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Proc Natl Acad Sci U S A. Dec 11, 2007; 104(50): 19931–19935.
Published online Dec 6, 2007. doi:  10.1073/pnas.0706792104
PMCID: PMC2148400
Genetics

Input–output robustness in simple bacterial signaling systems

Abstract

Biological signaling systems produce an output, such as the level of a phosphorylated protein, in response to defined input signals. The output level as a function of the input level is called the system's input–output relation. One may ask whether this input–output relation is sensitive to changes in the concentrations of the system's components, such as proteins and ATP. Because component concentrations often vary from cell to cell, it might be expected that the input–output relation will likewise vary. If this is the case, different cells exposed to the same input signal will display different outputs. Such variability can be deleterious in systems where survival depends on accurate match of output to input. Here we suggest a mechanism that can provide input–output robustness, that is, an input–output relation that does not depend on variations in the concentrations of any of the system's components. The mechanism is based on certain bacterial signaling systems. It explains how specific molecular details can work together to provide robustness. Moreover, it suggests an approach that can help identify a wide family of nonequilibrium mechanisms that potentially have robust input–output relations.

Keywords: signal transduction, systems biology

Consider a system that produces an output, such as the level of a phosphorylated protein, based on an input signal. The level of output as a function of input signal strength is called the system's input–output relation. One may ask whether this input–output relation is sensitive to variations in the concentrations of the components that make up the system. Because the concentrations of many proteins vary from cell to cell by tens of percents as a result of inherent stochastic processes (13), and the concentrations of metabolite components such as ATP may vary by as much as a factor of 10 in different conditions (4), it might be expected that the input–output relation will also vary. If this is the case, different cells exposed to the same input signal will display different outputs. Although such variability may be desirable in some cases, it might be deleterious in systems where survival critically depends on accurate match of output to input.

One may thus ask whether signaling systems can provide an input–output relation independent of the levels of all components in the system, including proteins and ATP. Such a mechanism can be said to have a robust input–output relation. Complete robustness of this type is difficult to achieve; most known models have an output that depends on the level of at least one of the system's components. For example, a model of the bacterial osmotic-stress signaling system EnvZ/OmpR, although robust to variations in the levels of some components, is not robust to ATP levels (5). Known biochemical models of robust adaptation, including the well studied example of bacterial chemotaxis, also do not seem to have this property (615). In these models, the output level is sensitive to variations in the protein concentrations (7, 8). Thus, the goal of the present paper is to suggest a way in which signaling systems can provide an input–output relation that is robust to the levels of all of the components of the system§.

To clarify the notion of input–output robustness, consider a simple counterexample based on allosteric control mechanisms. The system is composed of a protein denoted A (Fig. 1a) that undergoes equilibrium transitions between an active state A1 and an inactive state A0. The output of the system is the concentration A1 of the active state.

Fig. 1.
A signaling scheme that is not robust to variations in protein levels. (a) Protein A is found in an inactive state A0 and an active state A1, whose concentration is the system's output. The input signal s affects one of the rate constants k(s) (similar ...

The partitioning of protein molecules between the A0 and A1 forms is governed by the reaction

equation image

where s is the input signal (for example, the concentration of an effector molecule), k(s) is the forward rate constant, and k′ is the backward rate constant. Thus, by modulating k(s), the input s determines the output A1.

As in many signaling systems, the transient times for transitions between A0 and A1 are much faster than the changes in the input signal. Thus, one can consider the output of the system at steady state, which is determined by the equilibrium condition k(s)A0 = k′A1, and the conservation equation A0 + A1 = AT, where AT is the total concentration of protein A. Using the former equation in the latter yields the input–output relation of the system, A1 = AT/(1 + k′/k(s)). This input–output relation is not robust, because the output A1 depends on the total concentration AT. Thus, cells with high AT will show an input–output curve that is different from that of cells with low AT (Fig. 1b).

How can input–output robustness emerge in biochemical signaling systems? Here, we suggest a robust input–output mechanism that is based on properties found in a class of bacterial signaling systems. We also suggest a simple approach for analyzing the robustness properties of more complex circuits. Finally, we discuss experimentally testable predictions that the present mechanism makes.

Results

A Mechanism for Robust Input–Output Relation.

