, stochasticity at the local level of the network and global physiology are synergistically involved in cell response determination.
doi:10.1038/msb.2010.12
PMCID: PMC2872608  PMID: 20393577
adaptation; bistability; differentiation; lac operon; stochastic gene expression
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.
doi:10.1038/msb.2010.2
PMCID: PMC2858441  PMID: 20212522
bistable switch; coupling modules; genetic sequential logic circuit; NOR gate; push-on push-off switch
3.  Adaptation by stochastic switching of a monostable genetic circuit in Escherichia coli 
Stochastic switching of a bistable genetic circuit represents a potential cost-saving strategy for adaptation to environmental challenges. This study reports that stochastic switching of a monostable circuit can be sufficient to mediate reversible adaptation in E. coli.
Stochastic switching of a monostable circuit mediated the adaptation of the engineered OSU12-hisC Escherichia coli strain to histidine starvation.The population shift of OSU12-hisC was accompanied by growth recovery and was reversible upon histidine addition. This is the first report of adaptation mediated by stochastic switching based on a monostable structure.Weak directionality in stochastic switching initiated the population shift and the fast growth of the occasionally appearing fit cells drove the later stages of adaptation.Adaptation of OSU12-hisC was resulted from the enhanced expression of the structural genes within the native His operon, along with the transcriptional reorganization of a large number of genes.
The fundamental mechanisms underlying adaptations can be divided into responsive switching and stochastic switching (Kussell and Leibler, 2005). Responsive switching is generally considered as resulting from evolved regulatory units, such as operons and regulons, which enable immediate adaptation (Jacob and Monod, 1961). However, as cells are subject to a wide range of both genetic and environmental perturbations that damage the specificity or efficiency of regulatory systems (Carroll, 2005; Crombach and Hogeweg, 2008), the limited number of regulatory units that can evolve and remain functional may not be sufficient to completely protect cell populations from the danger of extinction. Whether and how cells are able to survive external perturbations, when the corresponding regulatory units are absent or have been genetically disrupted, is an open question of great importance.
Recent studies showed the stochastic switching provided cells a huge potential for sustenance under severe conditions via a so-called ‘bet-hedging' strategy. The experimental evidence was generally based on a bistable genetic structure that fixed stochastically appearing fit state thus limiting further random switching (Kussell and Leibler, 2005; Acar et al, 2008). In contrast to bistable gene expression, monostable gene expression is much more common (Newman et al, 2006) and does not rely on a specific complex genetic architecture. Since a monostable structure has no fixation effect, the fit cells that would appear stochastically tend to return to the original steady state (i.e., unfit state). To achieve a population shift from a maladaptive state (but stable) to an adaptive state (but unstable), a significant increase in fitness (i.e., growth rate) of the fit cells is necessary. Otherwise, the random switching will mask occasionally occurring adaptive transitions and lead to an unchanged population at the stable but maladaptive state. Whether adaptation can be achieved by stochastic switching based on a monostable structure is however an open issue.
To address this question, we applied an engineered E. coli strain, OSU12-hisC, carrying a foreign gene circuit encompassing a physiologically functional gene, hisC, replaced from its native chromosomal locus (Figure 1A). Consequently, hisC in OSU12-hisC is no longer responsive to the native regulation (His operon) that senses histidine depletion. Instead, the foreign gene circuit provided a monostable structure for hisC's stochastic switching. The green fluorescent protein (gfpuv5) was co-expressed with hisC for the quantitative evaluation of HisC in single cells. The upstream regulation of TetR, whose expression level was reported by the red fluorescent protein (dsred.T4), was introduced to achieve the inducible GFP (HisC) level. The full induction of TetR by IPTG was applied to avoid any possible upstream noise that caused by the abundance of endogenous LacI.
Microscopic observation revealed that the OSU12-hisC cells showed stronger green fluorescence after histidine depletion (Figure 1B), which suggested an increased expression level of hisC. Population analysis using flow cytometry showed that the distributions of both GFP concentration and GFP bias (GFP/RFP ratio) in OSU12-hisC shifted towards a higher level in histidine-free conditions (Figure 1C and D), whereas, the depletion caused only a slight change in distributions of OSU11, a control strain carrying both the same engineered genetic circuit and an intact His operon, including the hisC gene in its native context. Repeated experiments revealed that the increases in both GFP concentration (∼2.1 folds) and GFP bias (∼1.5 folds) due to histidine depletion were highly significant (P<0.005, N=6) in OSU12-hisC. In particular, the increased GFP bias strongly suggested that the change in gene expression occurred specifically in the rewired hisC (i.e., GFP) but not in all genes (e.g., RFP). Furthermore, both the growth recovery accompanied population shift and the stress relaxation triggered restoration were clearly observed. It strongly indicated that the adaptation was mediated by stochastic switching of hisC under the monostable control.
Analysis on microcolonies' formation (Figure 4A) showed stochastic behaviour and directionality in individual cells. Variation in cellular GFP level was clearly observed in individual cells. Stochastic switching of hisC was verified according to the random changes in GFP bias along with the cell division under histidine-rich conditions (Figure 4B). On the other hand, the microcolonies formed under the histidine-free conditions tended to the higher level of GFP bias were observed (Figure 4B). The directional tendency favoured the high GFP (HisC) level was evidently detected in the first 2 h after histidine depletion, which resulted in a population shift (Figure 4C). In contrast, the distributions of microcolonies grown in histidine-rich conditions kept steady, due to the randomized directions of stochastic switching (Figure 4C). Further analysis showed that the stochastic fluctuations in the initial state had an important role not only in fate decision (i.e., whether to grow) but also in the directionality of the stochastic switch.
Microarray analysis showed the adaptation of OSU12-hisC was resulted from the enhanced expression of the structural genes within the native His operon, along with the transcriptional reorganization of a large number of genes. In summary, in contrast to bistable structures, the monostable structure used here did not fix the phenotype but allowed the cells to decide where to go. Taken together, the findings suggest that bacteria do not necessarily need to evolve signalling mechanisms to control gene expression appropriately, even for essential genes.
Stochastic switching is considered as a cost-saving strategy for adaptation to environmental challenges. We show here that stochastic switching of a monostable circuit can mediate the adaptation of the engineered OSU12-hisC Escherichia coli strain to histidine starvation. In this strain, the hisC gene was deleted from the His operon and placed under the control of a monostable foreign promoter. In response to histidine depletion, the OSU12-hisC population shifted to a higher HisC expression level, which is beneficial under starving conditions but is not favoured by the monostable circuit. The population shift was accompanied by growth recovery and was reversible upon histidine addition. A weak directionality in stochastic switching of hisC was observed in growing microcolonies under histidine-free conditions. Directionality and fate decision were in part dependent on the initial cellular status. Finally, microarray analysis indicated that OSU12-hisC reorganized its transcriptome to reach the appropriate physiological state upon starvation. These findings suggest that bacteria do not necessarily need to evolve signalling mechanisms to control gene expression appropriately, even for essential genes.
doi:10.1038/msb.2011.24
PMCID: PMC3130557  PMID: 21613982
adaptation; gene regulation; monostability; stochastic switching; transciptome
4.  Origin of bistability underlying mammalian cell cycle entry 
Mammalian cell cycle entry is controlled at the restriction point by a bistable and resettable switch, which is shown to emerge from a minimal gene circuit containing a mutual-inhibition feedback loop between Rb and E2F modules, coupled with a feed-forward loop between Myc and E2F modules.
A minimal gene circuit is identified underlying bistable cell cycle entry in mammalian cells by analyzing all possible link combinations in a simplified Rb–E2F signaling network.This minimal gene circuit contains a mutual-inhibition feedback loop between Rb and E2F modules, coupled with a feed-forward loop between Myc and E2F modules, which forms an AND-gate control of the E2F activation.Experimental disruption of this minimal gene circuit abolishes maintenance of the activated E2F state, supporting its importance for the bistability of Rb–E2F system.This minimal gene circuit suggests basic design principles for the robust control of the bistable cell cycle entry at the R-point.
The Rb–E2F pathway plays a critical role in controlling cell cycle entry and progression in mammalian cells. Deregulation of the RB–E2F pathway is implicated in most, if not all, human cancers (Nevins, 2001; Weinberg, 2007). Recently, we have demonstrated that the Rb–E2F pathway controls cell proliferation at the restriction point (R-point) by functioning as a bistable switch. This bistable switch converts graded and transient growth signals into an all-or-none activation of E2F activity. Once switched ON by growth stimulation, the E2F activity remains ON, even when the growth stimulation is diminished (Yao et al, 2008). Interestingly, the Rb–E2F bistable switch is intrinsically resettable: the steady-state E2F level is at the monostable OFF state at low-serum conditions (Yao et al, 2008). This resettability can provide a tight control over cell growth responses by limiting spontaneous cell cycle entry. Meanwhile, by creating a wide hysteresis loop and correspondingly long time delay, the bistability of the Rb–E2F switch can drive temporally irreversible R-point transition and subsequent cell cycle progression.
The resettable bistability of the Rb–E2F switch provides a mechanistic explanation of the R-point control between quiescence and cell proliferation. However, the design features that underlie this switching property have not been elucidated. Defining these features can provide insights into the essential control mechanism underlying mammalian cell cycle entry. Such a control mechanism could be conserved across various cell types while disrupted in most, if not all, cancer cells.
To this end, we constructed and analyzed a library of mathematical models that encompass all possible circuit designs derived from a simplified Rb–E2F network (Figure 1). We identified a minimal gene circuit that is uniquely robust in generating resettable bistability. This minimal circuit consists of a mutual-inhibition feedback loop between the Rb (RP) and E2F (EE) modules (Figure 1, links 5, 6) and a feed-forward loop between the Myc (MD) and E2F (EE) modules (Figure 1, links 7, 3, 6). These two regulatory motifs form an AND-gate control of E2F activation (system output). Our modeling analysis suggested that the mutual-inhibition feedback loop between the Rb and E2F modules is critical for generating a robust bistable switch. Meanwhile, the feed-forward loop between the Myc and E2F modules and the AND-gate control are critical for the resettability. Underscoring the importance of this model-predicted minimal circuit, targeted disruption of this circuit abolishes maintenance of the activated E2F state, supporting its necessity for the bistability of the Rb–E2F system (Figure 3).
The unique topology of the minimal circuit, by combining a mutual-inhibition feedback loop and a feed-forward loop into an AND-gate control of system output, also contributes to its structural flexibility in creating resettable bistability. This structural flexibility is manifest in the ability of the minimal circuit to often maintain resettable bistability, despite alterations in its network topology. This property could facilitate the system evolvability. Altogether, our study suggested a minimal gene circuit underlying the origin of the resettable bistability in the Rb–E2F network, which controls normal cell cycle entry of mammalian cells. Consistent with this notion, this minimal gene circuit appears targeted and disrupted by frequent mutations in human cancers.
