Tài liệu Báo cáo khoa học: Steady-state kinetic behaviour of functioning-dependent structures docx

Steady-state kinetic behaviour of functioning-dependent structures Michel Thellier1,3, Guillaume Legent1, Patrick Amar2,3, Vic Norris1,3 and Camille Ripoll1,3 ´ ´ ´ ´ Laboratoire ‘Assemblages moleculaires: modelisation et imagerie SIMS’, Faculte des Sciences de l’Universite de Rouen, Mont-Saint-Aignan Cedex, France ´ Laboratoire de recherche en informatique, Universite de Paris Sud, Orsay Cedex, France Epigenomics Project, GenopoleÒ, Evry, France Keywords enzyme kinetics; metabolic or signalling pathways; mathematical modelling; protein associations Correspondence M Thellier, Laboratoire Assemblages ´ ´ moleculaires: modelisation et imagerie SIMS ´ FRE CNRS 2829, Faculte des Sciences de ´ l’Universite de Rouen, F-76821 Mont-SaintAignan Cedex, France Fax: +33 35 14 70 20 Tel: +33 35 14 66 82 E-mail: Michel.Thellier@univ-rouen.fr (Received 12 January 2006, revised 26 June 2006, accepted 20 July 2006) A fundamental problem in biochemistry is that of the nature of the coordination between and within metabolic and signalling pathways It is conceivable that this coordination might be assured by what we term functioning-dependent structures (FDSs), namely those assemblies of proteins that associate with one another when performing tasks and that disassociate when no longer performing them To investigate a role in coordination for FDSs, we have studied numerically the steady-state kinetics of a model system of two sequential monomeric enzymes, E1 and E2 Our calculations show that such FDSs can display kinetic properties that the individual enzymes cannot These include the full range of basic input ⁄ output characteristics found in electronic circuits such as linearity, invariance, pulsing and switching Hence, FDSs can generate kinetics that might regulate and coordinate metabolism and signalling Finally, we suggest that the occurrence of terms representative of the assembly and disassembly of FDSs in the classical expression of the density of entropy production are characteristic of living systems doi:10.1111/j.1742-4658.2006.05425.x Numerous studies have shown that proteins involved in metabolic or signalling pathways are often distributed nonrandomly, as multimolecular assemblies [1–15] Such assemblies range from quasi-static, multienzyme complexes (such as the fatty acid synthase or the a-oxo acid dehydrogenase systems [5]) to transient, dynamic protein associations [2,3,7,15,16] Comparison of yeast and human multiprotein complexes has shown that conservation across species extends from single proteins to protein assemblies [11] Multi-molecular assemblies may comprise proteins but also nucleic acids, lipids, small molecules and inorganic ions Such assemblies may interact with membranes, skeletal elements and ⁄ or cell organelles [3,4,15,17] They have been termed metabolons, transducons and repairosomes in the case of metabolic pathways [3,10,18–23], signal transduction [24] and DNA repair [12], respectively, or, more generally, hyperstructures [17,25–28] According to Srere [3], metabolons are enzyme assemblies in which intermediates are channelled from each enzyme to the next without diffusion of these intermediates into the surrounding cytoplasm [2– 7,9,15,23,29–33] Potential advantages of channelling [7,9,15,30,31,34,35] are (i) reduction in the size of the pools of intermediates (a point, however, contested by some authors [36,37]), (ii) protection of unstable or scarce intermediates by maintaining them in a proteinbound state, (iii) avoidance of an ‘underground’ metabolism in which intermediates become the substrates of other enzymes [38], and (iv) protection of the cytoplasm from toxic or very reactive intermediates The terms static and dynamic channelling have been used to describe, respectively, the channelling in a quasipermanent metabolon and in a transient association between two enzymes occurring while the intermediate metabolite is transferred from the first enzyme to the second [39,40] Abbreviation FDS, functioning-dependent structure FEBS Journal 273 (2006) 4287–4299 ª 2006 The Authors Journal compilation ª 2006 FEBS 4287 Functioning-dependent structures M Thellier et al We propose here to generalize the concept of dynamic channelling or, more precisely, the concept of a structure that dynamically and transiently forms to carry out a process, into that of functioning-dependent structure (FDS) [41] In other words, an FDS is a dynamic, multimolecular structure that assembles when functioning and that disassembles when no longer functioning, and thus is created and maintained by the very fact that it is in the process of accomplishing a task The lifetime of such a structure may be short or long, depending only on the duration of the process that is catalysed by the FDS An FDS catalyses efficiently the processes that have allowed this FDS to form It can therefore be viewed as a self-organized structure Published examples of transient, dynamic multimolecular assemblies, that only form in an activitydependent manner include: the role of the bifunctional protein complex cysteine synthetase in the synthesis of cysteine in Salmonella typhimurium [42]; the metabolite-modulated formation of complexes (especially binary complexes) of sequential glycolytic enzymes [4,43,44]; the functional coupling of pyruvate kinase and creatine kinase via an