Product dependence and bifunctionality compromise the ultrasensitivity of signal transduction cascades

Abstract

Covalent modification cycles are ubiquitous. Theoretical studies have suggested that they serve to increase sensitivity. However, this suggestion has not been corroborated experimentally in vivo. Here, we demonstrate that the assumptions of the theoretical studies, i.e., irreversibility and absence of product inhibition, were not trivial: when the conversion reactions are close to equilibrium or saturated by their product, “zero-order” ultrasensitivity disappears. For high sensitivities to arise, not only substrate saturation (zero-order) but also high equilibrium constants and low product saturation are required. Many covalent modification cycles are catalyzed by one bifunctional ‘ambiguous’ enzyme rather than by two independent proteins. This makes high substrate concentration and low product concentration for both reactions of the cycle inconsistent; such modification cycles cannot have high responses. Defining signal strength as ratios of modified (e.g., phosphorylated) over unmodified protein, signal-to-signal response sensitivity equals 1: signal strength should remain constant along a cascade of ambiguous modification cycles. We also show that the total concentration of a signalling effector protein cannot affect the signal emanating from a modification cycle catalyzed by an ambiguous enzyme if the ratio of the two forms of the effector protein is not altered. This finding may explain the experimental result that the pivotal signal transduction protein PII plus its paralogue GlnK do not control steady-state N-signal transduction in Escherichia coli. It also rationalizes the absence of strong phenotypes for many signal-transduction proteins. Emphasis on extent of modification of these proteins is perhaps more urgent than transcriptome analysis.

Living organisms excel in selective responsiveness. To some alterations in their environment, they do not respond at all. To other such alterations, they respond by little changes in some, and strong changes in other molecular activities; i.e., by homeostasis and adaptive functional changes, respectively (1). The subtlety and diversity of the responses may explain why the functions of many signal-transformation mechanisms are incompletely understood. Intracellular signal transduction may have properties akin to those of neural networks (2). One such property, signal amplification, can be brought about by co-operative mechanisms in single protein molecules. In 1981, Goldbeter and Koshland (3) proposed additional, multienzyme-based mechanisms that should produce strong variations in some system variables as a response to minor changes in parameter values. One of these comprises covalent enzyme conversion (3–7), catalyzed by additional converting enzyme(s) that ‘oppose’ each other (8, 9): they return the interconvertible enzyme to its original state entailing a free-energy dissipating reaction such as the hydrolysis of ATP (10). Goldbeter and Koshland showed that such a mechanism could give rise to a higher sensitivity than the more traditional co-operative or allosteric enzyme mechanism. ‘Zero-order’ ultrasensitivity was obtained when the modified protein exists at a concentration far above the Michaelis constants for both modifying enzymes. A cascade of such enzyme cycles could lead to even higher amplification factors, as quantified in a multiplication theorem (11, 12, 13). The conditions necessary for high sensitivity when each of the steps is catalyzed by completely independent modifier enzymes that follow irreversible Michaelis-Menten rate equations have been revealed (14–17).

Experimental proof of ultrasensitivity achieved by enzyme interconversion has been slow in coming. Early on, quite a few regulatory systems were shown to contain opposing enzyme conversion cycles. Examples include the (in)activation of glutamine synthetase and PII (18), the interconversion of fructose bisphosphatase-2 and phosphofructokinase-2 in some organisms (19), and, more recently, the interconversion of protein kinases and phosphatases in the MAP-kinase cascades (20). In vivo, the responsiveness of glutamine synthetase to changes in PII concentration has been extremely low (21). In cell populations, the MAP-kinase cascade lacked ultrasensitivity, although ultrasensitivity did occur at the single-cell level, but perhaps through a different mechanism: a feedback loop in the regulatory network (22). The absence of experimental validation of zero-order ultrasensitivity vis-à-vis the omnipresence of interconversion cycles in cell regulation puts into question whether such cycles do generate ultrasensitivity as readily as has been suggested.

