J. theor. Biol. (1975) 52,285-297
Facilitated Diffusion: The Elasticity of Oxygen Supply A. N.
STOKES
C.S. I. R.U. Division of Environmental Mechanics, P.O. Box 821, Canberra City, A.C.T. 2601, Australia (Received 29 January 1974, and in revisedform
18 November 1974)
It is well established that the presence of oxygen-carrying proteins such as haemoglobin can facilitate the diffusion of oxygen through a solution. In this paper, it is shown that some properties of a facilitated flow are substantiahy different from those of unfacilitated flux, including especially the stability of the tension at which the oxygen arrives at the end of the
diffusion path. The concept of the “output resistance” of the supply is introduced, and a facilitated pathway is shown to have a lowered resistance. A role for the storage capacity of the bound oxygen reservoir is also developed; it is shown that delivery oxygen tensions are stabilized
against transient changes in oxygen demand. In treating the equations of facilitated diffusion, a simplified approach is used to take account of boundary layers in the solution where deviations from oxygen-protein
equilibrium
are significant. A measure of the thickness and importance
of such boundary layers is calculated.
1. Introduction The flux of oxygen through a solution is affected in a number of ways by the presence of oxygen-carrying proteins such as haemoglobin or myoglobin. It is, of course, enhanced, and much recent work has been done on the significance of this enhancement, both experimental (Scholander, 1960; Wittenberg, 1959, 1966, 1970) and theoretical (Kutchai, Jacquez & Mather, 1970; Goddard, Schultz & Bassett, 1970; Mutray, 1971; Kreuzer & Hoofd, 1970). The proteins also store oxygen, and this property may have importance, for example, in connection with the function of myoglobin in muscle. The model of Wyman (1966), covering the diffusion of oxygen through protein solutions over small distances, has been used frequently. It leads to a non-linear, second-order equation governing the steady-state concentration of oxygen. Since the concentrations are usually near equilibrium levels, the theoretical analyses referred to above have used this as a starting point, and have usually made allowances for deviations from equilibrium 285
286
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STOKES
near the endpoints. The paper of Kutchai et al. (1970) is an exception; here numerical solutions are obtained by a quasilinearization technique. Analysis of the facilitated diffusion model shows that the actual enhancement of flux in biological situations is variable, and often not large. For example, if oxygen diffuses from a source at 20 mm Hg through a solution of 1 mM myoglobin, the facilitated component of the flux cannot, according to the data of Wittenberg (1970), be more than about twice the unfacilitated component. Furthermore, myoglobin has a high affinity for oxygen and can therefore only be effective after the oxygen tension has dropped to low levels. In soybean root nodules, the high-affinity oxygen-carrying protein leghaemoglobin is present and it has often been held that its function is to facilitate the flow of oxygen to nitrogen-fixing bacteroids. It seems, however, that the leghaemoglobin is present only in the immediate neighbourhood of the bacteroids (Bergersen & Goodchild, 1973) and that while nitrogen fixation is in progress the average extent of its oxygenation is only about 20 % (Appleby, 1969). These observations indicate that leghaemoglobin can facilitate only the very last part of the diffusion process, which would do little to increase the overall flow. The suspicion that leghaemoglobin may do more than just increase oxygen flux is confirmed by some results reported by Wittenberg, Bergersen, Appleby & Turner (1974). Here, Rhizobium bacteroids from soybean root nodules, when suspended in a leghaemoglobin solution in contact with 20 mm Hg oxygen in nitrogen, show active nitrogen-fixing ability. If there is no leghaemoglobin, normal respiration is retarded somewhat, whereas nitrogen fixation is reduced greatly, and cannot be restored by supplying oxygen at a higher tension. Were the amount of oxygen supplied the sole concern, an increase in supply tension would soon raise the flux to the required level. In this paper, possible roles, other than simple facilitation are demonstrated for oxygen-carrying proteins. In a static situation, partially oxygenated proteins act to buffer oxygen tensions in the same way that acidbase systems act as pH buffers. This effect influences the diffusion of oxygen through such a protein solution and tends to stabilize the tension at the point of delivery at values in the neighbourhood of pt, the oxygen tension at which the protein is half saturated with oxygen. The storage capacity of the protein offers the second and different ability to stabilize oxygen tensions against transient peaks of oxygen demand. The magnitudes of these effects are calculated for a model system, and are shown to be numerically greater than the facilitation of flux. In the discussion, an analogy is developed between the diffusion of oxygen and
ELASTICITY
OF
OXYGEN
287
SUPPLY
the flow of electricity. It is useful because some of the parameters governing the flow of electricity through non-linear devices are of value; in particular, the notion of input/output resistance corresponds to the inverse of our steady-state stabilization factor. In this way, the effect of a haemoglobin layer may be compared with that of a voltage regulator, or resistancelowering device. 2. The Theoretical Equation
The usual model of one-dimensional steady diffusion of oxygen, with simplified kinetics and constant rate coefficients, is treated here. The notation of Wyman (1966) is used : C CP
Y m k’ k X t
0, D,
= = = = = = = = = =
concentration of dissolved oxygen concentration of protein in solution fractional saturation of protein number of molecules of oxygen per binding site “on” constant for the oxygen-protein reaction off constant space variable time diffusivity of oxygen diffusivity of protein
D, and D, are assumed constant, and D, is assumed to be unaffected by the attachment of oxygen. The partial differential equations for the concentrations are: D, 2
= mk’cc,(l
D, %;$ Dp $
- Y) - mkc, Y + ;
= kc, Y - k’cc,(l
[c,(l - Y)] = k’cc,(l
- Y) + & (cp Y)
- Y) - kc, Y + & (cp-cp
Y).