enzyme–product–enzyme complex in muscle [45]; the interaction between serine acetyl-transferase and O-acetylserine(thiol)-lyase in higher plants [46,47]; the ATP- and pH-dependent association ⁄ dissociation of the V1 and V0 domains of the yeast vacuolar H+- ATPases [48–50]; the promotion by substrate binding of the assembly of the three components of protein-mediated exporters involved in protein secretion in Gram-negative bacteria [51]; the first step of glycogenolysis in vertebrate muscle tissues by the sequential formation of a phosphorylase–glycogen complex followed by the binding of phosphorylase kinase to this previously formed complex [18]; the clustering of the anchoring protein gephyrin with glycine receptors following glycine receptor activation in postsynaptic regions of spinal neurons [52–55]; the clustering of antigen receptors followed by binding of intracellular proteins, such as protein tyrosine kinases, to the cytoplasmic portion of the receptors in the case of signalling through lymphocyte receptors (reviewed in [56]); the organization of functional rafts in the plasma membrane upon T-cell activation [57]; the glycine decarboxylase complex in higher plants [58]; the assembly of water-soluble, cytosolic proteins with the membrane-anchored flavocytochrome b558 for the catalysis of the NADPH-dependent reduction of O2 into the superoxide anion O2– in stimulated phagocytic cells [59]; the dynamic association of HSP90 with the RPM1 disease resistance protein in the response of Arabidopsis plants to infection by Pseudomonas syringae [60]; the association of protein complexes with 4288 assembling actin molecules in the lamellipodium tip of moving cells [61]; the clustering of glutamate receptors opposite the largest and most physiologically active sites of presynaptic release [62]; the differential nucleotide-dependent binding of Bfp proteins in the transduction of mechanical energy to the biogenesis machine of Escherichia coli [63] Even the Golgi apparatus of Saccharomyces cerevisiae can be viewed as a dynamic structure with a size that depends on its functioning such that it grows when it is secreting and shrinks when it is not [64–67] It is striking that these cellular systems that have very different structures and functions nevertheless exhibit the common behaviour of assembling into transient complexes or FDSs when functioning Why? A fundamental problem in biochemistry is that of coordination The functioning of a protein in a metabolic or signalling pathway in vivo is coordinated with that of the other proteins in the same pathway, and the functioning of the pathway itself is coordinated with that of the other pathways within the cell In metabolic pathways, the regulation needed for such coordination comes in part from the sigmoidal kinetics provided by allosteric enzymes, due to the fact that subunit–subunit interactions are added to the classical enzyme–substrate interactions [68] It is therefore tempting to speculate that FDSs are involved in the coordination within and between metabolism and signalling If FDSs are to have a central role in coordination, they should be predicted to generate regulatory kinetics via the enzyme–enzyme interactions that constitute them In the following, we have endeavoured to test this prediction by numerically studying the steady-state kinetics of a model system of two sequential monomeric enzymes, E1 and E2, which, when free, are of the Michaelis–Menten type (i.e., with a single substratebinding site and no regulatory site) Our results show that the metabolite-induced association of these two enzymes into an FDS [20] may, under steady-state conditions, confer to the FDS basic regulatory kinetic features, that the individual enzymes lack These include the full range of input ⁄ output characteristics found in electronic circuits such as a linear relationship between input and output, an output limited to a narrow range of inputs, a constant output whatever the input, and even switch-like behaviours (Fig 1) Hence a metabolite-induced FDS could generate a wide variety of kinetics that could serve as signals Modelling a two-enzyme FDS The different substances and reactions that can possibly take place when an FDS is involved in the overall trans- FEBS Journal 273 (2006) 4287–4299 ª 2006 The Authors Journal compilation ª 2006 FEBS M Thellier et al Functioning-dependent structures output A B output input input output C D output (b) (a) input input Fig Classical input ⁄ output relationships in electrical circuits (A) Linear response: this behaviour is obtained when a generator is connected to a load (resistor) (B) Constant response: this behaviour is obtained when a source of current is connected to a load; whatever the value of the load, and therefore whatever the value of the potential difference, the current is unchanged (C) Impulse response: the output is non-null only for a particular value (or a narrow range of values) of the input (D) curve (a): Step response: this behaviour corresponds to a switch from low or null current to high current when the potential difference exceeds a threshold; curve (b): Inverse step response: this behaviour corresponds to a switch from high current to low or null current when the potential difference exceeds a threshold formation of an initial substrate, S1, into a final product, S3, via reactions catalysed by two enzymes, E1 and E2, are represented in Fig In total, 29 