Many of the experimental examples of enzyme conversion have a characteristic that has not yet been dealt with in the theoretical analyses: the two opposing reactions are often catalyzed by one and the same bifunctional protein. Such an ambiguous enzyme can carry out either one of two catalytic cycles. Additional molecules binding to the protein make its action unequivocal. In the glutamine synthetase (GS) regulatory cascade for instance, GS inactivation through adenylylation is catalyzed by the form of adenylyltransferase (ATase) that is generated by interaction with PII. The activating (deadenylylation) reaction is catalyzed by the ATase that interacts with PII-UMP (23, 24). The interconversion of PII and PII-UMP is catalyzed by yet another bifunctional enzyme, i.e., uridylyltransferase. The GS cascade also has a branch that regulates gene expression. Here, the bifunctional NRII appears to function as both the kinase and the phosphatase of the transcription factor NRI, depending on interactions with PII and GlnK.

The fact that many of the enzymes engaged in signal transduction are ambiguous puts into question one of the more pervasive assumptions in the theoretical treatments of ultrasensitive enzyme cycles, i.e., that the modification-reaction rates be independent of the concentration of their product. For, in these cases, the product of either reaction is the substrate of the same enzyme in the other (opposing) reaction. Hence, it is unrealistic to assume that the enzyme is completely oblivious of its product.

The present study has two aims. The first one is to extend the sensitivity analysis to interconvertible systems where the conversions are product sensitive. We investigate two different factors that can result in product dependence: reversibility and product saturability. This study was performed using the tools of metabolic control analysis, MCA (25–29) and metabolic control design, MCD (30–35). It is a generalization of the case of product competitive inhibition in irreversible reactions studied by Goldbeter and Koshland (6).

The second aim is to examine whether systems where the two enzymes catalyzing the two opposing reactions are not independent can still be highly sensitive. In particular, we are interested in the case where both steps in the conserved cycle are catalyzed by the same ambiguous protein. We shall show that zero-order ultrasensitivity disappears when the modifying enzyme is strongly product sensitive or ambiguous. The conclusions achieved are compared with the experimental results reported by Van Heeswijk (21) for the glutamine synthetase (de)adenylylation cascade of Escherichia coli.

Theoretical Background: Sensitivity Properties

The relationship between the change of a systemic steady-state variable (Y) and the modification in the effector that ultimately caused this change is represented by the response coefficient (25). This coefficient is defined as the relative change of a system variable (Y) divided by the relative, infinitesimal change in the effector (p):

This is a systemic property. In contrast, the π-elasticity coefficient is a property of an individual reaction process in the system rather than of the system as a whole (36, 37):

where v is the rate of the isolated process, or “step.” Here, it is assumed that the concentrations directly affecting the rate (represented by “X”) are maintained at constant levels when p is changed. The response coefficient is related to the π-elasticity coefficient through the response theorem (1, 25, 38, 39):

where the summation is over all process rates in the system. The π-elasticity coefficients (π) depend on the particular effector that is changed. The control coefficients C (C = (v_i/Y)(∂Y/∂v_i) describe the effect that the change in the rate (v_i) of the process has on the system variable (Y), being independent of the particular effector (p) that is changed. They represent intrinsic properties of the network (40, 41).

The Enzyme Conversion Cycle

The analysis of the sensitivity properties will be performed for an enzyme conversion cycle (also called monocyclic cascade) where the enzyme E alternates between an inactive form (E_α) and an active, modified form (E_β). In Fig 1A, the enzyme conversion is catalyzed by two independent modifier enzymes, e₁ and e₂. In Fig. 1B, both steps in the interconvertible cycle are catalyzed by the same, ambiguous enzyme, e. The total concentration of the interconvertible enzyme E is assumed to be constant (E_T = E_α + E_β). This is a simplification, as in reality, part of the enzyme may be sequestered and the extent of sequestration may depend on the state of the enzyme (42).

Figure 1.

Scheme of a monocyclic cascade where E_α and E_β are the different forms of the interconvertible enzyme E. It is assumed that the metabolite concentrations A, B, C, and D are constant. In Fig. 1A, we represent the case where the interconvertion reactions are catalyzed by two independent modifier enzymes, e₁ and e₂, and in Fig. 1B, we represent interconvertion reactions catalyzed by two forms, eG and eH, of a single bifunctional enzyme e.

For Fig. 1, the response theorem may be combined with the concentration summation relationship (C + C = 0) to obtain:

This equation shows that the system variable E_β should exhibit a high sensitivity to changes in the effector p if both the “intrinsic factor,” C, and the “extrinsic factor,” π − π, are sufficiently high. When C is zero, the system variable E_β does not change, whatever the effect of the effector p on the isolated rate is. Similarly, if the π-elasticity coefficients are equal, R is zero, irrespective of the magnitude of C.

