Robustness of MetaNet graph models:

BioSystems 65 (2002) 61 – 78 www.elsevier.com/locate/biosystems

Robustness of MetaNet graph models: Predicting control of urea production in humans Michael C. Kohn a,*, Abdul S. Tohmaz b, Karen J. Giroux b, Gregory M. Blumenthal a, Michael D. Feezor c, David S. Millington c a

Laboratory of Computational Biology and Risk Analysis, National Institute of En6ironmental Health Sciences, Mail Drop A3 -06, P.O. Box 12233, Research Triangle Park, NC 27709, USA b BioKinetics, Inc., P.O. Box 14650, Research Triangle Park, NC 27709, USA c Mass Spectrometry Facility, Medical Genetics Program, Duke Uni6ersity Medical Center, P.O. Box 14991, Research Triangle Park, NC 27709, USA Received 3 December 2001; received in revised form 11 December 2001; accepted 26 December 2001

Abstract Urea production in human liver was described by a MetaNet graph, a flowchart-like representation of metabolic pathways that includes parameters for the kinetic constants of the constituent enzymes. Formal operations on the graph facilitate the identification of ligand-binding equilibria that participate in feedback regulation in the network of biochemical reactions. The state of the biochemical network is specified by the concentrations of the intermediates. At any particular time, the influence of an identified locus of regulation is proportional to the respective fractional saturation of the corresponding binding site. Enzymes that make or consume the feedback chemicals share in the control of the strength of the feedback signal in proportion to their fractional saturation. This model predicts control of urea production by the processes that deliver amino groups to the urea cycle enzymes more than by the cycle enzymes themselves. Mitochondrial membrane transport processes are important for transmission of information through the network, but irreversible enzymes and processes far from equilibrium control the strength of the feedback signal. Systematic variation of the parameter values by amounts comparable to the expected variability of their measured values indicated a high probability of invariance in the identities of the predicted control points. The properties of the model are consistent with those of error-tolerant scale-free networks. These results demonstrate the robustness of a MetaNet model’s predictions with respect to uncertainties in the values of its parameters. Published by Elsevier Science Ireland Ltd. Keywords: Graph models; Robustness; Metabolic regulation; Urea cycle

1. Introduction * Corresponding author. Tel.: + 1-919-541-4929; fax: +1919-541-1479. E-mail address: [email protected] (M.C. Kohn).

Biochemical pathways typically comprise many enzyme-catalyzed reactions, as well as several spontaneous features (e.g. molecular diffusion or

0303-2647/02/$ - see front matter. Published by Elsevier Science Ireland Ltd. PII: S 0 3 0 3 - 2 6 4 7 ( 0 2 ) 0 0 0 0 2 - 3

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metal ion chelation equilibria). Rather than forming linear sequences of elementary steps, such pathways may include feedback loops whereby two or more enzymes, remote from each other in the pathway, may bind the same ligand. That is, the concentration of a specific chemical intermediate influences the rate of a biochemical process remote from that intermediate in the biochemical pathway. Feedback loops have historically been considered to contain the loci of metabolic control. The existence of several such loops in a pathway complicates the identification of a given system’s sites of regulation. Furthermore, the nonlinear kinetics that describe the temporal behavior of biochemical systems may require more quantitative data than are actually available. Predicting the behavior of multiple overlapping feedback loops is difficult and, in most cases, impossible for the unaided human mind. Yet, coherent regulation of the biochemical network emerges from the properties of such structures, making the identification of those processes vital for understanding metabolic regulation. Over the past 30 years, there have been several attempts to develop qualitative tools for the dynamic analysis of regulatory networks (Kauffman, 1973; Thomas et al., 1995). Graph– theoretic methods can be used to represent the essential structure and regulatory properties of a biochemical network. In this context, a graph is a collection of geometric objects called nodes that are connected by (possibly directed) lines called arcs. The connectivity of the graph often indicates the flow of some resource among the objects constituting the model. Such a diagram is a model that reflects the relationships among the items represented by the nodes. Its resemblance to familiar biochemical pathway flowcharts makes it intuitively appealing to biological scientists. Kohn and coworkers devised a generalized graph model for metabolic systems (Kohn and Bedrosian, 1985; Kohn and Lemieux, 1991; Kohn and Letzkus, 1983) and extended it to gene expression systems (Kohn, 1986) and evaluation of signal propagation along biochemical pathways (Kohn, 1992). The graph, referred to as a MetaNet (Metabolic Network), includes nodes for metabolite pools, biopolymers, and their ligand-binding

relationships. The graphical representation is rich enough to represent known synergistic and inhibitory effects of ligand binding. A MetaNet is similar to a ‘hybrid Petri net,’ which includes both discrete and continuous elements (Matsuno et al., 2000; McAdams and Shapiro, 1995; Kohn and Lemieux, 1991), but the biochemical meaning of the nodes is more intuitive. Complex biochemical networks can also be represented by ‘molecular interaction maps’ (Kohn, 1999), which have an extremely complex set of arcs to represent interactions among proteins. These can be represented easily in the MetaNet format. Analysis of the network involves a set of logical rules and mathematical calculations. This approach separates local regulation of an enzyme by its ligands from the coherent control of a pathway by feedback regulation, which was defined as the flow of information between remote portions of the pathway. Here, we report an extension of the MetaNet model and test the reliability of its qualitative predictions. This work includes a description of the MetaNet graph model representation, its application to urea production in human liver, and a demonstration of the robustness of the model’s predictions to parameter variations. The implications of the graphical analysis for treatment of patients with inborn errors of metabolism are discussed.

