Mathematical model supporting the superoxide theory of oxygen toxicity

bYeeRadical Biology & Medicine, Vol. 16, pp. 63-72, 1994 Printed in the USA.All rightsreserved.

0891-5849/94 $6.00 ~-.00 Copyright © 1993PergamonPress Ltd.

Review Article MATHEMATICAL MODEL SUPPORTING THE SUPEROXIDE THEORY OF OXYGEN TOXICITY YUICHIRO J. SUZUKI* and GEORGE D. FORDt *Department of Molecular & Cell Biology, University of California, Berkeley, CA, USA tDepartment of Physiology, Medical College of Virginia, Richmond, VA, USA (Received 4 December 1992; Revised and Accepted 17 May 1993)

Abstract--The discovery of superoxide dismutase was followed by a proposal that superoxide anion radical (02"-) is a major factor in oxygen toxicity. The knowledgeof superoxide chemistry, however, led some chemists to conclude that since 02°- is not very reactive in aqueous solution, the more reactive hydroxyl radical (HO") was most likely to be the major damage causing species. Some have defended the superoxide theory by emphasizing that nonindiscriminate and selective reactivity could provide more toxicity than would high, indiscriminate reactivity. In the present study, network thermodynamic simulation was used to create a situation in which O2"- would selectively react with a substrate in a hypothetical sequence of subreactions supporting biologicalprocesses. In this situation, when the simulation of the chemical reactions was carried out using reasonable parametric values found in the literature, the selectivereaction of O2"- to one molecule in the sequence caused a 95% disruption of the observableprocess, whereas indiscriminately targeted HO" attack caused only 0 to 35% inhibition. The major cause of the weak effect of HO" was found, in this particular model, to be a lack of sufficient availability of HO" due to both its slow generation by the Fenton reaction and a large demand for reactions with inconsequential targets. This model supports the superoxide theory of oxygentoxicity by demonstrating that a simple set of circumstances can quantitatively lead to the proposed selective superoxide toxicity. The present study also advocates the use of novel network thermodynamic simulation techniques for solving problems concerning biological oxidants and antioxidants. Keywords--Computer simulation, Free radicals, Mathematical modeling, Network thermodynamics, Oxy-radicals, Reactive oxygen species, SPICE, Superoxide anion radical, Superoxide theory of oxygen toxicity

INTRODUCFION

cal (02"-) is a m a j o r factor i n oxygen toxicity a n d that superoxide d i s m u t a s e s provide the p r i m a r y defense against such toxicity.l-3 This superoxide theory o f oxygen toxicity, however, has b e e n q u e s t i o n e d because, i n a c h e m i c a l sense, 02"- is k n o w n as a p o o r o x i d a n t i n a q u e o u s s o l u t i o n 4'5 a n d the stronger oxidant, the hydroxyl radical (HO"), was felt to be a p o t e n t i a l l y m o r e toxic agent. W h e n o n e recognizes the very high reactivity o f H O ' , which c a n react with m o s t organic c o m p o u n d s with reaction rates o f 107-101° M - I s -~ (ref. 6), it does a p p e a r likely that HO" is the deleterious d a m a g i n g species i n oxygen toxicity. M c C o r d a n d Russell, 7 however, r e e m p h a s i z e d the superoxide theory o f oxygen toxicity stating that: "superoxide is n o t a highly reactive free radical. T h e hydroxyl radical, b y contrast, has a l m o s t u n l i m i t e d reac-

T h e discovery o f superoxide d i s m u t a s e in aerobic cells led to a proposal that the superoxide a n i o n radiAddress correspondence to: Yuichiro J. Suzuki, 251 Life Sciences Addition, Department of Molecular & Cell Biology, University of California, Berkeley, CA 94720 USA. Yuichiro Justin Suzuki is an American Heart Association California AffiliatePostdoctoral Research Fellow at the Department of Molecular & Cell Biology, University of California, Berkeley. After studying biology at Ohio Northern University, he received his M.S. and Ph.D. degrees in physiologyand biophysics at the Medical College of Virigina. He is a member of the Oxygen Society, Biophysical Society and American Chemical Society. He is also a cofounder of the Bay Area Oxygen Club with Prof. Lester Packer, and is currently serving as a corresponding organizer. His research interests in biological oxidation include the role of biological oxidants and antioxidants on signal transduction and gene regulation. George D. Ford is a Professor of Physiology and a Director of the School of Basic Health Sciences at the Medical College of Virginia, Virginia Commonwealth University. His major research interests are the mechanisms of excitation-contraction coupling in vascular smooth muscle and pertubations to these mechanisms caused by intercellular magnesium ions or reactive oxygen species. In 1982-83, he was a Fulbright Research Scholar at the University

of Leuven in Leuven, Belgium. He is also the director of the departmental teaching laboratory and codeveloper of a video disk-based instructional laboratory program in cardiovascular physiology.

