lipid peroxidation in mitochondrial inner membranes. i. an integrative kinetic model

ELSEVIER

Free Radical Biology & Medicine, Vol. 21, No. 7, pp. 917-943, 1996 Copyright © 1996 Elsevier Science Inc. Printed in the USA. All rights reserved

0891-5849/96 $15.00 + .00

PII S0891-5849(96)00185-2

Original Contribution

LIPID PEROXIDATION IN MITOCHONDRIAL INNER MEMBRANES. I. AN INTEGRATIVE KINETIC MODEL

FERNANDO ANTUNES,* ARMINDO SALVADOR,* H. SUSANA MARINHO, *re

RUI ALVES,* and RUY E. PINTO **~

*Grupo de Bioqufmica e Biologia Te6ricas, Instituto de Investiga~5o Cientffica Bento da Rocha Cabral, Lisboa, Portugal; *Departamento de Qufmica e Bioquimica da Faculdade de CiSncias da Universidade de Lisboa, Lisboa, Portugal; and *Centro de

Estudos de Bioqufmica e Fisiologia da Faculdade de CiSncias da Universidade de Lisboa, Lisboa, Portugal

(Received 12 December 1995; Revised 4 April 1996; Accepted 11 April 1996)

Abstract--An integrative mathematical model was developed to obtain an overall picture of lipid hydroperoxide metabolism in the mitochondrial inner membrane and surrounding matrix environment. The model explicitly considers an aqueous and a membrane phase, integrates a wide set of experimental data, and unsupported assumptions were minimized. The following biochemical processes were considered: the classic reactional scheme of lipid peroxidation; antioxidant and pro-oxidant effects of vitamin E; pro-oxidant effects of iron; action of phospholipase A2, glutathione-dependent peroxidases, glutathione reductase and superoxide dismutase; production of superoxide radicals by the mitochondrial respiratory chain; oxidative damage to proteins and DNA. Steady-state fluxes and concentrations as well as half-lives and mean displacements for the main metabolites were calculated. A picture of lipid hydroperoxide physiological metabolism in mitochondria in vivo showing the main pathways is presented. The main results are: (a) perhydroxyl radical is the main initiation agent of lipid peroxidation (with a flux of 10-7Ms ~); (b) vitamin E efficiently inhibits lipid peroxidation keeping the amplification (kinetic chain length) of lipid peroxidation at low values (~ 10); (c) only a very minor fraction of lipid hydroperoxides escapes reduction via glutathione- dependent peroxidases; (d) oxidized glutathione is produced mainly from the reduction of hydrogen peroxide and not from the reduction of lipid hydroperoxides. Copyright © 1996 Elsevier Science Inc.

Keywords--Free radicals, Antioxidants, Multicompartment model, Mathematical model, Simulation, Perhydroxyl radical, Iron, Vitamin E

INTRODUCTION

Understanding the metabolism of lipid hydroperoxides in vivo is a difficult task:

t. The process occurs in an heterogeneous open system and under the influence of a multitude of enzyme activities. Experiments in vitro are far from reproducing such conditions. So, their results cannot be straightforwardly extrapolated on to the biological environment.

2. Many chemical species involved react nonspecifi- cally and, owing to their low concentrations and

Address correspondence to: Fernando Antunes, Grupo de Bio- qufmica e Biologia Te6ricas, Instituto de Investiga~o Cientffica Bento da Rocha Cabral, C 9. Bento da Rocha Cabral, 14, P-1250 Lisboa, Portugal.

.

half-lives, are difficult to detect experimentally in vivo. The number and complexity of the interactions taking place hinders an intuitive comprehension of the process based on the available data.

Mathematical models help the understanding of complex systems, 1 and so, they are an adequate meth- odology to study lipid peroxidation. Indeed, several works apply mathematical models to the study of lipid peroxidation and associated processes (see, e.g., ref. 2 and references therein, and refs. 3 and 4). In a previous work, simple chemical experiments in vitro concerning lipid peroxidation were simulated. 2 A quantitative to semiquantitative agreement with the experimental results was observed. This verification encouraged using

917

918 F. ANTUNES et aL

mathematical models of lipid peroxidation in vivo as a useful tool to improve understanding of this process in vivo and worked as a partial validation of the present model.

The current work presents a moderately detailed model of lipid peroxidation in the matrix side of rat hepatocyte mitochondrial inner membranes and surrounding matrix environment that aims to simulate the physiological state in vivo. A communication concerning some methodological aspects of this work was pre- viously published. 5 The following considerations guided the approach taken in this study: (a) The un- certainty about the relevance of several factors makes the setup of simple models liable to subjective mis- judgements. More comprehensive models minimize this problem and allow to integrate a wider set of information. Recent developments in computer hardware and software 6"7 made this approach practicable and in- expensive. (b) The intracellular space contains a variety of environments, each with characteristic physical- chemical properties. Averaging these properties over the different environments can be misleading, as biochemical processes are usually nonlinear. Therefore, reliable models must refer either to one or to a set of precisely defined compartments. (c) Mitochondria are a logical choice to study the metabolism of lipid hydroperoxides: the mitochondrial respiratory chain is a continuous source of active oxygen species; 8 the mitochondrial inner membrane lipids are also very unsaturated; 9 deleterious oxidative events taking place both at the mitochondrial inner membrane and at the matrix play a key role on radical damage associated to ischemia-reperfusion, 1°-13 were implicated in several degenerative diseases 14 and are the basis 15-18 of several ageing theories.

Among the theoretical works referred above, Babbs and Steiner's 4 is probably the most similar to this work, owing to its integrative aims and general approach (simulation of a multicompartment kinetic model). However, these two works differ in fundamental aspects. (a) In ref. 4 the simulations do not focus on a precisely defined biological compartment. (b) The present work studies the physiological situation, while ref. 4 is centered on ischemia-reperfusion. (c) The present work is centered on the long-term behavior of the system (a steady state), as this is expected to most closely resemble the physiological conditions (discussed in ref. 2). On the contrary, in ref. 4 the simulations do not converge for a stationary attractor, because, for example, lipid peroxidation products accumulate and antioxidants are completely depleted. (d) The present work overcomes technical limitations of ref. 4, as discussed below.

The present model is based as much as possible on

available experimental data, and unsupported assumptions were minimized. Some of these necessary assumptions were validated. 2 In this way, discrepancies between predicted responses and experimentally observed physiological responses may help to identify de- ficiencies in our knowledge about lipid peroxidation. So, we intend the model to be a heuristic tool and not a description. It aims mainly to test the consistency of different hypotheses about the metabolism of lipid hydroperoxides with available kinetic data. Such analysis may also assess the relevance of different mechanisms, help the interpretation of experiments, and suggest new experimental approaches. This was illustrated in a study of the reduction of hydroperoxides by glutathione-dependent peroxidases in vivo. 19

This work describes the estimation of the parameters of the model, characterizes the reference (physiological) steady state and discusses biochemical-relevant aspects of these results. Owing to uncertainties in the parameters, the robustness of the results is evaluated and limitations of the model are discussed.

DESCRIPTION OF THE MODEL

Whenever several experimental measurements for a parameter were available in literature, the determinations that most closely approached the physiological conditions were chosen.

For convenience, the description of the model is ar- ranged into logical modules.

Modeling of the unsaturated lipid species

In mitochondria, unsaturated moieties are found mainly in 1-saturated,2-unsaturated diacylphospholipids (SL-LH), and in cardiolipin which is unsaturated at both positions. 2° The SL-LH species were aggregated as a single species. All the rate constants that depend on the nature of the unsaturated moiety (Table 1) were calculated from the rate constants for each unsaturated moiety and from the relative concentration of each of these moieties. The validity of the aggregation relies on two conditions: (a) The relative concentrations of the different moieties should be constant. Otherwise, the value of the aggregated rate constant must be up- dated at each time to reflect changes in the composition of the aggregated variable. This model treats the concentrations of unsaturated phospholipids as (constant) parameters. (b) All reactions must be either zeroth or first order in the lumped species: if two species (say two peroxyls LOOa and LOO~) are lumped as one (say LOO~ot), then a first-order rate expression only requires knowledge of the concentration of the lumped species. However, that is not true if the reaction is, for example,

Mathematical model of lipid peroxidation in vivo

Table 1. Aggregated Rate Constants Considered in the Model

919

UFA

Hydrogen Abstraction from UFA by Perhydroxyl Radicals

(M-' s -1)

02 Addition to Carbon-Centered

Radicals (s -l) Hydrogen Abstraction from UFA

by Peroxyl Radicals ( M - I s I)

Hydrogen Abstraction from UFA by Alkoxyl Radicals

( M ' s -l)

18:1 No reaction" 1.0 X 109hc 1.1 × 10 2d 3.8 × 106e 18:2 1.2 × 103~ 3 × 1088 1.9 × 10 TM 8.8 × 106~ 18:3 1.7 x 1034 3 × 1088 4.1 × 10 TM 1.3 × l0 TM

20:4 3.0 × 103a 2 × 108b 5.1 × 10 Id 2.05 × 10 TM

22:6 5 × 103f 1 × l0 st 9.6 × 10 la 4 × 107f Aggregated value

for SL-LH g 2.0 × 103 4.4 ) 10 8h 3.4 × 10 ~ 1.5 × 107 Aggregated value

for Lyso-LH i 1.7 × 103 4.6 x 108h 2.9 x 101 1.3 × 107

The values of the aggregated rate constants were calculated based on the individual values of the rate constants for each unsaturated moiety and on the relative concentration of these moieties.

~Ref. 21. b Ref. 22.

Ref. 23. d See text. e Ref. 24. f Extrapolated from the experimental values determined for 18:1; 18:2; 18:3, and 20:4. s The relative proportion of moieties considered was 29, 20, 41, and 10% for 18:1, 18:2, 20:4, and 22:6, respectively (estimated from data

in ref. 9 for the composition of phosphatidylcholine and phosphatidylethanolamine). h It was assumed that the relative proportion of both carbon-centred radicals and unsaturated acyl moieties are similar. ~The relative proportion of moieties considered was 30, 33, 30, and 7% for 18:1, 18:2, 20:4, and 22:6, respectively (estimated from data

in ref. 9 for the composition of cardiolipin, phosphatidylcholine, and pbosphatidylethanolamine).

second order in the lumped species. This can be illustrated by comparing the rate expressions for propaga- t i o n - kp[LH]([LOOa] + [LOO~]) = kp [LH] [LOOtot] - - a n d for terminat ion--kt × ([LOO~] 2 + [LOO~] 2 + [LOO~][LOO~]) =# kdLOOtot] e. In the model, the only reaction that is nonfirst order in a lumped species is the reaction of termination between two peroxyls. The rate of this reaction is several orders of magnitude lower than the first-order termination via vitamin E under physiological conditions. So the error involved in the aggregation is very low.

The peroxidation of cardiolipin (a diphosphatidyl- glycerol) was modeled based on experimental data for dilinoleoylphosphatidylcholine, as better data are lacking. That is, the two phosphatidyl moieties were considered to behave independently. About 80% of the unsaturated acyl moieties of cardiolipin are linoleoyl moieties, and the remaining 20% are mainly monounsaturated fatty acyl moieties. 2° For the sake of simplic- ity, linoleoyl moieties were the only unsaturated moieties considered in the model, and they were assumed to be present as dilinoleoylphosphatidyl moieties. So, discarding the monounsaturated moieties underesti- mates cardiolipin oxidisability. On the other hand, we assumed that all linoleoyl moieties may undergo intramolecular propagation, when only 56 -66% of the peroxyls may undergo this reaction, as estimated from ref. 20. This overestimates oxidizability. The error associated to these assumptions is, however, less important than the discrepancies between experimental estimates

of cardiolipin oxidizability. Cardiolipin is strongly resistant to peroxidation in intact mitochondria. 25 Yet, in experiments with isolated mitochondrial inner membranes, cardiolipin oxidisability is close to that expected from fatty acyl composition. 25 Due to these doubts, we decided to consider the two situations.

Lysophospholipids were also considered in this work. They constitute about 1% of the inner membrane phospholipids 9 and were, like SL-LH, aggregated as one new species. The composition of these phospholipids in unsaturated moieties is unknown as far as we are aware. Therefore, we considered a composition similar to the other phospholipids in mitochondrial inner membranes. The aggregated rate constants are shown in Table 1.

The basic scheme of lipid peroxidation

The set of reactions proposed by Bolland 26 for the basic scheme of lipid peroxidation was adopted.

Initiation. The primary initiation agents considered in the model are hydroxyl, perhydroxyl, and o~-tocopheroxyl radicals. The generation of alkoxyl radicals by iron-mediated decomposition of hydroperoxides and the chain propagation by these radicals is also included.

The hydroxyl radical reacts nonselectively with unsaturated fatty acids, by addition to double bonds or by abstraction of an hydrogen atom. In both cases, a carbon centered radical is f o r m e d . 23"27-28 A rate constant

920 F. ArCrUNES et al.

of 5 X 108 M -~ s -~ was determined for these reactions in lecithin bilayers. 29 All lipid radicals formed were assumed to enter into the lipid peroxidation propagating reactions with the same efficiency as pentadienyl radicals do. This may represent a small overestimation of the initiation ability of hydroxyl radicals. 3° Reactions of the hydroxyl radical with proteins and DNA were also included in the model (see below) providing a "shielding effect."