Precise control of cell proliferation is fundamental to tissue homeostasis and differentiation. Mammalian cells commit to proliferation at the restriction point (R-point). It has long been recognized that the R-point is tightly regulated by the Rb–E2F signaling pathway. Our recent work has further demonstrated that this regulation is mediated by a bistable switch mechanism. Nevertheless, the essential regulatory features in the Rb–E2F pathway that create this switching property have not been defined. Here we analyzed a library of gene circuits comprising all possible link combinations in a simplified Rb–E2F network. We identified a minimal circuit that is able to generate robust, resettable bistability. This minimal circuit contains a feed-forward loop coupled with a mutual-inhibition feedback loop, which forms an AND-gate control of the E2F activation. Underscoring its importance, experimental disruption of this circuit abolishes maintenance of the activated E2F state, supporting its importance for the bistability of the Rb–E2F system. Our findings suggested basic design principles for the robust control of the bistable cell cycle entry at the R-point.
doi:10.1038/msb.2011.19
PMCID: PMC3101952  PMID: 21525871
bistable switch; cell cycle checkpoint; design principle; Rb–E2F pathway; robustness
5.  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.
doi:10.1038/msb.2011.49
PMCID: PMC3202791  PMID: 21811230
feedforward motifs; gene dosage and noise; mammalian cells; microRNAs; negative autoregulation
6.  Quantitative analysis of regulatory flexibility under changing environmental conditions 
Day length changes with the seasons in temperate latitudes, affecting the many biological rhythms that entrain to the day/night cycle: we measure these effects on the expression of Arabidopsis clock genes, using RNA and reporter gene readouts, with a new method of phase analysis.Dusk sensitivity is proposed as a simple, natural and general mathematical measure to analyse and manipulate the changing phase of a clock output relative to the change in the day/night cycle.Dusk sensitivity shows how increasing the numbers of feedback loops in the Arabidopsis clock models allows more flexible regulation, consistent with a previously-proposed, general operating principle of biological networks.The Arabidopsis clock genes show flexibility of regulation that is characteristic of a three-loop clock model, validating aspects of the model and the operating principle, but some clock output genes show greater flexibility arising from direct light regulation.
The analysis of dynamic, non-linear regulation with the aid of mechanistic models is central to Systems Biology. This study compares the predictions of mechanistic, mathematical models of the circadian clock with molecular time-series data on rhythmic gene expression in the higher plant Arabidopsis thaliana. Analysis of the models helps us to understand (explain and predict) how the clock gene circuit balances regulation by external and endogenous factors to achieve particular behaviours. Such multi-factorial regulation is ubiquitous in, and characteristic of, living systems.
The Earth's rotation causes predictable changes in the environment, notably in the availability of sunlight for photosynthesis. Many biological processes are driven by the environmental input via sensory pathways, for example, from photoreceptors. Circadian clocks provide an alternative strategy. These endogenous, 24-h rhythms can drive biological processes that anticipate the regular environmental changes, rather than merely responding. Many rhythmic processes have both light and clock control. Indeed, the clock components themselves must balance internal timing with external inputs, because circadian clocks are reset daily through light regulation of one or more clock components. This process of entrainment is complicated by the change in day length. When the times of dawn and dusk move apart in summer, and closer together in winter, does the clock track dawn, track dusk or interpolate between them?
In plants, the clock controls leaf and petal movements, the opening and closing of stomatal pores, the discharge of floral fragrances, and many metabolic activities, especially those associated with photosynthesis. Centuries of physiological studies have shown that these rhythms can behave differently. Flowering in Ipomoea nil (Pharbitis nil, Japanese morning glory) is controlled by a rhythm that tracks the time of dusk, to give a classic example. We showed that two other rhythms associated with vegetative growth track dawn in this species (Figure 5A), so the clock system allows flexible regulation.
The relatively small number of components involved in the circadian clockwork makes it an ideal candidate for mathematical modelling. Molecular genetic studies in a variety of model eukaryotes have shown that the circadian rhythm is generated by a network of 6–20 genes. These genes form feedback loops generating a rhythm in mRNA production. A single negative feedback loop in which a gene encodes a protein that, after several hours, turns off transcription is capable of generating a circadian rhythm, in principle. A single light input can entrain the clock to ‘local time', synchronised with a light–dark cycle. However, real circadian clocks have proven to be more complicated than this, with multiple light inputs and interlocked feedback loops.
We have previously argued from mathematical analysis that multi-loop networks increase the flexibility of regulation (Rand et al, 2004) and have shown that appropriately deployed flexibility can confer functional robustness (Akman et al, 2010). Here we test whether that flexibility can be demonstrated in vivo, in the model plant, A. thaliana. The Arabidopsis clock mechanism comprises a feedback loop in which two partially redundant, myb transcription factors, LATE ELONGATED HYPOCOTYL (LHY) and CIRCADIAN CLOCK ASSOCIATED 1 (CCA1), repress the expression of their activator, TIMING OF CAB EXPRESSION 1 (TOC1). We previously modelled this single-loop circuit and showed that it was not capable of recreating important data (Locke et al, 2005a). An extended, two-loop model was developed to match observed behaviours, incorporating a hypothetical gene Y, for which the best identified candidate was the GIGANTEA gene (GI) (Locke et al, 2005b). Two further models incorporated the TOC1 homologues PSEUDO-RESPONSE REGULATOR (PRR) 9 and PRR7 (Locke et al, 2006; Zeilinger et al, 2006). In these circuits, a morning oscillator (LHY/CCA1–PRR9/7) is coupled to an evening oscillator (Y/GI–TOC1) via the original LHY/CCA1–TOC1 loop.
These clock models, like those for all other organisms, were developed using data from simple conditions of constant light, darkness or 12-h light–12-h dark cycles. We therefore tested how the clock genes in Arabidopsis responded to light–dark cycles with different photoperiods, from 3 h light to 18 h light per 24-h cycle (Edinburgh, 56° North latitude, has 17.5 h light in midsummer). The time-series assays of mRNA and in vivo reporter gene images showed a range of peak times for different genes, depending on the photoperiod (Figure 5C). A new data analysis method, mFourfit, was introduced to measure the peak times, in the Biological Rhythms Analysis Software Suite (BRASS v3.0). None of the genes showed the dusk-tracking behaviour characteristic of the Ipomoea flowering rhythm. The one-, two- and three-loop models were analysed to understand the observed patterns. A new mathematical measure, dusk sensitivity, was introduced to measure the change in timing of a model component versus a change in the time of dusk. The one- and two-loop models tracked dawn and dusk, respectively, under all conditions. Only the three-loop model (Figure 5B) had the flexibility required to match the photoperiod-dependent changes that we found in vivo, and in particular the unexpected, V-shaped pattern in the peak time of TOC1 expression. This pattern of regulation depends on the structure and light inputs to the model's evening oscillator, so the in vivo data supported this aspect of the model. LHY and CCA1 gene expression under short photoperiods showed greater dusk sensitivity, in the interval 2–6 h before dawn, than the three-loop model predicted, so these data will help to constrain future models.
The approach described here could act as a template for experimental biologists seeking to understand biological regulation using dynamic, experimental perturbations and time-series data. Simulation of mathematical models (despite known imperfections) can provide contrasting hypotheses that guide understanding. The system's detailed behaviour is complex, so a natural and general measure such as dusk sensitivity is helpful to focus on one property of the system. We used the measure to compare models, and to predict how this property could be manipulated. To enable additional analysis of this system, we provide the time-series data and experimental metadata online.
The circadian clock controls 24-h rhythms in many biological processes, allowing appropriate timing of biological rhythms relative to dawn and dusk. Known clock circuits include multiple, interlocked feedback loops. Theory suggested that multiple loops contribute the flexibility for molecular rhythms to track multiple phases of the external cycle. Clear dawn- and dusk-tracking rhythms illustrate the flexibility of timing in Ipomoea nil. Molecular clock components in Arabidopsis thaliana showed complex, photoperiod-dependent regulation, which was analysed by comparison with three contrasting models. A simple, quantitative measure, Dusk Sensitivity, was introduced to compare the behaviour of clock models with varying loop complexity. Evening-expressed clock genes showed photoperiod-dependent dusk sensitivity, as predicted by the three-loop model, whereas the one- and two-loop models tracked dawn and dusk, respectively. Output genes for starch degradation achieved dusk-tracking expression through light regulation, rather than a dusk-tracking rhythm. Model analysis predicted which biochemical processes could be manipulated to extend dusk tracking. Our results reveal how an operating principle of biological regulators applies specifically to the plant circadian clock.
doi:10.1038/msb.2010.81
PMCID: PMC3010117  PMID: 21045818
Arabidopsis thaliana; biological clocks; dynamical systems; gene regulatory networks; mathematical models; photoperiodism
7.  Spatial Epigenetic Control of Mono- and Bistable Gene Expression 
PLoS Biology  2010;8(3):e1000332.
Changes in the spatial distribution of regulatory protein binding elements relative to gene coding sequences is sufficient to change gene expression patterns from graded to switch-like.
Bistability in signaling networks is frequently employed to promote stochastic switch-like transitions between cellular differentiation states. Differentiation can also be triggered by antagonism of activators and repressors mediated by epigenetic processes that constitute regulatory circuits anchored to the chromosome. Their regulatory logic has remained unclear. A reaction–diffusion model reveals that the same reaction mechanism can support both graded monostable and switch-like bistable gene expression, depending on whether recruited repressor proteins generate a single silencing gradient or two interacting gradients that flank a gene. Our experiments confirm that chromosomal recruitment of activator and repressor proteins permits a plastic form of control; the stability of gene expression is determined by the spatial distribution of silencing nucleation sites along the chromosome. The unveiled regulatory principles will help to understand the mechanisms of variegated gene expression, to design synthetic genetic networks that combine transcriptional regulatory motifs with chromatin-based epigenetic effects, and to control cellular differentiation.