2. Elements of a MetaNet model Two categories of nodes are defined (Table 1) in these models. ‘Chemnodes’ represent the chemicals in the network, and ‘relnodes’ represent the relationships among chemicals. Circular chemnodes indicate metabolite and ligand pools and square chemnodes represent biopolymers with catalytic properties. If an arc (an arrow in Table 1) intersects an edge of a square node, the chemnode represents a protein (enzyme). If it intersects the square’s corner, the chemnode represents a nucleic acid. The kinetics for these two catalyst types are treated identically in the model; this convention is just to facilitate visual identification of the role of the chemical species in the

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pathway. Triangles for relnodes are drawn conventionally with their apex pointed in the forward reaction direction. A chemnode’s immediate descendant node is always a relnode, but a relnode’s immediate descendant can be either a chemnode or another relnode. Relnodes contain symbols (Table 1) to indicate the nature of the functional relationship between their respective ancestor and descendant nodes. If an activator or inhibitor relnode’s immediate descendant node is a square chemnode, the modifier is noncompetitive, i.e. it affects the catalytic capacity of the enzyme represented by the ‘target’ chemnode. If the descendant is another relnode, the modifier is competitive, i.e. it affects the binding affinity represented by the target relnode. A relnode containing a zero represents a spontaneous (i.e. uncatalyzed) reaction. The previous version of the graph model was extended by allowing multiple ancestor and descendant chemnodes of relnodes for spontaneous reactions. Relnodes containing a zero are the only nodes that are not required to have a unique ancestor and a unique descendant. The graph’s nodes are joined by directed arcs (the arrows in Table 1). Chemnodes for the reactants of an enzyme-catalyzed transformation are connected to the square node that represents the enzyme via relnodes that point toward the square node, and chemnodes for the reaction products are connected to this square node via relnodes

Table 1 Nodes of the graph model

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that point away from the square node. Similarly, the chemnode ancestors of a relnode containing a zero denote the reactants and the chemnode descendants denote the products. Because a relnode’s apex points in the forward reaction direction, the arcs’ arrowheads are redundant and are usually omitted. Enzyme and spontaneous reaction subgraphs are linked into a network when a chemnode for an enzyme’s product serves as the next enzyme’s reactant chemnode (see Kohn and Lemieux, 1991, for a simple example). Chemnodes are associated with a number, c, the chemical concentration. Square chemnodes are also associated with a pair of numbers, Vf and Vr, the forward and reverse turnover numbers (a new feature of MetaNet to be used in analyzing signal propagation). Relnodes not containing a zero are associated with three numbers, K, the binding constant (equilibrium constant for dissociation of the enzyme-ligand complex), n, the ‘cooperativity index’ (essentially a Hill exponent), and M, the stoichiometry of the bound chemical in the reaction catalyzed by the enzyme. By convention, K= 0 for an irreversibly released product. Relnodes containing a zero are associated with an equilibrium constant Keq of the spontaneous reaction and a table of stoichiometries for the reactants and products (another new feature of MetaNet). If a reaction consumes r molecules of a particular reactant, M= − r. If a reaction produces p molecules for a particular product, M=p. A modifier always has a stoichiometry of zero. A binding constant can be given by an algebraic equation. Owing to electrostatic interactions, the membrane potential can influence binding of an ion to a membrane carrier protein. The Nernst equation gives the binding constant as a function of transmembrane voltage as K= K0e − h(V − V0)/RT where K0 is the binding constant at the reference voltage V0 and h is the gating charge (net charge on the occupied binding site) on the carrier protein.

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Each relnode is also associated with a ‘signal strength’ function S=

1 1+ (K%/c)n

%

%

(Kk /ck )nk +

activators

activators

k

h

(ch /Kh )nh



1 −Q/Keq

Y= %



(K%i /ci )ni 1 +

reactants k

%

(cj /K%j )nj



products k

Although this is analogous to w/Vmax in enzyme kinetics, MetaNet makes no attempt to reproduce the actual enzyme kinetics. Rather, the calculated quantity Y is a rough estimate of the fraction of the enzyme’s catalytic capacity that is being utilized in the current metabolic state. This estimate is used to qualitatively assess the enzyme’s contribution to coherent control of the pathway being modeled. The effective binding constants are as given above. Q is the mass-action ratio, and Keq is an approximation to the equilibrium constant of the overall reaction. 5

Q=

cj Mj

products i

5

Kj

reactants ci Mi i

If a binding constant is zero (only possible for an irreversibly released product), the entire term in which that binding constant appears should be deleted from the formula for the saturation function. The effect of noncompetitive modifiers on the saturation function is given by Y

Y% =

As binding of competitive modifiers may also be subject to competition from other ligands, the corresponding binding constants also may have to be replaced by effective binding constants as given above. Each square chemnode is associated with an approximate saturation function Y.