63

64

5 . J. SUZUKI AND G. D. FORD I able I. R a t e C o n s t a n t s Used in lhe M o d e l

HO2 ~ O~, ---- H:O2 ~ O: Fe 3+ + O~ ~ Fe > ~ Oe Fe 2+ + 0 2 ~ Fe 3. + O ; Fe '+ + H202 ~ Fe 3. ~ - t t O " + Ot-I HO" + Fe 2' ~ Fe ~* ~- O H HO'+H202-+O ~ ~ H,O+ H ~ 2HO" --* H2Oz H O ' + O~ --+ O: + O t t

tivity, but cn a mole-for-mole basis this does not necessarily imply greater toxicity. A cell may recover very, well after losing 5% of every, enzyme activity, but it may not recover at all after losing 95% of one particularly vital enzyme activity. Reactivity and toxicity are, to a large degree, inversely related." Although the statement, in general, seems realistic, the notion stating the inverse relationship between indiscriminate reactivity and toxicity is certainly not a well-accepted concept in biochemical toxicology. The present study used mathematical modeling and computer simulation to test the viability of such a concept. This is important, as some biological molecules have been shown to bc directly affected by superoxide. 7'8 Examples are epinephrine, 9 catalase, m lactic dehydrogenase bound NAI)H, ~ creatine phosphokinase, v Ca 2 +-ATPase of vascular smooth muscle sarcoplasmic reticulure, ~2 and [4Fe-4S] cluster-containing acid dehydratases su,:h as aeonitase, ~3'~4 6-phosphogluconate dehydratas¢, '5 and a,/~-dihydroxyisovalerate dehydratase. ~6 In addition, 02"- has been shown to influence some of the key processes in cell regulation: it stimulates I P3-induced Ca 2+ release from the sarcoplasmic reticulum of smooth muscle 17 and also induces c-fos proto-oncogene expression in proximal tubular epithelium cells.~8 Mathematical modeling and computer simulation have proven to be useful in free radical research. ~9-2' However, numerical solutions of the nonlinear partial differential equations arising in reaction-diffusion problems challenge the speed and storage capabilities of computer systems as well as mathematical sophistication and programming skills. The present study applied the more novel network thermodynamic method which is an easy to use, yet powerful, modeling technique. = The most unique feature of this technique, which is derived from the theories of Oster, Perelson, and Katchalsky 23 and permits coupled flows and driving forces to be analyzed in terms of circuits, is that the model is based on a topological network instead of a series of equations. In addition, this approach does not require excessive computer time or storage. The network thermodynamic method has

Rate C o n s t a n t s (M J s i)

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been applied to a variety of biological systems including glomerular dynamics, 24 sodium transport across epithelial tissue, 25'26 the effect of insulin on adipocyle hexose transport, 27 and action of anticancer agents. -'2 We have previously used this approach to simulate free radical reactions to estimate products in an experimental system in which the effects of reactive oxygen species generated by hypoxanthine plus xanthine oxidase on Ca2+-ATPase were investigated. 2s The results from the present simulation support the superoxide theory of oxygen toxicity by demonstrating a situation in which the selective effect of the less reactive superoxide causes a significant toxicity, using a reaction scheme and parameters that could easily occur in physiological systems.

MATERIALS

AND METItODS

The model begins with the assumption that some observable biological "process" is represented as a pool of accumulated elements resulting from an ordered sequence of processes, each of which may be represented as a first order chemical reaction, i.e., A --~ B. Such a model would be represented as fol-