If steady conditions are attained, the time derivatives vanish and D
a2c cax2
D --
m P
a2(cPy> ax2
-=
D,-$n(c,-e,Y).
(4)
Then mD,c,Y(l-Y)-D,c
=4x+1,4
where 4 is the total oxygen flux and $ is a constant determined
(5) by the
288
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STOKES
boundary conditions. These are assumed to be a boundary membrane at x = 0 acting as a source of oxygen, and a membrane at x = L where oxygen is delivered. Both membranes are taken to be permeable to oxygen but not to protein. There is no net flux of protein, so cP is constant, and the condition of zero protein Flux at the boundaries means that dY d;=O
and
DC:;=-4
at x = 0 and x = L. The equations are normalized by taking a characteristic oxygen concentration c,, conventionally that at the source boundary, and putting:
Then dZW --= uU(yW+ 1)-u, dy2 /IV-W = Fy+A, W(0) = I,
(8)
dW & = -F,
aty = Oandy = 1. In the types of protein facilitation problems normally arising, a is large. This implies that either d2 W/dy2 is large, or that U(y W+ 1) N 1. The latter equation, corresponding to equilibrium conditions, prevails almost everywhere, and W(y) is the solution of a simple quadratic equation. Wyman (1966) examines this solution, showing that near y = 0 oxygen transport was mainly by dissolved oxygen, but as y increases, protein is an increasingly important carrier. The equilibrium solution does not fit all the boundary conditions. The conflict is most serious at y = 1, where transport at equilibrium predicts that protein is at its most important as a carrier, so i?U/ay reaches a maximum, while our boundary condition is that aU/ay = 0 at y = 1. At y = 0 there is usually no problem, since here, under the equilibrium hypothesis, most oxygen is transported in the free state, and this is con-
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sistent with the boundary conditions. Some correction may be needed if the protein is substantially unsaturated near y = 0. Kreuzer & Hoofd (1970) divided the diffusion path into three zones: a mid-zone, where equilibrium conditions apply, and the two end-zones, where deviations from equilibrium may occur. Goddard et al. (1970) applied the singular perturbation technique of matching asymptotic expansions to derive higher order terms. Our approach is more like that of Kreuzer BEHoofd, but simpler and, we believe, slightly more accurate.We use the approximation, true to second order in y, that near y = 1, V(y) is constant, and derive a resulting approximate solution near y = 1. Then (9) becomesa linear equation in W, with solutions 1-U(l) WY) = yuo
+ P ev d~w(W--1))
+ Q exp (Ja@(l)(l
-y)).
(lo)
Since the correction to W(1) must decreaseas y decreases,Q must be zero. From (9), -. p = - JqIJ(l)
(11)
In fact, JarV(l) is usually very large, so the boundary layer where equation (10) applies is very thin. However, it can produce a significant variation in the calculated value of W(1). Where CIis large, the boundary layer solution (10) can be incorporated in a global solution by writing
We do not, however, need this overall solution for our later calculations. Instead, all we seek is the relationship between the three quantities w, = m9, w, = W(1) and F, any two of which determine the third. Usually W, is given, and so we seeka relationship between WI and F. We would be on uncertain ground in calculating W, without making a correction for the deviation from equilibrium near y = 1. Assuming equilibrium at y = 0, we then have the required relationship implicit in the equations /?U,w,=-
1 ---I1 Y ( Ul
W, = A+F, --
>
&&9
(12)
(13)
290
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where k A=------r%+l
wo
and U1 = U(1). If F is given, elimination of Ui yields a quartic equation for W,. A simpler computational scheme is to start with WI = 0, calculate U, from (12), then a new WI from (13), and repeat. If fit’, is large compared with W,, convergence is rapid. If W, is given, a cubic equation for F results. 3. Numerical Application
We adopt the values of the constants used in the first example of Kutchai et al. (1970). These are chosen because concentration profiles have been calculated, and the results have been used as a basis for comparison by other authors (Goddard et al., 1970; Mitchell & Murray, 1973). The numbers used apply to a 100 u thick layer of 160 gm/l solution of human haemoglobin at 25°C with oxygen supplied to one surface at 20 mm Hg tension. The molar concentration used has been adjusted to allow for the ability of haemoglobin to carry four oxygen molecules; the model used assumes that each site has the same kinetic constants regardless of the state of oxygenation of the whole molecule. Once this adjustment has been made, the analysis is equivalent to that of a protein with one oxygen-binding site, and for simplicity the discussion proceeds as if that were so. The values were: DP DC t77cp
CO k’ k L
3 x 10e7 cm’/sec 1.5 x lo-’ cm’/sec 1-8x lo-’ mole/ml 3.6 x 1O-* mole/ml 16x lo9 ml/mole.sec 52 set- ’ 0.01 cm.
From these we derive a=1*733x105 p = 10 y = 11.08.
ELASTICITY
0
01
02
OF OXYGEN
0.3 0.4 05
291
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0.6 0.7 0.6 09
0
Fro. 1. Dimensionless flux F as a function of WI, the relative oxygen tension at delivery.
60 70 60 50 40
-gin
facilitated
case
30 20
~
‘O, 0
0.1 0.2 0.3 0.4 05
O-6 O-7 O-6 O-9 I.0
FIG. 2. Gradient of the curve in Fig. 1. -(dE/d W,) is a measure of the stability of the oxygen tension against flux changes. T.B.