1+

5 i

(K%)%% =



Kj

products i

reactants

equal to the fraction of the corresponding binding sites occupied by the ligand whose concentration is c. The effective binding constant, which reflects the activities of competitive modifiers, is given by

K m 1+

5

Vf Keq = Vr

1+

%

activators k

(Kk/ck )nk +

%

(ch /Kn )nh

inhibitors k

Because binding of noncompetitive modifiers may be subjected to competition, effective binding constants for the modifiers may be required in the above equation. The enzymatic rate is approximated by Y% ×Et, where Et is the total concentration of the enzyme. If Q exceeds Keq, the reaction is proceeding in the reverse direction. In that case, the roles of reactants and products must be reversed and Vf and Vr also switch roles. Then Y% must be re-evaluated and its value negated. Note that the saturation function varies from 0 to 1 for an irreversible enzyme and from − 1 to 1 for a reversible enzyme. If an enzyme catalyzes more than one reaction (i.e. has alternative substrates), each substrate (or reversibly released product) is presumed to act as a competitive inhibitor of all other substrates (products) except for common cofactors. Cases where an enzyme catalyzes more than one reaction or where one enzyme serves as the substrate for another enzyme (i.e. an enzymic cascade) introduce ambiguity because it is not clear which relnode connected to a given enzyme node represents which reaction. To avoid this problem, relnodes are labeled with the index number of the corresponding reaction. The signal strength for a spontaneous reaction is given by the normalized displacement from equilibrium

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S = 1 − Q/Keq which is analogous to the numerator of an enzymatic saturation function. The mass-action ratio Q is constructed the same way as for enzymatic reactions. Note that if the spontaneous reaction is irreversible, Keq =0, the second term in the expression for S is ignored, and S 1 at all reactant concentrations.

3. Formal operations on the graph Deletion of an enzyme node and all relnodes denoting binding of ligands to that enzyme creates a ‘gap’ in the residual graph between the deleted enzyme’s reactant and product chemnodes. An acyclic path that connects a reactant ‘gap–boundary’ node to a product gap– boundary node forms a feedback loop when the deleted nodes are replaced (see Kohn and Lemieux, 1991, for an example). Feedback loops reflect the topology of the network as a whole; they are emergent systemic properties. Clearly, the regulatory properties of the deleted enzyme have no effect on either the identity of the feedback loop or the strength of the feedback signal flowing along the ‘gap – spanning’ paths. Thus, feedback of information around the gap does not depend on local regulation at the level of the deleted enzyme. The union of all the gap– spanning paths obtained by deletion of each enzyme in sequence (with subsequent replacement) constitutes the routes by which feedback information flows through the network. A set of relnodes whose deletion would simultaneously sever all the gap– spanning paths is termed a ‘cut’ set. The cut set with the fewest members represents the ligand-binding processes that are the critical feedback events. The immediate chemnode ancestors of the relnodes in the smallest cut set are the feedback metabolites. These chemicals share in the global feedback regulation of the pathway in proportion to the values of their signal strength functions. Determination of the minimal cut set qualitatively identifies the factors controlling the flow

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of feedback information in the pathway being modeled. There may be more than one cut set with the same minimal number of members. This outcome is termed a ‘degenerate’ solution. The cut set in a degenerate solution that has the largest sum of signal strengths (i.e. carries the largest total feedback signal) for the state of the system being analyzed is selected as the critical cut set under those conditions. This quantitative comparison of degenerate cuts removes the degeneracy of the solution and identifies the most important feedback processes. The activities of the enzymes that produce or consume the feedback metabolites set the concentrations of those chemical species and thus determine the signal strengths associated with the cut set relnodes. These enzymes share in the control of feedback in proportion to the values of their saturation functions. Because enzymes that are near equilibrium or are highly inhibited have saturation function values close to zero, these enzymes are less effective in regulating the concentrations of the feedback metabolites than are enzymes that are closer to saturation. The metabolites that are potential feedback regulators and the enzymes that control their concentrations can be identified from the structure of the graph alone. Calculating the effectiveness of each feedback metabolite and controlling enzyme requires estimates of the parameters and the intermediate concentrations. The instantaneous state of the system is specified by the concentrations of the intermediates, which determine the values of the signal strengths. As the state changes, a different degenerate cut set can become dominant or the altered fractional saturation of binding sites can switch the relative importance of relnodes within a cut set. Consequently, the predicted locus of feedback regulation and the identities of the controlling enzymes can change as a system moves along a trajectory in state space or switches between alternative steady states. This was demonstrated with a MetaNet model of prokaryotic gene expression (Kohn, 1986) and a MetaNet model of glycolysis in Ehrlich ascites cells (Kohn and Lemieux, 1991).