reactions act on 17 substances (free substances and complexes) and, to account for a formation of the FDS solely dependent on its activity, the reaction E1+E2 ¼ E1E2 does not exist in this scheme Note that the symbols used in Fig to describe the complexes are such that E1S2E2 and E1E2S2 mean that S2 is bound to the catalytic site of E1 or of E2, respectively, within the FDS, etc To write down the steady-state conditions of functioning of the system (further details given in Appendix), (i) we assume that external mechanisms supply S1 and remove S3 as and when they are consumed and produced, respectively, such that S1 is maintained at a constant concentration and S3 at a zero concentration, and (ii) we use the set of algebraic equations obtained by writing down the mass balance of the 15 other species involved For convenience, we have reasoned using dimensionless variables (note that capital letters are used for chemical species and small letters for dimensionless concentrations) We have also taken into account the fact that the law of mass action has to be satisfied whatever the pathway from S1 to S3 When all calculations are carried out for any given value of the concentration, s1, of S1, the steady-state rate of transformation of S1 into S3 is calculated as corresponding to both the rate of consumption of S1, v(s1), and the rate of production of S3, v(s3), and the shape of the curves {s1, v(s1)} is examined in cases involving either free enzymes alone or an FDS with free enzymes It is worth noting that it would only be necessary to add a few more reactions to Fig to describe the interaction of these enzymes with other proteins or molecules and hence study systems in which, for example, small proteins contribute to the formation of the enzyme–enzyme complexes [15]; the theoretical treatment would be longer but otherwise essentially the same as that followed here Results Kinetics of the overall reaction of transformation of S1 into S3 The system with only the free enzymes, E1 and E2 The overall rate of functioning of two free sequential enzymes of the Michaelis–Menten type involved in a metabolic pathway has already been computed as a function of the concentration of initial substrate under FEBS Journal 273 (2006) 4287–4299 ª 2006 The Authors Journal compilation ª 2006 FEBS 4289 Functioning-dependent structures M Thellier et al Fig The scheme of the reactions involved in the functioning of our model of a two-enzyme FDS The system comprises 17 different chemical species (free enzymes, free substrates or products, and binary, ternary or quaternary complexes) indicated in the green circles These species are linked to one another by 29 chemical reactions numbered R1 to R29 as indicated in the rectangles steady-state conditions [69] The results are summarized in Fig 3A Briefly, curves monotonically increasing up to a plateau and exhibiting no inflexion points were obtained for all parameter values tested Occasionally, the shape of these curves was close to that of a hyperbola Cases existed (with the smallest K2 values in Fig 3A) in which the overall rate of reaction became a quasi-linear function of the concentration of initial substrate, s1, almost up to the plateau (which never occurs when a single enzyme is involved) Hence, under certain conditions, free enzymes can generate signals or other behaviours corresponding to a linear relationship between input (concentration of first substrate) and output (rate of production of final product) (Fig 1A) The system with an FDS At some parameter values, in the case of an FDS, the {s1, v(s1)} curves were similar to those obtained with the free enzymes, i.e., they increased monotonically without an inflexion point up to a plateau and sometimes exhibited an extended linear response with v(s1) proportional to s1 over a large range of s1 values (Fig 3B, curves c and d) However, at other parameter values, the {s1, v(s1)} curves exhibited a variety of forms that were not found with the free enzymes For instance, in Fig 3B, the curves (a) and (b) exhibited substrate-inhibition behaviour, i.e., with increasing s1, the rate of consumption of S1 initially increased then, after reaching a maximal value, decreased The occurrence of {s1, v(s1)} curves with a substrateinhibition shape was examined further (Fig 4) At some parameter values, with increasing s1, the rate of 4290 consumption of S1 decreased to almost zero (Fig 4A) This means that this FDS system exhibited a sort of inversed behaviour in which it was active at low s1 values (except at the very lowest s1 values) and inactive at the high s1 values This corresponds to the scenario in Fig 1C in which an increasing input leads to an output in the form of a spike or impulse Another case in which an increasing input leads to an output in the form of an impulse (i.e., corresponding to the scenario in Fig 1C) is depicted in Fig 4B At other values of the parameters, with increasing s1, the rate of consumption of S1 again increased, reached a maximal value, then decreased, whilst at saturating values of s1 the rate of consumption of S1 reached a plateau (instead of decreasing to zero) (Fig 4C) Moreover, at the largest K1 values (K1 ¼ 104), the rate of consumption of S1 almost immediately reached the plateau (Fig 4C, curve d), which means that the response of the system became effectively independent of s1 (except again at the very lowest s1 values) This corresponds to the