Báo cáo khoa học: Mathematical modelling of the urea cycle A numerical investigation into substrate channelling docx

Mathematical modelling of the urea cycle A numerical investigation into substrate channelling Anthony D. Maher 1 , Philip W. Kuchel 1 , Fernando Ortega 2 , Pedro de Atauri 2 , Josep Centelles 2 and Marta Cascante 2 1 School of Molecular and Microbial Biosciences, University of Sydney, Australia; 2 Department de Bioquimica i Biologia Molecular, Universitat de Barcelona, Spain Metabolite channelling, the process in which consecutive enzymes have confined substrate transfer in metabolic pathways, has been proposed as a biochemical mechanism that has evolved because it enhances catalytic rates and protects unstable intermediates. Results from experiments on the synthesis of radioactive urea [Cheung, C., Cohen, N.S. & Raijman, L (1989) J. Biol. Chem. 264, 4038–4044] have been interpreted as implying channelling of arginine between argininosuccinate lyase and arginase in permeabi- lized hepatocytes. To investigate this interpretation further, a mathematical model of the urea cycle was written, using Mathematica it simulates time courses of the reactions. The model includes all relevant intermediates, peripheral metabolites, and subcellular compartmentalization. Analy- sis of the output from the simulations supports the argument for a high degree of, but not absolute, channelling and offers insights for future experiments that could shed more light on the quantitative aspects of this phenomenon in the urea cycle and other pathways. Keywords: arginase; mathematical modeling; metabolite channeling; urea cycle. There has been considerable debate in recent years over the phenomenon of metabolite channelling [1–3]. Channelling has been defined as Ôthe process by which two or more sequential enzymes in a pathway interact to transfer an intermediate from one active site to another without allowing free diffusion into the bulk systemÕ [4]. It has been suggested that channelling plays a fundamental role in regulation of certain reaction schemes, and in protection of substrates that are subject to both biochemical and spon- taneous chemical transformation. Despite this, some theor- etical studies have given evidence that channelling would not decrease pool sizes of metabolites [5,6]. The urea cycle is the means whereby a uriotele eliminates the potentially harmful ammonia from the body by converting it to urea prior to excretion. The pathway of ureogenesis was elucidated by Krebs and Henseleit in 1932 [7]. While considerable excitement surrounded this publication the proposal for the cycle was not universally accepted in the biochemical commu- nity. Indeed Bach et al. [8] in 1944 argued that the so-called Ôornithine cycleÕ was insufficient to account for the rate of synthesis of urea from ammonia in the liver, so, there must be another pathway. Their conclusion was based on the observation that the known inhibition of arginase by high ornithine concentrations did not consid- erably diminish the synthesis of urea by isolated liver slices. Their incorrect interpretation of the data was surmised by Krebs to be a consequence of the fact that arginine and ornithine do not rapidly penetrate liver slices [9]. Hence the ornithine would not have reached its equilibrium concentration by a large margin, and thus it would not have exerted its inhibitory effect. On the other hand, the computer simulation study of the urea cycle by Kuchel et al. [10] showed that even if the ornithine in the cytoplasm had reached the concentrations used by Bach et al. [8] the cycle flux would not have been affected. In other words, the flux control coefficient of arginase, even in the presence of high ornithine concentrations, was small compared with that of other enzymes in the cycle. Many reports have provided evidence of spatial organization of enzymes and proteins involved in urea synthesis in the liver. Cohen et al. in 1987 [11] showed that in the matrix of mitochondria isolated from rat hepatocytes, ornithine carbamoyltransferase preferentially uses cytoplasmic ornithine as a substrate. In further work, Cheung et al. [12] incubated permeabilized rat hepatocytes with radiolabelled bicarbonate and measured Correspondence to P. W. Kuchel, School of Molecular and Microbial Biosciences, University of Sydney, NSW 2006, Australia. Fax: + 61 2 9351 4726, Tel.: + 61 2 9351 3709, E-mail: p.kuchel@mmb.usyd.edu.au Abbreviations: ASL, argininosuccinate lyase; ASS, argininosuccinate synthase; OCT, ornithine carbamoyltransferase; CP, carbamoyl phosphate; MCA, metabolic control analysis. Enzymes: arginase (EC 3.5.3.1); argininosuccinate lyase (EC 4.3.2.1); argininosuccinate synthase (EC 6.3.4.5); carbamoyl phosphate synthase (ammonia) (EC 6.3.4.16); ornithine carbamoyltransferase (EC 2.1.3.3). Note: A web site is available at: http://www.mmb.usyd.edu.au Note: The model, with all rate equations and initial conditions, as used in the present work, is available (in Mathematica notebook form) from the authors at a.maher@mmb.usyd.edu.au or p.kuchel@mmb.usyd.edu.au. Note: The mathematical model described here has been submitted to the Online Cellular Systems Modelling Database and can be accessed free of charge at: http://jjj.biochem.sun.ac.za/database/maher/ index.html (Received 17 June 2003, accepted 7 August 2003) Eur. J. Biochem. 