## Abstract

A mathematical model describing facilitation of O_{2} diffusion by the diffusion of myoglobin and hemoglobin is presented. The equations are solved numerically by a finite-difference method for the conditions as they prevail in cardiac and skeletal muscle and in red cells without major simplifications. It is demonstrated that, in the range of intracellular diffusion distances, the degree of facilitation is limited by the rate of the chemical reaction between myglobin or hemoglobin and O_{2}. The results are presented in the form of relationships between the degree of facilitation and the length of the diffusion path on the basis of the known kinetics of the oxygenation-deoxygenation reactions. It is concluded that the limitation by reaction kinetics reduces the maximally possible facilitated oxygen diffusion in cardiomyoctes by ∼50% and in skeletal muscle fibers by ∼ 20%. For human red blood cells, a reduction of facilitated O_{2} diffusion by 36% is obtained in agreement with previous reports. This indicates that, especially in cardiomyocytes and red cells, chemical equilibrium between myoglobin or hemoglobin and O_{2} is far from being established, an assumption that previously has often been made. Although the “O_{2} transport function” of myoglobin in cardiac muscle cells thus is severely limited by the chemical reaction kinetics, and to a lesser extent also in skeletal muscle, it is noteworthy that the speed of release of O_{2} from MbO_{2}, the “storage function,” is not limited by the reaction kinetics under physiological conditions.

- myoglobin
- hemoglobin

myoglobin (Mb) and hemoglobin (Hb) can facilitate O_{2} diffusion by diffusion of their oxygenated forms that occurs in parallel to the diffusion of dissolved oxygen. This has been demonstrated first for Hb by Scholander (41), Hemmingsen and Scholander (18), and Wittenberg (45, 46) and thereafter by Wittenberg for both Hb and Mb (47) in solutions of these hemoproteins forming layers of 150-μm thickness. In a theoretical study, Wyman (49) showed that chemical reaction in conjunction with rotational protein diffusion could not be the mechanism of this facilitation because the chemical reaction rates are too slow but that the reaction rates are sufficiently fast to allow a facilitation by translational protein diffusion. This mechanism was confirmed later by experimental determinations of the hemoglobin diffusion coefficient in concentrated Hb solutions (31, 39).

As pointed out very early on by Moll (30) and Wyman (49), facilitated O_{2} diffusion can only be effective if the relaxation time of the chemical reaction between O_{2} and Hb or Mb, respectively, is considerably shorter than the relaxation time of diffusion. Since the latter depends on the length of the diffusion path, or the layer thickness, it follows that hemoprotein-mediated facilitated O_{2} diffusion will begin to decrease when the diffusional path length falls below a certain limit, whose value depends on the kinetics of the chemical reaction. This limit has been studied theoretically for the case of Hb-facilitated O_{2} diffusion by Kutchai et al. (23), who showed that the process starts to become limited by the speed of the chemical reaction, when the thickness of the layer of hemoglobin solution decreases below ∼100 μm. Consequently, they predicted for the average thickness of a red cell (1.6 μm) a strong limitation of facilitation by the reaction kinetics. Although the necessity of establishing also for Mb a relation between diffusion path length and degree of facilitation has been recognized by Wittenberg and Wittenberg (48) and by Gros et al. (17), no such calculation has been published so far. The information is clearly needed since in many estimations in the literature of the role of Mb for O_{2} transport in cardiac and skeletal muscle cells, the tacit assumption has been made that the chemical reaction rate is not limiting in this process, and full chemical equilibrium exists within cells or layers of solution (5, 6, 25, 28, 34). In the present paper, therefore, the question is addressed whether Mb-facilitated O_{2} diffusion in cardiac and skeletal muscle cells is limited by the speed of the chemical reaction for the diffusion path lengths given by the dimensions of these cells. The mathematical model consists of a complete description of the process of facilitated O_{2} diffusion, which is solved numerically in a straight-forward fashion using a finite-difference method. This simple approach was not feasible 40 yr ago due to prohibitive computation times but is now viable as a result of the drastically enhanced computation speed of personal computers, even though the computation of one set of conditions as reported here still requires several days, and in special cases even weeks. The present paper assesses the question of reaction-limitation in muscle cells by treating the problem as a one-dimensional or cylindrical one. The one-dimensional approach has also been applied by Moll (32) and Kutchai et al. (23) to the case of red cells. In contrast, Clark et al. (8) modeled O_{2} unloading from red cells as a three-dimensional, time-dependent problem by using an elaborate approximate analytical solution on the basis of a boundary layer concept by matched asymptotic expansions. To compare the present numerical method with that of Kutchai et al. (23) as applied to Hb solutions and red cells, computations of facilitated diffusion by Hb diffusion for the conditions of human red blood cells have been included in this study.

## METHODS

#### The equations.

In the simpler case of Mb, the reaction of O_{2} with the heme protein is given by:
_{2}] is the concentration of oxymyoglobin, [Mb] is the concentration of deoxymyoglobin, *t* is time, k_{A} is the second-order rate constant of association of Mb and O_{2}, and k_{D} is the first-order rate constant of dissociation of MbO_{2}.