Author Summary
In the simplest scenario, a gene is expressed when an activator protein binds to its regulatory sequence, and is silenced when the regulatory sequence is bound by a repressor. Many genes are regulated by both activators and repressors, with the response determined by the combined influence of both factors. When the response is monostable graded, expression is finely tuned to a level that reflects the proportion of the bound activator to the bound repressor. Monostable graded systems allow cells to respond precisely to stimuli. If the response is bistable, the response of each cell depends on whether the activator or the repressor wins. Bistable regulation results in the same gene being expressed in some cells and silenced in others, an outcome that promotes cellular differentiation. It remains unclear, however, how different genetic regulatory structures code for monostable graded and bistable responses. We modeled mathematically the behavior of repressors as they bind to and spread their inhibitory effect along genes and found that the spatial distribution of the binding sites determines which response is chosen. If repressors bind both upstream and downstream of the coding sequence, the response is bistable. If they bind only to one side of the coding sequence, the response is monostable. We confirmed our theoretical findings using synthetic genetic constructs in yeast. These findings help to explain how variations in the location of regulatory elements can lead to cellular differentiation and adaption to varying environments.
doi:10.1371/journal.pbio.1000332
PMCID: PMC2838748  PMID: 20305717
8.  Synthetic in vitro transcriptional oscillators 
A fundamental goal of synthetic biology is to understand design principles through engineering biochemical systems.Three in vitro synthetic transcriptional oscillators were constructed and analyzed: a two-node-negative feedback oscillator, an amplified negative-feedback oscillator, and a three-node ring oscillator.The in vitro oscillators are governed by similar design principles as previous theoretical studies and synthetic oscillators in vivo.Because of unintended reactions that arise even without the complexity of living cells, several challenges remain for predictive and robust oscillator performance.
Fundamental goals for synthetic biology are to understand the principles of biological circuitry from an engineering perspective and to establish engineering methods for creating biochemical circuitry to control molecular processes—both in vitro and in vivo (Benner and Sismour, 2005; Adrianantoandro et al, 2006). Here, we make use of a previously proposed class of in vitro biochemical systems, transcriptional circuits, that can be modularly wired into arbitrarily complex networks by changing the regulatory and coding sequence domains of DNA templates (Kim et al, 2006; Subsoontorn et al 2011). Using design motifs for inhibitory and excitatory regulations, three different oscillator designs were constructed and characterized: a two-switch negative-feedback oscillator, loosely analogous to the p53–Mdm2-feedback loop (Bar-Or et al, 2000); the same oscillator augmented with a positive-feedback loop, loosely analogous to a synthetic relaxation oscillator (Atkinson et al, 2003); and a three-switch ring oscillator analogous to the repressilator (Elowitz and Leibler, 2000).
DNA and RNA hybridization reactions (Figure 1B) can be assembled to create either an inhibitable switch (Figure 1A, right and bottom) with a threshold set by the total concentration of its DNA activator strand (Figure 1C, bottom), or an activatable switch (Figure 1A, left and top) with a threshold set by its DNA inhibitor strand concentration (Figure 1C, top). This threshold mechanism is analogous to biological threshold mechanisms such as ‘inhibitor ultrasensitivity' (Ferrell, 1996) and ‘molecular titration' (Buchler and Louis, 2008). Using these design motifs, we constructed a two-switch negative-feedback oscillator (Figure 1A, inset): RNA activator rA1 activates the production of RNA inhibitor rI2 by modulating switch Sw21, while RNA inhibitor rI2, in turn, inhibits the production of RNA activator rA1 by modulating switch Sw12. A total of seven DNA strands are used, in addition to the two enzymes, bacteriophage T7 RNA polymerase and Escherichia coli ribonuclease H. The fact that such a negative-feedback loop can lead to temporal oscillations can be seen from a mathematical model of transcriptional networks. Experimental results showed qualitative agreement with predicted oscillator behavior from simple model simulations.
The fully optimized system revealed five complete oscillation cycles with a nearly 50% amplitude swing (Figure 3A) until, after ∼20 h, the production rate could no longer be sustained in the batch reaction. Gel measurements verified oscillations in RNA concentrations and switch states (Figure 3B and C). However to our surprise, rather than oscillations with constant amplitude and constant mean, the RNA inhibitor concentration builds up after each cycle. An extended mathematical model that incorporated an interference reaction from ‘waste' product (Figure 3B and C) could qualitatively capture this behavior.
Using a new autoregulatory switch Sw11, we added a positive-feedback loop to the two-node oscillator to make an amplified negative feedback oscillator (Design II, Figure 1D). Further, we replaced the excitatory connection of Sw21 by a chain of two inhibitory connections, Sw23 and Sw31, to construct a three-switch ring oscillator (Design III, Figure 1D). All three oscillator designs could be tuned to reach the oscillatory regime in parameter space.
Reassuringly, our in vitro oscillators exhibit several design principles previously observed in vivo. (1) Introducing delay in a simple negative-feedback loop can help achieve stable oscillation (Novák and Tyson, 2008; Stricker et al, 2008). (2) The addition of a positive-feedback self-loop to a negative-feedback oscillator provides access to rich dynamics and improved tunability (Tsai et al, 2008). (3) Oscillations in biochemical ring oscillators (such as the repressilator) are sensitive to parameter asymmetry among individual components (Tuttle et al, 2005). (4) The saturation of degradation machinery and the management of waste products could play an important role.
However, several significant difficulties remain for predictive and robust oscillator performances: limited lifetime of closed batch reactions, interference from waste products, and asymmetry of switch components make quantitative modeling and predictio difficult. As a complementary approach to top-down view of systems biology, cell-free in vitro systems offer a valuable training ground to create and explore increasingly interesting and powerful information-based chemical systems (Simpson, 2006). In vitro oscillators could be used to orchestrate other chemical processes such as DNA nanomachines (Dittmer and Simmel, 2004) and to provide embedded controllers within prototype artificial cells (Noireaux and Libchaber, 2004; Griffiths and Tawfik, 2006).
The construction of synthetic biochemical circuits from simple components illuminates how complex behaviors can arise in chemistry and builds a foundation for future biological technologies. A simplified analog of genetic regulatory networks, in vitro transcriptional circuits, provides a modular platform for the systematic construction of arbitrary circuits and requires only two essential enzymes, bacteriophage T7 RNA polymerase and Escherichia coli ribonuclease H, to produce and degrade RNA signals. In this study, we design and experimentally demonstrate three transcriptional oscillators in vitro. First, a negative feedback oscillator comprising two switches, regulated by excitatory and inhibitory RNA signals, showed up to five complete cycles. To demonstrate modularity and to explore the design space further, a positive-feedback loop was added that modulates and extends the oscillatory regime. Finally, a three-switch ring oscillator was constructed and analyzed. Mathematical modeling guided the design process, identified experimental conditions likely to yield oscillations, and explained the system's robust response to interference by short degradation products. Synthetic transcriptional oscillators could prove valuable for systematic exploration of biochemical circuit design principles and for controlling nanoscale devices and orchestrating processes within artificial cells.
doi:10.1038/msb.2010.119
PMCID: PMC3063688  PMID: 21283141
cell free; in vitro; oscillation; synthetic biology; transcriptional circuits
9.  Noise Contributions in an Inducible Genetic Switch: A Whole-Cell Simulation Study 
PLoS Computational Biology  2011;7(3):e1002010.
Stochastic expression of genes produces heterogeneity in clonal populations of bacteria under identical conditions. We analyze and compare the behavior of the inducible lac genetic switch using well-stirred and spatially resolved simulations for Escherichia coli cells modeled under fast and slow-growth conditions. Our new kinetic model describing the switching of the lac operon from one phenotype to the other incorporates parameters obtained from recently published in vivo single-molecule fluorescence experiments along with in vitro rate constants. For the well-stirred system, investigation of the intrinsic noise in the circuit as a function of the inducer concentration and in the presence/absence of the feedback mechanism reveals that the noise peaks near the switching threshold. Applying maximum likelihood estimation, we show that the analytic two-state model of gene expression can be used to extract stochastic rates from the simulation data. The simulations also provide mRNA–protein probability landscapes, which demonstrate that switching is the result of crossing both mRNA and protein thresholds. Using cryoelectron tomography of an E. coli cell and data from proteomics studies, we construct spatial in vivo models of cells and quantify the noise contributions and effects on repressor rebinding due to cell structure and crowding in the cytoplasm. Compared to systems without spatial heterogeneity, the model for the fast-growth cells predicts a slight decrease in the overall noise and an increase in the repressors rebinding rate due to anomalous subdiffusion. The tomograms for E. coli grown under slow-growth conditions identify the positions of the ribosomes and the condensed nucleoid. The smaller slow-growth cells have increased mRNA localization and a larger internal inducer concentration, leading to a significant decrease in the lifetime of the repressor–operator complex and an increase in the frequency of transcriptional bursts.
Author Summary
Expressing genes in a bacterial cell is noisy and random. A colony of bacteria grown from a single cell can show remarkable differences in the copy number per cell of a given protein after only a few generations. In this work we use computer simulations to study the variation in how individual cells in a population express a set of genes in response to an environmental signal. The modeled system is the lac genetic switch that Escherichia coli uses to find, collect, and process lactose sugar from the environment. The noise inherent in the genetic circuit controlling the cell's response determines how similar the cells are to each other and we study how the different components of the circuit affect this noise. Furthermore, an estimated 30–50% of the cell volume is taken up by a wide variety of large biomolecules. To study the response of the circuit caused by crowding, we simulate the circuit inside a three-dimensional model of an E. coli cell built using data from cryoelectron tomography reconstructions of a single cell and proteomics data. Correctly including random effects of molecular crowding will be critical to developing fully dynamic models of living cells.
doi:10.1371/journal.pcbi.1002010
PMCID: PMC3053318  PMID: 21423716
10.  Dissecting specific and global transcriptional regulation of bacterial gene expression 
An experimental-computational approach is applied to dissect the contribution of specific transcription factor-mediated versus global growth-dependent regulation to bacterial gene expression, and obtain a quantitative understanding of dynamic adaptations in arginine biosynthesis of E. coli.
We present a model-based approach to quantitatively dissect simultaneous contributions from specific transcription factors and the global growth status to bacterial gene expression, based on parameter inference from GFP-based promoter activity measurements.We show that growth rate can be used to predict the unregulated expression baseline of a gene, since growth rate dependence of global regulation occurs both in steady state and during transient changes in growth rate.We obtain a quantitative understanding of both specific and global regulation in arginine biosynthesis, as demonstrated by accurate model-based predictions of complex transient gene-expression responses to simultaneous perturbation in growth rate and arginine availability.We uncover two principles of joint regulation of the arginine biosynthesis pathway: (i) specific regulation by repression dominates in steady metabolic states and (ii) global regulation sets the maximal expression reachable during transition between steady metabolic states.
Gene expression is regulated by specific transcriptional circuits but also by the global expression machinery as a function of growth. Simultaneous specific and global regulation thus constitutes an additional—but often neglected—layer of complexity in gene expression. Here, we develop an experimental-computational approach to dissect specific and global regulation in the bacterium Escherichia coli. By using fluorescent promoter reporters, we show that global regulation is growth rate dependent not only during steady state but also during dynamic changes in growth rate and can be quantified through two promoter-specific parameters. By applying our approach to arginine biosynthesis, we obtain a quantitative understanding of both specific and global regulation that allows accurate prediction of the temporal response to simultaneous perturbations in arginine availability and growth rate. We thereby uncover two principles of joint regulation: (i) specific regulation by repression dominates the transcriptional response during metabolic steady states, largely repressing the biosynthesis genes even when biosynthesis is required and (ii) global regulation sets the maximum promoter activity that is exploited during the transition between steady states.
doi:10.1038/msb.2013.14
PMCID: PMC3658269  PMID: 23591774
expression machinery; modelling; synthetic biology; transcriptional circuit; transcriptional regulation
11.  Transcriptional Dynamics of the Embryonic Stem Cell Switch 
PLoS Computational Biology  2006;2(9):e123.
