Information Transfer in Metabolic Pathways:
Effects of Irreversible Steps in Computer Models

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This page contains part 2 of the full text of the following paper: Athel Cornish-Bowden and María Luz Cárdenas (2001) "Information Transfer in Metabolic Pathways: Effects of Irreversible Steps in Computer Models" Eur. J. Biochem. 268, 6616–6624.

RESULTS

Fig. 1. Linear pathway with an irreversible step. The kinetic equations for the component enzymes are listed in Table 1.

Linear pathway with feedback

Table 1. Kinetic parameters for the model illustrated in Fig. 1. The symbols refer to those in eqn. (1) of the text, unless otherwise noted. Metabolite concentrations are shown by lower-case italic letters that correspond to the metabolite names, i.e. s₁ is the concentration of S₁, etc. When numerical values are given for concentrations (i.e. for x₀ and x₆) these were treated as independent of the system, i.e. as constants; where no values are given they were treated as dependent variables and calculated by the simulation program. In all cases units are arbitrary (but consistent), but concentrations can be taken to be in mM if this is considered helpful for understanding.

^aWhen E₁ was assumed to be feedback-inhibited by S₅ a term (s₅/K_fb)² was added to the denominator of the rate expression. ^bThe rate for E₂ was calculated from eqn. (2) of the text, with K_eq = 5, V = 120, a₁ ≡ s₁, a₂ ≡ [ATP] = 8 (constant), p₁ = s₂, p₂ ≡ [ADP] = 2 (constant), K_m1 = 7, K_m2 = 5, K_p1 = 3, K_p2 = 0.8. Note that as the concentrations of ATP and ADP are fixed these metabolites were treated as external to the model. They could not be treated as components of a conservation relationship because the model as defined does not conserve the sum of their concentrations. ^cThe rate for E_3a was calculated from a modified form of eqn. (1) in which p in the disequilibrium term represents s_3as_3b and the denominator is written as a product of three terms (1 + s₂/K_m)(1 + s_3a/K_p1)(1 + s_3b/K_p2), with K_m = 4, K_p1 = 9 and K_p2 = 7. ^dIn some simulations the equilibrium constant of E₄ was treated as infinite (i.e. the disequilibrium term in the numerator was treated as unity), and in some K_p, the Michaelis constant constant of the product, was also treated as infinite, i.e. the term in s₄ was omitted from the denominator. ^eVariation in demand for the end-product S₅ was simulated by varying the value of the limiting rate V for E₆ in the range 0 to 2.4.

The pathway shown in Fig. 1 is more elaborate than the simple one examined earlier [19], and incorporates more of the complications found in real metabolic pathways, but it has the same essential character as a linear biosynthetic pathway with a single end-product S₅, which is consumed by a demand pathway represented in vestigial form by the final step catalysed by E₆. To examine the importance of allowing for the reversibility of nearly irreversible steps the reaction catalysed by E₄ has an equilibrium constant of 1000 in favour of the forward direction, which was treated as infinite in some simulations (i.e. the back reaction was ignored). Feedback inhibition of E₁ by S₅ was likewise considered in some simulations and not in others. Ordinary product inhibition of E₄ by S₄ is shown explicitly in the Figure because although such inhibition is an inevitable property of any reversible enzyme-catalysed reaction it is sometimes ignored in steps considered to be irreversible. Details of the kinetic equations are given in Table 1. Variation in metabolic demand was simulated by varying the limiting rate of E₆. Inclusion of ATP and ADP in the model at constant concentrations is in a sense superfluous, as their concentrations could, of course, be subsumed in the rate constants of a simpler model. Nonetheless, we include them to emphasize that the sort of conclusions we reach here do not depend on the use of oversimplified models in which every enzyme has just one substrate and just one product.

