J Appl Physiol 112: 1466–1473, 2012.
First published February 23, 2012; doi:10.1152/japplphysiol.00835.2011.
The rate of the deoxygenation reaction limits myoglobin- and hemoglobinfacilitated O2 diffusion in cells
Volker Endeward
Zentrum Physiologie, Vegetative Physiologie 4220, Medizinische Hochschule Hannover, Hannover, Germany
Submitted 6 July 2011; accepted in final form 22 February 2012
myoglobin; hemoglobin
MYOGLOBIN (Mb) and hemoglobin (Hb) can facilitate O2 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-␮m 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 O2 diffusion can only be effective if the relaxation
time of the chemical reaction between O2 and Hb or Mb,
respectively, is considerably shorter than the relaxation time of
diffusion. Since the latter depends on the length of the diffu-
sion path, or the layer thickness, it follows that hemoproteinmediated facilitated O2 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 O2 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 ␮m. Consequently, they predicted
for the average thickness of a red cell (1.6 ␮m) 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 O2 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 O2 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 O2 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 onedimensional 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 O2 unloading from red cells as a threedimensional, 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
Address for reprint requests and other correspondence: V. Endeward, Medizinische
Hochschule Hannover, Vegetative Physiologie 4220 Carl-Neubergstr. 1, 30625 Hannover, Germany (e-mail: [email protected]).
1466
The equations. In the simpler case of Mb, the reaction of O2 with
the heme protein is given by:
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Endeward V. The rate of the deoxygenation reaction limits myoglobin- and hemoglobin-facilitated O2 diffusion in cells. J Appl
Physiol 112: 1466 –1473, 2012. First published February 23, 2012;
doi:10.1152/japplphysiol.00835.2011.—A mathematical model describing facilitation of O2 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 O2. 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
O2 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 O2 is far from
being established, an assumption that previously has often been made.
Although the “O2 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 O2 from MbO2, the “storage function,” is not limited by
the reaction kinetics under physiological conditions.
Limitation of Facilitated O2 Diffusion by Chemical Reaction
Mb ⫹ O2 ↔ MbO2
(1)
The rate of the oxygenation-deoxygenation reaction then is described by:
d关MbO2兴/dt ⫽ kA关Mb兴关O2] ⫺ kD关MbO2]
•
Endeward V
1467
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
Hb4 ⫹ 4O2 ↔ Hb4(O2)4
(9)
(2)
where [MbO2] is the concentration of oxymyoglobin, [Mb] is the
concentration of deoxymyoglobin, t is time, kA is the second-order
rate constant of association of Mb and O2, and kD is the first-order rate
constant of dissociation of MbO2.
The diffusion processes of O2 and oxymyoglobin are described by
Fick’s second law of diffusion:
where the equilibrium constant and the kinetic constants differ between each of the single oxygenation steps. The calculation here is
simplified by 1) considering the chemical reaction to occur with
identical hemoglobin monomers, 2) using the half-saturation pressure
of the O2 binding curve in combination with the overall kinetic
constants kA and kD to approximate the kinetics of the Hb-O2 reaction
in a fashion identical to that of Mb (see Eq. 2):
⭸ 关O2兴/ ⭸ t ⫽ DO2 · ⭸2关O2] ⁄ ⭸ x2
(3)
d[Hb1O2] ⁄ dt ⫽ kA[Hb1][O2] ⫺ kD[Hb1O2]
⭸ 关MbO2兴/ ⭸ t ⫽ DMbO2 · ⭸2关MbO2] ⁄ ⭸ x2
(4)
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 O2 diffusion in red cells (23, 32).
Parameters. In the case of myoglobin, the reaction rate constant kD
of 11–12 s⫺1 reported for 20°C by Antonini (1) and Gibson et al. (13)
is used. With ⌬Hoff of 19 kcal/mol, one obtains a value of kD 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 (P50) of Mb of 2.4 –2.8 Torr
(1, 40) and an O2 solubility ␣O2 in muscle tissue of 1.5 ⫻ 10⫺6
mol·l⫺1·Torr⫺1 (cf. Ref. 10), one obtains from this kD value a kA
value of ⬃15.4 ⫻ 106 M⫺1/s. These rate constants imply half-times of
MbO2 dissociation of 12 ms and, depending on the concentrations of
the reaction partners, of ca. 0.5 ms for the association of Mb with O2.
