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Factors influencing facilitated diffusion of oxygen in the presence of hemoglobin and myoglobin
Respiration Physiology (1972) 15, 104-124;
North-Holland Publkhing Company, Amsterdam
FACTORS INFLUENCING FACILITATED DIFFUSION OF OXYGEN IN THE PRESENCE OF HEMOGLOBIN AND MYOGLOBIN’
F. KREUZER
and L. J. C. HOOFD
Department of Physiology, University of Nijmegen, Nijmegen, The Netherlands
Abstract. The experimental results concerning the facilitation of oxygen diffusion in the presence of hemoglobin or myoglobin may be explained by the simultaneous diffusion of oxyhemoglobin or oxymyoglobin. In order to prove this assumption it should be possible to show that there is quantitative agreement between the experimental values and those computed using the physical parameters obtained independently. The present authors previously presented an approximate solution for the basic equations of diffusion including chemical reactions; an improved solution is provided in the present paper. Since the agreement between experiments and computer solutions was much less satisfactory in the system Mb + 0, than in the system Hb + O,, we investigated the effects of various other factors of possible importance for both systems of Hb and Mb. These computations led to the following conclusions: 1. The values of the pigment diffusion coefficients are of crucial importance; 2. The knowledge of the actual diffusion path length is also important; a quantitative theoretical treatment is presented; 3. Concerning the chemical reaction rates the dissociation constant is more likely to be limiting; 4. The effect of a possible and even minute oxygen back pressure on the low pressure side may be crucial in the case of the system Mb+O,; 5. The nonequilibrium approach suggested here is valid more generally than the equilibrium assumption made in most previous cases; the latter is apt to provide too high computed fluxes; 6. Contrary to previous presumption there is a decrease of the facilitated oxygen flux with high values of Po,, particularly in the system Mb+O,. These results might provide new insight in the mechanism of facilitated oxygen diffusion and suggest more appropriate experimental arrangements in future investigations. Facilitated diffusion Hemoglobin
Myoglobin Oxygen
The experimental results of Scholander (1960) and Wittenberg (1966) concerning the facilitation of oxygen diffusion in the presence of Hb or Mb may be explained on the basis of the assumption that HbOz or MbOz diffuses simultaneously with oxygen. Acceptedfor publication 30 December 1971.
’ Part of the results of this study has been presented at the Alfred Benzon Symposium IV: Oxygen atlinity of hemoglobin and red cell acid-base status, Copenhagen, Denmark, May 17-22, 1971. and at the Symposium on Passive permeability of cell membrane, Rotterdam. The Netherlands, July 2&22, 1971. 104
FACILITATED
DIFFUSION
OF OXYGEN
105
In order to prove this assumption it should be shown that there is quantitative agreement between the experimental values and those computed using the physical parameters obtained independently. Previous attempts assuming chemical equilibrium between oxygen and carrier protein throughout the layer were not conclusive. Kreuzer and Hoofd (1970a) presented an approximate solution for the basic equations of diffusion including chemical reactions and applied the resulting expressions to numerical computer solutions for the system Hb + 0, in a film at steady state similar to the experimental situation of Wittenberg (1966) using recent numerical values for the diffusion coefficients of oxygen and Hb in Hb solutions of various concentrations. The facilitated fluxes thus computed agreed well with the experimental data of Wittenberg (1966) when choosing well-founded values of the Hb diffusion coefficients. The authors concluded that the mathematical treatment of facilitated diffusion of oxygen based on simultaneous diffusion of oxygen and oxyhemoglobin provides a quantitative description of the experimental results for the system Hb+ 0, if the chemical reaction rates were taken into account. A subsequent similar study by Kreuzer and Hoofd (1970b) for the system Mb+ 0, however revealed a much less satisfactory agreement between computation and experimental data. These findings prompted us to further compare the two systems Hb+ O2 and Mb+ O2 with particular reference to the influence of all possibly important factors. It might be expected also that one or the other factor could be of different weight in the two systems in view of their inherent differences. This might shed new light on the importance of the various factors in a more general sense. We therefore investigated, always comparing the systems Hb + 0, and Mb + O,, the effects of varying the following parameters: 1) the diffusion coefficients of HbOz or MbOl, D,, and D,, resp.; 2) the respective chemical dissociation constant k; 3) the back pressure of oxygen on the low pressure side ; 4) the layer thickness L ; 5) the presence or absence of chemical equilibrium throughout the layer; 6) the oxygen pressure on the high pressure side. It will be shown that these comparisons lead to reasonable predictions concerning the discrepancies still prevailing, also with respect to details of the experimental arrangement to be considered in future investigations. Mathematical methods The basic approach was the same as in our previous paper (Kreuzer and Hoofd, 1970a). We assumed again the presence of three regions in the layer: a middle bulk very near equilibrium and two marginal regions not at equilibrium. The solutions for these two boundary layers were improved as described below. It may be shown that this approach is also applicable to the case of Mb + OZ. The two basic equations are at steady state:
(14
D,,
w =
k’ [R] [0,] - k [RO,]
F. KREUZER
106
(lb)
D
P
!!t&ki! = k[RO
2
dx2
AND
L. 3. C. HQOFD
]_k’[R][()
2]
where p] = pigment con~ntration~ PO,] = oxygenate pigment ~n~ntration and [ 0,] = oxygen concentration ; Do, and D, =diffusion coefficient of 0, and pigment, resp. ; k’ and k = chemical association and dissociation constant, resp. In the boundary regions of x near 0 and x near L we may write : (2)
Lo21 = W21°
-A(x)
where [02]0=solution for the oxygen concentration in the middle part of the slab, extended to the boundary regions. Superscript zero always indicates the concentration in the core of the slab. A(x) is a correction to [O,]’ at the boundaries, both A and [O,]’ being functions of position x. A(x) disappears when x is “far’” from x =0 and x = L but may be important at x near 0 and x near L. Since the addition of eqs. (la) and (1b) is zero we have for [RO,] : [RO,] = [RO,]”
+ +
A(x). P
Since PO,] W
+ @] = [Ro] = total pigment ~n~ntration
(with Dao, = D, = DJ :
[R] = [RI0 - +-’ A(x). P
After inserting into eq. (la) we get : D
‘=
d2i?i?_ dx2
D
d’dfx)=
[R]O
k’
O2 dx2
I
WI
-
- 2
k { [RO,]’
P
A(x)]
+ 2
P
jTO,]‘A(x))
A(x) ) .
A(x) is a rapidly varying function whereas the marginal extensions of CO,]‘, CR]“, and @O,]” are practically constant in the regions where A(x) is important, and there is equilibrium among them so that [RI0 [O,]’ = [RO,]‘. Thus the equation above may be rearranged to give:
(41
,-, WE O2 dx2
k’~R]“~k’~~O*]o+k~~A(x~-k~~A(x)2. P
P
P
There is an exact solution for eq. (4) in the case where [RI” and [O,]” are nearly constant and if there is a region where both A(x) and dA(x)/dx tend to zero, i.e.: for x near 0: 6iu$Dpqoe-“oX A(x) = (5.1) k’(1 +q0e-“0”)2 where
FACILITATED
DIFFUSlON
OF OXYGEN
107
for x near L ;
where 16.2j
~,
~
c
k’[gKI:= k+$&, L
02
+
P
J* /
The values of q. and q1 and, from these, of [&jr= o etc., follow from the boundary cmditions, in particular from those requiring that there is no flux of HbOz across the surlkc, i.e. :
Therefore the total flux of oxygen must be carried by free oxygen only, i.e.: {‘7c,d) For x= 0 inserting eqs. (2) and (5.1) into (7~) gives:
whereas (7a) for x=0 may be written as :
10s
F. KREUZER AND L. J. C. HOBFD
For the respective oxygen pressures we get at x = 0 :
and at x=L: (11.2)
YR, = [021xo=L+
6a:q,D, k’(1 -q1)2 *
Thus the six eqs. (6), (lo), and (11) together with the exact solution for the total flux of 0,:
(14
F =
DC&‘- PO)+ L
943021x=,L
l3021,=0)
contain seven unknown quantities: Q, xi, q,, qi, [O,]:,,, [O,]E=t, and F, which may be determined in a computer program. Since q,, and qi are small quantities the terms (1 + q0e-o*)2 and (1- qi e-u1(L-X))2in the denominator of eq. (5.1) and eq. (5.2) resp. introduce only small corrections to A(x) in most cases. These terms become important only if either q. and q, are not small and/or A(x) is comparable to [OJO, or if even a small change in [OJ’ (as a virtual extension to the bunco corresponds to a large change in fRO,]“, as in the case of Mb with its steep dissociation curve. We will see below what the computed consequences are for Hb and for Mb. The solution presented here is superior to that offered previously in that it provides a nearly exact description of the oxygen concentration in the boundary layers because even relatively large variations of [R] are taken into account whereas [R] was kept constant previously. In a similar manner as in our previous paper the course of [O,]” and fRO,]” throughout the whole membrane may be determined from: (13)
Fx + B = Do2 [0,] + D, PO,]
which becomes with eqs. (2) and (3): (13’)
Fx + B = D,, [OJ”+
D, [RO,]”
where B = Do,[O,]:=,+D,~O,]x”=,
.