Fig. 2. Simulation of the model in Fig. 1. In each row the left-hand panel (acegi) shows flux control coefficients and (in the inset) the flux through E₆ as a function of the demand expressed by the value of V and the right-hand panel (bdfhj) shows the metabolite concentrations in the same range of con-ditions. The different rows represent different versions of the model, and in each case a schematic version of Fig. 1 is added where there is room for it to give a quick indication of the properties considered, which were as follows: (ab) full model as illustrated in Fig. 1, with feedback inhibition of E₁ by S₅ and E₄ treated as reversible; (cd) E₄ treated as irreversible; (ef) E₄ reversible but no feedback inhibition of E₁; (gh) no feedback inhibition of E₁, and E₄ irreversible, but still inhibited by its product S₄; (ij) no feedback inhibition of E₁, E₄ irreversible and no effect of S₄ on E₄. Note that a more expanded scale for metabolite concentrations is used in the top two rows (bd) in order to make the much smaller variations in concentrations that occur in the presence of feedback inhibition more easily visible.

As seen in Fig. 2a, flux control in the complete model is concentrated in E₆ at low demand; as the demand increases it changes smoothly to majority control by E₁, with a small contribution from E₂. This means that at low demand the synthetic flux is essentially equal to the demand, falling significantly below it only when the demand starts to exceed what the supply can support (inset to Fig. 2a). This satisfactory degree of flux control is achieved without requiring enormous changes in metabolite concentrations (Fig. 2b), even though the feedback inhibition incorporated in the model is weak compared with what occurs in many real systems. The general implications of this sort of behaviour for the regulatory design of pathways in living organisms have been discussed previously [24, 25], and we shall not consider them here.

As long as there is a feedback loop the small degree of reversibility in E₄ is irrelevant. Treating this reaction as strictly irreversible has no perceptible effect on the flux control coefficients (Fig. 2c), on the flux (inset to Fig. 2c), or on the metabolite concentrations (Fig. 2d). It is also irrelevant in the absence of a feedback loop if the irreversible enzyme is subject to ordinary product inhibition, as may be seen by comparing Fig. 2ef with the virtually indistinguishable Fig. 2gh. Thus reversibility as such is no more important in the absence of a feedback loop than it is in the presence of one.

The major change in the model of Fig. 1 comes on eliminating not only the feedback loop and the reversibility of step 4 but also the inhibitory action of S₄ on E₄ (Fig. 2ij). The flux and flux control coefficients then do not change at all at high levels of demand (Fig. 2i), but at low demand there is no steady state at all; the concentrations of the late metabolites rise even more steeply than in Fig. 2f as the demand falls close to the critical level (Fig. 2j), and their uncontrolled increases at this level explains the loss of steady state. In simple terms it occurs because the supply steps produce S₅ faster than the limiting rate (or "maximum velocity" in the terminology often used) of the demand pathway.

These results are consistent with previous observations (e.g. [24]) that apparently similar flux behaviour in different models may well conceal substantial differences in the behaviour of the metabolite concentrations. They suggest that if one is forced by lack of experimental information or by the need to avoid unnecessarily complicated rate equations to use models that are simpler than reality it will normally be better to ignore reversibility in near-irreversible steps than to ignore feedback loops that go around such steps.

The fact that product inhibition must always be possible at high product concentrations does not exclude more complicated behaviour at low concentrations, such as product activation in nitrite reductase [26], but such behaviour is rarely reported and is probably rare in nature, so it is unlikely to be an important consideration in the design of metabolic models; by contrast, ordinary product inhibition is not only theoretically necessary but is also frequently detected experimentally, and must certainly be an important consideration in the design of metabolic models.

Branched pathway with feedback inhibition and cross-activation

Fig. 3. Branched pathway with feedback inhibition and cross-activation. The model is shown in the central part of the Figure, and kinetic parameters are listed in Table 2. Each of the panels (a) to (e) shows the response to changes in the demand in step 4a of the control coefficients for flux through branch a; in each case the inset shows the corresponding changes in the three fluxes over the same range of demand. In all cases the axes and scales are as shown explicitly in Panel c. The grey curves linking the panels to the scheme in the centre of the Figure show which special effects apply to each panel. In Panel (a) there were no special effects, i.e. no reversibility in step 3a, no feedback inhibition of step 2a and no cross activation of step 2b. Each of the other panels shows the effect of adding just one line of communication from S_3a to E₁. Panel (b) shows the effect of allowing for the reverse reaction in step 3a (but no feedback or activation of the other branch); Panel (c) shows the effect of feedback inhibition of E_2a by S_3a (while keeping step 3a irreversible); Panel (d) shows the effect of moderate activation of E_2b by S_3a (with no feedback inhibition of E_2a or reversibility in step 3a); and Panel (e) shows the effect of almost unlimited activation of E_2b by S_3a (again, without any other effects).