The diffusion processes of O_{2} and oxymyoglobin are described by Fick's second law of diffusion:
_{O2} is the diffusion coefficient of O_{2} in the medium considered, D_{MbO2} is the diffusion coefficient of oxymyoglobin (identical to that of myoglobin), and *x* is the diffusion path.

Since facilitated diffusion is a process of simultaneous diffusion and reaction of free and bound O_{2}, it can be stated that, in a certain infinitesimal volume element, at *x* the change of O_{2} concentration with time is equal to the net amount of O_{2} diffusing into this element, D_{O2}·∂^{2}[O_{2}]/∂*x*^{2}, minus the net amount of O_{2} consumed within the element by reaction with Mb:

The analogous consideration applies to the change of MbO_{2} concentration with time:

A layer of Mb solution of defined thickness ℓ (treated as a plane sheet) is considered, and, by numerically solving *Eqs. 5* and *6*, temporal and spatial courses of [O_{2}] and [MbO_{2}] within this layer are obtained. After the calculations have been continued for a sufficient time interval, steady state is reached, the concentration profiles of O_{2} and MbO_{2} in the layer become time-independent, and the fluxes of free and Mb-bound O_{2} as well as the total O_{2} flux have assumed their final steady-state values. The total flux of O_{2} across the entire layer is obtained from the fluxes of O_{2} leaving and entering, respectively, the layer at the boundaries. The average free flux of O_{2} across the layer is calculated by using the boundary concentrations of O_{2}, [O_{2}]_{x=0} and [O_{2}]_{x=ℓ}, from D_{O2}·([O_{2}]_{x=0} − [O_{2}]_{x=ℓ}). The difference gives the average facilitated flux across the layer. Free and facilitated O_{2} fluxes are used to calculate the percentage facilitations shown in Figs. 3⇓–5.

In view of the approximately cylindrical geometry of muscle cells, we compared the results obtained with the one-dimensional approach of *Eqs. 5* and *6* with numerical solutions for the case of a cylindrical geometry. For this situation, the following equations were applied (9):
*r* signifies the radial axis in the polar coordinate system. Finite differences were formulated as described by Crank (9). The overall procedure was analogous to the above one-dimensional case. The steady-state average fluxes of O_{2} (free) and MbO_{2} (facilitated) were again calculated as described above from the total O_{2} flux across the cylinder surface and from the boundary concentrations of dissolved O_{2} using *Eq. 5.5* in Ref. 9 (p. 62). Values of *r* varied between 0 and *a* (equal to the radius of the cylinder considered).

In the case of hemoglobin, the equations describing facilitated diffusion are identical to those for Mb. However, the correct reaction equation has to be written as
*1*) considering the chemical reaction to occur with identical hemoglobin monomers, *2*) using the half-saturation pressure of the O_{2} binding curve in combination with the overall kinetic constants k_{A} and k_{D} to approximate the kinetics of the Hb-O_{2} reaction in a fashion identical to that of Mb (see *Eq. 2*):

This treatment of the oxygenation-deoxygenation kinetics of Hb is identical to that used by previous investigators of the reaction- and path length-dependence of facilitated O_{2} diffusion in red cells (23, 32).

#### Parameters.

In the case of myoglobin, the reaction rate constant k_{D} of 11–12 s^{−1} reported for 20°C by Antonini (1) and Gibson et al. (13) is used. With ΔH_{off} of 19 kcal/mol, one obtains a value of k_{D} at 37°C of 60 s^{−1} (1), which is used in the present calculations that all pertain to 37°C. With a half-saturation pressure (P_{50}) of Mb of 2.4–2.8 Torr (1, 40) and an O_{2} solubility α_{O2} in muscle tissue of 1.5 × 10^{−6} mol·l^{−1}·Torr^{−1} (cf. Ref. 10), one obtains from this k_{D} value a k_{A} value of ∼15.4 × 10^{6} M^{−1}/s. These rate constants imply half-times of MbO_{2} dissociation of 12 ms and, depending on the concentrations of the reaction partners, of ca. 0.5 ms for the association of Mb with O_{2}. These numbers indicate that dissociation is considerably slower than association and thus will be the primary cause when the rate of chemical reaction becomes rate limiting in the facilitated diffusion process. The Mb diffusion coefficient inside cardiac and skeletal muscle cells was taken to be 2 × 10^{−7} cm^{2}/s (20, 35, 36, 37), and an additional calculation was performed with the D_{Mb} of 8 × 10^{−7} cm^{2}/s derived from NMR measurements for rat cardiac cells by Lin et al. (26, 27). D_{O2} in muscle tissue was taken to be D_{O2} = K_{O2}/α_{O2} = 1.3 × 10^{−9} mmol·cm^{−1}·min^{−1} mmHg^{−1}/1.5 × 10^{−6} mol·l^{−1}·Torr^{−1} = 1.4 × 10^{−5} cm^{2}/s, where K_{O2} is Krogh's diffusion constant describing the gas transport rate per area and partial pressure gradient (cf. 10). Myoglobin concentration used for cardiac muscle tissue was 0.19 mM (42). The boundary Po_{2} values were set to be 10 Torr at *x* = 0 and 0.1 Torr at *x* = ℓ, thus approximating the most favorable conditions for facilitated O_{2} diffusion conceivable in physiological situations. The thickness of the layer considered in the calculations was the major variable, and ranges used were intended to encompass cardiac and skeletal muscle fiber radii and diameters, respectively.

