Multidimensional Diffusion Modes and Collision Frequencies of Cytochrome c with Its Redox Partners*

We have determined the modes and rates of cyto- chrome c diffusion as well as the collision frequencies of cytochrome c with its redox partners at the surface of the isolated, mitochondrial inner membrane over a broad range (0-150 mM) of ionic strengths. Using fluorescence recovery after photobleaching, resonance energy transfer, and direct binding assay, we determined that the diffusion coefficient of cytochrome c is independent of its concentration and quantity bound to the inner membrane, that the distance of cytochrome c from the membrane surface increases with increasing ionic strength, and that there is no significant immobile fraction of cytochrome c on the membrane regardless of ionic strength. The rate of cytochrome c diffusion increases while its mode of diffusion changes progres-sively from lateral to three-dimensional with increas- ing ionic strength. At physiological ionic strength (100-160 mM), the diffusion of cytochrome c is three- dimensional with respect to the surface of the inner membrane with a coefficient of 1.0 X 1O“j cm2/s, and little, if any cytochrome c is bound to the membrane regardless of its concentration. Furthermore, as ionic strength is raised from zero to 150 mM, the cytochrome c k,, for the inner membrane increases, its mean occu- pancy time on the inner membrane to collide with a redox partner (T) decreases, and its diffusion-based collision frequencies with its redox partners decrease. These data reveal the significance of both diffusion and concentration (affinity) of cytochrome c near the surface of the inner membrane in the control of the collision frequency of cytochrome c with its redox partners. DPH only in H, medium in the presence of cytochrome c was due to RET by the Forster mechanism. Therefore, the two components of DPH life-time were not resolved. Since the life-time of DPH remained unchanged in the presence and absence of cytochrome c at 150 mM KCl, the fluorescence intensity of DPH at 150 mM KC1 was used in the proximity studies as a reference, indicating no RET. Materials-Horse heart cytochrome c (type VI) and FITC was purchased from Sigma; DPH was purchased from Molecular Probes (Eugene, OR); and tetrahydrofuran (gold label) was purchased from Aldrich. All chemicals were reagent-grade.


Multidimensional Diffusion Modes and Collision Frequencies of
Cytochrome c with Its Redox Partners* (Received for publication, March 27,1987) Sharmila Shaila Gupte and Charles R. Hackenbrock From the Laboratories for Cell Bwlogy, Department of Cell Biology and Anatomy, University of North Carolina School of Medicine, Chapel Hill, North Carolina 27599 We have determined the modes and rates of cytochrome c diffusion as well as the collision frequencies of cytochrome c with its redox partners at the surface of the isolated, mitochondrial inner membrane over a broad range (0-150 mM) of ionic strengths. Using fluorescence recovery after photobleaching, resonance energy transfer, and direct binding assay, we determined that the diffusion coefficient of cytochrome c is independent of its concentration and quantity bound to the inner membrane, that the distance of cytochrome c from the membrane surface increases with increasing ionic strength, and that there is no significant immobile fraction of cytochrome c on the membrane regardless of ionic strength. The rate of cytochrome c diffusion increases while its mode of diffusion changes progressively from lateral to three-dimensional with increasing ionic strength. At physiological ionic strength (100-160 mM), the diffusion of cytochrome c is threedimensional with respect to the surface of the inner membrane with a coefficient of 1.0 X 1O"j cm2/s, and little, if any cytochrome c is bound to the membrane regardless of its concentration. Furthermore, as ionic strength is raised from zero to 150 mM, the cytochrome c k,, for the inner membrane increases, its mean occupancy time on the inner membrane to collide with a redox partner ( T ) decreases, and its diffusion-based collision frequencies with its redox partners decrease. These data reveal the significance of both diffusion and concentration (affinity) of cytochrome c near the surface of the inner membrane in the control of the collision frequency of cytochrome c with its redox partners.
