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 cycl
Trang 1Mathematical modelling of the urea cycle
A numerical investigation into substrate channelling
Anthony D Maher1, Philip W Kuchel1, Fernando Ortega2, Pedro de Atauri2, Josep Centelles2
and Marta Cascante2
1
School of Molecular and Microbial Biosciences, University of Sydney, Australia;2Department 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
Mathematicait 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 the 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 system [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 ornithine cycle 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 (EC3.5.3.1); argininosuccinate lyase (EC4.3.2.1);
argininosuccinate synthase (EC6.3.4.5); carbamoyl phosphate
synthase (ammonia) (EC6.3.4.16); ornithine carbamoyltransferase
(EC2.1.3.3).
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)
Trang 2the 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 downstream metabolites Specifically, 1 mM
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
H14CO3 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: NH4Cl, ornithine, aspartate and H14CO3 [12] At saturating sub-strate concentrations (15 mMHCO3, 5 mMaspartate, 5 mM
NH4Cl, 5 mM ornithine), the permeabilized hepatocytes synthesized urea at rates comparable with that of intact cells (4 nmolÆmin)1Æmg)1 dry weight compared with 13 nmolÆ min)1Æmg)1dry weight) However, at physiological ammonia and ornithine concentrations (0.5 mMand 0.2 mM, respect-ively), urea was formed at 12.1 nmolÆmin)1ÆmL)1of cells [12] Incubations (2 mL, final volume) had been terminated
by adding 1 mL 5MHClO4 Unreacted HCO3 (including, presumably, all unreacted H14CO3) was evaporated as CO2
by heating the deproteinized supernatants for 90 min at
70C 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 NH3, a small amount of radioactivity was typically recovered as urea, arginine or citrulline due to small amounts of endo-genous ornithine and NH3, along with counts fixed in compounds other than those of the urea cycle Thus the total counts fixed after HCO3 removal were corrected for counts fixed independently of ornithine and NH3 The results were then tabulated as total counts fixed in urea, arginine and citrulline after 1 min and expressed as
a percentage of (NH3+ 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 ims, mat
and cyt denote intermembrane space, mitochondrial matrix and
cytoplasm, respectively.
Trang 3which 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
MetabolicControlAnalysis (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 small 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] H14CO3 ) was added to the cells This
results in all of the label ending up in urea However, for
simplicity, the first 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 radiolabelled substrate 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· 108d.p.m.Ælmol)1 It is stated in the
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 for14Cwas 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 mM but the total concentration of
NH4C l was 0.5 mM So we defined a concentration of CP as 0.5· 10)3molÆL)1and initial concentration of labelled CP
as 2.26· 10)5molÆL)1 Steady-state urea production was assigned a value of 2.02· 10)7molÆs)1ÆL per cells that corresponds to the observed 12.1 nmolÆmin)1ÆmL 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 lM 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 Mathematicafunction, 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
Trang 4concentration (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 (Vmax) of the arginase
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
(NH3 + ornithine)-dependent counts found in these
metabolites It was assumed that the remainder of
(NH3+ 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 mMunlabelled 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 simulated value column gives values predicted by the simulation
of the arginase-loading 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
Experimental value column.
Simulated value Experimental value
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
Trang 5next step was to simulate the effect on this labelling pattern of
the addition of a 1-mMexcess of arginine From a modelling
point of view, this was accomplished by creating a separate
pool of exogenous arginine (called argQ 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 The data listed in Tables 1 and 2 were generated from 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 free mixing factor ( fm) to the rate equation for exogenous arginine hydrolysis by arginase in the Mathematicaprogram; 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 100% free mixing Decreasing this free mixing factor 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 When fm¼ 1, the exogenous arginine is most rapidly used up; and it is constant at 1 mMwhen fm¼ 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 mMexogenous 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
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
0 to 1 All values are those that the simulation predicted after 1 min of incubation See the text for further details.
Urea 10.9 12.2 13.5 14.7 15.7 16.5 17.2 17.7 18.2 18.5 18.8
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
Trang 6distribution 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 fine-tune
this model to generate data 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 mM 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 mM
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 mMexcess 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 tool
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 free-mixing 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 preference factor 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.