Concerning the validity of these solutions see Appendix. comptl?f program The program was written in a slightly modified version of FORTRAN adapted to the WATFIV compiler used at the University of Nijmegen. The input list included, expressed by the appropriate symbols: thickness of membrane (L) in em, reaction Hb concentrations in g%, diffusion coefftconstants k in set- ’ and k’ in ml/M/~ cients of Hb and 0, in cm’/sec at the respective Hb concentrations, initial 0, pressure
FACILITATED
DIFFUSION
OF OXYGEN
109
(PO at x =0) and 0, pressure on the other side (P at x = L). The physical parameters were those at 25 ‘C. The output list contained, again expressed by the appropriate symbols: total flux of 0, (F), O2 flux by plain diffusion, facilitated O2 flux, and the virtual pressures at x=O,L([OJ”/r at x=0 and x = L), followed by a list of O2 pressures and pigment saturations at various points within the membrane as chosen before. The construction of the program included three parts. The first part was made to read and check the input data. The second part contained an iteration process to calculate ao, a,, q,, q,, [OJ~,o, [O&‘=,_, [ROJXEo, [ROJx=,_ and F in the following order : from eqs. (1 l), and from these [RO&‘=,,, and [R]z=o,L since there 4 PzlxO=o.L is equilibrium among them ; b) WJ~=O.Lfrom eq. (3a) and eqs. (5) at x=O,L; c) F from eq. (12); d) x0, ai, from eq. (6); e) q,, q,, from eqs. (10) using the approximation (1 fq)3 ‘v 1 f 3 q + 3 q2 (since q is small) and solving the resulting quadratic equation for q. The iteration process was started with the values q. = q i = 0 and continued until the computed results for q,, qi changed by less than 0.01 ‘A. In the third part the values of [O,] and [ROJ were computed at certain points of the membrane. Results and discwsion When plotting the facilitated flux against Hb concentration (Kreuzer and Hoofd, 197Oa, fig. 9), there is good agreement between the experimental values of Wittenberg (1966) and the computed line when using the D,, values of our compromise curve for the computations (see fig. 3 in Kreuzer and Hoofd, 1970a). In a similar plot for Mb however the computed values of the facilitated flux are almost twice as large as the experimental points of Wittenberg (1966) when using the Dhtb data of Riveros-Moreno and Wittenberg (1968). A similar conclusion was reached by Murray (1971). We also examined the computed curve obtained by using DM, values calculated from our D,, compromise curve by the ratio of the square roots or of the cubic roots of the molecular weights (the latter probably being more justified for high-molecular substances). These computed lines come closer to the experimental points, particularly when using the cubic root ratio, but the agreement remains poor, the computed curve still being too high. We shall try now to get an impression concerning the. factors possibly influential in rationalizing the difference in behavior between Hb and Mb. Wittenberg (1966) pointed out that the rate of dissociation should be particularly important in limiting the rate of facilitation since pigments of appropriate molecular size (ascaris body wall Hb and succinyl ascaris perienteric fluid Hb) but with very small rate of O2 dissociation do not facilitate O2 diffusion. We therefore investigated the effect of changes in the’dissociation constant k (the association constant k’ being held unchanged) on the facilitated flux.
110
F. KREUZER AND L. 1. C. HOOFD facilitated flux (10-10 M /cm2/sec)
Fig. 1. Influence of the dissociation constant k on the facilitated O;, ffux in Mb solutions of various conantrations. The figure shows two sets of computed curves. The three curves on top were computed using the DHb values of Riveros-Moreno and Wittenberg (1968) and the three lower curves by applying the D,, values obtained from the DHbcompromise curve of Kreuzer and Hoofd (1970a) by the ratio ot’the cubic roots of the molecular weights. The values of k of 17.5, 11. and 5.5 set-’ used in both sets of curves are indicated in the figure. L= 180 p. PO,= 160 us 0 mm Hg. k’= 14 x IO9ml/M/set.
The general effect of changes in k is very small for the system Hb + O2 {not shown here). The situation is quite different for Mb+& (fig. 1). A decrease in k greatly reduces the facilitated flux as expected, the absolute values of k being smaller than in the case of Hb+ Oz. The generally accepted value of k is 11 see- * for Mb but the closest approach to the experimental points is found &ith k of about 5.5 see- ‘, Dhtb having been calculated from the compromise D Hbby the cubic root ratio as described above.
FACILITATED Hb+ 02 facilitated flux (lo-” M /cm*/ set
DIFFUSION
111
OF OXYGEN
)
1.5
01 0
5
10
15
20 25 PO2 back pressure (mm HQ )
Fig. 2. Influence of back pressure (abscissa) on the facilitated 0, flux in a Hb solution D,,=2.016x10~‘~2/sec.Computedcurveobtainadwithk’=3x109ml/M/secandk=42.5sec~‘.L= 186 p. Po, at high pressure side = 160 mm Hg.
of
18 g/100 ml with
Wittenberg (1969) and Kreuzer (1970) have discussed the possible influence of O2 back pressure on the low pressure side concerning the experimental facilitated flux or, in other words, the possible interference by inadequate stirring in the low pressure gas compartment where PO, was supposed to be zero in the preceding cases. Figure 2 shows the general trend of the influence of back pressure for Hb + OZ. A similar pattern was found by Kutchai, Jacquez and Mather (1970, Tab. IV) ; it should be noted however that the two approaches are quite different; we have developed an analytical treatment whereas that of Kutchai et al. (1970) is purely numerical. The finding that there should still be some facilitation at back pressures where experimentally a complete disappearance of any enhancement was found (i.e. above 10 mm Hg Po, back pressure) again points to the discrepancy mentioned by Hemmingsen and Scholander (1960) that there was no facilitation with a back pressure of 10 mm Hg although the Hb on the low pressure side would be only 5565% saturated at this partial pressure and a considerable saturation gradient would thus remain across the layer (Kreuzer, 1970). Wittenberg (1970) explains this discrepancy by the fact that the local oxygen pressure adjacent to the low-pressure face was much higher than measured in these experiments due to insufficient stirring of the gas phase. It is calculated that a back pressure of 2.8 mm Hg would be required for a fit of the computed curve with the experimental points when using the D,, values of Riveros-Moreno and Wittenberg (1968). Such a back pressure could easily be detected and is unlikely to have been present in the experiments of Wittenberg (1966). The general course of the influence of the back pressure for Mb + O2 is presented in fig. 3 (note the difference in scale on the abscissa!). Figure 4 shows that a back pressure of only 0.12 mm Hg is required for a good tit of the best curve, calculated with Dh(,,values obtained from our compromise
112
F. KREUZER Mb*02 facilitated (1CI-’ lo
AND
L. J. C. HOOFD
f Iux
M /cd/set
1
2.c
1.5
1.0
0.5
0
oc
5
1
1.5
2
PO2 back Fig. 3. Influence
of back pressure
with D,, = 6.79 x IO-’ cm’!sec lo-’
cm2/sec
from D,,
(abscissa)
compromise
cubic roots of the molecular
on the facilitated
from Riveros-Moreno
weights
and Hoofd
of 15 g/l00
ml
(1968) in top curve or D,, = 3.98 x (19%)
obtained
curve. L= 180 p. PO, at high pressure
k’= 14x log ml/M/set.
Hg )
(mm
flux in a Mb solution
and Wittenberg
curve of Kreuzcr in bottom
0,
2.5 pressure
by the ratio of the side=
160 mm Hg.
k= 11 sect.
D,, by the ratio of the cubic roots of the molecular weights, and of 0.6 mm Hg for a fair agreement when using the D,, values of Riveros-Moreno and Wittenberg (1968). Thus the experimental values of the facilitated flux are much more sensitive to back pressure in the case of Mb+ OZ. The assumption ofa specific value for the effective thickness L of the layer is another factor affecting the computed results of the facilitated flux. The actual thickness of the Millipore filters considered here is 150 ,Ubut only SO’%are liquid space ; therefore Wittenberg (1966) as well as Kreuzer and Hoofd (1970a) have worked with a corrected, thickness of 180 p. Murray (1971) allowed for the tortuosity of the channels and assumed a thickness of 220 ~1or about 50% more than 150 ~1.We have reconsidered this problem in the following way. Assume that the diffusion path across the layer consists of N partial paths 6i running under an angle of Bi to the direct path. Then : N
(14)
L = c hi cos pi. i=l
FACILITATED
facilitated ( lo-‘O
0
DIFFUSION
113
OF OXYGEN
flux
M /cm2/sec
1
5
10
0
15
20
25
30 35 Ck(b(g/lOO ml
)
Fig. 4. Influence of back pressure on the facilitated O2 flux in Mb solutions. The two top curves were calculated with a back pressure of zero for comparison, with D,, from Riveros-Moreno and Wittenberg (1968) or from the compromise curve of Kreuzer and Hoofd (1970a) by the ratio of the cubic roots of the molecular weights respectively, the two bottom curves for a back pressure of I mm Hg with the same respective sets of D,, values. The two middle curves were computed to give the best possible agreement with the experimental points of Wittenberg (1966) by choosing appropriate values of back pressure, i.e. 0.6 mm Hg when using the Du, values from Riveros-Moreno and Wittenberg (1968) and as little as 0.12 mm Hg when using the D,, values from Kreuzer and Hoofd (1970a). L = 180 p. PO, at high pressure side = 160 mm Hg. k’= 14x109ml/M/sec. k=ll set-‘.
The length L’ of the diffusion path is:
(15)
L’=
; i=
6,.
1
Assuming that pi may have any value between -n/2 and + 7r/2and is proportionately
114
F. KREUZER AND L. J. C. HOOFD facilitated flux ( 10-‘” M /cm’/sec
0
)
I
10
5
15
20
25
30
35
CH~ (g/100
ml
)
Fig. 5. Influence of dilfusion path length on the facilitated O2 flux in Hh solutions. The path length of 150 p + 57 % = 235 p used here was obtained as described in the text. The three computed curves refer from top to bottom to the D,, values according to Riveros-Moreno and Wittenberg (1%8), Keller and Friedlander (1966). and Kreuzer and Hoofd (197Oa). PO,= 160 DS0 mm Hg. k’=3 x 10’ ml/M/set. k=42.5 se-‘.
distributed between these limits (i.e. the diffusing particles move around obstacles arranged at random) eq. (14) becomes: + n/2 L=i;I;j
cosj?dfi=L'
f
-n/Z
whence (16)
L=qL.