In the case of hemoglobin, a k_{A} value of 3.5 × 10^{6} M^{−1}·s^{−1} has been reported by Gibson et al. (12) for pH 7.1 and 37°C, which is similar to that obtained by Mochizuki et al. (29). For the present calculations, we have used the values reported by Bauer et al. (3) for the in vivo conditions inside red cells (pH 7.2; [2,3-bisphosphoglycerate] 6.5 mM; 37°C). Their k_{D} is 250 s^{−1} and k_{A} is 6.1 × 10^{6} M^{−1}·s^{−1} (with an O_{2} solubility of 1.5 × 10^{−6} mol·l^{−1}·mmHg^{−1} as mentioned above, and the ratio of k_{D}/k_{A} corresponds to an O_{2} half-saturation pressure for hemoglobin of P_{50} = 27 Torr). This k_{D} for Hb is markedly faster than that used in the case of Mb. D_{O2} inside the red cell is taken to be the same as in muscle tissue. D_{Hb} within the red cell is 6.4 × 10^{−8} cm^{2}/s (31). Intraerythrocytic hemoglobin concentration is 20 mM Hb monomer. The boundary Po_{2} values were set to 40 and 100 Torr, respectively, which represents the physiological range experienced by a red cell in the lung (and in the tissue) under resting conditions.

#### Numerical solution.

Using the finite difference method as implemented in MATLAB 2008b, *Eqs. 5* and *6* were solved numerically. In the case of Mb in a muscle cell, the boundary Po_{2} values given above were assigned to *x* = 0 (Po_{2} of 10 Torr) and *x* = ℓ (Po_{2} of 0.1 Torr), respectively. Throughout the interior of the layer, a homogeneous initial Po_{2} of 0.1 Torr was set, with the exception of *x* = 0, which was set to 10 Torr. In the case of hemoglobin, *x*_{0} was set to 100 Torr, and the interior of the layer as well as *x* = ℓ were set to 40 Torr. The layers were usually divided into *N* = 200 sections, i.e., Δ*x* = ℓ/200. In several cases, *N* was raised to 500 or 1,000. The time interval inserted into *Eqs. 5* and *6* (after conversion of the differential equations into difference equations) was determined in dependence of the inserted value of Δ*x* from the following relation (9):
*t* was sufficiently small by making sure that a further reduction of Δ*t* did not affect the results. These values of Δ*x* and Δ*t*, pertaining to a given value of ℓ, were used in conjunction with the following boundary conditions: *1*) the boundary Po_{2,x=0} was set to 10 Torr (as mentioned above, for the case of Mb); *2*) the boundary Po_{2,x=ℓ} was set to 0.1 Torr (as mentioned above, for the case of Mb); *3*) the sum of [Mb] and [MbO_{2}] was set to be [Mb_{tot}] everywhere in the layer, where [Mb_{tot}] is the total Mb concentration and a constant (0.19 mM in the case of Mb).

The boundary concentrations of Mb and MbO_{2} were obtained from the calculation, but initial estimates were derived from [Mbtot] and the boundary Po_{2} values on the basis of the assumption of chemical equlibrium. The boundary conditions in the case of Hb were formulated analogously.

For these conditions, *Eqs. 5* and *6* were numerically integrated and solved for the profiles of Po_{2}, [Mb], and [MbO_{2}] throughout the layer. The calculation cycles were repeated until time-independent concentration profiles of [O_{2}] and [MbO_{2}], or [HbO_{2}], respectively, were obtained in the layer. For each calculation cycle, %Facilitation (at *x* = ℓ), defined as ratio of facilitated flux over free flux in the last volume segment adjacent to *x* = ℓ, F_{MbO2}/F_{O2}, or F_{HbO2}/F_{O2}, was calculated in the following way: in the last segment, adjacent to *x* = ℓ, the total flux of O_{2} leaving this segment was determined for each point of time from the balance between the O_{2} entering the segment and the O_{2} consumed/produced by chemical reaction. After steady-state conditions have been established, the total flux leaving this element is identical to the total O_{2} flux occurring across the entire layer, because total flux must then be the same everywhere in the layer. Facilitated flux averaged over the entire layer is obtained as the difference between this total O_{2} flux and the free O_{2} flux as determined from the overall Po_{2} difference Po_{2,x=0} − Po_{2,x=ℓ} and D_{O2}. It follows from this way of calculation that %Facilitation is representative for the entire cell/tissue layer only after steady-state conditions have been reached. Figure 1 shows for the case of *x* = 5 μm that %Facilitation (*x* = ℓ) starts with a value of 0, a consequence of the initial conditions described, then increases with time, and reaches a plateau after ∼0.2 ms, indicating that steady state is established in the layer after this time. This plateau value represents the %Facilitation averaged over the entire layer under steady-state conditions. It is this quantity that is given on the ordinates of Figs. 3⇓–5. These calculations were repeated for a large number of layer thicknesses ℓ, as is apparent on the abscissae of Figs. 3⇓–5.