Oxygen addition. Carbon-centered radicals, resulting from both the initiation and propagation reactions, undergo molecular rearrangements and are, subsequently, able to react in several different ways. The reversible reaction with dioxygen forming a peroxyl radical is probably the most important pathway at physiological concentrations of molecular oxygen. In the model, this is the sole destination of the carbon-centered radicals.

Rate constants for/3-scission of a set of unsaturated fatty acid peroxyl radicals were determined in benzene solutions. 31'32 Because these rate constants are not significantly different, the average value, 1.5 X 102 M-~s -1, has been assumed for all peroxyl radicals.

Propagation. Propagation rate constants are strongly dependent on the nature of the fatty acid loosing the hydrogen atom. Unfortunately, these constants are not known for all unsaturated fatty acids. However, many determinations can be found for linoleic acid. The determination by Barclay et a l . 33 for pure dilinoleoylphosphatidylcholine bilayers, 36.1 M-~s -~, is the near- est to the physiological conditions. From this value, a rate constant of 19.25 M-~s -~ per linoleoyl chain is obtained (see Appendix 1). Propagation rate constants for the other unsaturated fatty acids were calculated (see Table 1) based on the value for linoleoyl and assuming that as the number of double bonds increases from 1 to 6, the relative value of the rate constant increases as 0.00057, 1, 2.11, 2.65, 3.81, and 4.97. The latter series was estimated from data for hydrogen abstraction from different acyl esters by tocopheroxyl radicals in an homogeneous benzene solution, an

Termination. Only the reaction between two peroxyl radicals was considered. Under physiological dioxygen pressures, this reaction strongly predominates over termination both between carbon centered radicals or between a carbon centered radical and a peroxyl radical (simulation results not shown). In dilinoleoylphosphatidylcholine bilayers, a rate constant for termination has been determined as 2k, = 1.32 X 105 M - 1 s - 1 . 33 At these experimental conditions (760 Torr dioxygen), this value is representative of the reaction between two peroxyls. So, the rate constant for the latter reaction was

estimated from this value as 8.8 x 10 4 M-is -1 (see Appendix 1).

Hydroperoxide decomposition. Perhydroxyl radicals are able to remove an hydrogen atom from an hydroperoxide group (reaction 27, Table 2). The value for this rate constant, 2 x l0 s M-is -1, was estimated considering that: (a) abstraction of an hydrogen atom from an unsaturated moiety does not compete significantly with reaction 2735 in experiments where the concentration of hydroperoxides was about 5% of the concentration of unsaturated fatty acids. So, a lower bound for the rate constant for reaction 27 is 10 x 2 x 103/0.05 = 4 X 105 M-is -~. (b) Thomas et al.36 were not able to detect reaction 27 in competition with perhydroxyl radical dismutation (kaism = 1.3 X 106 M-Is-l) . Thus, an upper bound for the rate constant is 0.1 X 1.3 X 10 6 =

1.3 X 105 M-is -1. The discrepancy between these values can be ex-

plained, at least in part, by the different experimental conditions used by the two groups.

Fe 2+ can also decompose lipid hydroperoxides, with formation of alkoxyl radicals. The median of three similar determinations in homogeneous solut ion--2.4 X 102, 37 3.2 X 10z, 38 and 1.5 X 103 M-is -1 (referred in ref. 39) was adopted for this rate constant.

Alkoxyl radicals are able to remove any hydrogen atom from a polyunsaturated fatty acyl chain with the following preferences bis-allylic > allylic > secondary. 24 In the model, all the radicals formed in this reaction were assumed to be pentadienyls (or, in the case of oleic acid, allyls).

Correction factors accounting for sterical effects originated from the presence of two adjacent acyls in diacylphospholipids were considered (Appendix 1). So, the rate constants appearing in Table 2, both for reactions of termination and propagation involving diacylphospholipids, already include the corrective factor.

Reactions involving iron

Modelling the pro-oxidant effects of iron in biological systems is a difficult task, owing to the complex chemistry of this transition metal and to experimental uncertainties about the nature and concentrations of its biological pro-oxidant forms. For the sake of simplic- ity, a single "average" iron species was assumed. The amount of "chelatable" iron pool observed in whole liver is 22.2 nmol/g of wet tissue, a° This amount of iron should represent an overestimation of t h e " active" iron towards oxidative processes. Owing to the uncertainties about the level of this species we studied the effect of changing the iron concentration by several orders of

M a t h e m a t i c a l m o d e l o f l i p i d p e r o x i d a t i o n i n v i v o 9 2 1

T a b l e 2 . A b b r e v i a t e d L i s t o f t h e R e a c t i o n s I n c l u d e d i n t h e M o d e l

N r R e a c t i o n P a r a m e t e r N r R e a c t i o n P a r a m e t e r

1 P - L H + H O " - - P-L" kt = 5 .0 X 108 M -~ s -~ 41 P - L O O " + T o c O H - - P - L O O H + k41 ~ 5 .8 X 10 3 M -I s i

T o c O " 2 " L H " + H O 2 ;n - - " L " ' + H 2 0 2 k2 = 1.2 X 103 M -~ s ~ 4 2 P -LO" + T o c O H ~ P - L O H + T o c O " 1(42 = 1 X 108 M -I s -I 3 S L - L H + HO2 m - - SL-L" + H 2 0 2 k3 = 2 .0 X 103 M -~ s -1 43 T o c O " + U b q H 2 - - T o c O H + U b q H " k43 = 2 .2 X 105 M ~ s -~

4 L y s o L H + HO2 ,~, ~ L y s o L " + H 2 0 2 k4 = 1.7 × 103 M -L s L 4 4 P - L H + T o c O " - - P-L" + T o c O H k44 = 1 × 10 -1 M - t s -1 5 " L " ' + O2m ~ " L O O " ' ks = 3 .0 × 108 M -~ s -1 45 P - L O O H + T o c O " - - P - L O O " + 1 ~ = 1 M - t s -~

T o c O H 6 SL-L" + O2~, ~ S L - L O O " k6 = 4 .4 x 108 M t s -I 4 6 T o c O " + O2" - + H + - - T o c O H + k,6 = 4 .5 x l 0 s M -~ s

0 2 aq 7 L y s o L " ~- O2m ~ L y s o L O O " k7 = 4 .6 × 108 M t s - i 47 T o c O H + H O z m ~ H 2 0 2 + T o c O " k47 = 2 X 105 M -~ s - j

8 P - L O O " - - P-L" + O2m k4 = 1.5 × 102 s L 4 8 T o c O H + 0 2 " - + H ÷ - H 2 0 2 + k48 = 6 M -~ s t

T o c O " 9 " L H " + " ' L O O * " ~ " L ' " + k9 = 1.4 × 10 ~ M -1 s -~ 4 9 P L A 2 k49 = 2.3 X 10 -5 s t

" L O O H " X - Y ~ L y s o X + Y f r e e

10 " L H " + S L - L O O " ~ " L " ' + kl0 = 1.4 X 10 ~ M -~ s t 5 0 P L A 2 kso = 2 .3 × 10 5 s-~ S L - L O O H S L - L O O H • L O O H f i ' e e

11 " L H " + L y s o L O O " ~ " L " ' + k u = 1.9 × 10 ~ M -1 s -~ 51 L O O H f r e e + GPXrd + H + ~ G P X o ks~ = 2.1 × 107 M -~ s - t L y s o L O O H + L O H f r e e

12 S L - L H + " L O O " ' ~ S L - L " + kt2 = 2 .6 X 10 t M - t s - ' 52 G P X o + G S H ~ G S G P X + H 2 0 k52 = 4 x 104 M -~ s " L O O H "

13 S L - L H + S L - L O O " - - S L - L " + k~3 = 2 .6 X 10 ~ M -1 s i 53 G S G P X + G S H - - GPX~d + G S S G k53 = 1 x 107 M ' s -1

S L - L O O H + H +

14 S L - L H + L y s o L O O " ~ SL -L " + kt4 = 3 .4 × 101 M - ' S -1 5 4 H202 + GPXrd + H + - - G P X o + k~4 = 2.1 × 107 M -~ s -~ L y s o L O O H H z O

15 L y s o L H + " L O O " ' - - L y s o L " + k~5 = 2 .2 X 10 t M -~ s - t 55 L O O H f r e e + P H G P X ~ + H + ~ k55 = 3 × 107 M - ' S - 1

" L O O H " P H G P X o + L O H f r e e 16 L y s o L H + S L - L O O " ~ L y s o L " + k16 = 2 .2 × 101 M -~ s -~ 5 6 L O O H - L O O H + P H G P X ~ + H + - - k~6 = 1.2 × 107 M -~ S -1

S L - L O O H P H G P X o + L O O H - L O H 17 L y s o L H + L y s o L O O " - - L y s o L " + k i t = 2 .9 X 10 ~ M ~ s - t 57 L O O H - L O O H + P H G P X , ~ + H ÷ ~ k57 = 1.2 x 107 M -~ s -~

L y s o L O O H P H G P X o + L O H - L O O H 18 L O O ' - L H - - L O O H - L " k~8 = 1.0 × 10' s -1 58 P - L O O H + PHGPX~a + H ÷ ~ k58 = 1.0 X 107 M - t s 1

P H G P X o + P - L O H

19 L H - L O O " - - L ' - L O O H kt9 = 1.0 X 101 s - ' 59 P H G P X o + G S H - - G S P H G P X + k59 = 1 x 10 ~ M ~ s -~

H 2 0 2 0 " L O O " ' + " L O O " ' - - p r o d u c t s k2o = 6 .6 X 104 M - t s - ' 6 0 G S P H G P X + G S H ~ P H G P X ~ + 1%o = 1 × 10 r M ~ s -~

G S S G + H ÷ 21 " L O O " " + S L - L O O " ~ p r o d u c t s k2~ = 6 .6 x 104 M 1 s-~ 61 H202 + PHGPX~d + H ÷ - - P H G P X o k6, = 3 .2 X 106 M -~ s 1

+ H z O 2 2 " L O O " ' + L y s o L O O " ~ p r o d u c t s k22 = 6 .6 × 104 M -~ s -~ 62 G S S G + N A D P H + H ÷ ~ 2 G S H V = 3 .2 × 10 -~ M s - t

+ N A D P + K m = 5 X 10 - ~ M 23 S L - L O O ° + S L - L O O " - - p r o d u c t s k23 = 6 .6 X 104 M -~ s -1 63 - - G S H k63 = 7 . 0 4 x 10 -8 M s -1 2 4 S L - L O O " + L y s o L O O " - - p r o d u c t s k~4 = 6 .6 x 104 M -~ s -~ 6 4 G S H - - k6~ = 6 .4 × 10 -6 s ~1

25 L y s o L O O " + L y s o L O O " - - p r o d u c t s k~s = 8.8 × 10 ~ M -~ s -~ 65 L i p i d - - k ~ = 2 .3 × 10 ~ s - t

2 6 L O O ' - L O O " - - p r o d u c t s k:~ 6 = 4 .4 × 10 ~ s -~ 6 6 O2m + e - ~ O2" k6~ = 8 x 10 -~ s -~

27 P - L O O H + HOz,~ - - P - L O O " + k.t7 = 2 × 10 ~ M %-~ 67 S O D - M n 3÷ + O2"- ~ S O D - M n 2+ + k67 = 4 .7 X 109 M - t s -~

H202 Oz 2 8 P - L O O H + F e 2+ ~ P - L O " + Fe 3÷ + I<28 = 3 .2 x 102 M ~ s -~ 68 S O D - M n 2+ + 0 2 " + 2 H ÷ - - k68 = 4 .7 × 109 M -I s - l

H O - S O D - M n 3+ + H 2 0 2 2 9 " L H " + " L O " ' - - " L " ' + k29 = 6 .6 x 106 M -1 s i 69 H O ~ q + HOEa q - - H202 + 0 2 aq k69 = 7 .6 × 105 M -1 s t

" L O H "

3 0 " L H " + S L - L O " - - " L " ' + k3o = 6 .6 X 106 M -~ s ~ 7 0 HO2 t~ + HO2 ~ ~ H 2 0 2 + O2m k70 = 5 X l08 M -I s 1 S L - L O H

31 " L H " + L y s o L O " ~ " L " ' + k31 = 8.8 X 106 M 1 s--t 71 H O ~ q Jr O2" - -L H ÷ _ H20~ + O2 ,q kTi =: 8.5 × 107 M t s - t L y s o L O H

3 2 S L - L H + " L O " ' - - SL -L " + k32 = 1.1 × 107 M -~ s -1 72 HO2~q = 0 2 " - + H + K72 = 2 X 10 ~ M " L O H "

33 S L - L H + S L - L O " ~ S L - L " + ka3 = 1.1 X 107 M -1 s I 73 HO2~q = HO2 ~ K~3 = 1 S L - L O H

3 4 S L - L H + L y s o L O " ~ SL -L " + k3, = 1.5 × 107 M -1 s -1 74 O2" - + p r o t e i n ~ - - p r o d u c t s k7.~ = 0 .31 M -~ s -~ L y s o L O H

35 L y s o L H + " L O " ' ~ L y s o L " + k35 = 1.0 x 107 M -~ s t 75 H O ~ q + pro te in ,q - - p r o d u c t s kTs = 12 M 1 s " L O H "

36 L y s o L H + S L - L O " ~ L y s o L " + k36 = 1.0 × 107 M -~ s -~ 76 HO2m + prote inm ~ p r o d u c t s k76 = 12 M - t s S L - L O H

37 L y s o L H + L y s o L O " ~ L y s o L " + k37 = 1.3 x 107 M - I S -1 77 HO" + prote in ,q ~ p r o d u c t s k77 = 4 .6 X 108 M ' s - ' L y s o L O H

3 8 L O ' - L H - - L O H - L " k38 = 4 .4 X 106 s -I 78 0 2 * - + D N A - - p r o d u c t s k78 = 0 .31 M -1 s

3 9 L H - L O " - - L ' - L O H k39 = 4 .4 × 106 s -1 7 9 HOzaq + D N A - - p r o d u c t s k79 = 12 M ~ s -~ 4 0 F e 2÷ + H202 ~ Fe 3+ + H O - + H O " k4o = 2 X 104 M - t s -~ 80 H O " + D N A - - p r o d u c t s ks0 = 5.1 × 108 M - t s -1

P , a n y p h o s p h o l i p i d ( c a r d i o l i p i n , 1 - s a t u r a t e d , 2 - u n s a t u r a t e d - d i a c y l p h o s p h o l i p i d o r l y s o p h o s p h o l i p i d ) ; " X ' ' , a m o i e t y s t e r i f i e d t o c a r d i o l i p i n ;

X - Y , t w o m o i e t i e s s t e r i f i e d t o c a r d i o l i p i n ; s u b s c r i p t s a q a n d m r e f e r t o s p e c i e s i n a q u e o u s a n d m e m b r a n e p h a s e , r e s p e c t i v e l y .