A mechanism for a robust input–output relation can be suggested based on biochemical features that are found in a class of bacterial two-component signaling systems. Table 1 lists six such systems, each of which is composed of two proteins, an input-sensitive sensor denoted X and a response regulator denoted Y. The sensor X senses the input signal and acts to phosphorylate the diffusible response-regulator Y. The phosphorylated form of Y, denoted YP, activates the expression of relevant genes (16). Thus, the input signals of these systems affect gene expression by setting the output, which is the concentration YP.

Table 1.
Bacterial two-component signaling systems that underlie the present model

One well studied example is the EnvZ/OmpR two-component signaling system of E. coli. Its primary input is the osmolarity of the medium in which the bacterium grows, and its output is the concentration of phosphorylated response-regulator OmpR, which controls the expression level of genes. In this system, a correlate of the output is the ratio between the transcription levels of two genes controlled by YP. This correlate, which is a continuous function of the input signal, has been experimentally shown by Batchelor et al. (17) to have a high degree of robustness; at a given input level, the correlate changes by <5% when protein levels are varied by 2-fold and by ≈20% for 10-fold changes (5). This approximate robustness breaks down only at high over- or underexpression of the proteins.

The signaling systems of Table 1 all share certain specific biochemical features (Fig. 2a).The first feature concerns the sensor kinase activity. In these systems, the sensor is not a simple kinase that binds Y and ATP to phosphorylate Y. Rather, the sensor works in two steps. First, it phosphorylates itself by binding and hydrolyzing ATP (16). The rate constant of this autophosphorylation, va(s), is controlled by the input signal s. The phosphorylated sensor, denoted XP, then performs a phosphotransfer step; it transfers the phosphoryl group to Y, thereby forming YP (16).

Fig. 2.
Mechanism for robust signaling based on a class of bacterial signaling systems. (a) Sensor X bound to ATP phosphorylates itself to form XP. The phosphorylated sensor XP then transfers the phosphoryl group to the response-regulator Y, thereby forming Y ...

This two-step process is found in virtually all bacterial two-component systems. The following features, however, appear to be particular only to a class of systems, such as those in Table 1. In these systems, the sensor X is a bifunctional enzyme; it catalyzes not only the phosphorylation of Y but also the dephophosphorylsphorylation of YP (16). Finally, ATP, which is used as the phosphoryl donor for the autophosphorylation reaction, is also required as a cofactor for the dephosphorylation reaction (1820). This ATP dependence occurs despite the fact that ATP is not used as an energy source in the dephosphorylation step.

The features above were used to construct a signaling mechanism whose reactions are shown in Fig. 2b. To find the input–output relation of the mechanism and study its robustness, it is necessary to solve for the fixed points of the seven nonlinear differential equations that describe the mass-action kinetics of the model. This is done in the supporting information (SI). However, another way to obtain the input–output relation presents itself when the system is viewed as a black box that breaks down ATP and releases phosphoryl groups. Consider the fluxes of phosphoryl into and out of the system. The influx of phosphoryl groups from the cell's ATP pool is equal to the rate of the autophosphorylation reaction:

equation image

The outflux of phosphoryl groups is equal to the rate of the dephosphorylation reaction:

equation image

To proceed, one can compute the concentration of the complex X·ATP·YP. This complex is formed in the model by the binding of X·ATP to YP and is lost when the constituents dissociate or when the dephosphorylation reaction takes place:

equation image

Thus, at steady state, the concentration of the complex is proportional to the product of its component concentrations:

equation image

Using this in Eq. 2 yields

equation image

At steady state, Ji = Jo; using Eqs. 1 and 5, one finds that:

equation image

When [X·ATP] is nonzero, one can divide it out from both sides of Eq. 6. This results in a robust input–output relation, in which YP depends only on kinetic rate constants:

equation image

Hence, the output YP does not depend on the level of any of the proteins in the system, or on the level of ATP. The output is responsive to the input signal via the rate constant va(s). This mechanism thus shows a robust input–output relation (Fig. 2c).