Numerous observations support the concept that electron transfer between specific, nonstoichiometric redox components of the mitochondrial inner membrane is coupled to, i.e. preceded by, random, diffusion-based collisions between these components (1)(2)(3)(4)(5)(6)(7). Applying the technique of fluorescence recovery after photobleaching (FRAP)l to a homogenous (twodimensional) diffusion system, we have previously shown that the redox protein complexes and ubiquinone, all integral to the inner membrane, diffuse laterally in the membrane plane * This work was supported in part by National Institutes of Health Grant GM28704 and National Science Foundation Grant PCM84-02569. The costs of publication of this article were defrayed in part by the payment of page charges. This article must therefore be hereby marked "Odvertisement" in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.
The abbreviations used are: FRAP, fluorescence recovery after photobleaching; RET, resonance energy transfer by Forster mechanism; FITC, fluorescein isothiocyanate; DPH, 1,5-diphenyl-1,3,5-hexatriene; Hepes, 4-(2-hydroxyethyl)-l-piperazineethanesulfonic acid. at rates which are independent of ionic strength (7). In contrast to these redox components, cytochrome c is a basic redox protein that associates with the surface of the mitochondrial inner membrane (8,9). At neutral pH and low ionic strengths, cytochrome c interacts electrostatically at the membrane surface with its redox partners, cytochrome bc, and cytochrome oxidase, as well as with membrane lipids (10). Thus, unlike other redox components, cytochrome c can be removed from the mitochondrial inner membrane simply by washing in 150 mM KC1, i.e. at physiological ionic strength (11). For these reasons, the physical and functional interactions of cytochrome c with the mitochondrial inner membrane are understandably complex.
In this report, we report results which reveal that the rate of cytochrome c diffusion and its diffusion mode vary with ionic strength but are independent of cytochrome c concentration irrespective of ionic strength. We also report on the relative binding and proximity of cytochrome c to the membrane as a function of ionic strength using a direct binding assay and resonance energy transfer by the Forster mechanism (RET) (12). The data show that cytochrome c functions as a three-dimensional diffusant at a physiological (100-150 mM) ionic strength. From these data, we present a new treatment to calculate the diffusion-based collision frequencies for a heterogenous diffusion system where cytochrome c diffuses laterally, pseudo-laterally, or three-dimensionally while its redox partners diffuse in two dimensions at various ionic strengths, including physiological. The data reveal the significance of both the diffusion and concentration (affinity) of cytochrome c near the inner membrane surface in controlling its collision frequency with its redox partners. In the accompanying article (46), we report on the rates of electron transfer mediated by cytochrome c for each of its three diffusion modes and present the significance of its three-dimensional diffusion mode in the electron transport process in isolated inner membranes as well as in intact, whole mitochondria at physiological ionic strength.
FRAP and Binding Assays-As previously described (7). FRAP was carriedout on KC1-washed, Ca2+-fused mitochondrial inner membranes in a glass chamber. In the first binding assay, the glassattached membranes were labeled using 20 +1 of various concentrations of FITC-cytochrome c in appropriate buffer. The quantity of FITC-cytochrome c bound to the glass-attached membranes was detected fluorometrically (7). In the second binding assay, 2 or 20 nmol of native or FITC-horse heart cytochrome c were added to KClwashed membranes (5.0 mg/ml protein) in 0.3 mM Hepes (0.3 mM ionic strength), 10 mM KPi (23 mM ionic strength), and 25 mM KPi 5241 This is an Open Access article under the CC BY license.
(56 mM ionic strength) buffer in 1.0-ml total volume, incubated at room temperature for 5 min, and then centrifuged in a Beckman Microfuge Model B for 4 min at 9OOO X g. The pellets were solubilized with a final concentration of 0.8% d u m cholate and 100 mM K P , pH 7.4. The cytochrome c content of the pellet material and supernatants was determined by obtaining the difference spectra of ferricyanide-oxidized versus dithionite-reduced cytochrome c.