922 F. ANTUNES et al.

magnitude on several relevant fluxes. The reference concentration for iron in the model w a s 10 -7 M.

In the studies by Yegorov et al., 4° all iron was found in the reduced form, suggesting that the reductive capacity of liver is high enough for perma- nent reduction of weakly bound iron. So, in the model the redox cycle of iron was not considered. Instead, Fe 2+ was treated as a parameter with a constant concentration of 10 -7 M.

Reactions 28 and 40 in Table 2 account for lipid peroxidation-related pro-oxidant effects of iron. The first reaction decomposes lipid hydroperoxides to alkoxyl radicals; in the second reaction hydroxyl radicals (or other high-reactive species) that initiate lipid peroxidation are produced. The rate constant for reaction 40 was taken from ref. 41. These authors used EDTA as an iron chelating agent. At least as far as hydroxyl radical production is concerned, the reactivities of Fe( I I I ) -EDTA, Fe( I I I ) -ADP, and free Fe(III) (with iron being reduced by superoxide radical) at #M concentrations seem alike. 42

The reactions of i ron-oxygen complexes were ne- glected, because there is lack of consistent quantitative kinetic data about the formation and reactions of these species.

Reactions involving vitamin E or vitamin E radicals

Vitamin E was considered to be exclusively com- posed of a-tocopherol because it is both the predomi- nant component within cells and the most potent antioxidant. 43

Chain-breaking reactions, a-Tocopherol may act as a chain-breaking antioxidant in several ways: (a) reduction of peroxyl radicals (reaction 41): rate constants for this reaction in homogeneous solution are in the range 105 to 106 M - I S - I . 44 However, in dilinoleoylphosphatidylcholine bilayers the value 5.8 × 103 M-~s -l was obtained. 45 The latter value was used in the model. (b) Reduction of alkoxyl radicals (reaction 42): rate constants for the reduction of tert-butoxyl radicals by a- tocopherol of 3.8 × 109 and 6.6 × 108 M -~ s -~ were determined in 1:1 di-tert-butyl peroxide:benzene and " w e t " acetonitrile (10% water), respectively. 46 We assumed that the rate constant for this reaction is lower in lipid bilayers than in solution, as it is observed in the reduction of peroxyl radicals. Therefore, a rate constant of 108 M-~s -~ was considered. (c) Reactions with carbon-centered radicals: Vitamin E reacts with alkyl radicals but not with allyl or pentadienyl radicals. 47 In the model, the formation of alkyl radicals has not been considered. Hence, these reactions are not taken into account. (d) Addition of the a-tocopheroxyl radical to

a peroxyl radical. This reaction was not included in the model, as it represents a sink for a-tocopherol. Because a-tocopheroxyl radicals are a very small fraction of a- tocopherol (see Results), the chain-breaking effect of this reaction is not important.

Tocopheroxyl reduction. In vivo, a-tocopherol is re- generated from a-tocopheroxyl radicals. The involvement of a variety of agents in this process has been proposed. Among these, the importance of ascorbate is well established. 48-53 However, we found no data in the literature concerning the presence of ascorbate in the mitochondrial matrix. So, this metabolite was not considered in the present model. There is also increasing evidence 54-56 that ubiquinols, which are abundant in the mitochondrial inner membrane, 57 may be important in the regeneration of vitamin E. Therefore, reaction 43 of Table 2 was considered. Its rate constant was determined in ethanol for ubiquinol-10. 55

Pro-oxidant effects of vitamin E have been reported in a variety of systems (see, e.g., refs. 34, 37, 58-60). Reactions 44 and 45 were pointed as cause of that behavior.

Rate constants for reaction 44 were determined in benzene solution as 7.5 × 10 -2 M-is -l, for methyl linolenate and 8.2 × l 0 -2 M ~s -~, for methyl linolenate. 6j In the model, a rate constant of 10 -~ M-~s -~ was used.

Concerning the reduction of tocopheroxyl radical by hydroperoxides, the only kinetic data found in literature was for the reaction between alkyl hydroperoxide and 5,7-diisopropyl-tocopheroxyl (an analogue of tocopheroxyl) in benzene solution. 62 The rate constants obtained were 0.134, 0.242, and 0.365 M-~s -~, for n-butyl, sec-butyl, and tert-butyl hydroperoxides, respectively. However, tocopheroxyl radicals are more unstable than the analogue used, and sterical factors also make the latter less react ive) 8 Rate constants for reaction 44 obtained with this analogue 34 are about five times lower than those obtained with tocopheroxyl radicals. 6~ Also, rate constants for 5,7-diisopropyl-tocopheroxyl reduction by several ubiquinols are about one order of magnitude lower than those for tocopheroxyl reduction. 55 Therefore, a rate constant of 1 M-~s -~ was assumed for reaction 45.

The rate constant for the reaction of superoxide radicals with tocopheroxyl radicals (reaction 46) was determined, by using Trolox C in aqueous solution (pH 7), as 4.5 X108 M-Is 1.63

a- Tocophe rol oxidation by perhydroxyl and superoxide radicals. Arudi et al. 64 measured rate constants for reactions 47 and 48 using a-tocopherol in ethanol solution: 2 × 105 and 6 M-~s l, respectively. The charge of the lipid bilayer can strongly influence the rate con-

Mathematical model of lipid peroxidation in vivo 923

stants of reactions of membrane components (such as vitamin E) with superoxide radical (or other ions). Much higher rates for reaction 48 in positively charged membranes as compared with those obtained in nega- tively charged membranes were obtained. 68 In the model, this effect was not taken into account.

Detoxification of oxidized lipids

Reactions involving phospholipase A2 (PLA2). PLA2 (E.C.3.1.1.4) activity is usually assumed to be important in the detoxification of phospholipid hydroperoxides, because "classic" glutathione peroxidase (E.C.I . l l . I .9) (GPX) is unable to reduce diacylphospholipid hydroperoxides. 66-7° However, a quantitative analysis of PLA2 action shows that in rat liver mitochondria its activity is too low to be of importance in the detoxification of phospholipid hydroperoxides. 19 Nevertheless, PLA2 action is included in the present model with kinetics as described in ref. 19. PLA2 is considered to catalyze the hydrolysis of 1-saturated- acyl,2-hydroperoxide phospholipid, producing free fatty acid hydroperoxides. It also acts on all species characterizing the peroxidation of cardiolipin, yielding a free fatty acid and a lysophospholipid (reactions 49 and 50 in Table 2).

Reactions involving glutathione. Several enzymes with a glutathione (GSH) dependent peroxidase activity have been described. In the model, only two of these enzymes, both selenium dependent, were considered: (1) GPX--d iscovered by Mills, 7t and (2) phospholipid hydroperoxide glutathione peroxidase (E.C. 1.11.1.12) (PHGPX) 72

Soluble GSH-S-transferases (E.C.2.5.1.18) were not considered because they show much lower fatty acid hydroperoxide peroxidase activities than GPX. 73

A membrane bound GSH-S-transferase (E.C. 2 . 5 . 1 . 1 8 ) 74-76 with non-Se-dependent phospholipid hydroperoxide peroxidase activity has been described. 77 Both suitable kinetic data for modeling this activity and convincing evidence of its importance against accu- mulation of lipid hydroperoxides in mitochondrial inner membranes are missing. So, the action of this enzyme was not considered in the model.

Glutathione peroxidases (GPX and PHGPX). The kinetics of GPX and PHGPX used in this model are those described in ref. 19.

GSSG reductase (E.C.1.6.4.2). In the model, NADPH concentration is assumed as constant and sat- urating for GSSG reductase. This is not realistic upon an intense and prolonged oxidative stress. However, in

near-to-physiological situations, it is probably a reasonable approximation: the Henri-Michaelis-Menten constant of the enzyme for N A D P H - - 3 x 10 -6 M 7 8 -

is three orders of magnitude lower than the concentration of this metabolite in rat liver mitochondria--5 X 10 -3 M. 79 Under these circumstances, the enzyme shows hyperbolic steady-state kinetics towards GSSG, with Km (GSSG) = 5 X 10 -5 M. 78 The maximum ve- locity for this kinetics can be estimated from an activity of 19.3 nmol/min/mg mitochondrial protein. 79 Assum- ing a concentration of 1 mg mitochondrial protein//A osmotically active matrix (see Appendix 2), that activity converts to 3.2 X 10 -4 M s - l . The quasi-steady-state kinetics was used, as the available data do not allow a more detailed treatment.

Glutathione influx and effiux. For the sake of sim- plicity, a constant influx (7.04 X 10 -8 Ms -t) and a first- order efflux of GSH (with a rate constant of 6.4 x 10 -6

s -1) were assumed. These parameters are such that the steady-state concentration of GSH matches the experimental value of 1.1 x 10 -2 M obtained for rat liver mitochondria 85 and that the half-life of the metabolite matches the experimental value of 2 3 0 h. 86

GSSG is retained by mitochondria 87-89 and we did not find any reports of a GSSG influx into these organelles. So, exchange of GSSG between mitochondria and cytosol was not included in the model.

Lipid metabolism. The fate of the products of lipid peroxidation is still unclear. We assumed that a process with the same time scale determined for PLA2 (8.4 h) acts as a sink for all lipid species (including free fatty acids) considered in the model (reaction 65 in Table 2).

Production and interconversion of active oxygen species

Mitochondria are an important source of cellular hydrogen peroxide. 8'9° In these organelles, hydrogen peroxide is produced mainly by dismutation of superoxide radicals. 91'92 In the model, the monovalent reduction of dioxygen is first order towards dioxygen. Provided the redox state of the respiratory chain is kept constant, this seems a good approximation in a large range of dioxygen concentrations. 93 An apparent rate constant of 8 x 10 -t s -I was calculated for this reaction to obtain a 1.6 x 10 -s Ms i rate of superoxide radical production at [02]rnem b . . . . = 1 X 10 -4 M. (Compartmentation was taken into account.) The rate of superoxide production by the respiratory chain is estimated as twice the rate of mitochondrial hydrogen peroxide production in con- trolled state 4 - - 0 . 5 0 nmol/mirdmg protein. 9° The au-


thors considered this value representative of the physiological condition. This value may be, at one hand, overestimated since state 4 is the state of greatest production of hydrogen peroxide. 8 On the other hand, it may be underestimated, because mitochondrial GPX was not inhibited during these measurements of hydrogen peroxide release from "intact" mitochondria.

In the model, hydrogen peroxide is produced by superoxide dismutation (reactions 68-71), among other (nonzeroth order) reactions.

The mechanism proposed by Fielden et al. 94 for the catalytic action of superoxide dismutase (E.C. 1.15.1.1) (SOD), with the rate constants obtained for the chicken liver mitochondrial (Mn-dependent) enzyme at pH 7 .8 , 95 w a s used in the model. Noncatalyzed dismutation of superoxide radicals was also considered (reactions 69, 70, and 71 in Table 2) with rate constants in aqueous phase obtained from ref. 96 and in membrane phase estimated from ref. 97.

Perhydroxyl and superoxide radicals were considered to be in fast equilibrium (K = 2 × 10 -5 M ref. 98), because no values for the rate constants of the reactions HO2---O~ + H + and 0 ~ - + H+~HO~ were found in the literature.

Reactions of active oxygen species with proteins and DNA (reactions 74-80). Excepting water, proteins are the main constituent of mitochondria. The rate constants of reactions between proteins and radicals were estimated from the rate constants for individual amino acids, weighted by their frequency of occurrence in proteins. Bielski and Shiue 99 refer, based on studies with the hydroxyl radical, that the reactivity of perhydroxyl and superoxide radicals towards aminoacyl residues in proteins should not be drastically different from that towards free amino acids. One of the main factors that can change the rate constant is the accessibility of the radicals to the aminoacyl residues in proteins. So, we corrected the individual rate constant for free amino acids by the mean solvent-accessible fraction of area of each residue in proteins (see Table 3). The aggregated rate constant (per aminoacyl residue) was calculated as:

Z kRadical+amino acid/ i

× Fractional accessibilityi

× Occurrence in proteinsi

with the sum over the 20 aminoacyl residues. The fractional accessibility of a residue was calculated as its mean accessibility in proteins divided by the standard- state accessibility. The standard accessibility is defined as the average surface area of a residue X in Gly-X-

Gly tripeptides. The mean accessibility of the amino acid residues in proteins was calculated for 4410 residues taken from 23 x-ray elucidated proteins.1°l

Rate constants for reactions between hydroxyl radicals and DNA have been determined in several works. ~°2 An average value of 5.1 × 108 M-~s -1 (per nucleotide residue) was used in the model. Unfortu- nately, rate constants for the reactions of superoxide and perhydroxyl radicals with DNA, individual nucle- otides, nucleosides, or nitrogen bases were not found in literature.l°2 The upper limit of 106 M-~s - ~ (ref. 11 l) is rather high, taking into account the reactivity of these radicals. As such, and somewhat arbitrarily, the reactivity of perhydroxyl and superoxide radicals towards DNA was assumed to be the same as that observed with proteins. A weak support for this supposition is that the reactivity of hydroxyl radicals towards both proteins and DNA is similar (----5 × 108 M-Is ~).