270, 3953–3961 (2003) Ó FEBS 2003 doi:10.1046/j.1432-1033.2003.03783.x the distribution of radioactive urea, arginine and citrul- line after 1 min of incubation. Dilution of the label with specific, unlabelled intermediates at several steps in the pathway had only minor effects on the specific activities of ÔdownstreamÕ metabolites. Specifically, 1 m M unla- belled arginine had little effect on the amount of radioactive urea produced by the cells in 1 min. The results were interpreted to imply that added metabolites in the bulk solvent do not mix freely with the endo- genous cytoplasmic intermediates and are preferentially used by urea cycle enzymes. A similar protocol was used to demonstrate the preference of matrix ornithine carbamoyltransferase for endogenously formed car- bamoyl phosphate [13]. In addition, immunocytochemical studies [14] support the hypothesis that some of the urea cycle enzymes are spatially organized in vivo. A detailed mechanistic-kinetic model of the urea cycle was previously written by one of us (P. W. Kuchel) [10], but aspects of channelling and the effects of subcompartmen- tation of the cycle were not explored. Hence, the aims of the current work were to: (a) extend this model in the widely available and readily modifiable program, Mathematica; (b) expand the model so that it could distinguish between events in subcellular compartments such as the mitochon- dria; and (c) include equations for additional, distinct radioactive, and exogenous substrates (Fig. 1). We also aimed to (d) predict the pattern of distribution of radio- activity in cytoplasmic urea cycle intermediates that would be expected following addition of particular radioactive substrates to the cells. Finally, we aimed to (e) study the effect that addition of unlabelled intermediates would have on this distribution, assuming various proposed mecha- nisms of urea synthesis. To investigate the latter aim we paid particular attention to the results of an experiment pub- lished by Cheung et al. [12] in which unlabelled (cold) arginine was added to suspensions of permeabilized hepatocytes that were synthesizing radioactive urea from H 14 CO 3 – . The mathematical model described here has been submitted to the Online Cellular Systems Modelling Database and can be accessed free of charge at: http:// jjj.biochem.sun.ac.za/database/maher/index.html Premise The simulations were designed to reflect as closely as possible the experimental set up described by Cheung et al. [12]. Isolated hepatocytes were prepared from fresh rat livers by treatment with collagenase and then exposed for a short time to the membrane-active a-toxin from Staphylococcus aureus. This toxin permeabilized the plasma membranes of the hepatocytes to low molecular weight compounds such as arginine, citrulline, ornithine and lysine, yet largely main- tained inside the cell compounds with molecular masses greater than 5000, such as larger proteins including the enzymes involved in ureogenesis. However, 13% of the total arginase in the suspension appeared outside the cells prior to the incubations, with this figure rising to about 20% after 1 min [12]. Incubations had been performed in a buffer supplemented with the substrates necessary for urea synthesis: NH 4 Cl, ornithine, aspartate and H 14 CO 3 – [12]. At saturating sub- strate concentrations (15 m M HCO 3 – ,5 m M aspartate, 5 m M NH 4 Cl, 5 m M ornithine), the permeabilized hepatocytes synthesized urea at rates comparable with that of intact cells (4 nmolÆmin )1 Æmg )1 dry weight compared with 13 nmolÆ min )1 Æmg )1 dry weight). However, at physiologicalammonia and ornithine concentrations (0.5 m M and 0.2 m M , respect- ively), urea was formed at 12.1 nmolÆmin )1 ÆmL )1 of cells [12]. Incubations (2 mL, final volume) had been terminated by adding 1 mL 5 M HClO 4 . Unreacted HCO 3 – (including, presumably, all unreacted H 14 CO 3 – ) was evaporated as CO 2 by heating the deproteinized supernatants for 90 min at 70 °C. The total counts of radioactivity fixed in the remain- der of the suspension were then determined, as were the total counts fixed specifically in urea, arginine and citrulline [12]. In the absence of added ornithine and NH 3 ,asmall amount of radioactivity was typically recovered as urea, arginine or citrulline due to small amounts of endo- genous ornithine and NH 3 , along with counts fixed in compounds other than those of the urea cycle. Thus the total counts fixed after HCO 3 – removal were corrected for counts fixed independently of ornithine and NH 3 . The results were then tabulated as total counts fixed in urea, arginine and citrulline after 1 min and expressed as a percentage of (NH 3 + ornithine)-dependent counts (those from urea, arginine, citrulline and argininosucci- nate). It was then possible to compare differences (or similarities) between the distributions of radioactivity in these intermediates across a range of experiments in Fig. 1. Schematic representation of the urea cycle used as the basis for the computer model showing metabolites and compartmentation. An asterisk indicates a radiolabelled counterpart of the metabolite. Metabolites: CP, carbamoyl phosphate; Orn, ornithine; Cit, citrulline; ATP, adenosine triphosphate; Asp, aspartate; AMP, adenosine monophosphate; PPi, pyrophosphate; AS, argininosuccinate; Fum, fumarate; Arg, endogenous arginine; ArgQ, exogenous arginine. Enzymes: 1, ornithine carbamoyltransferase; 2, argininosuccinate synthase; 3, argininosuccinate lyase; 4, arginase. Subscripts ÔimsÕ, ÔmatÕ and ÔcytÕ denote intermembrane space, mitochondrial matrix and cytoplasm, respectively. 