scenario in Fig 1B in which the output is independent of the input A curve is shown (Fig 4D) that over a wide range of low values of s1 has a relatively constant and high rate of consumption of S1 but that with higher values of s1 drops rapidly to a constant and low rate of consumption This resembles the switch shown in Fig 1D curve (b) Curves with a sigmoid shape, i.e., resembling the switch shown in Fig 1D curve (a), were sometimes obtained (Fig 5A) At the parameter values tested, however, the adjustment of the curve to a Hill function v(s1) ¼ vmaxỈ(s1)n ⁄ [(k)n+(s1)n] (in which n is the Hill FEBS Journal 273 (2006) 4287–4299 ª 2006 The Authors Journal compilation ª 2006 FEBS M Thellier et al Functioning-dependent structures v(s1) v(s1) 0.2 0.18 d e d b c 0.1 c b 0.12 a a 0.06 A B 0 0.01 0.02 s1 0.05 0.1 s1 Fig Examples of computed {s1, v(s1)} curves (A) Case of a system made of two free enzymes: the parameter values are e1t ¼ e2t ¼ 0.5, K ¼ 100, k1r ¼ (Eqn A6), k2r ¼ 100, k3r ¼ k4r ¼ k9r ¼ k10r ¼ 1, k4f calculated according to Eqn (A25), K1 ¼ 10, K3 ¼ 100, K9 ¼ K10 ¼ and K2 ¼ 0.10 (curve a), 0.05 (curve b), 0.01 (curve c), 0.001 (curve d) and 0.0001 (curve e) Modified from [69] (B) Case of a two-enzyme FDS: the parameter values are e1t ¼ e2t ¼ 0.5, K ¼ 100, k1r ¼ (Eqn A6), k2r ¼ 100, k3r ¼ k4r ¼ k5r ¼ k6r ¼ k7r ¼ k8r ¼ k9r ¼ k10r ¼ k11r ¼ k12r ¼ k13r ¼ k14r ¼ k15r ¼ k16r ¼ k17r ¼ k18r ¼ k19r ¼ k20r ¼ k21r ¼ k22r ¼ k23r ¼ k24r ¼ k25r ¼ k26r ¼ k27r ¼ k28r ¼ k29r ¼ 1, K1 ¼ 10, K2 ¼ 0.01, K5 ¼ 1000, K3 ¼ K10 ¼ K11 ¼ K12 ¼ K13 ¼ K15 ¼ K17 ¼ K29 ¼ 1, K27 ¼ 100, K9 ¼ 10 (curve a), 102 (curve b), 103 (curve c), 104 (curve d) and all the other Kj calculated as indicated in Eqns (A25) to (A27) and Table A2 coefficient, vmax is the maximal rate of reaction and k is the value of s1 that gives v(s1) ẳ 0.5ặvmax) was not entirely satisfactory because a perfect straight line was not obtained (r2 ¼ 0.985) when using the Hill system of coordinates, {log s1, log [v(s1) ⁄ (vmax–v(s1))]} (Fig 5B); moreover, the sigmoidicity was rather weak (Hill coefficient equal to only 1.47) There were cases in which even more complicated responses occurred For example, in Fig in which K10 was varied from to 103 and in which all the other parameters have the values given in the figure caption, a {s1, v(s1)} curve similar to those in Fig 4C and with a low plateau value was observed with the smallest K10 values (Fig 6, curve a) while the substrate-inhibition effect was less and the plateau was higher with increasing K10 values (Fig 6, curve b) Finally, with the highest values of K10 (Fig 6, curves c and d), the {s1, v(s1)} curves increased monotonically to a plateau but with two inflexion points that conferred on them a dual-phasic aspect Dual-phasic kinetic curves are often exhibited by both natural and artificial enzymatic and transport systems [70–72]; although the functional advantage of such kinetics is not clear, it is interesting that this complex behaviour can be revealed by an FDS with as few as two enzymes Discussion The consequences of channelling on metabolism have been extensively explored by modelling In channelling, the intermediate metabolites are confined to very small volumes within a metabolon and have short half-lives It may therefore be invalid to assume that the local statistical distribution of any molecule is Poissonian and therefore that the classical macroscopic law of kinetics can be used to describe the reaction rates [29,73–75] Indeed, certain models based on this invalid assumption may even lead to an apparent violation of the second law of thermodynamics [73] The model developed here is based on the classical macroscopic laws of kinetics but, importantly, is self-consistent in the sense that it uses the same assumptions to determine and compare the kinetics of two enzymes freely diffusing or assembled into a FDS Numerous command or control devices used in engineering are made from elements with input ⁄ output functions as shown in Fig In electronics, these functions include the linear function obtained when a source of potential difference is connected to a resistor (Fig 1A), the constant function obtained when a current source is connected to a resistor (Fig 1B), the impulse function (Fig 1C) and the increasing (Fig 1D, curve a) or decreasing (Fig 1D, curve b) step function We have shown here that the assembly of only two enzymes can result in a variety of input ⁄ output relationships including, importantly, those with characteristics similar to these basic functions Hence, the assembly of just two enzymes could provide a macromolecular mechanism for control processes This is illustrated by the following examples The substrate concentration could be encoded in a linear response (Fig 1A) (Note that we occasionally obtained linear responses from a system of FEBS Journal 273 (2006) 4287–4299 ª 2006 The Authors Journal compilation ª 2006 FEBS 4291 Functioning-dependent structures M Thellier et al v(s1) v(s1) 0.04 0.0008 B A 0.0004 0.02 0.0000 0.05 s1 0.1 0 v(s1) 0.08 s10.1 v(s1) a b 0.04 05 C c 0.02 D 0.01 d 0 0.02 0.04 s1 0 0 s10.1 Fig Various types of substrate-inhibition {s1, v(s1)} curves computed in the case of a two-enzyme FDS (A) Example of an almost total inhibition at high s1 values (impulse behaviour): the parameter values are e1t ¼ e2t ¼ 0.5, K ¼ 100, k1r ¼ (Eqn A6), k2r ¼ 104, k3r ¼ k4r ¼ k5r ¼ k6r ¼ k7r ¼ k8r ¼ k9r ¼ k10r ¼ k11r ¼ k12r ¼ k13r ¼ k14r ¼ k15r ¼ k16r ¼ k17r ¼ k18r ¼ k19r ¼ k20r ¼ k21r ¼ k22r ¼ k23r ¼ k24r ¼ k25r ¼ k26r ¼ k27r ¼ k28r ¼ k29r ¼ 1, K1 ¼ 10, K2 ¼ 0.0001, K5 ¼ 106, K3 ¼ K9 ¼ K10 ¼ K11 ¼ K12 ¼ K13 ¼ K17 ¼ 1, K15 ¼ K27 ¼ 100, K29 ¼ 1000 and all the other Kj calculated as indicated in Eqns (A25) to (A27) and Table A2 (B) Another example of an impulse behaviour: the parameter values are e1t ¼ e2t ¼ 0.5, K ¼ 1000, k1r ¼ (Eqn A6), k2r ¼ 104, k3r ¼ k4r ¼ k5r ¼ k6r ¼ k7r ¼ k8r ¼ k9r ¼ k10r ¼ k11r ¼ 1, k12r ¼ 103, k13r ¼ k14r ¼ k15r ¼ k16r ¼ k17r ¼ k18r ¼ k19r ¼ k20r ¼ k21r ¼ k22r ¼ k23r ¼ k24r ¼ k25r ¼ k26r ¼ k27r ¼ k28r ¼ 1, k29r ¼ 104, K1 ¼ 10, K2 ¼ 0.0001, K3 ¼ 1000, K5 ¼ 106, K9 ¼ K10 ¼ K11 ¼ 1, K12 ¼ 0.001, K13 ¼ 100, K15 ¼ 1000, K17 ¼ 1, K27 ¼ 100, K29 ¼ 10000 and all the other Kj calculated as indicated in Eqns (A25) to (A27) and Table A2 (C) Examples of an only partial inhibition at high s1 values: the parameter values are e1t ¼ e2t ¼ 0.5, K ¼ 100, k1r ¼ (Eqn A6), k2r ¼ 100, k3r ¼ k4r ¼ k5r ¼ k6r ¼ k7r ¼ k8r ¼ k9r ¼ k10r ¼ k11r ¼ k12r ¼ k13r ¼ k14r ¼ k15r ¼ k16r ¼ k17r ¼ k18r ¼ k19r ¼ k20r ¼ k21r ¼ k22r ¼ k23r ¼ k24r ¼ k25r ¼ k26r ¼ k27r ¼ k28r ¼ k29r ¼ 1, K1 ¼ 10 (curve a), 102 (curve b), 103 (curve c) and 104 (curve d), K2 ¼ 0.01, K5 ¼ 103, K9 ¼ 10, K3 ¼ K10 ¼ K11 ¼ K12 ¼ K13 ¼ K15 ¼ K17 ¼ K29 ¼ 1, K27 ¼ 100 and all the other Kj calculated as indicated in Eqns (A25) to (A27) and Table A2 (D) Example of an inversed step response: the parameter values are e1t ¼ e2t ¼ 0.5, K ¼ 1000, k1r ¼ (Eqn A6), k2r ¼ 104, k3r ¼ k4r ¼ k5r ¼ k6r ¼ k7r ¼ k8r ¼ k9r ¼ k10r ¼ k11r ¼ 1, k12r ¼ 103, k13r ¼ k14r ¼ k15r ¼ k16r ¼ k17r ¼ k18r ¼ k19r ¼ k20r ¼ k21r ¼ k22r ¼ k23r ¼ k24r ¼ k25r ¼ k26r ¼ k27r ¼ k28r ¼ 1, k29r ¼ 104, K1 ¼ 10, K2 ¼ 0.0001, K3 ¼ 75, K5 ¼ 106, K9 ¼ K10 ¼ K11 ¼ 1, K12 ¼ 0.001, K13 ¼ 100, K15 ¼ 1000, K17 ¼ 1, K27 ¼ 100, K29 ¼ 10000 and all the other Kj calculated as indicated in Eqns (A25) to (A27) and Table A2 two enzymes that diffused freely, i.e., without FDS.) Homeostasis results when, despite the concentration of the initial substrate, s1, varying, the rate of production of the final product is constant (Fig 1B) An impulse that could constitute a signal, results when, at a narrow range of low concentrations of substrate s1, the rate of production of the final product takes the form represented in Fig 1C (Fig 4A,B show a more realistic representation) A switch as represented in Fig 1D (curve a) could be based on the sigmoid curve in the production rate A switch from a high rate to a low rate of production occurs 4292 when s1 exceeds the threshold s0 at the inflection point (Fig 4D) and this could correspond to a substrate-inhibition behaviour Hence the assembly of two enzymes into an FDS could allow a switch behaviour Alternatively, it could allow this enzyme system to be efficient at a low substrate concentration but not at a high concentration where the substrate would become available for enzymes in a different metabolic pathway A strongly sigmoid curve from low to high rates of production was not revealed by our calculations (see above) Weakly sigmoid curves from low to high rates FEBS Journal 273 (2006) 4287–4299 ª 2006 The Authors Journal compilation ª 2006 FEBS M Thellier et al Functioning-dependent structures v(s1) log z1 0.08 A B 0.04 -3 log s1 -2 -1 -1 -2 0 0.1 -3 s1 Fig Example of a sigmoid {s1, v(s1)} curve computed in the case of a two-enzyme FDS The parameter values are e1t ¼ e2t ¼ 0.5, K ¼ 100, k1r ¼ (Eqn A6), k2r ¼ 10, k3r ¼ k4r ¼ k5r ¼ k6r ¼ k7r ¼ k8r ¼ k9r ¼ k10r ¼ k11r ¼ k12r ¼ k13r ¼ k14r ¼ k15r ¼ k16r ¼ k17r ¼ k18r ¼ k19r ¼ k20r ¼ k21r ¼ k22r ¼ k23r ¼ k24r ¼ k25r ¼ k26r ¼ k27r ¼ k28r ¼ k29r ¼ 1, K1 ¼ K2 ¼ 0.1, K3 ¼ 10, K5 ¼ 1000, K9 ¼ K10 ¼ K11 ¼ K12 ¼ K13 ¼ K15 ¼ K17 ¼ K29 ¼ 1, K27 ¼ 100 and all the other Kj calculated as indicated in Eqns (A25) to (A27) and Table A2 (A) Curve represented using the direct system of coordinates, {s1, v(s1)} (B) Curve represented using the Hill system of coordinates, {log s1, log z1} with z1 ¼ {v(s1) ⁄ [vmax–v(s1)]}; from the slope of the dashed regression line fitted to the curve, the Hill coefficient was estimated to be of the order of 1.47 v(s1) 0.4 d c 0.2 b a 0 0.1 s1 0.2 Fig Examples of dual-phasic {s1, v(s1)} curves computed in the case of a two-enzyme FDS The parameter values are e1t ¼ e2t ¼ 0.5, K ¼ 100, k1r ¼ (Eqn A6), k2r ¼ 100, k3r ¼ k4r ¼ k5r ¼ k6r ¼ k7r ¼ k8r ¼ k9r ¼ k10r ¼ k11r ¼ k12r ¼ k13r ¼ k14r ¼ k15r ¼ k16r ¼ k17r ¼ k18r ¼ k19r ¼ k20r ¼ k21r ¼ k22r ¼ k23r ¼ k24r ¼ k25r ¼ k26r ¼ k27r ¼ k28r ¼ k29r ¼ 1, K1 ¼ K9 ¼ 10, K2 ¼ 0.01, K3 ¼ K11 ¼ K12 ¼ K13 ¼ K15 ¼ K17 ¼ K29 ¼ 1, K5 ¼ 1000, K27 ¼ 100, K10 ¼ (curve a), 10 (curve b), 102 (curve c) and 103 (curve d) and all the other Kj calculated as indicated in Eqns (A25) to (A27) and Table A2 of production were sometimes observed with Hill coefficients of less than (Fig 5) but these could not constitute switches Compared with the sigmoidicity of allosteric enzymes [68], that of a two-enzyme FDS – the only type tested here – is poor Experimental results are consistent with this because the formation of a protein–protein complex of serine acetyl transferase with O-acetylserine(thiol)-lyase strongly modifies the kinetic properties of the first enzyme and results in a transition from a typical Michaelis–Menten behaviour to a behaviour displaying positive cooperativity with respect to serine and acetyl-CoA with a Hill coefficient in the range of 