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Transport of oxygen in solutions of hemoglobin and myoglobin
Transport of Oxygen in Solutions of Hemoglobin and Myoglobin JOSE M. GONZALEZFERNANDEZ AND
SUSIE E. ATTA Muthematid Reseurch Brunch, Nutional Institute of Arthritrs. Metuholism Discuses, Nutronui Institutes of Heulth, Bethesda Mrrq&nxJ .?0.?05
und Digestive
Recelced .! December 1980
ABSTRACT A method to solve numerically the diffusion-reaction equations that describe the steady state transport of oxygen through solutions of hemoglobin and myoglobin is presented. The method has a rather general range of applicability; in particular, it does not depend on the assumption of near chemical equilibrium. In the first part of the paper the chemical kinetics are described by the single step reaction P+ X,- PX. The nature of the method allows for its extension to more complex situations. In the latter part of the paper, it is extended to take into account the cooperative properties induced by the four step reaction (Adair) of the hemoglobin-oxygen chemical kinetics. The method to include the nonhomogeneous spatial parameters analysis of the available experimental information
will be extended in reports forthcoming as they occur in the striated muscle. An is presented here. The results confirm
those from previous analysis that the experimental facilitated transport of oxygen can be accounted for by the simultaneous diffusion of oxygen, the protein, and the oxygen-protein compound plus their chemical interaction. The magnitude of the facilitated transport of oxygen for the model that includes the cooperative properties of hemoglobin is compared with the one for the model associated with the single step reaction, Some features contained in the former.
that are not available
in the single
step model,
are shown
to favor
the
facilitated transport. The results for hemoglobin as it occurs in the red cells make it plausible that facilitation may contribute to the transport of oxygen in the red cell inside the systemic capillaries.
INTRODUCTION Wittenberg [29] and Scholander [25, 261 found experimentally that the steady state transport of oxygen in solutions of hemoglobin and myoglobin inside a millipore slab is larger than in water or saline. This phenomenon was called facilitated diffusion of oxygen. For a presentation of the subject up to 1970 the reader is referred to Wittenberg’s [3 l] comprehensive review. MATHEMATICAL
BIOSCIENCES
54: 265-290
0Elsevier North Holland, Inc., 198 1 52 Vanderbilt Ave., New York, NY 10017
265
(1981)
00255564/81/040265+26$02.50
266
JOSE M. GONZALEZ-FERNANDEZ
AND SUSIE E. ATTA
A model that has been proposed [3 1, 321 to represent this transport is based on the assumption that at each point the phenomenon involves the diffusion of and the chemical interaction between the substrates oxygen (X), protein (P), and the compound PX. The total transport of oxygen would then result from two additive fluxes: the diffusion of oxygen in solution and the diffusion of oxygen combined with the protein. The chemical kinetics involved were represented by the single step reaction P+xgPx, k,
where k, and k, are the recombination and dissociation constants respectively. Although this representation is correct for myoglobin, it is only an approximation for hemoglobin; some of the limitations of its use have been discussed [ 181. Several methods to obtain numerical solutions for this model have been presented in the literature. One approach [32, 331 assumed chemical equilibrium throughout the slab. Other authors 18, 17, 21, 24) assumed near chemical equilibrium inside the body of the slab, away from the boundaries. By using singular perturbation techniques they obtained expressions that were used for the computation of the solutions. Other numerical methods [3, 181 without the assumption of near-equilibrium have also been presented. The numerical method developed here starts with the observation that at each point the nonlinear chemical kinetics can be approximated locally by its linear approximation. Thus one has analytic solutions associated locally with the problem. A technique is developed to patch these solutions throughout the slab and up to the boundaries, where it is required that the boundary conditions of the problem be satisfied. The method has a rather general range of applicability; in particular it does not depend on the assumption of near chemical equilibrium. The qualitative features of the transport were examined. For the single step model the dissociation of oxygen from the carrier at the low oxygen boundary appears to limit the facilitation. The experimental observations of Wittenberg [30] on the dependence of facilitation on heme concentration and his finding that the values of facilitation per mole of heme by myoglobin and hemoglobin are the same have been analyzed previously. For hemoglobin satisfactory agreement was obtained [ 14, 171. For myoglobin, however, the facilitated fluxes computed [ 161 were about twice the experimental ones. The problem for myoglobin became amenable to the singular perturbation solution of Rubinow et al. [24]. However, some quantitative discrepancies remained. For heme concentrations larger than 5 m M their results for hemoglobin are lower than the experimental ones, and their results for myoglobin are higher. By incorporat-
267
TRANSPORTOFOXYGEN
ing the partial inactivation of myoglobin that took place during the course of the experiments as reported by Wittenberg [30], it will be shown that equality of facilitated flow per mole of heme by hemoglobin and myoglobin is obtained. For the chemical reaction of hemoglobin with oxygen a more correct representation than the one mentioned above would be the Adair [I] four step binding model PX,_, +&*,, k:,
i=l ,..., 4 ,
where PX, stands for the compound formed by one molecule of hemoglobin and i molecules of oxygen. It will be shown that by a natural averaging the four step process can be mapped into a process formally similar to the single step reaction. The attending new overall recombination (k, ) and dissociation (k,)coefficients are some averages of the Adair recombination constants (kb) and dissociation (k,) constants respectively. Those average coefficients are not constants. Their structure incorporates the cooperative properties of the reaction of hemoglobin with oxygen. Since every local analytic solution involved in the numerical method depends only on the local values of the chemical coefficients, then the method extends readily to the case of nonconstant chemical coefficients. It will be shown that the calculated facilitated transport by hemoglobin in solution, pso= 10 Torr, could be significantly larger than the one calculated for the single step model. This difference is in part related to the change from a hyperbola to a sigmoid of the associated oxygen dissociation curve. It will be shown that the dissociation of oxygen from the carrier inside the slab is confined no longer to a boundary layer. Benesch et al. [4, 51 have shown that the naturally occurring 2,3diphosphoglycerate in the red cell decreases the affinity of the hemoglobin for oxygen. Gibson [7] has shown that this effect can be described by essentially an increment in the value of the dissociation constants. For the hemoglobin in the red cell, psO --24 Torr, these effects enhance the facilitated flow by promoting the dissociation of oxygen in the vicinity of the low oxygen concentration boundary. By using values for the boundary conditions motivated by the partial pressures of oxygen found in the systemic capillaries by Duling et al. [6] and the dissociation constants adjusted for a psO ~23.4 Torr, it will be shown that for a slab one micron thick the facilitated flow contributes 50 to 65% of the total transport. One micron is likely to be within the order of magnitude of the diffusion path in the red cell inside the capillary. This result suggests the possibility that facilitation may contribute to the transport of oxygen in the systemic capillaries. The single step model representation and its associated kinetics constants were originally proposed by Hartridge and Roughton [10,111 in 1923. A
JOSE M. GONZALEZ-FERNANDEZ
268
AND SUSIE E. ATTA
result of independent interest developed in the present paper establishes the place that this approximation occupies in relation to the Adair [ 11 four step binding model. Motivated by the presence of membranes in the striated muscle, the numerical method has been extended to take into account the presence of multiple membranes impermeable to the carrier on the diffusion path. The results will be presented in a forthcoming report. THE MODEL Consider the region of space 0 G z < 1. Inside this slab let x, and x z be the concentrations of X and PX respectively. The chemical kinetics involved are assumed to be represented by the single step reaction
Let 9(x,, x1) be the net amount of oxygen per unit volume per unit time moving from the combined PX into the dissolved phase X. Let D, and D, be the constant diffusion coefficients of X and PX respectively. The model [30, 3 1, 321 for the steady state transport is described by the two differential equations d2x,
D,--
fd(x,,x,)=O,
Ll2$
-Q(x,,
dz2
with boundary
x2)‘o
(1)
conditions
(2) The boundary conditions on dx,/dz represent the constraint that the protein-oxygen compound cannot go through the boundaries at z = 0 and z = f. For the chemical kinetics
P+ X 2 PX.
9(x,,x,)=--k,x,(C,-x,)+k2x,,
(3)
where C, is the concentration of P plus that of PX. For the steady state conditions when the diffusion coefficient of P equals that of PX, it can be shown [ 121 that C, is constant across the slab.
TRANSPORT
269
OF OXYGEN
From (l), D, d2x,/dz2
t-D, d2x2/dz2 =O. Then (4)
is a constant across the slab. One recognizes that F is the total flow of oxygen. Let Fd denote the flow of dissolved oxygen (X), and F, denote the flow of combined oxygen ( PX). Then Fd = - D, dx, /dz, F, = - D, dx, /dz. From (4) OGZGl.
F=&(z)+&.(z),
(5)
Equation (5) states that at every point the total flow of oxygen is apportioned between the flow of dissolved oxygen and the flow of combined oxygen. The changes in this apportionment are brought about by the chemical reactions Pt X= PX. The rate of this change with respect to z is given by &+(x,,x,). At every point along z one can talk about the longitudinal fluxes Fd and F,., and the “transversal” flux Cp. The latter motivates the following representation for $(x1, x2). First consider the relationship between x, and x2 at chemical equilibrium, i.e., for
By using (3) and (6) one obtains
Cpk,x,
x2= k,x,
tk,
(7)
’
Call this function x;~(x,). Its Cartesian graph is usually called the oxygen dissociation curve. By rearranging terms in (3) one obtains ~(~,,XZ)=(k,x,+k2)[X2-x2eq(x,)].
(8)
Thus the transversal flux + can be expressed as the product of a “gradient” -xzq(x,) that is equal to the difference between the actual x2 and the x2 that would be in equilibrium with the prevailing x,, times a “chemical conductance” k,x, + k,. One can integrate (4) between z=O and z=l, and use (2) to obtain
x2
0
D,v+D,
I
x2(0) -x*(f) I
= F.