3954 A. D. Maher et al. (Eur. J. Biochem. 270) Ó FEBS 2003 which comparatively large amounts of unlabelled urea cycle intermediates were added to the cells, in order to assess the influence of these added substrates on the rate of urea synthesis. Model Written in Mathematica, the basic model simulates the time- dependent flux of metabolites through the urea cycle (see Fig. 1) using a general metabolic simulation package called ÔMetabolicControlAnalysisÕ (MCA) developed by Mulqui- ney and Kuchel [15], with all the features of MCA described by Heinrich and Schuster [16]. The model includes steady- state enzyme-kinetic equations to describe the multisub- strate reactions (see Appendix, part 1) of the enzymes and distinguishes between reactions that occur in different compartments (e.g., mitochondrial matrix, intermembrane space and cytoplasm). It also contains parallel equations for separately identifiable radioactive substrates. Parameters in the model have been assigned to fit as closely as possible with those in the experimental set up. It was assumed that changes in compartment volumes during the course of the incubations were insignificant. The previous model constructed by Kuchel et al.[10] was based on numerous references but the key ones [17–22] were used as the starting point for these simulations, which was then extended as follows. (a) Compartmentalization: reactions of the urea cycle are known to take place in both the cytoplasm and the mitochondrial matrix. Metabolites such as ornithine and citrulline must traverse the inter- membrane space that separates the former two compart- ments. Another compartment to be considered was the extracellular medium, since the plasma membranes were made permeable to ÔsmallÕ molecules with a-toxin. Fig. 1 indicates the metabolites considered in this simulation, together with their respective compartments. Note that all cytoplasmic metabolites have a spatially distinguishable, yet chemically identical and rapidly exchangeable equiva- lent in the extracellular medium; these metabolites are omitted from Fig. 1 for clarity. The cytoplasmic urea cycle enzymes argininosuccinate synthase (ASS) argininosucci- nate lyase (ASL) were modelled as though they are confined to the cytoplasm, whereas % 13% of the total arginase was found outside the cells [12], thus it is capable of hydrolysing extracellular arginine. (2) Addition of radioactive substrates: in the experiments described by Cheung et al. [12] H 14 CO 3 ) was added to the cells. This results in all of the label ending up in urea. However, for simplicity, the ÔfirstÕ reaction in the cycle [that performed by carbamoyl phosphate synthase (ammonia)] was modelled as the instantaneous conversion of bicarbonate to car- bamoyl phosphate; in other words, for simulation purpo- ses, the Ôradiolabelled substrateÕ added to the cells was carbamoyl phosphate. The simulation requires that initial values (in molÆL )1 ) be entered for both the nonradioactive and radioactive species. The following procedure was used to select initial concentrations for unlabelled and labelled carbamoyl phosphate. We assumed that the stock bicarbonate was 100% labelled. According to the methods section in [12] the original specific activity was 55 mCiÆmmol )1 , equi- valent to 1.221 · 10 8 d.p.m.Ælmol )1 .Itisstatedinthe legend to Table 2 of the paper by Cheung et al. [12] that the specific activity in Experiment 1 was 530 c.p.m.Ænmol )1 [12]. This is equal to 552.1 d.p.m.Ænmol )1 (because the stated counting efficiency for 14 C was 96%). Thus we assume there were 0.00452 mol labelled bicarbonate per mol total bicar- bonate. In the experiments, the total concentration of bicarbonate was 15 m M but the total concentration of NH 4 Clwas0.5 m M . So we defined a concentration of CP as 0.5 · 10 )3 molÆL )1 and initial concentration of labelled CP as 2.26 · 10 )5 molÆL )1 . Steady-state urea production was assigned a value of 2.02 · 10 )7 molÆs )1 ÆL per cells that corresponds to the observed 12.1 nmolÆmin )1 ÆmL per cells produced in the experiments. Initial concentrations of substrates in the extracellular milieu were given values according to the Methods section of Cheung et al. [12]. Other intracellular and mitochond- rial metabolites were assigned a value of 1 l M .The unitary rate constants and rate equations for the four relevant enzymes, and their initial concentrations, were taken from the original urea cycle simulation [10] (Appendix, part 3). Rate constants for membrane exchange of metabolites were assigned values consistent with transport through the outer mitochondrial membrane and the plasma membrane being faster than transport through the inner mitochondrial membrane, and very rapid exchange across the cytoplasmic membrane. Rate equations were also included for the removal of meta- bolites from the system (mimicking the realistic scenario that their concentrations remain relatively constant in the cells); while pools were set up for the input of CP, ATP and aspartate, each being given a value designed to result in the desired steady-state rate of