Intuitive explanation for bistability
A classic bifunctional TCS cannot show bistability, unless an independent Ph that acts upon the phosphorylated RR is present. A classic monofunctional TCS can show a bistable response on a post-translational level. What is the underlying mechanism for this bistability? It is impossible to explain bistability definitively without a rigorous mathematical or numerical analysis of the equations that govern the dynamic behaviour of the system. This analysis can, however, be translated into real biological components and interactions that constitute a mechanism capable of exhibiting bistability.
One requirement for the existence of bistability is the formation of a dead-end complex between the unphosphorylated forms of SK and RR. Our structural analysis of the EnvZ/OmpR TCS suggests that binding of the unphosphorylated RR (OmpR) to one subunit of the SK (EnvZ) dimer may protect the His residue of this subunit from trans-autophosphorylation by the other subunit. If this is the case, then a dead-end complex will be formed; its dissociation is required in order to resume the cycle of SK autophosphorylation → phosphotransfer to RR → dephosphorylation of RR. Moreover, the formation of this complex is self-enhancing, thereby creating a positive feedback loop. The more SK.RR complex formed, the smaller the fraction of phosphorylated RR. In turn, the smaller the fraction of phosphorylated RR, the higher the probability that unphosphorylated RR binds to an unphosphorylated SK prior to autophosphorylation of the SK, thereby forming more dead-end complex.
This positive feedback loop leads us to the second requirement for the existence of bistability: dead-end complex formation should affect the phosphatase reaction less than the autophosphorylation reaction. Otherwise, if the SK is bifunctional, this positive feedback loop will be counteracted by a negative feedback loop: the formation of the dead-end complex also decreases the amount of unphosphorylated SK available to dephosphorylate RR∼P. If no alternative Ph is present and if spontaneous dephosphorylation of RR∼P is much slower than SK-enhanced dephosphorylation of RR∼P, then the two feedback loops exactly counteract one another. Thus, a pathway for RR∼P dephosphorylation that carries a large fraction of the dephosphorylation flux and is independent of the SK also is required for the existence of bistability.
Our simulations indicate that formation of the SK.RR complex does not affect the concentration of RR∼P in the bifunctional system. This observation is consistent with the results of Batchelor and Goulian (2003). Although their model did not include formation of the SK.RR complex, they also found that the concentration of RR∼P is not sensitive to the total concentration of SK if [SK] << [RR]. The formation of the dead-end complex in our model decreases the effective concentration of SK, but this decrease does not affect the systemic output RR∼P. We note that when a strong alternative Ph is present, the contribution of the negative feedback loop created by the formation of the dead-end complex is diminished. Therefore, although a bifunctional SK is capable of achieving a bistable response (data not shown), the system is effectively monofunctional because the phosphatase activity of SK is insignificant.
Physiological consequences of bistability
At the population level, hysteretic bistability has three fundamental physiological consequences. (i) Hysteric switches tend to display ALL-or-NONE steady-state output characteristics. This means that precise continuous tuning of the systemic output to environmental input is not possible. Therefore, we do not expect to see bistability in TCS that are required to respond continuously to a wide range of environmental conditions through a continuous variation of cellular gene expression. Consistent with this idea, bifunctionality of the SK EnvZ prevents bistability in the EnvZ/OmpR system. On the other hand, such ALL-or-NONE responses might be important for developmental switches that require in a digital (yes or no) commitment to differentiation. (ii) The existence of two different switching thresholds and bistability between these can lead to irreversible transitions between the two stable steady states in response to transient signals. Again, such a design would be deleterious if a graded response were required of the sensor, but may be advantageous for switches controlling differentiation and development. (iii) The slow response of bistable switches to signals having amplitudes near the switching threshold allows for filtering of transient fluctuations that otherwise would lead to transitions that could be either physiologically detrimental or metabolically very costly. On the other hand, outside the range of bistability, these systems exhibit a fast response to changes in input signal and a fast recovery from spurious fluctuations in the phosphorylation level of its proteins.
