Analysis of Cellular Exposure to Peroxynitrite in Suspension Cultures

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Chem. Res. Toxicol. 2003, 16, 920-932

Analysis of Cellular Exposure to Peroxynitrite in Suspension Cultures Nitesh Nalwaya† and William M. Deen*,†,‡ Department of Chemical Engineering and Biological Engineering Division, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 Received November 18, 2002

A mathematical model was developed to predict the intracellular concentrations of NO, O2-, and peroxynitrite in suspension cell cultures exposed to NO and/or peroxynitrite. Oxygen and CO2 were also considered. Steady state concentrations were computed as a function of radial position within an idealized spherical cell, with a distinction being made between cytosolic and mitochondrial values. Spatial variations in the intracellular concentrations of O2, CO2, and NO were found to be negligible. The extremely low membrane permeabilities for O2(estimated from lipid bilayer data) caused O2- to be consumed in the compartment in which it was generated (mitochondria or cytosol) and resulted in concentrations that depended on the generation rate and the concentrations of superoxide dismutase and NO in the individual compartments. Special attention was paid to the origins of intracellular peroxynitrite. Potential sources of peroxynitrite include intracellular generation in mitochondria and cytosol and (depending on the type of experiment) diffusion of extracellular peroxynitrite into the cell. The relative importance of extracellular and intracellular sources was estimated for a wide variety of conditions. The calculated mitochondrial concentrations were generally 5-10 times higher than the cytosolic values, and it was found that mitochondria may act either as sources or sinks for cytosolic peroxynitrite, depending on the experimental conditions. For the baseline conditions, including an NO concentration of 1 µM and no peroxynitrite in the medium, the cytosolic peroxynitrite concentration was estimated as ∼2 nM. The extracellular peroxynitrite concentration required to double the cytosolic level was ∼25 nM, and an extracellular concentration of ∼100 nM was needed to effect a 5-fold increase. For extracellular concentrations smaller than 25 nM, intracellular generation predominated.

Introduction Peroxynitrite (ONOO-) is a potent oxidant formed by the rapid reaction of NO with O2- (1). Nitric oxide is synthesized by endothelial cells, epithelial cells, macrophages, and neurons and is involved in the modulation of blood pressure, neural activity, and other physiological functions. It participates also in the immune response to infections. Activated macrophages synthesize NO at high rates (2, 3), creating local NO concentrations toxic enough to help kill invading microorganisms. Superoxide is produced as a byproduct of mitochondrial respiration and also by cytosolic enzymes such as xanthine oxidase (4-6). The frequent coexistence of NO and O2- implies that ONOO-, which is cytotoxic and mutagenic (7), will be formed throughout the body. Peroxynitrite can cause irreparable damage to DNA, including strand breaks and DNA-DNA cross-links (8, 9). It can initiate lipid peroxidation (10) and hydroxyl radical attack (11) and cause Tyr nitration (12). Peroxynitrite also leads to the opening of the permeability transition pore in the inner mitochondrial membrane (13, 14), an event linked to necrosis and apoptosis (15, 16). Because of these effects, ONOOhas been linked to cancer and to neurodegenerative disorders such as Alzheimer’s disease (17). * To whom correspondence should be addressed. Tel: (617)253-4535. Fax: (617)258-8224. E-mail: [email protected]. † Department of Chemical Engineering. ‡ Biological Engineering Division.

The probable involvement of NO and ONOO- in various pathophysiological processes has motivated numerous studies of their effects on cultured “target cells”, cells that synthesize little or no NO themselves. In such studies, NO has been supplied by diffusion through gas permeable membranes (18), decomposition of donor compounds (19), addition of NO-saturated solutions (20), or synthesis by “generator cells” (macrophages) in cocultures (21). The controlled delivery of peroxynitrite is made difficult by its extremely short half-life (∼0.04 s) (22). Nonetheless, preformed ONOO- has been introduced as a bolus dose (23, 24) or by continuous infusion of a stock solution into the culture medium (19, 23). Alternatively, ONOO- has been synthesized continuously in situ by combining NO delivery with enzymatic generation of O2(19, 25). In some systems, the spatial and temporal variations in the concentrations of NO, O2-, and ONOO- in the extracellular fluid are becoming fairly well-understood. For example, detailed analyses of diffusion coupled with NO oxidation chemistry are available for activated macrophages suspended using carrier beads (26) or grown on plates under stagnant media (27). In either arrangement, NO was found to be distributed widely within the liquid, whereas the reactivity of O2- and ONOO- caused them to be localized within a few micrometers of the cells where they were produced. However, there appear to have been no attempts to compute the corresponding

10.1021/tx025664w CCC: $25.00 © 2003 American Chemical Society Published on Web 07/02/2003

Cellular Exposure to Peroxynitrite in Suspension Cultures

Figure 1. Schematic of model system. (a) Representative target cell surrounded by culture medium, with enlargement showing a mitochondrion; (b) idealized spherical target cell with spherical mitochondria; and (c) model system at the mitochondrial length scale.

intracellular concentration profiles, either in macrophages or in target cells. For O2- and ONOO- especially, which have limited membrane permeabilities, the intracellular concentrations might be very different from those in the culture medium. Because DNA damage and many other pathophysiological events result from intracellular reactions, methods are needed to relate intracellular concentrations to the conditions of a cell culture experiment. The technical difficulty of measuring intracellular concentrations makes mathematical models an important tool in such efforts. In this study, we developed a model to describe intracellular reaction and diffusion rates of NO, O2-, and ONOO- in target cells suspended in a stirred culture medium. The reaction and diffusion of O2 and CO2 were also considered. Steady state concentrations were computed as a function of position within an idealized cell, with a distinction being made between cytosolic and mitochondrial values. To focus on intracellular events, the concentrations of the five key species in the bulk medium were assumed to be known. Special attention was paid to the origins of intracellular ONOO-. Potential sources of peroxynitrite include intracellular generation in mitochondria and cytosol and (depending on the type of experiment) diffusion of extracellular peroxynitrite into the cell. The relative importance of extracellular and intracellular sources was estimated for a variety of hypothetical conditions.

Model Formulation Overview. The model involves steady state processes occurring on two length scales, cellular and subcellular. Figure 1a is a schematic of a typical cell suspended in a stirred medium. The model geometry for the coarser length scale is shown in Figure 1b, in which the cell is idealized as a sphere of radius a () 10 µm) with cytosolic concentration C h i(r) for species i at radial position r. To account for the resistance to mass transfer between the cell and the bulk liquid (concentration C°i ), the cell is assumed to be surrounded by a stagnant film of thickness δ (e a) with concentration Ci(r). The finer length scale corresponds to a mitochondrion, of which there are ∼103 per cell. Mitochondria are important here because they act as localized sources for O2- and peroxynitrite and have internal concentrations which may differ greatly from those in the cytosol. As shown in Figure 1c, a mitochondrion is idealized as a sphere of radius b () 0.6 µm) with internal concentration C ˆ i(r). The external compartment in this case is the cytosol, and it is assumed

Chem. Res. Toxicol., Vol. 16, No. 7, 2003 921

that C h i is nearly constant on the mitochondrial length scale (as shown later). Critical to both the cellular and the mitochondrial models is that the respective spheres are bound by infinitesimally thin (∼0.01 µm) membranes of limited permeability. The cytosol and mitochondria together comprise some 80% of the cell volume (28), so that the other organelles are neglected. The cellular volume in our model was taken to be 20% mitochondria and 80% cytosol. Mitochondria are actually capsules 1-2 µm long and 0.5-1 µm wide (29), consisting of a matrix enclosed by an inner membrane and a space between the inner and the outer membranes. Because the outer membrane is freely permeable to molecules up to ∼5000 Da (28), by “intramitochondrial volume” we mean the matrix and by “mitochondrial membrane” we mean the inner membrane. The remainder of this section is organized as follows. The reactions considered are summarized first; the key reactants are O2, CO2, NO, O2-, and peroxynitrite. (The sum of ONOO- and ONOOH is denoted as Per and referred to as “total peroxynitrite” or just “peroxynitrite”. Similarly, “total superoxide” is the sum of O2- and HO2; however, at the pH values of interest, the amount of HO2 is negligible and a separate symbol for total superoxide is usually not needed. When it is essential to distinguish between O2- anion and total superoxide, such as with membrane transport, total superoxide is denoted as Sup.) The equations that govern the concentrations of the key species in each of the regions (mitochondria, cytosol, and film) are then stated, completing the general formulation. It is shown next that most of the 15 differential equations (five species in three regions) can be reduced to algebraic expressions and that only four differential equations (two each for the intracellular space and film) need be solved numerically. The section closes with a description of the numerical methods applied to the simplified model. Chemical Reactions. The most important reactions for our purposes are (i) respiratory consumption of O2 and generation of CO2 in mitochondria; (ii) production of O2- in mitochondria and cytosol; (iii) consumption of O2- by superoxide dismutase (SOD) in mitochondria and cytosol; (iv) reaction of NO with O2- throughout, to yield peroxynitrite; (v) decomposition of peroxynitrite throughout, catalyzed largely by CO2; and (vi) consumption of NO and O2 by dioxygenase activity in cytosol. The consumption of O2 occurs mainly in the electron transfer steps catalyzed by enzymes at the inner surface of the inner mitochondrial membrane. Because of the extensive invaginations of this membrane (cristae) that project into the mitochondrial matrix (29), we modeled O2 consumption as being uniformly distributed inside the mitochondria. Under ordinary culture conditions, there is sufficient oxygen to make the rate of O2 consumption independent of its concentration. Likewise, CO2 production was assumed to be spatially uniform within the mitochondria, with a rate equal to that of respiratory O2 consumption. The formation of O2- in mitochondria occurs in the respiratory electron transfer chain (30), so that it was assumed to be uniformly distributed, like O2 consumption and CO2 formation. Superoxide production in the cytosol by enzymes such as xanthine oxidase (4) was modeled as a uniform source in that region. We also allowed for the possibility of continuous (e.g., enzymatic) O2- generation in the culture medium, a feature of certain experi-