Requirements for High Intrinsic Response

To examine when the intrinsic factor is high (|C| ≫ 1), we will use an additional coefficient, describing the sensitivities of the individual isolated rates to changes in the variable concentrations, i.e., the ɛ-elasticity coefficients. This coefficient is defined as the relative change in the rate (v) of an individual enzyme-catalyzed reaction divided by the relative change in an internal metabolite (S) that uniquely causes the former (25):

when all the other internal metabolites and parameters are kept constant. Using MCA, control coefficients (C) can be expressed in terms of the ɛ-elasticity coefficients (9, 27–29, 39, 43–45, 46). The inverse relationship is achieved using MCD, i.e., the ɛ-elasticity coefficients required to produce a given pattern of systemic responses can be expressed in terms of the independent control coefficients associated to those responses (30–32, 34).

E_T = E_α + E_β reduces the number of independent concentration variables to one, for which we choose E_β. The corresponding reduced scheme is (9)

The relationship of the elasticities of the reduced scheme with the ones of the original system can be derived (28). At the steady state v = v ≡ J. Using MCD, one obtains the following relationships between the control coefficients and the ɛ-elasticity coefficients:

1

Because the two reactions are here assumed to be catalyzed by independent enzymes, in principle, the two elasticities of these equations can be altered independently (e.g., through site-directed mutagenesis).

How could one design an interconversion cycle that shows a high intrinsic factor, i.e., |C| ≫ 1? Since E_β is a substrate of v₂, we would normally expect that an increase in v₂ would produce a decrease in E_β, and hence C < 0. Our target of design, therefore, is −C ≫ 1. The design consists in finding the ɛ-elasticity coefficients (and rate equations) that would produce such a systemic response. We will focus on enzymes that show normal kinetic behavior, i.e., where substrate activates and product inhibits. These requirements are equivalent to −ɛ > 0 and ɛ > 0. Introducing these restrictions into Eq. 1, we immediately find that 0 < C < 1. As a consequence, to design |C| ≫ 1, the absolute values of the two ɛ-elasticity coefficients should be much smaller than unity. This conclusion is valid independently of the particular rate equations of the modifier enzymes that catalyze the reactions. It should also apply when an ambiguous enzyme catalyzes both reaction 1 and reaction 2, although it may then be more difficult to vary the two ɛ-elasticity coefficients independently of each other. A similar conclusion has been drawn by Small and Fell (14) for the particular case of irreversible rate equations.

Independent Modifying Enzymes: Implications of Product Dependence

At this point, we will consider the scheme of Fig. 1A, where the steps are governed by independent reversible Michaelis–Menten rate equations (see Appendix, which is published as supporting information on the PNAS web site, www.pnas.org.). With the conservation relationship E_T = E_α + E_β and the independent variable z = E_β/E_T we obtain:

2

Here, V_si are the maximal rates, K_ei are the apparent equilibrium constants, and K_si and K_pi are the Michaelis–Menten constants for the substrate and product, respectively.

We shall analyze first the special case of a completely symmetric system:

Adding up the two general expressions of the ɛ-elasticity coefficient (Eq. 1):

Dropping indices 1 and 2, computing the ɛ-elasticity coefficients as the normalized derivatives of the rate laws (Eq. 2), and substituting in the previous equation using z = ½, one obtains, for the intrinsic sensitivity of the (symmetric) system:

3

where α ≡ K_P/K_S (and K_e > 1).

This equation shows that three conditions should be fulfilled simultaneously for the system to present a high sensitivity. The first condition is the classical one (6), i.e., the concentration of the modified enzyme exceeding the Michaelis–Menten constant. The second condition, of a high equilibrium constant, stresses the requirement of substantial free-energy input into the cyclic reaction. In many actual systems, the cycle is coupled to ATP hydrolysis, giving rise to a high effective equilibrium constant. The third condition is that the product inhibition constant be substantially higher than the Michaelis constant. This condition requires appreciable specificity of the modifying enzymes, i.e., an ability to discriminate between the two forms of the interconvertible enzyme. When the first and third conditions adopt values favoring high sensitivity (E_T/K_S and α very high), then the value of C is bounded by the equilibrium constant, i.e., C < K_e. For a symmetric model to have a sensitivity of 100, the equilibrium constants need to exceed 100. On the other hand, if the equilibrium constant is very high (as is expected, for example, for phosphorylation–dephosphorylation reactions), the resulting sensitivity depends on the combined effect of the other two conditions. The contour plot of Fig. 2 represents how the sensitivity increases with E_T/K_S and α = K_P/K_S values.