19
292
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STOKES
As described in the previous section, F was then determined as a function of W,. It is graphed in Fig. 1, and its derivative with respect to W,, which is a measure of stabilization of W,, is graphed in Fig. 2. The total oxyprotein content G of the solution for each value of W, was calculated assuming equilibrium throughout for the value of F previously calculated at the given value of W,. This equilibrium assumption very slightly underestimates the oxyprotein content, which in turn is slightly less than the total oxygen content of the solution. To three significant figures, these three quantities are identical. 6 is graphed against W, in Fig. 3, and its derivative dfi/dlV, in Fig. 4. The numerical results used in the graphs are given in Table I.
0
0 I
0 2
0.3
0.4
0.5
0 6
0.7
0.8
O-9
I.0
w,
FIG. 3. Total oxygen content (moles) of the diffusion path per cm2 (C), as a function of WI.
In the absence of protein the following values would have been obtained: F=i-W, -..=dF 1 dK Is = (1 -$W,)x ~~~ = -1-S x lo-” d W,
3.6 x 10-i’ mole/cm2.
mole/cm’
ELASTICITY
OF OXYGEN
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SUPPLY
0.1 0.2 0.3 0.4 O-5 O-6 0.7 O-8 09 I.0 u; FIG. 4. Gradient of the curve in Fig. 3, being a measure of the amount of stored oxygen which a solution will yield in response to a change in oxygen tension at delivery. The stored oxygen stabilizes this oxygen tension against transient flux changes. 0
TABLE
1
Numerical results used in the graphs
F
0
0.02 0.05 O-08 0.10 o-15 0.20 o-35 x:: 1
9444 7*8&l 6.302 5234 4.695 3.711 3*040 l-858 l-196 0.495 0
--dF
u(x 10’)
dW1
(mole/cnP)
92.97 65.46 42.48 29.84 24.36 15.92 11.32 5.565 3.559 2.269 l-759
0.94 l-08 1.21 l-30 l-34 1 a42 l-47 l-56 1.59 1 a629 1.65
$(X 107) im~le,kma) 8.015 5.562 3.533 2.428 l-951 1.224 O-833 0.358 O-203 0.106 0.068
4. Discussionof Results
In our example, F reaches its maximum level when W, = 0, and is then ten times the unfacilitated flux level. The presence of protein increases
294
A.
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the level of -dF/dW, by a maximum factor of 94.4, and of -dij/dW, by a maximum factor of 4400, both when W, = 0. The total amount of oxygen that is given up when WI changes from I to 0 is 7.1 x IO-* mole/cm2 in the presence of protein; this is 394 times the value corresponding to the absence of protein. These calculations show that the parameters governing the stability of oxygen concentration levels with respect to flux changes are far more drastically altered by the presence of protein than is the flux level itself. The results of Wittenberg et al. (1974) seem to show that in the absence of facilitation, cells can suffer from high oxygen concentrations. Even where the oxygen intake of cells appears to be uniform, stabilization of oxygen tension can provide an advantage. The rate of consumption of oxygen will probably be regulated by feed-back mechanisms. In such systems, one must look for possible instabilities, and the stabilization of oxygen concentration against both transient and sustained changes in flux will be one of the factors governing overall stability. More insight into the steady-state stabilization properties of haemoglobin is available with an analogy from electrical circuit theory. If F corresponds to a current, and W to a voltage, then dW,/dF corresponds to an output conductance, and its inverse dF/dW, to an output resistance. An unfacilitated, diffusive pathway corresponds to an ohmic conductor. A facilitated pathway corresponds to a resistance-lowering device, since the output resistance is low. We have regarded W, as fixed, corresponding to a voltage source, or zero-resistance supply. This is not necessary. If W, is regarded as a function of F and W,, then differentiating by the chain rule gives
The first term on the right is a measure of the stabilization potential against flux or consumption changes, and the change in W, or the external tension. If the source has resistance dW,/dF, so that the source tension varies with resistance is transmitted but diminished :
of the delivery second against a finite output flux, then this
since
Generally (S/a W&,
is of moderate size and not much affected by change
ELASTICITY
OF
OXYGEN
SUPPLY
295
in W,, while (c!%V,/C~F), is small if stabilization is working well. In our example, (LV/W,) lies between 164 and 1 a76 for W, between 0 and 1. If W, = 0.05, then
dK = -0.0235+@0398 dF
‘2.