Since n/2 = 1.57 this implies that the mean diffusion path is 57% longer than the membrane thickness of 150 p(, i.e. 235 cc. Figure 5 shows a plot of the facilitated flux against Hb concentration with three sets of data for D,, and with a membrane thickness of 150 p+ 57 % =235 p. The facilitated fluxes are reduced by some 20x, and the D, data of Keller and Friedlander (1966) now provide the best fit. Similarly fig. 6 demonstrates the influence of membrane thickness on the facilitated flux for Mb + O2 ; here the fluxes have been computed for membrane thicknesses of 180 ~(150 p + 20 %) and 235 p (150 p + 57 %) using the D,, values of Riveros-Moreno and Wittenberg (1968) and those obtained
FACILITATED facilitated (lo-‘O
DIFFUSION
115
OF OXYGEN
flux
M/cm2/sec)
2.5-
L -150~1+20%
Wittenbeq(1966)
c*(b (g/loo
ml )
Fig. 6. Influence of diffusion path length on the facilitated 0, flux in Mb solutions. With both the D,, values from Riveros-Moreno and Wittenberg (1968; two top curves) and those from Kreuzer and Hoofd (197Oa; two bottom curves), the facilitated 0, fluxes were compared when using a path length of 150 p + 20 % = 180 p (as applied hitherto) shown by the curves ~ and -.-.--e, or a path length of 150 p + 57 % = 235 I( (see text) represented by the curves -----and --.--.-.The best tit with the experimental points of Wittenberg (1966) is obtained when using the D ub values from Kreuzer and Hoofd (1970a) and a path length of 235 p. PO, = 160 vs 0 mm Hg. k’= 14 x lo9 ml/M/set. k = 11 se-c- i.
from the D,, compromise curve by the ratio of the cubic roots of the molecular weights. The best agreement with the experimental points is obtained with the membrane of 235 p using the “compromise” D,, values. Figure 7 presents the gradients of PO, and saturation across the layer for Mb+ 0, with a PO, of 10 mm Hg on the high pressure side (and no back pressure) for three membrane thicknesses of 5, 50, and 500 cc. These plots confirm the pronounced influence of the membrane thickness on the computed facilitated fluxes, particularly in the range of thin layers, as also shown for Hb by Kutchai et al. (1970, fig. 1).
116
F. KREUZER AND L. J. C. HOOFD Mb+02 saturation (01.) 100
75
50
25
0
Fig. 7. Influence ofdiffusion path lengths of $50, and 500 ~1(from top to bottom) on the computed gradients of PO, (three bottom curves) and percent saturation (three top curves) across the layer ofa Mb solution of 15 g/100 ml with a D,, value of 3.98 x IO-’ cm’/sec. The three pairs of curves are indicated by the same line symbols. The curves are drawn to scale for the flux contribution. PO, - 10 us 0 mm Hg. k’ = 14 x IO9 ml/ M/xc. k=lI set-I. Kutchai (1970) computed, for the non-steady-state situation of the system Hb + O2 (at 37 Y’Cand pH 7.4), the half-saturation times tSO and plotted their logarithms against the logarithms of the reciprocal of the layer thickness L. For L greater than 1.6 p the graph was linear with slope -2, so that in this region t,, was proportional to Lz as to be expected if the diffusion of oxygen was the limiting process. For L less than 1.6 p the graph was curvilinear, probably because in this range diffusion of oxygen was so rapid that chemical reaction rates were more important in determining the overall rate of oxygen uptake. He also found that the influence of HbOt diffusion
FACILITATED
Hb+Oz facilitated flux (lO-‘” M/cm’/sec
DIFFUSION
117
OF OXYGEN
1
5-
2-
l-
.5-
.210
20
50
loo
Fig. 8. Difference betwee.n chemical equilibrium ing the calculated 100 ml with a D,,
200 500 1000 thickness of membrane
(top curve) and nonequilibrium
2cK3 L (u)
(bottom curve) conccm-
facilitated 0, flux as plotted against the diffusion path length in a Hb solution of 18 g/ of 2.016 x lo-’
cm2/sec from Kreuzer and Hoofd (1970a). PO, = 160 DS0 mm Hg. k’= 3 x lo9 ml/M/xc. k=42.5 set-‘.
rose with increasing L from very small values in extremely thin layers to a maximum above about 10 cc,presumably again reflecting the importance of reaction rates in the thinner layers and the primacy of diffusion in the thicker layers. However, for layers of the same thickness the contribution by Hb02 diffusion was less here than in the case of steady state. In our previous paper (Kreuzer and Hoofd, 1970a) we had obtained good agreement with the experimental results of Wittenberg (1966) when using the D,, values of our compromise curve and assuming absence of chemical equilibrium at the boundaries, whereas most previous calculations by other authors were based on the assumption of equilibrium throughout. The question arises now as to what influence on the computed facilitated flux is exerted by the presence or absence of chemical equilibrium. When plotting the computed curve of the facilitated fluxes in the presence of equilibrium as compared with the nonequilibrium values using the compromise DHb values, the equilibrium curve is located somewhat higher (as expected) but the difference of about 5 % is not important with respect to the experimental points ; thus it may be justitIed indeed to deal with the system Hb+ O2 equally well with the assumption of equilibrium. The difference between equilibrium and nonequilibrium is
118
F. KREUZER AND L. J. C. HOOFD
tocilitated
flux
(10-X)M/cm2/sec)
2.5
,/--_. /’ I
‘yquillbrium
I’
‘\ ‘\\
/
‘\
i
:
/
\
\
nonequlibrlum -. 1. 1.
.A’
\
\
\
\
1 ‘\
\
.
I
\
‘1. .
\
\
\
\
\. 1,
‘\\ \
.
‘\\ ‘\
‘\ ‘\ ‘\
.
Fig. 9. Comparison between chemical nonequilibrium and equilibrium for the facilitated 0, flux in Mb solutions of various concentrations with D,, values obtained from D,, compromise curve of Kreuzer and Hoofd (1970a) by the ratio of the cubic roots of the molecular weights. Top curve= equilibrium, bottom 14~10~ml/M/sec.k=Ilse.~~. curve = nonequilibrium. L = I80 p. Po, = 16OusOmmHg.k’=
a function of the membrane thickness, as also shown by Kutchai et al. (1970, table I and iig. 3); there is no difference with thick layers and the difference increases with decreasing thickness (fig. 8). The importance of assuming equilibrium or nonequilibrium becomes much more pronounced with the system Mb+ 0, as demonstrated by figs. 9 and 10. The inadequacy of the equilibrium approach is much more evident here than in the case of Hb + OZ. Figure 11 shows the P,,gradients for equilibrium, nonequilibrium, and equilibrium between the virtual pressures at the boundaries providing the same facilitated flux as in the case ofequilibrium for the system Mb + OZ. It had been concluded from all experimental determinations that when plotting the facilitated fluxes against Po2, a constant value was reached over a wide range of Po, above the region of low Po, where the facilitated flux has to drop to zero at a P,, =0 of course. Figures 12 and 13 (for Hb+ 0, and Mb+ O2 resp.) however shoti
FACILITATED Mb+02 facilitated flux (lo-lo M/cm2/sec)
DIFFUSION
119
OF OXYGEN
,
IO-
.2’
I
5
\ 10
20
50
100
200 thickness
500 of
1000 membrane
I
2000 L (JJ)
Fig. 10. Difference between chemical nonequilibrium (bottom curve) and equilibrium (top curve) conceming the calculated facilitated 0, flux as plotted against the diffusion path length in a Mb solution of 15 g/ 100 ml with a D,, of 3.98 x IO-’ cm’/sec obtained from the D,, compromise curve of Kreuzer and Hoofd (1970a) by the ratio of the cubic roots of the molecular weights. Po, = 10 OS0 mm Hg. k’= 14 x IO9 ml/M/ sec. k=ll set’.
that in caSe of the nonequilibrium approach a decline of the facilitated fluxes is to be expected with high values of Po2, particularly with the system Mb + Oz. Inspection of figure 1 in Wittenberg (1966) concerning the system Hb+ 02, where the fluxes are plotted against Po2, shows that the last of seven experimental points at a PO, of about 600 mm Hg does seem to have a tendency of lying a little lower than the straight regression line (no data of this kind are available for Mb+,02). Our curve of fig. 12 would predict a decline of about 2 % from the maximum value at a PO, of 600 mm Hg. Conclusioos 1. The values of the physical parameters are of crucial importance. The pigment diffusion coefficients determined by several authors in particular show considerable deviations and need to be known more accurately, especially also fon Mb where only one set of experimental data is available (Riveros-Moreno and Wittenberg, 1968); 2. The knowledge of the actual diffusion path length is also important. The use of Millipore filters may not be the method of choice due to their internal structure and the uncertainties as to their tortuosity; 3. Concerning the chemical reaction rates it is particularly the dissociation constant of the gas-pigment complex which is important for the flux ;
120
F. KREUZER
AND
_--.
-
Mb+02
P (mm
L. J. C. HOOFD
Hg)
8-
6-
2__/* _____-.-_d __....---. _______--.-_____.___---.. I 2.5
0 0
._-*
_’,’
,’
,’
i 5
x (u) Fig
I I.
Difference
between equilibrium
layer of 5 p thickness containing
and nonequilibrium
Ox pressures of IO rs0 mm Hg on the two sides. ----equilibrium
between IO and 0 mm Hg Po,. -----
I.5 mm Hg (extrapolated
concerning
the PO2 gradients in a very thin
a Mb solution of IS g!lOO ml with D,,=3.98 = nonquilibrium =quilibrium
from bulk at equilibrium!).
x IO-’
between virtual
facilitated
exposed to
=
0s pressures of 9.1 and
k’= I4 x IO9 ml/M/set.
Hb* 02 -
k = I I seC’.
1
flux
(10 - lo M /cm2/sec
cm’sec
as usual in the present paper.