production of urea. Furthermore, arginine was considered a competitive inhibitor of argininosuccinate synthase [12]. Once all of the rate laws for each biochemical and membrane-transport reaction had been defined, a numerical solution to the system was obtained using the built-in Mathematica function, NDSolve. In our simulations using the add-on package MCA, the stoichiometry of each individual reaction was first defined, and from this, three matrices were generated, called the stoichiometry matrix, the substrate matrix, and the velocity matrix. A function called NDSolveMatrix uses the built-in NDSolve function to solve the system of differential equations using these matrices. Results Simulation in the absence of added metabolites The Mathematica program stores the numerical solution of the differential equations as a set of interpolating functions for each variable (metabolite) modelled in the system (see Fig. 1). In order to be useful, any simulation must approach as near as possible to available experimental data. The output from a model can take a number of forms and Fig. 2 shows some of the graphs generated for the time dependence of selected metabolites modelled in our system. The ornithine concentration in the cytoplasm (Fig. 2A) is seen to decline within the first 300 s of starting the reactions, with a corresponding increase in cytoplasmic citrulline (Fig. 2B). The curve of the argininosuccinate Ó FEBS 2003 Modelling to predict urea cycle kinetic mechanisms (Eur. J. Biochem. 270) 3955 concentration (Fig. 2C) is seen to increase within the first 100 s of simulated time, and then decrease as it is converted to arginine. Arginine (Fig. 2D) exhibits a similar flux pattern to argininosuccinate, except that its concen- tration decreases within the first few seconds of the simulation, this effect is ascribed to the high catalytic capacity (V max )ofthearginase. Relevant Mathematica functions were written to extract the distribution of radioactivity, and the total measurable radioactivity in labelled metabolites from the simulations. Results are presented in [12] both as c.p.m. measured in urea, arginine and citrulline, along with the percentage of (NH 3 + ornithine)-dependent counts found in these metabolites. It was assumed that the remainder of (NH 3 + ornithine)-dependent counts was in argininosuc- cinate. Table 1 shows the output from the simulation for the distribution of radioactivity in urea, arginine, citrulline and argininosuccinate as a percentage, alongside the corresponding values obtained in the experiments by Cheung et al. [12]. Below this the predicted c.p.m. in urea, arginine and citrulline is also listed for the simulation and the experiment by Cheung et al. [12]. While the simulated values did not exactly match those of the experiments, the pattern of distribution of radioactivity is similar, with most being in citrulline, followed by urea, argininosuccinate and arginine. Simulating the effect of the addition of 1 m M unlabelled arginine on the distribution of radioactivity in cytoplasmic intermediates of the urea cycle As the pattern of distribution of radioactivity predicted by the simulation was similar to that found by experiment, the Fig. 2. Examples of graphical output from the computer model of the urea cycle. Time course graphs for cytoplasmic ornithine, cytoplasmic citrulline, argininosuccinate, arginine and urea are presented in A–E, respectively. AS, argininosuccinate. Table 1. Simulated and experimental values for the distribution of ‘(NH 3 + ornithine)-dependent’ metabolites [12] with no added arginine. The Ôsimulated valueÕ column gives values predicted by the simulation of the Ôarginase-loadingÕ experiment described by Cheung et al. 12], after 1 min of simulation, as a percentage of the total radioactivity in the listed metabolites. Also presented is the total c.p.m. predicted in urea, arginine and citrulline for the same simulation. The data are juxtaposed with the values obtained by experiment [12] in the ÔExperimental valueÕ column. Simulated value Experimental value Urea (%) 22.0 27 Arginine (%) 5.2 7 Citrulline (%) 56.6 46 Argininosuccinate (%) 16.2 20 c.p.m. in urea 5486 3750 c.p.m. in arginine 1305 1020 c.p.m. in citrulline 14 145 6470 3956 A. D. Maher et al. (Eur. J. Biochem. 270) Ó FEBS 2003 next step was to simulate the effect on this labelling pattern of the addition of a 1-m M excess of arginine. From a modelling point of view, this was accomplished by creating a separate ÔpoolÕ of exogenous arginine (called ÔargQÕ in the program, and in Fig. 1) defined as being chemically indistinguishable from the endogenous arginine with respect to its ability to be hydrolysed by cytoplasmic or extracellular arginase. The pattern of distribution of radioactivity presented in Table 2, which assumes that all intermediate metabolites are free to mix with the bulk solvent, follows the pattern predicted in the absence of exogenous arginine. There is a slight decrease in the radioactivity predicted in urea, with corresponding increases in arginine and citrulline. The total counts found in urea were slightly decreased in this simulation, with slight increases in arginine and citrulline. It was reported by Cheung et al. [12] that 44% of the exogenous arginine had been hydrolysed by the end of the incubations (after 60 s). Fig. 3 plots the predicted time