So the protein solution layer not only delivers oxygen at a stable low tension from a high tension source, but greatly reduces the effect of any output resistance present in the supply. In contrast, without facilitation an oxygen tension drop can be achieved only at the expense of a corresponding increase in output resistance, and, additionally, any output resistance present in the source is transmitted without reduction. To derive its oxygen from a low-resistance source may not always be desirable for a respiring cell vulnerable to high oxygen tensions. If there is an increased amount of oxygen in its environment, then, short of putting up some physical barrier, the cell can only maintain a low tension by increasing its rate of consumption. The lowering of oxygen tension produced by this response is proportional to the supply resistance. In other words, it may be desirable to have the oxygen supply tension unresponsive to flux changes caused by changes in cell requirements but responsive to changes needed to deal with oxygen excess. No unfacilitated diffusive supply can vary its responsiveness in such a way, but as Fig. 2 shows, facilitated diffusion has exactly this property, with the resistance, or responsiveness, increasing as the delivery oxygen tension increases. Another argument is sometimes available to reinforce the hypothesis that stabilization of oxygen tension is the major function of oxygencarrying proteins. This is based on the occurrence of proteins with a high oxygen afhnity (Wittenberg, 1970). Leghaemoglobin, for example, is a high-afhnity protein occurring in legume root nodules. It has kinetic constants k = 118 x 10’ moles/cm’/sec, k = 44/set and consequently p+ = 0.02 mm Hg (Wittenberg et al., 1974). If haemoglobin were replaced by ieghaemoglobin in the above example, the oxygen flux would be less for every value of WI, approaching equality only as WI becomes very small. Even when W, = 0, F = 6.97. However, for small values of W, leghaemoglobin gives very much better stabilization ; when WI = 0, dF/d W = 2890, a 30-fold increase over the corresponding value for haemoglobin. Some recent experiments have indicated that in the root nodules of legumes the symbiotic Rhizobium bacteroids do have an optimum oxygen concentration for nitrogen fixation and that this optimum is near the p+ of the leghaemoglobin which is present in their neighbourhood. Bergersen & Turner (1975) have shown markedly increased nitrogenase activity when
296
A.
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oxygen is supplied from leghaemoglobin at low tension, although they were not prepared to attribute the increase solely to the effect of oxygen concentrations. Stokes (1975) advanced an explanation of the behaviour of Rhizobium bacteroids in stirred suspension which postulated such an optimum. In these nodules stabilization of oxygen tensions seems likely to be an important function of leghaemoglobin. 5. Thickness
of the Boundary Layer
The boundary layer is a local phenomenon at one boundary, and is better dealt with in absolute units, since the solution layer thickness is not relevant. From equations (10) and (1 I), the boundary layer correction function is
with u1 = c,,[l- Y(l)], and 4 the absolute flux, defined previously. JDc/k’uI is the distance constant of the exponential term, and is a measure of the thickness of the boundary-..- layer. . For efficient facilitation of flux, u1 is close to cP signifying that JDJk’c, 1s a reasonable estimate of the constant in these circumstances. The constant rises as the flux diminishes, and reaches a maximum of J(D,/k’c,)(l + y) when the flux is zero. -__ The term +/dk’Dc u1 measures the total change in terminal oxygen tension caused by the boundary layer. It is large when the flux is high; this is the reason for the finding of Kutchai et al. (1970) that the equilibrium transport hypothesis fails for thin layers. But it is also relatively more significant when high-affinity proteins are being considered. This is because the oxygen tension change AC can become large with respect to p+ for the protein; in that case oxygen transport outside the boundary can only take place at tensions higher than AC, and thus higher than pt. This is outside the zone of efficient facilitation. The boundary layer itself is a region where facilitation is poor, so that overall the efficiency of facilitation is limited. In our example, the values of the parameters were: cD
c1 = OmmHg 2mmHg 4mmHg 10mmHg 20 mm Hg
J--- k’u, 0.106 p o-150 p 0.184 p 0.261 p 0.355 p
-_. &=Ac
0.200 mm Hg 0.141 mm Hg 0*112 mm Hg 0.0626 mm Hg OmmHg
ELASTICITY
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OXYGEN
SUPPLY
297
Because the boundary layer is so thin, it will be necessary in trying to reach a solution by numerical methods to use a procedure with a variable step size, with small steps in the boundary layer. For this reason we have some doubts about the accuracy of the solutions of Kutchai et al. (1970) for longer diffusion paths. In the example where they calculate profiles from the data above, but with c,, = 200 mm Hg, the upper limit to the distance constant is l-08 ~1; but when L = 100 p, for example, their step size is 3.125 p. 6. Conclusion
Our conclusion is that the most important function of oxygen-carrying proteins may be the stabilization of the tension at which oxygen is supplied; rather than simply increasing the amount of oxygen that can be carried. The author thanks Dr J. B. Wittenberg and Dr C. A. Appleby for many helpful discussions, and for stimulating his interest in the topic. REFERENCES APPLEBY, C. A. (1969). Biochim. biophys. Actu 188, 222. BERGERSEN, F. J. & GOODCHILD, D. J. (1973). Ausr. J. biol. Sci. 26,741. BERGERSEN, F. J. & %NER, G. L. (1975). Submitted for publication. GODDARD, J. D., SCHULTZ, J. S. & BASSETT, R. J. (1970). Gem. Engng Sci. 25,665. KREUZER, F. & Hoom, L. J. C. (1970). Rap. Physiol. 8,280. KUTCHAI, H., JACQUEZ, J. A. & MATHER, F. J. (1970). Biophys. J. 10, 38. MITCHELL, P. J. & MURRAY, J. D. (1973). Biophysik 9, 177. MURRAY, J. D. (1971). Proc. R. Sot. Ser. B 178, 95. SCHOLANDER, P. F. (1960). Science, N. Y. 131,585. STOKES. A. N. (1975). In preparation. Wmxmm~, J. B. (1959). Bioi. Bull. mar. biol. Lab., Woods Hole 117,402. Wrrmmmo, J. B. (1966). .I. biof. Chem. 241, 104. WI~NBERO, J. B. (1970). Physiol. Rev, 50,559. WII-IENBERG, J. B., BERQENSEN, F. J., APPLEBY, C. A. & TURNER, 0. L. (1974). J. Chem. (in press). WYMAN, J. (1966). J. biol. Chem. 241, 115.
biol.
Facilitated Diffusion: The Elasticity of Oxygen Supply A. N.