)
1.5 I
I
facilatatlon
____ _ __ - - - - - - - -
with
Hbq
UltuWtlOn
l!l.Y’l”“:P!“l’_‘“_“l”_‘l’l”i______--_______
‘II PO2 (mm Hg) Fig. 12. Plot of the facilitated solution
0,
flux against the Po, at.the high pressure side (no back pressure) in a Hb
of I8 g:lOO ml with a D,
contribution
at a saturation
of 2.016x
difference
present approach
lO~‘cms.;sec.
of lOO%. Hyperbolic
L= 180~
Horizontal
curve==nonquilibrium
(k’ = 3 Y I O9 ml,IM/sec. k = 42.5 sec. ‘).
line on top=flux according
to the
FACILITATED
OJ
0
50
100
DIFFUSION
150
121
OF OXYGEN
200
250 PO2 (mml-lg)
Fig. 13. Plot of the facilitated 0, flux against the PO, at the high pressure side (no back pressure) in a Mb solution of 15 g/lo0 ml with a D,, of 3.98 x lo-’ cm’/sec. L= 180 k Horizontal line on top=flux contribution at a saturation difference of 100 %. Solid curve = nonequilibrium according to the present approach (k’= 14 x lo9 ml/M/set, k= 11 set- ‘). There is a pronounced decrease of the facilitated 0, flux in the presence of high 0, pressures and a more marked difference between the two lines than in the case of Hb (fig. 12).
4. The effect of a possible back pressure on the low pressure side indicates that the gas concentrations on both sides of the membrane must be known accurately. The real absence of any back pressure on the low pressure side is critically important in the case of Mb and may be the main factor in the relatively poor agreement between computation and experimental data in this case ; 5. The nonequilibrium approach is valid more generally than the equilibrium assumption. The latter is apt to provide computed fluxes which are too high; 6. Contrary to previous presumption there is a decrease of the facilitated oxygen flux with high values of PO, when assuming chemical nonequilibrium, particularly in the system Mb+ Oz. Appendix. Validity of the sohths
There are two conditions under which the approach of this paper holds: 1. a2, defining the terms in parenthesis that multiply A(x) in the first right-side term of eq. (4), from which t1,,and cur(eqs. (6.1) and (6.2)) were derived, must be approximately constant in the regions where A(x) is important; 2. the term D,,d’ [0,1°/dx2 in eq. (4’) must be negligible. Concerning 1. Suppose that x2 changes, i.e., that [O,]” and [RI0 near x =0 and
122
F. KREUZER
x= L are not approximately (Al)
k’lR1’ D + 02
AND
L. J. C. HOOFD
constant with position. Then we may write near x=0:
k+k;)[ozll
‘v ai(l+s(v+aex))
P
where [O,]’ and [R]’ are the new “equilibrium solutions” with condition 2 still being fulfilled, and v allows for the (small) differences between [O,]j=e and [O,]p=e as well as [R]i=e and @]f=e. Therefore : (A2)
LO,1 = COJ’-
A(x)(l -&f(x))
a,, and A(x) are the previous solutions (eqs. (6.1) and (6.2)). E indicates the extent of deviation from these solutions. With the aid of eq. (A2) we may also compute [RO,] ’ and CR]’ as in eqs. (3a, b). We can solve f(x) by inserting into eq. (la), neglecting the terms with s2 (arising from the product [p] [O,]), and using eq. (4) which gives: (A3)
f”(x) +{ z}f(x)
+ {a; - !$$if(x)=
-ai(v+aOx).
We may now approximate the solution for f(x) by stating that A(x) = (6 dD,q,/ k’)e-lox (see eq. (5.1)), which must hold in the region where f(x) is important: (A4
f(x)=f(O)+~a,x(l+2v+a,x)
where f(0) is a constant of integration. Application of all boundary conditions and of eq. (12) provides a very complex equation for f(0) and for v, with an upper limit of f(0) =a ; one may calculate however that f(0) is much smaller than this limiting value in most cases pertaining here. We can also compute E by differentiating eq. (Al) and inserting [O,]’ z [O,]‘:
(A5)
&=
It appears that &f(O)is small in all cases covered here and (1 +&f(x)) is unimportant where &f(x) becomes large. Concerning 2 . The condition that Do,d2[02]‘/dx2 must be small was already checked thoroughly in our previous paper (Kreuzer and Hoofd, 1970a, pp. 288-289). Suppose that in the important cases the flux F is predominantly due to the diffusion of oxygen, and that DoId [02]‘/d x2 may amount to not more than two percent of the other terms (eq. (4’)), then for the pigment concentrations pertaining here and if the facilitation is not large : (‘46)
for Hb: P/L< 19;
for Mb: P/L< 2.1
(P/L in mm Hg/lc).
This is important for the cases covered in the following figures: Figure 8 : With Po, = 160 us 0 mm Hg we find from eq. (A6) L 2 9 p ; with L = 9 p three quarters of the flux are due to the diffusion of oxygen.
FACILITATED
DIFFUSION
OF OXYGEN
123
Figure 10 : With Po, = 10 us 0 mm Hg we find similarly L > 4.8 p ; at L = 5 ~1only one quarter of the flux is due to the diffusion of oxygen ; therefore a further check is needed here as done concerning fig. 11 (see below). Figure 12 : Here L = 180 p, so that the critical Po, < 3400 mm Hg which is far above the range covered (about 700 mm Hg). Figure 13 : Here again L = 180 cc, so that the critical P,, < 380 mm Hg which also exceeds the range covered (about 280 mm Hg). If however Do~dz[O,]o/d x2 exceeds 2% of the other terms of eq. (4’) in the core region the calculated profiles of [0,] and [PO,] in the core of the slab will not be exact, i.e., there also will be a region in the core where A(x) # 0. This however is of no influence on the computed flux F since the solutions for A(x) at x=0 and x= L, and for [O,]” remain unchanged ; for in computing the flux F no use has been made of the condition that there is equilibrium everywhere in the cafe of the membrane. Thus we may conclude that for thicker membranes PO, may be. larger than indicated in eq. (A6). For thinner membranes however this may probably no longer be allowed, so that the limits of eq. (A6) should be taken into account ; this has always been done. A(x) is particularly important in the region where PO2is low. This may be illustrated by the case of fig. 11 (see also the layer thickness of 5 p in fig. 7). It may be calculated here that D,,d2[02]‘/d x2 is largest for x = 4.7 p, i.e. close to the opposite boundary of x = L and in the region where A(x) should approximate zero. The influence of this however is small ; even if we assume that A(x) at x = L is 20 % larger, the flux F would be smaller by only 1.5 %. In order to be sure the solution near x = L was furthermore checked with an analog computer which is able to simulate exact solutions of differential equations such as (la) and (lb) though only over a distance of about 1 p so that only insignificant deviations may be seen ; the flux F thus computed was always correct to within 1%. This check justifies a posteriori also fig. 10. Figure 11 is furthermore impcrtant because here [O,]p=,_- [O,]~=, is only 76% ofyP-YP, or, with eq. (2), (A(x=O)-A(x=L))/(yP-yPo)=24% (note that -A(x= L) is positive since there is a minus sign in eq. (5.2)). This figure of 24 % is the highest value computed in this study but there is no reason why in other cases this term cannot be larger. Of course A(x = 0) - A(x = L) may never be larger than y(P - PO) since in this case the flux F would be negative (see eqs. (12), (2), and (3)) with a positive pressure difference (P - PO).With the present approach this could never occur because it follows from F< Oandeq.(lO) that q,, q, < Oand witheq.(5)A(x=O)-A(x=L)< 0, which contradicts the assumption A(x = 0) - A(x = L) > y( P - PO). Acknowledgement
We wish to gratefully acknowledge the stimulating discussions with Professor Jerome S. Schultz, visiting professor, during the last phase of writing this paper. References Hemmingsen,
E. and P. E. Scholander
Science 132: 1379-1381.
(1960). Specilic transport of oxygen through hemoglobin
solutions.
124
F. KREUZER AND L. J. C. HOOFD
Keller, K. H. and S. K. Friedlander (1966). Diffusivity measurements of human methemoglobin. J. Gen. Physiol. 49 : 681-687. Kreuzer, F. (1970). Facilitated diffusion of oxygen and its possible significance; a review. Respir. Physiol. 9: I-30.
Kreuzer, F. and L. J. C. Hoofd (197Oa). Facilitated dilfusion of oxygen in the presence of hemoglobin. Respir. Physiol.
8: 28CL302.
Kreuzer, F. and L. J. C. Hoofd (1970b). Facilitated diffusion of oxygen in the presence of hemoglobin and myoglobin. AGARD Conference Proc. No. 65 : Fluid dynamics of blood circulation and respiratory flow, p. 30/l-5. Kutchai. H. (1970). Numerical study of oxygen uptake by layers of hemoglobin solution. Respir. Physiol. 10: 273-284. Kutchai, H., J. A. Jaquex and F. J. Mather (1970). Nonquilibrium facilitated oxygen transport in hemoglobin solutions. Biophys. J. 10: 38-54. Murray, J. D. (1971). On the molecular mechanism of facilitated oxygen diffusion by hemoglobin and myoglobin. Proc. Roy. Sot. B 178: 95-l 10. Riveros-Moreno, V. and J. B. Wittenberg (1968). The self-diffusion coeflicients of hemoglobin and myoglobin in concentrated sohttions. Manuscript, see also: J. Biol. Chem. 247: 895-901 (1972). Scholandcr, P. F. (1960). Oxygen transport through hemoglobin solutions. Science 131: 585-590. Wittenberg, J. B. (1966). The molecular mechanism of hemoglobin-facilitated oxygen diffusion. J. Biol. Chem. 241: 104-l 14. Wittenbcrg, J. B. (1969). Personal communication. (Letter of May 7, 1969). Wittenberg J. B. (1970). Myoglobin-facilitated oxygen diffusion : role of myoglobin in oxygen entry into muscle. Physiol. Rev. 50: 559436.