Variations in the concentrations of the sensor or ATP do not affect the input–output relation. The only loss of robustness occurs if the total level of protein Y, denoted YT, falls below the expected YP level for a given input signal (Fig. 2c). In this case, there is not enough Y protein to reach the YP value given by Eq. 7. If this happens, a complete analysis of the model (see SI) shows that all of the Y molecules are phosphorylated, and YP = YT. Hence, the system cannot respond at all to the input signal. It therefore follows that both robustness and responsiveness to the signal require that YT exceeds a certain threshold, given by the maximal desired output level YP in expected physiological conditions.

Note that all three biochemical features of the mechanism are required for input–output robustness. First, ATP dependence of dephosphorylation is essential. Indeed, in a model without this feature, one finds that Yp~ATP (ref. 5; see SI). Hence, the output is sensitive to fluctuations in the level of ATP (4). Similarly, if the sensor was not bifunctional, and dephosphorylation was carried out by a separate phosphatase protein Z, the balance of phosphoryl influx and outflux would require that [X·ATP] ~ [Z·YP] ~ ZYP. This would result in a steady-state level YP ~ [X·ATP]/Z that depends on the intracellular levels of both the sensor X and phosphatase Z. Robustness would be lost.

Finally, the two-step nature of the kinase is also essential for robustness. If the sensor was a simple kinase that directly transfers a phosphoryl group from ATP to the substrate without first phosphorylating itself, the ATP breakdown rate would depend on the concentration of the complex X·ATP·Y, which at steady state is proportional to [X·ATP]Y. This would balance with the dephosphorylation flux, so that [X·ATP]Y ~ [X·ATP]YP. As a result, the output YP would be proportional to the level Y of free response regulator and would thus depend on the total level of response-regulator YT, thereby abolishing robustness. In summary, robustness over a wide range of parameters in the present mechanism seems to require the combined effects of all three biochemical features.

We also studied the effect of adding reactions to the model. For example, spontaneous dephosphorylation of YP is known to occur on a much slower timescale than the response time of the system (Table 1), and accordingly we find it has a negligible effect on robustness (SI). As a second example, we find that adding ADP as a cofactor for the dephosphorylation activity of the sensor [in addition to ATP and with similar efficiency (21)] has a negligible effect on robustness (SI). This is primarily because the ADP concentration is much lower than the ATP concentration in the cell (22). We also studied other reactions, including spontaneous dephosphorylation of XP, reverse autophosphorylation, phosphotransfer, and dephosphorylation steps, as well as an alternative way to form the ternary complex in the system. In the supporting information (SI), we show that when the rates of these additional reactions are small compared with the rates of the present model reactions, they have only a small effect on robustness.

Remarks on the Black Box Approach.

To analyze the properties of the mechanism presented above, the system was considered as a black box that breaks down ATP. More generally, the black box approach can be used to suggest a class of systems that have the potential to show robust input–output relations. It can also point to system characteristics that may rule out such robustness.

Consider a system with one reaction that breaks down ATP at rate Ji and another reaction that releases phosphoryl ions at rate Jo (Fig. 3a). Under what conditions might component Y in the system be robust? This can happen if, at steady state, Ji and Jo depend in the same way on the concentrations of all components except Y but depend in different ways on the concentration of Y itself. Thus, the steady-state influx and outflux can be expressed as follows: Ji = f(X1…, XN, Y, ATP)g(Y) and Jo = f(X1XN,Y, ATP)h(Y), where g(Y) and h(Y) are different functions of Y that intersect at only one point. These functions also depend on kinetic rate constants, some of which are signal-sensitive. If the system reaches a stable steady state, then Ji = Jo, and one has fg = fh. Assuming the fluxes are nonzero, the function f can be eliminated from both sides of the equation, which results in g = h. This can be solved to yield Y as a function of rate constants only, which makes the relation between the input signal and the output Y robust with respect to all component concentrations.

Fig. 3.
Black box approach and conditions that prohibit robust input–output relations. (a) Considering a system as a black box that breaks down ATP at rate Ji and releases phosphoryl groups at rate Jo suggests that the concentration of component Y is ...

Note that such robust systems can include any number of reactions within the black box, such as multiple phospho-transfer cascades, as long as a stable steady state is reached, and the influx and outflux of phosphoryl groups are as described above. Thus, many variants of the model of Fig. 2b, which add reactions inside the black box, can in principle be formed, and all such variants can display the robustness property.