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4. Computer implementation Some have suggested that the kinetic parameters of biochemical networks must have specific values to permit proper functioning of the system (Segal et al., 1986). Others believe that systemic performance may be a robust property of the network rather than depending on ‘fine tuning’ of the parameter values (Barkai and Leibler, 1997). The staff of BioKinetics, Inc. (http:// www.biokinetics.net) implemented the above strategy in a computer program (METANET™) to facilitate drawing and analysis of a MetaNet graph. The program calculates all signal strengths and saturation functions. The software identifies all cut sets and ranks them in order of their total signal strengths. This program was used to investigate the sensitivity of the identity of the most important ligandbinding processes in the minimal cut set to the values of the parameters in a large-scale network. High sensitivity would be consistent with the proposal that parameters are fine-tuned for their function. Low sensitivity would be consistent with robustness of network performance with respect to parameter variations and would increase confidence in the predictions of a MetaNet model. The hepatic pathways for urea production have been studied intensively and are well understood. We addressed the question of robustness by constructing a MetaNet model of the urea pathway and analyzing its regulatory properties with the METANET™ computer program.

5. Modeling urea production Humans excrete excess nitrogen derived from protein metabolism as urea formed in the liver. Urea is produced by a cycle of one mitochondrial (OCT in Fig. 1) and three cytosolic enzymes (ASS, ASL, A in Fig. 2). One of the urea nitrogens comes from the amino group of cytosolic aspartate. The source of the other nitrogen has been controversial. Ammonia is fixed as carbamoyl phosphate, which is condensed with ornithine to produce citrulline in the urea cycle. Because a low rate of ammonia production by

glutamate dehydrogenase had been observed in isolated liver mitochondria and the activity of liver glutaminase is low, it had been claimed that ammonia is produced in the intestinal mucosa from glutamine and transported to the liver (Lund and Watford, 1976). The low hepatic glutaminase activity is supported by the finding that when 13N-(amide) glutamine was injected into the portal vein of rats, less than 2% of the label appeared in other metabolic intermediates in the liver and only a minute amount of the label appeared in blood urea (Cooper et al., 1988). However, glutamine uptake accounts for only 9% of the ammonia released from the gut (Ha¨ ussinger and Gerok, 1986). The observed low rate of ammonia production from glutamate was attributed to the low rate of glutamate uptake by isolated mitochondria, and hepatic glutamate dehydrogenase was identified as the origin of ammonia in liver (Brosnan et al., 1981). Therefore, the origin of the second nitrogen was attributed in this model to ammonia produced by glutamate dehydrogenase. Other processes represented in the MetaNet model (Figs. 1 and 2) include: “ enzymes that transfer amino groups among aspartate, glutamate, and ornithine; “ mitochondrial membrane carriers that transport aspartate, glutamate, ornithine, and citrulline between mitochondria and cytosol; and “ production of D1 pyrroline-5-carboxylate, produced by deamination of ornithine, and creatine, derived from arginine. Concentrations of intermediates (Table 2) and the biochemical parameters (Table 3) were estimated from the literature. Human data were used for the maximal velocities of the enzymes because these values can be highly variable among species. Where concentrations or binding constants were not available for human liver, data from another species were used as these quantities tend to be less variable among species. The average value was used when several values were reported for the same parameter (variations of 20–30% about the mean are typical). Enzyme concentrations were calculated as the ratio of the maximal velocity to the turnover number. All cooperativity indices were set to 1, as

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Fig. 1. Mitochondrial metabolism associated with urea production in human liver. The OAT pathway is an alternative route for disposition of excess nitrogen. Abbreviations are given in Tables 2 and 3. Chemnodes with thick borders indicate connections to the cytosolic portion of the network.

there are no data demonstrating cooperative binding. All enzyme reactants and products were assigned unit stoichiometries except for carbamoyl phosphate synthetase, which hydrolyzes two molecules of MgATP. The fully assembled model was encoded as a METANET™ graph and analyzed to identify the sites of feedback regulation of urea production.