dependence of the concentration of added arginine for the first 60 s. It can be seen that in this simulation almost all the added arginine is hydrolysed within the first 60 s. Thus the current model had to be altered in some way in order to match the experimental results. Simulation of channelling of arginine from argininosuccinate lyase to arginase ThedatalistedinTables1and2weregeneratedfrom simulations in which it was assumed that all intermediates in the pathway were free to mix throughout the bulk solvent. As all of the data generated by our simulations did not follow the pattern observed in the experiments of Cheung et al. [12], we investigated what effect the assumption of channelling would have on our simulated data. This is in spite of there being no detectable binding between the two enzymes that catalyse the consecutive reactions in the urea cycle under (admittedly) in vitro conditions [19]. As stated above, channelling is essentially the direct transfer of a metabolite from one enzyme to another, without allowing diffusion into the bulk solvent; to simulate this situation access of the exogenous arginine to its active site on arginase was restricted. Mathematically this was achieved by introducing a Ôfree mixing factorÕ ( fm) to the rate equation for exogenous arginine hydrolysis by arginase in the Mathematica program; fm can take any value between 0 and 1. Table 3 shows output from 11 simulations when fm was increased from 0 to 1 in increments of 0.1. The right- hand column, with an fm value of 1.0, shows that the distribution of radioactivity is identical (by definition) to that given in Table 2 because this simulation had the assumption of Ô100% free mixingÕ. Decreasing this Ôfree mixing factorÕ reduced the percentage of radioactivity predicted in urea, arginine and argininosuccinate, with the proportion in citrulline increasing. Figure 4 gives a combined plot of the time dependence of the exogenous arginine concentrations for the 11 simula- tions described here. Each curve on this graph corresponds to the time-dependent concentration of added arginine in a simulation, each with a different value of fm.Whenfm ¼ 1, the exogenous arginine is most rapidly used up; and it is constant at 1 m M when fm ¼ 0. It can be seen from Fig. 4, that for this set of simulations a value for fm of % 0.1 would result in 44% hydrolysis of the added arginine, which is consistent with the experimental results. Table 3 shows that setting fm to 0.1 gives a radioactivity distribution after 60 s of simulated time of 12.2% in urea, 3.9% in arginine, 73.1% in citrulline and 10.8% in argininosuccinate. While this simulated effect of the addi- tion of 1 m M exogenous arginine is not identical to the effect seen experimentally, the pattern of the alteration in the Table 2. Simulated and experimental [12] values for the distribution of ‘(NH 3 + ornithine)-dependent’ metabolites with 1 m M ‘added’ arginine. See legend to Table 1 for explanation of numbers and symbols. The simulated values are the same as the column corresponding to fm ¼ 1.0 in Table 3. Simulated value Experimental value Urea (%) 18.8 24 Arginine (%) 6.6 7 Citrulline (%) 58.3 55 Argininosuccinate (%) 16.3 14 c.p.m. in urea 4696 3950 c.p.m. in arginine 1651 1070 c.p.m. in citrulline 14 573 8930 Fig. 3. Computer simulated time course of the concentration of added arginine in a liver cell preparation. In this simulation, 100% free mixing of the arginase pools was assumed (i.e., fm ¼ 1). Table 3. Simulated values for the distribution of (NH 3 + ornithine)-dependent metabolites with 1 m M added arginine with fm values ranging from 0to1.All values are those that the simulation predicted after 1 min of incubation. See the text for further details. fm 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Urea 10.9 12.2 13.5 14.7 15.7 16.5 17.2 17.7 18.2 18.5 18.8 Arginine 2.7 3.9 4.8 5.4 5.8 6.1 6.3 6.4 6.5 6.6 6.6 Citrulline 78.6 73.1 68.8 65.6 63.3 61.7 60.5 59.7 59.1 58.6 58.3 Argininosuccinate 7.9 10.8 12.9 14.3 15.2 15.7 16.0 16.2 16.2 16.3 16.3 Ó FEBS 2003 Modelling to predict urea cycle kinetic mechanisms (Eur. J. Biochem. 270) 3957 distribution is similar in terms of an increase in citrulline at the expense of the other labelled metabolites. This can be explained by the fact that argininosuccinate synthase is inhibited by arginine. Simulations at lower values of fm retain cytoplasmic arginine for longer than simulations with high values of fm (Fig. 4), and therefore have increased counts recovered in citrulline. Discussion For this paper, we developed a mathematical model of the urea cycle in which all metabolic reactions are confined to specific cellular subcompartments, and we have included relevant membrane transport reactions such that all metabolites are both chemically and spatially identifiable. A specific aim of this work was to develop and Ôfine-tuneÕ this model to generate ÔdataÕ for time-course simulations that are comparable to those obtained experimentally. The intention was to use this model to assist in making conclusions that might explain the molecular mechanisms behind these observations. The output from the simulations presented above are consistent with an interpretation that endogenous arginine is preferentially used by arginase. When in the simulations we assume that the exogenous arginine can access only 10% of the