It was tested whether the spatial resolution chosen by the value of *N* yields sufficient precision by comparing a series of calculations with varying *N* values. In the case of a 5-μm-thick Mb solution, the %Facilitation obtained was 8.94% with *N* = 100, and 9.06%, 9.15%, and 9.17% with *N* values of 200, 500, and 1,000, respectively. These numbers allow one to estimate that the value calculated with *N* = 200 deviates from the true value by ∼1%. Thus a number of 200 steps appears entirely sufficient.

On order to study diffusion of O_{2} and MbO_{2} in radial direction in a cylinder, *Eqs. 7* and *8* were solved by an analogous finite difference method in MATLAB 2008b. The Po_{2} at the outer circumference of the cylinder (*r* = *a*) was set to 10 Torr. Po_{2} of 0.1 Torr was set to be either in the center of the cylinder at *r* = 0 (implying the O_{2} sink of the muscle fiber to be located in the center), or, as a reasonable approximation of the true situation, at an intermediate position in the cylinder at *r* = *a*/2.

## RESULTS AND DISCUSSION

The kinetics of the reaction between hemoprotein and O_{2} causes a dependence of the degree of facilitation of O_{2} diffusion upon the length of the diffusion path.

When the oxygenated hemoprotein and O_{2} diffuse alongside through a cell or layer of solution, albeit at different concentration gradients and diffusivities of protein and O_{2}, facilitated diffusion can develop by association reaction between hemoprotein and O_{2} (predominantly in the first part of the entire diffusion path to be overcome) and thereafter by dissociation reaction (predominantly in the later part of the entire diffusion path). In this way, the chemical reactions make possible the carriage of O_{2} by the hemoprotein over a certain diffusion distance, a process called facilitated diffusion, as well as its timely release when and where the free O_{2} is needed, e.g., at the mitochondrion. This mechanism can therefore operate only when the kinetics of the association as well as the dissociation reaction are fast enough to permit reaction times sufficiently short compared with the time required for O_{2} to cross the entire diffusion distance. This condition for a facilitation of O_{2} diffusion has been recognized and formulated early on by Moll (30) and Wyman (49). Wyman showed theoretically that the given reaction kinetics of Mb and O_{2} allows facilitation by translational diffusion of Mb to occur in thicker layers of Mb solution, but clearly not by rotational diffusion, where the effective “diffusion distance” is equal to the diameter of the carrier protein, ∼42 Å in the case of Mb and 62 Å in the case of Hb (43, 17). This general concept was later confirmed by Gros et al. (15, 16). They showed that facilitated diffusion can occur by rotational carrier diffusion in addition to translational diffusion in the case of the protonation reaction, which is extremely fast with half-times in the nanosecond range. Interestingly, even with such fast reaction kinetics, facilitated proton diffusion by rotational protein diffusion was demonstrable (15, 16) only with two very large proteins, earthworm hemoglobin (d = 276 Å, molecular weight of ∼3.7 × 10^{6} Da) and apoferritin (d = 146 Å, molecular weight of 450,000 Da), but was absent in the two significantly smaller proteins myoglobin (d = 42 Å, molecular weight of 17,000) and serum albumin (d = 72 Å, molecular weight of 67,000 Da). These experimental observations clearly demonstrate the positive correlation between “diffusion distance” and the effectiveness of facilitated diffusion at a given reaction kinetics. The results of the calculations presented in the following show that this relationship is also an important parameter when one considers facilitated O_{2} diffusion by Mb or Hb translational diffusion within the limited diffusion paths as they prevail in cardiac and skeletal muscle or in red blood cells.