1.3–2.0 [47] It is probable that many more types of FDSs exist than those found so far experimentally (see above) Indeed, many FDSs may have escaped detection precisely because they tend to dissociate as the substrate concentration decreases, as generally occurs during in vitro studies It may even turn out that most enzymes and other proteins such as those involved in signalling assemble into FDSs in vivo when functioning These FDSs may be connected to more permanent structures such as membranes and the cytoskeleton They may even be connected to one another to form a network integrating FDSs responsible for metabolism and for signal transduction [11,76] Such a vision of intracellular organization is supported by many studies showing the recruitment of proteins into functional structures (reviewed in [3–5]) and the coordination of multiple functions via the formation of networks of signalling complexes [11,16,77–79] More than 50 different types of protein assemblies, containing up to 35 proteins, have been identified in functions that include transcription regulation, cell-cycle ⁄ cell-fate control, RNA processing, and protein transport [13] It could be argued that the concept of FDS should not be limited to the intracellular level Indeed, a concept similar to that of the FDS has been employed at the multicellular level to explain how neurones participate in FEBS Journal 273 (2006) 4287–4299 ª 2006 The Authors Journal compilation ª 2006 FEBS 4293 Functioning-dependent structures M Thellier et al different assemblies at different times depending on the task to be carried out [80] Biochemists are familiar with the Structure fi Function relationship with respect to proteins or other active molecules or cell substructures They are less familiar with the idea that the very functioning of these cellular components may result in their assembling into a dynamic structure from which a better or even a new functioning emerges In this case, the relationship above must be changed into Structure Function This leads to the intuition that the very existence of such a self-organizing relationship in a system is an indication that this system is a living one To try to express this quantitatively, consider the density of entropy production in a process involving an FDS According to the second law of thermodynamics, the functioning of any system entails a positive production of entropy that can be written as a bilinear form of the flux densities of the processes and their conjugated driving forces [81] Whichever reaction pathway in our system is chosen to connect S1 and S3 (Fig 2), under steady-state conditions the only molecules that undergo transformation are S1 and S3 while the other molecules remain unchanged Hence, the corresponding density of entropy production, r, is that of the overall reaction of transformation of S1 into S3, and r does not depend on whether the system is catalysed via free enzymes or an FDS Out of steady state, however, the situation is different because the free enzymes, E1 and E2, can act immediately on their substrates whereas the FDS enzymes must assemble into an FDS before they can act Consequently, if r is expressed in the standard way, terms representing the entropic cost of FDS assembly ⁄ disassembly are present only in the description of living systems Acknowledgements We thank Jacques Ricard and Derek Raine for helpful comments and criticisms References Mowbray J & Moses V (1976) The tentative identification in Escherichia coli of a 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Emergent properties of networks of biological signalling pathways Science 283, 381–387 79 Bhalla S & Iyengar R (2001) Robustness of the bistable behaviour of a biological signalling feedback loop Chaos 11, 221–226 80 Singer W (2003) Oscillations and synchrony: time as coding space in neuronal processing Nova Acta Leopoldina NF 88 332, 35–56 81 Nicolis G & Prigogine I (1977) Self-Organization in Non-Equilibrium Systems Wiley Interscience, New York Table A1 The various reactions possibly taking place in the system under study Reactions R1 to R4 correspond to the formation of enzyme–substrate complexes, reactions R5 to R8 and R22 to R25 correspond to the formation of the FDS, reactions R9 to R11, R13 and R26 to R29 correspond to the transformation of S1 into S2 by enzyme E1, or S2 into S3 by enzyme E2, reaction R12 corresponds to the channelling of S2 from E1 to E2 within the FDS and reactions R14 to R21 correspond to the fixation of a second substrate by the FDS For any of these reactions, j, k’jf is the rate constant of the reaction written left to right and k’jr is the rate constant of the reaction written right to left Reference number Reaction Rate constants Appendix R1 R2 R3 R4 R5 R6 R7 R8 R9 R10 R11 R12 R13 R14 R15 R16 R17 R18 R19 R20 R21 R22 R23 R24 R25 R26 R27 R28 R29 E1+S1 ¼ E1S1 E1+S2 ¼ E1S2 E2+S2 ¼ E2S2 E2+S3 ¼ E2S3 E1S1+E2 ¼ E1S1E2 E1S2+E2 ¼ E1S2E2 E2S2+E1 ¼ E1E2S2 E2S3+E1 ¼ E1E2S3 E1 S1 ¼ E1 S2 E2 S2 ¼ E2 S3 E1S1E2 ¼ E1S2E2 E1S2E2 ¼ E1E2S2 E1E2S2 ¼ E1E2S3 E1S1E2+S2 ¼ E1S1E2S2 E1S2E2+S2 ¼ E1S2E2S2 E1S2E2+S3 ¼ E1S2E2S3 E1E2S2+S1 ¼ E1S1E2S2 E1E2S2+S2 ¼ E1S2E2S2 E1S1E2+S3 ¼ E1S1E2S3 E1E2S3+S1 ¼ E1S1E2S3 E1E2S3+S2 ¼ E1S2E2S3 E1S1+E2S2 ¼ E1S1E2S2 E1S1+E2S3 ¼ E1S1E2S3 E1S2+E2S2 ¼ E1S2E2S2 E1S2+E2S3 ¼ E1S2E2S3 E1S1E2S2 ¼ E1S2E2S2 E1S1E2S2 ¼ E1S1E2S3 E1S2E2S2 ẳ E1S2E2S3 E1S1E2S3 ẳ E1S2E2S3 kÂ1f, kÂ1r kÂ2f, k¢2r k¢3f, k¢3r k¢4f, k¢4r k¢5f, k¢5r