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Nalwaya and Deen

ments. A major pathway for intracellular O2- consumption is dismutation, which is catalyzed by manganese SOD (MnSOD) in mitochondria and copper SOD (CuSOD) in cytosol. The dismutation reaction was written as k1,Mn/k1,Cu

2O2- + 2H2O 98 O2 + H2O2 + 2OH-

(1)

where the respective rate constants for the two forms of SOD are k1,Mn and k1,Cu. The concentrations of the SODs in the respective compartments were assumed to be uniform. When NO is present, O2- will also react with it to form peroxynitrite (1, 31) k2

NO + O2- 98 ONOO-

(2)

where the rate constant is k2. Because NO readily penetrates cells, reaction 2 will occur in the cytosol, in the mitochondria, and (if there is extracellular O2-) in the culture medium. The intracellular competition between reactions 1 and 2 depends on the relative concentrations of SOD and NO. It was assumed that NO is supplied mainly by the medium. The nature of the actual NO source (physical addition, chemical release, or synthesis by other cells in a coculture) was not considered; rather, the NO concentration in the bulk liquid was simply specified. On the basis of reports of a mitochondrial NO synthase (32, 33), a constant mitochondrial source term was also included. The peroxynitrite anion formed by reaction 2 is in equilibrium with peroxynitrous acid (ONOOH). The fraction of total peroxynitrite that is present as the anion is

f)

1 1 + 10pKPer-pH

(3)

where KPer is the acid dissociation constant; the fraction in the acid form is 1 - f. The acid decomposes readily to NO3- and NO2-: k4

ONOOH 98

{

NO3- + H+ NO2- +

1 2 O2

+ H+

has been reported recently. The reaction is represented as k6

2NO + 2O2 + NAD(P)H 98 2NO3- + NAD(P)+ + H+ (6) The oxygen-dependent cellular consumption of NO was also noted in hepatocytes by Thomas et al. (20), who proposed a rate expression that is first order in both NO and O2. Thus, intracellular oxidation of NO to NO3- may compete with peroxynitrite formation. Although there is also an uncatalyzed reaction of NO with O2 that leads to NO2- (sometimes referred to as autoxidation), it is too slow to affect NO or O2 concentrations on cellular or subcellular length scales (27). Governing Equations. With the assumption of a steady, spherically symmetric concentration field, the conservation equation for species i in the cytosol is

(

The formation of NO3 accounts for 70% of the total ONOOH decomposed (i.e., the rate constant for that pathway is 0.7k4) (34). Peroxynitrite anion also decomposes to NO3- but via a different pathway: k5

ONOO- + CO2 + H2O 98 NO3- + CO32- + H+ (5) Under physiological conditions, the CO2-catalyzed pathway (reaction 5) predominates (22). In addition, peroxynitrite reacts with NO to form NO2- (34), but under the conditions of interest here, that reaction is much slower than reactions 4 and 5 and can be neglected. Peroxynitrite will also react with cellular constituents such as lipids and proteins. Although difficult to quantify, this was included in some simulations by increasing the rate constant for peroxynitrite consumption above what is implied by reactions 4 and 5. The cellular consumption of NO by a heme- and flavindependent dioxygenase (35), with NO3- as the product,

(7)

where D h i is the effective cytosolic diffusivity (adjusted for obstructions due to intracellular organelles) and mi is the rate of entry by diffusion from the mitochondria, per unit cell volume (i.e., mol s-1 m-3). The remaining term is the rate of formation by chemical reaction. Because R h i is based on cytosolic volume, it is multiplied by the fraction of cell volume occupied by cytosol; φ is the mitochondrial volume fraction. R h i has contributions (positive for sources and negative for sinks) from all homogeneous reactions involving species i. Using mi to represent cytosolmitochondrial exchange relies on the fact that mitochondria are much smaller than cells, allowing them to be approximated as continuously distributed sources or sinks. The differential equations for the mitochondria and the liquid film are analogous to eq 7 but without mi or the factor 1 - φ. Equation 7 and its analogues were applied to O2, CO2, NO, total superoxide, and total peroxynitrite. At the cell center, symmetry requires that

dC hi (0) ) 0 dr

(4)

-

)

hi D h i d 2 dC r + mi + (1 - φ) R hi ) 0 2 dr dr r

(8)

An analogous boundary condition applies at the center of a mitochondrion. Matching of concentrations where the liquid film meets the bulk solution gives

Ci (a + δ) ) C°i

(9)

If no reactions occur where a mitochondrion contacts the cytosol or the cell meets the liquid film (i.e., neglecting heterogeneous reactions), species fluxes across those spherical surfaces must be continuous. At the cell membrane, that equality of fluxes leads to

-D hi

dC hi dCi (a) ) P h i[C (a) (10) h i(a) - κjiCi(a)] ) - Di dr dr

where P h i is the cell membrane permeability, κji is the ratio of intracellular to extracellular concentrations at equilibrium, and it is assumed that there are no intramembrane reactions. For the dissolved gases (O2, CO2, and NO), it was assumed that κji ) 1. Superoxide and peroxynitrite were assumed to diffuse across the mem-


branes only in their electrically neutral forms (HO2 and h i for ONOOH, respectively). Hence, the values of κji and P total superoxide and total peroxynitrite were influenced by transmembrane pH differences. Expressions for κji and P h i are derived in the Appendix. A relation analogous to eq 10 was written also for the flux at the membrane of a mitochondrion situated at a given distance from the center of the cell. The relation was written in mitochondrial coordinates (Figure 1c), with the radius of the cell in eq 10 replaced by the radius of the mitochondria (i.e., a replaced by b). Also, the cytosolic quantities were replaced by the corresponding mitochondrial ones and the film region quantities were replaced by cytosolic ones. Thus, for a mitochondrion situated at a distance r′ from the center of the cell, the external concentration was that in the cytosol at radial position r′ in the cellular/cytosolic coordinates. The reaction rate expressions inside the mitochondria were

R ˆ O2 ) - kˆ resp ) - R ˆ CO2

(11)

ˆ NO C ˆ O2R ˆ NO ) kˆ g,NO - kˆ 2 C

(12)

ˆ NO C ˆ O2- - k1,Mn C ˆ SOD C ˆ O2- (13) R ˆ O2- ) kˆ g,O2- - k2 C R ˆ Per ) k2 C ˆ NO C ˆ O2- - k4(1 - ˆf) C ˆ Per - k5 ˆf C ˆ CO2 C ˆ Per (14) where kˆ resp is the rate of O2 consumption by respiration, kˆ g,NO and kˆ g,O2- are the generation rates of NO and O2-, respectively, and ˆf is obtained from eq 5 by using the mitochondrial pH. In the cytosol, where the dioxygenase reaction was included, the rates for O2 and NO were

h NO C h O2 R h O2 ) -k6 C

(15)

R h NO ) -k2 C h O2- C h NO - k6 C h NO C h O2

(16)

In this region, CO2 production was neglected, and the rate equations for O2- and peroxynitrite were similar to the mitochondrial expressions (eqs 13 and 14). In the film region, the reaction rates for O2 and CO2 were zero, NO and O2- each had the usual consumption term corresponding to peroxynitrite formation, and the peroxynitrite expression was again like eq 14. When it was desired to represent enzymatic generation of O2- in the culture medium (film region and bulk solution), a term kg,O2- was added to the O2- expression. Inspection of the governing equations reveals that within any region, the problems for the five species are coupled by the reaction rates, and for any given species, the problems in the three regions are coupled by the membrane flux conditions. Thus, computing the concentration fields requires, in principle, that all 15 boundary value problems be solved simultaneously. One approach would be to first solve the mitochondrial-scale problems based on an assumed set of cytosolic concentrations and to use the resulting C ˆ i values to provide first estimates of mi. The cytosol and film problems could be solved then, and the updated C h i used at the mitochondrial level to improve the estimates of mi. The need to compute position-dependent mi values for all five species would make such an iterative approach extremely cumbersome,