Figure 2 illustrates the mechanism of the reaction limitation at shorter diffusion distances for the case of Mb-facilitated O_{2} diffusion in muscle. The example was calculated for a diffusion path length of 3.5 μm, the constants as defined above for facilitated O_{2} diffusion in muscle cells, and the boundary conditions listed above. As discussed below, 3.5 μm represents a reasonable estimate of the effective diffusion path in cardiomyocytes. Figure 2 shows the gradients of [O_{2}], [MbO_{2}], and [Mb] established under steady-state conditions for a case in which facilitation is markedly limited by the rate of MbO_{2} deoxygenation. The boundary concentrations of O_{2} of course reflect the inserted boundary Po_{2} values mentioned above. In the curve representing the [O_{2}] gradient calculated with k_{D} = 60 s^{−1} (continuous line), there is a slight bend in the middle part, the curve being somewhat flatter in the second half compared with the first half. The O_{2} gradient in the second half is less steep because there, as is apparent from the middle curve of Fig. 2, the oxymyoglobin gradient (and thus facilitated diffusion) is greater than in the first half. This behavior reflects the fact that the total flux of O_{2} under steady-state conditions must be identical everywhere along the diffusion path. The dashed curve shows the [O_{2}] gradient calculated for values of k_{D} and k_{A} that are both 1,000-fold greater than those employed for the continuous curve. It is apparent that this results in a more pronounced bend in the course of the [O_{2}] gradient, which is due to the increase in the oxymyoglobin gradient in the second half of the (dashed) curve in Fig. 2, *middle*. The oxymyoglobin gradient shown in the middle panel directly illustrates the mechanism of reaction limitation: with the factual (low) reaction velocity, [MbO_{2}] at the right-hand boundary does not fall to the low level of 0.007 mM predicted for chemical equilibrium but remains at 0.061 mM, ∼10 times higher. This indicates a build-up of undissociated MbO_{2} at the right-hand boundary due to a limitation of MbO_{2} deoxygenation by the slow deoxygenation kinetics. When the reaction is accelerated by a factor of 1,000, this build-up almost disappears (dashed curve). It is noted that facilitation, when calculated with the true kinetic constants, in this example is reduced to a little over 50% of its maximum at chemical equilibrium. In conjunction with this, the MbO_{2} concentration difference (continuous line) between the boundaries of the layer is reduced to the same percentage. Thus facilitation is reduced because, and to the extent that, the oxymyoglobin concentration difference is reduced due to slow deoxygenation kinetics.

#### Dependence of myoglobin-facilitated O_{2} diffusion on path length: comparison with effective diffusion paths in cardiac and skeletal muscle cells.

Figure 3 shows the results of calculations determining the degree of Mb-facilitated diffusion for various layer thicknesses, i.e., diffusion distances ℓ. The intracellular Mb diffusion coefficient employed is 2 × 10^{−7} cm^{2}/s. All data points represent steady-state conditions and are taken from the plateaus of figures of the type of Fig. 1. It is apparent that %Facilitation depends strongly on the length of the diffusion path. Maximum facilitation for the conditions representing the situation in a muscle cell amounts to ∼14%, a figure that is reached only when the reaction kinetics is infinitely fast and does not limit the diffusion-reaction process at all (horizontal line in Fig. 3). At a path length of 1 μm, %Facilitation is <2%, and half-maximal facilitation is attained at an ℓ slightly below 3 μm. Figure 3 also shows that full maximal facilitation is not yet reached at ℓ = 25 μm, where it amounts to 12%, i.e., 6/7 of the maximum. Figure 4 shows that the maximum is not even completely established at ℓ = 100 μm.

#### Facilitation in cardiomyocytes.

What is the implication of these results for Mb-facilitated O_{2} diffusion in cardiac muscle cells? In the case of human heart tissue, Armstrong et al. (2) determined an average radius of cardiomyocytes in intact cardiac tissue of 7 μm. From published capillary densities, Endeward et al. (10) derived the radius of the Krogh cylinder used to describe O_{2} supply to cardiac tissue to be 10 μm, i.e., of similar magnitude. In applying the results of Figs. 1, 3, and 4, the simplification is made that the muscle fibers are “plane sheets,” and they are considered as cuboids rather than cylinders. This implies that the diffusion process is taken to occur across two opposite sides of the cuboid into its interior. The maximal diffusion distance that has to be overcome by O_{2} within the cardiomyocyte is expected to be of the order of 7 μm, and often, most pronounced in the case of subsarcolemmal mitochondria, considerably less. Using a path length of 7 μm, a %Facilitation of 10% is read from Fig. 3, i.e., only ∼2/3 of the facilitation expected when no reaction limitation occurs. If one assumes the average diffusion path length of O_{2} in the cardiomyocyte to be half the fiber radius, 3.5 μm, %Facilitation is only 7.6% or slightly more than one-half of the facilitation in the absence of reaction limitation. In conclusion, facilitated O_{2} diffusion in heart tissue is markedly limited by the speed of the Mb deoxygenation reaction and will amount to approximately one-half of the value predicted from Po_{2} gradients and O_{2} and Mb diffusivities ignoring the finite speed of the O_{2}-Mb kinetics.