Preparation of Asolectin Vesicles-Asolectin (1 g) was hydrated in 5 ml of H,, medium for 1 h on ice and then sonicated on ice for 30 min at 10-min intervals using a Branson sonicator at a power setting of 5 (output = 35 watts) to form small, unilamellar vesicles (14). These vesicles were centrifuged at 235,000 X g for 60 min to remove titanium pieces and large, multilamellar vesicles. The supernatant containing small, unilamellar vesicles was used within 48 h.
Incorporation of DPH into Mitochondrial Inner Membranes-The concentration of DPH in tetrahydrofuran was determined spectrophotometrically using an extinction coefficient of 80,000 M" cm" at 350 nm. DPH was added slowly in 1-pl aliquots to inner membranes while stirring. DPH in aqueous medium is nonfluorescent; therefore, it was unnecessary to remove unbound DPH. The conditions for the DPH incorporation into inner membranes were: DPH:phospholipid ratio, 1:loO; concentration of the solvent tetrahydrofuran, <0.5%. Labeling was carried out at 0 "C while stirring for 60 min. The succinate oxidase activity of DPH-labeled inner membranes was approximately 85% compared to unlabeled inner membranes. DPHlabeled inner membrane were stored overnight and used within 24 h after labeling.
RET Memurements-RET measurements were performed essentially as described by Gupte and Lane (15). The intensity of DPH emission incorporated into inner membranes was monitored digitally at 410 nm using a Perkin-Elmer fluorescence spectrophotometer 650-40 in the ratio mode at 23 "C. The excitation wavelength for these measurements was 365 nm, and the slit width of both excitation and emission monochromators was 5 nm. For the determination of percent RET from membrane-bound DPH to cytochrome c as a function of ionic strength, KCl/Hm medium (300 mosM solution containing various concentrations of KC1 and H m medium buffered with 2 mM Hepes, pH 7.4) was utilized. The percent RET was calculated as (1 -(fluorescence intensity of DPH at a given KC1 concentration/fluorescence intensity of DPH at 150 mM KC1 concentration)) X 100 in the presence of cytochrome c. For an individual set of experiments, the percent RET for each KC1 concentration was measured in triplicate. For each cuvette, the digital output of fluorescence intensity was signal-averaged for 9 s. All samples in an individual set were statistically randomized. The distance (R,) for 50% RET and the actual distance R between the donor-acceptor pair of DPH-cytochrome c was calculated as described by Stryer (16). The spectral overlap integral ( J ) for this pair was calculated to be 7 X lo-", the refractive index was 1.4, and the orientation factor (E) was assumed to be 2/3 for the random orientation of donor and acceptor molecules. The quantum yield of DPH incorporated into the membrane was taken to be 0.8 (17).
The life-time of DPH was determined using a SLM 4800 phase modulation fluorescence spectrophotometer as described by Barrow and Lentz (18) using the following settings: excitation wavelength = 365 nm; emission monitored using a Schott KV 450 broad-band cutoff filter; slit width = 3 nm; and modulation frequency = 18 MHz. The decrease in the fluorescence life-time of membrane-bound DPH in the presence of cytochrome c was noted in H, medium, but not in 150 mM KCl. These observations were sufficient to verify that the decrease in the fluorescence intensity of membrane-incorporated DPH only in H, medium in the presence of cytochrome c was due to RET by the Forster mechanism. Therefore, the two components of DPH life-time were not resolved. Since the life-time of DPH remained unchanged in the presence and absence of cytochrome c at 150 mM KCl, the fluorescence intensity of DPH at 150 mM KC1 was used in the proximity studies as a reference, indicating no RET.
Materials-Horse heart cytochrome c (type VI) and FITC was purchased from Sigma; DPH was purchased from Molecular Probes (Eugene, OR); and tetrahydrofuran (gold label) was purchased from Aldrich. All chemicals were reagent-grade.