Characterization and values of the variables of the model

The variables of the model are time and both metabolite and enzyme concentrations. Some of these chemical species, however, were considered as external metabolites whose concentrations were treated as parameters. These concentrations must be given as input, along with the kinetic parameters, at the time of run- ning the simulations. They were estimated as follows (all concentrations are referred to the compartment where the species is present):

1. [H+]: The pH of the mitochondrial matrix is about 7.5.1~2 However, if a significant surface potential ex- ists, the pH at the membrane surface is different from that in the bulk medium. '3 For biomembranes the pH is generally assumed to be between 2 and 3 units lower than in the bulk aqueous phase. 114 There- fore, reactions in the matrix and in the interface between membrane and aqueous phase were considered to proceed, respectively, at pH 7.5 and 5.

2. [02]: The average concentration of dioxygen in liver is estimated at 3.5 × 10 -5 M. ~15 For the lipid phase, a threefold higher concentration was assumed, as a partition coefficient of 3 for dioxygen between di- miristoylphosphatidylcholine bilayers (above the main transition temperature) and water was measured, l ~6

3. [UbqH2]: The forms of coenzyme Q with 9 and 10 isoprene units are the most common in mammals. Only the reduced forms were considered. These were assumed to behave similarly when regenerating c~-tocopherol. In rat liver mitochondria, ubiquinol-9 and ubiquinol-10 are approximately six-


Table 3. Estimation of the Rate Constants of the Reactions of HO'; HO~, and O~ with Aminoacyl Residues in Proteins

925

Occurrence in Fractional Aminoacyl Residue Proteins (%) ~ Accessibility b k~. + o~- (M -j s ~) ~ k~ + noi ( M-I s ~) a k.. + HO" (M-I s - l ) e

Ala 9.0 0.26 -<0.06 _<44 6.5 X 1 0 7 ref. 103 Arg 4.7 0.36 -<0.13 -<63 3.5 × 109 ref. 104 Asn 4.3 0.37 -<0.16 -<54 4.9 × 107 ref. 105 Asp 5.5 0.38 -<0.18 -<12 7.5 × 1 0 7 ref. 104 Cys 2.8 0.09 -<15 -<600 4.1 × 10 ~° ref. 106 Gin 3.9 0.38 -<0.25 -<23 5.4 × 108 ref. 105 Glu 6.1 0.38 -<0.39 -<30 2.3 × 10 ~ ref. 105 Gly 7.8 0.28 -<0.42 -<49 1.7 × 1 0 7 ref. 104 His 2.0 0.22 <-1 -<95 5.7 × 1 0 9 ref. 103 Ile 4.6 0.12 -<2 -<39 1.8 × 10 '~ ref. 105 Leu 7.4 0.15 -<0.21 -<23 1.7 × 10 ') ref. 104 Lys 7.0 0.48 -<3.3 -<13 3.5 × 108 ref. 105 Met 1.7 0.15 -<0.33 -<49 8.5 × 10 '~ ref. 107 Phe 3.5 0.12 -<0.36 -<180 6.5 × 1 0 9 ref. 108 Pro 4.6 0.36 -<0.16 -<17 4.8 × 10 ~ Ser 7.1 0.34 -<0.53 -<55 3.2 × 10 s ref. 104 Thr 6.0 0.30 -<0.21 -<13 5.1 × 108 ref. 105 Trp 1.1 0.15 -<0.24 1.3 × 10 j° ref. 108 Tyr 3.5 0.24 -<10 1.3 × 10 I° ref. 110 Val 6.8 0.14 -<0.18 -<11 7.6 × 108 ~ Aggregated rate

constants 0.31 12 4.6 x 108

Frequency of occurrence of each aminoacyl residue in the primary structures of 207 unrelated proteins of known b Ref. 101. c pH 10 (ref. 99). d pH 1-2 (ref. 99). pH 6.0-7.5, taken from ref. 102.

f Average between the rate constants taken from ref. 109 and ref. 105.

sequenceJ ~

fold and twofold more abundant than a-tocopherol , respectively. 57 Therefore a concentrat ion of 8 ×

[TocOH] = 1.6 × 10 -3 M (see above) was consid-

ered for UbqH2 in the model. 4. [LH-LH]: Cardiolipin constitutes 17% of P-lipid in

mitochondria, 9 and linoleic moieties constitute 77% of total acyl 2° moieties. Assuming a concentrat ion

of P-lipid of 0.6 M (see Appendix 3) we estimated [LH-LH] at 0.6 × 0.17 × 0.77 = 0.08 M.

5. [SL-LH]: Phosphatidylcholine and phosphatidyleth-

anolamine are the main 1-saturated,2-unsaturated

diacylphospholipids in mitochondrial inner mem- b r a n e s - a b o u t 80% of total P-lipids. 9 Therefore, a

concentrat ion of 0.6 × 0.8 = 0.48 M may be estimated.

6. [LysoLH]: Lysophospholipids constitute about 1% of P-lipid in mitochondrial inner membranes. 9 Thus, its concentrat ion may be estimated at 0.6 × 0.01 = 0.006 M.

7. [Fe z+] = 10 -7 M (see text).

8. [protein]aq, [protein]m: In the model, these variables stand for the total concentrat ion of aminoacyl residues, respectively, in the aqueous phase and buried

in the membrane. The concentrat ion of (total) protein in mitochondria is about 1 mg/#l of osmotically active matrix (see Appendix 2). The mitochondrial

.

matrix contains ~ 6 7 % of total mitochondrial protein. 83 By weighting the molar mass of aminoacyl

residues by their frequency of occurrence in proteins

an average molar mass of 110 is obtained. Consid- ering this value, a concentrat ion of 6.1 M was obtained for aminoacyl residues in the matrix. In the

membrane plane, proteins account for 1/3 to 1/2 of the volume 1~8-~2° (assuming this is proportional to

the area). Considering also a specific volume for proteins of 0.74 cm3/g and the same average molar

mass for aminoacyl residues as above, a concentration of 4.1 to 6.1 M was calculated for aminoacyl

residues buried in the membrane. A value of 5.0 M was used in the model.

[DNA]: In mammals, the size of each molecule of mitochondrial DNA is about 16500 base pairs. ~2~

There are between 4.3 × 109 mitochondfia/mg protein 122 and 7.2 × 109 mitochondria/mg protein, 123'124

with about 5 - 1 0 DNA molecules TM per organelle.

Considering also 1 #1 osmotically active matrix/rag

total mitochondrial protein (see Appendix 2), the concentration of nucleotide residues was estimated as 1.2

to 3.9 raM. In the model, a value of 2.6 mM was used.

The concen t ra t ions of all other species were

treated as dependent variables. The model deter-


mines their evolution once parameters, independent variables and a suitable set of initial conditions are given. This evolution is, however, constrained by several conservation relationships, each corresponding to a further independent variable whose value must be given ab initio:

1. [TocOH] + [TocO']: The level of vitamin E in mitochondria is in the range 0.08-0.22 nmol/mg of protein. 57'125 Only about one-fifth of the vitamin E of mitochondria is in the mitochondrial inner membrane. 125 In mitoplasts, the level of respectively vitamin E and phospholipid is 0.074 nmol/mg of protein 125 and 0.20 #mol/mg of protein. 9 So, the molar ratio between phospholipids and vitamin E is estimated at 2700, which corresponds to a vitamin E concentration of 2 × 10 -4 M if a 0.6 M concentration is assumed for phospholipids (Appendix 3).

2. [SOD-Mn 3+] + [SOD-Mn2+]: Tyler 126 estimated a concentration of 1.1 x 10 _5 M in rat liver mitochondrial matrix.

3. [GPX~d] + [GPXo] + [GSGPX]: The concentration of an enzyme in a cellular fraction can be estimated from the following data: (a) total protein concentration in the fraction; (b) ratio of specific activities between purified enzyme and the enzyme in the cellular fraction; (c) molar mass of the enzyme. Owing to incomplete purification and to loss of activity during purification, the specific activity of the purified enzyme is usually underestimated. The specific activity of purified GPX can be estimated more rig- orously by using both the data from ref. 127 and the estimated concentration of Se as GPX in rat liver cytosol - -0 .46 ppm. ~28 (----5.8 X 10 6 M). Assuming a cytosolic protein concentration of 0.22 mg/#l, we obtained the value 740 units/mg purified enzyme, which is 2.7 times as much as the original value. From this value, considering the specific activities for the other fractions, and assuming a protein concentration of 1 mg mitochondrial protein//A osmotically active matrix, the GPX (tetramer) concentration in mitochondria was estimated at 2.5 X 10 -6 M.

4. [PHGPXrd] + [PHGPX0] + [GSPHGPX]: A PHGPX total concentration of 3.8 × 10 -8 M was calculated from the GPX concentration estimated above, as described in ref. 19.

Compartmentation

The model considers an aqueous and a membrane compartment each with a characteristic composition and reactivity. Owing to lack of precise data about the partition of the different species between the two com-

partments, the approximation of assuming most species to be present in only one of the compartments was made. All lipid species, vitamin E, ubiquinol, and PLA2 are present in lipid phase only. Dioxygen, proteins, and perhydroxyl radical are considered in the two compartments. All other species are considered to be present only in aqueous phase. Some interfacial reactions are allowed. For perhydroxyl radical a fast equilibrium between the two compartments and a partition coefficient of 1 were assumed. The ratio between the mitochondrial matrix and the mitochondrial inner membrane volumes was estimated at 5 (Appendix 4).

The calculation of the rates of production or con- sumption due to interface reactions takes into account the volumes of each compartment, as discussed in Ap- pendix 5. All concentrations and rates are referred either to the compartment where the species is present or to the compartment where the reaction takes place. Rates of interfacial reactions are referred to the membrane phase.

Overall, the model contains 776 reactions that are represented in a brief form in Table 2.

MATERIALS AND METHODS

Setup and analysis of the model

The reactional scheme described above was modeled by a system of differential equations. Conservation relationships and fast equilibria allow the elimination of several differential equations. This increases the efficiency of numerical simulations and avoids singular or ill-conditioned matrices in sensitivity analysis. To minimize numerical errors the differential equations corresponding to the variables with the highest values in each conservation relat ionship--TocOH, GPXrd, PHGPXrd, and SOD-MnZ+--were eliminated. A fast equilibrium between perhydroxyl and superoxide radicals was considered in both compartments. The three differential equations corresponding to these metabolites were eliminated. A new differential equation corresponding to an auxiliary dependent variable, defined as ~7 = [O~-] + [HOz ' ] J5 + [HO2"]aq, was introduced.

The resulting system of 58 differential equations was integrated numerically by using the algorithm LSODA} 31 The results were confirmed with the algorithm STIFBS. m These two algorithms are adequate to solve "s t i f f" sets of differential equations (equation where very different time scales coexist) as those of the present model.

The nonaided setup and analysis of such a complex model would be cumbersome and error-prone. There- fore, we used PARSYS (Package for Analysis of Reac-


tional SYStems). v PARSYS operates on an algebraic manipulation environment. 6 Starting from a symbolic description of the chemical reactions and from the kinetic data, this program is able to set up and simplify the kinetic differential equations, to generate a C + + routine for the simulation, and to analyze the model.

Calculation of the half-life times

The turnover times for the dependent variables were estimated from the diagonal elements of the exponential matrix of the jacobian. These elements are formed by a linear combination of the modes of the s y s t e m - - exponential functions of the form exp(hit) (where t is time). Typically one, or a set with very close his, of these modes predominates in each diagonal element of the exponential matrix. The Xi of the most important mode is, therefore, a good estimate of the turnover number.

The half-life time was obtained by multiplying the inverse of the turnover number by In 2.

Calculation of the molecular mean displacements

The mean displacement of a molecule in brownian motion, [[xb[, can be calculated from ILx[I =

~/2 x M x D x t, ~33'~34 where I[xhl is the displacement vector, M is the dimensionality of the system (2 for the membrane, 3 for the aqueous phase), D is the diffusion coefficient, and t is time. The average net distance that a molecule travels from the local of production was calculated with this expression considering t to be the half-life time.

Diffusion coefficients cannot be taken from experiments in water or in pure phospholipid membranes, because the viscosity of fluids increases in presence of macromolecules.

Lateral diffusion coefficients in fluid phase bilayers are typically in the range 10 -12-10 -H mZs-l. 135 In mitochondrial inner membranes a value of 5 × 10 -~3 mZs 1 was measured for a lipid analogue, ~36 and so we used it in the model.