The first summand on the left corresponds to the oxygen flux that would exist in the absence of the carrier. It is called the nonfacilitated flux and
270
JOSE M. GONZALEZFERNANDEZ
denoted by F,,. The second summand denoted by F,. Thus F= F,, + Ff PIECEWISE Consider
ANALYTIC
the facilitated
flux and
APPROXIMATION
the linear approximation
+(X:,X:; to +(x,,x*) system
is called
AND SUSIE E. ATTA
x,,X2)=-k,(Cp-X:)X,+(kZ+k,X:)X2-k,XTX:
(9)
about the point (x:, x;), for x: GO and O
d2y,+$4x:,x:;
D,---
Y,tYz)‘O,
D2+$(x:r*f;
y,,y,)=O
dz2
approximates the original system (1) in a region about (x:, x2*). The error involved in this approximation will be considered later on. The following construction is started by specializing the points (x y, xz) to be used. Let (x:, XT) belong to the set defined by (6). The equation x,, x,)=0 defines a straight line tangent to the locus (6) at the #(x:,x;; point (x:, XT). Consider now a collection (XT’, xz’), i= 1,2,..., I, of points belonging to the locus (6), with O
x1 such that x;“”
9
i=1,2 ,...) I,
and xT’
(10) Associate to the interval
(11)
the linear system
(12) This system will be used to approximate (I) on the interval (11). Let x, be in the i th interval defined by (11). Since in this interval
TRANSPORT
271
OF OXYGEN
x;
X;
Xl
Xl
x1 (concentration of oxygen) FIG. I. The function #(s;‘. .xr’; x ,, x,) is the linear approximation (9) to +(x1. x2) about (XT’, .xI’). The point (XT’. XT’) is the intersection of $(x7’, I;‘; T,, x2)=0 and $(,r;‘-‘, x2’-‘; x,, x2)=0. The function $(xf’, XT’; x,, x,) is used as the linear approxi-
mation
to+(x,,x2)
on the regionXitL’
then by using (l), (3) and (12), (9) one finds that the error between the correct and the approximate second derivatives associated with the point (x,,x,)isboundedbyk,lx:‘-x,)1x:‘xz /. For a given grid (10) define the grid size 6 by 6=
max
r=2,..., I
Then those errors are bounded
uniformly on x, . Next, the piecewise analytic the grid (10) the I- 2 variables
approximation
and the I-2 PI 9
-xTI).
by
Z I)
with O
(xr(-’
i=2,...,I-1,
will be defined. Associate
to
(14)
variables i=2,...,
(15)
JOSE M. GONZALEZ-FERNANDEZ
212 Consider functions
the functions
y,, yZ defined
on a piecewise manner
by the set of
i=2,...,1-1,
Y;tY;,
that satisfy the linear prescribed as follows:
AND SUSIE E. ATTA
system (12) with domains
and boundary
conditions
for i=2, z, =O
,Y:h)=x?*?
4; dz
=o,
Yzz(Z*)‘P2i
:=o
for i=3,4 ,..., I-1, z,_,
y;(z,)=xY’, Y;(z,>=P,;
and for i=I, z,_,
Y;(z,-I)=PI-l~ The general solution
=I, Y:(z,)=xL
4: =o. dz :,
of (12) can be given by
y;=r;k~+7;k;(z-z,_,)+~;exp[~,(z-z,_,)]+~~exp[-X,(z-r,_,)], y;=r;k;+~+7jk;(z-z,_,)-~~exp[X,(z-z,_,)]-~~exp[-h,(z-z,_,)], 2 (16)
where k; =k,(C, k;=k,x:‘+k,, a’ =k,x:‘xT’
-x2*‘), (17)
TRANSPORTOFOXYGEN
273
and
(18) The coefficients $, j= I,. . . ,4, are determined from the boundary conditions of the corresponding interval. Motivated by the physical nature of the problem, the matching flow conditions ( $jz,-(zj;,=o,
j=l,2,
i=2,3 ,..., I-l,
(19)
are demanded. Observe that the number of unknowns (14) (15) equals the number of conditions (19). Since the boundary conditions at z=O and at z=I coincide with the ones given by (2), one can expect that this piecewise analytic solution will tend to the solution of the original problem (1). (2) as the error bound (13) tends to zero with 6. A numerical solution to the approximating system is equivalent to finding the 2(1-2) numbers z, [Eq. (14)] and p, [Eq. (15)] that satisfy the 2( 1-2) conditions (19). One can solve this nonlinear system by some numerical approximation such as the Newton method. Advantage can be taken of the fact that the matrices involved in the linear steps of the Newton method have analytic expressions. For the grid interval xF’+’
OF THE SOLUTIONS
FOR THE SINGLE
In order to study the dependence of the solutions on the values of the chemical parameters a series of computations covering a wide range of those
274
JOSE M. GONZALEZ-FERNANDEZ TABLE
C,,(heme) k, kz D, DZ /
AND
SUSIE E. ATTA
I
8.929X IO-’ M 3.0X IO” M ’ set -1 4.25X10 5+‘rsecc’ 1.5X IO -’ cmZsec -’ [31] 3.0XlO~‘cm”sec -‘[31] 1.0X10 ‘cm
values was carried out. The numerical values for the parameters used are the ones listed in Table I, for n = 1,. . . ,9. The values for n = 6 correspond to single step approximation to hemoglobin [IO, 11, 171. To follow the terminology used in the literature the concentration of oxygen will be expressed in Torr, and the concentration of the protein-oxygen compound in volumes of oxygen STP per volume of solution. The solubility coefficient of oxygen was taken as 1.6X 10 -’ MTorr -’ [28]. The values of XI: and xi in (2) were 190 and 0 Torr respectively. In Fig. 2 the solutions y,(z) and yz(z) are plotted. For the smallest value of k ,, k, considered, n= 1, the graph of y, approximates the straight line
10.20 x2 ti WI 0.15
i
- 0.10
- 0.05
-0
FIG. 2. This graph shows the dependence of v,(z) and r2( z) on the values of k, and The indexing digit stands for the value of n in the assigned k, =3.0X IO” A4 ‘set ’ ‘. The unptimed index corresponds to y,, and the primed and X-,=4.25X10 “” set index corresponds to pa. The values of the other parameters are listed in Table 1. k2.
TRANSPORT OF OXYGEN
275
joining the points z=O, X, =xp and z=/, x,=x:, and the graph of y, approximates a constant. This gives a negligible value of facilitated transport. As n increases the absolute value of the slope of y, increases near z=O and z = 1, and decreases in an intermediate region. The change of slope near z = I is not apparent for the scales used. This change can be appreciated in Fig. 5, where the solution y,( z) for n = 6 is plotted. As n increases, the value of y,(O) increases toward xsq(xy) and the value of yz(/) decreases toward xsq(x{) 1181. Some relationships between the physicochemical processes taking place inside the slab and the form of the solutions will be considered next. Consider first the case n=3 of the preceding sequence. The results are represented in Fig. 3. One observes that there is a value z* of z such that y,xSq(yr) for z*z* there is a net transfer of oxygen from the combined to the dissolved phase, i.e., the oxygen is unloaded from the carrier [8]. One can also observe that in the loading region the differences betweeny> and xr‘J(y,) are not negligible and that the larger differences take place inside the body of the slab. For n=6 the absolute values of the gradient
-
F 0.10
F”l
-10.05
Ft -0
FIG. 3. This graph illustrates the solutions when chemical equilibrium is not attained inside the body of the slab. The chemical kinetics coefficients are k, =3.0X IO’ M ’ set ’ and k, =4.25X 10 ’ set ‘. A large part of the loading process occurs inside the body of the slab away from the boundary.
276
JOSE M. GONZALEZ-FERNANDEZ
AND SUSIE E. Al-l-A
were found to be negligibly small inside the body of the slab, (Fig. 4). However, the increment of the chemical conductance k,y, + k, was proportionately larger than the decrement of Iyz -~;~(_y,)(. The net result was an increase in the transversal flow of oxygen and in the maximum values of (d’y,/dz’(. Therefore, for hemoglobin the assumption of near chemical equilibrium inside the body of the slab [ 17, 21, 24, 321 for this and larger values of 1 finds here a numerical justification. The profiles of F, and Fd for n = 5,. . ,9 show sharp changes in layers located at z=O and at z = 1; for n =6 see Fig. 4. The details of the changes near z=I for n=6 are shown on a magnified scale in Fig. 5. The width (J of a boundary layer will be defined as the length of the interval on which 0.98 of the rapid change in Fd (or F, ) takes place. To analyze the structure of the solution on the boundary layers the function +(x,, x2) of the original problem (1) was replaced by the linear approximation (9). Then the solution (16) was evaluated by solving (12) for the interval z, _ , , z, where the rapid change in Fd takes place. The appropriate boundary values for the interval were obtained from the numerical solution already computed. It was found that on this interval
y, -x;~(Y,)
Fd~-_,T;k;+D,?~h,exp[-X,(z-z,_,)]
'0.10
1 F”f
6
0.05
Jo FIG. 4. This graph illustrates the solution for hemoglobin. n=6. Inside the body of the slab the values of v2 and x’;4(,,) differ negligibly from one another. 73% of the loading occurs at the right of the midpoint of the slab.
TRANSPORT
277
OF OXYGEN 0.15 FC
x2
Fd
x;"
$104
0.025 yclJ VOI 0.10
0.020 Xl torrx 102
e-v*-~'x,O_,
0.015
0.05
0.010
0.005
0 0.98
FIG. 5.