Inactive pool [

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lows: A --~ B --~ C "~ D "~ E "~ F "-~ G "~ H --~ I -"~ J ~ observed biological process. The number of"steps" in the model was arbitrarily chosen as 10 and the first order kinetic rate constant for the reaction C ~ D was also arbitrarily chosen to be 100 times lower than all other such rate constants (i.e., the rate-limiting step). Such a model may be considered as a representation of any event requiring a sequential set of events, i.e., it could be considered a metabolic pathway in which a prescribed sequence of events leads to the formation o f some product (the observed parameter) or it could be considered an ordered sequence o f subreactions in an overall enzymatic reaction, i.e., an ATPase must first bind its substrate, break the high energy bond, and then liberate the breakdown products (the observed parameters). In our simulation, the initial concentration o f ' A ' was arbitrarily chosen as 10 ~M, and those o f other elements started at zero. The next assumption is that reactive oxygen species inhibit the biological process under observation

by reacting with one or more o f the elements catalyzing the reactions and thereby reducing the concentration of subsequent intermediates. In practice, only two such reactive oxygen species, O2"- and H O ' , were considered. Extension of the model to allow for other reactive oxygen species could be done if so desired. The generation of reactive oxygen species was "coupled" to the sequence by providing two or more pathways for the reaction o f each element of the sequence; one pathway would be the normal sequence, e.g., A --~ B while other pathways would react A with a c o m m o n pool o f reactive oxygen species, e.g., A + 02"- --~ inactive pool. Note this coupling allows for scavenging effects, i.e., once O2"- reacts with A, it is removed from the c o m m o n pool of reactive oxygen species so that it cannot react with other substrates in the sequence nor with the substrates generating more reactive oxygen species. Again, in practice, the comm o n HO" pool was allowed to react with every element in the step sequence, but only one element in the chain was allowed to react with 02"-. All elements

66

Y . J . SUZUKI AND G. D. FORt>

in the chain (A through J) were presumed to react with HO" with a bimolecular rate constant of l 0 9 u n l e s s otherwise stated. When Oz'- reacted, a bimolecular rate constant of 107 was assumed unless otherwise stated. The reactive oxygen species were calculated using the reactions summarized in Table 1. All kinetic constants of these reactions were taken from the literature as referenced in the table. The initial Oz'- needed to key these reactions was calculated using a xanthinexanthine oxidase generator, i.e., a Michaelis-Menton reaction with K m = 1.0 #M 33 V m a x - - 1.0 ttM/s, and initial values of 50 uM and 0.25 m M for xanthine and molecular oxygen, respectively. The concentration of ferric ion, an ingredient necessary to catalyze the production of H O ' , was assumed to be 3 #M unless otherwise noted. A descriptive representation of the model is shown in Fig. 1. The complete model was assembled using the theory of network thermodynamics (see Fig. 2 for circuit diagram) and the solutions at 0.1 rain were obtained using SPICE2 (Simulation Program with Integrated Circuit Emphasis, Version 2) running on a VAX 8650. The simulation was done with an initial xanthine concentration of zero to produce control values for the biological response. All subsequent values were expressed in percent of this "control" biological process. (See Appendix 1 for a detailed introduction to network thermodynamic modeling and simulation using SPICE2, and Appendix 2 for a complete SPICE2 program for the model used in the present study.) RESULTS

The results of the model when the HO" alone was allowed to react with every element in the reaction sequence are shown in the right-most column of Fig. 3. No inhibition resulted even though the each element in the reaction sequence reacted with the HO" at a rapid rate (k = 109 M - I s - 1). However, as shown in the middle column, if just one of elements in the reaction sequence (in this case, the third element or 'C'), was allowed to react with the Oz'- at a lower rate (k = 107 M - I S - I ) , significant inhibition resulted. The leftmost column of Fig. 3 demonstrates that when the two previous conditions were both allowed to exist, there was no significant synergism of the Oz'--induced inhibition. One could argue that this model does not adequately reflect the reactivity of the H O ' . However, increasing the rate constant with which the HO" interacts with each species in the reaction sequence as high as 1015 still produces no significant inhibition, despite a million-fold increase in apparent reactivity (data not

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HO Fig. 3. Inhibition o f biological process. For 02 ° -induced inhibition, the element "C' was reacted with O2"- with a bimolecular kinetic rate constant of 107 M ~s -~ to produce irreversibly inactive species. For HO"-induced inhibition all the elements (A J ) were reacted with HO" with a bimolecular kinetic rate constant ~f 10" M - t s ~ to produce irreversibly inactive species. The initial concentration of ferric ion was 3 ~tM, while the bimolecular kinetic rate constant for the Fenton reaction was 76 M ~s t

shown). Likewise, the inhibitory effect of O~'- is not particularly dependent on the 02"- reactivity. As shown in Fig. 4, reducing the rate constant for the reaction between 02"- and the single element in the reaction sequence as low as 105 still results in significant inhibition of the overall process. This result is not too surprising if one inspects the equations generating the H O ' . The only source of HO" is the Fenton reaction (the fourth reaction in Table 1). This reaction, under ideal circumstances. produces a very low level of H O ' . This reaction also absolutely requires the presence of trace amounts of iron. However, once present, the iron merely serves as a catalyst being rapidly converted back into the ferrous form by the 02"-. As a result, the ability of" HO" to produce inhibition, in our model, is not a function of the available 'free' iron. The simulation result shows that increasing the 'free' iron concentration by at least two orders of magnitude (up to 0.3 raM) still does not result in sufficient HO" production to produce inhibition of the biological process (data not shown).