The diffusion coefficient for dioxygen was estimated at (1.7 to 2.7) × 10 -9 m2s -1 from data in ref 137 for solubility and tissue dioxygen permeability (solubility × diffusivity). This value is about half the dioxygen diffusion coefficient in wa te r - -3 .8 × 10 -9 m2s -1, estimated from data in ref. 138. This is expected, because the diffusivity of small molecules (Mr < 700) in cells was reported to be 1/5 to 1/2 of that in water. ~39'14° Hydroxyl radical, superoxide radical, and hydrogen peroxide were considered to have the same diffusion coefficient as dioxygen.

The diffusion coefficients for GSH and GSSG were estimated from diffusion coefficients measured in the cytoplasm for metabolites with similar molar mass. For sucrose (M~ = 324) and methylene blue (Mr = 320) the values 2.0 × 10 -~° mZs -1 and 1.5 × 10 -~° m2s -~ were obtained. ~39 In the model we used the intermediate value of 1.8 × 10 -~° m2s -~ for GSH. For eosin (Mr = 648) a diffusion coefficient of 8 x 10 --H mZs i was obtained, 139 value that was used in the model for GSSG.

RESULTS

General results

For every set of plausible initial conditions tested the simulations always converged to the same physiologically meaningful steady state. Except for GSH, which was assumed to have a half-life of approximately 30 h, 86 all other variables of the system relax to this steady state within a time scale of seconds or faster (see below). For mathematical convenience, we assumed that the physiological system stays in steady state.

In Table 4, steady-state results obtained from the model for the set of parameters of reference are presented. In general, the results agree with the current views on lipid peroxidation. As expected, the relationship [carbon-centered radicals] < [peroxyl radicals] < [lipid hydroperoxides] is observed (Table 4, rows 1, 2, and 3). A previous theoretical work reported that [peroxyl radicals[ < [carbon-centered radicals].4 If we use the same rate constants used in ref. 4 we would still observe, both by a simple steady-state analysis or by simulation, an higher concentration for peroxyl radicals compared with C-centered radicals. This anomalous result obtained in ref. 4 may be due to shortcomings of the simulation algorithm used (Euler's method). This method has poor stability and poor accuracy. 132 Fur- thermore, the stiffness of the system of differential equations in ref. 4 makes Euler 's method use imprac- ticable, owing to long computation time and to accu- mulation of rounding errors. So, Babbs and Steiner 4 had to make the approximation that fast-turnover species were at a quasi-steady state and used a hybrid method for solving the system of coupled differential and algebraic equations. The solution of the algebraic equations does not exclude the possibility of reaching an unstable steady state or other physiologically mean- ingless steady state. Further evidence for large numerical errors being present in the work by Babbs and Steiner is the values they obtain for amplification (kinetic chain length): 1.71 for a two-compartment model simulating strong oxidative conditions (no antioxidants present), and 0.0014 for the homogeneous system.

928 F. ANTUNES et aL

Table 4. Results Obtained from the Model

Full Model Model without Cardiolipin

1 [carbon-centered radicals[ (m) 3.9 × 10 -l° M 3.0 × 10 -1° M 2 [peroxyl radicals] (m) 9.6 × 10 -8 M 8.0 × 10 -s M 3 [lipid hydroperoxides moieties] (m) 4.3 × 10 -6 M 2.9 × 10 -6 M 3 [lipid hydroperoxides moieties] (m) 4.3 × 10 -6 M 2.9 × 10 -6 M 4 [TocO']/[TocOH] 1.7 × 10 -6 1.4 × 10 -6 5 ([GPXo[ + [GSGPX])/[GPX]tot 1.8 × 10 -3 1.8 × 10 -3 6 ([PHGPXo] + [GSPHGPX])/[PHGPX]tot 8.1 × 10 -3 5.4 × 10 -3 7 [O~-] in the matrix (aq) 3.1 × 10 -1° M 3.1 × 10 -]° M 8 [H202] (aq) 3.8 × 10 -8 M 3.8 × 10 -8 M 9 2 × [GSH]/[GSSG] 1.7 × 104 1.7 × 104 10 total rate of initiation (m) 1.1 × 10 -7 M s -1 9.3 × 10 -8 Ms -~ 11 total rate of lipid hydroperoxide production (m) 1.6 x 10 -6 M s -1 1.1 × 10 -6 M s -I 12 amplification (kinetic chain-length) 15 12 13 initiation via hydroxyl radical (m) 8.6 × 10 -12 Ms -I 6.5 × 10 -12 Ms -I 14 initiation via perohydroxyl radical (m) 1.1 × 10 -7 M s -I 9.3 × 10 -8 Ms -] 15 initiation via tocopheroxyl radical (m) 2.2 × 10 -1~ Ms -l 1.4 × 10 -11 Ms -1 16 rate of lipid hydroperoxide decomposition via iron (m) 1.4 × 10 -1° Ms -1 9.2 × 10-" Ms -t 17 rate of lipid hydroperoxide decomposition via perphydroxyl radical (m) 8.3 × 10 -11 Ms -1 5.6 × 10 -11 Ms -~ 18 rate of lipid hydroperoxide decomposition via tocopheroxyl radical (m) 1.4 × 10 -13 M s - t 8.1 × 10 -16 Ms -1 19 rate of GSSG production dependent of hydrogen peroxide (aq) 8.0 × 10 -6 Ms -I 8.0 × 10 -6 Ms -] 20 rate of GSSG production dependent of lipid hydroperoxide (aq) 3.1 × 10 -7 M s -~ 2.2 × 10 --7 Ms -1 21 rate of hydroxyl radical production (aq) 7.6 × 10 -11 Ms -1 7.6 × 10 -11 Ms -1 22 rate of reaction of hydroxy radical with matrix proteins (aq) 7.5 × 10-11 Ms- 1 7.5 × 10-11 Ms- 1 23 rate of reaction of superoxide radical with matrix proteins (aq) 5.8 × 10 -1° Ms -1 5.8 × 10 -1° Ms -j 24 rate of reaction of perhydroxyl radical with matrix proteins (aq) 3.6 × 10 -1~ Ms -~ 3.6 × 10 -lz Ms -1 25 rate of reaction of perhydroxyl radical with membrane proteins (m) 5.8 × 10 -9 MS -] 5.8 × 10 -9 Ms -] 26 rate of reaction of hydroxyl radical with DNA (aq) 3.5 × 10 -14 Ms -j 3.5 × 10 -~4 Ms -a 27 rate of reaction of superoxide radical with DNA (aq) 2.5 × 10 -1-~ Ms -I 2.5 × 10 -~3 Ms -~ 28 rate of reaction of perhydroxyl radical with DNA (aq) 1.5 × 10 -14 Ms -1 1.5 × 10 -~4 Ms -~ 29 fraction of lipid hydroperoxide that escapes GSH-peroxidases 2.1 × 10 -4 2.0 × 10 -4 30 fraction of hydrogen peroxide that escapes GSH-peroxidases 9.4 × 10 -6 9.4 × 10 -6 31 fraction of superoxide/perhydroxyl radicals that escapes SOD 1.5 × 10 -3 1.3 × 10 -3

The full model considers cardiolipin to have an oxidizability similar to that predicted from its acyl composition while the model without cardiolipin assumes that this lipid is fully resistant to peroxidation.

(m) and (aq) referred, respectively, to membrane phase and aqueous phase. To compare fluxes referred to different compartments it should be taken into account that the volume of aqueous phase is five times the volume of membrane phase.

T h e s e va lues (par t icu la r ly the last one) are c h e m i c a l l y

un feas ib l e and are in con t ras t w i th the va lues w e ob-

t a ined for the ampl i f i ca t ion : 12 to 15 in a w e l l - p r o t e c t e d

s y s t e m (Tab le 4, r o w 12). As obse rved in the exper iments , 91'92 in the present

w o r k it is obse rved that h y d r o g e n pe rox ide is p roduced

s to ich iomet r i ca l ly f r o m superox ide radicals: less than

0 .2% o f superox ide radicals escape S O D (Table 4, r o w

31). A l s o the concen t ra t ion o f hyd rogen pe rox ide is

c lose to es t imat ions f r o m expe r imen ta l data: its concen -

t rat ion was es t imated in rat l ive r in the range 10 - 9 - 1 0 -7

M, TM which is s imi lar to the va lue w e ob ta ined (4 ×

10 -8 M; Tab le 4, r o w 8). T h e G S H - d e p e n d e n t e n z y m e s

are a lmos t all in their r educed f o r m (Table 4, rows 5 and

6) and so the rate o f hyd rope rox ide reduc t ion ca ta lysed

by these e n z y m e s is l inear ly dependen t on hyd rope rox -

ide concent ra t ion , wh ich is the expec ted s i tuat ion under condi t ions in v ivo . 14z A h igh rat io b e t w e e n G S H and

G S S G (Table 4, r o w 9) was obta ined as obse rved ex-

per imenta l ly . T h e va lue o b t a i n e d - - l . 7 × 1 0 a - - i s

h ighe r than the h ighes t expe r imen ta l rat ios repor ted in

hepa tocy te cytosol , wh ich are in the range o f ( 3 - 7 ) × 102.143'144 This d i sc repancy m a y be due to two factors:

the ox ida t ion o f g lu ta th ione in expe r imen ta l sys tems is

ve ry diff icul t to avoid; and processes not cons ide red in

the present m o d e l m a y oxid ise glutathione.

Half- l ives and mean displacements

T h e e s t ima ted ha l f - l i fe t imes and the m e a n d i sp lace-

m e n t for key spec ies are p r e sen t ed in T a b l e 5. T h e s e

pa rame te r s w e r e ca l cu la t ed as de sc r ibed in Mate r i a l s

and Me thods . T h e va lues ind ica ted in Tab l e 5 fo r su-

p e r o x i d e rad ica l w e r e those ob ta ined fo r the aux i l i a ry

va r i ab le that cha rac te r i zes the r e s e r v o i r f o r m e d bo th by

p e r h y d r o x y l and supe rox ide radicals . T h e s e va lues

cha rac te r i ze the b e h a v i o r o f supe rox ide rad ica l be-

cause : (a) the t u r n o v e r o f this r e se rvo i r is m a i n l y de-

t e r m i n e d by the inf lux o f supe rox ide radica ls f r o m the

resp i ra to ry cha in and by supe rox ide d i smutase ; and (b)

this r e se rvo i r is m a i n l y c o m p o s e d by supe rox ide

radicals .

H y d r o x y l rad ica l and a lkoxy l rad ica ls o n c e f o r m e d

reac t a l m o s t i m m e d i a t e l y w i thou t d i f fus ing f r o m thei r

f o r m a t i o n site. A l s o c a r b o n - c e n t e r e d radica ls can dif-

fuse on a v e r a g e on ly 6 n m in the m e m b r a n e b e f o r e


Table 5. Half-Lives and Mean Displacements Estimated from the Model

929

Species Diffusion Coefficient ( m 2 S -1) Half-Life (s) Mean Displacement (#m)

Alkoxyl radicals 5 X 10 -13a (6.4-12.0) X 10 8 (3.6--4.6) X 10 4 Carbon-centered radicals 5 x 10 -13a (1.5-2.3) X 10 -5 (5.5-6.8) x 10 -3 Peroxyl radicals 5 X 10 -133 (2.7-59.0) X 10 -z (2.3-11.1) X 10 -~ Non-sterified hydroperoxides 5 x 10 -133 3.3 x 10 -3 8.1 x 10 -2 Phospholipid hydroperoxides 5 x 10 -133 1.8 1.9 TocO" 5 X 10 -10b 2.0 X 10 -3 2.0 HO" 2 X 10 9c 2.4 X 10 --1° 1.7 X 10 -3

O~- 2 x 10 9c 1.3 x 10 -5 4.0 X 10 -I H202 2 X 10 -9c 3.3 x 10 3 6.3 GSSG 8 x 10 -H~ 1.1 x 10 -~ 7.4 GSH 1.8 x 10 -1°c 1.1 x 105 1.1 x 104

a Ref. 136. b Ref. 145. c See Materials and Methods.

reacting with dioxygen. Because the area per phospholipid in a membrane is about 0.74 nm 2 (from ref. 146 considering acyl chains with 18 carbons) the carbon- centered radical may travel a net distance of about 6 to 7 phospholipid molecules. Peroxyl radicals may diffuse significant distances--about 0.5 #m. Considering that the area of mitochondrial inner membranes is 40 m2/g of mitochondrial protein 129'13° and that each mitochondrion has about 1 0 -13 g of protein, 122-124 then we can

estimate the membrane area in a mitochondrion at about 4 x l 0 -12 m E (that is, a square with length 2 #m). Hence, a peroxyl radical can travel on average through about one-quarter of the length of the membrane, and, thus, it is able to perform well its key role as a chain- propagating species. Lipid hydroperoxides are also able to travel long distances through the membrane. The high mean displacements of peroxyl radicals and lipid hydroperoxides are an indication against the formation of highly peroxidized membrane domains. This remark is obviously dependent on an homogeneous membrane composition in unsaturated moieties.

Hydrogen peroxide has a high mean displacement: about 6 #m. Data from electron microscopy show that mitochondria have a diameter of 0 .5-1 .0 #m and a length of a few micrometers (more rarely, up to 10 #m). 147 Thus, this species is able to diffuse throughout a mitochondrion. As proposed for peroxisomes, 148 part of the hydrogen peroxide may also escape to cytosol. Superoxide radical has a lower mean displacement-- 0.4 # m - - a n d so probably only a very minor fraction (that must be in the protonated form) may escape mitochondria. However, this mean displacement is high enough for superoxide to be able to damage cellular components (e.g., DNA and proteins) not very close to the local of production of these species.