k¢6f, k¢6r k¢7f, k¢7r k¢8f, k¢8r k¢9f, k¢9r k¢10f, k¢10r k¢11f, k¢11r k¢12f, k¢12r k¢13f, k¢13r k¢14f, k¢14r k¢15f, k¢15r k¢16f, k¢16r k¢17f, k¢17r k¢18f, k¢18r k¢19f, k¢19r k¢20f, k¢20r k¢21f, k¢21r k¢22f, k¢22r k¢23f, k¢23r k¢24f, k¢24r k¢25f, k¢25r k¢26f, k¢26r k¢27f, k¢27r k¢28f, k¢28r k¢29f, k¢29r The basis of the model of a two-enzyme FDS For computing purpose, it is convenient to list in a table all the different reactions, Rj, appearing in Fig (Table A1) The rate constants, k¢jf and k¢jr, of the forward and reverse reactions are also indicated in the table Note that, depending on the molecularity of the terms in the left-hand side of the reactions, k¢1f to k¢8f and k¢14f to k¢25f are expressed in mol)1ặs)1ặm3, while the other rate constants (kÂ9f to kÂ13f, kÂ26f to k¢29f and all the k¢jr) are expressed in s)1 In the following, when any reaction, Rj, in the table proceeds left to right or right to left, it is written jf or jr, respectively With these conventions, the chain of reactions ‘1f-9f-2r-3f10f-4r’ corresponds to the classical case in which the free enzymes, E1 and E2, transform S1 into S3 via the liberation of S2 by E1, the diffusion of S2 in the reaction medium and the recapture of S2 by E2 Any other chain of reactions equivalent to S1 fi S3 (e.g 12f-17f26f-28f-16r) implicates an FDS Definition of dimensionless quantities For easier analysis, we treat our problem using dimensionless variables and parameters If the concentration of any substance, X, is written [X], a dimensionless concentration, x, may be obtained by normalizing [X] to the total concentration of enzymes ([E1]t+[E2]t), x ẳ ẵX=ẵE1 t ỵ ẵE2 t ị A1ị s ¼ k01r Á t Similarly, the dimensionless expression, kj, of the rate constants, k¢j, will be obtained by normalization to k¢1r for the rate constants that are expressed in s)1, and by normalization to k¢1r ⁄ ([E1]t+[E2]t) for those that are expressed in mol)1ặs)1ặm3, e.g., e.g., k9f ẳ k09f =k01r ; k9r ¼ k09r =k01r ; k5r ¼ k05r =k01r ; etc: e1t ẳ ẵE1 t =ẵE1 t ỵ ẵE2 t ị; e1 ẳ ẵE1 =ẵE1 t ỵ ẵE2 t ị; A2ị e2 ẳ ẵE2 =ẵE1 t ỵ ẵE2 t ị; A4ị and k1f ẳ ẵE1 t ỵ ẵE2 t Þ Á k01f =k0lr ; e1 s1 e2 s3 ¼ ẵE1 S1 E2 S3 =ẵE1 t ỵ ẵE2 t ị; etc: Because k¢1r is expressed in s)1, a dimensionless expression, s, of the time, t, may be written as ðA3Þ k5f ẳ ẵE1 t ỵ ẵE2 t ị k05f =k01r ; etc: ðA5Þ With these conventions, it should be noted that k1r is always expressed as FEBS Journal 273 (2006) 4287–4299 ª 2006 The Authors Journal compilation ª 2006 FEBS 4297 Functioning-dependent structures M Thellier et al k1r ¼ k01r =k01r ẳ A6ị The basic equations of the steady-state problem It is apparent in Table A1 that the 29 reactions under consideration (R1 to R29) involve 17 different chemical species (E1, E2, S1, S2, S3, E1S1, E1S2, E2S2, E2S3, E1S1E2, E1S2E2, E1E2S2, E1E2S3, E1S1E2S2, E1S2E2S2, E1S1E2S3 and E1S2E2S3) Assuming that external mechanisms supply S1 and remove S3 as and when they are consumed and produced, respectively, such that S1 is maintained at a constant concentration and S3 at a zero concentration, the steady-state condition of functioning of the system is obtained by writing down the mass balance of the 15 other species involved Using the dimensionless quantities, this is written de1 =ds ¼ k1r Á e1 s1 À k1f e1 s1 ỵ k2r e1 s2 k2f e1 s2 ỵ k7r e1 e2 s2 k7f e1 e2 s2 ỵ k8r Á e1 e2 s3 À k8f Á e1 Á e2 s3 ẳ A7ị de2 =ds ẳ k3r e2 s2 k3f e2 s2 ỵ k4r e2 s3 k4f e2 s3 ỵ k5r Á e1 s1 e2 À k5f Á e2 Á e1 s1 ỵ k6r e1 s2 e2 k6f e2 e1 s2 ẳ A8ị de1 s1 e2 =ds ¼ k5f Á e2 Á e1 s1 À k5r Á e1 s1 e2 À k11f Á e1 s1 e2 ỵ k11r e1 s2 e2 k14f s2 e1 s1 e2 ỵ k14r e1 s1 e2 s2 À k19f Á s3 Á e1 s1 e2 ỵ k19r e1 s1 e2 s3 ẳ A14ị de1 s2 e2 =ds ¼ k6f Á e2 Á e1 s2 À k6r Á e1 s2 e2 À k12f Á e1 s2 e2 ỵ k12r e1 e2 s2 k15f s2 e1 s2 e2 ỵ k15r e1 s2 e2 s2 À k16f Á s3 Á e1 s2 e2 ỵ k16r e1 s2 e2 s3 ẳ A15ị de1 e2 s2 =ds ẳ k7f e1 Á e2 s2 À k7r Á e1 e2 s2 À k13f e1 e2 s2 ỵ k13r e1 e2 s3 k17f s1 e1 e2 s2 ỵ k17r Á e1 s1 e2 s2 À k18f Á s2 e1 e2 s2 ỵ k18r e1 s2 e2 s2 ẳ A16ị de1 e2 s3 =ds ẳ k8f Á e1 Á e2 s3 À k8r Á e1 e2 s3 ỵ k13f e1 e2 s2 k13r e1 e2 s3 À k20f Á s1 Á e1 e2 s3 ỵ k20r e1 s1 e2 s3 k21f s2 e1 e2 s3 ỵ k21r e1 s2 e2 s3 ẳ A17ị de1 s1 e2 s2 =ds ¼ k14f Á s2 Á e1 s1 e2 À k14r e1 s1 e2 s2 ỵ k17f s1 Á e1 e2 s2 À k17r Á e1 s1 e2 s2 ỵ k22f e1 s1 e2 s2 k22r Á e1 s1 e2 s2 À k26f Á e1 s1 e2 s2 ỵ k26r e1 s2 e2 s2 k27f e1 s1 e2 s2 ỵ k27r e1 s1 e2 s3 ẳ A18ị de1 s1 e2 s3 =ds ¼ k19f Á s3 Á e1 s1 e2 À k19r Á e1 s1 e2 s3 ds2 =ds ¼ k2f e1 s2 ỵ k2r e1 s2 k3f e2 s2 ỵ k3r e2 s2 k14f s2 e1 s1 e2 ỵ k14r Á e1 s1 e2 s2 À k15f Á s2 e1 s2 e2 ỵ k15r e1 s2 e2 s2 ỵ k20f s1 e1 e2 s3 k20r e1 s1 e2 s3 ỵ k23f e1 s1 Á e2 s3 À k23r Á e1 s1 e2 s3 k18f s2 e1 e2 s2 ỵ k18r Á e1 s2 e2 s2 À k21f Á s2 e1 e2 s3 ỵ k21r e1 s2 e2 s3 ẳ ỵ k27f e1 s1 e2 s2 À k27r Á e1 s1 e2 s3 À k29f Á e1 s1 e2 s3 ỵ k29r e1 s2 e2 s3 ẳ A19ị A9ị de1 s2 e2 s2 =ds ¼ k15f Á s2 Á e1 s2 e2 À k15r Á e1 s2 e2 s2 de1 s1 =ds ¼ Àk1r e1 s1 ỵ k1f e1 s1 ỵ k5r e1 s1 e2 ỵ k18f s2 e1 e2 s2 À k18r Á e1 s2 e2 s2 þ k24f Á e1 s2 Á e2 s2 À k24r Á e1 s2 e2 s2 À k5f Á e2 Á e1 s1 k9f e1 s1 ỵ k9r e1 s2 À k22f Á e1 s1 Á e2 s2 þ k22r Á e1 s1 e2 s2 À k23f Á e1 s1 e2 s3 ỵ k23r e1 s1 e2 s3 ẳ A10ị de1 s2 =ds ẳ k2f Á e1 Á s2 À k2r Á e1 s2 À k6f e2 e1 s2 ỵ k6r e1 s2 e2 ỵ k9f e1 s1 k9r e1 s2 À k24f Á e1 s2 Á e2 s2 þ k24r Á e1 s2 e2 s2 À k25f Á e1 s2 e2 s3 ỵ k25r e1 s2 e2 s3 ¼ de1 s2 e2 s3 =ds ¼ k16f Á s3 Á e1 s2 e2 À k16r Á e1 s2 e2 s3 ðA11Þ À k7f Á e1 Á e2 s2 k10f e2 s2 ỵ k10r e2 s3 À k22f