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 cm2/s (20, 35, 36, 37), and an
additional calculation was performed with the DMb of 8 ⫻ 10⫺7 cm2/s
derived from NMR measurements for rat cardiac cells by Lin et al.
(26, 27). DO2 in muscle tissue was taken to be DO2 ⫽ KO2/␣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 cm2/s, where KO2 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 PO2 values were set to be 10 Torr at x ⫽ 0 and 0.1
Torr at x ⫽ ᐉ, thus approximating the most favorable conditions for
facilitated O2 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 kA value of 3.5 ⫻ 106 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 kD is 250 s⫺1 and kA is 6.1 ⫻ 106
M⫺1·s⫺1 (with an O2 solubility of 1.5 ⫻ 10⫺6 mol·l⫺1·mmHg⫺1 as
mentioned above, and the ratio of kD/kA corresponds to an O2
half-saturation pressure for hemoglobin of P50 ⫽ 27 Torr). This kD for
Hb is markedly faster than that used in the case of Mb. DO2 inside the
red cell is taken to be the same as in muscle tissue. DHb within the red
cell is 6.4 ⫻ 10⫺8 cm2/s (31). Intraerythrocytic hemoglobin concentration is 20 mM Hb monomer. The boundary PO2 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 PO2 values given above
were assigned to x ⫽ 0 (PO2 of 10 Torr) and x ⫽ ᐉ (PO2 of 0.1 Torr),
respectively. Throughout the interior of the layer, a homogeneous
⭸ 关O2兴/ ⭸ t ⫽ DO2 · ⭸2关O2] ⁄ ⭸ x2 ⫺ kA[Mb][O2] ⫹ kD[MbO2] (5)
The analogous consideration applies to the change of MbO2 concentration with time:
⭸ 关MbO2兴/ ⭸ t ⫽ DMbO2 · ⭸2关MbO2] ⁄ ⭸ x2
(6)
⫺ kA[Mb][O2] ⫹ kD[MbO2]
A layer of Mb solution of defined thickness ᐉ (treated as a plane
sheet) is considered, and, by numerically solving Eqs. 5 and 6,
temporal and spatial courses of [O2] and [MbO2] within this layer are
obtained. After the calculations have been continued for a sufficient
time interval, steady state is reached, the concentration profiles of O2
and MbO2 in the layer become time-independent, and the fluxes of
free and Mb-bound O2 as well as the total O2 flux have assumed their
final steady-state values. The total flux of O2 across the entire layer is
obtained from the fluxes of O2 leaving and entering, respectively, the
layer at the boundaries. The average free flux of O2 across the layer
is calculated by using the boundary concentrations of O2, [O2]x⫽0 and
[O2]x⫽ᐉ, from DO2·([O2]x⫽0 ⫺ [O2]x⫽ᐉ). The difference gives the
average facilitated flux across the layer. Free and facilitated O2 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):
⭸ 关O2兴/ ⭸ t ⫽ DO2 · ⭸2关O2] ⁄ ⭸ r2 ⫹ DO2/r · ⭸ [O2兴/ ⭸ r
(7)
⫺ kA[Mb][O2] ⫹ kD[MbO2]
⭸ 关MbO2兴/ ⭸ t ⫽ DMbO2 · ⭸2关MbO2] ⁄ ⭸ r2 ⫹ DMbO2/r · ⭸ [MbO2兴/ ⭸ r
(8)
⫹ kA[Mb][O2] ⫺ kD[MbO2]
where 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 O2 (free) and MbO2 (facilitated) were
again calculated as described above from the total O2 flux across the
cylinder surface and from the boundary concentrations of dissolved
O2 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).