Details
exp[ -A,( /-z)]
of the solution
illustrates
width of the boundary
an
0.99
I cmxl02
0 1.00
of Fig. 4 close to the boundary
opriori approximation
at ;=/.
to the relative changes layer at z= I; A, was evaluated for s;’ =x{ =O.
The graph
of
in Fd and to the
and F, m-D02~;k;-D2~~h,exp[-h,(z-z,_,)]. Since I- exp( -4) =0.98, which is the number then one is led to the condition X,a=4
used in the definition
of u,
(20)
for the evaluation of a. For the boundary layer at z=O the value of A, may be approximated u priori by letting x;’ = xp in (17) and for the boundary layer at z= I by letting XT’ =x{. Good numerical approximations to the changes in FJ (or 4,) and to a were obtained at both boundaries (see Fig. 5 for n=6). The same good agreement was obtained for the runs n = 4 to n = 9. At the place where the condition of near-equilibrium is satisfied, then the ratio Fc/ Fd is close to ( D, /D, ) dx;‘l/dx,, a resealed slope of the oxygen dissociation curve (7). When X: is large enough, then the shape of the oxygen dissociation curve determines that the larger values of F, occur in the body of the slab away from the high oxygen boundary; see Fig. 4 [27, 331. Near this boundary one expects a smaller value of e,, and since by (2) F, =0 at z =O, then this determines only a small jump in I$ there. This jump induces the existence of a boundary layer. The loading process, in this instance, is not a boundary layer process. The value of the loading is therefore not directly related to the resulting value of n,(O) which enters in the expression for the
278
JOSE M. GONZALEZFERNANDEZ
AND SUSIE E. ATT’A
facilitated flow, D,[ x2(O) - x2( /)I/[; thus on the above conditions the loading does not play a limiting role in the facilitated flow. Closer to the low oxygen boundary the opposite situation holds. The condition of near equilibrium and the smaller values of x, imply the existence of larger values of F,., and since by (2) Fc =O at z=l, one then expects a large jump of F, at this boundary. The direction of this change is opposite the one conditioned by the oxygen dissociation curve. Thus the unloading process is essentially a boundary layer process (Figs. 4 and 5). The condition (20) with (18) expresses the dependence of the width of the boundary layer on the physiocochemical parameters k , , k,, D,, and D,. Within the range of validity of the above considerations the width of the boundary layer is then independent of the thickness of the slab [21]. To verify this assertion the thickness of the slab was varied from 1 to 300 microns. For the chemical kinetics that correspond to n=6 it was found that for I between 30 and 300 microns the solution x2 was near equilibrium inside the body of the slab and that the width of the boundary layer at z=I was constant and equal to 0.91 microns. This value coincided with the one given by (20) where X, was computed a priori by using xy’=x{=O and x;‘=x;q(x{) in (17). From the second equation in (1) the unloading of oxygen equals the integral of (8) on this interval. For the decreasing values of I, increasing values of unloading were attended on this interval by larger values of the gradient by larger values of x2 at z=I, thus limiting the x2 -x2 ‘q and in particular increment of the facilitated flow. It can be shown that the exponents X, obtained a priori as indicated above coincide with the exponents y and 6 for the boundaries at z = 0 and at z = I, respectively, of the singular perturbation expansion developed by Rubinow et al. [24].
ANALYSIS OF EXPERIMENTAL RESULTS: RELATION BETWEEN FACILITATED TRANSPORT PROTEIN CONCENTRATION
AND
Wittenberg [30] performed a series of experiments with solutions of hemoglobin and myoglobin that covered a wide range of concentrations of these proteins. The width of the millipore filter used was 1SO X 10 -’ cm. The area of each face exposed to gases was 11.5 cm2; the fractional space of porosity equaled 0.79. To proceed with the numerical computations one needs the values of the experimental parameters. The values of the self-diffusion coefficients used for hemoglobin and myoglobin as functions of the protein concentration will be the ones obtained by Riveros-Moreno and Wittenberg [22, 311. A review of the results obtained by others is presented by Kreuzer [ 161 and Kreuzer et al.
TRANSPORT OF OXYGEN
279
[ 171. The ones for free oxygen were compiled by Kreuzer et al. [ 171. These values have been essentially confirmed by other experimental techniques [9]. The chemical kinetics coefficients for the single step approximation to hemoglobin were taken as k, =3.0X lo6 M-‘see-’ and k, =42.5 set-’ [IO, 11, 17],andformyog1obinask,=1.40X10’M~’sec~’andk,=11.0sec~’ [2]. The partial pressure of oxygen was taken to be 300 Torr on one side of the slab [24] and 0 on the other side. The effective value of 1 is geometrically uncertain due to the tortuosity of the millipore channels. Keller and Friedlander [ 151 reported a tortuosity factor of 1.63. From the results on the diffusion of oxygen and nitrogen through a solution of ferric hemoglobin reported by Wittenberg [30, Fig. I] one calculates an average effective I of 2.45 X 10 -2 cm, and a tortuosity factor of 1.63. From the results [30, Figs. 3, 41 for filter widths ranging from 2.5 X 10 -3 to 3.3 X 10 -’ cm one obtains a tortuosity factor of 1.34. From the discrepancy between the apparent diffusion coefficient of Nz and the expected one reported in Fig. 11 of Wittenberg’s review [3 I], one may infer a tortuosity factor of 1.57. The numerical results obtained for hemoglobin will be discussed first. To begin with, the value chosen for I was 2.45X 10 -’ cm; this value was also used by Rubinow et al. [24]. The facilitated fluxes computed showed the same qualitative dependence on heme concentration as the experimental ones. However, they were consistently lower than the experimental ones. On the other hand, the maximum possible facilitated flow for the model considered here can be obtained by settingy2(0)=x;q(xy) and y2(C)=x;q(x:)=0. The maximum facilitated flow as a function of heme concentration is only slightly larger than the one computed, but smaller than the experimental one; see Fig. 6. The fact that the experimental facilitated flow is consistently larger than the theoretical maximum may be taken as indirect evidence that the value ascribed to I was too large. If one assumed a tortuosity factor of 1.43, then an adequate quantitative agreement was obtained [ 141; see Fig. 6. Wittenberg reported [30] that at the termination of each experiment 10 to 25% of the myoglobin was found to be oxidized to the ferric state. Therefore, the experimental values of facilitated flow that Wittenberg [30] reported as functions of total heme concentration were replotted in Fig. 6 as functions of the chemically available heme concentration, which is taken to be 0.85 times the total heme concentration reported. On the other hand, for the numerical calculations the self-diffusion coefficient of the carrier and the diffusion coefficient of oxygen were both chosen to correspond to (0.85))’ times the heme concentration of the chemically available form. The tortuosity factor of 1.43 will be retained. The computed facilitated flow is plotted against the heme concentration of the chemically available form in the same Fig. 6. Quantitively the agreement between the theoretical and experimental results
280
JOSE M. GONZALEZ-FERNANDEZ 2.0 r
n
AND SUSIE E. ATTA
0.
.
0
,I’
‘\
. .‘;---.
I’
0
I
/’
81 0 I
Of....f.-I.. 5
‘\ *
‘\
‘\
o‘\, ‘\ ‘\
.
“9
. ’
,‘..,,‘...’ 10 15 20 HEME CONCENTRATION (mM)
FIG. 6. Relationship between facilitated circles represent the experimental values
8
25
flow of oxygen and heme concentration. Solid for hemoglobin. Open circles represent the
experimental values for myoglobin when the abscissae stand for the heme concentration of the chemically available form. The line of dashes represents for hemoglobin the maximum possible theoretical facilitated flow for 1=2.45X IO -’ cm. The solid line represents the numerical results for hemoglobin for 1=2.14X lO-2 cm (tortuosity factor is 1.43). The dot-dash line represents the results for myoglobin, /=2.14X IO -’ cm. For the values of the other parameters
see the text.
is satisfactory. This confirms Rubinow et al.‘s conjecture [24] that when the correction for oxidation of myoglobin is introduced, a satisfactory agreement with the experimental observations could be expected. The numerical results reproduce the experimental finding [30] that the facilitation for hemoglobin (solid line) and for myoglobin (dot-dash line) are approximately the same as functions of heme concentration, not of molar concentration (Wittenberg’s paradox [30, 3 I]); see Fig. 6. It was conjectured by Wittenberg [30, p. 108 (footnote); 31, p. 5671 that the equality of facilitated transports could be due to the balancing of two opposing effects. On the one hand, the value of the self-diffusion coefficient for myoglobin is larger than the one for hemoglobin, and on the other hand, the dissociation coefficient, which determines in large part the unloading of oxygen, is larger for hemoglobin than for myoglobin. Rubinow et al. [24] confirmed, as part of their analysis, that Wittenberg’s conjecture resolves the paradox. Observe that the results for hemoglobin test essentially only the relation between the concentration and the self-diffusion coefficient of the protein, since the values of x,(O) and x,(l) are very close to the ones for equilibrium. In contrast, the results for myoglobin depend also on the limitation that the chemical kinetics places on the unloading of oxygen. The latter resulted in a fairly large departure of x,(f) from equilibrium. Thus the above findings constitute added indirect evidence on the validity of the model.