Chem. Res. Toxicol., Vol. 16, No. 7, 2003 923 Table 1. Physicochemical Constants (at 37 °C) quantity

value

ref

109 M-1 s-1

2.5 × 1.8 × 109 M-1 s-1 1.6 × 1010 M-1 s-1 4.5 s-1 2.9 × 104 M-1 s-1 4.8 6.75 2.8 × 10-9 m2/s 2.3 × 10-9 m2/s 5.1 × 10-9 m2/s 2.8 × 10-9 m2/s 2.6 × 10-9 m2/s 0.4 m/s 0.054 m/s 0.9 m/s 4.2 × 10-8 m/s 4.2 × 10-9 m/s 1.6 × 10-5 m/s 2.4 × 10-6 m/s 0.40 9.94 0.51 6.76

k1,Cu k1,Mn k2 k4 k5 pKSup pKPer DO2 DCO2 DNO DO2DPer PO2 PCO2 PNO P h Sup P ˆ Sup P h Per P ˆ Per κjSup κˆ Sup κjPer κˆ Per

65 66 31 67 22 68 67 69 see text 70 26 26 71 72 73 see text see text see text see text see text see text see text see text

however. Fortunately, as discussed next, the magnitudes of the lengths, diffusivities, reaction rate constants, and other parameters justify some major simplifications. Parameter Values and Approximations. 1. Dimensionless Groups. The approximations used were based on dimensionless parameters which measure rates of reaction relative to diffusion (Damko¨hler numbers) or rates of membrane permeation relative to diffusion (membrane Biot numbers). For the cell of radius a, the Damko¨hler number for an nth-order reaction (Dan) and the Biot number (Bi) were defined as

Dan )

kn(κj Cext)n-1 a2 D h

, Bi )

P ha D h

(17)

where kn is the reaction rate constant, Cext is the concentration immediately outside the cell, and the subscripts to denote species i have been omitted. For a mitochondrion, b replaced a as the length scale and the physicochemical quantities for the mitochondrion replaced those for the cytosol and cell membrane. For the film, aqueous diffusivities and rate constants were needed, but a remained a suitable length scale. The most pertinent Damko¨hler numbers are Da0 and Da1, because the important reactions turn out to be zeroorder, first-order, or pseudo-first-order. As shown in the Appendix, for steady reaction and diffusion in a sphere surrounded by a membrane, if Da0 , 1, Da0 , Bi, Da1 , 1, and Da1 , Bi, then the concentration inside the sphere will be uniform and equal to κCext. We term this result, which is the same as if both reactions were absent, the “slow reaction limit”. If the first-order reaction is slow (Da1 , 1) but the zero-order reaction is not, a constant internal concentration will result if Bi , 1 (eq A4). Even if neither reaction is slow compared to diffusion, the internal concentration will be uniform, such that C ) k0/ k1, if the permeability is low enough (eq A5). We term this the “low permeability limit”. If Da0 and Da1 are not small but Bi . xDa1, the internal concentration is not uniform but there is no transmembrane gradient, a more modest but still useful simplification. 2. Parameter Values. Except where noted, all calculations were based on the parameter values in Tables 1

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Table 2. Parameters for Target Cell Suspension Culture parameter C°O2 C°CO2 C°NO C h SOD C ˆ SOD k6

value 210 µM 1.07 mM 1 µM 30 µM (3-60 µM) 6 µM (1-15 µM) 1 × 105 M-1 s-1

parameter kˆ resp kˆ g,NO k h g,O2kˆ g,O2kg,O2-

value 100 µM/s 0.5 µM/s 0.3 µM/s (0-1 µM/s) 1.0 µM/s (0.5-3 µM/s) 0 (0-3 µM/s)

and 2. Table 1 lists values of the reaction rate constants, diffusivities, and membrane permeabilities. In the absence of specific data, the aqueous diffusivities for O2and peroxynitrite were assumed to equal those of O2 and NO3-, respectively (26). The diffusivity of CO2 reported by Gros and Moll (36) at 22 °C was increased by a factor of 1.34 to correct it to 37 °C; this correction was obtained by assuming that Diµ/T is constant, where µ is the viscosity of water. Cytosolic and mitochondrial diffusivities for the dissolved gases were taken to be two-thirds of their aqueous values, as has been found for the intracellular diffusivity of O2 (37). However, for O2- and peroxynitrite in the cytosol, where mitochondria act as obstructions (mitochondrial Bi , 1, as discussed below), the diffusivity was reduced by an additional 30%, based on the effective diffusivity in a suspension of impermeable spheres with φ ) 0.2 (38). The membrane permeability coefficients were based on data for lipid bilayers. The permeability for total superoxide (PSup) at pH 7.3 is 2.1 × 10-8 m/s (39), which, from eq A9, gave PHO2 ) 6.7 × 10-6 m/s. Likewise, the permeability for total peroxynitrite (PPer) at pH 7.4 is 8 × 10-6 m/s (40), which gave PONOOH ) 4.4 × 10-5 m/s. These values of PHO2 and PONOOH were substituted in eq A9 to get the superoxide and peroxynitrite permeabilities of the cytosolic and mitoh Per, P ˆ Sup, and P ˆ Per). In chondrial membranes (i.e., P h Sup, P calculating the anion fractions in the three compartments (as used in the permeability and reaction rate expressions), the pH values in the medium, cytosol, and mitochondria were generally assumed to be 7.4, 7.0, and 8.0 (28), respectively. At these pH values, 81.7, 64.0, and 94.7% of total peroxynitrite was present as the anion (ONOO-), respectively. The corresponding values for superoxide were 99.7, 99.4, and 99.9%. The κ values for total superoxide and total peroxynitrite were calculated from eq A8. Baseline values for the more biological and/or experiment-specific parameters are given in Table 2. For several of the more uncertain parameters, ranges are also shown. The bulk concentrations of O2 and CO2 correspond to PBS in equilibrium with 21% O2 and 5% CO2 at 37 °C. The NO concentration is representative of values achieved using either suspension cultures of macrophages (41) or a delivery system employing gas-permeable tubing (42). The values chosen for the SOD concentrations in cytosol and mitochondria are both within the reported range of ∼4 to 60 µM (6, 43-45). The 5:1 ratio of cytosolic to mitochondrial SOD concentration follows from the finding that ∼90% of intracellular SOD activity is cytosolic (CuSOD) (45) and the relative volumes of the two compartments. The rate for the dioxygenase-catalyzed consumption of NO varies greatly among mammalian cells; the highest reported is that for rat hepatocytes (20, 35). The value shown for k6 is that for the hepatocyte data (20), adjusted for cell volume. The O2 consumption rate used, kˆ resp ) 100 µM/s (based on mitochondrial volume), is in the typical range for mammalian cells (46,

Nalwaya and Deen Table 3. Dimensionless Parameters for Mitochondriaa species O2 CO2 NO O2Per

Da0 10-4

10-5 10-4 1 to ∞ 10-3-10-2

Da1

Bi

0 0 10-4-10-3 1-10 4 × 10-3

100 20 200 10-6 7 × 10-4

a The dimensionless parameters (Da , Da , and Bi) are as 0 1 defined in eq 17 but using the mitochondrial radius and physicochemical constants. Da0 and Da1 are the intrinsic rates of reaction (zero-order and first-order, respectively) relative to the rate of diffusion inside a mitochondrion. Likewise, Bi is the rate of membrane permeation relative to the rate of diffusion. Small or large values indicate that certain rate processes, or certain resistances, have a negligible effect on the mitochondrial concentrations (see text).

Table 4. Dimensionless Parameters for Cytosola species

Da0

Da1

Bi

O2 CO2 NO O2Per

10-3 10-4 10-3 103 to ∞ 10-2 to ∞

10-2 0 1 103-104 1

2 × 103 300 3 × 104 10-4 10-1

a The dimensionless parameters (Da , Da , and Bi), which are 0 1 as defined in eq 17, compare various rate processes (reactions, diffusion, and membrane permeation) on the cellular length scale. See Table 3 and text for additional explanation.