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where DO2 is the diffusion coefficient of O2 in the medium considered,
DMbO2 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 O2, it can be stated that, in a certain
infinitesimal volume element, at x the change of O2 concentration with
time is equal to the net amount of O2 diffusing into this element,
DO2·⳵2[O2]/⳵x2, minus the net amount of O2 consumed within the
element by reaction with Mb:
(10)
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Limitation of Facilitated O2 Diffusion by Chemical Reaction
•
Endeward V
initial PO2 of 0.1 Torr was set, with the exception of x ⫽ 0, which was
set to 10 Torr. In the case of hemoglobin, x0 was set to 100 Torr, and
the interior of the layer as well as x ⫽ ᐉ were set to 40 Torr. The layers
were usually divided into N ⫽ 200 sections, i.e., ⌬x ⫽ ᐉ/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 ⌬x from the following relation (9):
⌬t Ɱ (⌬x)2 ⁄ (2DO2)
(11)
Fig. 1. %Facilitation in a layer of 0.19 mM myoglobin solution of a thickness (ᐉ)
of 5 ␮m as a function of time. The boundary PO2 values are held at 10 and 0.1 Torr,
and the diffusion process starts at time ⫽ 0. DMb ⫽ 2.10⫺7 cm2/s. %Facilitation
initially is 0, because the calculation is based on the total flux of O2 leaving the
layer at ᐉ, a value which is initially 0, because initially there is no PO2 gradient
between the interior of the layer and the boundary ᐉ. After ⬃0.2 ms, a plateau is
reached, indicating that a steady state with time-independent concentrations of Mb,
MbO2, and O2 inside the layer is established. The plateau values were used for all
values of %Facilitation shown in Figs. 3–5.
imation 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 O2
causes a dependence of the degree of facilitation of O2 diffusion upon the length of the diffusion path.
When the oxygenated hemoprotein and O2 diffuse alongside
through a cell or layer of solution, albeit at different concentration gradients and diffusivities of protein and O2, facilitated
diffusion can develop by association reaction between hemoprotein and O2 (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 O2 by the hemoprotein over a certain diffusion
distance, a process called facilitated diffusion, as well as its
timely release when and where the free O2 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 O2 to cross the entire
diffusion distance. This condition for a facilitation of O2
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 O2 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
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where it was ascertained that ⌬t was sufficiently small by making sure
that a further reduction of ⌬t did not affect the results. These values
of ⌬x and ⌬t, pertaining to a given value of ᐉ, were used in
conjunction with the following boundary conditions: 1) the boundary
PO2,x⫽0 was set to 10 Torr (as mentioned above, for the case of Mb);
2) the boundary PO2,x⫽ᐉ was set to 0.1 Torr (as mentioned above, for
the case of Mb); 3) the sum of [Mb] and [MbO2] was set to be [Mbtot]
everywhere in the layer, where [Mbtot] is the total Mb concentration
and a constant (0.19 mM in the case of Mb).
The boundary concentrations of Mb and MbO2 were obtained from the
calculation, but initial estimates were derived from [Mbtot] and the
boundary PO2 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 PO2, [Mb], and [MbO2] throughout the layer.
The calculation cycles were repeated until time-independent concentration profiles of [O2] and [MbO2], or [HbO2], respectively, were
obtained in the layer. For each calculation cycle, %Facilitation (at x ⫽
ᐉ), defined as ratio of facilitated flux over free flux in the last volume
segment adjacent to x ⫽ ᐉ, FMbO2/FO2, or FHbO2/FO2, was calculated in
the following way: in the last segment, adjacent to x ⫽ ᐉ, the total flux
of O2 leaving this segment was determined for each point of time from
the balance between the O2 entering the segment and the O2 consumed/produced by chemical reaction. After steady-state conditions
have been established, the total flux leaving this element is identical
to the total O2 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 O2 flux and the free O2 flux as determined from the overall
PO2 difference PO2,x⫽0 ⫺ PO2,x⫽ᐉ and DO2. 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 ␮m that %Facilitation (x ⫽ ᐉ)
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 ᐉ, 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-␮m-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 O2 and MbO2 in radial direction in
a cylinder, Eqs. 7 and 8 were solved by an analogous finite difference
method in MATLAB 2008b. The PO2 at the outer circumference of the
cylinder (r ⫽ a) was set to 10 Torr. PO2 of 0.1 Torr was set to be either
in the center of the cylinder at r ⫽ 0 (implying the O2 sink of the
muscle fiber to be located in the center), or, as a reasonable approx-
Limitation of Facilitated O2 Diffusion by Chemical Reaction
Fig. 2. Concentration profiles of O2, MbO2, and Mb across a 3.5-␮m-thick
layer of muscle tissue, calculated with the conditions and constants specified in
METHODS. The diffusion path chosen is representative for cardiomyocytes.