#### The cardiomyocyte as a cylinder.

Considering the cardiomyocte as a cuboid with two rather than four sides accessible to O_{2} diffusion is of course a major simplification. The calculation was therefore repeated assuming the cardiac muscle fiber to be represented by a cylinder of radius *a* = 7 μm. In this case, it had to be assumed that O_{2} has access to the cell on the entire cylinder surface, i.e., Po_{2} was set to 10 Torr everywhere at the cylinder surface and O_{2} diffusion occurred from the surface toward the interior of the cylinder. This is of course also a major simplification since, physiologically, O_{2} supply is not present equally at the entire circumference of the cylinder. Two cases were considered. First, the “O_{2} sink” was assumed to be in the center of the cylinder, at *r* = 0. This results in an extremely small degree of facilitation of O_{2} diffusion by 2.2%. This is due to the cylindrical geometry, which implies the available “effective diffusion area” to decrease drastically toward the center of the cylinder. Therefore, the gradient of O_{2} is very shallow in the outer regions of the cylinder, and Mb there remains almost fully saturated. The gradients of Po_{2} and MbO_{2} begin to become steeper just before the center of the cylinder where Po_{2} = 0.1 Torr is reached. Thus facilitation can only occur on the last fractions of a micrometer from the center, and on this short diffusion distance reaction limitation is severe. Of course, it is entirely unrealistic to base this calculation on a localization of all mitochondria in the center of the cylinder. The second approach was therefore based on the more reasonable assumption of the O_{2} sink being localized in the cylinder halfway between *r* = 0 and *r* = *a*, i.e., at *r* = *a*/2. This yields a facilitation of intracellular O_{2} diffusion of 7.4%, i.e., a %Facilitation almost identical to what has been derived above for the cuboid model. It may be noted that %Facilitation will decrease again as the O_{2} sink moves toward the cylinder surface because of the decreasing diffusion path length. As can be appreciated from Fig. 3, O_{2} transport to subsarcolemmal mitochondria, ∼1 μm below the surface membrane, will hardly be supported by facilitated diffusion.

#### Facilitation in skeletal muscle.

The dimensions of most skeletal muscle fibers are considerably greater than those of cardiomyocytes, often with radii between 25 and 50 μm. O_{2} supply occurs essentially in a radial direction (19). Jürgens et al. (19) derived a Krogh cylinder radius of 24 μm for the human quadriceps femoris muscle. The diffusion distance to be overcome by O_{2} then will be ∼25 μm maximally, corresponding to 12% facilitation (6/7 of the maximal 14%) according to Fig. 3. The figure implies, as in the case of cardiac muscle cells considered above, the assumption of diffusion into a plane sheet. If the average diffusion path is again taken to be half the fiber radius, 12 μm, %Facilitation becomes 11% or 4/5 of the maximum predicted for infinite reaction speed. With the same Mb diffusion coefficient of 2 × 10^{−7} cm^{2}/s, maximal facilitation is not even reached with the radius of a very thick skeletal muscle fiber of 50 μm, as is apparent from the lower half of Fig. 4. The cylinder approach described above, now applied to skeletal muscle, again yields a very similar facilitation, 10.7% for *r* = *a*/2. In conclusion, even in the much larger skeletal muscle fibers, some reaction-limitation of facilitated O_{2} diffusion remains, reducing %Facilitation to 4/5 to 6/7 of the maximal value. Nevertheless, it is clear that the dimensions of skeletal muscle fibers are considerably more favorable for Mb-facilitated O_{2} diffusion than those of cardiomyocytes.

#### Effect of D_{Mb} on facilitation.

Figure 4 illustrates the effect of D_{Mb} on these calculations. Lin et al. (26, 27) reported a D_{Mb} in cardiac tissue of 8 × 10^{−7} cm^{2}/s (37°C) from NMR measurements, whereas Baylor and Pape (4), Jürgens et al. (20), and Papadopoulos et al. (35–37), using classical techniques, consistently found values of ∼2 × 10^{−7} cm^{2}/s for 37°C. As recently discussed in a joint review (17), the higher D_{Mb} obtained by NMR might be due to much shorter (but yet to be determined precisely) effective diffusion distances of the Mb inside the muscle cell when observed by NMR, which might prevent some of the more widely spaced intracellular obstacles hindering Mb diffusion over larger distances to become effective in the NMR measurement. The upper half of Fig. 4 has been obtained with D_{Mb} = 8 × 10^{−7} cm^{2}/s and shows, first, that maximal facilitation of course is fourfold greater than with D_{Mb} = 2 × 10^{−7} cm^{2}/s. Second, it is apparent that, with increasing speed of Mb diffusion, the limitation by the speed of the chemical reaction becomes more pronounced. For an assumed average diffusion distance of 3.5 μm, as used above for the heart, one finds 7.6% facilitation with D_{Mb} = 2 × 10^{−7} cm^{2}/s, or 54% of the maximum, whereas with D_{Mb} = 8 × 10^{−7} cm^{2}/s one reads a %Facilitation of 26% from the upper curve of Fig. 4, corresponding to 46% of the maximal value given by the horizontal line. This reflects the fact that the relation between the speed of diffusion and the speed of the chemical reaction is the critical parameter determining reaction limitation of facilitation: the faster the diffusion process, the greater the requirement for rapid chemical reaction rates. For either D_{Mb} value, the conclusion remains that for ℓ = 3.5 μm, roughly only about one-half of the maximal facilitation is achieved. It is concluded from these considerations that Mb-facilitated diffusion in the heart is expected to be severely limited by reaction velocity, by ∼50%, whereas in skeletal muscle this limitation is also present but of lesser importance.