47) and corresponds to ∼6 nmol/min/106 cells. The generation rate for NO in mitochondria was chosen as 0.5% of the O2 consumption rate (48). The data on H2O2 production in liver cells (5) were used to estimate the fraction of oxygen that is converted to superoxide in the mitochondria and cytosol. For mitochondria, it was assumed that all of the H2O2 generated was from superoxide, whereas the cytosolic O2- generation rate was estimated by assuming that 20% of the H2O2 generated was from superoxide. The latter value is based on a study which showed that at neutral pH, only 20% of the H2O2 generated by xanthine oxidase is formed with O2- as an intermediate (49). The final estimates for mitochondrial and cytosolic superoxide generation were 1.0 and 1.2% of oxygen consumption, respectively. In our base case, O2- generation in the medium was assumed to be absent (i.e., kg,O2- ) 0). As already mentioned, we chose a ) 10 µm, b ) 0.6 µm (the radius of a sphere of equivalent volume), and φ ) 0.2. We also chose δ ) a, which corresponds to gentle stirring. Using the baseline parameter values, Da0, Da1, and Bi were estimated for each of the five key species in the mitochondria and cytosol, as shown in Tables 3 and 4. The resulting simplifications are discussed next, together with the expressions for mi. 3. Oxygen and Carbon Dioxide. For O2 in mitochondria, Da0 ) 10-4, Da1 ) 0, and Bi ) 100 (Table 3), making the slow reaction limit easily applicable. This implies that respiratory consumption notwithstanding, the O2 concentration in mitochondria will equal that in the adjacent cytosol. The mitochondria-cytosol exchange term in eq 7 must match respiration, so that

mO2 ) -φ kˆ resp

(18)

In the cytosol, O2 loss to the mitochondria creates an apparent zero-order rate constant of k0 ) φ kˆ resp. The corresponding value of Da0 is ∼10-3 (Table 4). The value of Da1 in the cytosol, based on the pseudo-first-order rate



constant k1 ) k6C°NO, is also small (∼10-2). Together with Bi ∼ 103, this indicates that the slow reaction limit applies again; that is, the O2 concentration in the cytosol is uniform and equal to that just outside the cell. In the film, the O2 reactions (including that with NO) are negligible. Also, the oxygen transport into the cell produces a negligible concentration drop in the film (∼5 nM). Hence, the concentration is inferred to be uniform once more. Thus, the O2 concentration throughout the system will equal that in the bulk medium, and no differential equations need to be solved for O2. The same conclusions apply to CO2, which also has small Damko¨hler numbers and large Biot numbers at both length scales (Tables 3 and 4). In all of the estimates of Da0 for O2 and CO2, the concentration in the bulk medium was used to evaluate Cext, which is justified a posteriori by the finding that the concentrations are uniform. 4. Nitric Oxide. For NO in mitochondria, the pseudoˆ O2-. As discussed below, first-order rate constant is k2 C C ˆ O2- is uniform and ranges from 10 to 100 pM, depending on the assumed O2- generation rate and mitochondrial SOD concentration. It follows that Da1 for NO is at most 10-3 (Table 3). Because Da0 (based on mitochondrial NO generation) also does not exceed 10-4 and Bi ) 200, the NO concentration in mitochondria is uniform and equal to that in the adjacent cytosol, as for O2 and CO2. The rate of NO addition to cytosol from mitochondria equals the net rate of NO generation in mitochondria, so that

C ˆ O2- ) 37 pM. Another consequence of the low O2permeability is that mO2-) 0, to good approximation. At the cytosolic length scale, where Table 4 shows that Da0 (>103) and Da1 (>103) are both large but Bi is small (10-4), the low permeability limit (eq A5) applies again. The expression for C h O2- is similar to eq 20, except with cytosolic values:

mNO ) φ (kˆ g,NO - k2 C ˆ O2- C ˆ NO)

(19)

In the cytosol, the apparent zero-order rate constant for NO is k0 ) φ kˆ g,NO, which yields Da0 ∼ 10-3 (Table 4). The most important contribution to the overall first-order rate constant tends to be oxidation to NO3-, so that Da1 for NO in Table 4 was based on k1 ) k6C°O2. Because the upper bound for k6 (Table 2) yields Da1 ∼ 1, whereas Bi ) 3 × 104, the cytosolic NO concentration is not always uniform. For such situations, the differential equations for the cytosol and film were solved numerically (see below), the only approximation being that the transmembrane concentration difference was neglected. For smaller Da1, the intracellular NO concentration could be equated with the bulk value, as for O2 and CO2. Although the NO concentration is not quite constant at the cellular level, the variations are small enough that setting Cext ) C°NO, as done here, only slightly underestimates the mitochondrial or cytosolic values of Da0. 5. Superoxide. The reactions of O2- with NO and SOD in mitochondria can be lumped into a single pseudo-firstorder term. The resulting values of Da1 range from 1 to 10 (Table 3), depending on the concentration of MnSOD, which could range from ∼1 to 15 µM. The low permeability makes Bi small (10-6), whereas Da0 is large (g1). (How much Da0 might exceed unity depends on the cytosolic concentration of O2-, which is estimated to be 10 µm) for three values of k6, the rate constant for the dioxygenase reaction. As already mentioned, k6 ) 1 × 105 M-1 s-1 corresponds to the highest NO consumption rate reported in the literature (for hepatocytes). Even with that upper bound, there was just an 8% drop in the NO concentration across the film and a further 10% drop

Nalwaya and Deen

Figure 3. Superoxide concentrations in mitochondria and cytosol as a function of the respective SOD concentrations, in the absence of NO or with a bulk NO concentration of 1 µM.

within the cell (curve a). In other cell types that have been studied, the NO consumption rates are roughly 1050% of those in hepatocytes (35), suggesting that for most cells the NO concentration profiles would fall between curves b and c in Figure 2. Accordingly, in the remaining calculations, the intracellular NO concentration was simply equated with that in the bulk culture medium, and only the peroxynitrite problems were solved. Other factors, such as the uncertainty in SOD concentrations, have much more important effects on the conclusions than does this approximation for NO. Superoxide Concentration. As discussed in the model formulation section, the O2- concentrations in the mitochondria and cytosol are each predicted to be spatially uniform, although different from one another. Equations 20 and 21, with the cytosolic NO concentration equated with that in the bulk medium, were used to determine the mitochondrial and cytosolic O2- concentrations, respectively. Figure 3 shows the dependence of the O2- concentrations on the respective SOD concentrations, at two bulk NO concentrations. Because it speeds O2consumption, an increase in the SOD concentration in either compartment tends to reduce the corresponding O2- concentration. If NO is absent (solid curves), the O2concentration is calculated to be quite large for SOD concentrations < 1 µM. If NO is present (dashed curves), the O2- consumed in forming peroxynitrite lowers the O2concentration in either compartment. However, if the SOD levels are sufficiently high, NO can no longer compete with SOD for O2-, and the effect of NO disappears. The amount of O2- converted to peroxynitrite depends on the relative rates of its reaction with SOD and NO, which in turn depend on the relative concentrations of SOD and NO. For the baseline concentrations (Table 2), 60% of the O2- in mitochondria is converted to peroxynitrite, but just 18% percent is converted in the cytosol. Overall, 37% of the intracellular O2- is converted to peroxynitrite. Also, for every mole of peroxynitrite generated in the cytosol, 2.8 mol are calculated to be generated in the mitochondria. These results stress the importance of distinguishing mitochondrial from cytosolic reactions. As mentioned earlier, the membrane permeability values used here are those for lipid bilayers. To the extent that anion-specific channels are present in a given cell type, they may cause the actual permeabilities for total


Figure 4. Peroxynitrite concentration profiles when there is no peroxynitrite present in the bulk culture medium. Results are shown for the baseline parameters and for three multiples of the baseline cell membrane permeability.

Figure 5. Peroxynitrite concentration profiles assuming peroxynitrite generation in the culture medium at a rate of 1 µM/ s. Results are shown for the baseline parameters and for three multiples of the baseline cell membrane permeability.