Figure 2.

Contour map of C as a function of E_T/K_S and α. The plot was calculated from Eq. 3 for K_e = 10¹⁰. The curves represent pairs of values (E_T/K_S and α) for which the sensitivity, given by C, is the same.

The increase in reversibility and product saturation both compromise the possibility to obtain ultrasensitivity. However, analysis of Eq. 3 shows that their effects are rather different. With respect to reversibility, we find that even if we are in the extreme most favorable conditions of complete substrate saturation and no product saturation, a sufficiently low value of the equilibrium constant would abolish ultrasensitivity. In contrast, if we consider complete irreversibility, an increase in substrate saturation may always compensate for the unfavorable effect of product saturation.

In the Appendix (see supporting information), we show that these conclusions are not confined to the special symmetrical system. We prove that again the intrinsic factor associated to the sensitivity of E_β, |C|, has the equilibrium constant K_e2 as an upper limit (see Eq. A4, which is published as supporting information on the PNAS web site). Again, the substrate saturation required to generate a given sensitivity decreases both with an increase in the equilibrium constant (see Eq. A10, which is published as supporting information on the PNAS web site) and with an increase in the product Michaelis–Menten constant (see Eq. A11, which is published as supporting information on the PNAS web site). The general asymmetric case has the novel feature that an unfavorable value (to obtain high sensitivity) in a parameter of one of the enzymes may be compensated by a favorable value in a parameter of the other enzyme (47). For instance, low substrate saturation of one of the modifier enzymes may be compensated by high substrate saturation in the other.

The fact that these three conditions mentioned above should be fulfilled simultaneously for the system to present high sensitivity doesn't mean that ultrasensitivity is unlikely to be observed. Some degree of ultrasensitivity is still quite likely, especially for some covalent modifications cycles such as phosphorylation-dephosphorylation cycles (which are very common in eukaryotes). An example is found in the regulation of glycogen phosphorylase in vitro, where the sensitivity of phosphorylase a to changes in the kinase/phosphatase activity ratio shows a moderate increase with the total phosphorylase concentration (48). A more recent example is the mitogen-activated protein kinase (MAPK) cascade (49). In this case, it is reported that MAPK activation in Xenopus oocytes extracts shows a response/stimulus curve with a sensitivity equivalent to a cooperative enzyme with a Hill coefficient of 4–5.

As the sensitivity of the interconversion cycle is described by the response coefficient, i.e., by the product of the intrinsic factor (|C|) and the extrinsic factor (π − π), it is relevant to analyze if the extrinsic factor can overcome the decrease on cascade sensitivity caused by the product dependence of the modifier enzymes. Irrespective of how the modifier enzymes are inhibited, i.e., by a competitive, uncompetitive, or mixed inhibitor, the π-elasticity coefficients will reside between 0 and −1. For simple activators, the π-elasticities assume values between 0 and 1. Consequently the extrinsic factor will assume absolute values between 0 and 2. At most, this can double the sensitivity accomplished by the intrinsic factor (refs. 50, 51 and references therein).

No Zero-Order Ultrasensitivity for an Ambiguous Modifier Enzyme

Here, the modification reactions are not independent, because a single enzyme is involved in both of them (Fig. 1B):

where G and H are the effectors that modify the enzyme. We assume that the protein does not show significant catalytic activity in the absence of effector, and that it presents only one type of activity when is associated to either effector. The steady-state rate equations of the opposing processes read:

4

where: V_m1 = k₂e_T, V_m2 = k₄e_T, K = K_m1(1 + G/K_G + H/K_H)/(G/K_G), K = K_m2(1 + G/K_G + H/K_H)/(H/K_H), K_m1 = (k₋₁ + k₂)/k₁, K_m2 = (k₋₃ + k₄)/k₃, K_G = k₋₅/k₅, and K_H = k₋₆/k₆. e_T represents the total concentration of the modifier enzyme e. K_m1 and K_m2 are the substrate Michaelis–Menten constants. K_G and K_H are the dissociation equilibrium constants of the enzyme-effector complexes.