Membrane
Binding of Cytochrome c-When FITC-labeled cytochrome c was eluted on a CM52 column (19), five major peaks were obtained. Addition of peak IV FITC-cytochrome, which eluted immediately prior to the native cytochrome c peak, to inner membranes resulted in same succinate oxidase activity as native cytochrome c. Thus, peak IV appeared to consist of cytochrome c with a FITC-labeled lysine which is not significantly involved in either the electron transport process or binding to its redox partners; hence, peak IV was selected for FRAP studies.
The quantity of FITC-cytochrome c bound to inner membranes was determined by two separate methods (Table I). The first method consisted of a fluorometric measurement of the percentage of the total FITC-cytochrome c bound to the membranes in the FRAP chamber. The second method consisted of incubation of either FITC-cytochrome c or native cytochrome c with the membranes and subsequent centrifugation, followed by spectroscopic measurement of cytochrome c bound to the pelleted membranes. The data from these two diverse types of assays show that the amount of FITC-cytochrome c and native cytochrome c bound to the membranes decreased to the same extent as the ionic strength of the medium was increased.
Lateral and Pseudo-luteral Diffusion of Cytochrome c-We have previously determined that the rate of lateral diffusion of cytochrome c on the inner membrane is ionic strengthdependent (7). In this study and contrary to the results of Vanderkooi et al. (20), we determined that the rate of lateral diffusion of cytochrome c at different ionic strengths <56 mM is independent of the quantity of cytochrome c bound to the membrane, including physiological, and its concentrations (Table I). In addition, in our study, the percent fluorescence recovery of FITC-cytochrome c after photobleaching indicated no significant immobile fraction of cytochrome c on the membrane surface at any ionic strength or concentration. Experiments designed to measure the FRAP of FITC-cytochrome c at ionic strengths >56 mM were not possible since, as with native cytochrome c, very few FITC-cytochrome c molecules remain bound to the membranes for sufficient times at these ionic strengths to be detected by FRAP.

Assessment of Cytochrome c-Membrane Proximity
by RET-The proximity of cytochrome c to, or its average distance from, the membrane over a broad range of ionic strengths was determined by measuring the RET from the fluorescent probe DPH incorporated into the bilayer to the cytochrome c heme. DPH was utilized as a donor because of its ease of incorporation as well as high quantum yield in membranes (21). We determined the spectral overlap between DPH and the cytochrome c heme to be excellent and the extinction coefficient of the reduced and oxidized cytochrome c to be equal at 410 nm. Thus, the increase in intensity of DPH emission at 410 nm, using an excitation wavelength of 365 nm, was measured in the presence of cytochrome c at various ionic strengths to calculate the RET and hence, the average distance of cytochrome c from the membrane.
Initially, the RET from DPH, incorporated into the sonicated asolectin vesicle bilayer, to cytochrome c heme at various ionic strengths from 0 to 150 mM was measured to characterize the RET. We found the RET to decrease progressively with increasing ionic strength. The physiological concentration of free K+ and C1-ions in the cytoplasm is 100-150 mM (22), which was accepted as the physiological ionic strength in our studies. The KC1 concentration and ionic strength in the range of 0-150 mM KC1 were treated as equivalent by assuming the activity coefficients of K+ and C1to be equal to 1. The change in RET was identical for KC1 and NaCl at equivalent ionic strengths, which reveals that the change in the proximity of cytochrome c to the membrane is ionic strength-dependent and is not due to a specific ion effect. In addition, it was determined that RET was independent of the DPH:cytochrome c ratio (data not shown).
Life-time of DPH in the Bilayer-The fluorescence life-time of DPH, measured for various conditions, was identical when measured at zero or 150 mM ionic strength in the absence of cytochrome c, which showed that the life-time of DPH incorporated into the vesicle bilayer is independent of ionic strength (Table 11). In addition, the life-time of DPH remained unchanged at 150 mM ionic strength when cytochrome c was present in the aqueous medium, indicating that no RET from DPH occurred. In contrast Three-dimensional Diffusion of Cytochrome c at the Mitochondrial Inner Membrane Surface-The RET from DPH incorporated into isolated mitochondrial inner membranes to cytochrome c (cytochrome c concentration, 2 p~; cytochrome c:phospholipid ratio, 1:83) was monitored at various ionic strengths (Fig. 1). For this ratio, the RET decreased, and the average distance of cytochrome c from the inner membrane increased with increasing ionic strength.