281
TRANSPORTOFOXYGEN EXTENSION PROPERTIES
OF THE MODEL TO INCLUDE OF HEMOGLOBIN
THE COOPERATIVE
A more precise representation than the one used above for the chemical reaction between hemoglobin and oxygen would involve a four step binding model (Adair [ 1I), px,_,+xZpx,, k:,
i=1,...,4,
where PX, stands for the compound formed by one molecule of hemoglobin and i molecules of oxygen. An analysis of the facilitated transport will be carried out to include the influence of the different constants kk, kb. The resulting model will retain the simple description in terms of the two variables x, and x2. Let J, be the increment of the molar concentration of PX, per unit of time contributed by the chemical reactions. Then the second equation of (1) generalizes to d*PX D,-z dz
+J,=o,
(21)
i=1,...,4,
where J,=kXPX,_,X-k’,PX,-kh+‘PX,X+k’,i’PX,+,,
i=1,2,3,
(22)
and J,=k; Multiply
PX,X-k;PX,.
(21) by i and sum over i= I,...,4 4
to obtain
d2PX.
4X+-&
(23) i=l
i=l
Let 4 x2=
iPX,,
x i=l
and as customary
define the intrinsic kk=(4-i-t
l)ki,
rate constants k,=iki,
ki and ki by i=1,...,4.
282
JOSE M. GONZALEZ-FERNANDEZ
AND
SUSIE E. ATTA
Then from (23) and (22) d2x2 D2-
+X
dz2
ki+‘(4-ii)
i
PX, -
i
r=O
k;iPX,
1=l
=O.
(24)
Since 2:=,(4-i) PX, equals the concentration of unoccupied sites (reduced hemes) and Cp=, i PX, equals the concentration of occupied sites (oxygenated hemes), it is natural to define an average recombination coefficient by
; k:+‘(i-i) r=a
k;,=
j.
and an average dissociation
(4-i)
coefficient
i k,,=
(25) PX,
by
k;iPX,
‘=’
(26)
,i,iPX,
Since C, =4X:=,
PX,
.
PX,, then from (24) d’x, D2-
In a similar manner
dz2
+k:&-x,)x,
-Lx2
=O.
(27)
one obtains d2x, -k:,(C, D,-dz2
-x2)x,
+k,,x,=O.
Equations (27) and (28) are formally similar to (1) plus (3). However, the chemical kinetic coefficients kiv and k,, are not constants. Their values depend on the prevailing families PXi, i=O, I,. . ,4, which themselves are part of the solution to the problem in question. As a first approximation, the families (PX;) associated with the hemoglobin oxygen dissociation curve will be used in (25), (26). For this choice one defines k,(x2)=kLv and k2(x2)=kav to be used in (27) and (28). Equation (3) is then replaced by +(x17x2)=
-k,(x,)(Cp
-x2)x,
+&(x2)x2.
(29)
TRANSPORT
OF OXYGEN
283
The difference between the linear x7, XT and $(x,, x2) is given by
approximation
4(x:,
xz; x,, x2) about
stand for the second-order partial derivatives of $J where%,x2ad +.x2.x, evaluated between xi,xz and x7,x;. Observe that &_+,=O. The error between the correct and the approximate second derivatives of solutions to (1) at x,, x2 is then 0(x, -xz) uniformly in x,, x2. Because of the form of this error bound one needs to change the grid used in the preceding numerical approximation. Instead of the set (10) one may consider the set x:‘, i= 1,. . , I, and may associate to the interval x:’
TABLE 2 I
k: (Mm’secm’)
k; (set _ ‘)
1 2
9.00x IO6 4.0x IOh
1.080X IO3 2.44x 102
3 4
3.5 x IO6 4.0x IO’
2.8X IO 4.80X IO
284
JOSE M. GONZALEZ-FERNANDEZ
AND
SUSIE E. AT-I-A
The functions k,( x2) and k2( x2) were calculated by using the values of ki and k; obtained by Ilgenfritz et al. (Table 2). To describe them numerically it is convenient to consider them to be functions of f, where f =x2/CP. The value of k, is approximately equal to 8.5X IO6 M-‘see-’ for O
Parameter
C’=8.93X
10e3
TABLE 3 Values Used for the Computations
M heme;
D, =3.0X
IO-’
cm’/sec;
Pertaining
to Fig. 7
D, = 1.5X 10 -5 cm2/sec;
xp=
190
Torr; x( =O Torr
A.
Model with cooperative properties The values for k,( x2) and k2(x2) are derived from Ilgenfritz (Table 2) and PX, in equilibrium; P5c = 10 Torr; O2 sol. coeff.=1.7X10-6
et al.% values for k;, k:,
M Torr-‘. Single step model
B. C. D.
k,=3.00X106 M-‘set-‘; k,=42.5 set-‘; P5n=8.9Torr. k,=7.00X106 M-‘set-‘; kzz99.16 set-‘; Ps,,=8.9Torr; O2 sol. coeff.= 1.6X 10e6 M Torr-‘. Jacquez et al. 1141: C,=9,375XlO~-‘Mheme; D2=3.6X10-‘cm*/sec; D,=l.4X10~5cm2/sec; xy =200 Torr; xi =0.05 Torr; k,=3.00X106 M-‘set-‘; k2=40.0 O2 sol. coeff.=l.8X10-6 M Torr-‘.
set-‘;
P,,=7.4Torr;
TRANSPORT
285
OF OXYGEN
2
6
10
20
30
40
60
70
80
FIG. 7. Dependence of F,/FfmaX on the length of the diffusion path for hemoglobin solution. Curve A represents results from the cooperative model; curves B, C, and represent
results from
the single step model. The values used for the parameters
in D
are listed in
Table 3.
Equations (27) and (28) with the boundary conditions (2) were solved for different lengths of the diffusion path 1. The values used for the parameters are listed in Table 3. To express the magnitude of the facilitated transport the ratios F,/F,““, where Ffmax= D2[x~4(x~)-x~4(x~)J, are graphed in Fig. 7. For diffusion lengths larger than 35.75 microns this ratio is nearly constant. By using the tortuosity factor 1.43 these lengths correspond to values of slab thickness larger than 25 microns. Therefore, the values of facilitated flow are approximately of the form constant/l, as was observed experimentally in this range by Wittenberg [30]. For comparison purposes the values of F,/F;“” for several single step chemical representations are included. The profile of for the cooperative model contrasts with the ones for the single step F, /F,“” models. For the former the value of this ratio is larger than for the latter ones for values of I down to 10 microns. However, for values of I smaller than 10 microns it decreases rapidly with decreasing I, to eventually become smaller than the values associated with the single step model. One may note that the overall apparent second order rate constant k’ obtained in conditions of excess ligand by Ilgenfritz et al. was 5X lo6 M-‘set-’ [13]. This is significantly smaller than the least value 8.5X IO6 M - ’ set - ’ taken by k, in the model as above. order to this discrepancy the results by Gibson were analyzed in terms the Adair using the values the constants obtained by Ilgenfritz et Gibson the course of of deoxyhemoglobin 10e5 M oxygen,
286
JOSE M. GONZALEZFERNANDEZ
AND
SUSIE E. A’ITA
and (D) 1.55X 10 p4 M oxygen. It was verified that the numerical solutions agree closely with the experimental results. The values of kX and k,, were computed and their relationships with the values of j were studied. It was found that at each j the values of kiv are smaller (k,, are larger) for faster transients. For. instance, at j=0.3 kj, =7.7X lo6 M p’sec-’ for curve C (at equilibrium k iv =8.2X10h M-‘see-‘), kh,=6.2X10h M-‘set-’ for curve B, and ki,=5.6X106 Mp’secm-’ for curve A. The latter is very close to the overall k’ quoted above. It was found that these different values reflect the different populations PX,, i=O, 1,. . , 4. The relative amounts of the intermediate species PX, and PX, are larger for faster transients. One may examine the meaning of the values of k’= 3 X 10” M ~ ’ set- ’ and k=42.5sec-’ that Hartridge and Roughton [ 10, 1 I] proposed in 1923 for the single step model. From the above analysis the value of k’ appears to reflect the values of kf and k:, and the value of k appears to reflect the value of k:. These values embody the low range values of the recombination and of the dissociation constants respectively. Thus, in terms of kinetics the values of Hartridge and Roughton yield a lower bound for the speed of the chemical kinetics of hemoglobin with oxygen. Those values would be too small when the process studied corresponded to either a slow transient or a situation where the species were close to chemical equilibrium and their use resulted in a solution away from equilibrium. Therefore, the results for hemoglobin reported in the previous section are still correct. On the other hand, one could expect the values of curve A in Fig, 7 to be correct for values of F, / Ffmax larger than 0.85. For smaller values of this ratio one could expect, because of the departure from equilibrium, that the correct solution will decrease with decreasing 1 faster than the computed solution of Fig. 7. The results of Moll [20] and Kutchai et al. [19] provided experimental evidence of facilitated transport of oxygen through 300 and 165 micron layers of packed erythrocytes. Since the dimensions of a single red cell are much smaller, Kutchai et al. [ 181 studied theoretically the transport in layers of thickness commensurate with that of the red cell; a single step chemical kinetics model was used. The problem will be reexamined here by using a fuller description of the chemical kinetics as it pertains to hemoglobin in the red cell. The half saturation point psO for whole blood, 37°C pH 7.4, is approximately 24 Torr [23]. The effect of 2,3-diphosphoglycerate on the oxygen affinity of hemoglobin has been elucidated by Benesch et al. [4, 51. Its chemical kinetic interpretation has been obtained by Gibson [7]. This author showed that the change in the Adair rate constants of hemoglobin in solution from no phosphates ( psO = 2 Torr) to about 0.5 mole DPG per mole Hb ( pso = 10 Torr) was effected fundamentally by increments in the dissociation constants. One may extrapolate to the situation in the erythrocite in which one mole
TRANSPORT
287
OF OXYGEN TABLE 4 Parameters for the Cooperative Model for Hemoglobin in the Red Cell C,=2.20XlO-‘Mheme &=6.80X IO-‘cm2sec D,=7.00XIO~-hcm2sec~’
-’
Ps,,= 23.45 Torr 02sol.cocff.=1.7X10~6MTorr~’
DPG is present per mole Hb and the pSO=24 Torr. As a first approximation the values of k,(x,) were kept as the ones described above. Then, the values of k,(x,) were calculated so that the associated oxygen dissociation curve, pSO=23.45 Torr, was a good fit to the one for whole blood. The calculated values of k, decrease from 2.28X103 set-’ at j=O to 1.05X103 sect’ at f= 0.15; then they decrease more gradually to 4.1 X 10 2 set - ’ at f= 0.4; and for larger values off they decrease at about a constant rate to 9.6 X 10 set - ’ at f= 1.0. For these functions k, and k, one may calculate approximately the associated intrinsic chemical constants. The values of ki, i= 1,. . . ,4, are essentially the ones listed in Table 2. The values of k; are as follows: k: =2.28X lo3 set-‘, k; =2.90X IO2 set-‘, k: = 1.71 X IO2 set-‘, and kj = 9.6X 10 set-‘. Again, Eqs. (27) and (28) with the boundary conditions (2) were solved for different lengths of the diffusion path, [. The values of the other parameters are listed in Table 4. The ratios Ff/Fnnf and F,/F,““” for the boundary values xp = 100 Torr and xi =O Torr are listed in Table 5. The magnitude of the facilitated flow, F,, is about the same as the one of the nonfacilitated flow, Fnr, for I= 1 micron. For comparison purposes the results for the single step
TABLE x1’= 100 Torr
5
Torr 1x’=O I F,/FfmaX
F,/F,r I (microns) 0.50 0.75 1.00 2.00 5.00 15.00 25.00
Single step
Cooperative
model
model
0.3609 0.4769 0.5566 0.7278 0.8848 0.9774 0.9982
0.591 0.845
Single step model
Cooperative model
1.206
0.3499 0.4620 0.5396 0.7053 0.8577
0.485 0.693 0.845 0.960 0.990
1.217
0.9475 0.9677
0.996
1.03I 1.160
JOSE M. GONZALEZ-FERNANDEZ
AND
SUSIE E. A’ITA
2 MICRONS
FIG. 8. Details of the unloading of oxygen in the cooperative model for hemoglobin the red cell. Boundary values: x,” - I00 Torr ( X{ =O Torr; I= IO microns.