Additionally, the existence of bistability has consequences on a single-cell level. Bistability implies that, within a given population of cells and for a new signal, two alternative states may coexist. It is hard to imagine a situation in which this type of response could have evolved for systems in which a decisive adaptation of the cell to a new state is fundamental for immediate survival (such as osmotic stress). In such a situation, a homogeneous response of the population is expected to maximize survival and therefore bistability should be deleterious. However, one can imagine a situation in which the cell is repeatedly subjected to large changes in the environment, requiring a large and rapid shift in metabolism to ensure survival. Spontaneous transitions that are uncorrelated with the environmental signal in a fraction of the population may be a beneficial strategy when changes in the environment are random and fast (Thattai and van Oudenaarden, 2004; Kussell and Leibler, 2005). Multistable switches in general allow noise-driven transitions between steady states. However, our stochastic simulations reveal a low probability of stochastic transition between the two steady states that result from post-translation regulation in TCS. This observation is in agreement with the observation that transcription is the major source of noise in bacteria (Thattai and van Oudenaarden, 2001; Ozbudak et al., 2002; Kaern et al., 2005). Accordingly, we expect that population-level effects of bistability in TCS will be of the most physiological relevance.
Biological relevance of bistability in two-component systems
Batchelor et al. (2004) have demonstrated a graded response for the EnvZ/OmpR TCS. Our model of the system predicts the same type of behaviour, but only when the dephosphorylation of the RR that is independent of the SK phosphotase is insignificant. The autophosphatase activity of OmpR is small (Qin et al., 2001) and no alternative Ph acting on OmpR has been reported. However, alternative views on the role of EnvZ in the dephosphorylation of OmpR∼P have been expressed in the literature. Mattison and Kenney (2002) suggested that the affinity of OmpR∼P and EnvZ is small and that there is an alternative Ph that acts on OmpR∼P in vivo. However, subsequent findings by Yoshida et al. (2002a,b) questioned these conclusion and presented evidence that both phosphorylated and unphosphorylated forms of OmpR interact strongly with EnvZ and with similar affinities. We view our predictions regarding the interactions within the TCS that are required for bistability, taken together with the graded response of the EnvZ/OmpR system shown by Batchelor et al. (2004), as further evidence against the existence of an alternative Ph for OmpR∼P. Moreover, our results illustrate an additional role for the bifunctional design in TCS – to ensure a graded system response over large signal variation regardless of dead-end complex formation.
Are there TCS in which post-translational bistability can take place? We conducted a literature survey of TCS, for which their biochemistry has been characterized (Table S1). Few of these are characterized in full detail. Those with the most extensive determination of their parameters are the CheA/CheY system (Baker et al., 2006), the NtrB/NtrC system (Jiang et al., 2000a,b; Pioszak and Ninfa, 2003), the Spo0 system (Piggot and Hilbert, 2004), and the VanS/VanR system (Wright et al., 1993; Fisher et al., 1996). The data available for these and other TCS (Kato and Groisman, 2004) suggest that the equilibrium and rate constants for the corresponding reactions in different classical TCS are very similar and mostly within an order of magnitude. A recent survey of all the TCS for E. coli provides further support for this claim (Yamamoto et al., 2005). Several of the well-characterized TCS – NtrB/NtrC, VanS/VanR, PmrD/PmrB – are bifunctional with no alternative Ph and are therefore not expected to show post-tran bistability. However, the NtrB/NtrC system includes a third protein whose regulatory role is important, so a more complex model needs to be built in order to fully analyse the possibility of bistability for this TCS.
Our simulations suggest that bistability is possible when there is cross-talk between two bifunctional TCS. For example, the SK, VanS, from Enterococci has been demonstrated in vitro to interact not only with its cognate RR, VanR, but also with the RR, PhoB, from E. coli (Fisher et al., 1996). In our computer model of this system (not shown), the dead-end complex that PhoB forms with VanS is able to sequester VanS from the VanS–VanR phosphorylation cycle, thus creating a positive feedback loop similar to the one discussed above. Furthermore, PhoB can also interact with its cognate bifunctional SK, PhoR, which provides an alternative to the VanS pathway for PhoB dephosphorylation. However, we must stress that this model was set up on the basis of in vitro experiments with proteins from different bacteria (Fisher et al., 1996). Thus, the VanS/PhoB interaction is unlikely to be physiologically meaningful in vivo. Nevertheless, these simulations suggest that a TCS that is similar to this may be an adequate initial design for building a synthetic TCS that acts as a bistable switch.