A shorthand notation for relnodes was developed to simplify reference to particular subgraphs and was used in Table 3. The symbols \ , +\, −\, and 0\ represent relnodes for substrate, activator, or inhibitor binding or a spontaneous reaction, respectively. These symbols appear between the names of the ancestor and descendant chemnodes. When the descendant node is another

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relnode (i.e. for competitive binding) the relationship is denoted by structures such as I −\ (A \ D). I is an inhibitor and A and D are the ancestor and descendant nodes, respectively, of the relnode that is the descendant of the inhibitory relnode. Parentheses can be nested recursively to represent modifiers affecting the binding of other modifiers. Compartmentation was indicated by the prefixes m- for mitochondrial and cfor cytosolic. The computer analysis of the urea production model identified a single cut set comprising 13

members (Table 4). No other cut sets of 13 members or fewer were found, indicating the absence of degeneracy in this particular model. That is, any other state of the system, characterized by different intermediate concentrations, would have the same minimal cut set but different signal strengths for the constituent relnodes. Only two of the relnodes in the minimal cut set were for binding to enzymes of the urea cycle itself. Five relnodes were for enzymes that provide amino groups to cycle enzymes as aspartate or carbamoyl phosphate. Five relnodes were for trans-

Fig. 2. Cytosolic metabolism associated with urea production in human liver. Creatine production is an alternative pathway for disposition of excess nitrogen. Abbreviations are given in Tables 2 and 3. Chemnodes with thick borders indicate connections to the mitochondrial portion of the network.

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Table 2 Concentrations of metabolic intermediates in the urea production model Metabolite

Concentration (mM) Cytosolica,b

Coenzymes Coenzyme A (CoA)b,d Acetyl coenzyme A (AcCoA)b,d AMPb,d,e MgATPb,d,e,f NADb,d,e NADHb,d,e Inorganic chemicals Orthophosphate (PI)b,d,e Ammonia (NH3)g h Bicarbonate (HCO− 3 ) Amino acids and related compounds Carbamoyl phosphate (CP)i,j Citrulline (Cit)j,k Ornithine (Orn)j,k Arginine (Arg)j,l Arginosuccinnate (ArgSucc)k Glutamate (Glu)b,d Aspartate (Asp)b,d Glycine (Gly)m Acetylglutamate (AcGlu)l,n,o

0.218 4.54

7.4

0.085 0.51 0.05 0.034 10.6 1.20 2.0

Mitochondrialb,c

0.38 0.24 2.6 1.8 1.58 0.36 4.1 0.089 7.0 0.085 0.15 0.42

2.57 0.40 0.68

Carboxylic acids a-Ketoglutarate (aKG)b,d,p Oxaloacetate (OAA)b,d Fumarate (Fum)q

2.0 0.015 1

Production of sinks for nitrogen Guandinyl acetate (GuaAc)r Creatine (Cr)q S-adenosyl methionine (AdoMet)s D1 Pyrroline-5-carboxylate (P5C)t Ureak,u

0.042 1 0.046 0.02 6.0

0.31 0.07

Abbreviations are those used in the pathway schemes. a Assumed to comprise 80% of the cellular space. b Soboll et al. (1976). c Assumed to comprise 20% of the cellular space. d Siess et al. (1982)). e So¨ ling (1982). f Calculated from cytosolic Mg2+ = 0.4 mM (Siess et al., 1982) and mitochondrial Mg2+ =0.83 mM (Panov and Scarpa, 1996). g Duszynski et al. (1978). h West (1991). i Keppler and Holstege (1982). j Meijer and Hensgens (1982). k Raijman (1976). l Cathelineau et al. (1982). m Set equal to the Km of arginine:glycine amidinotransferase. n Cohen (1976). o Powers-Lee and Meister (1988). p Zuurendonk et al. (1976). q Sink chemical set to program default value as it has no effect on model. r Mikami et al. (1982). s Mudd et al. (1980). t Assumed equal to 0.2×Km for ornithine aminotransferase. u Saheki et al. (1982a).

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Table 3 Kinetic constants in the urea production model

a

Enzyme, concentration (mM) Carbamoyl phosphate synthesis N-acetylglutamate synthetase (AGS), 2.23×10−5b,c

Carbamoyl phosphate synthetase (CPS), 0.003c,e

Urea cycle enzymes Ornithine carbamoyltransferase (OCT), 0.00231i

Vf, Vr (min−1)

Relnode

2230d 0

m-Glu\AGS AcCoA\AGS AGS\CoA AGS\AcGlu AcGlu+\CPS HCO− 3 \CPS NH3\CPS m-MgATP\CPS CPS\CP CPS\AMP CPS\m-Pi

1550f 0

33 120f 0

Arginosuccinate synthetase (ASS), 0.00193e

777f 0

Arginosuccinate lyase (ASL), 0.000778e

4710f 3340

Arginase (A), 0.00102e

1.41×106f 0

Mitochondrial membrane translocases Ornithine–citrulline translocase (OTL), 0.00616o

Glutamate translocase (GTL), 0.0001p Glutamate–aspartate translocase (GATL), 0.00862s

Transaminases and glutamate metabolism Ornithine aminotransferase (OAT), 0.0331v

Mitochondrial glutamate:oxaloacetate transaminase (m-GOT), 0.001u

7440o 7440o 7440o 10 200q 10 200q 29 000t 29 000t

8450v 74w

32 180z 16 400z, A

m-Orn\OCT CP\OCT OCT\m-Cit OCT\m-Pi c-Cit\ASS c-MgATP\ASS c-Asp\ASS ASS\ArgSucc ASS\c-AMP ASS\c-Pi ArgSucc\ASL ASL\Fum ASL\Arg Arg\A A\c-Orn A\Urea c-Orn−\A