#### Effect of O_{2} consumption on facilitation.

The presented computational approach does not take into account the fact that, in muscle tissue, intracellular O_{2} transport is combined with intracellular O_{2} consumption. Can O_{2} consumption affect the role of Mb-facilitated diffusion? We have incorporated into the above equations an O_{2} consumption that is homogeneously distributed across the sarcoplasm. To estimate the effect of O_{2} consumption on facilitation, some calculations were performed for the following altered conditions: *1*) Po_{2} at *x* = 0 was held at 10 Torr, as above; *2*) a maximal specific cardiac O_{2} consumption of 600 ml O_{2}·min^{−1}·kg^{−1} (10) was inserted; *3*) the total flux of O_{2} at *x* = ℓ was set to zero. The latter condition reflects the fact that, in a plane sheet (i.e., the cuboid model), the entire half-thickness of the sheet is supplied with O_{2} from one side of the sheet, and no O_{2} diffuses into the other half of the sheet, i.e., all O_{2} entering the sheet under steady-state conditions at *x* = 0 is consumed within this half of the sheet. The path length (equal to half-thickness) in these calculations was 7 μm. This yielded the gradients of [O_{2}] and [MbO_{2}] under conditions of maximal O_{2} consumption of cardiomyocytes, which were used to calculate free and facilitated fluxes. Accordingly, in the maximally respiring cardiomyocyte at capillary Po_{2} = 10 Torr, %Facilitation amounts to only 4.7% (1/3 of the maximum) when the cuboid model is applied and to 3.3% (1/4 of the maximum) when the cylinder model is used. Thus facilitated O_{2} diffusion in this case is of even lesser significance than when we consider merely O_{2} diffusion without O_{2} consumption. Besides by the reaction limitation of facilitation, this is caused by boundary Po_{2} values at *x* = ℓ that turn out to be ∼4 Torr (cuboid) and ∼7 Torr (cylinder), respectively. This implies that, even at maximal O_{2} consumption, in conjunction with Po_{2,x=0} = 10 Torr, a Po_{2,x=ℓ} substantially above 0.1 Torr is maintained in the cardiomyocyte, thus further reducing the contribution of facilitated O_{2} diffusion.

#### Dependence of hemoglobin-facilitated O_{2} diffusion on path length: comparison with the effective diffusion path in red cells.

Only in the case of red blood cells, efforts have been made previously to determine to which extent facilitated O_{2} diffusion inside cells is limited by the speed of the Hb-O_{2} kinetics. The present computational approach was applied to this problem to compare the results with these previous reports. For a red cell in its resting discoid shape, the effective thickness for considerations of gas uptake is 1.6 μm, and, since gas is taken up from both sides of the disc, the relevant diffusion path for O_{2} is half of this value, 0.8 μm (11). For this diffusion path, Fig. 5 predicts a facilitation of ∼13% for the conditions given above including the boundary Po_{2} values of 40 and 100 Torr. The maximum %Facilitation under these conditions is calculated to be 20.3%. Thus the limitation by the speed of the Hb-O_{2} reaction decreases facilitation to 64% of the maximum. Similar calculations for the facilitation of steady-state O_{2} diffusion through layers of 33 g% hemoglobin solution have been reported by Kutchai et al. (23), who obtained for a layer thickness of 0.75 μm and similar boundary Po_{2} of 50 and 125 Torr, a %Facilitation of 16.8%, a figure that represented 59% of the maximum facilitation possible in the absence of reaction limitation. Thus the present results on red cells agree quite well with those of Kutchai et al. (23).

The measurements of steady-state O_{2} fluxes across layers of packed red cells by Moll (33) and Kutchai and Staub (24) touch on the question of reaction limitation inside red cells but seem to be at variance with both theoretical results just mentioned. Both groups performed these measurements in the absence and presence of sufficient CO to completely block the hemoglobin and found a major reduction of O_{2} flux in the presence of CO. Moll (33) estimated that the CO-dependent O_{2} flux corresponds roughly to what is expected from measured intraerythrocytic Hb diffusion coefficients, and Kutchai and Staub (24) observed very similar facilitated fluxes in packed red cells and in hemoglobin solution. Thus both of these experimental studies did not obtain evidence for a reaction limitation of facilitated O_{2} diffusion, which should have been apparent in 165- to 300-μm-thick layers of packed intact red cells but not in layers of Hb solution of identical thickness. This seems to disagree with the theoretical results by Kutchai et al. (23) and those reported in the present study. It should be noted, however, that in layers of packed red cells with O_{2} diffusion occurring from one side of the layer to the other, the effective intracellular diffusion distance is 1.6 μm in red cells oriented in parallel to the plane of layer rather than the half-thickness of 0.8 μm and may be as great as 8 μm in red cells oriented perpendicularly to the plane of the layer. Other experimental and theoretical evidence suggests, however, that some reaction limitation should be expected in these experiments. Wittenberg (47) reported measurements of the dependence of Hb-facilitated O_{2} fluxes on the thickness of the layer of Hb solution. Kutchai et al. (23) replotted Wittenberg's data to show that facilitation in % of the maximum is independent of layer thickness between 300 and 60 μm but is markedly decreased at a thickness of ∼25 μm. The theoretical results of Kutchai et al. (23) are in good agreement with these data and show that some reaction limitation begins to appear below a layer thickness of 100 μm and causes a decrease in %Facilitation to 45% of the maximum at a layer thickness of 5 μm. (It may be noted that these data and calculations pertain to boundary Po_{2} values of 100 and 0 Torr, which are quite different from the values considered in the present study.) Overall, one can conclude that a majority of studies demonstrates that facilitated O_{2} diffusion in Hb solutions starts to become reaction limited at layer thicknesses of <100 μm and that, under physiological conditions, this limitation cuts down facilitation inside red cells by roughly 40%.

