Sensitivity analysis of metabolic cascades catalyzed by bifunctional ...
Sensitivity analysis of metabolic cascades catalyzed by bifunctional enzymes Fernando Ortega(1), Måns Ehrenberg(2)*, Luis Acerenza(3), Hans V. Westerhoff(4), Francesc Mas(1), Marta Cascante(5)* (1) Centre de Química Teòrica, (CeRQT-PCB) and Departament de Química Física. Facultat de Química, Universitat de Barcelona (UB). C/ Martí i Franquès, 1. E-08028 BARCELONA (Catalonia, SPAIN). ([email protected]) and ([email protected]) (2) Department of Cell and Molecular Biology, BMC, Box 596, Uppsala University, SE-75 124 UPPSALA, (SWEDEN) ([email protected]) (3) Sección Biofísica. Facultad de Ciencias. Universidad de la República. Iguá 4225, MONTEVIDEO 11400, (URUGUAY) ([email protected]) (4) Stellenbosch Institute for Advanced Study, South Africa and BioCentrum Amsterdam, De Boelelaan 1087, NL-1081 HV AMSTERDAM, EU ([email protected]) (5) Departament de Bioquímica i Biologia Molecular. Facultat de Química and Centre de Química Teòrica, (CeRQT-PCB). Universitat de Barcelona (UB). C/ Martí i Franquès, 1. E-08028 BARCELONA (Catalonia, SPAIN) ([email protected]) * To whom correspondence should be addressed (e-mail [email protected] and [email protected]) Key words: metabolic cascades, bifunctional enzymes, sensitivity analysis, zero-order ultrasensitivity. Abstract Covalent modification/demodification cycles are common in metabolism. When the modification and demodification steps are carried out by two independent enzymes, the degree of modification can be ultrasensitive to the total concentration of either catalyst. We recently showed that the degree of modification of a target molecule cannot exhibit ultrasensitivity to the free concentrations of effectors that decide whether a bifunctional enzyme acts as modifier or demodifier. However, here we can now demonstrate that the degree of modification of a target molecule can display ultrasensitivity to the total, rather than free, concentrations of such effectors. Our results clarify some general aspects of ultrasensitive responses to effectors, including competitive inhibitors, in mono-cyclic cascades. Introduction Goldbeter and Koshland showed macroscopically that cyclic metabolic pathways can display ultrasensitivity in reactions where a target molecule is modified or demodified by either one of two different enzymes (Goldbeter & Koshland, 1981). When both enzymes have small Km-values and low concentrations in relation to the concentration of the target molecule, then near zero order kinetics arises and ultrasensitivity follows. Ultrasensitivity means that a small fractional change in the activity of one of the enzymes brings about a larger fractional change of the degree of modification of the target. The conditions under which such behaviour can arise with two different and independent enzymes of Michaelis-Menten type for the modification and demodification steps have been extensively characterised (Cárdenas & Cornish-Bowden, 1989; Small & Fell, 1990). Berg et al. analyzed such reactions in the mesoscopic perspective and
showed that large fluctuations can occur in the degree of modification of the target and that these can reduce considerably the average degree of ultrasensitivity in relation to the macroscopic approximations (Berg et al., 2000). In vitro experiments have confirmed the existence of ultrasensitivity in modification/demodification cycles, but experimental data corroborating ultrasensitivity in living cells are missing (Ferrell, 1996, van Heeswijk, 1998). Goldbeter and Koshland extended their original analysis to include cases where the modification or demodification enzymes are competitively inhibited (Goldbeter & Koshland, 1984). They showed that for this case a markedly different behaviour in the modification kinetics with respect to a non-competitive inhibitor. Subsequently, it was demonstrated that the fractional change of the degree of modification of the target molecule with respect to the fractional change in the concentration of a competitive inhibitor is always smaller than one (Cárdenas & CornishBowden, 1989). In a previous paper (Ortega et al., 2002), we analyzed commonly occurring schemes where the modifying and demodifying reactions are carried out by the same, bifunctional enzyme and where the direction of the reaction is determined by which one of two effectors that is bound to the enzyme. We concluded that the degree of modification of a target cannot respond with ultrasensitivity to a change in the free concentration of either one of the effectors. These theoretical results were used to interpret experiments (van Heeswijk, 1998) on how the glutamine synthetase (de)adenylylation cascade depends on the intracellular concentration of the pivotal protein PII of E. coli. Expressing the steady-state rate of enzymes in terms of free effector concentrations is a common practice in enzymology. Furthermore, it is often assumed that free and total effector concentrations are the same. In this work, we demonstrate for effectors, including competitive inhibitors, that when their free and total concentrations differ significantly, then the degree of modification of a target molecule can display zero-order ultrasensitivity to these total concentrations. Monocyclic cascades with bifunctional enzymes Systems, where a bifunctional enzyme, controlled by two effector molecules, carries out both modification and demodification reactions, are exemplified by the scheme in Fig 1. Here, the interconvertible enzyme E is either present in an unmodified (Eb) or in a modified (Ea ) form. Conversions between Ea and Eb are catalysed by the enzyme, e, which can be in complex with an effector G or H. Figure 1: Scheme of a monocyclic cascade where Ea and Eb are the different forms of the target enzyme E. It is assumed that the metabolite concentrations A, B, C and D are constant. We represent interconvertion reactions catalysed by two forms, eG and eH, of a single bifunctional enzyme e.