The couple GSSG/GSH is often considered to be the redox buffer of the cell, 149'15° GSH is assumed to be the homeostatic arm and GSSG is considered to be the

floating (sensor) arm. 150 That is, alterations of the redox state of the cell are sensed by alterations of the ratio GSSG/GSH. The results in Table 5 support this view. GSH has a high half-life while GSSG has a fast turnover. However, the turnover of GSSG should be slow enough for GSSG to diffuse throughout the mitochondrial matrix, signalling the change in the redox state. The mean displacement of 7 #m obtained for GSSG in mitochondria is appropriate for a signalling role, taking into account the dimensions of this organelle.

Cardiolipin peroxidation

The doubts about cardiolipin oxidasibility are a potential hurdle for the model. However, Table 4 shows that results are not much different if cardiolipin is considered either to be fully resistant to peroxidation or to have a similar oxidisability to dil inoleoylphosphatidylcholine.

If intramolecular reactions are discarded, peroxidation of the two arms of the diacylphospholipid proceeds almost independently in cardiolipin peroxidation (Fig. 1). That is, products arising from intermolecular abstraction of the bis-allylic hydrogen at the two arms are negligible. The reason is that diacyl moieties oxidized in one acyl do not accumulate, and, thus, do not compete effectively with nonoxidized diacyl moieties for radicals. Species as LOO'LH or LHLOO" are excep- tions (see Fig. 1). Unsaturated acyls of these species do compete effectively for LOO" through an intramolecular reaction. So, phospholipid dihydroperoxides (LOOHLOOH) are formed almost entirely through intramolecular oxidation. This result is very robust: it only depends on a small ratio between oxidised and nonoxidized acyl moieties, as observed experimentally and also as obtained in this model.

The rate of intramolecular propagation depends strongly on the first-order rate constant for intramolec-

930 F. ANTUNES e t al.

LHLH

8. 2

9.1"9

9.29e-9 LrLH

2.75e-ll

3.2~18

3.2o-18 LHLr - - ~ LrLr

2.75111 2.69e-21 li

q F 4.14¢t,

4.14e-17

LOOrLH ~ LOOHLH 7 s2~-a ~ LOI-ILH

4.7e-96~1 ~ 1.99e-721 1.58e4 -3

5.4 ~-16 2.3 -14 1.8 10

LOOrLr 4.17e-18). LOOHLr 5"77e'!2-~ LOHLr

2.61e-19 1.531-11 1.151-12

741 1, 23t 9

5.3e-17

LHLOOr 5.46e-16 LrLOOr

4"i~ N 2"61e'19ji 7.- ~" 4.175_18

LHLOOH 2.32e-14 ). LrLOOH

7.41e-15 q~ LOOrLOOr 2.76e-18

I 4.42~- 17

4.42e-17 *-LOOHLOOr 1.04¢-9_ LOHLOOr

2 . I9 2.15110

34 9

9.87e-15

~ L O O r L O O H 442~.a. L O O H L O 0 ~ LOHLOOH 5.81eM2

1 i i 1.53e-ll I~ 2.76f9 9.18 1.1~7 ,.0 9 .

LHLOH 1.84e-10 2.39e-9 3.43e-~) ,, LrLOH - LOOrLOH ~ LOOHLOH ~ LOHLOH

1.5~e-3 1.15e-12 2.15e- 10 1.26e-7 2.07e-3 / t

7.2~8 5.3~17 9.87~-15 5.8 12

Fig. 1. Simplified scheme showing the peroxidation of cardiolipin. The number below each species in the graph indicates the steady-state concentration (unit is M). The other numbers represent fluxes (unit is Ms-l). The character "r" symbolizes a radical dot.

ular propagation. Data about this rate constant are missing. Hence, the relative rate of intramolecular propagation compared with the intermolecular propagation depends strongly on our estimate (see Appendix 1) of this rate constant.

Pro-oxidant agents

The effect of iron concentration on lipid peroxidation is shown in Fig. 2a and b. The rate of lipid peroxidation is significantly affected by iron (curve 4) only for [Fe e+] > 1 X 10 -5 M. For concentrations lower than this level the iron-dependent influx of lipid radicals to the system (flux of initiation by hydroxyl radicals-- curve 3 - -p lus flux of decomposition of lipid hydroperoxide to alkoxyl radicals-- curve 2) is much lower than the initiation by perhydroxyl radicals (curve 1). At the reference concentration of iron used in this w o r k - - 10 -7 M - - t h e latter process is about three orders of magnitude higher than the former (Table 4, rows 13,

14, and 16). One weak support for using this concentration for iron is that with it the rate of hydroxyl radical production observed- -8 x 10 1~ Ms-m (Table 4, row 21) is within the range estimated by Chance et al. 8 - 10 12-10-9 Ms m--for this process. If the value estimated by Yegorov et al. 4° for the "chelatable iron" (2.2 X 10 _5 M) were used, the rate of this process would be 2 x 10 -8 Ms-L

The main mechanism for stimulation of lipid peroxidation by iron is the decomposition of lipid hydroperoxides catalyzed by ferrous ions because the flux of this reaction is higher than the initiation by hydroxyl radicals in all the range of iron concentration studied (compare curves 2 and 3 in Fig. 2a). Therefore, iron switches lipid peroxidation from a self-propagated cycle, where the number of lipid radicals is kept constant, to a chain-reaction where multiplication of the number of radicals takes place. Chain reactions are potentially much more difficult to control than self-propagated reactions. For iron concentrations higher than 5 x 10 -5

t- O "G t~

t t . . .

o

1.0E-02

1.0E-04

1.0E-06

1,0E-08

1,0E-10

1.0E-12

1.0E-14 1.0E-02

1.0E-03

1.0E-04

1.0E-05

1.0E-06 1.0E-07

1.0E-08

1.0E-09

1.0E-10

1.0E-11

1.0E-12

1.0E-13

1.0E-14

1.0E-15

1.0E-16

a

/ f J

f J J

J

b

' ' , , , , i , , , , , , , ,

c

J 6 j

J J

J

J J J 8 j t

J

. . . . , . , , i . . . . . , , , i . . . . . . . . J . . . . . . . .

1,0E-08 1.0E-07 1.0E.-06 1.0E-05 1.0E-04 [Fe2+] (M)

1.0E+05

1.0E+04

c- O

u 1.0E+03 ,=:

o .

E ,<

1.0E+02

1.0E+01

Fig. 2. Effect of ferrous ion concentration on lipid peroxidation (a and b) and on the oxidation of proteins and DNA (c). The following rates are shown: initiation of lipid peroxidation by perhydroxyl and hydroxyl radicals (curves 1 and 3, respectively); decomposition of lipid hydroperoxides to alkoxyl radicals (curve 2); rate of lipid peroxidation (curve 4); rates of reaction both of the superoxide/perhydroxyl couple and the hydroxyl radical with proteins in aqueous phase (curves 6 and 7, respectively) and DNA (curves 8 and 9, respectively). The amplification of lipid peroxidation is also shown (curve 5). The arrow indicates the reference value for the iron concentration used in the model.

931


1.0E-04 1E-15

A .1E-16

1.0E-05 ,..., =E --=

o .1E-17

~.. 1.0E-06 0 o

1.0E-07 . . . . . . . . . . . . . . . . . . . . . . . . . . , . . . . . . . . . . . . . . . . . 1E-19

1.0E-06 1.0E-05 1.0E-04 1.0E-03 1.0E-02 1.0E-01

["rocopherel]tot (M)

Fig. 3. Effect of e-tocopherol on the concentration of alkoxyl radicals (curves 1 and 2) and on the rate of l ipid peroxidation (curves 3 and 4) for two different conditions: reference system (curves 2 and 4); and, rate constant for the reaction between vitamin E and alkoxyl radicals set to zero (curves l and 3). The arrow indicates the reference value for the total concentration of ot-tocopherol used in the model.

M the system becomes very sensitive to this parameter (Fig. 2b): doubling the iron concentration to 1 x 10 -4

M increases both the rate of lipid peroxidation and the amplification by three orders of magnitude. At these levels of iron the system is no longer protected.

In Fig. 2c the effect of iron concentration on the oxidation of both proteins and DNA is shown. For iron concentrations below 10 -6 M the direct oxidation of DNA and proteins by the couple superoxide/perhydroxyl is higher than by the hydroxyl radical. The values of the rate constants involved in these processes are very doubtful and the results are not reliable. However, they are an indication that the direct oxidation of cellular components by the superoxide/perhydroxyl couple may not be negligible. Usually the direct attack of these species towards DNA and proteins is ignored based on the low reactivities of these species when compared with other oxidants such as the hydroxyl radical. It is commonly suggested that the pro-oxidant effects of these species are mediated by other species after Fenton or similar reactions. Direct evidence of DNA damage by hydroxyl radicals under close to physiological conditions was obtained only recently. 151

As with iron, the pro-oxidant effect of perhydroxyl radicals can be lipid hydroperoxide-dependent (reaction 27 in Table 2) or lipid hydroperoxide-independent (reactions 2, 3, 4, in Table 2) as discussed by Aikens and Dixfl 5 The results indicate that the latter pathway has a three order of magnitude higher flux than the former (Table 4, rows 14 and 17), in spite of the kinetic

rate constant for reaction 27 being two orders of magnitude higher than the kinetic rate constants for reactions 2, 3, and 4. This is explained by the efficiency of GSH-dependent peroxidases that keep lipid hydroperoxides at very low levels (about 1 0 - 3 % of the unsaturated acyls).

Action of a-tocopherol

a-Tocopherol at 2 × 10 -4 M decreases the rate of lipid hydroperoxide production by a factor of about 10 (see Fig. 3, curve 4). This high antioxidant efficiency is due to trapping of the chain-carrying (propagating) species by a-tocopherol.

a-Tocopherol is very effective against lipid peroxidation catalysed by decomposition of hydroperoxides to alkoxyl radicals. In this situation, ce-tocopherol not only inhibits the rate of lipid peroxidation by trapping the propagating species but also decreases the net influx of radicals to the system by decreasing the levels of lipid hydroperoxides. As can be observed in Fig. 3 curve 1, a-tocopherol is able to decrease the concentration of alkoxyl radicals under conditions where the rate constant between these two species was set to null. The effect of the direct trapping of alkoxyl radicals is never important at physiological o~-tocopherol concentrations (compare curves 1 and 2 in Fig. 3).

Pro-oxidant effects of c~-tocopherol were not observed in the present model (compare rows 15 and 18 with row 10 in Table 4). Only a very small fraction of


vitamin E is in the oxidized form (Table 4, row 4). This result is very robust because only one regenerating agent of c~-tocopherol was included in the present model (ubiquinol). Under physiological conditions other agents may contribute to a very high ratio between reduced and oxidised forms of vitamin E.

Answers to these questions were possible because the results obtained from the model for these processes indicate that one pathway predominates by orders of magnitude over the competing pathways. Therefore, the answers are given with a reasonable degree of confidence.

DISCUSSION

In this work we present an integrative mathematical model of lipid peroxidation in the mitochondrial inner membrane. Integrative models can give a reasonable picture of the behavior of a system in vivo from the knowledge of its reactions and respective kinetics. At present, this is the best way to conceive pictures of the situation in vivo because: (1) noninvasive methods can, at the moment, give only details about a flux of a reaction or a level of a metabolite in vivo, 152 and so they do not allow to obtain a broad perception of the physiological system; (2) the intuitive integration, as done during this century by biochemists, of the same data used to build up integrative models is limited both by the large amount of information and by the large number of relevant variables involved.

The parameters used in mathematical models of biochemical processes are usually obtained from experiments in vitro, often carried out under conditions far from the conditions in vivo. So the results obtained from integrative models should be interpreted as order of magnitude estimations of the physiological state. For processes where radicals are involved this can be very useful because experimental measures under physiological conditions are very difficult to carry out. The estimation of relative values of alternative pathways is more secure and probably more important than the estimation of absolute values. In particular, this work allowed to answer the following questions for physiological conditions:

1. What is the main initiator of lipid peroxidation? 2. Once lipid hydroperoxides are formed what is its

main destiny? Are they hydrolysed via PLA2 fol- lowed by reduction through "classic" GSH peroxidase? Are they reduced directly through PHGPX? Are they decomposed to alkoxyl radicals producing aldehydes?

3. Is the stimulating effect of perhydroxyl radical on lipid peroxidation dependent (via lipid hydroperoxide decomposition) or independent (via direct abstraction of a bis-allylic hydrogen) of lipid hydroperoxides?

4. What is the hydroperoxide (hydrogen or lipid hydroperoxide) mainly responsible for GSSG production?

Initiation of lipid peroxidation

In spite of the doubts associated with the levels of iron, the greatest importance of perhydroxyl radical compared with hydroxyl radical in the initiation of lipid peroxidation is a robust result (compare curves 1 and 3 of Fig. 2a). One limitation of the present model that may affect this result is the fact that site-specific reactions of hydroxyl radicals were not considered. In the model hydroxyl radical reacts predominantly with proteins and only 1/50 reacts with lipids (Table 4; rows 13, 21, and 22). If it were considered that all hydroxyl radical produced initiates lipid peroxidation, the initiation by hydroxyl radicals would be 50 times as much as those in Table 4 and Fig. 2. Therefore, if hydroxyl radicals were considered to react site specifically both with lipids and proteins the corrective factor involved (for the reaction with lipids) would be lower than 50. Hence, the result about the greater importance of perhydroxyl radicals to the initiation of lipid peroxidation would not change.