STOKES
C.S. I. R.U. Division of Environmental Mechanics, P.O. Box 821, Canberra City, A.C.T. 2601, Australia (Received 29 January 1974, and in revisedform
18 November 1974)
It is well established that the presence of oxygen-carrying proteins such as haemoglobin can facilitate the diffusion of oxygen through a solution. In this paper, it is shown that some properties of a facilitated flow are substantiahy different from those of unfacilitated flux, including especially the stability of the tension at which the oxygen arrives at the end of the
diffusion path. The concept of the “output resistance” of the supply is introduced, and a facilitated pathway is shown to have a lowered resistance. A role for the storage capacity of the bound oxygen reservoir is also developed; it is shown that delivery oxygen tensions are stabilized
against transient changes in oxygen demand. In treating the equations of facilitated diffusion, a simplified approach is used to take account of boundary layers in the solution where deviations from oxygen-protein
equilibrium
are significant. A measure of the thickness and importance
of such boundary layers is calculated.
1. Introduction The flux of oxygen through a solution is affected in a number of ways by the presence of oxygen-carrying proteins such as haemoglobin or myoglobin. It is, of course, enhanced, and much recent work has been done on the significance of this enhancement, both experimental (Scholander, 1960; Wittenberg, 1959, 1966, 1970) and theoretical (Kutchai, Jacquez & Mather, 1970; Goddard, Schultz & Bassett, 1970; Mutray, 1971; Kreuzer & Hoofd, 1970). The proteins also store oxygen, and this property may have importance, for example, in connection with the function of myoglobin in muscle. The model of Wyman (1966), covering the diffusion of oxygen through protein solutions over small distances, has been used frequently. It leads to a non-linear, second-order equation governing the steady-state concentration of oxygen. Since the concentrations are usually near equilibrium levels, the theoretical analyses referred to above have used this as a starting point, and have usually made allowances for deviations from equilibrium 285
286
A.
N.
STOKES
near the endpoints. The paper of Kutchai et al. (1970) is an exception; here numerical solutions are obtained by a quasilinearization technique. Analysis of the facilitated diffusion model shows that the actual enhancement of flux in biological situations is variable, and often not large. For example, if oxygen diffuses from a source at 20 mm Hg through a solution of 1 mM myoglobin, the facilitated component of the flux cannot, according to the data of Wittenberg (1970), be more than about twice the unfacilitated component. Furthermore, myoglobin has a high affinity for oxygen and can therefore only be effective after the oxygen tension has dropped to low levels. In soybean root nodules, the high-affinity oxygen-carrying protein leghaemoglobin is present and it has often been held that its function is to facilitate the flow of oxygen to nitrogen-fixing bacteroids. It seems, however, that the leghaemoglobin is present only in the immediate neighbourhood of the bacteroids (Bergersen & Goodchild, 1973) and that while nitrogen fixation is in progress the average extent of its oxygenation is only about 20 % (Appleby, 1969). These observations indicate that leghaemoglobin can facilitate only the very last part of the diffusion process, which would do little to increase the overall flow. The suspicion that leghaemoglobin may do more than just increase oxygen flux is confirmed by some results reported by Wittenberg, Bergersen, Appleby & Turner (1974). Here, Rhizobium bacteroids from soybean root nodules, when suspended in a leghaemoglobin solution in contact with 20 mm Hg oxygen in nitrogen, show active nitrogen-fixing ability. If there is no leghaemoglobin, normal respiration is retarded somewhat, whereas nitrogen fixation is reduced greatly, and cannot be restored by supplying oxygen at a higher tension. Were the amount of oxygen supplied the sole concern, an increase in supply tension would soon raise the flux to the required level. In this paper, possible roles, other than simple facilitation are demonstrated for oxygen-carrying proteins. In a static situation, partially oxygenated proteins act to buffer oxygen tensions in the same way that acidbase systems act as pH buffers. This effect influences the diffusion of oxygen through such a protein solution and tends to stabilize the tension at the point of delivery at values in the neighbourhood of pt, the oxygen tension at which the protein is half saturated with oxygen. The storage capacity of the protein offers the second and different ability to stabilize oxygen tensions against transient peaks of oxygen demand. The magnitudes of these effects are calculated for a model system, and are shown to be numerically greater than the facilitation of flux. In the discussion, an analogy is developed between the diffusion of oxygen and
ELASTICITY
OF
OXYGEN
287
SUPPLY
the flow of electricity. It is useful because some of the parameters governing the flow of electricity through non-linear devices are of value; in particular, the notion of input/output resistance corresponds to the inverse of our steady-state stabilization factor. In this way, the effect of a haemoglobin layer may be compared with that of a voltage regulator, or resistancelowering device. 2. The Theoretical Equation
The usual model of one-dimensional steady diffusion of oxygen, with simplified kinetics and constant rate coefficients, is treated here. The notation of Wyman (1966) is used : C CP
Y m k’ k X t
0, D,
= = = = = = = = = =
concentration of dissolved oxygen concentration of protein in solution fractional saturation of protein number of molecules of oxygen per binding site “on” constant for the oxygen-protein reaction off constant space variable time diffusivity of oxygen diffusivity of protein
D, and D, are assumed constant, and D, is assumed to be unaffected by the attachment of oxygen. The partial differential equations for the concentrations are: D, 2
= mk’cc,(l
D, %;$ Dp $
- Y) - mkc, Y + ;
= kc, Y - k’cc,(l
[c,(l - Y)] = k’cc,(l
- Y) + & (cp Y)
- Y) - kc, Y + & (cp-cp
Y).