In intact mitochondria, the concentration of cytochrome c is 100-700 p M depending on the mitochondrial configuration (23-25). However, due to the high absorbance of cytochrome c in the 410 nm region, the use of 100 or 700 p M exogenous cytochrome c with isolated inner membranes is not experi: mentally feasible for RET measurements. Therefore, various cytochrome c:phospholipid ratios were used with inner membranes for RET measurements in order to relate measurements on isolated inner membranes to the intact mitochondrion which contains a physiologically high cytochrome c concentration. A higher cytochrome c:phospholipid ratio resulted in a higher RET at each ionic strength (Fig. 2); and when these data were normalized such that RET at zero ionic strength was 100% for each cytochrome c:phospholipid ratio, the relative RET was identical for each ionic strength. These data show that the degree of RET is independent of the cytochrome c:phospholipid ratio and consequently, the cytochrome c concentration irrespective of ionic strength. Thus, by extrapolation, these results for the isolated inner membrane indicate that at physiological ionic strength, physiological concentrations of cytochrome c, and the physiological cytochrome c:phospholipid ratio as found in the intact mitochondrion, little, if any, cytochrome c is bound to the inner membrane. The percent binding of cytochrome c to inner membranes as determined by direct binding assay and the percent RET for the DPH-cytochrome c pair in inner membranes were compared as a function of ionic strength (Fig. 3). Again, the RET was maximum at zero ionic strength and decreased sharply when the ionic strength was increased. Significantly, at 50 mM ionic strength, RET revealed that 18% of the top1 cytochrome c molecules were within the RET range (100 A), whereas by direct binding assay, only 8% of the cytochrome c molecules were bound to the membrane surface (Fig. 3). therefore, 10% of the cytochrome c molecules within the RET range were not bound to the membrane but were diffusing near the membrane surface in three dimensions. Direct binding assays revealed that above approximately 100 mM ionic strength, virtually no cytochrome c was bound to the membrane surface; and for these ionic strengths, RET was negligible. These RET and direct binding data clearly show that cytochrome c diffusion is predominantly three-dimensional at physiological ionic strength. Significantly, it will be seen in the accompanying article (46) that electron transfer mediated by cytochrome c in the inner membrane is maximal at physiological ionic strength, i.e. when cytochrome c diffusion is three-dimensional.

DISCUSSION
Rates and Modes of Cytochrome c Diffusion-We determined previously that diffusion of cytochrome c at the mitochondrial inner membrane surface is characteristically lateral at low ionic strength and pseudo-lateral at intermediate ionic strength, with its rate of diffusion increasing with increasing ionic strength (7). All other redox components diffuse only laterally, the rates of which are independent of ionic strength (7). Studies from other laboratories (20, 26-28) specifically on the rate of lateral diffusion of cytochrome c are in reasonable agreement with ours. However, in contrast to our FRAP studies, Vanderkooi et al. (20) reported an approximately 50% immobile fraction for the lateral diffusion of a porphyrin cytochrome c at various ionic strengths. It was also reported that the rate of cytochrome c diffusion was concentrationdependent (20). The surprisingly high immobile fraction as well as concentration dependence are most likely due to an unnaturally strong binding (compared to native cytochrome c ) of the nonfunctional, perhaps denatured porphyrin cytochrome c to the membrane as determined in their work even at 150 mM salt concentration (20).
Based on measurements of lateral diffusion rates of cytochrome c similar to ours, but using estimations of cytochrome concentrations far lower than established by other laboratories (7, 29, 30), Ferguson-Miller and co-workers (26-28) concluded that lateral diffusion of cytochrome c is not sufficient to account for the rate of electron transport by a completely random diffusion mechanism. However, using their rates of lateral diffusion with correct cytochrome c concentrations,  have recently recalculated their diffusion data. These new calculations are consistent with our random collision model for electron transport (32).