K (mM)

3.0d 0.7d 0 0 0.1g 3.5g 0.25g 0.76h 0 0 0 0.4f 0.1j 0 0 0.041f,k 0.241f,k 0.0321f,k 0 0 0 0.052l 1.0 0.42 1.56m 18n 0 20.4n

m-Cit\OTL c-Orn\OTL OTL\m-Orn OTL\c-Cit c-Glu\GTL GTL\m-Glu c-Glu\GATL m-Asp\GATL GATL\m-Glu GATL\c-Asp

3.6o 0.16o 0.16o 3.6o 4.0q 1.75r 1.6u 0.4u 1.6u 0.4u

m-Orn\OAT m-aKG\OAT OAT\P5C OAT\m-Glu m-Glu\m-GOT m-OAA\m-GOT m-GOT\m-Asp m-GOT\m-aKG

1.1v 1.1v 0.1x 7.5y 7.5z, A 0.0242z, A 1.12z, A, B 0.7z, A, B

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Table 3 (Continued) Enzyme, concentration (mM)

Vf, Vr (min−1)

Relnode

Cytosolic glutamate:oxaloacetate transaminase (c-GOT), 0.001u

16 400z 32 180z, A

Glutamate dehydrogenase (GDH), 0.001u

64 850C 11 240C

c-Asp\c-GOT c-aKG\c-GOT c-GOT\c-Glu c-GOT\c-OAA m-Glu\GDH NAD\GDH GDH\m-aKG GDH\NH3 GDH\NADH

3.12z, A, B 0.31z, A, B 6.5z, A 0.048z, A 1.8z 0.071z 0.7z 0.5z, D 0.03z

Gly\GAT Arg\GAT c-Orn−\(Arg\GAT) GAT\c-Orn GAT\GuaAc AdoMet\GAMT GuaAc\GAMT GAMT\Cr

1.8F 1.3F 0.116H 0.253H 0.05I 0.09K 0.0014K 0

Creatine synthesis Arginine:glycine amidinotransferase (GAT), 0.0241E

Guanidinoacetate methyltransferase (GAMT), 0.00299J

2076E 44.2G

2120J 0

K (mM)

a The prefix m- indicates mitochondrial compartment. The prefix c- indicates cytosolic compartment. Abbreviations are those used in the pathway schemes. b Bachman et al. (1982). c Brusilow and Horwich (1995). d Tatibana et al. (1976). e Cohen (1976). f Powers-Lee and Meister (1988). g Elliott (1976). h Cohen et al. (1992). i Calculated as concentration of active sites from average of literature tissue capacities, 76.4 IU/g. j Average value from literature. k Saheki et al. (1982b). l Average from Murakami-Murofushi and Ratner (1982), Powers-Lee and Meister (1988) and Ratner (1976). m Sobero´ n and Palacios (1976). n Bedino (1977). o Indiveri et al. (1992) and Kra¨ mer and Palmieri (1989). p Assumed value; model is not sensitive to this parameter. q LaNoue and Schoolwerth (1984). r Calculated from mitochondrial membrane proton gradient and observed Km for external glutamate (LaNoue and Schoolwerth, 1984). s Assumed to comprise 0.1% of mitochondrial protein. Model is not sensitive to this parameter. t Kra¨ mer et al. (1986). u LaNoue et al. (1979). v Ohura et al. (1982). w Calculated from data of Strecker (1965). x Assumed owing to lack of data. y Set to average value for other mitochondrial aminotransferases owing to lack of data. z Barman (1969). A Braunstein (1973). B Tager et al. (1969). C Gonza´ lez et al. (1976). D Value selected to agree with equilibrium constant adjusted for mitochondrial pH (Barman, 1969; Williamson et al., 1976). E Similar kinetics in liver and kidney (Gross et al., 1986; Methfessel, 1976). F Walker (1979). G Calculated to reproduce equilibrium constant (Walker, 1979). H Estimated from data of (Walker, 1957) by formal optimization. I Assumed value, owing to lack of data. J Extrapolated from partially purified preparation of (Cantoni and Vignos, 1954). K Daly (1985).

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Table 4 Relnodes of the minimal cut set for the urea production model Relnode

Signal strength

GATL\c-Asp AcGlu+\CPS c-Orn\OTL c-Glu\GTL OTL\m-Orn c-Cit\ASS c-GOT\c-Glu GATL\m-Glu m-Glu\GDH m-Orn\OCT c-Orn−\(Arg\GAT) m-Glu\m-GOT m-aKG\OAT