The value of 10 -7 M s -1 for the flux of initiation by perhydroxyl radicals obtained in the model is credible because it depends on a number of parameters reasonably known: (a) the rate of production of superoxide radicals by the mitochondrial respiratory chain, the concentration of SOD and the rate constants for the SOD-catalyzed dismutation. These parameters determine the concentration of superoxide radicals and are known at the order of magnitude level (see description of the model); (b) the pH close to the membrane. The value used in this m o d e l - - p H = 5 - - s h o u l d be a close estimate to this parameter; I ~4 (c) the partition coefficient of perhydroxyl radicals between aqueous and lipid phase. The value used in the model was 1, as data about this parameter were not found in literature. This is probably accurate at the order of magnitude level, because for dioxygen this coefficient is 3 in favor of the membrane; ~16 (d) rate constant for the reaction of initiation by perhydroxyl radicals. Although this constant was determined in a 85% aqueous ethanol homogenous solution, 21 it should provide a reasonable estimate of the rate constant in biomembranes. Propagation rate constants for the peroxidation of linoleic acid are only two times higher in homogeneous solution ~53 than in bilayers. 33 On the other hand, in general, the hydro-


gen atom-abstracting ability of perhydroxyl radicals is greater in nonpolar environments, as biomembranes, than in aqueous solutions.~54

The question whether the rate of initiation of lipid peroxidation by perhydroxyl radicals provides a good estimate of the total rate of lipid peroxidation initiation depends on the existence of other important initiators in mitochondria that were not considered in the model. Perferryl ions have been suggested to be ini- tiating agents, 155'~56 but this proposal has been criti- cised 157 on thermodynamic grounds. Ferryl ion may be produced in the Fenton reaction as an alternative species to hydroxyl radical. ~58 However, the kinetic constraints observed for the production of hydroxyl radicals apply also to the production of ferryl ions. Complexes of ferric plus ferrous iron have been also suggested to be initiators of lipid peroxidation, 159'~6° but evidences against a role for these complexes 16~'162 have also been pointed out. Furthermore, if "chelatable" iron is predominantly in its reduced form in the cell, 4° then the formation of these complexes - -and consequently the initiation of lipid peroxidation by t h e m - - i s unlikely to be of importance. Similar initiators to those formed from iron may be formed from copper. If the copper pool is similar or lower than the iron pool, the initiators produced from it are probably also not quantitatively important in mitochondria under normal physiological conditions. Another possible initiator is singlet oxygen. However, this species does not lead to self-propagated cycles as it adds to the unsaturated moiety producing directly the hydroper- oxidef163 Moreover, the estimated rate of production of chemiluminescent species in submitochondrial particles is very low: 1 0 -17 Ms -~ is an estimate in the presence of tert-butyl hydroperoxide. 164 Semiquinone radicals participating in electron transport are also potentially initiators of lipid peroxidation, but direct evidence for that is lacking and Vladimirov et al. 38 have excluded this hypothesis. Radicals derived from the metabolism of xenobiotics, which can be important in endoplasmic reticulum, 165 are probably not important in mitochondria.

On the whole, and considering the available data, our estimated rate of initiation by perhydroxyl radicals (10 -7 Ms -l) should provide an order of magnitude estimation for the total rate of initiation of lipid peroxidation in mitochondrial inner membranes.

Amplification of lipid peroxidation

The amplification of lipid peroxidation (or kinetic chain length) obtained in this model depends mainly: (a) on the concentration of unsaturated moieties in the membrane, which is a known parameter; (b) on the

propagation rate constants for lipid peroxidation; these constants were estimated from data in liposomes, which are more fluid than biomembranes, 136 and so, our estimates are probably upper-limits; (c) on the chain- breaking capacity, which is underestimated in our model because the only chain-breaking antioxidant considered was vitamin E. In mitochondrial inner membranes, ubiquinols may also trap peroxyl radicals. 166 Ubiquinols are about seven times more concen- trated than vitamin E in mitochondrial inner membranes 57 but (in homogeneous solution) its inhibition rate constant ~67 is one order of magnitude lower than that of vitamin E. Therefore, if ubiquinols were considered the chain-breaking capacity would be approximately twice that we considered in this model. So, both the rate and the amplification of lipid peroxidation would be about half of those obtained (see Fig. 3)).

"Chain-branching" by decomposition of lipid hydroperoxides increases the amplification. However, only for very high--presumably nonphysiologicai-- concentrations of iron the amplification is affected by this process (Fig. 2b, curve 5).

In conclusion, the amplification obtained in the ab- sence of ubiquinol--lO to 15--should constitute an upper limit to this parameter. This upper limit should be about half if the ratio of the rate constants for the trapping of peroxyl radicals by ubiquinols and vitamin E in bilayers is similar to that in homogeneous solution.

Rate of lipid peroxidation

The estimate for the rate of lipid peroxidation depends on the rate of initiation and on the amplification. Considering a total rate of initiation of 10 -7 Ms -1 and an amplification of 5 to 10, the rate oflipidperoxida- tion is estimated at (5-10) × 10 -7 Ms -~ (referred to membrane phase). The rate of lipid peroxidation for the whole liver cell was estimated at 1 0 -6 M s - l , based on the rate of GSSG efflux from the liver. 8 Because mitochondria do not export GSSG to cytoplasm, 87-89 this value does not include the mitochondrial contribution. Therefore, by comparing these two estimates, and considering both that the volume ratio between the mitochondrial matrix and the mitochondrial inner membrane is 5 (Appendix 4) and that mitochondria represent between 1 0 and 20% of the total volume of an hepatocyte, 12~ we would conclude that lipid peroxidation in mitochondria accounts for 1 to 4% of total peroxidation in hepatocytes.

However, part of the GSSG exported from hepatocytes may be formed by other processes than the reduction of lipid hydroperoxides. One of these processes can be the reduction of hydrogen peroxide in cytosol

o . ' "~ ~ ~ I ,~ o , °° ~ o I ~°-

• ~o . ~ _ 1 ~ ," ~' I ~'

- ~ / / × ~ ' /~ I ~ - - ~ i ~ I_~

, . . o ~ , ~ e ~

. ~ , .

o S

-~ ~-~

~ ,-~ '~-~

~ ~.~

= N ''"

g x"

~-' r/3

~

.~ ~ - ~

~ . N

o~ ~'~

~ ~-~

935

936 F. ANTUNES et al .

via "classic" glutathione peroxidase. Hydrogen peroxide is produced in cytosol 9° and part of that produced in peroxisomes--between 2 % 148 and 30% 90 diffuses to the cytosol. Also, part of the mitochondrial hydrogen peroxide may diffuse to the cytosol because the permeability of biomembranes to hydrogen peroxide is similar to that of water, 8 and, a high mean square displacement (6.3 #m) for this species in the mitochondrial matrix was obtained in this work (Table 5). Therefore, the value of 10 -6 Ms -1 (ref. 8) may be overestimated and so mitochondria may account for a higher fraction of total lipid peroxidation in hepatocytes.

Lipid hydroperoxide reduction and decomposition

Once formed, lipid hydroperoxides can react through several pathways. As analyzed in ref. 19, reduction through PHGPX predominates over reduction through PLA2/GPX by four orders of magnitude. At the mitochondrial levels of PHGPX--4 × 10 -8 M - - lipid hydroperoxides are almost all reduced, and only a very minor fraction is decomposed. This is the reason why the pro-oxidant effect of perhydroxyl radicals on lipid peroxidation is mainly hydroperoxide independent (see Table 4, rows 14 and 17).

Concerning the decomposition of hydroperoxides by iron, if the iron pool "active" towards peroxidation is 10 -7 M, then only about 0.01% of the hydroperoxides escape reduction catalyzed by PHGPX (compare rows 11 and 16 in Table 4) producing alkoxyl radicals. One important implication of this result is that the decomposition of lipid hydroperoxides is unlikely to produce significant amounts of aldehydes, which are usually observed during peroxidation. 168 However, lipid peroxidation produces peroxides other than hydroperoxides, namely endoperoxides, 32 whose decomposition may originate aldehydes. 169 Preliminary results with another model of lipid peroxidation of mitochondrial inner membranes that includes the cyclization of peroxyl radicals indicates that endoperoxides constitute an important fraction of total lipid peroxides (about 50%). Be- cause dialkylperoxides are probably not reduced by Se-dependent peroxidases--as they are not reduced by GPX 17°'I7j - - a higher fraction of these species may decompose to alkoxyl radicals. These radicals would further decompose to secondary products of lipid peroxidation, like aldehydes and volatile products, and/or would abstract (bis-)allylic hydrogens increasing both the rate and the amplification of lipid peroxidation. To evaluate how important these processes could be it is fundamental to know what are the pathways of endo- peroxide detoxification. Free fatty-acid endoperoxides are reduced by GSH-S-transferases, 17°'171 but it is un-

known if there are enzymes able to catalyze the reduction of phospholipid endoperoxides. One possible candidate to this role is the membrane-bound GSH-S- transferase as membrane phospholipid hydroperoxides, 77 some epoxides, 172 and linoleic ozonide 77 are substrates for this enzyme. At least in mitochondria, it is important to have enzymes acting directly on pbospho- lipids because PLA2 activity is not high enough to avoid a high steady state of phospholipid peroxides. 19

Formation of GSSG

SOD and GPX act as a coupled enzymatic system reducing superoxide radicals to water, with a concom- itant production of GSSG. Therefore, the rate of GSSG production dependent on hydrogen peroxide is approximately half the rate of superoxide radical production (8 X 10 -6 MS -1, Table 4, row 19). The rate of GSSG production through reduction of lipid hydroperoxides is 3 × 10 -7 Ms -1 (referred to the aqueous phase) (Table 4, row 20) because it is similar to the rate of lipid peroxidation. Therefore, the oxidative challenge imposed by hydrogen peroxide is more than one order of magnitude higher than that imposed by lipid hydroperoxides.

FINAL REMARKS

Integrative modelling can contribute to a semiquantitative picture of complex processes under conditions in vivo. In spite of the large number of reactions included in the present model, its analysis allowed to set up a simplified scheme showing only the main pathways of the model (Fig. 4). According to the kinetic data considered in the model, these are the most important pathways for hydroperoxide metabolism in hepatocyte mitochondria under physiological conditions. Because the model was based on already known data, none of the pathways shown is a new pathway. The newness in this study is to analyze quantitatively, at the order of magnitude level, the integration of a large number of reactions. By doing that it was possible (1) to access the relative importance of competing pathways, thus answering the questions referred at the be- ginning of the Discussion, and (2) to estimate physiological fluxes and concentrations.

Because most metabolites and the few enzymes considered in the model are prevalent within the cell, the model can easily be adapted to other situations and some results are also relevant for other cellular compartments.

A c k n o w l e d g e m e n t s - - We are grateful to Professor Evgenii Volkov for kind and motivating discussions about his theoretical work on


lipid peroxidation. A.S. and F.A. acknowledge support from grants PRAXIS-XXI-BD/3457/94 and FMRH-BD-399-92 from JNICT- Portugal, respectively. We acknowledge the contribution of FACC (JNICT-Portugal) to the support of Grupo de Bioqulmica e Biologia Te6ricas.

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APPENDIX 1. CONSEQUENCES OF THE STERICAL HINDRANCE IN DIACYLPHOSPHOLIPIDS

ON THE RATE OF REACTIONS

The presence of two adjacent acyls in diacylphospholipids intro- duces a sterical factor that reduces the rate of intermolecular reactions involving these species. If each acyl in a membrane is considered to be in contact with n acyls, then the presence of an adjacent acyl moiety reduces the rate of reactions by a factor of l/n. Table 6 shows the correction factors considered in this work. Corrections were only introduced to reactions involving exclusively lipid species. In this work it was assumed that n = 4.

Estimation o f the propagation rate constant f rom

experimental data

If a diacylphospholipid is unsaturated at both positions (LH-LH) then both an intramolecular and an intermolecular reaction of propagation occur. The intermolecular and the intramolecular rate laws are, respectively, 2 x kp x [LH-LH] X [peroxyl] x ( l - l / n ) and kp' x [peroxyl] = kp x 2 X [LH-LH]max X (l/n) X [peroxyl] where [LH-LH]max is the concentration of LH-LH in a membrane formed only by LH-LH; the factor 2 X [LH-LH]m,, X (l/n) accounts for the concentration of the moiety LH in the same molecule of the peroxyl. The sum of these two rates for a membrane formed only by LH-LH is simply 2 x kp x [LH-LH]m~[peroxyl]. However, this value is an overestimation. A steady-state analysis of the following minimal scheme for lipid peroxidation of LH-LH:

LH-L" + 02 ~ LH-LO0"

LH-LH + LH-LOO" ~ LH-L" + LH-LOOH

LH-LOO" ~ L'-LOOH

L'-LOOH + 02 -- LOO'-LOOH

LH-LH + LOO°-LOOH ~ LH-L" + LOOH-LOOH

shows that the total rate of propagation is 2 x kp X [peroxyl] × [LHLH]m~x X (1 - (l/n)2). The reason for this is that the species produced from the intramolecular peroxidation (L'-LOOH or LOOH-L') cannot undergo intramolecular propagation.

The propagation rate constant measured by Barclay et al. (1989) 33 for pure dilinoleoylphosphatidylcholine bilayers is 36.1 M-~s ~. From the above expression we can estimate a rate constant of 19.25 M ~s -~ per linoleoyl chain, if we assume that n = 4. From this value the propagation rate constants for the other unsaturated fatty acids were calculated (see main text).