Continuous curves: calculated with kD ⫽ 60 s⫺1 and kA ⫽ 15.4 ⫻ 106
M⫺1·s⫺1; dashed curves: calculated with the 1,000-fold value of both constants.
Endeward V
1469
facilitated O2 diffusion in muscle cells, and the boundary
conditions listed above. As discussed below, 3.5 ␮m represents
a reasonable estimate of the effective diffusion path in cardiomyocytes. Figure 2 shows the gradients of [O2], [MbO2], and
[Mb] established under steady-state conditions for a case in
which facilitation is markedly limited by the rate of MbO2
deoxygenation. The boundary concentrations of O2 of course
reflect the inserted boundary PO2 values mentioned above. In
the curve representing the [O2] gradient calculated with kD ⫽
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 O2 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 O2 under steady-state conditions
must be identical everywhere along the diffusion path. The
dashed curve shows the [O2] gradient calculated for values of
kD and kA 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 [O2] 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, [MbO2] 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 MbO2 at the
right-hand boundary due to a limitation of MbO2 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 MbO2 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 O2 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 ᐉ. The intracellular Mb diffusion coefficient employed is 2 ⫻ 10⫺7 cm2/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 ␮m, %Facilitation is ⬍2%,
and half-maximal facilitation is attained at an ᐉ slightly below
3 ␮m. Figure 3 also shows that full maximal facilitation is not
yet reached at ᐉ ⫽ 25 ␮m, where it amounts to 12%, i.e., 6/7
of the maximum. Figure 4 shows that the maximum is not even
completely established at ᐉ ⫽ 100 ␮m.
Facilitation in cardiomyocytes. What is the implication of
these results for Mb-facilitated O2 diffusion in cardiac muscle
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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 ⫻ 106 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 O2 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 O2
diffusion in muscle. The example was calculated for a diffusion path length of 3.5 ␮m, the constants as defined above for
•
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Limitation of Facilitated O2 Diffusion by Chemical Reaction
cells? In the case of human heart tissue, Armstrong et al. (2)
determined an average radius of cardiomyocytes in intact
cardiac tissue of 7 ␮m. From published capillary densities,
Endeward et al. (10) derived the radius of the Krogh cylinder
used to describe O2 supply to cardiac tissue to be 10 ␮m, 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 O2
within the cardiomyocyte is expected to be of the order of 7
␮m, and often, most pronounced in the case of subsarcolemmal
mitochondria, considerably less. Using a path length of 7 ␮m,
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 O2 in the
cardiomyocyte to be half the fiber radius, 3.5 ␮m, %Facilitation is only 7.6% or slightly more than one-half of the facilitation in the absence of reaction limitation. In conclusion,
facilitated O2 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 PO2 gradients and O2 and Mb diffusivities ignoring the finite speed of the
O2-Mb kinetics.
The cardiomyocyte as a cylinder. Considering the cardiomyocte as a cuboid with two rather than four sides accessible to O2 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 ␮m. In this
case, it had to be assumed that O2 has access to the cell on
the entire cylinder surface, i.e., PO2 was set to 10 Torr everywhere at the cylinder surface and O2 diffusion occurred from
the surface toward the interior of the cylinder. This is of course
also a major simplification since, physiologically, O2 supply is
not present equally at the entire circumference of the cylinder.