The FRAP data presented in this study reveal that rates of lateral and pseudo-lateral diffusion of cytochrome c at nonphysiological, low ionic strengths become faster with increasing ionic strength irrespective of cytochrome c concentration. Furthermore, the equilibrium between the cytochrome c molecules at the surface of the inner membrane and those in the aqueous medium shifts toward free cytochrome c molecules in the aqueous medium, resulting in three-dimensional diffusion, as the ionic strength is raised to physiological.
Proximity of Cytochrome c to the Inner Membrane-RET has been utilized as a "spectroscopic ruler" toomeasure the distance between two chromophores up to 100 A (16). Moreover, RET has been used to determine changes in this distance due to changes in the interaction between the chromophores (33). We found that the RET from DPH incorporated into the mitochondrial inner membrane to cytochrome c decreases sharply as the ionic strength is raised; and at 50 mM ionic strength, the RET is only 18% of that at zero ionic strength. This decrease in RET as ionic strength is increased up to 150 mM is independent of different cytochrome c:phospholipid ratios for inner membranes. At physiological (100-150 mM) ionic strength, little, if any, cytochrome c molecules, regardless of cytochTome c concentration, are within the RET distance of 100 A. These RET data, combined with our cytochrome c direct binding assay data, reveal that cytochrome c molecules diffuse in three dimensions, colliding randomly with the inner membrane surface at physiological ionic strength.
In the intact, whole mitochondrion, the outer membrane is impermeable to cytochrome c (34) and represents a boundary, confining the three-dimensional diffusion of cytochrome c to the intermembrane space. The maximum concentration of cytochrome c in the intermembrane space is approximately 700 p~ (23-25). Even at this high concentration, cytochrome c physically occupies only 0.6% (v/v) of the intermembrane space. Three other intermembrane proteins have been identified at low concentrations in rat liver mitochondria; adenylate kinase, nucleoside diphosphokinase, and nucleoside monophosphokinase (9, 13). Considering these data, cytochrome c should be able to diffuse freely in the intermembrane space irrespective of its concentration or the presence of these proteins. The question of the physiological ionic strength of the intermembrane space of the intact mitochondrion is also important, but not usually considered. Since the outer membrane is freely permeable to most ions and low molecular weight metabolites, the ionic strength of the intermembrane space is most likely equal to that of the cytoplasm, i.e. 100-150 mM (22). Since with inner membranes, at ionic strength >90 mM, most of the cytochrome c, regardless of concentration, is outside the maximal RET distance of 100 A, the mode of cytochrome c diffusion should be three-dimensional in the intact mitochondrion at physiological, 100-150 mM ionic strength.
Rate of Three-dimensional Diffusion of Cytochrome c-The three-dimensional diffusion of cytochrome c in the mitochondrial intermembrane space raises the question of its role in electron transfer, e.g. is the rate of three-dimensional diffusion rate-limiting? The initial step to answer this question is to determine the rate of three-dimensional diffusion of cytochrome c at physiological ionic strength and then to calculate the frequency of its collisions with its redox partners. The collision frequency then can be compared with the electron transfer frequency. This comparison will be presented in the accompanying article (46).