Recent ChIP experiments of human and mouse embryonic stem cells have elucidated the architecture of the transcriptional regulatory circuitry responsible for cell determination, which involves the transcription factors OCT4, SOX2, and NANOG. In addition to regulating each other through feedback loops, these genes also regulate downstream target genes involved in the maintenance and differentiation of embryonic stem cells. A search for the OCT4–SOX2–NANOG network motif in other species reveals that it is unique to mammals. With a kinetic modeling approach, we ascribe function to the observed OCT4–SOX2–NANOG network by making plausible assumptions about the interactions between the transcription factors at the gene promoter binding sites and RNA polymerase (RNAP), at each of the three genes as well as at the target genes. We identify a bistable switch in the network, which arises due to several positive feedback loops, and is switched on/off by input environmental signals. The switch stabilizes the expression levels of the three genes, and through their regulatory roles on the downstream target genes, leads to a binary decision: when OCT4, SOX2, and NANOG are expressed and the switch is on, the self-renewal genes are on and the differentiation genes are off. The opposite holds when the switch is off. The model is extremely robust to parameter changes. In addition to providing a self-consistent picture of the transcriptional circuit, the model generates several predictions. Increasing the binding strength of NANOG to OCT4 and SOX2, or increasing its basal transcriptional rate, leads to an irreversible bistable switch: the switch remains on even when the activating signal is removed. Hence, the stem cell can be manipulated to be self-renewing without the requirement of input signals. We also suggest tests that could discriminate between a variety of feedforward regulation architectures of the target genes by OCT4, SOX2, and NANOG.
Synopsis
One key issue in developmental biology is how embryonic stem cells are regulated at the genetic level. Recent high throughput experiments have elucidated the architecture of the gene regulatory network responsible for embryonic stem cell fate decisions in human and mouse. In this work the authors develop a computational model to describe the mutual regulation of the genes involved in these networks and their subsequent effects on downstream target genes. They find that the core genetic network incorporates the functionality of a bistable switch, which arises due to positive feedback loops in the system. Also, this switch behaviour is very robust with respect to model parameters. The switch and architecture by which the genetic network regulates the downstream genes, is responsible for either maintaining the genes responsible for self-renewal on, and genes involved with differentiation off, or the opposite outcome, depending on whether the switch is on/off, respectively. The model also provides several predictions which can lead to further understanding of the network. The methods employed are fairly standard and transparent which facilitates further uncovering in future experimental investigations of genetic networks.
doi:10.1371/journal.pcbi.0020123
PMCID: PMC1570179  PMID: 16978048
12.  Bacterial adaptation through distributed sensing of metabolic fluxes 
We present a large-scale differential equation model of E. coli's central metabolism and its enzymatic, transcriptional, and posttranslational regulation. This model reproduces E. coli's known physiological behavior.We found that the interplay of known interactions in E. coli's central metabolism can indirectly recognize the presence of extracellular carbon sources through measuring intracellular metabolic flux patterns.We found that E. coli's system-level adaptations between glycolytic and gluconeogenic carbon sources are realized on the molecular level by global feedback architectures that overarch the enzymatic and transcriptional regulatory layers.We found that the capability for closed-loop self-regulation can emerge within metabolism itself and therefore, metabolic operation may adapt itself autonomously to changing carbon sources (not requiring upstream sensing and signaling).
Adaptations to fluctuating carbon source availability are of particular importance for bacteria. To understand these adaptations, it needs to be understood how a system's behavior emerges from the interactions between the characterized molecules (Kitano, 2002b). To attain such a system understanding of bacterial metabolic adaptations to carbon source availability, the coupling between the recognition and adjustment aspects and between the enzymatic and genetic regulatory layers must be understood. For many carbon sources, neither transmembrane sensors nor regulatory proteins with sensing function have been identified. Also, it remains unclear how multiple local regulations work together to accomplish a coherent adjustment on the systems level. In this paper, we show that (1) the interplay of the known interactions in E. coli's central metabolism is capable of recognizing carbon sources indirectly, and that (2) these molecular interactions can adjust E. coli's metabolic operation between growth on glycolytic and gluconeogenic carbon sources, and that (3) this adaptation is governed by general principles.
We hypothesized that the system-level adaptations between growth on glycolytic and gluconeogenic carbon sources are accomplished by a system-wide regulation architecture that emerges when the known enzymatic and transcriptional regulations become coupled through five transcription factor (TF)–metabolite interactions. To (1) assess whether such coupled molecular interactions can indeed work together to adapt metabolic operation, and if yes, (2) to understand this system-level adaptation in molecular-level detail, we constructed a large-scale differential equation model. The model topology comprises the Embden–Meyerhoff pathway, the tricarboxylic acid (TCA) cycle, the glyoxylate (GLX) shunt, the anaplerotic reactions, the diversion of carbon flux to the GLX shunt, the uptake of glucose, the uptake and excretion of acetate, enzymatic regulation, transcriptional regulation by four TFs, and the regulation of these TFs' activities through TF–metabolite interactions. We translated the topology into differential equations by assigning the most appropriate rate law to each interaction. The kinetic model comprises 47 ordinary differential equations and 193 parameters. Parameter values were estimated through application of the ‘divide-and-conquer approach' (Kotte and Heinemann, 2009) on published experimental steady state-omics data sets.
Model simulations reproduce E. coli's known physiological behavior in an environment with fluctuating carbon source availability. But how does the in silico cell recognize acetate without a transmembrane sensor for extracellular acetate or a TF binding to intracellular acetate? Similarly, it is unclear whether the glucose sensing function of the phosphotransferase system is the exclusive mechanism to recognize glucose, or whether this sensing function is integrated into a larger sensing architecture. The model suggests that the recognition is performed indirectly through a mechanism we termed distributed sensing of intracellular metabolic fluxes. This mechanism uses two distinct motifs, which we termed pathway usage and flux direction, to establish defined correlations between metabolic fluxes and the levels of certain, here termed flux-signaling metabolites. The binding of these metabolites to TFs propagates the flux information to the transcriptional regulatory layer. A molecular sensor for intracellular metabolic flux is thus defined as a system of regulations and enzyme kinetics, comprising (1) either of the two motifs pathway usage or flux direction and (2) the binding of the thus established flux-signaling metabolites to TF(s).
As the in silico cell establishes and uses sensors for several intracellular metabolic fluxes, the overall sensing architecture infers the present carbon sources from a pattern of metabolic fluxes and is as such of a distributed nature. The core of this sensing architecture is formed not by transmembrane sensors but by four flux sensors, which establish flux-signaling metabolites according to the two proposed general motifs. These flux sensors use intracellular metabolic flux as a means to correlate the presence of extracellular carbon sources with the levels of intracellular metabolites. The recognition of glucose through the PTS transmembrane complex is embedded as one flux sensor in this distributed sensing architecture; the other three flux sensors function without the help of transmembrane complexes.
The in silico cell achieves the coupling between recognition and adjustment through its TFs, whose activities respond to the available carbon sources and at the same time regulate the expression of target genes. This combined recognition and adjustment, centered on the four TFs, closes four global feedback loops that overarch the metabolic and genetic layers as illustrated in Figure 6. The adaptation of the in silico cell arises from the global feedback loop-embedded, flux sensor-adjusted transcriptional regulation of the four TFs, with each TF performing one part of the overall adaptation. This adaptation incorporates both the influence of the metabolic on the genetic layer, achieved through TF–metabolite interactions, and of the genetic on the metabolic layer, achieved through the impact of adjusted enzyme levels on metabolic fluxes.
The existence of the global feedback architectures challenges the conventional view that top-level regulatory proteins recognize environmental conditions and adjust downstream metabolic operation. It suggests that the capability for closed-loop self-regulation can emerge within metabolism itself and therefore, metabolic operation may adapt itself autonomously (not requiring upstream sensing and regulation) to changing carbon sources.
To conclude, the presented differential equation model of E. coli's central metabolism offers a consistent explanation of how a multitude of known molecular interactions fit into a coherent systems picture; the interactions work together like gear wheels that mesh with one another to adapt central metabolism between growth on the glycolytic substrate glucose and the gluconeogenic substrate acetate. The deduced general functional principles provide the missing link to understand system-level adaptations to carbon sources in molecular-level detail. The proposed principles fall under the umbrella of distributed flux sensing. The flux sensing mechanism entails the binding of TFs to flux-signaling metabolites, which are established through the motifs signaling of pathway usage and signaling of flux direction, and are embedded in global feedback loop architectures. These principles allow an autonomous adaptation of metabolic operation to growth in fluctuating environments.
The recognition of carbon sources and the regulatory adjustments to recognized changes are of particular importance for bacterial survival in fluctuating environments. Despite a thorough knowledge base of Escherichia coli's central metabolism and its regulation, fundamental aspects of the employed sensing and regulatory adjustment mechanisms remain unclear. In this paper, using a differential equation model that couples enzymatic and transcriptional regulation of E. coli's central metabolism, we show that the interplay of known interactions explains in molecular-level detail the system-wide adjustments of metabolic operation between glycolytic and gluconeogenic carbon sources. We show that these adaptations are enabled by an indirect recognition of carbon sources through a mechanism we termed distributed sensing of intracellular metabolic fluxes. This mechanism uses two general motifs to establish flux-signaling metabolites, whose bindings to transcription factors form flux sensors. As these sensors are embedded in global feedback loop architectures, closed-loop self-regulation can emerge within metabolism itself and therefore, metabolic operation may adapt itself autonomously (not requiring upstream sensing and signaling) to fluctuating carbon sources.
doi:10.1038/msb.2010.10
PMCID: PMC2858440  PMID: 20212527
computational model; metabolism; regulation; sensing; systems biology
13.  Extension of a genetic network model by iterative experimentation and mathematical analysis 
Molecular Systems Biology  2005;1:2005.0013.