Table 2. Kinetic parameters for the model illustrated in Fig. 3. The caption to Table 1 applies mutatis mutandis to this Table.

^aE₁ followed the reversible Hill equationÊ[22] with a Hill coefficient of 2.5, so the values listed as K_m and K_p are actually the half-saturation concentrations of substrate and product respectively. When feedback inhibition by S_3a was present the half-inhibition concentration of S_3a was 0.2. ^bWhen E_2a was inhibited by S_3a, i.e. in Fig. 3c, the inhibition was competitive with inhibition constant 0.5. ^cThe arbitrary appearance of the values for K_eq for E_2a and E_3b is an artifact of the transformation to the form of eqn. (1) of rate equations that were expressed in a different way at the time the simulations were done. ^dWhen E_3a was treated as irreversible, K_eq and K_p were infinite. ^eThe limiting rate of E_4a was varied in the range 0 to 2 to simulate variation in demand for the end-product S_3a of branch (a) of the pathway. ^fWhen E_2b was weakly activated by S_3a, i.e. in Fig. 3d, the specific activation constant was 12 and the other kinetic parameters were as follows: V = 450, K_m = 0.6, K_p = 1; when very strong activation was allowed, i.e. in Fig. 3e, V was increased to 45000 and the activation constant to 1200. Note that none of these changes affect the value of the equilibrium constant K_eq.

Fig. 3 illustrates a more complex model designed to show whether the conclusions drawn from the simple linear pathway apply more generally. It consists of a branched supply pathway that feeds two different demand processes. One end-product, S_3a, inhibits E_2a, the first committed enzyme in its branch, but it has no feedback effect on E₁, the first enzyme in the complete pathway; as the reaction that produces it, catalysed by E_3a, is virtually irreversible it has no direct effect through the pathway either unless the small degree of reversibility is allowed for. However, it may have an indirect effect on E₁ if it activates E_2b, the first committed enzyme of the other branch, because activation of this enzyme tends to increase the concentration of the other end-product, S_3b, which does exert feedback inhibition on E₁. If a flow of information from S_3a to E₁ is all that is required to ensure a satisfactory flux response to low demand in step 4a then one might expect that this circuitous route via activation of E_2b and inhibition of E₁ by S_3b could compensate for irreversibility of E_3a and lack of a feedback inhibition loop around it.

Results obtained with various versions of this model confirm this expectation to some degree, though they also indicate that reality is less simple than suggested by the simplest model. First of all, in the absence of any information transfer from S_3a to the beginning of the pathway—no reversibility in E_3a, no feedback inhibition of E_2a and no feedback activation of E_2b—there is no steady state at low demand and no variation in either flux or metabolite concentrations at high demand (Fig. 3a). Making E_3a reversible allows a steady state to exist at low demand, but the transition from one regime to the other is extremely abrupt (Fig. 3b). Introduction of feedback inhibition of E_2a by S_3a (without making the reaction catalysed by E_3a reversible) has qualitatively a similar effect (Fig. 3c) but the transition is now much smoother, closer to the way one would expect a real metabolic system to behave.

The question now is whether circuitous information transfer via activation of the other branch can compensate for lack of feedback inhibition. As seen in Figs. 3d–e, this type of effect can give a satisfactory response, mimicking feedback inhibition of E_2a by S_3a to some degree, but to maintain a steady state at very low demand the activation must be strong. If the maximum degree of activation is insufficient the region in which there is no steady state is not completely eliminated (Fig. 3d), but it can be made negligible by allowing very substantial activation (Fig. 3e). This result contrasts with those in Figs. 2b and 3c, in which weak feedback inhibition is seen to be sufficient to maintain a steady state at low demand.