The three-dimensional diffusion coefficient of cytochrome c in water, calculated from the Stokes-Einstein equation, is 1.5 X lo-' cm2/s. This diffusion coefficient compares closely to that of lysozyme, which is a small soluble protein ( A I r 14,400) similar to cytochrome c ( A I r 12,400). Studies on the three-dimensional diffusion of lysozyme, using interferometry, show that at low ionic strength and infinite dilution, its diffusion coefficient is 5.8 X lo-' cm2/s; at 700 p M concentration, it is 5.2 x lo-' cm'/s; at ionic strengths between 50 and 150 mM, it is 1.4 x cm'/s (35). Using FRAP, similar low cm'/s three-dimensional diffusion coefficients were obtained for lysozyme and other soluble proteins of similar size in phosphate-buffered saline, the ionic strength of which is approximately 150 mM (36). Thus, the lowest three-dimensional diffusion coefficient of cytochrome c at 150 mM ionic strength and at 700 p M concentration is approximately 1.0 X

Collision Frequency of Cytochrome c with Its Redox Partners-
We have previously measured the lateral diffusion of mitochondrial inner membrane redox components at ionic strengths lower than physiological and have calculated the bimolecular diffusion-controlled collision frequencies between specific redox partners from these measurements using the Hardt equation (37) for a homogenous diffusion system comprised of two-dimensionally diffusing redox partners (7). It should be pointed out that the Hardt equation for two dimensions assumes that all colliding reaction partners are inert, hard spheres confined to the membrane, but does not consider change in affinity of any reaction partner for the membrane surface at different ionic strengths. These assumptions are reasonable, and our previous calculations are adequate for the collision frequencies of cytochrome c at the lower ionic strengths where diffusion is essentially two-dimensional.
Since we have now determined the modes and rates of cytochrome c diffusion for ionic strengths from 0 to 150 mM, we can calculate the bimolecular diffusion-controlled collision cm2/s. frequencies of cytochrome c diffusing in two and/or three dimensions with its two-dimensionally diffusing redox partners, i.e. a heterogenous diffusion system, using the treatment of Astumian and Chock (38). This treatment also includes the affinities of cytochrome c for the inner membrane which apply at different ionic strengths. Accordingly, we can describe three collisional modes for cytochrome c, diffusing in three dimensions, with its membrane-restricted redox partners, diffusing laterally. 1) Some cytochrome c molecules collide directly with their redox partner molecules; 2) some cytochrome c molecules collide nonspecifically with the membrane and remain adsorbed long enough to diffuse laterally or pseudo-laterally to collide with their redox partner molecules; 3) some cytochrome c molecules, without collision with their redox partner molecules, dissociate from the membrane surface such that the location of a subsequent collision is approximately independent of their previous locations on the membrane. The relative contribution of each of these paths can be calculated by a branching method for the membrane surface dynamics of these paths.
The probability that a collision between cytochrome c and a redox partner can occur depends upon various factors: the fraction of membrane surface covered by the membranerestricted redox partner, the rates of lateral and three-dimensional diffusion of cytochrome c, the rates of lateral diffusion of the redox partners, the mean occupancy time of cytochrome c on the membrane, the rate of the nonspecific dissociation constant of cytochrome c with the membrane, the radius of the membrane, the concentration of cytochrome c and redox partner; and the radius of the reactive area of cytochrome c and redox partner. Collectively, these factors contribute to the probability of one of the collisional modes being more favorable for collision.
Specifically, the probably that a collision between a molecule of cytochrome c and a membrane-restricted redox partner molecule will occur is dependent on both 7 and kd (Table 111). 7 is defined as the mean occupancy time for cytochrome c on the membrane to collide with a redox partner (Table 111, sixth column) and is derived from the experimental diffusion coefficients of cytochrome c and its redox partner (fourth column), as well as from the effective quantity of the redox partner (second column) and other parameters (see Table 111, Footnote c ) . It has been shown that the kd of cytochrome c with various membrane types increases with increasing ionic strength (41-45). Compiling the data from these sources, we have selected the kd of cytochrome c for the inner membrane as 1 at zero mM, 1 X io3 at 50 mM, and 1 X io9 at 150 mM ionic strength.' At low ionic strength (Table 111, third column), 7 values are long (sixth column), and kd values are low (seventh column); thus, cytochrome c tends to remain adsorbed to the membrane, and the probability of collision with either of its redox partners is maximum (eighth column). At high ionic strength, T values are short, and kd values are high; thus, cytochrome c tends to dissociate from the membrane, and the probability of collision with either of its redox partners decreases (eighth column). Using the experimentally determined 7 values and the kd values selected above, the values for the collision frequencies between cytochrome c and either of its redox partners at the three ionic strengths are obtained (ninth column).