We extend the current model of the plant circadian clock, in order to accommodate new and published data. Throughout our model development we use a global parameter search to ensure that any limitations we find are due to the network architecture and not to our selection of the parameter values, which have not been determined experimentally. Our final model includes two, interlocked loops of gene regulation and is reminiscent of the circuit structures previously identified by experiments on insect and fungal clocks. It is the first Arabidopsis clock model to show such good correspondence to experimental data.Our interlocked feedback loop model predicts the regulation of two unknown components. Experiments motivated by these predictions identify the GIGANTEA gene as a strong candidate for one component, with an unexpected pattern of light regulation.*
This study involves an iterative approach of mathematical modelling and experiment to develop an accurate mathematical model of the circadian clock in the higher plant Arabidopsis thaliana. Our approach is central to systems biology and should lead to a greater, quantitative understanding of the circadian clock, as well as being more widely relevant to research into genetic networks.
The day–night cycle caused by the Earth's rotation affects most organisms, and has resulted in the evolution of the circadian clock. The circadian clock controls 24-h rhythms in processes from metabolism to behaviour; in higher eukaryotes, the circadian clock controls the rhythmic expression of 5–10% of genes. 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.
The relatively small number of components involved in the central circadian network makes it an ideal candidate for mathematical modelling of complex biological regulation. Genetic studies in a variety of model organisms have shown that the circadian rhythm is generated by a central network of between 6 and 12 genes. These genes form feedback loops generating a rhythm in mRNA production. One negative feedback loop in which a gene encodes a protein that, after several hours, turns off transcription is, in principle, capable of creating a circadian rhythm. However, real circadian clocks have proven to be more complicated than this, with interlocked feedback loops. Networks of this complexity are more easily understood through mathematical modelling.
The clock mechanism in the model plant, A. thaliana, was first proposed to comprise a feedback loop in which two partially redundant genes, 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 preliminary network and showed that it was not capable of recreating several important pieces of experimental data (Locke et al, 2005). Here, we extend the LHY/CCA1–TOC1 network in new mathematical models. To check the effects of each addition to the network, the outputs of the extended models are compared to published data and to new experiments.
As is the case for most biological networks, the parameter values in our model, such as the translation rate of TOC1 protein, are unknown. We employ here an optimisation method, which works well with noisy and varied data and allows a global search of parameter space. This should ensure that the limitations we find in our networks are due to the network structure, and not to our parameter choices.
Our final interlocked feedback loop model requires two hypothetical components, genes X and Y (Figure 4), but is the first Arabidopsis clock model to exhibit such a good correspondence with experimental data. The model simulates a residual short-period oscillation in the cca1;lhy mutant, as characterised by our experiments. No single-loop model is able to do this. Our model also matches experimental data under constant light (LL) conditions and correctly senses photoperiod. The model predicts an interlocked feedback loop structure similar to that seen in the circadian clock mechanisms of other organisms.
The interlocked feedback loop model predicts a distinctive pattern of Y mRNA accumulation in the wild type (WT) and in the cca1;lhy double mutant, with Y mRNA levels increasing transiently at dawn. We designed an experiment to identify Y based on this prediction. GIGANTEA (GI) mRNA levels fit very well to our predicted profile for Y (Figure 6), identifying GI as a strong candidate for Y.
The approach described here could act as a template for experimental biologists seeking to extend models of small genetic networks. Our results illustrate the usefulness of mathematical modelling in guiding experiments, even if the models are based on limited data. Our method provides a way of identifying suitable candidate networks and quantifying how these networks better describe a wide variety of experimental measurements. The characteristics of new putative genes are thereby obtained, facilitating the experimental search for new components. To facilitate future experimental design, we provide user-friendly software that is specifically designed for numerical simulation of circadian experiments using models for several species (Brown, 2004b).
*Footnote: Synopsis highlights were added on 5 July 2005.
Circadian clocks involve feedback loops that generate rhythmic expression of key genes. Molecular genetic studies in the higher plant Arabidopsis thaliana have revealed a complex clock network. The first part of the network to be identified, a transcriptional feedback loop comprising TIMING OF CAB EXPRESSION 1 (TOC1), LATE ELONGATED HYPOCOTYL (LHY) and CIRCADIAN CLOCK ASSOCIATED 1 (CCA1), fails to account for significant experimental data. We develop an extended model that is based upon a wider range of data and accurately predicts additional experimental results. The model comprises interlocking feedback loops comparable to those identified experimentally in other circadian systems. We propose that each loop receives input signals from light, and that each loop includes a hypothetical component that had not been explicitly identified. Analysis of the model predicted the properties of these components, including an acute light induction at dawn that is rapidly repressed by LHY and CCA1. We found this unexpected regulation in RNA levels of the evening-expressed gene GIGANTEA (GI), supporting our proposed network and making GI a strong candidate for this component.
doi:10.1038/msb4100018
PMCID: PMC1681447  PMID: 16729048
biological rhythms; gene network; mathematical modelling; parameter estimation
14.  Hidden hysteresis – population dynamics can obscure gene network dynamics 
Background
Positive feedback is a common motif in gene regulatory networks. It can be used in synthetic networks as an amplifier to increase the level of gene expression, as well as a nonlinear module to create bistable gene networks that display hysteresis in response to a given stimulus. Using a synthetic positive feedback-based tetracycline sensor in E. coli, we show that the population dynamics of a cell culture has a profound effect on the observed hysteretic response of a population of cells with this synthetic gene circuit.
Results
The amount of observable hysteresis in a cell culture harboring the gene circuit depended on the initial concentration of cells within the culture. The magnitude of the hysteresis observed was inversely related to the dilution procedure used to inoculate the subcultures; the higher the dilution of the cell culture, lower was the observed hysteresis of that culture at steady state. Although the behavior of the gene circuit in individual cells did not change significantly in the different subcultures, the proportion of cells exhibiting high levels of steady-state gene expression did change.
Although the interrelated kinetics of gene expression and cell growth are unpredictable at first sight, we were able to resolve the surprising dilution-dependent hysteresis as a result of two interrelated phenomena - the stochastic switching between the ON and OFF phenotypes that led to the cumulative failure of the gene circuit over time, and the nonlinear, logistic growth of the cell in the batch culture.
Conclusions
These findings reinforce the fact that population dynamics cannot be ignored in analyzing the dynamics of gene networks. Indeed population dynamics may play a significant role in the manifestation of bistability and hysteresis, and is an important consideration when designing synthetic gene circuits intended for long-term application.
doi:10.1186/1754-1611-7-16
PMCID: PMC3700772  PMID: 23800122
15.  Data assimilation constrains new connections and components in a complex, eukaryotic circadian clock model 
Integrating molecular time-series data resulted in a more robust model of the plant clock, which predicts that a wave of inhibitory PRR proteins controls the morning genes LHY and CCA1.PRR5 is experimentally validated as a late-acting component of this wave.The family of sequentially expressed PRR proteins allows flexible entrainment of the clock, whereas a single protein could not, suggesting that the duplication of clock genes might confer this generic, functional advantage.The observed post-translational regulation of the evening protein TOC1 by interaction with ZTL and GI remains consistent with an indirect activation of TOC1 mRNA expression by GI, which was previously postulated from modelling.
Circadian rhythms are present in most eukaryotic organisms including plants. The core genes of the circadian clock are very important for plant physiology as they drive the rhythmic expression of around 30% of Arabidopsis genes (Edwards et al, 2006; Michael et al, 2008). The clock is normally entrained by daily environmental changes in light and temperature. Oscillations also persist under constant environmental conditions in a laboratory. The clock gene circuit in Arabidopsis is based on multiple interlocked feedback loops, which are typical of circadian genetic networks in other organisms (Dunlap and Loros, 2004; Bell-Pedersen et al, 2005). Mechanistic, mathematical models are increasingly useful in analysing and understanding how the observed molecular components give rise to the rhythmic behaviour of this dynamic, non-linear system.
Our previous model of Arabidopsis circadian clock (Locke et al, 2006) presented the core, three-loop structure of the clock, which comprised morning and evening oscillators and coupling between them (Figure 1). The morning loop included the dawn-expressed LATE ELONGATED HYPOCOTYL (LHY) and CIRCADIAN CLOCK ASSOCIATED 1 (CCA1) genes, which negatively regulate their expression through activation of the inhibitor proteins, PSEUDO-RESPONSE REGULATOR 9 (PRR9) and PRR7. These were described by a single, combined model component, PRR9/7. The evening loop included the dusk-expressed gene TIMING OF CAB EXPRESSION 1 (TOC1), which negatively regulates itself through inhibition of a hypothetical activator, gene Y. The evening-expressed gene GIGANTEA (GI) contributes to Y function. The morning and evening loops were connected through inhibition of the evening genes by LHY/CCA1 and activation of LHY/CCA1 expression by a hypothetical evening gene X. Here, we extend the previous model of circadian gene expression (Locke et al, 2006) based on recently published data (Figure 1). The new model retains the good match of our previous model to the large volume of molecular time-series data, and improves the behaviour of the model clock system under a range of light conditions and in a wider range of mutants.
The morning loop was extended by adding a hypothetical clock component, the night inhibitor (NI), which acts together with PRR9 and PRR7 to keep the expression of LHY and CCA1 at low levels over a broad interval spanning dusk. This regulation is important to set the phase of LHY/CCA1 expression at dawn. Data from the literature suggested that the PRR5 gene was a candidate for NI, leading us to predict that the sequentially expressed PRR9, PRR7 and PRR5 proteins together formed a wave of inhibitors of LHY and CCA1. This hypothesis was tested under discriminating light conditions, in which the light interval is replaced with the dawn and dusk pulses of light to form a ‘skeleton photoperiod'. Combining this protocol with mutation of the PRR7 and/or PRR5 genes, our new experimental results validated the model predictions and confirmed that PRR5 contributes to the function that we modelled as NI. During revision of this paper, that result received further experimental support (Nakamichi et al, 2010).
Model simulations revealed the functional importance of the inhibitor wave in entraining the clock to the light/dark cycle. Separating PRR9 from the other inhibitors in the model showed how the strong light activation observed for this gene contributes to more rapid entrainment. The observed, post-translation regulation of all three inhibitor proteins by light (Farre and Kay, 2007; Ito et al, 2007; Kiba et al, 2007) was also included in the model. Light-regulated degradation provides a molecular mechanism to explain the later phase of LHY and CCA1 expression under long photoperiods compared with short photoperiods, in line with experimental observations.
The connection between evening and morning loops was revised by including the inhibition of the morning gene PRR9 by the evening component TOC1, based on the data on TOC1-overexpressing plants (Makino et al, 2002; Ito et al, 2005). This inhibition causes a delay of PRR9 expression relative to LHY/CCA1, which allows LHY/CCA1 to reach a higher expression level at dawn. Our simulations showed that a partial mutant that lacks this inhibition of PRR9 by TOC1 is sufficient to cause the higher level of PRR9 and the short circadian period observed in toc1 mutant plants.