superoxide and total peroxynitrite to exceed those in Table 1. There is evidence that both superoxide and peroxynitrite diffuse through the anion channels of erythrocytes (51-53). However, results for peroxynitrite suggest that anion channel blockers reduce its permeability by less than 1 order of magnitude (52), despite the fact that erythrocytes are unusually rich in band 3 proteins (anion channels). The “low permeability approximation” applied here to superoxide would remain accurate even if the actual permeability was increased 1000-fold over the lipid bilayer value. Thus, we concur with Fridovich (54) that neglecting the permeation of superoxide anion across biological membranes is a good approximation. The effects of increasing the peroxynitrite permeability are discussed below. Peroxynitrite Concentration. The O2- and NO concentrations were substituted into the peroxynitrite equations to determine the rates of generation of peroxynitrite in the mitochondria and cytosol. The resulting peroxynitrite concentration profiles in the cytosol for various cases are plotted in Figures 4 and 5. Figure 4 depicts situations in which peroxynitrite is absent from the bulk medium (i.e., no O2- or peroxynitrite generation in the bulk or film regions). For the base case (solid curve), the cytosolic concentration of peroxynitrite is


nearly uniform at ∼2.2 nM, and the extracellular concentration is negligible. The relative constancy of the cytosolic concentration and the small amount of peroxynitrite that escapes from the cell reflect the low permeability. Thus, for the base case, the peroxynitrite concentration in the cytosol is dictated by a balance between its generation and its consumption; the flux across the cell membrane is negligible. Increasing the peroxynitrite permeability, as shown by the dashed curves, lowers the cytosolic peroxynitrite concentration and accentuates its radial variations, while leading to more escape of peroxynitrite into the medium. The low permeability approximation remains fairly accurate for a 2-fold increase in the assumed permeability but not for 5- or 10-fold increases; with a 10-fold increase, the concentration in the liquid film is calculated to be ∼20% of the cytosolic value. Figure 5 shows peroxynitrite concentration profiles for an assumed bulk solution concentration of 37 nM, which could be obtained, in principle, by generating O2- in the bulk at a rate of 1 µM/s and simultaneously supplying NO. Also, activated macrophages have been calculated to generate concentrations of ∼60 nM in their vicinity (27). With a substantial concentration of peroxynitrite in the bulk solution, there is now an influx into the cytosol (i.e., opposite to what occurred in Figure 4). This influx elevates the cytosolic concentration to ∼5 nM for the baseline permeability, or roughly double what was calculated assuming no bulk peroxynitrite. Higher assumed permeabilities facilitate peroxynitrite entry into the cell, thereby increasing the cytosolic concentration, as shown. Diffusion of peroxynitrite into the cell requires that there be a concentration gradient in the film region, but the concentration there was found to vary by 0) or out of the cytosol (N h g,Per/k h Per. The flux is zero when C h Per ) κjPer C°Per ) k In addition to peroxynitrite exchange between the cytosol and the medium, there is exchange between the cytosol and the mitochondria. The concentration in the

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Figure 6. Peroxynitrite generation rates and transmembrane fluxes for two situations: (a) no bulk peroxynitrite (baseline condition of Figure 4) and (b) bulk peroxynitrite concentration of 37 nM (baseline condition of Figure 5). Cytosolic and mitochondrial generation rates are shown by vertical arrows, and fluxes are shown by horizontal arrows. The units in each case are 10-19 mol/s.

Figure 7. Cytosolic peroxynitrite concentration as a function of bulk NO concentration, at three values of total SOD concentration. The intermediate SOD concentration (25.2 µM) corresponds to the baseline concentrations in Table 2.

cytosol is determined collectively by cytosolic generation, fluxes to or from mitochondria, and the flux across the cell membrane. Figure 6 depicts the magnitude of these three rate processes for the base cases in Figures 4 and 5. The cytosolic and mitochondrial generation rates in both cases are 1.8 × 10-19 and 5.0 × 10-19 mol/s, respectively. When no bulk peroxynitrite is present (Figure 6a), peroxynitrite leaves the cell at a rate of 0.4 × 10-19 mol/s. The mitochondria collectively supply peroxynitrite at a rate of 0.3 × 10-19 mol/s. When bulk peroxynitrite is generated at a rate of 1 µM/s (Figure 6b), the fluxes reverse direction. Now peroxynitrite enters the cell at a rate of 2.7 × 10-19 mol/s. As discussed earlier, the cytosolic concentration rises due to this influx, and this rise leads to a reversal of the flux across the mitochondrial membranes. Thus, instead of supplying peroxynitrite, mitochondria consume it at a rate of 1.1 × 10-19 mol/s. These results show that mitochondria can be either net generators or net consumers of cytosolic peroxynitrite, depending on the experimental conditions. Effects of NO on Peroxynitrite Concentration. Figure 7 shows the dependence of cytosolic peroxynitrite concentration on bulk NO concentration for three levels of total SOD. The total cellular SOD concentration was computed by weighting the cytosolic and mitochondrial values by their respective volume fractions. The 1:5 ratio of MnSOD to CuSOD concentrations was maintained as total SOD was varied, and the bulk concentration of peroxynitrite was assumed to be zero. As shown in Figure

Nalwaya and Deen

7, the peroxynitrite concentration is predicted to become progressively less sensitive to NO as the NO concentration is increased. Eventually, at NO concentrations of ∼30 µM (well above those in Figure 7), peroxynitrite concentrations become nearly independent of NO. Because SOD competes with NO for O2-, higher SOD concentrations result in lower peroxynitrite levels. An interesting range of NO concentrations is 10-100 nM, which is sufficient to activate guanylate cyclase (43) but generally not toxic. In this range, the peroxynitrite concentration would be 0.04-0.4 nM for the base case SOD concentrations. If NO is not delivered to the culture medium and (as assumed in the target cell model) there is no NO synthase in the cytosol, then NO will be provided only by mitochondria. The concentrations of NO and cytosolic peroxynitrite calculated under these conditions were only 4 and 0.02 nM, respectively. Thus, mitochondrial NO production will be negligible under most experimental conditions and seems unlikely to have pathophysiological consequences. Effects of Bulk Peroxynitrite on Cytosolic Peroxynitrite. To analyze the extent to which peroxynitrite in the culture medium affects that in the cytosol, we compared spatially averaged cytosolic concentrations in the absence and presence of bulk peroxynitrite. From eq A1, the average cytosolic concentration when C°Per ) 0 is j avg h Per, where R j avg k h g,Per [1 - R Per]/k Per is the average value of R j Per(η). To obtain a simple relationship, normalized forms of the average cytosolic concentration (θ) and bulk concentration (θ°) were defined as

θ)

k h Per C h avg Per k h g,Per [1 - R j avg Per]

; θ° )

k h Per C°Per κjPer R j avg Per k h g,Per [1 - R j avg Per]

(29)

Thus, both concentrations were normalized with respect to the cytosolic value obtained in the absence of bulk peroxynitrite. By definition, θ ) 1 and θ° ) 0 when C°Per ) 0. It follows from eq A1 and these definitions that

θ ) 1 + θ°

(30)

It is seen that the cytosolic peroxynitrite concentration is a linear function of that in the bulk medium. To illustrate the dependence of cytosolic concentration on bulk concentration and also to show the sensitivity of the results to certain parameter values, we now consider seven situations: case A, base parameters; case B, 5-fold increase in cell membrane permeability to peroxynitrite; case C, 5-fold increase in mitochondrial membrane permeability to peroxynitrite; case D, doubling the rate constant for cytosolic consumption of peroxynitrite (an allowance for possible reactions with proteins and other constituents); case E, reduction of mitochondrial pH to 7.5; case F, increase in mitochondrial pH to 8.5; and case G, a 5-fold reduction in the superoxide generation rates in both mitochondria and cytosol. In the absence of bulk peroxynitrite, the cytosolic concentrations for these seven cases were found to be 2.2, 1.6, 2.5, 1.4, 3.6, 1.5, and 0.44 nM, respectively. Figure 8 shows the bulk peroxynitrite concentration that would effect a doubling of the cytosolic concentration, for the same seven situations. That is, the concentrations in Figure 8 are those that would make θ ) 2, for which θ° ) 1. More generally, eq 30 shows that θ ) n corresponds to θ° ) n - 1. Thus, the bulk concentration needed to triple the cytosolic level is twice


Figure 8. Bulk peroxynitrite concentrations which yield a cytosolic concentration that is twice that in the absence of external peroxynitrite. Case A corresponds to the baseline parameter values; see text for explanation of cases B-G.

the value in Figure 8, that needed to quadruple it is three times that in Figure 8, and so on. Figure 8 shows that for the base parameters (case A), a bulk peroxynitrite concentration of 27 nM is required to double the cytosolic concentration (from 2.2 to 4.4 nM). If the cell membrane permeability was five times higher (case B), the bulk concentration required to elevate the cytosolic concentration from 1.6 to 3.2 nM is just 6.3 nM. That is, a higher membrane permeability would make the cell more sensitive to external peroxynitrite. If the mitochondrial membrane permeability was five times higher (case C), the cytosolic concentration in the absence of external peroxynitrite would be elevated from 2.2 to 2.5 nM, as already indicated. This is because a higher mitochondrial permeability facilitates peroxynitrite entry into the cytosol from mitochondria. In this case, the bulk peroxynitrite concentration required to double the cytosolic value from 2.5 to 5 nM is 40 nM or 50% higher than for the baseline conditions. This increase is due to the fact that, as shown in Figure 6, mitochondria become consumers of cytosolic peroxynitrite when there is sufficient bulk peroxynitrite. Hence, part of the peroxynitrite that enters the cell is consumed by the mitochondria, and that process is accelerated by a higher mitochondrial membrane permeability. As a result, more bulk peroxynitrite is needed to double the cytosolic concentration. Case D in Figure 8 is a scenario in which additional reactions of peroxynitrite with cellular components are assumed to be as fast as the decomposition of peroxynitrite. As already indicated, this decreases the cytosolic concentration in the absence of bulk peroxynitrite from 2.2 to 1.4 nM. Nonetheless, the bulk peroxynitrite required to double the cytosolic concentration is nearly the same as for the base case (25 nM, as compared to 27 nM for case A). The effects of 0.5 unit decreases or increases in mitochondrial pH (cases E and F, respectively) are mediated mainly by the effects of pH on the parameter κˆ Per in the transmembrane flux expression. For case E, there is a 65% increase in the reference value of the cytosolic concentration (to 3.6 nM). This is because κˆ Per ) 2.3 at pH 7.5, as compared to the base value of κˆ Per ) 6.7 at pH 8.0. This decrease in κˆ Per results in a 4-fold increase in the peroxynitrite flux from mitochondria to cytosol and thus an increase in cytosolic concentration. To double the already high cytosolic concentra-