North-Holland Publkhing Company, Amsterdam
FACTORS INFLUENCING FACILITATED DIFFUSION OF OXYGEN IN THE PRESENCE OF HEMOGLOBIN AND MYOGLOBIN’
F. KREUZER
and L. J. C. HOOFD
Department of Physiology, University of Nijmegen, Nijmegen, The Netherlands
Abstract. The experimental results concerning the facilitation of oxygen diffusion in the presence of hemoglobin or myoglobin may be explained by the simultaneous diffusion of oxyhemoglobin or oxymyoglobin. In order to prove this assumption it should be possible to show that there is quantitative agreement between the experimental values and those computed using the physical parameters obtained independently. The present authors previously presented an approximate solution for the basic equations of diffusion including chemical reactions; an improved solution is provided in the present paper. Since the agreement between experiments and computer solutions was much less satisfactory in the system Mb + 0, than in the system Hb + O,, we investigated the effects of various other factors of possible importance for both systems of Hb and Mb. These computations led to the following conclusions: 1. The values of the pigment diffusion coefficients are of crucial importance; 2. The knowledge of the actual diffusion path length is also important; a quantitative theoretical treatment is presented; 3. Concerning the chemical reaction rates the dissociation constant is more likely to be limiting; 4. The effect of a possible and even minute oxygen back pressure on the low pressure side may be crucial in the case of the system Mb+O,; 5. The nonequilibrium approach suggested here is valid more generally than the equilibrium assumption made in most previous cases; the latter is apt to provide too high computed fluxes; 6. Contrary to previous presumption there is a decrease of the facilitated oxygen flux with high values of Po,, particularly in the system Mb+O,. These results might provide new insight in the mechanism of facilitated oxygen diffusion and suggest more appropriate experimental arrangements in future investigations. Facilitated diffusion Hemoglobin
Myoglobin Oxygen
The experimental results of Scholander (1960) and Wittenberg (1966) concerning the facilitation of oxygen diffusion in the presence of Hb or Mb may be explained on the basis of the assumption that HbOz or MbOz diffuses simultaneously with oxygen. Acceptedfor publication 30 December 1971.
’ Part of the results of this study has been presented at the Alfred Benzon Symposium IV: Oxygen atlinity of hemoglobin and red cell acid-base status, Copenhagen, Denmark, May 17-22, 1971. and at the Symposium on Passive permeability of cell membrane, Rotterdam. The Netherlands, July 2&22, 1971. 104
FACILITATED
DIFFUSION
OF OXYGEN
105
In order to prove this assumption it should be shown that there is quantitative agreement between the experimental values and those computed using the physical parameters obtained independently. Previous attempts assuming chemical equilibrium between oxygen and carrier protein throughout the layer were not conclusive. Kreuzer and Hoofd (1970a) presented an approximate solution for the basic equations of diffusion including chemical reactions and applied the resulting expressions to numerical computer solutions for the system Hb + 0, in a film at steady state similar to the experimental situation of Wittenberg (1966) using recent numerical values for the diffusion coefficients of oxygen and Hb in Hb solutions of various concentrations. The facilitated fluxes thus computed agreed well with the experimental data of Wittenberg (1966) when choosing well-founded values of the Hb diffusion coefficients. The authors concluded that the mathematical treatment of facilitated diffusion of oxygen based on simultaneous diffusion of oxygen and oxyhemoglobin provides a quantitative description of the experimental results for the system Hb+ 0, if the chemical reaction rates were taken into account. A subsequent similar study by Kreuzer and Hoofd (1970b) for the system Mb+ 0, however revealed a much less satisfactory agreement between computation and experimental data. These findings prompted us to further compare the two systems Hb+ O2 and Mb+ O2 with particular reference to the influence of all possibly important factors. It might be expected also that one or the other factor could be of different weight in the two systems in view of their inherent differences. This might shed new light on the importance of the various factors in a more general sense. We therefore investigated, always comparing the systems Hb + 0, and Mb + O,, the effects of varying the following parameters: 1) the diffusion coefficients of HbOz or MbOl, D,, and D,, resp.; 2) the respective chemical dissociation constant k; 3) the back pressure of oxygen on the low pressure side ; 4) the layer thickness L ; 5) the presence or absence of chemical equilibrium throughout the layer; 6) the oxygen pressure on the high pressure side. It will be shown that these comparisons lead to reasonable predictions concerning the discrepancies still prevailing, also with respect to details of the experimental arrangement to be considered in future investigations. Mathematical methods The basic approach was the same as in our previous paper (Kreuzer and Hoofd, 1970a). We assumed again the presence of three regions in the layer: a middle bulk very near equilibrium and two marginal regions not at equilibrium. The solutions for these two boundary layers were improved as described below. It may be shown that this approach is also applicable to the case of Mb + OZ. The two basic equations are at steady state:
(14
D,,
w =
k’ [R] [0,] - k [RO,]
F. KREUZER
106
(lb)
D
P
!!t&ki! = k[RO
2
dx2
AND
L. 3. C. HQOFD
]_k’[R][()
2]
where p] = pigment con~ntration~ PO,] = oxygenate pigment ~n~ntration and [ 0,] = oxygen concentration ; Do, and D, =diffusion coefficient of 0, and pigment, resp. ; k’ and k = chemical association and dissociation constant, resp. In the boundary regions of x near 0 and x near L we may write : (2)
Lo21 = W21°
-A(x)
where [02]0=solution for the oxygen concentration in the middle part of the slab, extended to the boundary regions. Superscript zero always indicates the concentration in the core of the slab. A(x) is a correction to [O,]’ at the boundaries, both A and [O,]’ being functions of position x. A(x) disappears when x is “far’” from x =0 and x = L but may be important at x near 0 and x near L. Since the addition of eqs. (la) and (1b) is zero we have for [RO,] : [RO,] = [RO,]”
+ +
A(x). P
Since PO,] W
+ @] = [Ro] = total pigment ~n~ntration
(with Dao, = D, = DJ :
[R] = [RI0 - +-’ A(x). P
After inserting into eq. (la) we get : D
‘=
d2i?i?_ dx2
D
d’dfx)=
[R]O
k’
O2 dx2
I
WI
-
- 2
k { [RO,]’
P
A(x)]
+ 2
P
jTO,]‘A(x))
A(x) ) .
A(x) is a rapidly varying function whereas the marginal extensions of CO,]‘, CR]“, and @O,]” are practically constant in the regions where A(x) is important, and there is equilibrium among them so that [RI0 [O,]’ = [RO,]‘. Thus the equation above may be rearranged to give:
(41
,-, WE O2 dx2
k’~R]“~k’~~O*]o+k~~A(x~-k~~A(x)2. P
P
P
There is an exact solution for eq. (4) in the case where [RI” and [O,]” are nearly constant and if there is a region where both A(x) and dA(x)/dx tend to zero, i.e.: for x near 0: 6iu$Dpqoe-“oX A(x) = (5.1) k’(1 +q0e-“0”)2 where
FACILITATED
DIFFUSlON
OF OXYGEN
107
for x near L ;
where 16.2j
~,
~
c
k’[gKI:= k+$&, L
02
+
P
J* /
The values of q. and q1 and, from these, of [&jr= o etc., follow from the boundary cmditions, in particular from those requiring that there is no flux of HbOz across the surlkc, i.e. :
Therefore the total flux of oxygen must be carried by free oxygen only, i.e.: {‘7c,d) For x= 0 inserting eqs. (2) and (5.1) into (7~) gives:
whereas (7a) for x=0 may be written as :
10s
F. KREUZER AND L. J. C. HOBFD
For the respective oxygen pressures we get at x = 0 :
and at x=L: (11.2)
YR, = [021xo=L+
6a:q,D, k’(1 -q1)2 *
Thus the six eqs. (6), (lo), and (11) together with the exact solution for the total flux of 0,:
(14
F =
DC&‘- PO)+ L
943021x=,L
l3021,=0)
contain seven unknown quantities: Q, xi, q,, qi, [O,]:,,, [O,]E=t, and F, which may be determined in a computer program. Since q,, and qi are small quantities the terms (1 + q0e-o*)2 and (1- qi e-u1(L-X))2in the denominator of eq. (5.1) and eq. (5.2) resp. introduce only small corrections to A(x) in most cases. These terms become important only if either q. and q, are not small and/or A(x) is comparable to [OJO, or if even a small change in [OJ’ (as a virtual extension to the bunco corresponds to a large change in fRO,]“, as in the case of Mb with its steep dissociation curve. We will see below what the computed consequences are for Hb and for Mb. The solution presented here is superior to that offered previously in that it provides a nearly exact description of the oxygen concentration in the boundary layers because even relatively large variations of [R] are taken into account whereas [R] was kept constant previously. In a similar manner as in our previous paper the course of [O,]” and fRO,]” throughout the whole membrane may be determined from: (13)
Fx + B = Do2 [0,] + D, PO,]
which becomes with eqs. (2) and (3): (13’)
Fx + B = D,, [OJ”+
D, [RO,]”
where B = Do,[O,]:=,+D,~O,]x”=,
.