Model of 02"- Theory of O2 Toxicity

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bimolecular kinetic constant Fig. 4. Effect of changing the bimolecular kinetic rate constant fox the reaction between 02 °- and the molecule 'C' in O2"--induced inhibition. The element 'C' was reacted with 02"- to produce irreversibly inactive species. The initial concentration of ferric ion was 3 uM, while the bimolecular kinetic rate constant for Fenton reaction was 76 M-~s-~.

There is one way to greatly increase the Fenton reaction's production of HO', i.e., to use an iron chelate such as EDTA or ADP. 6 This effectively increases the bimolecular rate constant for the Fenton reaction. As shown in Fig. 5, increasing the bimolecular rate constant for the Fenton reaction to 105 does result in sufficient production of HO" to produce significant inhibition of the modeled process. However, as shown in Fig. 6, even at this elevated level of HO" production, the less reactive O2"- still produces a much greater inhibition. Also, as shown in the left-most column, there is no significant synergistic action of HO" in the O2"--induced inhibition (an increase of only 0.03% in the predicted inhibition). DISCUSSION

By using the kinetic rate constants available in the literature for the reactions of reactive oxygen generation, we could simulate a situation in which a hypothetical chemical pathway leading to a biological process was inhibited when a molecule in the pathway was reacted with 02"- and was inactivated by this radical. The HO" generated in this model, despite its

67

much higher reactivity, was ineffective in inhibiting the biological process. This situation is very contradictory to the general belief that HO" is an important initiator of free radical-induced tissue injury, 4-6 yet it is consistent with a view that HO" is unlikely to reach a critical macromolecular target unless its generation occurs site-specifically at the target, s There are several reasons why reactive HO" can become ineffective in causing the damage. Subsequent generation of HO" from 02"- requires more reactions including Fenton reaction (the fourth reaction in Table 1), which requires an iron catalyst whose in vivo concentration recently has been questioned for biologically relevant HO" formation by some researchers. 34'35From the present model study, we have learned that the key reaction for HO" to become a damaging species is this Fenton reaction. This is consistent with the finding from another model study by Babbs and Steiner ~9in which 109 reactions were simulated, starting with xanthine oxidase reaction with subsequent free radical reactions leading to lipid peroxidation. Two fundamental differences between their model and our present model is that a) they used a higher rate constant for the Fenton reaction; and b) they treated 02"- as a species with low reactivity. The

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bimolecular kinetic constant Fig. 5. Effect of changing the kinetic rate constant for Fenton reaction in HO"-induced inhibition. All the elements (A-J) were reacted with HO" with a bimolecular kinetic rate constant of 109 M-~s-~ to produce irreversibly inactive species. The initial concentration of ferric ion was 3 #M.

68

Y.J. SUZUKI AND G. D. FORD 100"

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HO' Fig. 6. Inhibition of biologicalprocesswhen the bimolecular kinetic rate constant for Fenton reaction is 105 M-~s-j. For O2"--induced inhibition, the element 'C' was reacted with 02"- with a bimolecular kinetic rate constant of 10 7 M - I s -I to produce irreversiblyinactive species. For HO"-induced inhibition all the elements (A-J) were reacted with HO° with a bimolecular kinetic rate constant of 1 0 9 M ~s-~ to produce irreversibly inactive species. Initial concentration of ferric ion was 3 uM, while the bimolecular kinetic rate constant for Fenton reaction was 105 M-~s-~.