cytoplasmic arginase, the output is similar to that found in the experiments. Analysis of the output presented above, however, raises other points worthy of consideration. In the argument by Cheung et al. [12] that channelling was a necessary inter- pretation several predictions were made with regard to the outcome of the experiments in the absence of channelling. These included that the addition of 1 m M unlabelled arginine would decrease the total counts of radioactivity in urea to the extent that they would be undetectable, with a corresponding increase in the percentage recovered as arginine. It was also argued that with the addition of 5 m M arginine, the percentage of counts recovered as arginine would have been increased in the absence of channelling, rather than the observed increase in the percentage recov- ered as citrulline. In our simulations in which free mixing is assumed there is no predicted significant increase in the percentage of counts recovered as arginine, nor is there a decrease in the percentage of counts recovered as urea after 60 s of simulated time to the extent to which they would be undetectable. In our simulation the addition of 1 m M excess arginine, with the high maximal velocity of arginase, sees a large increase in the ornithine concentration in the cyto- plasm, which in turn, translates into a large increase in the concentration of ornithine in the mitochondrial matrix. The ornithine carbamoyltransferase reaction is then largely dependent on the rate at which CP is produced in the matrix. Since the specific activity of the carbamoyl phos- phate produced in the matrix is the same as that of the bicarbonate, only small changes are predicted in the distribution of labelled cytoplasmic urea cycle intermediates after 60 s of simulated time. The increase in the percentage of counts recovered as citrulline can be attributed to the inhibition of argininosuccinate synthase by the added arginine. Another approach that might at first sight seem to provide a plausible explanation for the fact that only 44% of the added arginine was used in 60 s would be to simply decrease the concentrations of the enzymes until this condition was met. However, when all the enzyme concen- trations were decreased to achieve this outcome, almost all of the radioactivity was recovered in citrulline. A very large number of simulations was run with different concentra- tions of enzymes, and the model of the unperturbed urea cycle, that best fits the corresponding experimental results [12], is the one presented here. While metabolic research continues to provide evidence of pathways that exhibit direct transfer of metabolites between consecutive enzymes, the concept of metabolite channelling in pathways mediated by enzymes free in solution remains debated. There are several criteria with which to establish the presence of substrate channelling [4], including the isotope dilution method examined here. This paper highlights the importance of taking care when predicting possible outcomes of such experiments, in particular for cyclic enzymatic pathways. Due to the relatively high level of complexity in such pathways (as opposed to shorter, linear pathways) expected results are not always intuitive. The construction of detailed, and necessarily complex, mathematical models serves as a ÔtoolÕ to facilitate analysis of channelling in biochemical pathways like the urea cycle. There is a range of possible molecular mechanisms that may facilitate channelling in the urea cycle and other pathways. For this paper we have introduced a means of modelling for channelling with the Ôfree-mixingÕ factor, fm. This is only one of several possible approaches to the problem; it was based on the hypothesis that in vivo,urea cycle enzymes are spatially organized in a way such that exogenous metabolites have their access restricted to the binding sites on the enzymes. On the other hand, endo- genous metabolites are directly transferred to the binding site from the previous enzyme in the pathway. This is consistent with cytochemical evidence for such close proxi- mity for argininosuccinate synthase and argininosuccinate lyase [14]. Other approaches to modelling channelling may be necessary to account for data from similar experiments to those by Cheung et al. [12] in other pathways; this could involve allocating a Ôpreference factorÕ that an enzyme may have for one subset of a type of molecule over another subset, be it a radiochemical or physical distinction. Fig. 4. Predicted concentration of arginine in the total extramito- chondrial medium. Eleven separate simulations of the reaction scheme in Fig. 1 were used with fm ranging from 0 to 1.0. The curve with the most rapidly decreasing arginine concentration was that generated from a simulation where fm was set to 1, the remainder of the curves have a slope, at a given time, that decreases with decreasing fm. 3958 A. D. Maher et al. (Eur. J. Biochem. 