Table S1 summarizes our survey of TCS for which kinetic and bifunctionality information exists in the literature. We include an analysis regarding the plausible detection of bistability in the different TCS based upon the existence of SK bifunctionality, upon the existence of Ph for the RR that are independent of the SK, and upon the function of the TCS. The TCS with a plausible possibility of post-translational bistability can be divided into three groups according to their physiological function. Some of the TCS function as environmental sensors (for example, PhoR/PhoP in B. subtilis, BaeS/BaeR in E. coli, and SphS/SphR in Synechocystis); these are not likely to benefit from bistability as continuous and reversible sensing is expected. Other TCS (for example, YycG/YycF and the phosphorelays DivJ/DivK and PleC/PleD in B. subtilis) are involved in differentiation and cell cycle transitions – functions that benefit from bistable hysteretic switching. The remaining TCS control virulence or growth in particular environments (for example VirA/VirG in Agrobacterium tumefaciens, CreC/CreB, NarX/NarL, NarQ/NarP and YehU/YehT in E. coli, and PrrB/PrrA in Mycobacterium tuberculosis). These TCS will benefit from bistable switching if controlled lifestyle changes are differentiation-like transitions. Intriguingly, several experimental studies suggest bistability in the VirA/VirG system of A. tumefaciens (Brencic et al., 2005; Goulian and van der Woude, 2006). However, it is not known whether bistability in this system is due to a post-translational mechanism or if it involves autocatalytic transcriptional feedbacks.
Another possibility of finding TCS for which bistability is physiologically relevant is among the group of monofunctional TCS. In these systems, dephosphorylation of the RR is independent of the SK, which fulfils one of the requirements for bistability. The other requirement that must be met is the formation of a dead-end complex between the dephosphorylated SK and the RR. One of the best characterized monofunctional TCS controls bacterial chemotaxis; this system includes, among other molecules, the SK, CheA, the RRs CheY and CheB and the phosphatase CheZ (Baker et al., 2006). However, based on structural analysis it is unlikely that CheA–CheY will be a dead-end complex. Indeed, the RR, CheY, binds a specialized domain of the kinase CheA; seven structures for this complex have been reported (entries 1UOS, 1BDJ, 1EYA, 1FFW, 1AO0, 1FFG and 1FFS of the Protein Data Bank, PDB). Another domain of CheA, denoted Hpt, contains a His residue that is autophosphorylated (entries 1TQG, 1B3Q and 1I5N of the PDB). The long stretch of approximately 50 amino acids that connects the Hpt domain to the CheY binding domain has not been crystallized, and it contains two Pro residues. Using the Phyre web server (Kelley et al., 2000), we predict that one of these two residues (Pro132) is in a region expected to be disordered (data not shown). It is conceivable that this disordered loop around Pro132 can act as a hinge that allows the Hpt domain to swing between the CheA catalytic domain and the CheY binding domain. If this were so, then binding of CheY would not prevent phosphorylation of the CheA Hpt domain and no dead-end complex would form.
Bistability could be physiologically desirable in TCS that are used for the realization of a developmental switch. In fact, it has been shown that bistability occurs in sporulating populations of B. subtilis (Maamar and Dubnau, 2005; Veening et al., 2005). Spore formation is a costly cell fate decision in response to severe nutrient depletion or other extreme challenges in the environment. The existence of heterogeneity in an isogenic population of cells under these conditions leads to several adaptive strategies that are beneficial for genotype survival (see Dubnau and Losick, 2006 for a review). This sporulation process is regulated by a non-classical TCS, the Spo phosphorelay, in which four steps of phosphate transfer occur, eventually resulting in phosphorylation of the RR, Spo0A. Intriguingly, deletion of the Spo0E phosphatase that acts directly on Spo0A abolishes bistability (Veening et al., 2005). However, several other auto-stimulatory loops affecting transcription and phosphorylation of Spo0A have been reported (Strauch et al., 1992; Strauch and Hoch, 1993; Fujita and Sadaie, 1998). Therefore, no conclusive inference can be made regarding the role of the ‘dead-end’ complex mechanism for Spo0AP bistability. In many bacteria, such as Myxococcus xanthus or Caulobacter crescentus, multiple TCS are associated with development, cell cycle progression and differentiation (Skerker et al., 2005; Biondi et al., 2006; Goldman et al., 2006; Holtzendorff et al., 2006; Pierce et al., 2006; Ueki and Inouye, 2006). We believe that these TCS are likely candidates to exhibit the proposed mechanism of bistability. We suggest that investigations of the complex between the unphosphorylated SK and RR, and of the potential RR∼P dephosphorylation reactions that are independent of the SK are key to determining the plausibility of a bistable response for any given TCS.