reason for using a higher rate constant for the Fenton reaction in their model is because they used Fe 2÷E D T A as a catalyst which speeds up the reaction. A value for E D T A - F e 2+ complex has been shown to be on the order of 105 M - | S - I , 36 and a m o r e likely n u m ber for physiological Fe 2÷ complex of 104 has been suggested. 37 This magnitude of kinetic constant m a y be reasonable when an iron-chelator such as A D P is present. 38 Thus, the behavior o f iron in vivo is critically dependent upon the nature of the chelator, and the kinetic constant value for chelator-dependent Fenton reaction should be considered. The HO" generated using this level of kinetic constant, however, still did not exhibit effectiveness in inhibiting the biological process comparing to the direct effect of 02"in the present model as shown in Fig. 6. The greater reactivity of HO" also presents a problem because H O ' , once formed, can react indiscriminately with m a n y molecules. Competing targets that are not part of the specific reaction sequence that is modeled would even lessen the effectiveness of this

radical to reach a critical target for inhibition. For example, structural proteins, sugars, and glycoproteins in vivo could readily scavenge hydroxyl radicals. Superoxide elimination reactions proposed by von Sonntag provide a nonenzymatic route for detoxifying products of HO" reactions with sugars and peptides to yield, in this case, relatively less reactive superoxide and nonradical products. 39'4° Superoxide does not require any additional reactions or iron catalysts, and since 02"- is generally a p o o r reactant to m a n y molecules, it is available to react with any susceptible molecules. If any functional molecule is reactive to 02"--, there can be specific damage caused by this reactive oxygen species. This m a y be particularly true in the extracellular c o m p a r t m e n t where histochemical studies d e m o n strate that O2"- is more abundant. 4n42 Intracellular c o m p a r t m e n t m a y be more protected from Oz'- damage due to the presence of superoxide dismutases. In addition, 02"- toxicity mediated through metal oxidation proposed by Czapski et al. 43 m a y further enhance the cellular toxicity by this radical. The present model, which is simple enough to be applicable to m a n y biological situation, demonstrates that direct influence by superoxide could result in significant alterations o f biological processes, supporting the view pointed out by Fridovich 8 and McCord and Russell. 7 The superoxide theory of oxygen toxicity needs to be seriously considered. Acknowledgements - - This work was supported in part by a Grantin-aid from the American Heart Association, Virginia Affiliate to GDF, and was done in part during the tenure of a research Fellowship from the American Heart Association, California Affiliate to YJS. We thank Professors Irwin Fridovich (Durham), Joe McCord (Denver), Donald Mikulecky (Richmond), Lester Packer (Berkeley), William Pryor (Baton Rouge), and Guido Zimmer (Frankfurt) for valuable suggestions.

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Model of 02"- Theory of 02 Toxicity

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70

Y . J . SUZUKi AND G. D. FORD APPENDIX 1

Networkthermodynamicmodelingand simulation usingSPICE2 Network thermodynamics is a theoretical approach for investigating the dynamics of interdependent flow-force relationships. These flow-force relationships are the heart of biological processes. Some examples of flowing materials are molecules, ionic charge, and volume of water. Examples of forces driving these flows are chemical potential, electrical potential, and pressure. Any biological processes have pathways through which these materials flow, and they are represented as networks. The direction in which these materials flow always follows the law of thermodynamics. This powerful tool provides a means to solve a complex problem, without requiring great mathematical sophistication, by simply generating a network representing the process of interest. Another system subject to the same conservation laws and, thus, the same analytical approach, are electrical circuits and networks. Fortunately for biologists, engineers have already developed techniques by which such networks can easily be analyzed and even empirically simulated. One such software package is SPICE2 (Simulation Program with Integrated Circuit Emphasis, Version 2), which was used for the present study. Also, packages are commercially available for the use on personal computers, for example, PSPICE. 44 Network thermodynamics is very suitable for modeling macroscopic chemical reactions. Any chemical reaction depends on the concentration of the reactant and the kinetic rate constant, i.e., - d C / d t = kC. In network thermodynamics, the concentration is the voltage and the kinetic rate constant is the constant coefficient of the voltage dependent current source. In this case, the flowing material is the molecules which flow on the electrical network pathway as current, and the driving force, the chemical potential, is translated in the electrical network as voltage. The voltage is defined as the V-element and the voltage dependent current source is defined as the G-element (see Fig. 7D). The voltage is stored in the capacitor (defined as C-element) of a given size, i.e., the volume of the chemical system. For example, if we are to model a simple first order reaction: S --~ P with the rate constant of k, we can draw a network as shown in Fig. 7A. CS and CP are the capacitors (storage sites) for S and P, respectively, in a given size or volume of the compartment. The amounts of S and P and the size of CS and CP will determine the concentrations of S and P in these compartments, and reflect as chemical potentials (i.e., volt-

ages) on the nodes 1 and 2. GK defines which driving forces the flow is depended on, and the rate constant of the flow. We may write a SPICE2 program of the network as: CS 10 10UIC =1 CP 2 0 10U IC=0 GK 1 2 1 00.1 The first line defines the capacitor CS. It is located between node 1 and node 0 (ground). The volume of the capacitor is 10 microliters. Initial concentration of S is 1 molar. The second line defines the capacitor CP, GK