Concerning the validity of these solutions see Appendix. comptl?f program The program was written in a slightly modified version of FORTRAN adapted to the WATFIV compiler used at the University of Nijmegen. The input list included, expressed by the appropriate symbols: thickness of membrane (L) in em, reaction Hb concentrations in g%, diffusion coefftconstants k in set- ’ and k’ in ml/M/~ cients of Hb and 0, in cm’/sec at the respective Hb concentrations, initial 0, pressure
FACILITATED
DIFFUSION
OF OXYGEN
109
(PO at x =0) and 0, pressure on the other side (P at x = L). The physical parameters were those at 25 ‘C. The output list contained, again expressed by the appropriate symbols: total flux of 0, (F), O2 flux by plain diffusion, facilitated O2 flux, and the virtual pressures at x=O,L([OJ”/r at x=0 and x = L), followed by a list of O2 pressures and pigment saturations at various points within the membrane as chosen before. The construction of the program included three parts. The first part was made to read and check the input data. The second part contained an iteration process to calculate ao, a,, q,, q,, [OJ~,o, [O&‘=,_, [ROJXEo, [ROJx=,_ and F in the following order : from eqs. (1 l), and from these [RO&‘=,,, and [R]z=o,L since there 4 PzlxO=o.L is equilibrium among them ; b) WJ~=O.Lfrom eq. (3a) and eqs. (5) at x=O,L; c) F from eq. (12); d) x0, ai, from eq. (6); e) q,, q,, from eqs. (10) using the approximation (1 fq)3 ‘v 1 f 3 q + 3 q2 (since q is small) and solving the resulting quadratic equation for q. The iteration process was started with the values q. = q i = 0 and continued until the computed results for q,, qi changed by less than 0.01 ‘A. In the third part the values of [O,] and [ROJ were computed at certain points of the membrane. Results and discwsion When plotting the facilitated flux against Hb concentration (Kreuzer and Hoofd, 197Oa, fig. 9), there is good agreement between the experimental values of Wittenberg (1966) and the computed line when using the D,, values of our compromise curve for the computations (see fig. 3 in Kreuzer and Hoofd, 1970a). In a similar plot for Mb however the computed values of the facilitated flux are almost twice as large as the experimental points of Wittenberg (1966) when using the Dhtb data of Riveros-Moreno and Wittenberg (1968). A similar conclusion was reached by Murray (1971). We also examined the computed curve obtained by using DM, values calculated from our D,, compromise curve by the ratio of the square roots or of the cubic roots of the molecular weights (the latter probably being more justified for high-molecular substances). These computed lines come closer to the experimental points, particularly when using the cubic root ratio, but the agreement remains poor, the computed curve still being too high. We shall try now to get an impression concerning the. factors possibly influential in rationalizing the difference in behavior between Hb and Mb. Wittenberg (1966) pointed out that the rate of dissociation should be particularly important in limiting the rate of facilitation since pigments of appropriate molecular size (ascaris body wall Hb and succinyl ascaris perienteric fluid Hb) but with very small rate of O2 dissociation do not facilitate O2 diffusion. We therefore investigated the effect of changes in the’dissociation constant k (the association constant k’ being held unchanged) on the facilitated flux.
110
F. KREUZER AND L. 1. C. HOOFD facilitated flux (10-10 M /cm2/sec)
Fig. 1. Influence of the dissociation constant k on the facilitated O;, ffux in Mb solutions of various conantrations. The figure shows two sets of computed curves. The three curves on top were computed using the DHb values of Riveros-Moreno and Wittenberg (1968) and the three lower curves by applying the D,, values obtained from the DHbcompromise curve of Kreuzer and Hoofd (1970a) by the ratio ot’the cubic roots of the molecular weights. The values of k of 17.5, 11. and 5.5 set-’ used in both sets of curves are indicated in the figure. L= 180 p. PO,= 160 us 0 mm Hg. k’= 14 x IO9ml/M/set.
The general effect of changes in k is very small for the system Hb + O2 {not shown here). The situation is quite different for Mb+& (fig. 1). A decrease in k greatly reduces the facilitated flux as expected, the absolute values of k being smaller than in the case of Hb+ Oz. The generally accepted value of k is 11 see- * for Mb but the closest approach to the experimental points is found &ith k of about 5.5 see- ‘, Dhtb having been calculated from the compromise D Hbby the cubic root ratio as described above.
FACILITATED Hb+ 02 facilitated flux (lo-” M /cm*/ set
DIFFUSION
111
OF OXYGEN
)
1.5
01 0
5
10
15
20 25 PO2 back pressure (mm HQ )
Fig. 2. Influence of back pressure (abscissa) on the facilitated 0, flux in a Hb solution D,,=2.016x10~‘~2/sec.Computedcurveobtainadwithk’=3x109ml/M/secandk=42.5sec~‘.L= 186 p. Po, at high pressure side = 160 mm Hg.
of
18 g/100 ml with
Wittenberg (1969) and Kreuzer (1970) have discussed the possible influence of O2 back pressure on the low pressure side concerning the experimental facilitated flux or, in other words, the possible interference by inadequate stirring in the low pressure gas compartment where PO, was supposed to be zero in the preceding cases. Figure 2 shows the general trend of the influence of back pressure for Hb + OZ. A similar pattern was found by Kutchai, Jacquez and Mather (1970, Tab. IV) ; it should be noted however that the two approaches are quite different; we have developed an analytical treatment whereas that of Kutchai et al. (1970) is purely numerical. The finding that there should still be some facilitation at back pressures where experimentally a complete disappearance of any enhancement was found (i.e. above 10 mm Hg Po, back pressure) again points to the discrepancy mentioned by Hemmingsen and Scholander (1960) that there was no facilitation with a back pressure of 10 mm Hg although the Hb on the low pressure side would be only 5565% saturated at this partial pressure and a considerable saturation gradient would thus remain across the layer (Kreuzer, 1970). Wittenberg (1970) explains this discrepancy by the fact that the local oxygen pressure adjacent to the low-pressure face was much higher than measured in these experiments due to insufficient stirring of the gas phase. It is calculated that a back pressure of 2.8 mm Hg would be required for a fit of the computed curve with the experimental points when using the D,, values of Riveros-Moreno and Wittenberg (1968). Such a back pressure could easily be detected and is unlikely to have been present in the experiments of Wittenberg (1966). The general course of the influence of the back pressure for Mb + O2 is presented in fig. 3 (note the difference in scale on the abscissa!). Figure 4 shows that a back pressure of only 0.12 mm Hg is required for a good tit of the best curve, calculated with Dh(,,values obtained from our compromise
112
F. KREUZER Mb*02 facilitated (1CI-’ lo
AND
L. J. C. HOOFD
f Iux
M /cd/set
1
2.c
1.5
1.0
0.5
0
oc
5
1
1.5
2
PO2 back Fig. 3. Influence
of back pressure
with D,, = 6.79 x IO-’ cm’!sec lo-’
cm2/sec
from D,,
(abscissa)
compromise
cubic roots of the molecular
on the facilitated
from Riveros-Moreno
weights
and Hoofd
of 15 g/l00
ml
(1968) in top curve or D,, = 3.98 x (19%)
obtained
curve. L= 180 p. PO, at high pressure
k’= 14x log ml/M/set.
Hg )
(mm
flux in a Mb solution
and Wittenberg
curve of Kreuzcr in bottom
0,
2.5 pressure
by the ratio of the side=
160 mm Hg.
k= 11 sect.
D,, by the ratio of the cubic roots of the molecular weights, and of 0.6 mm Hg for a fair agreement when using the D,, values of Riveros-Moreno and Wittenberg (1968). Thus the experimental values of the facilitated flux are much more sensitive to back pressure in the case of Mb+ OZ. The assumption ofa specific value for the effective thickness L of the layer is another factor affecting the computed results of the facilitated flux. The actual thickness of the Millipore filters considered here is 150 ,Ubut only SO’%are liquid space ; therefore Wittenberg (1966) as well as Kreuzer and Hoofd (1970a) have worked with a corrected, thickness of 180 p. Murray (1971) allowed for the tortuosity of the channels and assumed a thickness of 220 ~1or about 50% more than 150 ~1.We have reconsidered this problem in the following way. Assume that the diffusion path across the layer consists of N partial paths 6i running under an angle of Bi to the direct path. Then : N
(14)
L = c hi cos pi. i=l
FACILITATED
facilitated ( lo-‘O
0
DIFFUSION
113
OF OXYGEN
flux
M /cm2/sec
1
5
10
0
15
20
25
30 35 Ck(b(g/lOO ml
)
Fig. 4. Influence of back pressure on the facilitated O2 flux in Mb solutions. The two top curves were calculated with a back pressure of zero for comparison, with D,, from Riveros-Moreno and Wittenberg (1968) or from the compromise curve of Kreuzer and Hoofd (1970a) by the ratio of the cubic roots of the molecular weights respectively, the two bottom curves for a back pressure of I mm Hg with the same respective sets of D,, values. The two middle curves were computed to give the best possible agreement with the experimental points of Wittenberg (1966) by choosing appropriate values of back pressure, i.e. 0.6 mm Hg when using the Du, values from Riveros-Moreno and Wittenberg (1968) and as little as 0.12 mm Hg when using the D,, values from Kreuzer and Hoofd (1970a). L = 180 p. PO, at high pressure side = 160 mm Hg. k’= 14x109ml/M/sec. k=ll set-‘.
The length L’ of the diffusion path is:
(15)
L’=
; i=
6,.
1
Assuming that pi may have any value between -n/2 and + 7r/2and is proportionately
114
F. KREUZER AND L. J. C. HOOFD facilitated flux ( 10-‘” M /cm’/sec
0
)
I
10
5
15
20
25
30
35
CH~ (g/100
ml
)
Fig. 5. Influence of dilfusion path length on the facilitated O2 flux in Hh solutions. The path length of 150 p + 57 % = 235 p used here was obtained as described in the text. The three computed curves refer from top to bottom to the D,, values according to Riveros-Moreno and Wittenberg (1%8), Keller and Friedlander (1966). and Kreuzer and Hoofd (197Oa). PO,= 160 DS0 mm Hg. k’=3 x 10’ ml/M/set. k=42.5 se-‘.
distributed between these limits (i.e. the diffusing particles move around obstacles arranged at random) eq. (14) becomes: + n/2 L=i;I;j
cosj?dfi=L'
f
-n/Z
whence (16)
L=qL.