in
model with k, =5.4X IO6 M-‘see-’ and k, =202 set-‘, PSO=22 Torr, are also listed. Some details of the numerical solution for the cooperative model for f= 10 microns are graphed in Fig. 8. It can be observed that the unloading takes place in the interval 7.83- 10 microns. The steep phase occurs in the boundary layer, 9.85 10 microns; its width agrees with the a priori calculated u. The slow phase occurs in the interval 7.83-9.85 microns, and it is attended by
TABLE 6 xj’=lOOTorr x’= 40Torr I (microns) 0.50 0.75 l.O4l 2.00 4.00 5.00 10.00 20.00 25.00
F,/F,r
F, / Ffmu
x:’ =40 Torr x{=l5Torr G/F,,
F,/Ffm””
0.23 I 0.280 0.319 0.379
0.508 0.595 0.702 0.834
0.938 I.296 I.514 I .937
0.360 0.496 0.580 0.741
0.422
0.929
2.321 2.462
0.889 0.943
0.448
0.985
XI’ = 25 Torr xl= 5 Torr h/F,,
F,/ Ffmax
I.500 1.971 2.238 2.670 2.900
0.477 0.627 0.712 0.849 0.922
3.046 3.095
0.969 0.984
TRANSPORT
289
OF OXYGEN
near chemical equilibrium there. Its beginning corresponds to x, = 15.8 Torr, which is the inflection point of the oxygen dissociation curve used. Compare with the single step model profiles of Fig. 5. To take into account the ranges of partial pressures of oxygen in the pulmonary circuit and in the systemic capillaries [6], further computations were carried out. The ratios F,/F,, and F,/F;““” are listed in Table 6. One can observe that for diffusion paths of 0.75 to 1.00 micron the facilitated flow contributes 56 to 60% of the total flow for xp ~40 Torr, XI = 15 Torr, and 66 to 69% for xp = 25 Torr, xi =5 Torr. Since the length of the diffusion path in an erythrocite inside a capillary is likely to be of that magnitude, these results make it plausible that facilitated flow by hemoglobin may contribute to the transport of oxygen in the systemic capillaries. To further elucidate this question it will be pertinent to study models that include other features of the capillary transport, especially the transit time.
REFERENCES I
4
G. S. Adair. The hemoglobin system. VI. The oxygen dissociation curve of hemoglobin. J. Biol. Chem. 63~529-545 (1925). E. Antonini and M. Brunori. Hemoglobin and Mvoglohin in Therr Reactions with Ligunds, North-Holland, Amsterdam, I97 I. R. J. Basset and J. S. Schultz, Nonequilibrium facilitated diffusion of oxygen through membranes of aqueous cobaltohistidine, Biochim. Bioph_vs. Actu. 2 I I : 194-2 15 ( 1970). R. Benesch, R. E. Benesch, and C. I. Yu, Reciprocal binding of oxygen and diphos-
5
phoglycerate by human hemoglobin, Proc. Nut. Acud. Sci. U.S. A. 59:526-532 (1968). R. Benesch and R. E. Benesch. Intracellular organic phosphates as regulators of oxygen
6
release by hemoglobin, Nuture 22 I :6 18-622 ( 1969). B. R. Duling and R. M. Beme, Longitudinal gradients
7
C~rculutrot~ Res. 37:325-332 Q. H. Gibson, The reaction
2 3
8 9 10
11
12 13
(1970). of oxygen
in periarteriolar
with hemoglobin
oxygen tension,
and the kinetic
basis of the
effect of salt on binding of oxygen, J. Biol. Chem. 245:3285-3288 (1970). J. D. Goddard, J. S. Schultz, and R. J. Bassett, On membrane diffusion with near equilibrium reaction, Chem. Eng. Sci. 25:665-683 (1970). G. Gross, Concentration dependence of the self-diffusion of human and Lumhricus terrestris hemoglobin, Bi0ph.w. J. 22:453-468 (I 978). H. Hartridge and F. J. W. Roughton. The kinetics of hemoglobin. Part 11. The velocity with which oxygen dissociates from its combination with hemoglobin, Proc. Rqv. Sot. London Ser. A 104:395-415 (1923). H. Hartridge and F. J. W. Roughton, The kinetics of hemoglobin. Part III. The velocity with which oxygen combines with reduced hemoglobin, Proc. Roy. Sot. London Ser. A 107:654-683 (1925). J. 2. Hearon. Distribution of conserved species in diffusion-reaction systems, Bul/. M&h. Biol. 35x59-61 (1973). G. Ilgenfritz and T. M. Schuster, Kinetics of oxygen binding to human hemoglobin, J. Biol. Chem. 249~2959-2973 (1974).
290 14
J. A. Jacquez,
I5
interfacial and thickness effects, Respir. Physiol. 15: 166- I81 (1972). K. H. Keller and S. K. Friedlander, Diffusivity measurements of human
16
bin, /. Gen. Physic/. 49:681-687 (1966). F. Kreuzer, Facilitated diffusion of oxygen
and its possible
17
Resptr. Physiol. 9: I - 30 ( 1970). F. Kreuzer and L. J. C. Hoofd.
diffusion
IX 19 20 21 22
H. Kutchai,
and E. Daniels,
Hemoglobin-facilitated
Facilitated
diffusion
of oxygen: methemoglo-
significance;
of oxygen
a review,
in the presence
human erythrocytes, J. Cen. Physic/. 53:576-589 (1969). W. Mall. Measurements of facilitated diffusion of oxygen in red blood Pflugers Arch. Gesumte Phwiol. Metrschen Twe 305:269-278 (1969).
cells at 37°C.
J. D. Murray, On the molecular mechanism of facilitated oxygen diffusion of hemoglobin and myoglobin. Proc. Rqv. SW. London. B. EIO[. Sci. 178:95-I IO (1971). V. Riveros-Moreno and .I. B. Wittenberg. The self-diffusion coefficients of myoglobin and hemoglobin in concentrated solutions, J. Btol. Chem. 247:X95-901 (1972).
23
F. J. W. Roughton, Transport of oxygen and carbon dioxide, in Handbook Ph_vsiolog??. Respirarion, Vol. I, Amer. Physiol. Sot., Washington, D.C., 1964.
24
S. I. Rubinow and M. Dembo, The facilitated myoglobin, Biophys. J. 18:29-42 (1977).
25
P. F. Scholander.
26
(1960). P. F. Scholander. P. F. Scholander.
21 2x
of
hemoglobin, Resprr. Ph,wo/. 8:280- 302 ( 1970). H. Kutchai. J. A. Jacquez, and F. J. Mather, Nonequilibrium facilitated oxygen transport in hemoglobin solution, Biophvs. J. 10:38-54 (1970). H. Kutchai and N. C. Staub, Steady-state. hemoglobin facilitated O2 transport in
Oxygen
transport
Oxygen diffusion, Tension gradients
through
diffusion
hemoglobin
of oxygen solutions,
Screttce 132:368 (1960). accompanying accelerated
by hemoglobin
of and
Science I3 I :585-590
oxygen
transport
in a
membrane, Science 149:876-877 (1965). J. Sendroy, Jr.. R. T. Dillon, and D. D. van Slyke, Studies of gas and electrolyte equilibria in blood XIX. The solubility and physical state of uncombined oxygen in blood. J. RIO/. Chem. 105:597-632 (1934).