The evening loop was extended by introducing the observed, post-translational regulation of the TOC1 protein by the F-box protein ZEITLUPE (ZTL) and stabilization of ZTL by its interaction with GI in the presence of light (Kim et al, 2007). GI's function in the clock model has thus been revised according to the data: GI promotes an inhibition of TOC1 protein function through positive regulation of ZTL. This results, together with negative regulation of Y by TOC1, in indirect activation of TOC1 mRNA expression by GI, which agrees with our earlier experimental data (Locke et al, 2006). Simulations showed that the post-translational regulation of TOC1 by ZTL and GI results in the observed long period of the ztl mutant and fast dampening of rhythms in the lhy/cca1/gi triple mutant.
This is the first mathematical model that incorporates the observed post-translational regulation into the genetic network of the Arabidopsis clock. In addition to specific, mechanistic insights, the model shows a generic advantage from the duplication of clock genes and their expression at different phases. Such clock gene duplications are observed in eukaryotes with larger genomes, such as the mouse. Analogous, functional duplication can be achieved by differential regulation of a single clock gene in distinct cells, as in Drosophila.
Circadian clocks generate 24-h rhythms that are entrained by the day/night cycle. Clock circuits include several light inputs and interlocked feedback loops, with complex dynamics. Multiple biological components can contribute to each part of the circuit in higher organisms. Mechanistic models with morning, evening and central feedback loops have provided a heuristic framework for the clock in plants, but were based on transcriptional control. Here, we model observed, post-transcriptional and post-translational regulation and constrain many parameter values based on experimental data. The model's feedback circuit is revised and now includes PSEUDO-RESPONSE REGULATOR 7 (PRR7) and ZEITLUPE. The revised model matches data in varying environments and mutants, and gains robustness to parameter variation. Our results suggest that the activation of important morning-expressed genes follows their release from a night inhibitor (NI). Experiments inspired by the new model support the predicted NI function and show that the PRR5 gene contributes to the NI. The multiple PRR genes of Arabidopsis uncouple events in the late night from light-driven responses in the day, increasing the flexibility of rhythmic regulation.
doi:10.1038/msb.2010.69
PMCID: PMC2964123  PMID: 20865009
Arabidopsis thaliana; biological clocks; circadian rhythms; mathematical model; systems biology
16.  A Bistable Switch and Anatomical Site Control Vibrio cholerae Virulence Gene Expression in the Intestine 
PLoS Pathogens  2010;6(9):e1001102.
A fundamental, but unanswered question in host-pathogen interactions is the timing, localization and population distribution of virulence gene expression during infection. Here, microarray and in situ single cell expression methods were used to study Vibrio cholerae growth and virulence gene expression during infection of the rabbit ligated ileal loop model of cholera. Genes encoding the toxin-coregulated pilus (TCP) and cholera toxin (CT) were powerfully expressed early in the infectious process in bacteria adjacent to epithelial surfaces. Increased growth was found to co-localize with virulence gene expression. Significant heterogeneity in the expression of tcpA, the repeating subunit of TCP, was observed late in the infectious process. The expression of tcpA, studied in single cells in a homogeneous medium, demonstrated unimodal induction of tcpA after addition of bicarbonate, a chemical inducer of virulence gene expression. Striking bifurcation of the population occurred during entry into stationary phase: one subpopulation continued to express tcpA, whereas the expression declined in the other subpopulation. ctxA, encoding the A subunit of CT, and toxT, encoding the proximal master regulator of virulence gene expression also exhibited the bifurcation phenotype. The bifurcation phenotype was found to be reversible, epigenetic and to persist after removal of bicarbonate, features consistent with bistable switches. The bistable switch requires the positive-feedback circuit controlling ToxT expression and formation of the CRP-cAMP complex during entry into stationary phase. Key features of this bistable switch also were demonstrated in vivo, where striking heterogeneity in tcpA expression was observed in luminal fluid in later stages of the infection. When this fluid was diluted into artificial seawater, bacterial aggregates continued to express tcpA for prolonged periods of time. The bistable control of virulence gene expression points to a mechanism that could generate a subpopulation of V. cholerae that continues to produce TCP and CT in the rice water stools of cholera patients.
Author Summary
Most pathogenic microorganisms infect in a stepwise manner: colonization of host surfaces is followed by invasion and injury of host tissues and, late in the infectious process, dissemination to other hosts occurs. During its residence in the host, the pathogen produces essential virulence determinants and often replicates rapidly, leading to a vast expansion of its biomass. Although this scenario is well established also for Vibrio cholerae, the cause of a potentially fatal diarrheal illness, it has not previously been possible to identify precisely when or where virulence determinants are produced in the intestine. We addressed this question by investigating the expression of virulence genes by individual V. cholerae during infection of the small intestine. Virulence genes were found to be powerfully expressed early in the infectious process by bacteria in close proximity to epithelial surfaces. Increased replication rates were also localized to epithelial surfaces. During later stages of the infection, the population of V. cholerae bifurcates into two fractions: one subpopulation continues to express virulence genes, whereas these genes are silenced in the other subpopulation. The genetic program controlling the continued production of virulence genes may mediate the persistence of a hyper-infectious subpopulation of bacteria in the stools of cholera patients.
doi:10.1371/journal.ppat.1001102
PMCID: PMC2940755  PMID: 20862321
17.  An incoherent regulatory network architecture that orchestrates B cell diversification in response to antigen signaling 
B cell receptor signaling controls the expression of IRF-4, a transcription factor required for B cell differentiation. This study shows that IRF-4 regulates divergent B cell fates via a ‘kinetic-control' mechanism that determines the duration of a transient developmental state.
The intensity of signaling through the B cell receptor controls the level of expression of IRF-4, a transcription factor required for B cell differentiation. The rate of IRF-4 production dictates the extent of antibody gene diversification that B cells undergo upon antigen encounter before differentiating into antibody-secreting plasma cells.Computational modeling and experimental analyses substantiate a model, whereby IRF-4 regulates B cell fate trajectories via a ‘kinetic-control' mechanism.Kinetic control is a process by which B cells pass through an obligate state of variable duration that sets the degree of cellular diversification prior to their terminal differentiation.An incoherent regulatory network architecture, within which IRF-4 is embedded, is the basis for realization of kinetic control.
The generation of a diverse set of pathogen-specific antibodies, with differing affinities and effector functions, by B lymphocytes is essential for efficient protection from many microorganisms. Antibody gene diversification in B cells is mediated by two molecular processes termed class-switch recombination and somatic hypermutation (CSR/SHM) (F1A). The former enables the generation of antibodies with the same antigen-binding specificity, but different effector domains, whereas the latter results in a repertoire of antibodies with a range of affinities for a given antigen containing the same effector domain. CSR/SHM occurs in antigen-activated B cells before their terminal differentiation into plasma cells. The transcription factor IRF-4 is required for CSR/SHM as well as plasma-cell differentiation, with its highest levels of expression being necessary for the latter. IRF-4 acts in the context of a network of regulators that include Blimp-1, Pax5, Bach2 and Bcl-6 (F1B). Despite extensive characterization of these individual factors, how the network responds to sensing of antigen by the B cell antigen receptor (BCR, antibody molecule expressed on cell surface) to regulate the extent of antibody gene diversification and plasma-cell differentiation remains to be addressed.
To address this issue, we assemble a computational model. The model reveals two contrasting scenarios that can underlie B cell fate dynamics. In one case, the initial rate of IRF-4 production controls a binary cell fate choice that involves either going to the CSR/SHM state or to the plasma-cell state; the time spent in the CSR state is relatively insensitive to the initial rate of IRF-4 production (herein called ‘basic bistability'). In the other case, IRF-4 drives all cells through a transient CSR/SHM state, but the initial rate of IRF-4 production sets its duration (‘kinetic control'). Both scenarios predict that increasing the initial rate of IRF-4 production favors the generation of plasma cells at the expense of CSR/SHM, but they differ fundamentally with respect to the underlying gene expression patterns.
To distinguish between these two scenarios experimentally, we utilize two different genetic models. The first involves the B1-8i transgenic mouse whose B cells express a rearranged V187.2 VDJ Ig heavy chain gene segment that is specific for the hapten nitrophenol (NP). The second is a newly developed mouse model that allows exogenous control of IRF-4 expression in naive primary B cells using a tet-inducible allele. Using these models, we show that (i) BCR signal strength sets the initial rate of IRF-4 accumulation and (ii) the concentration of IRF-4 is sensed by an incoherent gene regulatory network architecture to regulate the extent of CSR/SHM prior to plasma-cell differentiation. Our results are consistent with the ‘kinetic-control model' in which the levels of BCR-induced IRF-4 expression control the duration of an obligate CSR/SHM state that enables B cell diversification before terminal differentiation into plasma cells. Evidence for the transient CSR/SHM state is corroborated by both patterns of gene expression and the presence of AID-dependent mutations in individual non-switched plasmablasts.
Our results provide a molecular framework for understanding how B cells balance the competing demands for Ig CSR and SHM with the secretion of antibodies during humoral immune responses. The key feature of the network architecture that allows IRF-4 to coordinate the two competing states of gene expression in a temporal manner is that it simultaneously but asymmetrically activates both sides of a bistable mutual repression circuit. Because the two effects of the primary regulator antagonize each other, we describe the circuit as being based on an ‘incoherent' regulatory motif. Other incoherent regulatory motifs in varied biological systems are also associated with the acquisition of transient cell states, and we consider how the kinetic-control mechanism proposed by us could more generally serve to translate developmental cues into elaborate morphogenetic patterns.
The B-lymphocyte lineage is a leading system for analyzing gene regulatory networks (GRNs) that orchestrate distinct cell fate transitions. Upon antigen recognition, B cells can diversify their immunoglobulin (Ig) repertoire via somatic hypermutation (SHM) and/or class switch DNA recombination (CSR) before differentiating into antibody-secreting plasma cells. We construct a mathematical model for a GRN underlying this developmental dynamic. The intensity of signaling through the Ig receptor is shown to control the bimodal expression of a pivotal transcription factor, IRF-4, which dictates B cell fate outcomes. Computational modeling coupled with experimental analysis supports a model of ‘kinetic control', in which B cell developmental trajectories pass through an obligate transient state of variable duration that promotes diversification of the antibody repertoire by SHM/CSR in direct response to antigens. More generally, this network motif could be used to translate a morphogen gradient into developmental inductive events of varying time, thereby enabling the specification of distinct cell fates.
doi:10.1038/msb.2011.25
PMCID: PMC3130558  PMID: 21613984
BCR signal strength; bistability; gene regulatory network; ghost of a fixed point; Irf4
18.  A genetic bistable switch utilizing nonlinear protein degradation 
Background
Bistability is a fundamental property in engineered and natural systems, conferring the ability to switch and retain states. Synthetic bistable switches in prokaryotes have mainly utilized transcriptional components in their construction. Using both transcriptional and enzymatic components, creating a hybrid system, allows for wider bistable parameter ranges in a circuit.