Estimation o f the termination rate constant f rom experimental data

Barclay et al., 1989 s3 measured the termination rate constant for pure dilinoleoylphosphatidylcholine b i layers - -2kt = 1.32 × 105 M -~ s ~. Simulation analysis (see results) shows that the main forms of peroxyl radicals are LH-LOO" and LOO'-LH. Therefore, the total rate of termination is given by 2kt X [peroxyl][peroxyl] × (1 - 1/n), and so a kt = 8.8 × 104 M -1 s i is estimated.

Table 6. Corrections Introduced in the Propagation and Termination Reactions Accounting for the Sterical Factors

Reaction Rate Law

LH-LOO" + LH-LH ~ LH-LOOH + LH-L" LH-LOO" + LH-LH ~ LH-LOOH + L'-LH SL-LOO ° + SL-LH ~ SL-LOOH + SL-L" LysoLOO ° + LysoLH ~ LysoLOOH + LysoL ° SL-LOO" + LysoLH ~ SL-LOOH + LysoL ° LysoLOO" + SL-LH ~ LysoLOOH + SL-L" LH-LOO" + LH-LOO" ~ products LysoLOO" + LysoLOO ° -- products LysoLOO" + SL-LOO ° -- products

kp x [LH-LOO'][LH-LH] X (1 - l/n) " kp x [LH-LOO'][LH-LH] X (1 - l/n) ~ kp x [SL-LOO'][SL-LH] X (1 l/n) b kp x [LysoLOO'][Lyso-LH] kp x [SL-LOO'][LysoLH] x (1 - l/n) " kp x [LysoLOO'][SL-LH] ~ kt x [LHLOO'][LHLOO'] x (1 - l/n)d kt X [LysoLOO'][LysoLOO °] kt × [LysoLOO'][SL-LOO'] x (1 - l/n)e

Analogous corrections were considered for alkoxyl radicals. The same applies: a TO LOOH'-LH or SL-LOO'. h To LOO'-LH or LH-LOO ° ' To LH-LH. d To any combination of LH-LOO', LOO'-LH, SL-LOO °. ° If SL-LOO" is replaced by LH-LOO ° or LOO°-LH.


APPENDIX 2. ESTIMATION OF THE OS M OT IC ALLY ACTIVE VOLUME OF THE MITOCHONDRIAL MATRIX

Beavis et al. s° estimated an osmotically active volume for mitochondrial matrix of 1.65 # l /mg protein at 115 mosm and of 0 . 66 - 0.72/zl/mg protein at 272 mosm. These authors discuss results suggesting that the degree of swelling in vivo is close to the value observed in a 115 mosm medium: (a) mitochondria exhibit little intermembranar space in vivo and most closely resemble isolated mitochondria suspended in a 115 mosm medium; (b) The level of K ÷ found in vivo sl is about 2.4 t imes that observed in freshly isolated mitochondria at 272 mosm. 8° Because K ÷ is the major os- moticum of the matrix, it can be inferred that matrix volume in vivo is 2.0 to 2.5 t imes that of mitochondria suspended in a 272 mosm medium.

The conversion of the matrix from the resting state to the actively respiring condensed state involves approximately a twofold change

in its total volume. 82 Estimating the total matrix volume as the sum of the volumes of osmotical-active water matrix (1.65 #l/mg protein), s° osmotical-inactive water (0.28 #l/mg protein) 8° and matrix proteins (0.50 #l/rag protein), a value of 2.43 #l /mg protein was obtained. (Matrix contains about 67% of the mitochondrial proteins. 83 Considering a specific volume for proteins of 0.74 lzl/mg, a volume for matrix proteins of 0.50 #l/rag of protein is obtained). A 50% contraction of this volume involves loosing 1.21 #1 of osmotically active matrix, which gives 0.44 #1 osmotically active matrix/mg protein under active respiring conditions.

In physiological situations mitochondria are expected to be at an intermediate state between resting and active respiration. 84 Therefore, an intermediate v a l u e - - 1 #1 osmotically active matrix/mg pro te in - - was used in the model.

APPENDIX 3. ESTIMATION OF THE CONCENTRATION OF P-LIPID IN MITOCHONDRIAL INNER MEMBRANES

The concentration of pure phosphatidylcholine from egg lecithin is 1 M. l j7 Phosphatidylcholine accounts for 40.5% of the P-containing lipids in mitochondria 9 and its density/molar mass ratio is similar to that of the other most abundant phospholipids in mitochondrial membranes. Sphingomyelin, which has a very different structure, accounts for only 2.4% of mitochondrial phospholipids, and cholesterol is almost absent from the inner membrane. 9 Therefore, 1 M seems a

reasonable estimate for the concentration of pure mitochondrial phospholipids in a membrane without other components. A variety of estimates 1Is-n° indicate that proteins account for 1/2 to 1/3 of the area of mitochondrial membranes. Therefore, the concentration of phospholipids in the mitochondrial inner membrane can be estimated as 0.5 to 0.7 M. Hence, a reference concentration of 0.6 M was used in the model.

APPENDIX 4. ESTIMATION OF THE VOLUME RATIO

A ratio between the volume of the mitochondrial inner membrane and the volume of osmotically active matrix around 0.2 can be estimated in two different ways: (a) The concentration of 1 mg total mitochondrial protein//.tl osmotically active matrix was estimated in Appendix 2. Twenty-one percent of mitochondrial protein is present in the inner membrane. 83 Hence, assuming a specific volume of 0.71 -0 .75 #l/rag for proteins, a volume of 0.15 - 0.16 #1 inner membrane protein/#l osmotically active matrix can be estimated. The volume of lipid in the membrane can be estimated as 0.074 #l//zl osmotically active matrix, assuming a concentration of

BETWEEN MEMBRANE AND AQUEOUS PHASES

0.34 #mol P-containing lipid/mg membrane protein, ° and a molar volume of 11 l i p id /mol - - a s for pure phosphatidylcholine from egg lecithin. "7 Therefore, considering the contributions of lipid and protein, the above-mentioned ratio was calculated as 0.22 to 0.23 #1 inner membrane/#l osmotically active matrix. (b) The area of the inner membrane of rat liver mitochondria has been estimated as 2 4 0 m2/g mitochondrial protein, t29'13° and its thickness as 2 5 nm. Considering again 1 mg of mitochondrial protein/#l osmotically active matrix, the volume ratio comes as 0.2 #1 inner membrane//A osmotically active matrix.

APPENDIX 5. TREATMENT OF COMPARTMENTATION

In an homogeneous reactional system, the rates of change of the concentrations can be related to the rates of the reaction by:

= NV, (1)

where X stands for the vector of concentrations of the dynamical (internal) species, N stands for the stoichiometric matrix, and V stands for the vector of rates of the reactions. We will now seek a generalisation of this expression for multicompartment systems.

For the ith species, equation (1) can be written as

J ( i = ~ y i j v i , i e { l , 2 . . . . . n} (2) j - I

with Yo standing for the stoichiometric coefficient of the ith species in thej th reaction, X~ standing for the concentration of the ith species, and vj standing for the rate of the j th reaction. Equation (2) may, in turn, be decomposed as

r_hL= ~ Ec~ l h 0 Y0 i c { 1,2, n }

Ec, j=t Ec, ToE~ . . . . . (3)

where r~ stands for the rate of change of the number of moles of the ith species in the whole system, n0 stands for the rate of change of the number of moles of the ith species due to reaction j , E,_, stands for the spatial extension of the compartment containing species i, and Efj stands for the spatial extension of the compartment where reaction j is occurring.

In the homogeneous case, Ec, = E~ is the volume of the reactional medium. However, equation (3) can be readily generalized for the case of a multicompartment system. In the latter case, the E, may be different and have dimensions of area or length (cj could be, for instance, the surface of contact between two phases and E v would stand for the respective surface area). In the general case,

Fl i

P~ = E-'-~,

may be viewed as a local molar density of i in compartment ci, and


1 ho~ VJ=-- ToE, )

can be equated to a specific rate of reaction j. Therefore, equation (3) can be written as

E,.. ~ ) i = ~ / i j ~ I J j , ie {1,2 . . . . . n} (4)

j.-i ~c,,

and the matricial form corresponding to equation (1) is

O=diag[E,.,l INdiag[E,)]~, ie{1,2 . . . . . n} , je{1 ,2 . . . . . r} (5)

with diag[Ec,] and diag[E,)] standing for the diagonal matrices whose diagonal elements are the spatial extensions (volumes, areas, lengths) of, respectively, the compartments containing each of the species and the compartments where each reaction is occurring. (Note that by "compartments" we are referring not only to bulk phases but also to interfaces where some reactions may occur.)

So, to generalize equation (1) it is necessary to replace the concentrations for the corresponding molar densities, the rates of reactions for the corresponding specific rates, and the stoichiometric matrix by the "compartmentation matrix"

N~=diag[E,,] INdiag[E~], ie{1,2 . . . . . n}, j e{1 ,2 . . . . . r} (6)

943

constant of the reaction for the membrane,

E3 k appa = --:- K = F31K (7)

El

includes the geometric factor 1"31 with dimensions length -~ . This factor can be identified with the reciprocal of the thickness of the membrane and is, thus, the same for any membrane of the same thickness. On the other hand, the apparent rate constant for the water phase can be written as:

kapp'2 E3t¢ = F32ff El = E= = ~ F31K = FI2F~IK = FI2 kapp't (8)

F~,, the ratio between the volumes of lipid and membrane phases, is usually easy to estimate. In turn, k ~pp.~ could be estimated by measurements of rates of change of the concentrations in the membrane or correcting the rates of change in water phase by 1/F~>

If we consider another system consisting of the same kind of lipid membrane in an aqueous phase, the terms of the kinetic equations for A, B, C, and D accounting for the same interfacial reaction could be written as

--I'."PP'I P A P o , - r ' 12k app'I PAPB,

kaPp'l PAP8 and F'I2 ka;'p'l PAPS,

If a system has a simple geometry, at least some of the geomet- rical factors

E,~ F,~c, = E,-~

can be determined without explicit knowledge of the spatial extensions. For instance, if species i is contained in a cubic compartment, ci, immersed in another compartment and reaction j occurs at the interface, cj, between both compartments, then F,.e, = 6/l, with I standing for the side length of the cube.

According to the discussion above, the rate constants usually appearing in mass action multicompartment systems should be written as kc,j = F,;,.~Kj where K i represents a specific rate constant such that vj = KII" p~", j c { 1,2, r}. Ifci :# cj, k,~ is an apparent rate constant.

) i=l "' "' '

Unfortunately, this is not always appropriately taken into account in experimental determinations of rote constants in heterogeneous systems.

If at least one of the compartments has a reproducible and well-defined geometry, experimentally determined apparent rate constants may avoid the need of knowing some details of the geometry of the system. Within reasonable approximation, this is the case for biomembranes.

To illustrate this point we will consider a system containing a lipid membrane (compartment 1) and an aqueous phase (compartment 2) and where some interracial reactions occur. Consider, for instance, a mass action reaction of the type A + B ~ C + D where A and C are assumed to be present in the membrane and B and D are assumed to be present in the aqueous phase. This would contribute to the kinetic equations for A, B, C, and D with the terms

E3 E3 E3 E3

respectively. Here E~ and E2 stand for the volumes of the lipid and water phase, respectively, E3 stands for the surface area of the interface, and PA and PB stand for the molar concentrations of A and B in the respective compartments. (It is implicitly assumed that the membrane behaves as a bulk phase for A and C.) The apparent rate

where F'12 stands for the ratio between the volumes of lipid and membrane phases in the new system. Therefore, it is not necessary to know the thickness of the membrane.

ABBREVIATIONS

G P X - - " c l a s s i c " g l u t a t h i o n e p e r o x i d a s e

G S H - - r e d u c e d g l u t a t h i o n e

G S S G - - o x i d i z e d g l u t a t h i o n e

G S X - - a d d u c t o f g l u t a t h i o n e w i t h X

L ' - - c a r b o n - c e n t e r e d r a d i c a l m o i t e y

L H - - u n s a t u r a t e d m o i e t y

L H - L H - - c a r d i o l i p i n

L O H - - h y d r o x y m o i e t y

L O O ° - - p e r o x y l r a d i c a l m o i t e y

L O O H - - h y d r o p e r o x i d e m o i e t y

L y s o L H - - u n s a t u r a t e d l y s o p h o s p h o l i p i d

P H G P X - - p h o s p h o l i p i d h y d r o p e r o x i d e g l u t a t h i o n e

p e r o x i d a s e

P L A 2 - - p h o s p h o l i p a s e A2

S L - L H - - 1 - sa tu ra t ed ,2 -unsa tu ra t ed d i a c y l p h o s p h o l i p i d

T o c O ' - - o ~ - t o c o p h e r o x y l r a d i c a l

T o c O H - - a - t o c o p h e r o l

U b q H 2 - - u b i q u i n o l - 9 p l u s U b i q u i n o l - 1 0

U F A - - u n s a t u r a t e d f a t t y a c i d

X o - - o x i d i z e d f o r m o f s p e c i e s X

X r d - - r e d u c e d f o r m o f s p e c i e s X

[ X ] t o t - - s u m o f t h e c o n c e n t r a t i o n s o f all t h e f o r m s c o n -

s i d e r e d f o r s p e c i e s X

lipid peroxidation in mitochondrial inner membranes. i. an integrative kinetic model

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