0.968 0.872 0.761 0.726 0.724 0.675 0.620 0.616 0.588 0.512 0.288 0.255 0.220

port of amino acids across the mitochondrial membrane. These results identify the loci of feedback regulation of urea production in the provision of amino groups rather than in the cycle enzymes themselves. Especially prominent among these processes is binding to translocases (four of the largest five signal strengths), indicating the importance of substrate translocation to the maintenance of communication among the metabolizing enzymes. These results are consistent with the finding that the rate of urea synthesis in vivo is controlled by substrate availability (Beliveau Carey et al., 1993). Five of the members of the minimal cut set are for glutamate binding; four are for ornithine bindTable 5 Enzymes controlling the strength of the feedback signal in the urea production model Enzyme

Saturation function

ASS GTL AGS OCT OAT c-GOT GDH A GATL OTL

0.640 0.231 0.197 0.190 0.095 0.066 0.043 0.030 0.030 0.006

ing; and one relnode each is for citrulline, a-ketoglutarate, acetylglutamate, and aspartate binding. These are the feedback chemicals for this network. The enzymes that make or consume these intermediates and their saturation functions are listed in Table 5. The cycle enzymes do appear prominently in this list. Arginase is less important because it has a large maximal velocity and substrate concentrations far below saturation are sufficient to drive the cycle flux through this enzyme. The mitochondrial membrane translocases and the transaminases are less important controllers as the reactions are mostly near equilibrium. An exception is the proton-compensated glutamate translocase, which depends on the transmembrane proton gradient for uptake of glutamate by mitochondria and is therefore further from equilibrium. Rapidly reversible enzymes with nearly saturated substrate binding sites dominate the flow of information in the network. This property of metabolic pathways has been attributed to the high sensitivity of the rate through an enzyme near equilibrium to variations in substrate concentration (Fell, 1996). However, enzymes near equilibrium have little control of the strength of the feedback signal because their saturation functions are near zero. Control is dominated, instead, by irreversible enzymes (especially argininosuccinate synthetase) or enzymes far from equilibrium (e.g. the proton-compensated glutamate translocator). A topological analysis of flux control (Sen, 1990) predicted control of a pathway by enzymes whose rates are insensitive to their products. Similar conclusions were reached from a sensitivity analysis of a large-scale simulation of intermediary metabolism in the rat heart (Kohn, 1983).

6. Robustness of MetaNet predictions for urea production The sensitivity of the predictions of the MetaNet for urea production was investigated by systematic variation of its parameter values and re-evaluation of the signal strength of each member of the minimal cut set. The nominal value of each parameter was considered to be the mean of

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Fig. 3. Frequency of changes in rank order of the cut set relnodes as specified by the values of their signal strengths. Obtained from a population of 100 000 variants with parameter values randomly selected from logarithmic normal distributions with the nominal value as the mean and an assumed standard deviation of 20% of the mean.

a logarithmic normal distribution with a standard deviation equal to 20% of the mean, comparable to the spread of values expected from experimental measurement. In each of 100 000 runs all the binding constants were replaced by a value randomly selected from its respective distribution function with a program written in MATLAB. In this process, the distributions of parameter values were treated as independent. It is possible that strong correlations among parameter values could alter the reordering of the cut set relnodes, but that was not the case in this study. Fig. 3 is a histogram of the frequency with which the rank order of each member of the cut set, as given by its signal strength function, changed from that in the nominal model as a result of this sampling strategy. Cut set relnodes constituting 56% of the random samples retained their rank orders after perturbation of the binding constants. Relnodes constituting 18% of the random samples switched rank order with a relnode in another set by one place. The vast majority of these cases were for pairs of relnodes with similar nominal signal strengths. For example, the relnodes OTL \ m-Orn and c-Glu\GTL have

nominal signal strengths of 0.724 and 0.726, respectively. Random alteration of the binding constants often caused small numerical differences of opposite sign in the values of these signal strengths, resulting in swapping the rank order of the corresponding relnodes. However, the numerical changes in signal strengths were too small to affect the significance of the binding process for feedback regulation of the network. A change in rank order by two or more places could alter the qualitative implications of a MetaNet model. Pairs of relnodes switched rank order with relnodes in another set by two or more places in only 4% of the cases. The relnodes in this set were mostly c-Orn−\(Arg \ GAT), m-Glu \ m-GOT, and m-aKG \ OAT, which have the smallest signal strengths (Table 4) and are on the linear portion of the binding versus concentration curve. The signal strengths of more saturated binding sites are less sensitive to deviations in the corresponding binding constant or concentration. Because the signal strengths in the more sensitive set are so small, the corresponding ligand-binding events are minor contributors to the total feedback

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and parameter uncertainty has little effect on the model’s qualitative predictions. This robustness is confirmed in Fig. 4, which shows the empirically generated probability density function for the relative deviations, D, of the signal strengths from their nominal values. D=

S− S* S*

where S* is the signal strength at the nominal parameter value. The distribution is centered strongly at zero, and at least 95% of the distribution has a deviation within 920% of the nominal value. This distribution is much steeper than either a normal or logarithmic normal distribution. Even increasing the standard deviation of the logarithmic normal distribution of binding constants to 50% of the mean does not introduce great sensitivity of the signal strengths to parameter variations (Fig. 4). Nearly identical results were obtained when metabolite concentrations were all replaced by values randomly selected from a similar logarithmic normal distribution.