If steady conditions are attained, the time derivatives vanish and D
a2c cax2
D --
m P
a2(cPy> ax2
-=
D,-$n(c,-e,Y).
(4)
Then mD,c,Y(l-Y)-D,c
=4x+1,4
where 4 is the total oxygen flux and $ is a constant determined
(5) by the
288
A.
N.
STOKES
boundary conditions. These are assumed to be a boundary membrane at x = 0 acting as a source of oxygen, and a membrane at x = L where oxygen is delivered. Both membranes are taken to be permeable to oxygen but not to protein. There is no net flux of protein, so cP is constant, and the condition of zero protein Flux at the boundaries means that dY d;=O
and
DC:;=-4
at x = 0 and x = L. The equations are normalized by taking a characteristic oxygen concentration c,, conventionally that at the source boundary, and putting:
Then dZW --= uU(yW+ 1)-u, dy2 /IV-W = Fy+A, W(0) = I,
(8)
dW & = -F,
aty = Oandy = 1. In the types of protein facilitation problems normally arising, a is large. This implies that either d2 W/dy2 is large, or that U(y W+ 1) N 1. The latter equation, corresponding to equilibrium conditions, prevails almost everywhere, and W(y) is the solution of a simple quadratic equation. Wyman (1966) examines this solution, showing that near y = 0 oxygen transport was mainly by dissolved oxygen, but as y increases, protein is an increasingly important carrier. The equilibrium solution does not fit all the boundary conditions. The conflict is most serious at y = 1, where transport at equilibrium predicts that protein is at its most important as a carrier, so i?U/ay reaches a maximum, while our boundary condition is that aU/ay = 0 at y = 1. At y = 0 there is usually no problem, since here, under the equilibrium hypothesis, most oxygen is transported in the free state, and this is con-
ELASTICITY
OF
OXYGEN
289
SUPPLY
sistent with the boundary conditions. Some correction may be needed if the protein is substantially unsaturated near y = 0. Kreuzer & Hoofd (1970) divided the diffusion path into three zones: a mid-zone, where equilibrium conditions apply, and the two end-zones, where deviations from equilibrium may occur. Goddard et al. (1970) applied the singular perturbation technique of matching asymptotic expansions to derive higher order terms. Our approach is more like that of Kreuzer BEHoofd, but simpler and, we believe, slightly more accurate.We use the approximation, true to second order in y, that near y = 1, V(y) is constant, and derive a resulting approximate solution near y = 1. Then (9) becomesa linear equation in W, with solutions 1-U(l) WY) = yuo
+ P ev d~w(W--1))
+ Q exp (Ja@(l)(l
-y)).
(lo)
Since the correction to W(1) must decreaseas y decreases,Q must be zero. From (9), -. p = - JqIJ(l)
(11)
In fact, JarV(l) is usually very large, so the boundary layer where equation (10) applies is very thin. However, it can produce a significant variation in the calculated value of W(1). Where CIis large, the boundary layer solution (10) can be incorporated in a global solution by writing
We do not, however, need this overall solution for our later calculations. Instead, all we seek is the relationship between the three quantities w, = m9, w, = W(1) and F, any two of which determine the third. Usually W, is given, and so we seeka relationship between WI and F. We would be on uncertain ground in calculating W, without making a correction for the deviation from equilibrium near y = 1. Assuming equilibrium at y = 0, we then have the required relationship implicit in the equations /?U,w,=-
1 ---I1 Y ( Ul
W, = A+F, --
>
&&9
(12)
(13)
290
A.
N.
STOKES
where k A=------r%+l
wo
and U1 = U(1). If F is given, elimination of Ui yields a quartic equation for W,. A simpler computational scheme is to start with WI = 0, calculate U, from (12), then a new WI from (13), and repeat. If fit’, is large compared with W,, convergence is rapid. If W, is given, a cubic equation for F results. 3. Numerical Application
We adopt the values of the constants used in the first example of Kutchai et al. (1970). These are chosen because concentration profiles have been calculated, and the results have been used as a basis for comparison by other authors (Goddard et al., 1970; Mitchell & Murray, 1973). The numbers used apply to a 100 u thick layer of 160 gm/l solution of human haemoglobin at 25°C with oxygen supplied to one surface at 20 mm Hg tension. The molar concentration used has been adjusted to allow for the ability of haemoglobin to carry four oxygen molecules; the model used assumes that each site has the same kinetic constants regardless of the state of oxygenation of the whole molecule. Once this adjustment has been made, the analysis is equivalent to that of a protein with one oxygen-binding site, and for simplicity the discussion proceeds as if that were so. The values were: DP DC t77cp
CO k’ k L
3 x 10e7 cm’/sec 1.5 x lo-’ cm’/sec 1-8x lo-’ mole/ml 3.6 x 1O-* mole/ml 16x lo9 ml/mole.sec 52 set- ’ 0.01 cm.
From these we derive a=1*733x105 p = 10 y = 11.08.
ELASTICITY
0
01
02
OF OXYGEN
0.3 0.4 05
291
SUPPLY
0.6 0.7 0.6 09
0
Fro. 1. Dimensionless flux F as a function of WI, the relative oxygen tension at delivery.
60 70 60 50 40
-gin
facilitated
case
30 20
~
‘O, 0
0.1 0.2 0.3 0.4 05
O-6 O-7 O-6 O-9 I.0
FIG. 2. Gradient of the curve in Fig. 1. -(dE/d W,) is a measure of the stability of the oxygen tension against flux changes. T.B.