Since n/2 = 1.57 this implies that the mean diffusion path is 57% longer than the membrane thickness of 150 p(, i.e. 235 cc. Figure 5 shows a plot of the facilitated flux against Hb concentration with three sets of data for D,, and with a membrane thickness of 150 p+ 57 % =235 p. The facilitated fluxes are reduced by some 20x, and the D, data of Keller and Friedlander (1966) now provide the best fit. Similarly fig. 6 demonstrates the influence of membrane thickness on the facilitated flux for Mb + O2 ; here the fluxes have been computed for membrane thicknesses of 180 ~(150 p + 20 %) and 235 p (150 p + 57 %) using the D,, values of Riveros-Moreno and Wittenberg (1968) and those obtained
FACILITATED facilitated (lo-‘O
DIFFUSION
115
OF OXYGEN
flux
M/cm2/sec)
2.5-
L -150~1+20%
Wittenbeq(1966)
c*(b (g/loo
ml )
Fig. 6. Influence of diffusion path length on the facilitated 0, flux in Mb solutions. With both the D,, values from Riveros-Moreno and Wittenberg (1968; two top curves) and those from Kreuzer and Hoofd (197Oa; two bottom curves), the facilitated 0, fluxes were compared when using a path length of 150 p + 20 % = 180 p (as applied hitherto) shown by the curves ~ and -.-.--e, or a path length of 150 p + 57 % = 235 I( (see text) represented by the curves -----and --.--.-.The best tit with the experimental points of Wittenberg (1966) is obtained when using the D ub values from Kreuzer and Hoofd (1970a) and a path length of 235 p. PO, = 160 vs 0 mm Hg. k’= 14 x lo9 ml/M/set. k = 11 se-c- i.
from the D,, compromise curve by the ratio of the cubic roots of the molecular weights. The best agreement with the experimental points is obtained with the membrane of 235 p using the “compromise” D,, values. Figure 7 presents the gradients of PO, and saturation across the layer for Mb+ 0, with a PO, of 10 mm Hg on the high pressure side (and no back pressure) for three membrane thicknesses of 5, 50, and 500 cc. These plots confirm the pronounced influence of the membrane thickness on the computed facilitated fluxes, particularly in the range of thin layers, as also shown for Hb by Kutchai et al. (1970, fig. 1).
116
F. KREUZER AND L. J. C. HOOFD Mb+02 saturation (01.) 100
75
50
25
0
Fig. 7. Influence ofdiffusion path lengths of $50, and 500 ~1(from top to bottom) on the computed gradients of PO, (three bottom curves) and percent saturation (three top curves) across the layer ofa Mb solution of 15 g/100 ml with a D,, value of 3.98 x IO-’ cm’/sec. The three pairs of curves are indicated by the same line symbols. The curves are drawn to scale for the flux contribution. PO, - 10 us 0 mm Hg. k’ = 14 x IO9 ml/ M/xc. k=lI set-I. Kutchai (1970) computed, for the non-steady-state situation of the system Hb + O2 (at 37 Y’Cand pH 7.4), the half-saturation times tSO and plotted their logarithms against the logarithms of the reciprocal of the layer thickness L. For L greater than 1.6 p the graph was linear with slope -2, so that in this region t,, was proportional to Lz as to be expected if the diffusion of oxygen was the limiting process. For L less than 1.6 p the graph was curvilinear, probably because in this range diffusion of oxygen was so rapid that chemical reaction rates were more important in determining the overall rate of oxygen uptake. He also found that the influence of HbOt diffusion
FACILITATED
Hb+Oz facilitated flux (lO-‘” M/cm’/sec
DIFFUSION
117
OF OXYGEN
1
5-
2-
l-
.5-
.210
20
50
loo
Fig. 8. Difference betwee.n chemical equilibrium ing the calculated 100 ml with a D,,
200 500 1000 thickness of membrane
(top curve) and nonequilibrium
2cK3 L (u)
(bottom curve) conccm-
facilitated 0, flux as plotted against the diffusion path length in a Hb solution of 18 g/ of 2.016 x lo-’
cm2/sec from Kreuzer and Hoofd (1970a). PO, = 160 DS0 mm Hg. k’= 3 x lo9 ml/M/xc. k=42.5 set-‘.
rose with increasing L from very small values in extremely thin layers to a maximum above about 10 cc,presumably again reflecting the importance of reaction rates in the thinner layers and the primacy of diffusion in the thicker layers. However, for layers of the same thickness the contribution by Hb02 diffusion was less here than in the case of steady state. In our previous paper (Kreuzer and Hoofd, 1970a) we had obtained good agreement with the experimental results of Wittenberg (1966) when using the D,, values of our compromise curve and assuming absence of chemical equilibrium at the boundaries, whereas most previous calculations by other authors were based on the assumption of equilibrium throughout. The question arises now as to what influence on the computed facilitated flux is exerted by the presence or absence of chemical equilibrium. When plotting the computed curve of the facilitated fluxes in the presence of equilibrium as compared with the nonequilibrium values using the compromise DHb values, the equilibrium curve is located somewhat higher (as expected) but the difference of about 5 % is not important with respect to the experimental points ; thus it may be justitIed indeed to deal with the system Hb+ O2 equally well with the assumption of equilibrium. The difference between equilibrium and nonequilibrium is
118
F. KREUZER AND L. J. C. HOOFD
tocilitated
flux
(10-X)M/cm2/sec)
2.5
,/--_. /’ I
‘yquillbrium
I’
‘\ ‘\\
/
‘\
i
:
/
\
\
nonequlibrlum -. 1. 1.
.A’
\
\
\
\
1 ‘\
\
.
I
\
‘1. .
\
\
\
\
\. 1,
‘\\ \
.
‘\\ ‘\
‘\ ‘\ ‘\
.
Fig. 9. Comparison between chemical nonequilibrium and equilibrium for the facilitated 0, flux in Mb solutions of various concentrations with D,, values obtained from D,, compromise curve of Kreuzer and Hoofd (1970a) by the ratio of the cubic roots of the molecular weights. Top curve= equilibrium, bottom 14~10~ml/M/sec.k=Ilse.~~. curve = nonequilibrium. L = I80 p. Po, = 16OusOmmHg.k’=
a function of the membrane thickness, as also shown by Kutchai et al. (1970, table I and iig. 3); there is no difference with thick layers and the difference increases with decreasing thickness (fig. 8). The importance of assuming equilibrium or nonequilibrium becomes much more pronounced with the system Mb+ 0, as demonstrated by figs. 9 and 10. The inadequacy of the equilibrium approach is much more evident here than in the case of Hb + OZ. Figure 11 shows the P,,gradients for equilibrium, nonequilibrium, and equilibrium between the virtual pressures at the boundaries providing the same facilitated flux as in the case ofequilibrium for the system Mb + OZ. It had been concluded from all experimental determinations that when plotting the facilitated fluxes against Po2, a constant value was reached over a wide range of Po, above the region of low Po, where the facilitated flux has to drop to zero at a P,, =0 of course. Figures 12 and 13 (for Hb+ 0, and Mb+ O2 resp.) however shoti
FACILITATED Mb+02 facilitated flux (lo-lo M/cm2/sec)
DIFFUSION
119
OF OXYGEN
,
IO-
.2’
I
5
\ 10
20
50
100
200 thickness
500 of
1000 membrane
I
2000 L (JJ)
Fig. 10. Difference between chemical nonequilibrium (bottom curve) and equilibrium (top curve) conceming the calculated facilitated 0, flux as plotted against the diffusion path length in a Mb solution of 15 g/ 100 ml with a D,, of 3.98 x IO-’ cm’/sec obtained from the D,, compromise curve of Kreuzer and Hoofd (1970a) by the ratio of the cubic roots of the molecular weights. Po, = 10 OS0 mm Hg. k’= 14 x IO9 ml/M/ sec. k=ll set’.
that in caSe of the nonequilibrium approach a decline of the facilitated fluxes is to be expected with high values of Po2, particularly with the system Mb + Oz. Inspection of figure 1 in Wittenberg (1966) concerning the system Hb+ 02, where the fluxes are plotted against Po2, shows that the last of seven experimental points at a PO, of about 600 mm Hg does seem to have a tendency of lying a little lower than the straight regression line (no data of this kind are available for Mb+,02). Our curve of fig. 12 would predict a decline of about 2 % from the maximum value at a PO, of 600 mm Hg. Conclusioos 1. The values of the physical parameters are of crucial importance. The pigment diffusion coefficients determined by several authors in particular show considerable deviations and need to be known more accurately, especially also fon Mb where only one set of experimental data is available (Riveros-Moreno and Wittenberg, 1968); 2. The knowledge of the actual diffusion path length is also important. The use of Millipore filters may not be the method of choice due to their internal structure and the uncertainties as to their tortuosity; 3. Concerning the chemical reaction rates it is particularly the dissociation constant of the gas-pigment complex which is important for the flux ;
120
F. KREUZER
AND
_--.
-
Mb+02
P (mm
L. J. C. HOOFD
Hg)
8-
6-
2__/* _____-.-_d __....---. _______--.-_____.___---.. I 2.5
0 0
._-*
_’,’
,’
,’
i 5
x (u) Fig
I I.
Difference
between equilibrium
layer of 5 p thickness containing
and nonequilibrium
Ox pressures of IO rs0 mm Hg on the two sides. ----equilibrium
between IO and 0 mm Hg Po,. -----
I.5 mm Hg (extrapolated
concerning
the PO2 gradients in a very thin
a Mb solution of IS g!lOO ml with D,,=3.98 = nonquilibrium =quilibrium
from bulk at equilibrium!).
x IO-’
between virtual
facilitated
exposed to
=
0s pressures of 9.1 and
k’= I4 x IO9 ml/M/set.
Hb* 02 -
k = I I seC’.
1
flux
(10 - lo M /cm2/sec
cm’sec
as usual in the present paper.