Trang 7In 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 (1993) Channelling can
affect concentrations of metabolic intermediates at constant net
flux: artefact or reality? Eur J Biochem 213, 87–92.
7 Krebs, H.A & Henseleit, K (1932) Untersuchungen uber die
harnstoffbildung im tierkorper Hoppe-Seyler’s Z Physiol Chem.
210, 33–66.
8 Bach, S.J., Crook, E.M & Williamson, S (1944) On arginase and
its participation in urea synthesis in the liver Biochem J 38,
325–332.
9 Krebs, H.A (1976) The discovery of the ornithine cycle In The
Urea Cycle (Grisolia, S., Baguena, R & Mayor, F., eds), pp 1–12.
John Wiley & Sons, New York.
10 Kuchel, P.W., Roberts, D.V & Nichol, L.W (1977) The simula-tion of the urea cycle: correlasimula-tion of effects due to inborn errors in the catalytic properties of the enzymes with clinical-biochemical observations Aust J Exp Biol Med Sci 55, 309–326.
11 Cohen, N.S., Cheung, C.W & Raijman, L (1987) Channeling of extramitochondrial ornithine to matrix ornithine transcarbamy-lase J Biol 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 Academic Press, 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 OTCto 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 kn,Enzymewhere 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
Trang 81 Ornithine carbamoyltransferase.The metabolites ornithine and citrulline in the OTCequations are labelled as ornmat(t) and citmat(t), respectively, to distinguish them from the same cytoplasmic intermediates referred to in subsequent rate equations Other metabolites in the OTCreactions are carbamoyl phosphate and inorganic phosphate (CP(t) and Pi(t), respectively) and radioactive citrulline in the matrix (citRmat(t))
vOTC¼k1;OTCk3;OTCk5;OTCk7;OTCcp(t) ornmat(t) k2;OTCk4;OTCk6;OTCk8;OTCcitmat(t) Pi(t)
denominatorOTC
[OTC]
where
denominatorOTC=k2;OTCk7;OTC(k4;OTC+k5;OTCÞ þ k1;OTCk7;OTC(k4;OTC+k5;OTC)(cp(t)+cpR(t))
þ k2;OTCk8;OTC(k4;OTC+k5;OTC)Pi(t)þ k3;OTCk5;OTCk7;OTCornmat(t)þ k2;OTCk4;OTCk6;OTC(citmat(t)+citRmat(t))
þ k1;OTCk3;OTC(k5;OTC+k7;OTC)(cp(t)þ cpR(t))ornmat(t)þk6;OTCk8;OTC(k2;OTC+k4;OTC)Pi(t)(citmat(t)
þ citRmat(t))+k1;OTCk4;OTCk6;OTC(cp(t)þ cpR(t))(citmat(t)+citRmat(t))þ k1;OTCk3;OTCk6;OTC(cp(t)
þ cpR(t))ornmat(t)(citmat(t)+citRmat(t))þ k3;OTCk5;OTCk8;OTCornmat(t)Pi(t)
þ k3;OTCk6;OTCk8;OTCornmat(t)Pi(t)(citmat(t)+citRmat(t))
2 Argininosuccinate synthase.All metabolites are cytoplasmic For the AAS reaction the symbols cit(t), citR(t), ATP(t), Asp(t), PPi(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 KI,Argis 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) 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 þArg(t) + ArgR(t) + ArgQ(t)