The sensitivity in the degree of modification of E to changes in effector concentrations is y quantified by the response coefficient R p = ( p y ) (∂y ∂y ) (Kacser & Burns, 1973). The modifying and demodifying reactions in Fig. 1 obey the following equations k5 e + G ¨æÆ eG k -5 k6 e + H ¨æÆ eH (1) k -6 k1 k2 eG + Ea ¨æÆ eGEa ææÆ eG + Eb k -1 k3 k4 eH + Eb ¨æÆ eHEb ææÆ eH + Ea k -3 G and H are the free concentrations of the effectors G and H, respectively. We assume that (i) the enzyme is catalytically inactive in the absence of an effector (ii) it has only one type of activity depending on if it is bound to G or H. Conservation of molecules in Eq. (1) gives
ET = Ea + Eb + eGEa + eHEb eT = e + eG + eH + eGEa + eHEb GT = G + eG + eGEa HT = H + eH + eHEb
(2)
ET is the total concentration of E, eT is the total concentration of e, HT and GT are the total concentrations of H and G, respectively. In the case of glutamine synthetase, H and G are the modified and unmodified form of the same molecule, which leads to the additional constraint that HT + GT = constant. Sensitivity analysis of a monocyclic cascade catalysed by bifunctional modifier enzyme Analytical solutions of Eq. (1) are out of reach. However, numerical steady-state analysis can reveal how the concentrations vary as functions of the parameters of the system. Plots of Eb/ET as a function of HT/K6 are shown in Fig. 2 for different values of GT/eT. When GT/eT is small, Eb/ET changes in a narrow range of HT values, but when GT/eT is large, the change is much more gradual. In fact, the transition from Eb to Ea displays zero-order ultrasensitivity when (i) KmEi/ET
showed that large fluctuations can occur in the degree of modification of the target and that these can reduce considerably the average degree of ultrasensitivity in relation to the macroscopic approximations (Berg et al., 2000). In vitro experiments have confirmed the existence of ultrasensitivity in modification/demodification cycles, but experimental data corroborating ultrasensitivity in living cells are missing (Ferrell, 1996, van Heeswijk, 1998). Goldbeter and Koshland extended their original analysis to include cases where the modification or demodification enzymes are competitively inhibited (Goldbeter & Koshland, 1984). They showed that for this case a markedly different behaviour in the modification kinetics with respect to a non-competitive inhibitor. Subsequently, it was demonstrated that the fractional change of the degree of modification of the target molecule with respect to the fractional change in the concentration of a competitive inhibitor is always smaller than one (Cárdenas & CornishBowden, 1989). In a previous paper (Ortega et al., 2002), we analyzed commonly occurring schemes where the modifying and demodifying reactions are carried out by the same, bifunctional enzyme and where the direction of the reaction is determined by which one of two effectors that is bound to the enzyme. We concluded that the degree of modification of a target cannot respond with ultrasensitivity to a change in the free concentration of either one of the effectors. These theoretical results were used to interpret experiments (van Heeswijk, 1998) on how the glutamine synthetase (de)adenylylation cascade depends on the intracellular concentration of the pivotal protein PII of E. coli. Expressing the steady-state rate of enzymes in terms of free effector concentrations is a common practice in enzymology. Furthermore, it is often assumed that free and total effector concentrations are the same. In this work, we demonstrate for effectors, including competitive inhibitors, that when their free and total concentrations differ significantly, then the degree of modification of a target molecule can display zero-order ultrasensitivity to these total concentrations. Monocyclic cascades with bifunctional enzymes Systems, where a bifunctional enzyme, controlled by two effector molecules, carries out both modification and demodification reactions, are exemplified by the scheme in Fig 1. Here, the interconvertible enzyme E is either present in an unmodified (Eb) or in a modified (Ea ) form. Conversions between Ea and Eb are catalysed by the enzyme, e, which can be in complex with an effector G or H. Figure 1: Scheme of a monocyclic cascade where Ea and Eb are the different forms of the target enzyme E. It is assumed that the metabolite concentrations A, B, C and D are constant. We represent interconvertion reactions catalysed by two forms, eG and eH, of a single bifunctional enzyme e.
The sensitivity in the degree of modification of E to changes in effector concentrations is y quantified by the response coefficient R p = ( p y ) (∂y ∂y ) (Kacser & Burns, 1973). The modifying and demodifying reactions in Fig. 1 obey the following equations k5 e + G ¨æÆ eG k -5 k6 e + H ¨æÆ eH (1) k -6 k1 k2 eG + Ea ¨æÆ eGEa ææÆ eG + Eb k -1 k3 k4 eH + Eb ¨æÆ eHEb ææÆ eH + Ea k -3 G and H are the free concentrations of the effectors G and H, respectively. We assume that (i) the enzyme is catalytically inactive in the absence of an effector (ii) it has only one type of activity depending on if it is bound to G or H. Conservation of molecules in Eq. (1) gives
ET = Ea + Eb + eGEa + eHEb eT = e + eG + eH + eGEa + eHEb GT = G + eG + eGEa HT = H + eH + eHEb
(2)
ET is the total concentration of E, eT is the total concentration of e, HT and GT are the total concentrations of H and G, respectively. In the case of glutamine synthetase, H and G are the modified and unmodified form of the same molecule, which leads to the additional constraint that HT + GT = constant. Sensitivity analysis of a monocyclic cascade catalysed by bifunctional modifier enzyme Analytical solutions of Eq. (1) are out of reach. However, numerical steady-state analysis can reveal how the concentrations vary as functions of the parameters of the system. Plots of Eb/ET as a function of HT/K6 are shown in Fig. 2 for different values of GT/eT. When GT/eT is small, Eb/ET changes in a narrow range of HT values, but when GT/eT is large, the change is much more gradual. In fact, the transition from Eb to Ea displays zero-order ultrasensitivity when (i) KmEi/ET