29
J. B. Wittenberg. 117:402 (1959).
30
J. B. Wittenberg, The molecular mechanism of hemoglobin-facilitated oxygen diffusion.J. Btol. Chem. 241:104-114 (1966). J. B. Wittenberg, Myoglobin-facilitated oxygen diffusion: role of myoglobin in oxygen
31 32 33
Oxygen
transport:
a new function
proposed
for myoglobin.
RIO/. Bull.
entry into muscle, Phystol. Rec. 50:559-636 (1970). J. Wyman, Facilitated diffusion and the possible role of myoglobin as a transport mechanism. J. Rtol. Chem. 241: Il5- 121 (1966). D. B. Zilversmit. Oxygen-hemoglobin system: a model for facilitated membranous transport, Scrence 149:874-876 (1965).
SUSIE E. ATTA Muthematid Reseurch Brunch, Nutional Institute of Arthritrs. Metuholism Discuses, Nutronui Institutes of Heulth, Bethesda Mrrq&nxJ .?0.?05
und Digestive
Recelced .! December 1980
ABSTRACT A method to solve numerically the diffusion-reaction equations that describe the steady state transport of oxygen through solutions of hemoglobin and myoglobin is presented. The method has a rather general range of applicability; in particular, it does not depend on the assumption of near chemical equilibrium. In the first part of the paper the chemical kinetics are described by the single step reaction P+ X,- PX. The nature of the method allows for its extension to more complex situations. In the latter part of the paper, it is extended to take into account the cooperative properties induced by the four step reaction (Adair) of the hemoglobin-oxygen chemical kinetics. The method to include the nonhomogeneous spatial parameters analysis of the available experimental information
will be extended in reports forthcoming as they occur in the striated muscle. An is presented here. The results confirm
those from previous analysis that the experimental facilitated transport of oxygen can be accounted for by the simultaneous diffusion of oxygen, the protein, and the oxygen-protein compound plus their chemical interaction. The magnitude of the facilitated transport of oxygen for the model that includes the cooperative properties of hemoglobin is compared with the one for the model associated with the single step reaction, Some features contained in the former.
that are not available
in the single
step model,
are shown
to favor
the
facilitated transport. The results for hemoglobin as it occurs in the red cells make it plausible that facilitation may contribute to the transport of oxygen in the red cell inside the systemic capillaries.
INTRODUCTION Wittenberg [29] and Scholander [25, 261 found experimentally that the steady state transport of oxygen in solutions of hemoglobin and myoglobin inside a millipore slab is larger than in water or saline. This phenomenon was called facilitated diffusion of oxygen. For a presentation of the subject up to 1970 the reader is referred to Wittenberg’s [3 l] comprehensive review. MATHEMATICAL
BIOSCIENCES
54: 265-290
0Elsevier North Holland, Inc., 198 1 52 Vanderbilt Ave., New York, NY 10017
265
(1981)
00255564/81/040265+26$02.50
266
JOSE M. GONZALEZ-FERNANDEZ
AND SUSIE E. ATTA
A model that has been proposed [3 1, 321 to represent this transport is based on the assumption that at each point the phenomenon involves the diffusion of and the chemical interaction between the substrates oxygen (X), protein (P), and the compound PX. The total transport of oxygen would then result from two additive fluxes: the diffusion of oxygen in solution and the diffusion of oxygen combined with the protein. The chemical kinetics involved were represented by the single step reaction P+xgPx, k,
where k, and k, are the recombination and dissociation constants respectively. Although this representation is correct for myoglobin, it is only an approximation for hemoglobin; some of the limitations of its use have been discussed [ 181. Several methods to obtain numerical solutions for this model have been presented in the literature. One approach [32, 331 assumed chemical equilibrium throughout the slab. Other authors 18, 17, 21, 24) assumed near chemical equilibrium inside the body of the slab, away from the boundaries. By using singular perturbation techniques they obtained expressions that were used for the computation of the solutions. Other numerical methods [3, 181 without the assumption of near-equilibrium have also been presented. The numerical method developed here starts with the observation that at each point the nonlinear chemical kinetics can be approximated locally by its linear approximation. Thus one has analytic solutions associated locally with the problem. A technique is developed to patch these solutions throughout the slab and up to the boundaries, where it is required that the boundary conditions of the problem be satisfied. The method has a rather general range of applicability; in particular it does not depend on the assumption of near chemical equilibrium. The qualitative features of the transport were examined. For the single step model the dissociation of oxygen from the carrier at the low oxygen boundary appears to limit the facilitation. The experimental observations of Wittenberg [30] on the dependence of facilitation on heme concentration and his finding that the values of facilitation per mole of heme by myoglobin and hemoglobin are the same have been analyzed previously. For hemoglobin satisfactory agreement was obtained [ 14, 171. For myoglobin, however, the facilitated fluxes computed [ 161 were about twice the experimental ones. The problem for myoglobin became amenable to the singular perturbation solution of Rubinow et al. [24]. However, some quantitative discrepancies remained. For heme concentrations larger than 5 m M their results for hemoglobin are lower than the experimental ones, and their results for myoglobin are higher. By incorporat-
267
TRANSPORTOFOXYGEN
ing the partial inactivation of myoglobin that took place during the course of the experiments as reported by Wittenberg [30], it will be shown that equality of facilitated flow per mole of heme by hemoglobin and myoglobin is obtained. For the chemical reaction of hemoglobin with oxygen a more correct representation than the one mentioned above would be the Adair [I] four step binding model PX,_, +&*,, k:,
i=l ,..., 4 ,
where PX, stands for the compound formed by one molecule of hemoglobin and i molecules of oxygen. It will be shown that by a natural averaging the four step process can be mapped into a process formally similar to the single step reaction. The attending new overall recombination (k, ) and dissociation (k,)coefficients are some averages of the Adair recombination constants (kb) and dissociation (k,) constants respectively. Those average coefficients are not constants. Their structure incorporates the cooperative properties of the reaction of hemoglobin with oxygen. Since every local analytic solution involved in the numerical method depends only on the local values of the chemical coefficients, then the method extends readily to the case of nonconstant chemical coefficients. It will be shown that the calculated facilitated transport by hemoglobin in solution, pso= 10 Torr, could be significantly larger than the one calculated for the single step model. This difference is in part related to the change from a hyperbola to a sigmoid of the associated oxygen dissociation curve. It will be shown that the dissociation of oxygen from the carrier inside the slab is confined no longer to a boundary layer. Benesch et al. [4, 51 have shown that the naturally occurring 2,3diphosphoglycerate in the red cell decreases the affinity of the hemoglobin for oxygen. Gibson [7] has shown that this effect can be described by essentially an increment in the value of the dissociation constants. For the hemoglobin in the red cell, psO --24 Torr, these effects enhance the facilitated flow by promoting the dissociation of oxygen in the vicinity of the low oxygen concentration boundary. By using values for the boundary conditions motivated by the partial pressures of oxygen found in the systemic capillaries by Duling et al. [6] and the dissociation constants adjusted for a psO ~23.4 Torr, it will be shown that for a slab one micron thick the facilitated flow contributes 50 to 65% of the total transport. One micron is likely to be within the order of magnitude of the diffusion path in the red cell inside the capillary. This result suggests the possibility that facilitation may contribute to the transport of oxygen in the systemic capillaries. The single step model representation and its associated kinetics constants were originally proposed by Hartridge and Roughton [10,111 in 1923. A
JOSE M. GONZALEZ-FERNANDEZ
268
AND SUSIE E. ATTA
result of independent interest developed in the present paper establishes the place that this approximation occupies in relation to the Adair [ 11 four step binding model. Motivated by the presence of membranes in the striated muscle, the numerical method has been extended to take into account the presence of multiple membranes impermeable to the carrier on the diffusion path. The results will be presented in a forthcoming report. THE MODEL Consider the region of space 0 G z < 1. Inside this slab let x, and x z be the concentrations of X and PX respectively. The chemical kinetics involved are assumed to be represented by the single step reaction
Let 9(x,, x1) be the net amount of oxygen per unit volume per unit time moving from the combined PX into the dissolved phase X. Let D, and D, be the constant diffusion coefficients of X and PX respectively. The model [30, 3 1, 321 for the steady state transport is described by the two differential equations d2x,
D,--
fd(x,,x,)=O,
Ll2$
-Q(x,,
dz2
with boundary
x2)‘o
(1)
conditions
(2) The boundary conditions on dx,/dz represent the constraint that the protein-oxygen compound cannot go through the boundaries at z = 0 and z = f. For the chemical kinetics
P+ X 2 PX.
9(x,,x,)=--k,x,(C,-x,)+k2x,,
(3)
where C, is the concentration of P plus that of PX. For the steady state conditions when the diffusion coefficient of P equals that of PX, it can be shown [ 121 that C, is constant across the slab.
TRANSPORT
269
OF OXYGEN
From (l), D, d2x,/dz2
t-D, d2x2/dz2 =O. Then (4)
is a constant across the slab. One recognizes that F is the total flow of oxygen. Let Fd denote the flow of dissolved oxygen (X), and F, denote the flow of combined oxygen ( PX). Then Fd = - D, dx, /dz, F, = - D, dx, /dz. From (4) OGZGl.
F=&(z)+&.(z),
(5)
Equation (5) states that at every point the total flow of oxygen is apportioned between the flow of dissolved oxygen and the flow of combined oxygen. The changes in this apportionment are brought about by the chemical reactions Pt X= PX. The rate of this change with respect to z is given by &+(x,,x,). At every point along z one can talk about the longitudinal fluxes Fd and F,., and the “transversal” flux Cp. The latter motivates the following representation for $(x1, x2). First consider the relationship between x, and x2 at chemical equilibrium, i.e., for
By using (3) and (6) one obtains
Cpk,x,
x2= k,x,
tk,
(7)
’
Call this function x;~(x,). Its Cartesian graph is usually called the oxygen dissociation curve. By rearranging terms in (3) one obtains ~(~,,XZ)=(k,x,+k2)[X2-x2eq(x,)].