Results
In this paper, we demonstrate a tunable family of hybrid bistable switches in E. coli using both transcriptional components and an enzymatic component. The design contains two linked positive feedback loops. The first loop utilizes the lambda repressor, CI, and the second positive feedback loop incorporates the Lon protease found in Mesoplasma florum (mf-Lon). We experimentally tested for bistable behavior in exponential growth phase, and found that our hybrid bistable switch was able to retain its state in the absence of an input signal throughout 40 cycles of cell division. We also tested the transient behavior of our switch and found that switching speeds can be tuned by changing the expression rate of mf-Lon.
Conclusions
To our knowledge, this work demonstrates the first use of dynamic expression of an orthogonal and heterologous protease to tune a nonlinear protein degradation circuit. The hybrid switch is potentially a more robust and tunable topology for use in prokaryotic systems.
doi:10.1186/1754-1611-6-9
PMCID: PMC3439342  PMID: 22776405
19.  A synthetic library of RNA control modules for predictable tuning of gene expression in yeast 
The authors describe a library of synthetic RNA control elements that provide programmable post-transcriptional regulation of gene expression in yeast. This toolkit is then used to study endogenous regulation of the ergosterol biosynthetic pathway.
Rnt1p hairpins can act as effective posttranscriptional gene regulatory elements in the yeast Saccharomyces cerevisiae.Modification of the cleavage efficiency box (CEB) region of an Rnt1p hairpin can modulate Rnt1p cleavage rates, and thus the resulting gene regulatory activities of the hairpin control elements.A library of Rnt1p hairpins can act as a set of synthetic control modules that provide predictable tuning of gene expression over a wide range of expression levels.The Rnt1p-based control elements can be combined with any promoter to support titration of regulatory strategies encoded in transcriptional regulators, including feedback control around endogenous proteins.
The design of complex biological systems encoding desired functions require the development of genetic tools for the precise control of protein levels in cells (Elowitz and Leibler, 2000; Gardner et al, 2000; Basu et al, 2004). For example, in the design of engineered metabolic networks, the tuning of enzyme levels is often critical for overcoming metabolic burden (Jones et al, 2000; Jin et al, 2003), the accumulation of toxic intermediates (Zhu et al, 2001; Pfleger et al, 2006) and detrimental consequences associated with the redirection of cellular resources from native pathways (Alper et al, 2005b; Paradise et al, 2008). Various examples of libraries of genetic control modules have been described that have been generated through the randomization of well-characterized gene expression control elements (Basu et al, 2004; Pfleger et al, 2006; Anderson et al, 2007). However, most of these studies have been conducted in Escherichia coli such that there is a lack of similar tools for other cellular chassis.
The budding yeast, Saccharomyces cerevisiae, is a relevant organism in industrial processes, including biosynthesis and biomanufacturing strategies (Ostergaard et al, 2000; Szczebara et al, 2003; Nguyen et al, 2004; Veen and Lang, 2004; Ro et al, 2006; Hawkins and Smolke, 2008). The majority of existing methods for tuning gene expression in yeast are through transcriptional control mechanisms in the form of inducible and constitutive promoter systems (Hawkins and Smolke, 2006; Nevoigt et al, 2006; Nevoigt et al, 2007). RNA-based control modules based on posttranscriptional mechanisms may offer an advantage in that they can be coupled to any promoter of choice, providing for enhanced control strategies and finer resolution tuning of protein expression levels. Although posttranscriptional control elements, such as internal ribosome entry sites and AU-rich elements, have been applied to regulate heterologous gene expression in yeast (Vasudevan and Peltz, 2001; Zhou et al, 2001; Lautz et al, 2010), these control elements have exhibited substantial variability in activity and have not been engineered as synthetic libraries exhibiting a wide range of predictable gene regulatory activities.
RNase III enzymes are a class of enzymes that cleave double-stranded RNA. The S. cerevisiae RNase III enzyme, Rnt1p, exhibits a number of unique features that allow it to recognize very specific RNA hairpin substrates that harbor a consensus AGNN tetraloop sequence. Despite extensive characterization of this enzyme and its demonstrated role in processing non-coding RNA and mRNA, neither natural nor synthetic Rnt1p substrates have been used to control gene expression levels in yeast. Therefore, we developed a genetic control system based on directed Rnt1p processing of a target transcript. Specifically, Rnt1p hairpins were immediately flanked by a clamp sequence (that insulates the hairpin structure from surrounding sequences) and placed downstream of a gene of interest, where they direct cleavage and thus inactivate the transcript, resulting in rapid transcript degradation. We validated this Rnt1p-based control system with two Rnt1p hairpins based on previous in vitro studies and demonstrated that Rnt1p hairpins can act as gene control modules in yeast.
Previous in vitro studies had identified three key regions in Rnt1p hairpins: the cleavage efficiency box (CEB), the binding stability box and the initial binding and positioning box (Lamontagne et al, 2003). The CEB region affects the processing of the hairpin stem by Rnt1p, such that nucleotide (nt) modifications in this region are expected to specifically modulate the cleavage rate. We created an Rnt1p hairpin library by randomizing the CEB region (12 nt). This library was placed downstream of a fluorescent reporter protein and a cell-based screening assay was used to identify functional members of the library that resulted in lowered fluorescence levels. The functional Rnt1p hairpin library comprises 16 unique sequences that span a large gene regulatory range—from 8 to 85% (Figure 3A)—and are fairly evenly distributed across this range. The negative controls for each sequence (constructed by mutating the required consensus tetraloop sequence) demonstrated that the majority of gene knockdown observed from each hairpin is due to Rnt1p processing (Figure 3B). A correlation analysis on the transcript and protein levels for each library hairpin construct indicated a strong positive correlation and a strong preservation of rank order between the two in vivo regulatory measurements (Figure 3C). Characterization of the hairpin library in a different genetic context supported the broader utility of these control modules for providing predictable gene control.
We applied the Rnt1p control modules to titrating a key enzyme component of the endogenous ergosterol biosynthesis network—the ERG9 genetic target. Squalene synthase, encoded by the ERG9 gene, is responsible for catalyzing the conversion of two molecules of farnesyl pyrophosphate to squalene, the first precursor in the ergosterol biosynthetic pathway in S. cerevisiae (Poulter and Rilling, 1981; Figure 6A). We integrated several members of the Rnt1p hairpin library downstream of the native ERG9 gene to cover the regulatory range of the library (Figure 6B). A strong positive correlation and preservation of rank order was observed between the ERG9 transcript levels and their yEGFP3 counterparts (Figure 6C). However, ERG9 expression levels did not fall below ∼40%, regardless of the Rnt1p hairpin strength, indicating that a previously identified endogenous feedback mechanism associated with the native ERG9 promoter acts to maintain ERG9 expression levels at that threshold value. In addition, most strains exhibited high relative ergosterol levels and growth rates, except for two strains harboring synthetic Rnt1p hairpins that resulted in the lowest expression levels, which exhibited a significant reduction in the amount of ergosterol produced and growth rate (Figure 6D and E). Our studies indicate that the endogenous feedback mechanism can be acting to increase ERG9 expression levels to the desired set point in the slow-growing strains, but the perturbations introduced in these strains may result in other impacts on the pathway that inhibit the endogenous control systems from restoring cellular growth to wild-type rates. These studies support the unique ability of the synthetic Rnt1p hairpin library to systematically titrate pathway enzyme levels by introducing precise perturbations around major control points while maintaining native cellular control strategies acting through transcriptional mechanisms.
Advances in synthetic biology have resulted in the development of genetic tools that support the design of complex biological systems encoding desired functions. The majority of efforts have focused on the development of regulatory tools in bacteria, whereas fewer tools exist for the tuning of expression levels in eukaryotic organisms. Here, we describe a novel class of RNA-based control modules that provide predictable tuning of expression levels in the yeast Saccharomyces cerevisiae. A library of synthetic control modules that act through posttranscriptional RNase cleavage mechanisms was generated through an in vivo screen, in which structural engineering methods were applied to enhance the insulation and modularity of the resulting components. This new class of control elements can be combined with any promoter to support titration of regulatory strategies encoded in transcriptional regulators and thus more sophisticated control schemes. We applied these synthetic controllers to the systematic titration of flux through the ergosterol biosynthesis pathway, providing insight into endogenous control strategies and highlighting the utility of this control module library for manipulating and probing biological systems.
doi:10.1038/msb.2011.4
PMCID: PMC3094065  PMID: 21364573
gene expression control; metabolic flux control; RNA controller; Rnt1p hairpin; synthetic biology
20.  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.
doi:10.1038/msb.2010.37
PMCID: PMC2913400  PMID: 20571531
adaptation; feedback; fluorescence resonance energy transfer (FRET); frequency response; Monod–Wyman–Changeux (MWC) model
21.  Loads Bias Genetic and Signaling Switches in Synthetic and Natural Systems 
PLoS Computational Biology  2014;10(3):e1003533.
Biological protein interactions networks such as signal transduction or gene transcription networks are often treated as modular, allowing motifs to be analyzed in isolation from the rest of the network. Modularity is also a key assumption in synthetic biology, where it is similarly expected that when network motifs are combined together, they do not lose their essential characteristics. However, the interactions that a network module has with downstream elements change the dynamical equations describing the upstream module and thus may change the dynamic and static properties of the upstream circuit even without explicit feedback. In this work we analyze the behavior of a ubiquitous motif in gene transcription and signal transduction circuits: the switch. We show that adding an additional downstream component to the simple genetic toggle switch changes its dynamical properties by changing the underlying potential energy landscape, and skewing it in favor of the unloaded side, and in some situations adding loads to the genetic switch can also abrogate bistable behavior. We find that an additional positive feedback motif found in naturally occurring toggle switches could tune the potential energy landscape in a desirable manner. We also analyze autocatalytic signal transduction switches and show that a ubiquitous positive feedback switch can lose its switch-like properties when connected to a downstream load. Our analysis underscores the necessity of incorporating the effects of downstream components when understanding the physics of biochemical network motifs, and raises the question as to how these effects are managed in real biological systems. This analysis is particularly important when scaling synthetic networks to more complex organisms.
Author Summary
Cells rely on complex networks of protein-protein interactions in order to carry out life functions. Scientists believe that these networks are organized in a modular fashion; that is they are made up of functionally distinct parts like an electronic circuit. Modularity implies that just as we put together electronic parts to make an amplifier that we can use in many different circuits, we can put together biochemical reactions to make an amplifier, or a switch or an oscillator, which perform the same function in different organisms. This assumption is important in synthetic biology, where we engineer and assemble synthetic genetic circuits in living organisms in a modular fashion. We show that for important modules like genetic and signaling switches, the assumption of modularity has a crucial limitation. We show that if one simply connects a biological switch to another downstream circuit, the presence of the connection changes the operation of the switch, which in some cases may stop behaving like a switch. Our work underscores the importance of taking into account these downstream connections and suggests that natural systems may be balancing some of these components in order to ensure that despite diversity, modules continue to work in different systems with fidelity.
doi:10.1371/journal.pcbi.1003533
PMCID: PMC3967935  PMID: 24676102
22.  Computational Models of the Notch Network Elucidate Mechanisms of Context-dependent Signaling 
PLoS Computational Biology  2009;5(5):e1000390.