tion to 7.2 nM, more bulk peroxynitrite is needed (39 nM, Figure 8, case E). An increase in mitochondrial pH has the opposite effects. The value of κˆ Per is increased to 20.5 at pH 8.5, which reverses the direction of transport across the mitochondrial membrane and causes mitochondria to become consumers of cytosolic peroxynitrite. That drops the reference cytosolic concentration to 1.5 nM. To elevate that concentration to 3.0 nM, a bulk concentration of only 19 nM is required (Figure 8, case F). If O2- generation rates vary among different cell types or under different conditions in a given cell type, peroxynitrite generation rates will vary in proportion. Hence, when the superoxide generation rate is reduced 5-fold in both mitochondria and cytosol (case G), k h g,Per also decreases 5-fold. This leads to a proportionate decrease in the cytosolic peroxynitrite concentration (from 2.2 nM in the base case to 0.44 nM). The bulk concentration required to double the cytosolic value (5.5 nM, Figure 8, case G) is also one-fifth of that required in the base case. The linear relationship between the cytosolic peroxynitrite concentration and the superoxide generation rates applies also if the latter is elevated. If, for example, mitochondrial and cytosolic superoxide generation both doubled, the bulk peroxynitrite concentration needed to double the cytosolic value would itself double. It has been reported that exposure to NO may increase O2- production by mitochondria (55, 56) and mitochondrial O2production may also increase under pathological conditions (57). In two experimental studies in which O2- and peroxynitrite were generated in the bulk media, the reported rates of bulk superoxide generation were ∼0.007-0.1 µM/s (19, 25). Those rates would create bulk peroxynitrite concentrations of ∼0.2-4 nM. Most of the results in Figure 8 suggest that such external concentrations would not significantly affect cytosolic levels in target cells. Only if the cell membrane permeability was much higher than the baseline value (case B) or if superoxide generation was much lower (case G) would those bulk concentrations significantly perturb the intracellular levels. Other parameters that affect the sensitivity of the cytosolic peroxynitrite concentration to the bulk concentration include the relative concentrations of SOD and NO. As discussed already in connection with Figure 7, an increase in NO concentration or decrease in SOD concentration could also increase peroxynitrite generation. Concerning the effects of SOD, there is some uncertainty regarding the values of k1,Cu and k1,Mn. Certain anions, including Cl-, competitively inhibit SOD activity (58). At typical intracellular Cl- concentrations of 5-10 mM (59), this is estimated to decrease k1,Cu by 10-20%. However, relative to the uncertainty in the CuSOD concentration, this effect is negligible. The possible effects on SOD kinetics of other prominent intracellular anions (e.g., organic phosphates) appear not to have been studied. Although the discussion has focused on cytosolic peroxynitrite concentrations, the model can be used also to predict peroxynitrite concentrations in mitochondria. In general, it is found that mitochondrial concentrations are substantially higher than cytosolic ones; for the base case, the concentration inside the mitochondria is 18 nM or ∼8 times the cytosolic value. Knowing the relative magnitudes of the peroxynitrite concentrations in the cytosol and mitochondria may be helpful, for example, in identifying pathways responsible for cell death.

930


Addition of Preformed Peroxynitrite. In certain suspension culture experiments, peroxynitrite has been added to the medium at a constant rate (19, 23) instead of generating it in situ. In such systems, the steady state concentration in the bulk solution can be calculated by equating the rate of infusion with the rate of consumption in the bulk. Thus, if q denotes the rate of infusion of peroxynitrite in mol/s, the concentration in the bulk solution is (q/V)/[k4 (1 - f) + k5 f C°CO2], where V is the solution volume. The bulk concentration calculated in this manner can be substituted in the model to determine the film region and intracellular concentration profiles. Because there is no generation of peroxynitrite in the film in this case, the consumption term makes the concentration nonuniform there. Assuming gentle stirring (δ ) a, as usual), there is a ∼25% drop in the concentration in going from the bulk solution to the cell surface. This implies that for the same bulk concentration, the actual extracellular exposure in the infusion system is 25% lower than with continuous synthesis in the medium. By modulating the infusion rate, peroxynitrite concentrations in the micromolar range can be achieved. At such high concentrations, the influx of peroxynitrite from the culture medium into the target cells would be much greater than its generation inside the cells. Experiments have also been done with bolus addition of peroxynitrite to the culture medium (23, 24), but our model cannot be used for such situations. Because the time scale for decomposition of peroxynitrite (∼0.05 s) is comparable to that for its diffusion throughout a target cell (∼0.03 s), the added peroxynitrite does not survive long enough to permit application of a pseudo-steady model. Conclusion. In summary, a model was developed to predict the intracellular concentrations of NO, O2-, and peroxynitrite in suspension cell cultures exposed to NO and/or peroxynitrite. Although there have been previous models of NO reaction and diffusion in cells and tissues (60-64) and in the extracellular fluid surrounding activated macrophages (26, 27), none have considered intracellular O2- and peroxynitrite. Another novel feature of the present model is the separate treatment of cytosolic and mitochondrial reactions. By providing a way to correlate intracellular concentrations with those in the culture medium, the model should be helpful in designing toxicity and mutagenicity experiments that achieve desired levels of exposure. In particular, it yields estimates of the amounts of peroxynitrite derived from internal and external sources. Although the superoxide and peroxynitrite concentration estimates given here are necessarily tentative, due to uncertainties in parameters such as the SOD concentrations and superoxide generation rates, the model provides a framework that can be used to refine those estimates as more data become available. A logical extension of this work would be to model the peroxynitrite exposure of target cells cocultured with macrophages on plates (21). A more ambitious goal would be to simulate the situation in inflamed tissues, where cells next to an activated macrophage would “see” preformed peroxynitrite, but more distant cells would be exposed only to NO, with consequent intracellular generation of peroxynitrite.

Appendix Prototypical ReactionsDiffusion Problem. An analysis of diffusion in a sphere with zero- and first-order

Nalwaya and Deen

reactions provides guidance as to when certain approximations in intracellular models will be valid. It is assumed that in a sphere of radius a, bounded by a thin membrane of permeability P, a constant rate of production (k0) of some solute is accompanied by first-order consumption (rate constant k1). The internal concentration is C(r), and that just outside the sphere is Cext. The governing equations are similar to eq 7 (with mi ) 0 and φ ) 0), eq 8, and the left-hand part of eq 10. The dimensionless groups are as defined in eq 17. Using relative radial position (η ) r/a) and setting λ2 ) Da1, the exact solution to this linear boundary value problem is

C(η) ) Cext κ R(η) + R(η) )

k0 [1 - R(η)] k1

(

(A1)

)

Bi sinh λη (A2) η λ cosh λ + (Bi - 1) sinh λ

Within the function R(η) are the effects of Da1 and Bi, as well as the dependence on radial position. Two special cases in which C is independent of η are an unreactive solute (Da0 ) Da1 ) 0), for which R ) 1 and C ) Cextκ, and an impermeable membrane (Bi ) 0), for which R ) 0 and C ) k0/k1. As will be shown, C is constant also in several other situations. If the first-order reaction is slow compared to diffusion (λ2 , 1), the hyperbolic functions in eq A2 can be expanded to obtain simpler expressions for R and 1 - R. Neglecting terms of order λ2 relative to unity, eq A1 becomes

C(η) ) Cext κ

[

]

6Bi + BiDa0(1 - η2) + 2Da0 2(3Bi + λ2)

(A3)

If the zero-order reaction is also slow compared to diffusion (Da0 , 1) and permeation is fast compared to reaction (Bi . Da0 and Bi . λ2), then C ) Cextκ, as for an unreactive solute. This is the situation for O2 and CO2 at either length scale and for NO in mitochondria (all with κ ) 1). If the permeability is small (Bi , 1), eq A3 reduces to

C ) Cext κ

[

]

3Bi + Da0 3Bi + λ2

(A4)