It can also be seen that robustness of the present type cannot generally occur if there is more than one reaction that introduces phosphoryl groups into the system. If two different influxes Ji and J′i exist (Fig. 3b), they generally cannot be canceled out with Jo (in the sense of Eq. 6 above), leading to a loss of robustness. Similar considerations apply to cases where there is more than one way for phosphoryl groups to exit the system.

Robustness of the present type thus depends on a single route for uptake and release of phosphoryl groups. However, in the SI, we find that in the case of the present model one can come close to robustness if the influx and outflux due to secondary reactions J′i and J′o are small in magnitude relative to Ji and Jo. If the relative magnitude of the secondary to primary fluxes is of order ε and steady-state stability is maintained, robustness is generally lost by only a factor of order ε.

Discussion

This study presents a mechanism that can make the input–output relation of a biochemical system robust with respect to variations in the concentrations of all of its components. The mechanism is based on biochemical reactions found in a class of bacterial signaling systems. The present approach also provides guidelines for constructing other robust mechanisms by imposing conditions on the fluxes of covalent modifier molecules, such as phosphoryl groups, into and out of the system.

The present mechanism makes experimental predictions for systems such as those of Table 1. The first is that there exists a threshold in the total response-regulator concentration YT such that if YT is less than the threshold, then no ATP is consumed at steady state, and the output YP is sensitive to changes in YT. If YT exceeds the threshold, then ATP is consumed at steady state, and the system has a robust input–output relation. Robustness to protein levels can be tested by using controlled expression of the proteins (5, 8, 35). The prediction that the output is insensitive to the ATP level can be tested by perturbations that alter the cell's ATP concentration (4, 22).

The biochemical implementation of the present mechanism, at least in the example treated here, relies on bifunctional enzymes to carry out phosphorylation and dephosphorylation. Moreover, these bifunctional enzymes use ATP as a phosphoryl donor for one reaction and as a coenzyme for the opposite reaction. This feature is found also in eukaryotic systems such as P-type ATPases (23). It would be of interest to extend this study also to other biological control systems that show similar features.

Not all bacterial signaling systems, however, show the hallmarks of the present mechanism. Important examples include bacterial chemotaxis and sporulation. Indeed, it is expected that some signaling systems will not have robust input–output relations. Such systems would display cell–cell variation in their output at a given input signal. This variation, also called “individuality,” has been described, for example, in bacterial chemotaxis (7, 24, 25). Similarly, signaling systems that make sharp decisions, such as the phosphotransfer cascade in the sporulation system of Bacillus subtilis (26), are not expected to show robust input–output relations and indeed exhibit behavior that varies between individual cells. A robust mechanism should perhaps be expected only in signaling systems in which there is a sufficiently heavy fitness penalty if the input–output relation is not precise.

In summary, the present study suggests mechanisms that respond to a signal in a way that is independent of variations in the concentrations of the system components. This mechanism for robust signaling depends on specific nonequilibrium reactions. It may thus provide an explanation for biochemical details that may otherwise appear arbitrary. Experiments can readily determine whether robust signaling occurs in additional systems, by varying protein and metabolite levels and measuring the effect on the input–output relation.

Supplementary Material

Supporting Information:

Acknowledgments

We thank N. Barkai, E. Batchelor, A. Bren, E. Dekel, G. Falkovich, M. Feinberg, E. Hornstein, W. Hsing, N. Kashtan, W. Liebermeister, B. Meerson, J. Rabinowitz, J. Rosenheim, F. Sahar, E. Schneidman T. Shimizu, T. Silhavy, and all members of our laboratory for helpful discussions and comments. This work was supported by the Kahn Family Foundation, the Israel Science Foundation, Minerva, and the Human Frontiers Science Project.

Footnotes

The authors declare no conflict of interest.

This article is a PNAS Direct Submission. J.J.T. is a guest editor invited by the Editorial Board.

§Here, we use the term “robustness” to mean insensitivity to component concentrations (3335). This is sometimes called “dynamic robustness.” In addition, the term robustness is sometimes used to denote insensitivity to mutations or component deletions (36), which are not considered here.

This article contains supporting information online at www.pnas.org/cgi/content/full/0706792104/DC1.

When YT is less than the threshold, no ATP is consumed at steady state (see SI). Therefore, in the present mechanism, ATP consumption and robustness appear to be linked.

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