The dissociation rate constant (kd) of cytochrome c for inner membranes at various ionic strengths was calculated from the equilibrium association and dissociation constants as well as from the association rates of cytochrome c for phospholipid vesicle membranes or for solubilized cytochrome oxidase reported in the literature cited above. Lateral diffusion coefficients of cytochrome c from Table I and its membrane-restricted redox partners (Ref. 7; D for cytochrome b c l = 4.3 X 10"O cmz/s, and D for cytochrome oxidase = 3.7 X cmz/s) are added (40) to calculate T . At zero ionic strength, the diffusion is treated as ideally lateral; at 50 mM ionic strength, the diffusion is pseudo-lateral. However, it is treated as an effective, two-dimensional, lateral diffusion (40). At 150 mM ionic strength, the diffusion of cytochrome c is essentially three-dimensional with D = 1.0 X lo-' cmz/s as determined in the text. This three-dimensional diffusion value of D represents the theoretical upper limit for lateral diffusion and therefore is used to calculate the highest collision frequencies possible. e Mean occupancy time for cytochrome c on the membrane. 7 = (1.1 ?HS/N&ZD)ln(1.2 ?HS/NB?B), where DZD = lateral diffusion of cytochrome c and its redox partners as described in Footnote b, rHs = average radius of the mitochondrial inner membrane (7.5 X cm), NB = number of membrane-restricted reactive redox partners as in the second column, and re = 2.5 X cm for both cytochrome bcl and cytochrome oxidase. Radii are from Gupte et al. (7).
Membrane surface affinity or nonspecific dissociation rate constant of cytochrome c. e Probability of collision of cytochrome c with either of its redox partners via three collisional modes as defined in text.p = (a + @(Ia ) ) / ( l -((1a ) ( l -@))y), where a = NBrsz/4rHsz, y = rHs/(rHs + re), and @ = 1/(1 + k d . oxidized cytochrome c molecules and A = 6.6 X 10'' reduced cytochrome c molecules). To understand better the effect of k d on the collision frequency as well as to verify the appropriateness of the k d values selected to determine the collision frequencies in Table 111, we have calculated the collision frequencies between the reduced cytochrome bcl-oxidized cytochrome c redox pair (Fig.  4) and the reduced cytochrome c-oxidized cytochrome oxidase redox pair (not shown) over a range of kd values for each of the three diffusion rates in Table I11 (fourth column). The data show that any kd below 1 X 10' results in a maximum collision frequency, whereas any above 1 x 108 results in a minimum collision frequency regardless of diffusion rate. This underscores the significance of kd in addition to the diffusion rate in determining diffusion-based collision frequencies. At higher ionic strengths, the higher k d values and shorter 7 values as well as the RET and binding data indicate that the lower collision frequencies are due to a lower concentration (affinity) of cytochrome c near the surface of the membrane. These relationships will be discussed further in the accompanying article (46) with respect to electron transport. These results also show the appropriateness of the three k d values (1, 1 X lo3, and 1 X 1 0 ' ) selected to calculate the collision frequencies in Table 111.
Our data reveal the significance of diffusion and concentration (affinity) of cytochrome c near the surface of the inner membrane in the control of its collision frequency with its redox partners at any ionic strength. Considering our experimental finding that cytochrome c diffuses in three dimensions at higher and physiological ionic strengths, the calculations reported in this study represent a more definitive approach to determine the collision frequencies for cytochrome c with its redox partners. Comparison of the theoretical collision frequency with the experimental rate of electron transport, reported in the accompanying article (46), will show the significance of the three-dimensional diffusion-based collision frequencies of cytochrome c with its redox partners in mediating electron transport in the isolated inner membrane as well as in the intact, whole mitochondrion at physiological ionic strength. Sinauer Associates Inc.,