(8)
Thus the transversal flux + can be expressed as the product of a “gradient” -xzq(x,) that is equal to the difference between the actual x2 and the x2 that would be in equilibrium with the prevailing x,, times a “chemical conductance” k,x, + k,. One can integrate (4) between z=O and z=l, and use (2) to obtain
x2
0
D,v+D,
I
x2(0) -x*(f) I
= F.
The first summand on the left corresponds to the oxygen flux that would exist in the absence of the carrier. It is called the nonfacilitated flux and
270
JOSE M. GONZALEZFERNANDEZ
denoted by F,,. The second summand denoted by F,. Thus F= F,, + Ff PIECEWISE Consider
ANALYTIC
the facilitated
flux and
APPROXIMATION
the linear approximation
+(X:,X:; to +(x,,x*) system
is called
AND SUSIE E. ATTA
x,,X2)=-k,(Cp-X:)X,+(kZ+k,X:)X2-k,XTX:
(9)
about the point (x:, x;), for x: GO and O
d2y,+$4x:,x:;
D,---
Y,tYz)‘O,
D2+$(x:r*f;
y,,y,)=O
dz2
approximates the original system (1) in a region about (x:, x2*). The error involved in this approximation will be considered later on. The following construction is started by specializing the points (x y, xz) to be used. Let (x:, XT) belong to the set defined by (6). The equation x,, x,)=0 defines a straight line tangent to the locus (6) at the #(x:,x;; point (x:, XT). Consider now a collection (XT’, xz’), i= 1,2,..., I, of points belonging to the locus (6), with O
x1 such that x;“”
9
i=1,2 ,...) I,
and xT’
(10) Associate to the interval
(11)
the linear system
(12) This system will be used to approximate (I) on the interval (11). Let x, be in the i th interval defined by (11). Since in this interval
TRANSPORT
271
OF OXYGEN
x;
X;
Xl
Xl
x1 (concentration of oxygen) FIG. I. The function #(s;‘. .xr’; x ,, x,) is the linear approximation (9) to +(x1. x2) about (XT’, .xI’). The point (XT’. XT’) is the intersection of $(x7’, I;‘; T,, x2)=0 and $(,r;‘-‘, x2’-‘; x,, x2)=0. The function $(xf’, XT’; x,, x,) is used as the linear approxi-
mation
to+(x,,x2)
on the regionXitL’
then by using (l), (3) and (12), (9) one finds that the error between the correct and the approximate second derivatives associated with the point (x,,x,)isboundedbyk,lx:‘-x,)1x:‘xz /. For a given grid (10) define the grid size 6 by 6=
max
r=2,..., I
Then those errors are bounded
uniformly on x, . Next, the piecewise analytic the grid (10) the I- 2 variables
approximation
and the I-2 PI 9
-xTI).
by
Z I)
with O
(xr(-’
i=2,...,I-1,
will be defined. Associate
to
(14)
variables i=2,...,
(15)
JOSE M. GONZALEZ-FERNANDEZ
212 Consider functions
the functions
y,, yZ defined
on a piecewise manner
by the set of
i=2,...,1-1,
Y;tY;,
that satisfy the linear prescribed as follows:
AND SUSIE E. ATTA
system (12) with domains
and boundary
conditions
for i=2, z, =O
,Y:h)=x?*?
4; dz
=o,
Yzz(Z*)‘P2i
:=o
for i=3,4 ,..., I-1, z,_,
y;(z,)=xY’, Y;(z,>=P,;
and for i=I, z,_,
Y;(z,-I)=PI-l~ The general solution
=I, Y:(z,)=xL
4: =o. dz :,
of (12) can be given by
y;=r;k~+7;k;(z-z,_,)+~;exp[~,(z-z,_,)]+~~exp[-X,(z-r,_,)], y;=r;k;+~+7jk;(z-z,_,)-~~exp[X,(z-z,_,)]-~~exp[-h,(z-z,_,)], 2 (16)
where k; =k,(C, k;=k,x:‘+k,, a’ =k,x:‘xT’
-x2*‘), (17)
TRANSPORTOFOXYGEN
273
and
(18) The coefficients $, j= I,. . . ,4, are determined from the boundary conditions of the corresponding interval. Motivated by the physical nature of the problem, the matching flow conditions ( $jz,-(zj;,=o,
j=l,2,
i=2,3 ,..., I-l,
(19)
are demanded. Observe that the number of unknowns (14) (15) equals the number of conditions (19). Since the boundary conditions at z=O and at z=I coincide with the ones given by (2), one can expect that this piecewise analytic solution will tend to the solution of the original problem (1). (2) as the error bound (13) tends to zero with 6. A numerical solution to the approximating system is equivalent to finding the 2(1-2) numbers z, [Eq. (14)] and p, [Eq. (15)] that satisfy the 2( 1-2) conditions (19). One can solve this nonlinear system by some numerical approximation such as the Newton method. Advantage can be taken of the fact that the matrices involved in the linear steps of the Newton method have analytic expressions. For the grid interval xF’+’
OF THE SOLUTIONS
FOR THE SINGLE
In order to study the dependence of the solutions on the values of the chemical parameters a series of computations covering a wide range of those
274
JOSE M. GONZALEZ-FERNANDEZ TABLE
C,,(heme) k, kz D, DZ /
AND
SUSIE E. ATTA
I
8.929X IO-’ M 3.0X IO” M ’ set -1 4.25X10 5+‘rsecc’ 1.5X IO -’ cmZsec -’ [31] 3.0XlO~‘cm”sec -‘[31] 1.0X10 ‘cm
values was carried out. The numerical values for the parameters used are the ones listed in Table I, for n = 1,. . . ,9. The values for n = 6 correspond to single step approximation to hemoglobin [IO, 11, 171. To follow the terminology used in the literature the concentration of oxygen will be expressed in Torr, and the concentration of the protein-oxygen compound in volumes of oxygen STP per volume of solution. The solubility coefficient of oxygen was taken as 1.6X 10 -’ MTorr -’ [28]. The values of XI: and xi in (2) were 190 and 0 Torr respectively. In Fig. 2 the solutions y,(z) and yz(z) are plotted. For the smallest value of k ,, k, considered, n= 1, the graph of y, approximates the straight line
10.20 x2 ti WI 0.15
i
- 0.10
- 0.05
-0
FIG. 2. This graph shows the dependence of v,(z) and r2( z) on the values of k, and The indexing digit stands for the value of n in the assigned k, =3.0X IO” A4 ‘set ’ ‘. The unptimed index corresponds to y,, and the primed and X-,=4.25X10 “” set index corresponds to pa. The values of the other parameters are listed in Table 1. k2.
TRANSPORT OF OXYGEN
275
joining the points z=O, X, =xp and z=/, x,=x:, and the graph of y, approximates a constant. This gives a negligible value of facilitated transport. As n increases the absolute value of the slope of y, increases near z=O and z = 1, and decreases in an intermediate region. The change of slope near z = I is not apparent for the scales used. This change can be appreciated in Fig. 5, where the solution y,( z) for n = 6 is plotted. As n increases, the value of y,(O) increases toward xsq(xy) and the value of yz(/) decreases toward xsq(x{) 1181. Some relationships between the physicochemical processes taking place inside the slab and the form of the solutions will be considered next. Consider first the case n=3 of the preceding sequence. The results are represented in Fig. 3. One observes that there is a value z* of z such that y,
-
F 0.10
F”l
-10.05
Ft -0
FIG. 3. This graph illustrates the solutions when chemical equilibrium is not attained inside the body of the slab. The chemical kinetics coefficients are k, =3.0X IO’ M ’ set ’ and k, =4.25X 10 ’ set ‘. A large part of the loading process occurs inside the body of the slab away from the boundary.
276
JOSE M. GONZALEZ-FERNANDEZ
AND SUSIE E. Al-l-A
were found to be negligibly small inside the body of the slab, (Fig. 4). However, the increment of the chemical conductance k,y, + k, was proportionately larger than the decrement of Iyz -~;~(_y,)(. The net result was an increase in the transversal flow of oxygen and in the maximum values of (d’y,/dz’(. Therefore, for hemoglobin the assumption of near chemical equilibrium inside the body of the slab [ 17, 21, 24, 321 for this and larger values of 1 finds here a numerical justification. The profiles of F, and Fd for n = 5,. . ,9 show sharp changes in layers located at z=O and at z = 1; for n =6 see Fig. 4. The details of the changes near z=I for n=6 are shown on a magnified scale in Fig. 5. The width (J of a boundary layer will be defined as the length of the interval on which 0.98 of the rapid change in Fd (or F, ) takes place. To analyze the structure of the solution on the boundary layers the function +(x,, x2) of the original problem (1) was replaced by the linear approximation (9). Then the solution (16) was evaluated by solving (12) for the interval z, _ , , z, where the rapid change in Fd takes place. The appropriate boundary values for the interval were obtained from the numerical solution already computed. It was found that on this interval
y, -x;~(Y,)
Fd~-_,T;k;+D,?~h,exp[-X,(z-z,_,)]
'0.10
1 F”f
6
0.05
Jo FIG. 4. This graph illustrates the solution for hemoglobin. n=6. Inside the body of the slab the values of v2 and x’;4(,,) differ negligibly from one another. 73% of the loading occurs at the right of the midpoint of the slab.