#### Implications of the results for the potential roles of myoglobin as a cellular O_{2} store vs. a cellular O_{2} transporter.

Applying a modified Krogh cylinder model, we previously have shown that, in the heart, myoglobin-facilitated O_{2} diffusion plays no role during diastole when coronary perfusion is high and quasi-steady-state conditions exist in the tissue (10). This is due to the absence of major gradients of oxygenated Mb during diastole, since almost all of the Mb in the heart are fully oxygenated even under a significantly elevated work state (21, 22, 50). An analogous observation of full saturation of Mb has been made for resting skeletal muscle (7, 44), again implying the absence of a role of Mb for intracellular O_{2} transport. During systole, when in major parts of the left ventricle wall, coronary perfusion almost ceases, and the amount of Hb present in the capillaries is greatly reduced, myocardial Po_{2} drastically falls, and substantial intracellular gradients of MbO_{2} develop (10). In this phase of the cardiac cycle, MbO_{2} plays a role both as an O_{2} store and as an intracellular O_{2} transporter. Endeward et al. (10) estimated that systolic O_{2} supply of the left ventricular myocardium is supported to the extent of ∼20% by Mb-facilitated O_{2} diffusion and of 10% by the storage function of Mb, whereas the remaining 70% is provided by O_{2} bound to capillary Hb and by O_{2} dissolved in the tissue. It should be noted that these calculations were based on the assumption that Mb-facilitated O_{2} diffusion in cardiomyocytes is not limited by Mb-O_{2} reaction kinetics. Therefore, the numbers reported by Endeward et al. (10) for the contribution of facilitated diffusion have to be revised downward, approximately cutting the number of 20% down to 10%. This would mean that the relative roles of the storage and transport function of Mb might actually be of about equal size, and both of them together would account for roughly only one-fifth of systolic O_{2} supply. Thus O_{2} sources in the tissue other than MbO_{2} would be even more dominating in this phase. A noteworthy aspect of these findings is that facilitated O_{2} diffusion in the heart occurs only in conjunction with a depletion of the MbO_{2} store and then presumably serves to support the delivery of the released O_{2} to the mitochondria. These combined and about equal effects of Mb as an O_{2} store and as an O_{2} transporter should then be responsible for the compensatory adaptations observed in Mb_{null} mice by Gödecke et al. (14). The situation is likely somewhat different during prolonged contractile activity of skeletal muscle, where, after an initial fall in tissue Po_{2} and MbO_{2} resembling the systolic phase in the heart, a new steady state is reached, which is characterized by a time-independent partial desaturation of Mb (7, 38, 44). This latter situation appears to represent an example in which some facilitation can occur under quasi-steady-state conditions, although calculations on the basis of a Krogh cylinder model predict that this facilitation is of minor importance for sarcoplasmic O_{2} transport even under conditions of heavy exercise (19).

It has been shown here that facilitated O_{2} diffusion is clearly limited by the rate of the (deoxygenation) kinetics of Hb or Mb when the cells considered are not exceptionally large. This concerns the transport function of Hb and Mb. Does it also compromise the O_{2} storage function of Mb in the heart? Above, estimates are given of the half-times of MbO_{2} dissociation (12 ms) and of Mb association with O_{2} (∼0.5 ms). Can the rate of MbO_{2} dissociation limit the rate at which O_{2} is released from MbO_{2} during systole? The shortest possible duration of the systole in the maximally working human heart is 150 ms (10). This is >10-fold the half-time of the dissociation reaction, and thus any significant reaction limitation of the release of O_{2} from the MbO_{2} store is not to be expected. It appears that the kinetics of the Mb oxygenation-deoxygenation reaction is much better adapted to the storage function than to the transport function of Mb, at least in the heart.

## GRANTS

This work was partly supported a grant from the Deutsche Forschungsgemeinschaft, EN 908/1-1.

## DISCLOSURES

No conflicts of interest, financial or otherwise, are declared by the author(s).

## AUTHOR CONTRIBUTIONS

Author contributions: V.E. conception and design of research; V.E. analyzed data; V.E. interpreted results of experiments; V.E. prepared figures; V.E. drafted the manuscript; V.E. edited and revised the manuscript; V.E. approved the final version of the manuscript.

## ACKNOWLEDGMENTS

I thank Prof. G. Gros (Hannover) for proposing this study, continuous discussion, and critical reading of the manuscript.

- Copyright © 2012 the American Physiological Society