Two cases were considered. First, the “O2 sink” was assumed
to be in the center of the cylinder, at r ⫽ 0. This results in an
Endeward V
extremely small degree of facilitation of O2 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 O2
is very shallow in the outer regions of the cylinder, and Mb
there remains almost fully saturated. The gradients of PO2 and
MbO2 begin to become steeper just before the center of the
cylinder where PO2 ⫽ 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 O2 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 O2 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 O2 sink moves toward the cylinder
surface because of the decreasing diffusion path length. As can
be appreciated from Fig. 3, O2 transport to subsarcolemmal
mitochondria, ⬃1 ␮m 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 ␮m. O2
supply occurs essentially in a radial direction (19). Jürgens et
al. (19) derived a Krogh cylinder radius of 24 ␮m for the
human quadriceps femoris muscle. The diffusion distance to be
overcome by O2 then will be ⬃25 ␮m 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 ␮m, %Facilitation becomes 11% or 4/5 of
the maximum predicted for infinite reaction speed. With the
Fig. 4. Steady-state %Facilitation in layers of 0.19 mM myoglobin solution as
a function of layer thickness. Boundary PO2 values are 10 and 0.1 Torr. Solid
lines show the calculated results for DMb ⫽ 2 ⫻ 10⫺7 cm2/s; dashed lines are
for DMb ⫽ 8 ⫻ 10⫺7 cm2/s. Maximal facilitations are proportional to DMb, but
the results for DMb ⫽ 8 ⫻ 10⫺7 cm2/s are more strongly limited by the
diffusion path length, indicating an increased requirement for reaction speed
when Mb diffusivity is higher.
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Fig. 3. Steady-state %Facilitation in layers of 0.19 mM myoglobin solution as
a function of layer thickness. Boundary PO2 values of 10 and 0.1 Torr, DMb ⫽
2 ⫻ 10⫺7 cm2/s. Maximal facilitation under these conditions is 14%; halfmaximal facilitation of 7% is obtained at a layer thickness of ⬃3 ␮m.
•
Limitation of Facilitated O2 Diffusion by Chemical Reaction
Endeward V
1471
conditions at x ⫽ 0 is consumed within this half of the sheet.
The path length (equal to half-thickness) in these calculations
was 7 ␮m. This yielded the gradients of [O2] and [MbO2]
under conditions of maximal O2 consumption of cardiomyocytes, which were used to calculate free and facilitated fluxes.
Accordingly, in the maximally respiring cardiomyocyte at
capillary PO2 ⫽ 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 O2 diffusion in this case is of even lesser
significance than when we consider merely O2 diffusion without
O2 consumption. Besides by the reaction limitation of facilitation,
this is caused by boundary PO2 values at x ⫽ ᐉ that turn out to be
⬃4 Torr (cuboid) and ⬃7 Torr (cylinder), respectively. This
implies that, even at maximal O2 consumption, in conjunction
with PO2,x⫽0 ⫽ 10 Torr, a PO2,x⫽ᐉ substantially above 0.1 Torr is
maintained in the cardiomyocyte, thus further reducing the contribution of facilitated O2 diffusion.
Dependence of hemoglobin-facilitated O2 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 O2
diffusion inside cells is limited by the speed of the Hb-O2
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 ␮m, and, since gas
is taken up from both sides of the disc, the relevant diffusion
path for O2 is half of this value, 0.8 ␮m (11). For this diffusion
path, Fig. 5 predicts a facilitation of ⬃13% for the conditions
given above including the boundary PO2 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-O2 reaction decreases facilitation to 64% of the maximum.
Similar calculations for the facilitation of steady-state O2
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 ␮m and similar boundary PO2 of 50 and 125
Fig. 5. Steady-state %Facilitation in a 20 mM hemoglobin solution (as it exists
inside red cells) as a function of layer thickness. Boundary PO2 values of 40
and 100 Torr, DHb ⫽ 6.4 ⫻ 10⫺7 cm2/s. For 0.8 ␮m, the half-thickness of the
human red cell, a facilitation of ⬃13% is calculated, which represents ⬃64%
of the maximal facilitation of 20.3%.
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same Mb diffusion coefficient of 2 ⫻ 10⫺7 cm2/s, maximal
facilitation is not even reached with the radius of a very thick
skeletal muscle fiber of 50 ␮m, 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
O2 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 O2 diffusion than those of cardiomyocytes.