270) Ó FEBS 2003 In conclusion, we present a more advanced and realistic model of the urea cycle than has been available hitherto. The model affords a means of studying the kinetic consequences of enzyme and metabolite compartmentalization and should serve as a basis for more extended analysis of control and regulation phenomena of this high-flux pathway. Acknowledgements This work was supported by a grant from the Australian National Health and Medical Research Council and the Australian Research Council to P. W. Kuchel. A. D. Maher is the recipient of a University of Sydney Postgraduate Award. We thank Prof. Natalie Cohen for information regarding the experimental set-up. References 1. Cascante, M., Sorribas, A. & Canela, E.I. (1994) Enzyme–enzyme interactions and metabolite channelling: alternative mechanisms and their evolutionary significance. Biochem. J. 298, 313–320. 2. Hawkins, A.R. & Lamb, H.K. (1995) The molecular biology of multidomain proteins. Selected examples. Eur. J. Biochem. 232, 7–18. 3. Mendes, P., Kell, D.B. & Westerhoff, H.V. (1992) Channelling can decrease pool size. Eur. J. Biochem. 204, 257–266. 4. Anderson, K.S. (1999) Fundamental mechanisms of substrate channeling. Methods Enzymol. 308, 111–145. 5. Cornish-Bowden, A. (1991) Failure of channelling to maintain low concentrations of metabolic intermediates. Eur. J. Biochem. 195, 103–108. 6. Cornish-Bowden, A. & Cardenas, M.L. 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Chem. 262, 203–208. 12. Cheung, C.W., Cohen, N.S. & Raijman, L. (1989) Channeling of urea cycle intermediates in situ in permeabilized hepatocytes. J. Biol. Chem. 264, 4038–4044. 13. Cohen, N.S., Cheung, C.W., Sijuwade, E. & Raijman, L. (1992) Kinetic properties of carbamoyl-phosphate synthase (ammonia) and ornithine carbamoyltransferase in permeabilized mitochon- dria. Biochem. J. 282, 173–180. 14. Cohen, N.S. & Kuda, A. (1996) Argininosuccinate synthetase and argininosuccinate lyase are localized around mitochondria: an immunocytochemical study. J. Cell. Biochem. 60, 334–340. 15. Mulquiney, P.J. & Kuchel, P.W. (2003) Modelling Metabolism with Mathematica. CRC Press, Boca Raton, FL. 16. Heinrich, R. & Schuster, S. (1996) The Regulation of Cellular Systems. Chapman & Hall, New York, NY. 17. Marshall, M. & Cohen, P.P. (1972) Ornithine transcarbamylase from Streptococcus faecalis and bovine liver. II. Multiple binding sites for carbamyl-P and 1-norvaline, correlation with steady state kinetics. J. Biol. Chem. 247, 1654–1668. 18. Rochovansky, O. & Ratner, S. (1967) Biosynthesis of urea. XII. Further studies on argininosuccinate synthetase: substrate affinity and mechanism of action. J. Biol. Chem. 242, 3839–3849. 19. Kuchel, P.W., Nichol, L.W. & Jeffrey, P.D. (1975) Physicochemi- cal and kinetic properties of beef liver argininosuccinase. Studies in the presence and absence of arginase. Biochim. Biophys. Acta 397, 478–488. 20. Kuchel, P.W., Nichol, L.W. & Jeffrey, P.D. (1975) Interpretation of the kinetics of consecutive enzyme-catalyzed reactions. Studies on the arginase-urease system. J. Biol. Chem. 250, 8222–8227. 21. Ratner, S. (1972) Argininosuccinases and adenylosuccinases. In The Enzymes (Boyer,P.D.,ed.),pp.167.AcademicPress, New York. 22. Greenberg, D.M. (1960) Arginase. In The Enzymes (Boyer, P.D., Lardy, H. & Myrba ¨ ck, K., eds), pp. 257. Academic Press, New York. Appendix Enzyme rate equations The method for deriving the rate equations and assigning values to rate constants is given in detail by Kuchel et al. [10], and more recently using an automated procedure, by Mulquiney and Kuchel [15]. Briefly, for ornithine carbamoyltransferase (OTC), the rate equation for a reversible Bi Bi ordered sequential mechanism was assumed. Four of the eight unitary rate constants were given realistic assumed values, while the other four were deduced by simultaneously solving equations for known (in the literature) steady-state kinetic parameters written in terms of the unitary rate constants. For ASS, an ordered, sequential Ter Ter mechanism was assumed, and a procedure was followed similar to that for OTC to designate values for the 12 unitary rate constants after assuming realistic values for four of them. The same procedure was repeated for ASL (ordered Uni Bi) and arginase. For the purposes of this simulation arginase was considered to be an irreversible reaction, with product inhibition by ornithine [10]. In the following equations the unitary rate constants are written in the form k n,Enzyme where n is a number assigned to the unitary rate constant for the associated enzyme. Each rate equation is expressed as the difference between the rate laws for forward and reverse reactions, which are functions of the concentrations of the relevant metabolites. Each rate equation is also a function of the concentration of the enzyme, which in our simulations was assumed to be constant. A feature of these equations is that they have lengthy denominators, which are given in a separate equation in each case for clarity. For brevity only the rate equation for reactions involving nonradioactive metabolites are given. However, it is important to note that the denominator in each case always contains terms for relevant corresponding radioactive molecules. Ó FEBS 2003 Modelling to predict urea cycle kinetic mechanisms (Eur. J. Biochem. 