Suggested experimental tests
Based on the preceding subsection we expect that our suggested bistability mechanism may be applicable to some naturally occurring TCS. Table S1 analyses several candidate systems and pinpoints the cases in which bistability might be possible. Experimental work is required to find out whether bistability exists or not in these systems. Below we suggest possible generic guidelines to in vitro and in vivo experiments that can be used to verify the existence of bistability. However, specific experimentalprocedures will certainly depend on the details of the particular TCS.
i Existence of a long-lived complex between unphosphorylated SK and RR, in which RR interferes with phosphorylation of the SK subunit to which it is bound, is crucial for bistability to exist. To test for existence of the dead-end complex in vitro we suggest measuring the autophosphorylation flux of the SK (e.g. by following hydrolysis of radioactive ATP) after pre-incubation with increasing amounts of the cognate RR. If the RR binds SK forming a dead-end complex, then the rate of SK autophosphorylation will be slower with increasing concentrations of RR.
ii In vitro reconstructed TCS can also be used to test for bistability by: (i) measuring the steady-state concentration of phosphorylated RR (if stable and reliably detectable), or (ii) measuring the flux of ATP hydrolysis as function of signals that modulate kinase and/or phosphatase activity of SK. If the physiological signal is unknown or unusable, it may be possible to modulate the kinase activity by varying the ATP/ADP ratio, by addition non-hydrolyzable ATP analogues to serve as competitive inhibitors or by varying the concentrations of Mg2+ ions, using chelating agents such as EDTA. If the TSC is bistable, the model predicts that different steady states for the system can be observed depending on whether most of the RR is initially phosphorylated or unphosphorylated. Note that this in vitro system is only likely to display bistability if the alternative Ph is purified and included in the assay or if the RR possesses significant autophosphotase activity.
iii If the natural inducing signal of a TCS is known, an in vivo assay for bistability can be set up by monitoring the expression of genes under RR∼P control as a function of inducer concentration. If the mean expression levels of the relevant genes is different between cells that were initially pre-induced and cells that were not pre-induced, then bistability is present. Moreover, a bimodal population distribution of expression levels is expected to be observed for inducer concentrations near the switching threshold.
iv Our model predicts that TCS are likely to display a graded response, unless an alternative (to SK) pathway that dephosphorylates RR exists. However, we foresee that introducing an additional Ph in the system will allow re-engineering of the system response. For example, one can imagine introducing an additional mutated copy of the SK gene in the bacterial genome. The mutated SK would have only phosphatase activity. Such a re-engineered system may show a bistable response if a long-lived SK.RR complex is formed.
Conclusions and predictions
Using the EnvZ/OmpR TCS as an example we have used structural modelling and in silico protein docking to analyse the potential binding of the SK (EnvZ) to the RR (OmpR). Our analysis suggests that conformational changes must occur in the SK between recurring trans-autophosphorylation and phosphotransfer reactions. In order for the phosphotransfer to occur, the RR must bind the dimer in such a way that the catalytic domain of the SK is unable to access its phosphorylatable His residue, thus effectively preventing His phosphorylation whenever the RR is bound. This also may lead to the formation of a dead-end complex between the SK and the RR if the conformational change of the SK is as dependent on the binding of the RR as it is on the autophosphorylation of the His residue. The formation of such a dead-end complex would be self-enhancing and would result in bistability for the monofunctional system. However, bistability is not possible when the RR is mainly dephosphorylated by the unphosphorylated SK, as in the case of EnvZ/OmpR system. Our analysis identifies three conditions that are necessary for the existence of bistability in classical TCS: (i) there must be a high affinity complex between the unphosphorylated RR and the SK; (ii) formation of this complex must prevent autophosphorylation of the RR; and (iii) the largest fraction of the RR desphosphorylation flux must be independent of the dephosphorylated SK.