Effect of DMb on facilitation. Figure 4 illustrates the effect of
DMb on these calculations. Lin et al. (26, 27) reported a DMb in
cardiac tissue of 8 ⫻ 10⫺7 cm2/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 cm2/s for 37°C. As
recently discussed in a joint review (17), the higher DMb
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 DMb ⫽ 8 ⫻ 10⫺7 cm2/s and shows, first,
that maximal facilitation of course is fourfold greater than with
DMb ⫽ 2 ⫻ 10⫺7 cm2/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 ␮m, as used above for
the heart, one finds 7.6% facilitation with DMb ⫽ 2 ⫻ 10⫺7
cm2/s, or 54% of the maximum, whereas with DMb ⫽ 8 ⫻ 10⫺7
cm2/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
DMb value, the conclusion remains that for ᐉ ⫽ 3.5 ␮m,
roughly only about one-half of the maximal facilitation is
achieved. It is concluded from these considerations that Mbfacilitated 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 O2 consumption on facilitation. The presented
computational approach does not take into account the fact
that, in muscle tissue, intracellular O2 transport is combined
with intracellular O2 consumption. Can O2 consumption affect
the role of Mb-facilitated diffusion? We have incorporated into
the above equations an O2 consumption that is homogeneously
distributed across the sarcoplasm. To estimate the effect of O2
consumption on facilitation, some calculations were performed
for the following altered conditions: 1) PO2 at x ⫽ 0 was held
at 10 Torr, as above; 2) a maximal specific cardiac O2 consumption of 600 ml O2·min⫺1·kg⫺1 (10) was inserted; 3) the
total flux of O2 at x ⫽ ᐉ 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 O2 from
one side of the sheet, and no O2 diffuses into the other half of
the sheet, i.e., all O2 entering the sheet under steady-state
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Limitation of Facilitated O2 Diffusion by Chemical Reaction
Endeward V
myocardial PO2 drastically falls, and substantial intracellular gradients of MbO2 develop (10). In this phase of the cardiac cycle,
MbO2 plays a role both as an O2 store and as an intracellular O2
transporter. Endeward et al. (10) estimated that systolic O2 supply
of the left ventricular myocardium is supported to the extent of
⬃20% by Mb-facilitated O2 diffusion and of 10% by the
storage function of Mb, whereas the remaining 70% is provided by O2 bound to capillary Hb and by O2 dissolved in the
tissue. It should be noted that these calculations were based on
the assumption that Mb-facilitated O2 diffusion in cardiomyocytes is not limited by Mb-O2 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 O2 supply. Thus O2 sources in the tissue other than
MbO2 would be even more dominating in this phase. A
noteworthy aspect of these findings is that facilitated O2
diffusion in the heart occurs only in conjunction with a depletion of the MbO2 store and then presumably serves to support
the delivery of the released O2 to the mitochondria. These
combined and about equal effects of Mb as an O2 store and as
an O2 transporter should then be responsible for the compensatory adaptations observed in Mbnull 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 PO2 and MbO2 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-steadystate conditions, although calculations on the basis of a Krogh
cylinder model predict that this facilitation is of minor importance for sarcoplasmic O2 transport even under conditions of
heavy exercise (19).
It has been shown here that facilitated O2 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 O2 storage function of Mb in the heart?
Above, estimates are given of the half-times of MbO2 dissociation (12 ms) and of Mb association with O2 (⬃0.5 ms). Can
the rate of MbO2 dissociation limit the rate at which O2 is
released from MbO2 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 O2 from the MbO2 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.
ACKNOWLEDGMENTS
I thank Prof. G. Gros (Hannover) for proposing this study, continuous
discussion, and critical reading of the manuscript.
GRANTS
This work was partly supported a grant from the Deutsche Forschungsgemeinschaft, EN 908/1-1.
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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 O2 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 O2 flux in the presence of CO.
Moll (33) estimated that the CO-dependent O2 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 O2 diffusion, which should have been apparent in 165- to
300-␮m-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 O2 diffusion occurring from
one side of the layer to the other, the effective intracellular
diffusion distance is 1.6 ␮m in red cells oriented in parallel to
the plane of layer rather than the half-thickness of 0.8 ␮m and
may be as great as 8 ␮m 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 O2
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 ␮m but is markedly decreased at a thickness of
⬃25 ␮m. 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 ␮m
and causes a decrease in %Facilitation to 45% of the maximum
at a layer thickness of 5 ␮m. (It may be noted that these data
and calculations pertain to boundary PO2 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 O2 diffusion in Hb solutions starts to become reaction limited at layer thicknesses of
⬍100 ␮m 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 O2 store vs. a cellular O2 transporter. Applying a modified
Krogh cylinder model, we previously have shown that, in the
heart, myoglobin-facilitated O2 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 O2 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,
•
Limitation of Facilitated O2 Diffusion by Chemical Reaction
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.
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