270) 3959 1. Ornithine carbamoyltransferase. The metabolites ornithine and citrulline in the OTC equations are labelled as orn mat (t) and cit mat (t), respectively, to distinguish them from the same cytoplasmic intermediates referred to in subsequent rate equations. Other metabolites in the OTC reactions are carbamoyl phosphate and inorganic phosphate (CP(t) and P i (t), respectively) and radioactive citrulline in the matrix (citR mat (t)). v OTC ¼ k 1;OTC k 3;OTC k 5;OTC k 7;OTC cp(t) orn mat (t) À k 2;OTC k 4;OTC k 6;OTC k 8;OTC cit mat (t) P i (t) denominator OTC [OTC] where denominator OTC =k 2;OTC k 7;OTC (k 4;OTC +k 5;OTC Þþk 1;OTC k 7;OTC (k 4;OTC +k 5;OTC )(cp(t)+cpR(t)) þ k 2;OTC k 8;OTC (k 4;OTC +k 5;OTC )P i (t) þ k 3;OTC k 5;OTC k 7;OTC orn mat (t) þ k 2;OTC k 4;OTC k 6;OTC (cit mat (t)+citR mat (t)) þ k 1;OTC k 3;OTC (k 5;OTC +k 7;OTC )(cp(t) þ cpR(t))orn mat (t) þk 6;OTC k 8;OTC (k 2;OTC +k 4;OTC )P i (t)(cit mat (t) þ citR mat (t))+k 1;OTC k 4;OTC k 6;OTC (cp(t) þ cpR(t))(cit mat (t)+citR mat (t)) þ k 1;OTC k 3;OTC k 6;OTC (cp(t) þ cpR(t))orn mat (t)(cit mat (t)+citR mat (t)) þ k 3;OTC k 5;OTC k 8;OTC orn mat (t)P i (t) þ k 3;OTC k 6;OTC k 8;OTC orn mat (t)P i (t)(cit mat (t)+citR mat (t)) 2. Argininosuccinate synthase. All metabolites are cytoplasmic. For the AAS reaction the symbols cit(t), citR(t), ATP(t), Asp(t), PP i (t), AMP(t), as(t), Arg(t), ArgR(t) and ArgQ(t) denote citrulline, radioactive citrulline, ATP, aspartate, pyrophosphate, AMP, argininosuccinate, arginine, radioactive arginine and exogenous arginine, respectively. K I,Arg is the inhibition constant for arginine. v ASS ¼ k 1;ASS k 3;ASS k 5;ASS k 7;ASS k 9;ASS k 11;ASS cit(t) ATP(t) Asp(t) À k 2;ASS k 4;ASS k 6;ASS k 8;ASS k 10;ASS k 12;ASS PP i (t) AMP(t) as(t)  1 þ Arg(t) + ArgR(t) + ArgQ(t) K I;Arg  denominator ASS [ASS] where denominator ASS = k 2;ASS k 4;ASS k 9;ASS k 11;ASS (k 6;ASS þ k 7;ASS )+k 1;ASS k 4;ASS k 6;ASS k 8;ASS k 11;ASS (cit(t) þ citR(t)) PP i (t) þ k 1;ASS k 4;ASS k 9;ASS k 11;ASS (k 6;ASS þ k 7;ASS ) (cit(t) þ citR(t)) þ k 2;ASS k 5;ASS k 7;ASS k 9;ASS k 12;ASS Asp(t) (as(t) þ asR(t)) þ k 2;ASS k 5;ASS k 7;ASS k 9;ASS k 11;ASS Asp(t) þ k 1;ASS k 3;ASS k 6;ASS k 8;ASS k 11;ASS (cit(t) þ citR(t)) ATP(t) PP i (t) þ k 1;ASS k 3;ASS k 9;ASS k 11;ASS (k 6;ASS +k 7;ASS ) (cit(t) þ citR(t)) ATP(t) þ k 1;ASS k 4;ASS k 6;ASS k 8;ASS k 10;ASS (cit(t) þ citR(t)) PP i (t) AMP(t) þ k 1;ASS k 5;ASS k 7;ASS k 9;ASS k 11;ASS ðcitðtÞþcitRðtÞÞAspðtÞþk 3;ASS k 5;ASS k 7;ASS k 9;ASS k 12;ASS ATP(t) Asp(t) (as(t) þ asR(t)) þ k 3;ASS k 5;ASS k 7;ASS k 9;ASS k 11;ASS ATP(t) Asp(t) þ k 2;ASS k 5;ASS k 7;ASS k 10;ASS k 12;ASS Asp(t) AMP(t) (as(t) þ asR(t)) + k 1;ASS k 3;ASS k 5;ASS (k 7;ASS k 9;ASS þ k 7;ASS k 11;ASS + k 9;ASS k 11;ASS ) (cit(t) þ citR(t)) ATP(t) Asp(t) þ k 1;ASS k 3;ASS k 5;ASS k 8;ASS k 11;ASS (cit(t) þ citR(t)) ATP(t) Asp(t) PP i (t) þ k 2;ASS k 4;ASS k 6;ASS k 8;ASS k 11;ASS PP i (t) þ k 1;ASS k 3;ASS k 5;ASS k 7;ASS k 10;ASS (cit(t) þ citR(t)) ATP(t) Asp(t) AMP(t) þ k 2;ASS k 4;ASS k 9;ASS k 12;ASS (k 6;ASS +k 7;ASS ) (as(t) þ asR(t))+k 1;ASS k 3;ASS k 6;ASS k 8;ASS k 10;ASS (cit(t) þ citR(t)) ATP(t) PP i (t) AMP(t) þ k 2;ASS k 4;ASS k 6;ASS k 8;ASS k 10;ASS PP i (t) AMP(t) þ k 3;ASS k 5;ASS k 7;ASS k 10;ASS k 12;ASS ATP(t) Asp(t) AMP(t) (as(t)+asR(t)) þ k 2;ASS k 4;ASS k 6;ASS k 8;ASS k 12;ASS PP i (t) (as(t) þ asR(t)) + k 3;ASS k 6;ASS k 8;ASS k 10;ASS k 12;ASS ATP(t) PP i (t) AMP(t) Â (as(t) + asR(t)) þ k 2;ASS k 4;ASS k 10;ASS k 12;ASS (k 6;ASS þ k 7;ASS ) AMP(t) (as(t) + asR(t)) þ k 2;ASS k 5;ASS k 8;ASS k 10;ASS k 12;ASS Asp(t) PP i (t) AMP(t) (as(t) + asR(t)) þ k 8;ASS k 10;ASS k 12;ASS (k 2;ASS k 4;ASS + k 2;ASS k 6;ASS þ k 4;ASS k 6;ASS )PP i (t) AMP(t) (as(t) + asR(t)) þ k 1;ASS k 3;ASS k 5;ASS k 8;ASS k 10;ASS (cit(t) þ citR(t)) ATP(t) Asp(t) PP i (t) AMP(t) þ k 3;ASS k 5;ASS k 8;ASS k 10;ASS k 12;ASS ATP(t) Asp(t) PP i (t) AMP(t) (as(t) + asR(t)) 3. Argininosuccinate lyase. All metabolites are cytoplasmic. For the ASL reaction the symbols as(t), asR(t), fum(t), Arg(t), ArgR(t) and ArgQ(t) denote argininosuccinate, radioactive argininosuccinate, fumarate, arginine, radioactive arginine, and exogenous arginine, respectively. ÔfmÕ is the free-mixing factor, given a value between 0 and 1. Note that the only term in the denominator that this effects is ArgQ(t), and that fm does not appear in the inhibition of the ASS reaction (above). v ASL = (k 1;ASL k 3;ASL k 5;ASL as(t)) À (k 2;ASL k 4;ASL k 6;ASL fum(t) Arg(t)) denominator ASL [ASL] 3960 A. D. Maher et al. (Eur. J. Biochem. 270) Ó FEBS 2003 denominator ASL = k 5 (k 2 + k 3 )+k 1 (k 3 + k 5 ) (as(t) þ asR(t)) + k 2 k 4 fum(t) + k 6 (k 2 + k 3 ) (Arg(t) þ ArgR(t) þ fm*ArgQ(t)) + k 4 k 6 fum(t) (Arg(t) þ ArgR(t) + fm*ArgQ(t)) þ k 1 k 4 (as(t) + asR(t)) fum(t) 4. Arginase. All metabolites in this reaction are cytoplasmic. Note, however, that an identical reaction exists for extracellular arginase in the model. Here Arg(t), ArgR(t), ArgQ(t) and orn(t) stand for arginine, radioactive arginine, exogenous arginine and cytoplasmic ornithine, respectively. v Arginase = k 1;Arginase k 3;Arginase k 4;Arginase Arg(t) denominator Arginase [Arginase] where denominator Arginase = k 4;Arginase (k 2;Arginase + k 3;Arginase ) þ k 5;Arginase (k 2;Arginase + k 3;Arginase ) orn(t) þ k 1;Arginase (k 3;Arginase + k 4;Arginase ) (Arg(t) þ ArgR(t) + fm*ArgQ(t)) Ó FEBS 2003 Modelling to predict urea cycle kinetic mechanisms (Eur. J. Biochem. 270) 3961 . Mathematical modelling of the urea cycle A numerical investigation into substrate channelling Anthony D. Maher 1 , Philip W. Kuchel 1 , Fernando Ortega 2 ,. facilitate analysis of channelling in biochemical pathways like the urea cycle. There is a range of possible molecular mechanisms that may facilitate channelling