A

B

3-

~

C~ 7

-4 FK2

~CC2

C C

CP

D

G-element~ Voltagedependentcurrent source V-element@ Constantvoltagesource C-element

V

Capacitor

F-element~ Ckn~ntdependentcurrent source Fig. 7. Networkof (A) a simple first order reaction, (B) a second order reaction, and (C) a Michaelis-Mentonreaction.

Model of 02"- Theory of 02 Toxicity

which is located between node 2 and 0 and whose initial concentration is 0. G K defines the nature of flow. It is located between 1 and 2, and flowing from 1 to 2. The flow depends on the chemical potential at node 1 with rate constant of 0.1. This is all that is needed to simulate this reaction. The complete program, which plots out the solution of the concentration of S at time from 0 to 10 × 103 min with a time interval of 1 × 103 min, is: SPICE P R O G R A M FOR A FIRST ORDER REACTION CS 10 10UIC =1 CP 2 0 1 0 U IC=0 GK12100.1 .TRAN 1M 10M UIC .PLOT TRAN V(1) .END Now, consider a second order reaction: A + B --~ C with a bimolecular kinetic rate constant of k2. Here we can use a two-dimensional polynomial for GK2. The two-dimensional polynomial is defined as:

f(2V) = P0 + PIA + P2B + P3A 2 + P4AB + P5B z + P6A3 + P7A2B + P8AB / + • • • Using this equation, we can define the second order reaction v = k2[AI[BI

71

In biological system, the Michaelis-Menton reaction describing enzyme catalysis plays a very important role. The Michaelis-Menton equation is: v = Vm,x[Sl/(Km/+ [S]) where v is the initial velocity, Vmax is the maximal is the Michaelis Constant which is the substrate concentration at 1/2Vmax, and [S] is the substrate concentration. This equation can be rewritten velocity, K m

as

Vmax[S] = vK~ + v[S].

We can incorporate this equation into a network and define a node whose voltage corresponds to v. As shown in Fig. 7C, node 3 is the node that is defined as v and causes the flow from CS to CP through GMM. GN corresponds to the left side of the equation, V~ax[S], by writing the SPICE2 program as: GN 0 3 1 0 Vmax V(1) corresponds to the potential created by [S]. GQD corresponds to the right side of the equation, VKm+V[S], by two-dimensional polynomial of SPICE2 program, GQD 3 0 POLY(2) 3 0 1 0 0

Km 0 0 1.

Here by the definition of two-dimensional polynomial, f=0+Km

v+0S+0v 2+ lvS=vKm+VS.

as: GK2121POLY(2)10300000k2 where 1 0 and 3 0 are the two pairs of controlling nodes, P0 through P3 are all zero, and P4 = k 2. Thus f = P4AB = k2[A][B ] = v. The network for this reaction is shown in Fig. 7B. The SPICE2 program for this network is: SPICE P R O G R A M FOR A SECOND O R D E R REACTION CA 10 volume IC= initial concentration of A CB 3 0 volume IC = initial concentration of B CC 2 0 volume IC=initial concentration of C C C 2 4 0 volume IC=initial concentration of C GK2121POLY(2) 10300000k2 VK22120 F K 2 3 4 VK21 .END.

By KirchholTs current law, defining that sum of the current at a node is equal to 0, GN must be equal to GQD, that is, Vm~[S ] = vKm + v[S]. GMM, which is a voltage dependent current source, flows from S to P and is now dependent on V(3) which is v: GMM12301. The complete SPICE2 program of a Michaelis-Menton reaction is: SPICE P R O G R A M FOR A MICHAELIS-MENTON REACTION CS 10 volume IC = initial concentration of a substrate CP 2 0 volume IC = initial concentration of a product GMM 12301 GN0310Vmax GQD 3 0 POLY(2) 3 0 1 0 0 K m 0 0 1 .END. For further instructions for the use of network thermodynamic modeling, refer to White and Mikulecky. 22

72

Y.J. SUZUKI AND G. D. FORD APPENDIX 2

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