7. Conclusions MetaNet analysis of the urea production pathway predicts that feedback regulation is dominated by provision of amino groups to the cycle enzymes. This prediction was shown to be robust with respect to variations in parameter values, supporting the origin of the regulatory properties in the topology of the network. It is useful to examine alternative notions of metabolic regulation and to investigate the possibility that the predicted behavior was a chance occurrence. Achievement of homeostasis has been attributed to the presence of sigmoid interactions (Glass and Kauffman, 1972; Koshland et al., 1982). As none of the cooperativity indices in this model exceeds 1, the kinetics do not exhibit sigmoidicity. Random alteration of the intermediate concentrations did not alter the pattern of feedback regulation in the urea production model. Therefore, a restricted dynamic range of the feedback metabolites cannot be the origin of the robustness of the predicted control properties.

Fig. 4. Density function of the relative change in signal strengths of the cut set relnodes obtained from a population of 100 000 variants with parameter values randomly selected from logarithmic normal distributions with the nominal value as the mean. Solid line is for an assumed standard deviation of 20% of the mean and broken line is for an assumed standard deviation of 50% of the mean.

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Fig. 5. Power law distribution of node degrees in the urea production model. This distribution is characteristic of scale-invariance; the slope is nearly independent of the size of the network although the intercept may vary. The graph is dominated by m-Glu, which participates in six reactions, consistent with the prediction that urea production is regulated mostly by delivery of amino groups to the cycle enzymes.

Kinetic analysis by continuous functions and by logical analysis can yield comparable qualitative descriptions of systemic regulation (Thomas and D’Ari, 1990), suggesting that systemic properties are derived from the topology of the network rather than from the numerical values of its parameters. Randomly connected graphs constructed from a single type of node tend to have a Poisson distribution of node degrees, k (Bolloba´ s, 1985). Metabolic networks constructed from chemical relationships among substrates exhibit a power law distribution of node degrees (Jeong et al., 2000). Such a structure is called a scale-free network, whose topology is dominated by a small number of nodes of high degree and whose properties are known to be robust to alterations of structure (Jeong et al., 2000). Addition of nodes

to the graph does not significantly alter the slope of a log –log plot of probability (k) versus k but merely extends the line to higher degrees. It was shown previously that interactions among enzymes in a network are not random (Kohn and Bedrosian, 1985). A graph in which the nodes for two enzymes are joined only if the proteins share a common ligand forms loosely connected clusters instead of the Poisson distribution of connections expected for a randomly connected graph (Kohn and Bedrosian, 1985). In a MetaNet model, the probability of a ligand participating in a reaction is approximated by the degree of its circular chemnode divided by the sum of the degrees of all circular nodes. The urea production model exhibits a power law distribution of node degrees (Fig. 5) with an exponent of − 1.8, similar to the value of − 2.2 found for

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much larger networks (Jeong et al., 2000). The connectivity of the MetaNet is dominated by mitochondrial glutamate (k =6). Removal of the node for this intermediate severs several paths of information flow. This causes the graph to disintegrate into isolated segments, consistent with the identification of supply of amino groups as the critical aspect of feedback regulation. These properties are not expected from a randomly connected network. Novel therapeutic interventions for the treatment of metabolic diseases are usually based on trial and error guided by the general features of the affected pathway. A deeper knowledge of the complex interactions within metabolic pathways is essential for a more systematic approach to design therapy. The complexity of metabolism may give rise to regulatory interactions that are unsuspected from the properties of the constituent enzymes. Furthermore, the values of biochemical parameters may vary significantly among individuals, and their measurement in humans is restricted by ethical considerations. A tool that is not excessively sensitive to parameter values can further our understanding of the interactions among the enzymes, substrates, and products of metabolic pathways and facilitate a more rational approach to the treatment of metabolic disease. Persons with an inherited deficiency in one of the urea cycle enzymes accumulate the intermediate that is the substrate for the deficient enzyme, and they can achieve toxic plasma concentrations of ammonia or glutamine. The MetaNet model of urea production identified provision of aspartate as an important regulatory process. Transaminases rapidly exchange nitrogen among amino acids (Cooper et al., 1988), suggesting that excretion of waste nitrogen could be enhanced by conjugation of amino acids in equilibrium with aspartate. Treatment of patients deficient in ornithine carbamoyl transferase with benzoate or phenyl acetate promotes nitrogen excretion as hippuric acid or phenylacetylglutamine, respectively (Brusilow et al., 1980). The MetaNet model also identifies binding of citrulline to argininosuccinate synthetase as a control point. Administration of arginine, a precursor of citrulline, can compensate for deficiencies in several cycle enzymes (Brusilow, 1984). Thus, enhanced provision of substrates of cycle

enzymes or redirecting those substrates to alternative products are effective therapies for such enzyme deficiencies. The predictions of the MetaNet analysis rationalize therapies arrived at empirically.

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