This observation is easily rationalized, and it illustrates a practical difference between inhibition and activation that may have important implications for the regulatory design of metabolic systems in nature. Enzyme inhibition is usually linear or complete, which means that the rate approaches zero at saturating concentrations of inhibitor; the more complicated phenomenon of hyperbolic or partial inhibition is reported much more rarely (though it is unclear if it is really rare in nature or just rarely noticed). By contrast, reported activation effects are often hyperbolic, in the sense that an enzyme subject to activation often shows some activity in the absence of activator, and in addition, the amount of activation possible is limited by the limiting rates of the enzymes involved. In the case of Fig. 3, if the limiting rate of E_3b is not large enough to raise the concentration of S_3b sufficiently to produce a strong effect on E₁ then no amount of activation of E_2b will avoid the loss of the steady state at low levels of demand for S_3a. This argument may appear very complicated, but the essential point is quite simple: effects that depend on linear inhibition can always decrease a flux to whatever is necessary to ensure a steady state, but effects that depend on activation may fail to do so if the fully activated pathway is not active enough to remove its intermediates as fast as they arrive.

Parallel routes

Fig. 4. Parallel routes between two metabolites. The reactions are shown in Panel (a), and the kinetic parameters in Table 3. Panel (b) shows the response of the intermediate concentrations to changes in the demand, represented by values in the limiting rate of E₅ in the range 0 to 2. Panel (c) shows the variations in the three distinct fluxes over the same range of demand. Note that for about half of this range the flux in branch b is negative, i.e. the pathway operates as a cycle rather than as two parallel routes. Panel (d) shows the control coefficients for the flux through branch a over the same range of demands.

Table 3. Kinetic parameters for the model illustrated in Fig. 4. The caption to Table 1 applies mutatis mutandis to this Table.

Fig. 4 illustrates how information may be transferred around an irreversible reaction without requiring any regulatory effects at all in the usual sense of feedback or feedforward loops. In this model (Fig. 4a) the end-product S₄ is produced from the same precursor X₀ by two parallel routes, one of which contains an irreversible reaction catalysed by E_2a. (This implies, of course, that the equilibrium constant around the cycle is not unity, which in turn is only possible if additional metabolites are involved, such as ATP, ADP, water and inorganic phosphate. However, the conditions necessary for such futile cycles to exist are well known [e.g. 27] and do not require discussion here). Although there is no feedback loop in the system excessive production of S₄ by the irreversible route can still be signalled to E₁ by the reversible steps in the parallel route, and so in principle one may expect this system to be capable of reaching a steady state even at low demand for the product. This proves to be the case, though the stability of the flux response at low demand is achieved at the expense of enormous increases in the concentrations of all the intermediates, especially S₄ (Fig. 4b). Moreover, the apparently smooth response of the common flux through E₁ and E₅ conceals rather more complicated behaviour in the two parallel routes (Fig. 4c), which are seen to be truly parallel only at high demand, because at low demand the flux through branch a is much larger than the common flux and has to be balanced by negative flux in branch b. In other words at low demand the flux in branch b is from right to left in the Figure, and maintenance of the cycling must imply continuous hydrolysis of ATP or a similar process. Moreover, over the entire range, even at high demand, the flux in branch a changes in the opposite direction to the change in demand, i.e. it decreases as the demand increases, and vice versa. The smooth behaviour of the common flux likewise conceals behaviour in the control coefficients for flux through branch a that is not at all smooth (Fig. 4d): all of the flux control coefficients show abrupt transitions at the point where the flux in branch b changes sign, and two of them are infinite at this point.

This example of forcing flow into a parallel pathway is interesting, because although the model in Fig. 4 was conceived as a simple example of parallel pathways, it succeeds in switching off the net flow at low demand only by reversing the flow in the second route. This means that the central part of the pathway ceases to be a pair of parallel routes but becomes a cycle instead, implying a plausible evolutionary mechanism for the appearance of cycles in metabolism. It is easy to conceive of how duplication of the genes coding for several successive enzymes in a linear pathway might produce several isoenzymes in succesion. Subsequent mutations could then convert such a succession of isoenzymes into two distinct parallel routes between a given pair of metabolites (S₁ and S₄ in the symbolism of Fig. 4). In this connection it is interesting to note that although the tricarboxylate cycle is commonly represented in textbooks as a cycle, there are organisms such as Methylococcus capsulatus [28] where it does not operate as a cycle because α-ketoglutarate dehydrogenase is missing.

Continued...