19
292
A.
N.
STOKES
As described in the previous section, F was then determined as a function of W,. It is graphed in Fig. 1, and its derivative with respect to W,, which is a measure of stabilization of W,, is graphed in Fig. 2. The total oxyprotein content G of the solution for each value of W, was calculated assuming equilibrium throughout for the value of F previously calculated at the given value of W,. This equilibrium assumption very slightly underestimates the oxyprotein content, which in turn is slightly less than the total oxygen content of the solution. To three significant figures, these three quantities are identical. 6 is graphed against W, in Fig. 3, and its derivative dfi/dlV, in Fig. 4. The numerical results used in the graphs are given in Table I.
0
0 I
0 2
0.3
0.4
0.5
0 6
0.7
0.8
O-9
I.0
w,
FIG. 3. Total oxygen content (moles) of the diffusion path per cm2 (C), as a function of WI.
In the absence of protein the following values would have been obtained: F=i-W, -..=dF 1 dK Is = (1 -$W,)x ~~~ = -1-S x lo-” d W,
3.6 x 10-i’ mole/cm2.
mole/cm’
ELASTICITY
OF OXYGEN
293
SUPPLY
0.1 0.2 0.3 0.4 O-5 O-6 0.7 O-8 09 I.0 u; FIG. 4. Gradient of the curve in Fig. 3, being a measure of the amount of stored oxygen which a solution will yield in response to a change in oxygen tension at delivery. The stored oxygen stabilizes this oxygen tension against transient flux changes. 0
TABLE
1
Numerical results used in the graphs
F
0
0.02 0.05 O-08 0.10 o-15 0.20 o-35 x:: 1
9444 7*8&l 6.302 5234 4.695 3.711 3*040 l-858 l-196 0.495 0
--dF
u(x 10’)
dW1
(mole/cnP)
92.97 65.46 42.48 29.84 24.36 15.92 11.32 5.565 3.559 2.269 l-759
0.94 l-08 1.21 l-30 l-34 1 a42 l-47 l-56 1.59 1 a629 1.65
$(X 107) im~le,kma) 8.015 5.562 3.533 2.428 l-951 1.224 O-833 0.358 O-203 0.106 0.068
4. Discussionof Results
In our example, F reaches its maximum level when W, = 0, and is then ten times the unfacilitated flux level. The presence of protein increases
294
A.
N.
STOKES
the level of -dF/dW, by a maximum factor of 94.4, and of -dij/dW, by a maximum factor of 4400, both when W, = 0. The total amount of oxygen that is given up when WI changes from I to 0 is 7.1 x IO-* mole/cm2 in the presence of protein; this is 394 times the value corresponding to the absence of protein. These calculations show that the parameters governing the stability of oxygen concentration levels with respect to flux changes are far more drastically altered by the presence of protein than is the flux level itself. The results of Wittenberg et al. (1974) seem to show that in the absence of facilitation, cells can suffer from high oxygen concentrations. Even where the oxygen intake of cells appears to be uniform, stabilization of oxygen tension can provide an advantage. The rate of consumption of oxygen will probably be regulated by feed-back mechanisms. In such systems, one must look for possible instabilities, and the stabilization of oxygen concentration against both transient and sustained changes in flux will be one of the factors governing overall stability. More insight into the steady-state stabilization properties of haemoglobin is available with an analogy from electrical circuit theory. If F corresponds to a current, and W to a voltage, then dW,/dF corresponds to an output conductance, and its inverse dF/dW, to an output resistance. An unfacilitated, diffusive pathway corresponds to an ohmic conductor. A facilitated pathway corresponds to a resistance-lowering device, since the output resistance is low. We have regarded W, as fixed, corresponding to a voltage source, or zero-resistance supply. This is not necessary. If W, is regarded as a function of F and W,, then differentiating by the chain rule gives
The first term on the right is a measure of the stabilization potential against flux or consumption changes, and the change in W, or the external tension. If the source has resistance dW,/dF, so that the source tension varies with resistance is transmitted but diminished :
of the delivery second against a finite output flux, then this
since
Generally (S/a W&,
is of moderate size and not much affected by change
ELASTICITY
OF
OXYGEN
SUPPLY
295
in W,, while (c!%V,/C~F), is small if stabilization is working well. In our example, (LV/W,) lies between 164 and 1 a76 for W, between 0 and 1. If W, = 0.05, then
dK = -0.0235+@0398 dF
‘2.