K I;Arg
denominator ASS
[ASS]
where
denominatorASS= k2;ASS k4;ASS k9;ASSk11;ASS(k6;ASSþ k7;ASS) + k1;ASSk4;ASS k6;ASS k8;ASSk11;ASS(cit(t)
þ citR(t)) PPi(t)þ k1;ASSk4;ASS k9;ASSk11;ASS(k6;ASSþ k7;ASS) (cit(t)þ citR(t))
þ k2;ASSk5;ASS k7;ASS k9;ASSk12;ASSAsp(t) (as(t)þ asR(t)) þ k2;ASS k5;ASS k7;ASSk9;ASSk11;ASS Asp(t)
þ k1;ASSk3;ASS k6;ASS k8;ASSk11;ASS(cit(t)þ citR(t)) ATP(t) PPi(t)þ k1;ASS k3;ASS k9;ASSk11;ASS(k6;ASS+k7;ASS) (cit(t)
þ citR(t)) ATP(t) þ k1;ASS k4;ASSk6;ASS k8;ASS k10;ASS(cit(t)þ citR(t)) PPi(t) AMP(t)
þ k1;ASSk5;ASS k7;ASS k9;ASSk11;ASSðcitðtÞ þ citRðtÞÞAspðtÞ þ k3;ASSk5;ASSk7;ASSk9;ASSk12;ASS ATP(t) Asp(t) (as(t)
þ asR(t)) þ k3;ASSk5;ASS k7;ASS k9;ASSk11;ASSATP(t) Asp(t)þ k2;ASSk5;ASSk7;ASS k10;ASSk12;ASSAsp(t) AMP(t) (as(t)
þ asR(t)) + k1;ASS k3;ASSk5;ASS(k7;ASSk9;ASS þ k7;ASS k11;ASS+ k9;ASSk11;ASS) (cit(t)þ citR(t)) ATP(t) Asp(t)
þ k1;ASSk3;ASS k5;ASS k8;ASSk11;ASS(cit(t)þ citR(t)) ATP(t) Asp(t) PPi(t)þ k2;ASSk4;ASS k6;ASSk8;ASSk11;ASS PPi(t)
þ k1;ASSk3;ASS k5;ASS k7;ASSk10;ASS(cit(t)þ citR(t)) ATP(t) Asp(t) AMP(t)
þ k2;ASSk4;ASS k9;ASS k12;ASS(k6;ASS+k7;ASS) (as(t)þ asR(t))+k1;ASS k3;ASSk6;ASS k8;ASS k10;ASS(cit(t)
þ citR(t)) ATP(t) PPi(t) AMP(t)þ k2;ASSk4;ASS k6;ASSk8;ASSk10;ASSPPi(t) AMP(t)
þ k3;ASSk5;ASS k7;ASS k10;ASSk12;ASS ATP(t) Asp(t) AMP(t) (as(t)+asR(t))
þ k2;ASSk4;ASS k6;ASS k8;ASSk12;ASSPPi(t) (as(t)þ asR(t)) + k3;ASS k6;ASSk8;ASSk10;ASS k12;ASSATP(t) PPi(t) AMP(t)
(as(t) + asR(t)) þ k2;ASSk4;ASSk10;ASSk12;ASS(k6;ASSþ k7;ASS) AMP(t) (as(t) + asR(t))
þ k2;ASSk5;ASS k8;ASS k10;ASSk12;ASS Asp(t) PPi(t) AMP(t) (as(t) + asR(t))
þ k8;ASSk10;ASS k12;ASS(k2;ASSk4;ASS+ k2;ASSk6;ASSþ k4;ASSk6;ASS) PPi(t) AMP(t) (as(t) + asR(t))
þ k1;ASSk3;ASS k5;ASS k8;ASSk10;ASS(cit(t)þ citR(t)) ATP(t) Asp(t) PPi(t) AMP(t)
þ k3;ASSk5;ASS k8;ASS k10;ASSk12;ASS ATP(t) Asp(t) PPi(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 fm 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)
vASL = (k1;ASL k3;ASL k5;ASL as(t)) (k2;ASL k4;ASL k6;ASL fum(t) Arg(t))
Trang 9denominatorASL= k5(k2+ k3) + k1(k3+ k5) (as(t)þ asR(t)) + k2k4fum(t) + k6(k2+ k3) (Arg(t)þ ArgR(t)
þ fm*ArgQ(t)) + k4k6fum(t) (Arg(t)þ ArgR(t) + fm*ArgQ(t)) þ k1k4(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
vArginase = k1;Arginasek3;Arginase k4;Arginase Arg(t)
denominatorArginase
[Arginase]
where
denominatorArginase= k4;Arginase(k2;Arginase + k3;Arginase)þ k5;Arginase (k2;Arginase + k3;Arginase) orn(t)
þ k1;Arginase (k3;Arginase+ k4;Arginase) (Arg(t)þ ArgR(t) + fm*ArgQ(t))