This approximation applies to peroxynitrite in mitochondria. A constant internal concentration is obtained also if neither reaction is slow, provided that Bi is small enough. If Bi , λcoth λ - 1 and Bi , (λcoth λ - 1) (Da0/Da1), then R , 1 and

C)

k0 k1

(A5)

which is the same as for an impermeable sphere. This case, in which the concentration is determined by a balance between formation and consumption, and diffusion across the membrane is negligible, applies to O2- in mitochondria and cytosol. In judging how small Bi must be for eq A5 to hold, it is helpful to notice that λcoth λ 1 f λ2/3 as λ f 0, and λcoth λ - 1 f λ as λ f ∞. Thus,


the first restriction amounts to Bi , λ2/3 for small λ and Bi , λ for large λ. Membrane Flux Expressions. If an acid (HA) is in equilibrium with its anion (A-), as with the ONOOH/ ONOO- and HO2/O2-pairs, then it is advantageous to deal with their total concentration (i.e., CT ) CHA + CA). Likewise, it is their total flux (e.g., N hT ) N h HA + N h A-) that is of greatest interest. Evaluating membrane fluxes as in eq 10, the total flux across the cell membrane is

N hT ) P h T(C h T - κjT CT)

(A6)

The value of κjT, which determines when equilibrium is reached, depends on the mechanism for diffusion. If transmembrane diffusion is entirely in the acid form, then

h T - κjT CT) ) P h HA(C h HA - CHA) P h T(C

(A7)

Using fractional anion concentrations (fh in cytosol and f in culture medium), it follows from eq A7 and the definition of total concentration that

κjT )

1-f 1 - hf

h HA(1 - hf ) P hT ) P

(A8) (A9)

where hf and f are calculated by using the appropriate pKa and pH values in eq 3. Because the cytosolic pH is somewhat lower than in the cell surroundings, hf < f and κjT < 1. The corresponding results for the mitochondrial memˆ T) are obtained by substituting mitochonbrane (κˆ T and P drial for cytosolic quantities and cytosolic for culture medium quantities in eqs A8 and A9.

Acknowledgment. An early version of the intracellular model (without separate mitochondrial and cytosolic compartments) was formulated by Stephanie Homer. This work was supported by a grant from the National Cancer Institute (PO1-CA26731).

References (1) Huie, R. E., and Padmaja, S. (1993) The reaction of NO with superoxide. Free Radical Res. Commun. 18, 195-199. (2) Hibbs, J. B., Taintor, R. R., Vavrin, Z., and Rachlin, E. M. (1988) Nitric oxide: a cytotoxic activated macrophage effector molecule. Biochem. Biophys. Res. Commun. 157, 87-94. (3) Marletta, M. A., Yoon, P. S., Iyengar, R., Leaf, C. D., and Wishnok, J. S. (1988) Macrophage oxidation of L-arginine to nitrite and nitrate: nitric oxide is an intermediate. Biochemistry 27, 87068711. (4) Fridovich, I., and Handler, P. (1961) Detection of free radicals generated during enzymic oxidations by the initiation of sulfite oxidation. J. Biol. Chem. 236, 1836-1840. (5) Boveris, A., Oshino, N., and Chance, B. (1972) The cellular production of hydrogen peroxide. Biochem. J. 128, 617-630. (6) Tyler, D. D. (1975) Polarographic assay and intracellular distribution of superoxide dismutase in rat liver. Biochem. J. 147, 493504. (7) Tamir, S., and Tannenbaum, S. R. (1996) The role of nitric oxide (NO) in the carcinogenic process. Biochim. Biophys. Acta 1288, F31-F36. (8) Tretyakova, N. Y., Burney, S., Pamir, B., Wishnok, J. S., Dedon, P. C., Wogan, G. N., and Tannenbaum, S. R. (2000) Peroxynitriteinduced DNA damage in the supf gene: correlation with the mutational spectrum. Mutat. Res. 447, 287-303. (9) Burney, S., C., N. J., Dedon, P. C., and Tannenbaum, S. R. (1999) DNA damage in deoxynucleosides and oligonucleotides treated with peroxynitrite. Chem. Res. Toxicol. 12, 513-520.

Chem. Res. Toxicol., Vol. 16, No. 7, 2003 931 (10) Radi, R., Beckman, J. S., Bush, K. M., and Freeman, B. A. (1991) Peroxynitrite induced membrane lipid peroxidation: the cytotoxic potential of superoxide and nitric oxide. Arch. Biochem. Biophys. 288, 481-487. (11) Beckman, J. S., Beckman, T. W., Chen, J., Marshall, P. M., and Freeman, B. A. (1990) Apparent hydroxyl radical production from peroxynitrite: implications for endothelial injury by nitric oxide and superoxide. Proc. Natl. Acad. Sci. U.S.A. 87, 1620-1624. (12) Beckman, J. S. (1996) Oxidative damage and tryosine nitration from peroxynitrite. Chem. Res. Toxicol. 9, 836-844. (13) Scarlett, J. L., Packer, M. A., Porteous, C. M., and Murphy, M. P. (1996) Alterations to glutathione and nicotinamide nucleotides during the mitochondrial permeability transition induced by peroxynitrite. Biochem. Pharmacol. 52, 1047-1055. (14) Packer, M. A., Scarlett, J. L., Martin, S. W., and Murphy, M. P. (1997) Induction of the mitochondrial permeability transition by peroxynitrite. Biochem. Soc. Trans. 25, 909-914. (15) Zoratti, M., and Szabo, I. (1995) The mitochondrial permeability transition. Biochim. Biophys. Acta 1241, 139-176. (16) Hirsch, T., Marzo, I., and Kroemer, G. (1997) Role of the mitochondrial permeability transition pore in apoptosis. Biosci. Rep. 17, 67-76. (17) Smith, M. A., Harris, P. L. R., Sayre, L. M., Beckman, J. S., and Perry, G. (1997) Widespread peroxynitrite mediated damage in alzheimer’s disease. J. Neurosci. 17, 2653-2657. (18) Tamir, S., Lewis, R. S., Walker, T. D., Deen, W. M., Wishnok, J. S., and Tannenbaum, S. R. (1993) The influence of delivery rate on the chemistry and biological effects of nitric oxide. Chem. Res. Toxicol. 6, 895-899. (19) Pfeiffer, S., Schmidt, K., and Mayer, B. (2000) Dityrosine formation outcompetes tyrosine nitration at low steady-state concentrations of peroxynitrite: Implications for tyrosine modification by nitric oxide/superoxide in vivo. J. Biol. Chem. 275, 6346-6352. (20) Thomas, D. D., Liu, Z. P., Kantrow, S. P., and Lancaster, J. R. (2001) The biological lifetime of nitric oxide: implications for the perivascular dynamics of NO and O2. Proc. Natl. Acad. Sci. U.S.A. 98, 355-360. (21) Zhuang, J. C., Lin, D., Lin, C., Jethwaney, D., and Wogan, G. N. (2002) Genotoxicity associated with NO production in macrophages and co-cultured target cells. Free Radical Biol. Med. 33, 94-102. (22) Lymar, S. V., and Hurst, J. K. (1995) Rapid reaction between peroxynitrite ion and carbon dioxide: Implications for biological activity. J. Am. Chem. Soc. 117, 8867-8868. (23) Espey, M. G., Miranda, K. M., Thomas, D. D., and Wink, D. A. (2002) Ingress and reactive chemsitry of nitroxyl derived species within human cells. Free Radical Biol. Med. 33, 827-834. (24) Di Stasi, A. M. M., Mallozi, C., Macchia, G., Maura, G., Petrucci, T. C., and Minetti, M. (2002) Peroxynitrite affects exocytosis and snare complex formation and induces tryosine nitration of synaptic proteins. J. Neurochem. 82, 420-429. (25) Kavdia, M., Stanfield, J. L., and Lewis, R. S. (2000) Nitric oxide, superoxide, and peroxynitrite effects on the insulin secretion and viability of beta tc3 cells. Ann. Biomed. Eng. 28, 102-109. (26) Chen, B., Keshive, M., and Deen, W. M. (1998) Diffusion and reaction of nitric oxide in suspension cell cultures. Biophys. J. 75, 745-754. (27) Chen, B., and Deen, W. M. (2001) Analysis of the effects of cell spacing and liquid depth on nitric oxide and its oxidation products in cell cultures. Chem. Res. Toxicol. 14, 135-147. (28) Alberts, B., Bray, D., Lewis, J., Raff, M., Roberts, K., and Watson, J. D. (1994) Molecular Biology of the Cell, pp 553, 656, 665-666, Garland Publishing, New York. (29) Tzagoloff, A. (1982) Mitochondria, pp 15-16, Plenum Press, New York. (30) Boveris, A., and Cadenas, E. (2000) Mitochondrial production of hydrogen peroxide: Regulation by nitric oxide and the role of ubisemiquinone. IUBMB Life 50, 245-250. (31) Nauser, T., and Koppenol, W. H. (2002) The rate constant of the reaction of superoxide and nitric monoxide: reaching the diffusion limit. J. Phys. Chem. A 106, 4084-4086. (32) Giulivi, C., Poderoso, J. J., and Boveris, A. (1998) Production of nitric oxide by mitochondria. J. Biol. Chem. 273, 11038-11043. (33) French, S., Giulivi, C., and Balaban, R. S. (2001) Nitric oxide synthase in porcine heart mitochondria: Evidence for low physiological activity. Am. J. Physiol. 280, H2863-H2867. (34) Pfeiffer, S., Gorren, A. C. F., Schmidt, K., Werner, E. R., Hansert, B., Bohle, D. S., and Mayer, B. (1997) Metabolic fate of peroxynitrite in aqueous solutions. J. Biol. Chem. 272, 3465-3470. (35) Gardner, P. R., Martin, L. A., Hall, D., and Gardner, A. M. (2001) Dioxygen-dependent metabolism of nitric oxide in mammalian cells. Free Radical Biol. Med. 31, 191-204.