The fact that a single enzyme carries out both activities confers special kinetic properties to these kinetic equations, as compared to the equations for two independent modifier enzymes (Eq. 2). The denominators of v₁ and v₂ are identical. Moreover with the ambiguous enzyme, the product inhibition constant in the one reaction equals the Michaelis–Menten constant of the opposing reaction. As we shall see, this feature eliminates the possibility of having zero-order ultrasensitivity.

To analyze the intrinsic and extrinsic contributions to the sensitivity of the system in detail, we solve, with Eq. 4, v₁ − v₂ = 0 for the steady-state value of E_β:

5

where: V = V_m1/V_m2. It is important to note that E_β depends on the ratio G/H. By differentiation of equation 5, one obtains the intrinsic factor associated to the sensitivity of E_β:

6

Because the fraction E_β/E_T is between zero and one, this coefficient (−C) is positive and smaller than one. Consequently, there is no intrinsic contribution to ultrasensitivity.

To examine how the extrinsic factor, here related to the external regulators G and H, can affect the global response of the system, we consider the effectors G and H to be independent. In this case, the contribution of the extrinsic factor to the sensitivity of E_β, as calculated by differentiation of the rate equations (Eq. 4), is given by:

7

Accordingly, the explicit expression for the overall response (combining the intrinsic and extrinsic factors) to the independent parameters G and H becomes:

8

This equation shows that for the rate equations under consideration, the concentrations of the interconvertible forms of the cascade cannot present highly sensitive responses to the independent parameters G and H, because neither the intrinsic nor the extrinsic factor can convey this property: there is no zero-order ultrasensitivity for the ambiguous enzyme.

In actual signal transduction cascades, G and H are often interdependent, because they are two different forms of the same protein. If their sum, G + H = T, is maintained constant, the analysis renders that R and −R can present values greater than unity (results not shown). However, if we express the response coefficient with respect to the ratio G/H we obtain:

9

This cannot exceed 1: high responses are impossible. When the ambiguous protein is inside a cascade of such enzymes, it makes more sense to consider the response of the E_β/E_α ratio to the G/H ratio. This yields the surprisingly simple result:

10

i.e., the ratio E_β/E_α is proportional to the ratio of G/H: there can be neither amplification nor attenuation.

GS in E. coli

When the effector exists in two interconvertible forms, i.e., G + H = T, its total concentration T can be experimentally changed in such a way that the ratio G/H does not change. Then the proportion of the two forms of the interconvertible enzyme, E_β and E_α, remains unaltered (Eq. 5). This is relevant for the glutamine synthetase (GS) of E. coli, which plays a central role in nitrogen anabolism catalyzing the synthesis of glutamine from glutamate and ammonia. This enzyme is regulated by covalent modification to yield adenylylated GS, a less active form (52–53). The two opposite adenylylation–de-adenylylation interconversion reactions are catalyzed by the GS adenylyltransferase (ATase), an ambiguous enzyme: adenylylation of GS is stimulated by the pivotal protein PII whereas the deadenylylation reaction is activated by the urydylylated form of PII (PII-UMP). The system PII–PII-UMP is also regulated by covalent interconversion, constituting with the GS–GS-AMP a bicyclic cascade. Previous in vivo experiments studied the sensitivity properties of the glutamine synthetase cascade to changes in the concentration of the pivotal protein PII (21). The authors varied the total concentration of PII. The nonadenylylated GS concentration was completely insensitive to changes in total PII protein level, also when the paralogue of PII, GlnK (54), was absent (55).

We have shown that if the effectors G and H are changed while maintaining their ratio constant, the concentration of E_β is completely insensitive to those changes. In the case of ambiguous enzyme, the response (to G plus H at constant G/H) is always zero independent of the concentrations of G and H. Thus, the insensitivity of deadenylylated GS concentration to changes in PII experimentally determined maintaining the ratio PII/PII-UMP constant is in accordance with the analysis in this paper.

In conclusion, even though if in this section we have considered the most favorable case, i.e., the irreversible case for the appearance of high sensitivity, the fact that ATase is ambiguous prevents highly sensitive responses.