The Notch signaling pathway controls numerous cell fate decisions during development and adulthood through diverse mechanisms. Thus, whereas it functions as an oscillator during somitogenesis, it can mediate an all-or-none cell fate switch to influence pattern formation in various tissues during development. Furthermore, while in some contexts continuous Notch signaling is required, in others a transient Notch signal is sufficient to influence cell fate decisions. However, the signaling mechanisms that underlie these diverse behaviors in different cellular contexts have not been understood. Notch1 along with two downstream transcription factors hes1 and RBP-Jk forms an intricate network of positive and negative feedback loops, and we have implemented a systems biology approach to computationally study this gene regulation network. Our results indicate that the system exhibits bistability and is capable of switching states at a critical level of Notch signaling initiated by its ligand Delta in a particular range of parameter values. In this mode, transient activation of Delta is also capable of inducing prolonged high expression of Hes1, mimicking the “ON” state depending on the intensity and duration of the signal. Furthermore, this system is highly sensitive to certain model parameters and can transition from functioning as a bistable switch to an oscillator by tuning a single parameter value. This parameter, the transcriptional repression constant of hes1, can thus qualitatively govern the behavior of the signaling network. In addition, we find that the system is able to dampen and reduce the effects of biological noise that arise from stochastic effects in gene expression for systems that respond quickly to Notch signaling.
This work thus helps our understanding of an important cell fate control system and begins to elucidate how this context dependent signaling system can be modulated in different cellular settings to exhibit entirely different behaviors.
Author Summary
The Notch signaling pathway is an evolutionarily conserved signaling system that is involved in various cell fate decisions, both during development of an organism and during adulthood. While the same core circuit functions in various different cellular contexts, it has experimentally been shown to elicit varied behaviors and responses. On the one hand, it functions as a cellular oscillator critical for somitogenesis, whereas in other situations, it can function as a cell fate switch to pattern developing tissue, for example in the Drosophila eye. Furthermore, malfunctioning of Notch signaling is implicated in various cancers. To better understand the underlying mechanisms that allow the network to function distinctly in different contexts, we have mathematically modeled the behavior of the Notch network, encompassing the Notch gene along with two of its downstream effector transcription factors, which together form a network of positive and negative feedback loops. Our results indicate that the qualitative and quantitative behavior of the system can readily be tuned based on key parameters to reflect its multiple roles. Furthermore, our results provide insights into alterations in the signaling system that lead to malfunction and hence disease, which could be used to identify potential drug targets for therapy.
doi:10.1371/journal.pcbi.1000390
PMCID: PMC2680760  PMID: 19468305
23.  Dilution and the theoretical description of growth-rate dependent gene expression 
Expression of a gene is not only tuned by direct regulation, but also affected by the global physiological state of the (host) cell. This global dependence complicates the quantitative understanding of gene regulation and the design of synthetic gene circuits. In bacteria these global effects can often be described as a dependence on the growth rate. Here we discuss how growth-rate dependence can be incorporated in mathematical models of gene expression by comparing data for unregulated genes with the predictions of different theoretical descriptions of growth-rate dependence. We argue that a realistic description of growth effects requires a growth-rate dependent protein synthesis rate in addition to dilution by growth.
doi:10.1186/1754-1611-7-22
PMCID: PMC3847955  PMID: 24041253
Genetic circuits; Modeling; Bacterial growth; Dilution; Growth-rate dependence
24.  A model of yeast cell-cycle regulation based on multisite phosphorylation 
Multisite phosphorylation of CDK target proteins provides the requisite nonlinearity for cell cycle modeling using elementary reaction mechanisms.Stochastic simulations, based on Gillespie's algorithm and using realistic numbers of protein and mRNA molecules, compare favorably with single-cell measurements in budding yeast.The role of transcription–translation coupling is critical in the robust operation of protein regulatory networks in yeast cells.
Progression through the eukaryotic cell cycle is governed by the activation and inactivation of a family of cyclin-dependent kinases (CDKs) and auxiliary proteins that regulate CDK activities (Morgan, 2007). The many components of this protein regulatory network are interconnected by positive and negative feedback loops that create bistable switches and transient pulses (Tyson and Novak, 2008). The network must ensure that cell-cycle events proceed in the correct order, that cell division is balanced with respect to cell growth, and that any problems encountered (in replicating the genome or partitioning chromosomes to daughter cells) are corrected before the cell proceeds to the next phase of the cycle. The network must operate robustly in the context of unavoidable molecular fluctuations in a yeast-sized cell. With a volume of only 5×10−14 l, a yeast cell contains one copy of the gene for each component of the network, a handful of mRNA transcripts of each gene, and a few hundreds to thousands of protein molecules carrying out each gene's function. How large are the molecular fluctuations implied by these numbers, and what effects do they have on the functioning of the cell-cycle control system?
To answer these questions, we have built a new model (Figure 1) of the CDK regulatory network in budding yeast, based on the fact that the targets of CDK activity are typically phosphorylated on multiple sites. The activity of each target protein depends on how many sites are phosphorylated. The target proteins feedback on CDK activity by controlling cyclin synthesis (SBF's role) and degradation (Cdh1's role) and by releasing a CDK-counteracting phosphatase (Cdc14). Every reaction in Figure 1 can be described by a mass-action rate law, with an accompanying rate constant that must be estimated from experimental data. As the transcription and translation of mRNA molecules have major effects on fluctuating numbers of protein molecules (Pedraza and Paulsson, 2008), we have included mRNA transcripts for each protein in the model.
To create a deterministic model, the rate laws are combined, according to standard principles of chemical kinetics, into a set of 60 differential equations that govern the temporal dynamics of the control system. In the stochastic version of the model, the rate law for each reaction determines the probability per unit time that a particular reaction occurs, and we use Gillespie's stochastic simulation algorithm (Gillespie, 1976) to compute possible temporal sequences of reaction events. Accurate stochastic simulations require knowledge of the expected numbers of mRNA and protein molecules in a single yeast cell. Fortunately, these numbers are available from several sources (Ghaemmaghami et al, 2003; Zenklusen et al, 2008). Although the experimental estimates are not always in good agreement with each other, they are sufficiently reliable to populate a stochastic model with realistic numbers of molecules.
By simulating thousands of cells (as in Figure 5), we can build up representative samples for computing the mean and s.d. of any measurable cell-cycle property (e.g. interdivision time, size at division, duration of G1 phase). The excellent fit of simulated statistics to observations of cell-cycle variability is documented in the main text and Supplementary Information.
Of particular interest to us are observations of Di Talia et al (2007) of the timing of a crucial G1 event (export of Whi5 protein from the nucleus) in a population of budding yeast cells growing at a specific growth rate α=ln2/(mass-doubling time). Whi5 export is a consequence of Whi5 phosphorylation, and it occurs simultaneously with the release (activation) of SBF (see Figure 1). Using fluorescently labeled Whi5, Di Talia et al could easily measure (in individual yeast cells) the time, T1, from cell birth to the abrupt loss of Whi5 from the nucleus. Correlating T1 to the size of the cell at birth, Vbirth, they found that, for a sample of daughter cells, αT1 versus ln(Vbirth) could be fit with two straight lines of slope −0.7 and −0.3. Our simulation of this experiment (Figure 7 of the main text) compares favorably with Figure 3d and e in Di Talia et al (2007).
The major sources of noise in our model (and in protein regulatory networks in yeast cells, in general) are related to gene transcription and the small number of unique mRNA transcripts. As each mRNA molecule may instruct the synthesis of dozens of protein molecules, the coefficient of variation of molecular fluctuations at the protein level (CVP) may be dominated by fluctuations at the mRNA level, as expressed in the formula (Pedraza and Paulsson, 2008) where NM, NP denote the number of mRNA and protein molecules, respectively, and ρ=τM/τP is the ratio of half-lives of mRNA and protein molecules. For a yeast cell, typical values of NM and NP are 8 and 800, respectively (Ghaemmaghami et al, 2003; Zenklusen et al, 2008). If ρ=1, then CVP≈25%. Such large fluctuations in protein levels are inconsistent with the observed variability of size and age at division in yeast cells, as shown in the simplified cell-cycle model of Kar et al (2009) and as we have confirmed with our more realistic model. The size of these fluctuations can be reduced to a more acceptable level by assuming a shorter half-life for mRNA (say, ρ=0.1).
There must be some mechanisms whereby yeast cells lessen the protein fluctuations implied by transcription–translation coupling. Following Pedraza and Paulsson (2008), we suggest that mRNA gestation and senescence may resolve this problem. Equation (3) is based on a simple, one-stage, birth–death model of mRNA turnover. In Supplementary Appendix 1, we show that a model of mRNA processing, with 10 stages each of mRNA gestation and senescence, gives reasonable fluctuations at the protein level (CVP≈5%), even if the effective half-life of mRNA is 10 min. A one-stage model with τM=1 min gives comparable fluctuations (CVP≈5%). In the main text, we use a simple birth–death model of mRNA turnover with an ‘effective' half-life of 1 min, in order to limit the computational complexity of the full cell-cycle model.
In order for the cell's genome to be passed intact from one generation to the next, the events of the cell cycle (DNA replication, mitosis, cell division) must be executed in the correct order, despite the considerable molecular noise inherent in any protein-based regulatory system residing in the small confines of a eukaryotic cell. To assess the effects of molecular fluctuations on cell-cycle progression in budding yeast cells, we have constructed a new model of the regulation of Cln- and Clb-dependent kinases, based on multisite phosphorylation of their target proteins and on positive and negative feedback loops involving the kinases themselves. To account for the significant role of noise in the transcription and translation steps of gene expression, the model includes mRNAs as well as proteins. The model equations are simulated deterministically and stochastically to reveal the bistable switching behavior on which proper cell-cycle progression depends and to show that this behavior is robust to the level of molecular noise expected in yeast-sized cells (∼50 fL volume). The model gives a quantitatively accurate account of the variability observed in the G1-S transition in budding yeast, which is governed by an underlying sizer+timer control system.
doi:10.1038/msb.2010.55
PMCID: PMC2947364  PMID: 20739927
bistability; cell-cycle variability; size control; stochastic model; transcription–translation coupling
25.  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.
doi:10.1371/journal.pcbi.1002960
PMCID: PMC3610654  PMID: 23555205

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