So the protein solution layer not only delivers oxygen at a stable low tension from a high tension source, but greatly reduces the effect of any output resistance present in the supply. In contrast, without facilitation an oxygen tension drop can be achieved only at the expense of a corresponding increase in output resistance, and, additionally, any output resistance present in the source is transmitted without reduction. To derive its oxygen from a low-resistance source may not always be desirable for a respiring cell vulnerable to high oxygen tensions. If there is an increased amount of oxygen in its environment, then, short of putting up some physical barrier, the cell can only maintain a low tension by increasing its rate of consumption. The lowering of oxygen tension produced by this response is proportional to the supply resistance. In other words, it may be desirable to have the oxygen supply tension unresponsive to flux changes caused by changes in cell requirements but responsive to changes needed to deal with oxygen excess. No unfacilitated diffusive supply can vary its responsiveness in such a way, but as Fig. 2 shows, facilitated diffusion has exactly this property, with the resistance, or responsiveness, increasing as the delivery oxygen tension increases. Another argument is sometimes available to reinforce the hypothesis that stabilization of oxygen tension is the major function of oxygencarrying proteins. This is based on the occurrence of proteins with a high oxygen afhnity (Wittenberg, 1970). Leghaemoglobin, for example, is a high-afhnity protein occurring in legume root nodules. It has kinetic constants k = 118 x 10’ moles/cm’/sec, k = 44/set and consequently p+ = 0.02 mm Hg (Wittenberg et al., 1974). If haemoglobin were replaced by ieghaemoglobin in the above example, the oxygen flux would be less for every value of WI, approaching equality only as WI becomes very small. Even when W, = 0, F = 6.97. However, for small values of W, leghaemoglobin gives very much better stabilization ; when WI = 0, dF/d W = 2890, a 30-fold increase over the corresponding value for haemoglobin. Some recent experiments have indicated that in the root nodules of legumes the symbiotic Rhizobium bacteroids do have an optimum oxygen concentration for nitrogen fixation and that this optimum is near the p+ of the leghaemoglobin which is present in their neighbourhood. Bergersen & Turner (1975) have shown markedly increased nitrogenase activity when
296
A.
N.
STOKES
oxygen is supplied from leghaemoglobin at low tension, although they were not prepared to attribute the increase solely to the effect of oxygen concentrations. Stokes (1975) advanced an explanation of the behaviour of Rhizobium bacteroids in stirred suspension which postulated such an optimum. In these nodules stabilization of oxygen tensions seems likely to be an important function of leghaemoglobin. 5. Thickness
of the Boundary Layer
The boundary layer is a local phenomenon at one boundary, and is better dealt with in absolute units, since the solution layer thickness is not relevant. From equations (10) and (1 I), the boundary layer correction function is
with u1 = c,,[l- Y(l)], and 4 the absolute flux, defined previously. JDc/k’uI is the distance constant of the exponential term, and is a measure of the thickness of the boundary-..- layer. . For efficient facilitation of flux, u1 is close to cP signifying that JDJk’c, 1s a reasonable estimate of the constant in these circumstances. The constant rises as the flux diminishes, and reaches a maximum of J(D,/k’c,)(l + y) when the flux is zero. -__ The term +/dk’Dc u1 measures the total change in terminal oxygen tension caused by the boundary layer. It is large when the flux is high; this is the reason for the finding of Kutchai et al. (1970) that the equilibrium transport hypothesis fails for thin layers. But it is also relatively more significant when high-affinity proteins are being considered. This is because the oxygen tension change AC can become large with respect to p+ for the protein; in that case oxygen transport outside the boundary can only take place at tensions higher than AC, and thus higher than pt. This is outside the zone of efficient facilitation. The boundary layer itself is a region where facilitation is poor, so that overall the efficiency of facilitation is limited. In our example, the values of the parameters were: cD
c1 = OmmHg 2mmHg 4mmHg 10mmHg 20 mm Hg
J--- k’u, 0.106 p o-150 p 0.184 p 0.261 p 0.355 p
-_. &=Ac
0.200 mm Hg 0.141 mm Hg 0*112 mm Hg 0.0626 mm Hg OmmHg
ELASTICITY
OF
OXYGEN
SUPPLY
297
Because the boundary layer is so thin, it will be necessary in trying to reach a solution by numerical methods to use a procedure with a variable step size, with small steps in the boundary layer. For this reason we have some doubts about the accuracy of the solutions of Kutchai et al. (1970) for longer diffusion paths. In the example where they calculate profiles from the data above, but with c,, = 200 mm Hg, the upper limit to the distance constant is l-08 ~1; but when L = 100 p, for example, their step size is 3.125 p. 6. Conclusion
Our conclusion is that the most important function of oxygen-carrying proteins may be the stabilization of the tension at which oxygen is supplied; rather than simply increasing the amount of oxygen that can be carried. The author thanks Dr J. B. Wittenberg and Dr C. A. Appleby for many helpful discussions, and for stimulating his interest in the topic. REFERENCES APPLEBY, C. A. (1969). Biochim. biophys. Actu 188, 222. BERGERSEN, F. J. & GOODCHILD, D. J. (1973). Ausr. J. biol. Sci. 26,741. BERGERSEN, F. J. & %NER, G. L. (1975). Submitted for publication. GODDARD, J. D., SCHULTZ, J. S. & BASSETT, R. J. (1970). Gem. Engng Sci. 25,665. KREUZER, F. & Hoom, L. J. C. (1970). Rap. Physiol. 8,280. KUTCHAI, H., JACQUEZ, J. A. & MATHER, F. J. (1970). Biophys. J. 10, 38. MITCHELL, P. J. & MURRAY, J. D. (1973). Biophysik 9, 177. MURRAY, J. D. (1971). Proc. R. Sot. Ser. B 178, 95. SCHOLANDER, P. F. (1960). Science, N. Y. 131,585. STOKES. A. N. (1975). In preparation. Wmxmm~, J. B. (1959). Bioi. Bull. mar. biol. Lab., Woods Hole 117,402. Wrrmmmo, J. B. (1966). .I. biof. Chem. 241, 104. WI~NBERO, J. B. (1970). Physiol. Rev, 50,559. WII-IENBERG, J. B., BERQENSEN, F. J., APPLEBY, C. A. & TURNER, 0. L. (1974). J. Chem. (in press). WYMAN, J. (1966). J. biol. Chem. 241, 115.
biol.