Discussion

That enzyme conversion cycles could be expected to lead to very high responses (zero-order ultrasensitivity) to signals has been proposed on the basis of theoretical analyses by Goldbeter and Koshland (3, 7) but never really substantiated experimentally. In this paper, we show that many of the existing enzyme conversion cycles have a property that precludes the ultrasensitivity: their reactions are product-inhibited. This is true because the product is (of necessity) highly similar to the substrate, or because a single modifying enzyme catalyzes both the forward and the reverse enzyme conversion.

Our kinetic analysis of the monocyclic cascades with a single bifunctional modifier enzyme allowed us to analyze the role of the pivotal regulator protein PII in E. coli from a new perspective. This protein is obviously crucial for intracellular signal transduction. Yet, it lacked control on the average adenylylation state of GS even in the absence of its paralogue GlnK (55). We found that, especially because the ATase is an ambiguous enzyme, altered expression of total PII protein should not affect the steady-state activity of GS, unless the ratio of PII/PII-UMP were to change in parallel. That, however, should be unlikely, because in the absence of alterations in the activity of GS, there should be no steady-state mechanism for that ratio to change. Therefore, we explained the paradoxical finding that the important protein PII (plus GlnK) has no effect on the steady-state activity of the protein that it regulates.

Of course, the PII/PII-UMP interconversion is also regulated in vivo, again through a conversion cycle. Therefore, the view is substantiated that for signal transduction, it is the ratio of the two forms in which regulatory proteins appear rather than their total concentration that is important for signal transduction. This view is particularly useful when considering steady-state effects of ambiguous enzymes. The total concentration of the regulator of the ambiguous enzyme can exert no control whatsoever (at steady state). It is also relevant, however, for whenever the modification enzymes are product sensitive.

The implications of this finding of zero control by important regulatory enzymes (including all the ambiguous kinases/phosphatases, the ATases, the UTases, phosphofructokinase-2) are wide. First, it predicts that the genes encoding or otherwise affecting those enzymes have little phenotype when tested under steady-state conditions. In turn, this feature may account for the absence of clear phenotypes of so many unknown and known genes in the sequenced genomes (56). It also may call for more sophisticated analyses (57). Second, it shows that for (the understanding of) signal transduction, the total concentrations of many of the pivotal proteins are much less important than are their extents of modification. This means that transcriptome analyses may be less important for the understanding of signal transduction than perhaps anticipated. Also the proteome may be less relevant unless one focuses on the state of modification of the signal transduction proteins, which is more difficult than to focus on their total concentration. Transcriptome and total protein analysis should be seen as not so much elucidating signal transduction per se, but rather its adaptation and perhaps neural network-like behavior (2, 58).

Having observed the low response with respect to the total concentration of the signaller, we turned to analyze the response to the signal ratio, defined as the concentration ratio of the modified vs. the unmodified form of the signaller (e.g., the phosphorylated over the unphosphorylated form of the protein). Also, this response was smaller than 1. This finding is also quite important, as one of the presumed features of signal transduction cascades, i.e., of series of enzyme conversion cycles affecting each other, is that the amplification factors of the individual cycles multiply (11–13), leading to substantial signal amplification as the biological function of such a cascade. Our results—that the individual response coefficients do not exceed 1—imply that the perceived amplification function of cascades of interconvertible enzymes is unrealistic, at least when the enzyme are ambiguous, and more generally, because they tend to be product-inhibited.

For ambiguous enzymes, the response coefficient of the ratio of active to inactive modified enzyme with respect to the ratio of modified over unmodified signaller equalled 1. The implication is that for cascades consisting of ambiguous enzymes, the signal strength change is conserved along such cascades. It is neither amplified as proposed previously nor attenuated. This conclusion is quite the opposite from the reigning concept of multiplicative ultrasensitivity as the raison-d'être for signal transduction cascades.

Our findings appear to weaken the importance of one of the proposed biological functions of enzyme conversion cascades. The question arises as to what the most likely function of such cascades or of the individual cycles might be. Because this issue is not the concern of this paper, we limit ourselves to mentioning that our analysis was for steady states only, and that we did not address the possible integration of a number of different signals.

Abbreviations

GS

glutamine synthetase

ATase

adenylyltransferase