Á e1 s1 Á e2 s2 þ k22r Á e1 s1 e2 s2 ðA12Þ de2 s3 =ds ¼ k4f Á e2 Á s3 À k4r Á e2 s3 k8f e1 e2 s3 ỵ k8r e1 e2 s3 ỵ k10f e2 s2 k10r Á e2 s3 À k23f Á e1 s1 Á e2 s3 ỵ k23r e1 s1 e2 s3 k25f e1 s2 e2 s3 ỵ k25r e1 s2 e2 s3 ẳ 4298 ỵ k21f s2 Á e1 e2 s3 À k21r Á e1 s2 e2 s3 ỵ k25f e1 s2 e2 s3 k25r e1 s2 e2 s3 ỵ k28f e1 s2 e2 s2 À k28r Á e1 s2 e2 s3 ỵ k29f e1 s1 e2 s3 k29r e1 s2 e2 s3 ẳ A21ị de2 s2 =ds ẳ k3r e2 s2 ỵ k3f e2 s2 ỵ k7r e1 e2 s2 k24f e1 s2 e2 s2 ỵ k24r e1 s2 e2 s2 ẳ ỵ k26f e1 s1 e2 s2 À k26r Á e1 s2 e2 s2 À k28f e1 s2 e2 s2 ỵ k28r e1 s2 e2 s3 ẳ A20ị Now, each of the 29 reactions, Rj, in Table A1 has an equilibrium constant, Kj, equal to the ratio of its forward to its reverse rate constant Kj ẳ kjf =kjr A13ị A22ị Using the maple software (Maplesoft Europe, Zug, Switzerland), the rank of the 29 · 17 matrix of the FEBS Journal 273 (2006) 4287–4299 ª 2006 The Authors Journal compilation ª 2006 FEBS M Thellier et al Functioning-dependent structures stoichiometric coefficients is shown to be equal to 14 This means that, to solve the set of Eqns (A7) to (A21), we are justified in fixing arbitrarily the values of 14 equilibrium constants (or linear combinations of equilibrium constants) and in using appropriate linear combinations of these basic constants to calculate the other 15 equilibrium constants The choice of this base of 14 independent equilibrium constants is to a large extent arbitrary We have chosen K1 ; K2 ; K3 ; K5 ; K9 ; K10 ; K11 ; K12 ; K13 ; K15 ; K17 ; ðA23Þ K27 ; K29 and K as our base of independent equilibrium constants In this base, K is the equilibrium constant of the overall reaction of transformation of S1 into S3 This constant K may be expressed by considering any reaction pathway whose balance is S1 fi S3, e.g., {1f-2r-3f-4r-9f -10f}, i.e., K ¼ ðk1f Á k2r Á k3f Á k4r Á k9f Á k10f Þ= ðk1r Á k2f Á k3r Á k4f Á k9r Á k10r ị ẳ K1 K3 K9 K10 Þ=ðK2 Á K4 Þ ðA24Þ Table A2 A set of 15 independent reaction circuits with a zero balance, and the calculation of the nonindependent equilibrium constants The expression of the 15 circuits (L1 to L15), which we have chosen as a base, is given in the second column Then the expressions of the 15 nonindependent equilibrium constants (third column) are calculated using Eqns (A22) and (A26) along with the values of the 14 independent equilibrium constants (base A23), the expression of K4 (Eqn A25) and the expression of the nonindependent equilibrium constants already calculated Reaction circuits with a zero balance Reference number Expression Calculated equilibrium constants L1 L2 L3 L4 L5 L6 L7 5f-6r-9r-11f 3f-6r-15r-24f 12r-15f-18r 2f-7r-18r-24f 1f-7r-17r-22f 9r-22f-24r-26f 10r-22f-23r-27f L8 L9 L10 4f-5r-19r-23f 11r-14f-15r-26f 9r-23f-25r-29f L11 L12 L13 L14 L15 11f-16f-19r-29r 26f-27r-28f-29r 7f-8r-10r-13f 13r-17f-20r-27f 13r-18f-21r-28f K6 ẳ (K5ặK11) K9 K24 ẳ (K5ặK11ặK15) (K3ặK9) K18 ẳ K15 K12 K7 ẳ (K2ặK5ặK11ặK12) (K3ặK9) K22 ẳ (K2ặK5ặK11ặK12ặK17) (K1ặK3ặK9) K26 ẳ (K1ặK9ặK15) (K2ặK12ặK17) K23 ẳ (K2ặK5ặK11ặK12ặK17ặK27) (K1ặK3ặK9ặK10) K19 ẳ (K11ặK12ặK17ặK27) K K14 ẳ (K2ặK11ặK12ặK17) (K1ặK9) K25 ẳ (K2ặK5ặK11ặK12ặK17ặK27ặK29) (K1ặK3ặ(K9)2ặK10) K16 ẳ (K12ặK17ặK27ặK29) K K28 ẳ (K2ặK12ặK17ặK27ặK29) (K1ặK9ặK15) K8 ẳ (K2ặK5ặK11ặK12ặK13) (K3ặK9ặK10) K20 ẳ (K17ặK27) K13 K21 ẳ (K2ặK17ặK27ặK29) (K1ặK9ặK13) or K4 ẳ k4f =k4r ẳ K1 K3 Á K9 Á K10 Þ=ðK2 Á KÞ ðA25Þ The remaining 15 equilibrium constants (K6, K7, K8, K14, K16, K18, K19, K20, K21, K22, K23, K24, K25, K26 and K28) can be calculated along independent reaction circuits with a zero balance For example, the reaction circuit {5f-6r-9r-11f}, the overall total of which is easily shown to be by combining the forward reactions R5 and R11 with the reverse reactions R6 and R9, is described by the equation ðK5 Á K11 Þ=ðK6 Á K9 Þ ¼ ðA26Þ K6 ¼ ðK5 Á K11 Þ=K9 ðA27Þ hence and similarly with the circuits L1 to L15, as indicated in Table A2 Again, using the maple software, the set of Eqns (A7) to (A21) is solved, depending on the values of the parameters of the problem (the 14 fixed equilibrium constants, one of the two rate constants, kjf or kjr, of each reaction, Rj, present in Table A1 and the relative concentrations of the enzymes E1 and E2), and Eqn (A6) which must always be satisfied For any given value of the concentration, s1, of S1, the absolute values (positive) of the steady-state rate of transformation of S1 into S3 is calculated as corresponding to both the rate of consumption of S1, v(s1), and the rate of production of S3, v(s3), i.e., Vðs1 Þ ẳ k1r e1 s1 ỵ k1f e1 s1 k17r e1 s1 e2 s2 ỵ k17f Á s1 Á e1 e2 s2 À k20r Á e1 s1 e2 s3 ỵ k20f s1 e1 e2 s3 Vs3 ị ẳ k4f e2 s3 ỵ k4r Á e2 s3 À k16f Á s3 Á e1 s2 e2 ỵ k16r e1 s2 e2 s3 k19f s3 e1 s1 e2 ỵ k19r e1 s1 e2 s3 FEBS Journal 273 (2006) 4287–4299 ª 2006 The Authors Journal compilation ê 2006 FEBS Vs1 ị ¼ Vðs3 Þ ðA28Þ ðA29Þ ðA30Þ 4299 ... Results Kinetics of the overall reaction of transformation of S1 into S3 The system with only the free enzymes, E1 and E2 The overall rate of functioning of two free sequential enzymes of the... rate of transformation of S1 into S3 is calculated as corresponding to both the rate of consumption of S1, v(s1), and the rate of production of S3, v(s3), and the shape of the curves {s1, v(s1)}... values of the parameters, with increasing s1, the rate of consumption of S1 again increased, reached a maximal value, then decreased, whilst at saturating values of s1 the rate of consumption of