)
1.5 I
I
facilatatlon
____ _ __ - - - - - - - -
with
Hbq
UltuWtlOn
l!l.Y’l”“:P!“l’_‘“_“l”_‘l’l”i______--_______
‘II PO2 (mm Hg) Fig. 12. Plot of the facilitated solution
0,
flux against the Po, at.the high pressure side (no back pressure) in a Hb
of I8 g:lOO ml with a D,
contribution
at a saturation
of 2.016x
difference
present approach
lO~‘cms.;sec.
of lOO%. Hyperbolic
L= 180~
Horizontal
curve==nonquilibrium
(k’ = 3 Y I O9 ml,IM/sec. k = 42.5 sec. ‘).
line on top=flux according
to the
FACILITATED
OJ
0
50
100
DIFFUSION
150
121
OF OXYGEN
200
250 PO2 (mml-lg)
Fig. 13. Plot of the facilitated 0, flux against the PO, at the high pressure side (no back pressure) in a Mb solution of 15 g/lo0 ml with a D,, of 3.98 x lo-’ cm’/sec. L= 180 k Horizontal line on top=flux contribution at a saturation difference of 100 %. Solid curve = nonequilibrium according to the present approach (k’= 14 x lo9 ml/M/set, k= 11 set- ‘). There is a pronounced decrease of the facilitated 0, flux in the presence of high 0, pressures and a more marked difference between the two lines than in the case of Hb (fig. 12).
4. The effect of a possible back pressure on the low pressure side indicates that the gas concentrations on both sides of the membrane must be known accurately. The real absence of any back pressure on the low pressure side is critically important in the case of Mb and may be the main factor in the relatively poor agreement between computation and experimental data in this case ; 5. The nonequilibrium approach is valid more generally than the equilibrium assumption. The latter is apt to provide computed fluxes which are too high; 6. Contrary to previous presumption there is a decrease of the facilitated oxygen flux with high values of PO, when assuming chemical nonequilibrium, particularly in the system Mb+ Oz. Appendix. Validity of the sohths
There are two conditions under which the approach of this paper holds: 1. a2, defining the terms in parenthesis that multiply A(x) in the first right-side term of eq. (4), from which t1,,and cur(eqs. (6.1) and (6.2)) were derived, must be approximately constant in the regions where A(x) is important; 2. the term D,,d’ [0,1°/dx2 in eq. (4’) must be negligible. Concerning 1. Suppose that x2 changes, i.e., that [O,]” and [RI0 near x =0 and
122
F. KREUZER
x= L are not approximately (Al)
k’lR1’ D + 02
AND
L. J. C. HOOFD
constant with position. Then we may write near x=0:
k+k;)[ozll
‘v ai(l+s(v+aex))
P
where [O,]’ and [R]’ are the new “equilibrium solutions” with condition 2 still being fulfilled, and v allows for the (small) differences between [O,]j=e and [O,]p=e as well as [R]i=e and @]f=e. Therefore : (A2)
LO,1 = COJ’-
A(x)(l -&f(x))
a,, and A(x) are the previous solutions (eqs. (6.1) and (6.2)). E indicates the extent of deviation from these solutions. With the aid of eq. (A2) we may also compute [RO,] ’ and CR]’ as in eqs. (3a, b). We can solve f(x) by inserting into eq. (la), neglecting the terms with s2 (arising from the product [p] [O,]), and using eq. (4) which gives: (A3)
f”(x) +{ z}f(x)
+ {a; - !$$if(x)=
-ai(v+aOx).
We may now approximate the solution for f(x) by stating that A(x) = (6 dD,q,/ k’)e-lox (see eq. (5.1)), which must hold in the region where f(x) is important: (A4
f(x)=f(O)+~a,x(l+2v+a,x)
where f(0) is a constant of integration. Application of all boundary conditions and of eq. (12) provides a very complex equation for f(0) and for v, with an upper limit of f(0) =a ; one may calculate however that f(0) is much smaller than this limiting value in most cases pertaining here. We can also compute E by differentiating eq. (Al) and inserting [O,]’ z [O,]‘:
(A5)
&=
It appears that &f(O)is small in all cases covered here and (1 +&f(x)) is unimportant where &f(x) becomes large. Concerning 2 . The condition that Do,d2[02]‘/dx2 must be small was already checked thoroughly in our previous paper (Kreuzer and Hoofd, 1970a, pp. 288-289). Suppose that in the important cases the flux F is predominantly due to the diffusion of oxygen, and that DoId [02]‘/d x2 may amount to not more than two percent of the other terms (eq. (4’)), then for the pigment concentrations pertaining here and if the facilitation is not large : (‘46)
for Hb: P/L< 19;
for Mb: P/L< 2.1
(P/L in mm Hg/lc).
This is important for the cases covered in the following figures: Figure 8 : With Po, = 160 us 0 mm Hg we find from eq. (A6) L 2 9 p ; with L = 9 p three quarters of the flux are due to the diffusion of oxygen.
FACILITATED
DIFFUSION
OF OXYGEN
123
Figure 10 : With Po, = 10 us 0 mm Hg we find similarly L > 4.8 p ; at L = 5 ~1only one quarter of the flux is due to the diffusion of oxygen ; therefore a further check is needed here as done concerning fig. 11 (see below). Figure 12 : Here L = 180 p, so that the critical Po, < 3400 mm Hg which is far above the range covered (about 700 mm Hg). Figure 13 : Here again L = 180 cc, so that the critical P,, < 380 mm Hg which also exceeds the range covered (about 280 mm Hg). If however Do~dz[O,]o/d x2 exceeds 2% of the other terms of eq. (4’) in the core region the calculated profiles of [0,] and [PO,] in the core of the slab will not be exact, i.e., there also will be a region in the core where A(x) # 0. This however is of no influence on the computed flux F since the solutions for A(x) at x=0 and x= L, and for [O,]” remain unchanged ; for in computing the flux F no use has been made of the condition that there is equilibrium everywhere in the cafe of the membrane. Thus we may conclude that for thicker membranes PO, may be. larger than indicated in eq. (A6). For thinner membranes however this may probably no longer be allowed, so that the limits of eq. (A6) should be taken into account ; this has always been done. A(x) is particularly important in the region where PO2is low. This may be illustrated by the case of fig. 11 (see also the layer thickness of 5 p in fig. 7). It may be calculated here that D,,d2[02]‘/d x2 is largest for x = 4.7 p, i.e. close to the opposite boundary of x = L and in the region where A(x) should approximate zero. The influence of this however is small ; even if we assume that A(x) at x = L is 20 % larger, the flux F would be smaller by only 1.5 %. In order to be sure the solution near x = L was furthermore checked with an analog computer which is able to simulate exact solutions of differential equations such as (la) and (lb) though only over a distance of about 1 p so that only insignificant deviations may be seen ; the flux F thus computed was always correct to within 1%. This check justifies a posteriori also fig. 10. Figure 11 is furthermore impcrtant because here [O,]p=,_- [O,]~=, is only 76% ofyP-YP, or, with eq. (2), (A(x=O)-A(x=L))/(yP-yPo)=24% (note that -A(x= L) is positive since there is a minus sign in eq. (5.2)). This figure of 24 % is the highest value computed in this study but there is no reason why in other cases this term cannot be larger. Of course A(x = 0) - A(x = L) may never be larger than y(P - PO) since in this case the flux F would be negative (see eqs. (12), (2), and (3)) with a positive pressure difference (P - PO).With the present approach this could never occur because it follows from F< Oandeq.(lO) that q,, q, < Oand witheq.(5)A(x=O)-A(x=L)< 0, which contradicts the assumption A(x = 0) - A(x = L) > y( P - PO). Acknowledgement
We wish to gratefully acknowledge the stimulating discussions with Professor Jerome S. Schultz, visiting professor, during the last phase of writing this paper. References Hemmingsen,
E. and P. E. Scholander
Science 132: 1379-1381.
(1960). Specilic transport of oxygen through hemoglobin
solutions.
124
F. KREUZER AND L. J. C. HOOFD
Keller, K. H. and S. K. Friedlander (1966). Diffusivity measurements of human methemoglobin. J. Gen. Physiol. 49 : 681-687. Kreuzer, F. (1970). Facilitated diffusion of oxygen and its possible significance; a review. Respir. Physiol. 9: I-30.
Kreuzer, F. and L. J. C. Hoofd (197Oa). Facilitated dilfusion of oxygen in the presence of hemoglobin. Respir. Physiol.
8: 28CL302.
Kreuzer, F. and L. J. C. Hoofd (1970b). Facilitated diffusion of oxygen in the presence of hemoglobin and myoglobin. AGARD Conference Proc. No. 65 : Fluid dynamics of blood circulation and respiratory flow, p. 30/l-5. Kutchai. H. (1970). Numerical study of oxygen uptake by layers of hemoglobin solution. Respir. Physiol. 10: 273-284. Kutchai, H., J. A. Jaquex and F. J. Mather (1970). Nonquilibrium facilitated oxygen transport in hemoglobin solutions. Biophys. J. 10: 38-54. Murray, J. D. (1971). On the molecular mechanism of facilitated oxygen diffusion by hemoglobin and myoglobin. Proc. Roy. Sot. B 178: 95-l 10. Riveros-Moreno, V. and J. B. Wittenberg (1968). The self-diffusion coeflicients of hemoglobin and myoglobin in concentrated sohttions. Manuscript, see also: J. Biol. Chem. 247: 895-901 (1972). Scholandcr, P. F. (1960). Oxygen transport through hemoglobin solutions. Science 131: 585-590. Wittenberg, J. B. (1966). The molecular mechanism of hemoglobin-facilitated oxygen diffusion. J. Biol. Chem. 241: 104-l 14. Wittenbcrg, J. B. (1969). Personal communication. (Letter of May 7, 1969). Wittenberg J. B. (1970). Myoglobin-facilitated oxygen diffusion : role of myoglobin in oxygen entry into muscle. Physiol. Rev. 50: 559436.