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(36) Gros, G., and Moll, W. (1974) Facilited diffusion of CO2 across albumin solutions. J. Gen. Physiol. 64, 356-371. (37) Ellsworth, M. L., and Pittman, R. N. (1984) Heterogeneity of oxygen diffusion through hamster striated muscles. Am. J. Physiol. 246, H161-H167. (38) Deen, W. M. (1998) Analysis of Transport Phenomena, pp 194, Oxford University Press, New York. (39) Takahashi, M., and Asada, K. (1983) Superoxide anion permeability of phospholipid membranes and chloroplast thylakoids. Arch. Biochem. Biophys. 226, 558-566. (40) Marla, S. S., Lee, J., and Groves, J. T. (1997) Peroxynitrite rapidly permeates phospholipid membranes. Proc. Natl. Acad. Sci. U.S.A. 94, 14243-14248. (41) Lewis, R. S., and Deen, W. M. (1994) Kinetics of the reaction of nitric oxide with oxygen in aqueous solutions. Chem. Res. Toxicol. 7, 568-574. (42) Wang, C., and Deen, W. M. (2003) Nitric oxide delivery system for cell culture studies. Ann. Biomed. Eng. 31, 65-79. (43) Beckman, J. S., and Koppenol, W. H. (1996) Nitric oxide, superoxide, and peroxynitrite: the good, the bad, and the ugly. Am. J. Physiol. 40, C1424-C1437. (44) Freeman, B. A., and Crapo, J. D. (1982) Free radicals and tissue injury. Lab. Invest. 47, 412-426. (45) Peetersjoris, C., Vandevoorde, A. M., and Baudhuin, P. (1975) Subcellular localization of superoxide dismutase in rat liver. Biochem. J. 150, 31-39. (46) Ruffieux, P. A., Stockar, U. V., and Marison, I. W. (1998) Minireview: Measurement of volumetric (OUR) and determination of specific (qO2) oxygen uptake rates in animal cell cultures. J. Biotechnol. 63, 85-95. (47) Metzen, E., Wolff, M., Fandrey, J., and Jelkmann, W. (1995) Pericellular Po2 and O2 consumption in monolayer cell cultures. Respir. Physiol. 100, 101-106. (48) Valdez, L. B., Arnaiz, S. L., Bustamante, J., Alvarez, S., Costa, L. E., and Boveris, A. (2000) Free radical chemistry in biological systems. Biol. Res. 33, 65-70. (49) Fridovich, I. (1970) Quantitative aspects of the production of superoxide anion radical by milk xanthine oxidase. J. Biol. Chem. 245, 4055-4057. (50) Press, W. H., Flannery, B. P., Teukolsky, S. A., and Vetterling, W. T. (1993) Numerical Recipes in C: the Art of Scientific Computing, pp 358-359, 762-772, Cambridge University Press, New York. (51) Lynch, R. E., and Fridovich, I. (1978) Permeation of erythrocyte stroma by superoxide radical. J. Biol. Chem. 253, 4697-4699. (52) Denicola, A., Souza, J. M., and Radi, R. (1998) Diffusion of peroxynitrite across erythrocyte membranes. Proc. Natl. Acad. Sci. U.S.A. 95, 3566-3571. (53) Macfadyen, A. J., Reiter, C., Zhuang, Y. X., and Beckman, J. S. (1999) A novel superoxide dismutase-based trap for peroxynitrite used to detect entry of peroxynitrite into erythrocyte ghosts. Chem. Res. Toxicol. 12, 223-229. (54) Fridovich, I. (1995) Superoxide radical and superoxide dismutases. Annu. Rev. Biochem. 64, 97-112. (55) Sarkela, T. M., Berthiaume, J., Elfering, S., Gybina, A. A., and Giulivi, C. (2001) The modulation of oxygen radical production by nitric oxide in mitochondria. J. Biol. Chem. 276, 6945-6949. (56) Poderoso, J. J., Carreras, M. C., Lisdero, C., Riobo, N., Schopfer, F., and Boveris, A. (1996) Nitric oxide inhibits electron transfer

Nalwaya and Deen

(57)

(58) (59) (60) (61) (62) (63) (64)

(65)

(66) (67)

(68) (69) (70) (71) (72)

(73)

and increases superoxide radical production in rat heart mitochondria and submitochondrial particles. Arch. Biochem. Biophys. 328, 85-92. Packer, M. A., and Murphy, M. P. (1995) Peroxynitrite formed by simultaneous nitric oxide and superoxide generation causes cyclosporin: a sensitive mitochondrial calcium efflux and depolarization. Eur. J. Biochem. 234, 231-239. Rigo, A., Stevanato, R., and Viglino, P. (1977) Competetive inhibition of Cu, Zn superoxide dismutase by monovalent anions. Biochem. Biophys. Res. Commun. 79, 776-783. Maffly, R. H. (1981) The body fluids: volume, composition, and physical chemistry. in The Kidney (Brenner, B. M., and Rector, F. C., Eds.) Vol. 1, pp 98, W. B. Saunders Company, Philadelphia. Lancaster, J. R. (1994) Simulation of the diffusion and reaction of endogenously produced nitric oxide. Proc. Natl. Acad. Sci. U.S.A. 91, 8137-8141. Lancaster, J. R. (1996) Diffusion of free nitric oxide. Methods Enzymol. 268, 31-50. Vaughn, M. W., Kuo, L., and Liao, J. C. (1998) Estimation of nitric oxide production and reaction rates in tissue by use of a mathematical model. Am. J. Physiol. 43, H2163-H2176. Vaughn, M. W., Kuo, L., and Liao, J. C. (1998) Effective diffusion distance of nitric oxide in the microcirculation. Am. J. Physiol. 43, H1705-H1714. Wood, J., and Garthwaite, J. (1994) Models of the diffusional spread of nitric oxide: Implications for neural nitric oxide signaling and its pharmacological properties. Neuropharmacology 33, 1235-1244. Fielden, E. M., Roberts, P. B., Bray, R. C., Lowe, D. J., Mautner, G. N., Rotilio, G., and Calabres, L. (1974) Mechanism of action of superoxide dismutase from pulse radiolysis and electron paramagnetic resonance: Evidence that only half active sites function in catalysis. Biochem. J. 139, 49-60. Forman, H. J., and Fridovich, I. (1973) Superoxide dismutase: A comparison of rate constants. Arch. Biochem. Biophys. 158, 396400. Koppenol, W. H., Moreno, J. J., Pryor, W. A., Ischiropoulos, H., and Beckman, J. S. (1992) Peroxynitrite: A cloaked oxidant formed by nitric-oxide and superoxide. Chem. Res. Toxicol. 5, 834-842. Fridovich, I. (1978) The biology of oxygen radicals. Science 201, 875-880. Goldstick, T. K., and Fatt, I. (1970) Diffusion of oxygen in solutions of blood proteins. Mass Transfer Biol. Syst. 66, 101-113. Wise, D. L., and Houghton, G. (1968) Diffusion coefficients of neon, xenon, carbon monoxide, and nitric oxide in water at 10-60 °C. Chem. Eng. Sci. 23, 1211-1216. Stroeve, P., Smith, K. A., and Colton, C. K. (1976) Steady-state diffusion of oxygen in red blood cell and model suspensions. AIChE J. 22, 1133-1142. Gutknecht, J., Bisson, M. A., and Tosteson, D. C. (1977) Diffusion of carbon dioxide through lipid bilayer membranes. Effects of carbonic anhydrase, bicarbonate, and unstirred layers. J. Gen. Physiol. 69, 779-794. Subczynski, W. K., Lomnicka, M., and Hyde, J. S. (1996) Permeability of nitric oxide through lipid bilayer membranes. Free Radical Res. 24, 343-349.

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