TRANSPORT
277
OF OXYGEN 0.15 FC
x2
Fd
x;"
$104
0.025 yclJ VOI 0.10
0.020 Xl torrx 102
e-v*-~'x,O_,
0.015
0.05
0.010
0.005
0 0.98
FIG. 5.
Details
exp[ -A,( /-z)]
of the solution
illustrates
width of the boundary
an
0.99
I cmxl02
0 1.00
of Fig. 4 close to the boundary
opriori approximation
at ;=/.
to the relative changes layer at z= I; A, was evaluated for s;’ =x{ =O.
The graph
of
in Fd and to the
and F, m-D02~;k;-D2~~h,exp[-h,(z-z,_,)]. Since I- exp( -4) =0.98, which is the number then one is led to the condition X,a=4
used in the definition
of u,
(20)
for the evaluation of a. For the boundary layer at z=O the value of A, may be approximated u priori by letting x;’ = xp in (17) and for the boundary layer at z= I by letting XT’ =x{. Good numerical approximations to the changes in FJ (or 4,) and to a were obtained at both boundaries (see Fig. 5 for n=6). The same good agreement was obtained for the runs n = 4 to n = 9. At the place where the condition of near-equilibrium is satisfied, then the ratio Fc/ Fd is close to ( D, /D, ) dx;‘l/dx,, a resealed slope of the oxygen dissociation curve (7). When X: is large enough, then the shape of the oxygen dissociation curve determines that the larger values of F, occur in the body of the slab away from the high oxygen boundary; see Fig. 4 [27, 331. Near this boundary one expects a smaller value of e,, and since by (2) F, =0 at z =O, then this determines only a small jump in I$ there. This jump induces the existence of a boundary layer. The loading process, in this instance, is not a boundary layer process. The value of the loading is therefore not directly related to the resulting value of n,(O) which enters in the expression for the
278
JOSE M. GONZALEZFERNANDEZ
AND SUSIE E. ATT’A
facilitated flow, D,[ x2(O) - x2( /)I/[; thus on the above conditions the loading does not play a limiting role in the facilitated flow. Closer to the low oxygen boundary the opposite situation holds. The condition of near equilibrium and the smaller values of x, imply the existence of larger values of F,., and since by (2) Fc =O at z=l, one then expects a large jump of F, at this boundary. The direction of this change is opposite the one conditioned by the oxygen dissociation curve. Thus the unloading process is essentially a boundary layer process (Figs. 4 and 5). The condition (20) with (18) expresses the dependence of the width of the boundary layer on the physiocochemical parameters k , , k,, D,, and D,. Within the range of validity of the above considerations the width of the boundary layer is then independent of the thickness of the slab [21]. To verify this assertion the thickness of the slab was varied from 1 to 300 microns. For the chemical kinetics that correspond to n=6 it was found that for I between 30 and 300 microns the solution x2 was near equilibrium inside the body of the slab and that the width of the boundary layer at z=I was constant and equal to 0.91 microns. This value coincided with the one given by (20) where X, was computed a priori by using xy’=x{=O and x;‘=x;q(x{) in (17). From the second equation in (1) the unloading of oxygen equals the integral of (8) on this interval. For the decreasing values of I, increasing values of unloading were attended on this interval by larger values of the gradient by larger values of x2 at z=I, thus limiting the x2 -x2 ‘q and in particular increment of the facilitated flow. It can be shown that the exponents X, obtained a priori as indicated above coincide with the exponents y and 6 for the boundaries at z = 0 and at z = I, respectively, of the singular perturbation expansion developed by Rubinow et al. [24].
ANALYSIS OF EXPERIMENTAL RESULTS: RELATION BETWEEN FACILITATED TRANSPORT PROTEIN CONCENTRATION
AND
Wittenberg [30] performed a series of experiments with solutions of hemoglobin and myoglobin that covered a wide range of concentrations of these proteins. The width of the millipore filter used was 1SO X 10 -’ cm. The area of each face exposed to gases was 11.5 cm2; the fractional space of porosity equaled 0.79. To proceed with the numerical computations one needs the values of the experimental parameters. The values of the self-diffusion coefficients used for hemoglobin and myoglobin as functions of the protein concentration will be the ones obtained by Riveros-Moreno and Wittenberg [22, 311. A review of the results obtained by others is presented by Kreuzer [ 161 and Kreuzer et al.
TRANSPORT OF OXYGEN
279
[ 171. The ones for free oxygen were compiled by Kreuzer et al. [ 171. These values have been essentially confirmed by other experimental techniques [9]. The chemical kinetics coefficients for the single step approximation to hemoglobin were taken as k, =3.0X lo6 M-‘see-’ and k, =42.5 set-’ [IO, 11, 17],andformyog1obinask,=1.40X10’M~’sec~’andk,=11.0sec~’ [2]. The partial pressure of oxygen was taken to be 300 Torr on one side of the slab [24] and 0 on the other side. The effective value of 1 is geometrically uncertain due to the tortuosity of the millipore channels. Keller and Friedlander [ 151 reported a tortuosity factor of 1.63. From the results on the diffusion of oxygen and nitrogen through a solution of ferric hemoglobin reported by Wittenberg [30, Fig. I] one calculates an average effective I of 2.45 X 10 -2 cm, and a tortuosity factor of 1.63. From the results [30, Figs. 3, 41 for filter widths ranging from 2.5 X 10 -3 to 3.3 X 10 -’ cm one obtains a tortuosity factor of 1.34. From the discrepancy between the apparent diffusion coefficient of Nz and the expected one reported in Fig. 11 of Wittenberg’s review [3 I], one may infer a tortuosity factor of 1.57. The numerical results obtained for hemoglobin will be discussed first. To begin with, the value chosen for I was 2.45X 10 -’ cm; this value was also used by Rubinow et al. [24]. The facilitated fluxes computed showed the same qualitative dependence on heme concentration as the experimental ones. However, they were consistently lower than the experimental ones. On the other hand, the maximum possible facilitated flow for the model considered here can be obtained by settingy2(0)=x;q(xy) and y2(C)=x;q(x:)=0. The maximum facilitated flow as a function of heme concentration is only slightly larger than the one computed, but smaller than the experimental one; see Fig. 6. The fact that the experimental facilitated flow is consistently larger than the theoretical maximum may be taken as indirect evidence that the value ascribed to I was too large. If one assumed a tortuosity factor of 1.43, then an adequate quantitative agreement was obtained [ 141; see Fig. 6. Wittenberg reported [30] that at the termination of each experiment 10 to 25% of the myoglobin was found to be oxidized to the ferric state. Therefore, the experimental values of facilitated flow that Wittenberg [30] reported as functions of total heme concentration were replotted in Fig. 6 as functions of the chemically available heme concentration, which is taken to be 0.85 times the total heme concentration reported. On the other hand, for the numerical calculations the self-diffusion coefficient of the carrier and the diffusion coefficient of oxygen were both chosen to correspond to (0.85))’ times the heme concentration of the chemically available form. The tortuosity factor of 1.43 will be retained. The computed facilitated flow is plotted against the heme concentration of the chemically available form in the same Fig. 6. Quantitively the agreement between the theoretical and experimental results
280
JOSE M. GONZALEZ-FERNANDEZ 2.0 r
n
AND SUSIE E. ATTA
0.
.
0
,I’
‘\
. .‘;---.
I’
0
I
/’
81 0 I
Of....f.-I.. 5
‘\ *
‘\
‘\
o‘\, ‘\ ‘\
.
“9
. ’
,‘..,,‘...’ 10 15 20 HEME CONCENTRATION (mM)
FIG. 6. Relationship between facilitated circles represent the experimental values
8
25
flow of oxygen and heme concentration. Solid for hemoglobin. Open circles represent the
experimental values for myoglobin when the abscissae stand for the heme concentration of the chemically available form. The line of dashes represents for hemoglobin the maximum possible theoretical facilitated flow for 1=2.45X IO -’ cm. The solid line represents the numerical results for hemoglobin for 1=2.14X lO-2 cm (tortuosity factor is 1.43). The dot-dash line represents the results for myoglobin, /=2.14X IO -’ cm. For the values of the other parameters
see the text.
is satisfactory. This confirms Rubinow et al.‘s conjecture [24] that when the correction for oxidation of myoglobin is introduced, a satisfactory agreement with the experimental observations could be expected. The numerical results reproduce the experimental finding [30] that the facilitation for hemoglobin (solid line) and for myoglobin (dot-dash line) are approximately the same as functions of heme concentration, not of molar concentration (Wittenberg’s paradox [30, 3 I]); see Fig. 6. It was conjectured by Wittenberg [30, p. 108 (footnote); 31, p. 5671 that the equality of facilitated transports could be due to the balancing of two opposing effects. On the one hand, the value of the self-diffusion coefficient for myoglobin is larger than the one for hemoglobin, and on the other hand, the dissociation coefficient, which determines in large part the unloading of oxygen, is larger for hemoglobin than for myoglobin. Rubinow et al. [24] confirmed, as part of their analysis, that Wittenberg’s conjecture resolves the paradox. Observe that the results for hemoglobin test essentially only the relation between the concentration and the self-diffusion coefficient of the protein, since the values of x,(O) and x,(l) are very close to the ones for equilibrium. In contrast, the results for myoglobin depend also on the limitation that the chemical kinetics places on the unloading of oxygen. The latter resulted in a fairly large departure of x,(f) from equilibrium. Thus the above findings constitute added indirect evidence on the validity of the model.
281
TRANSPORTOFOXYGEN EXTENSION PROPERTIES
OF THE MODEL TO INCLUDE OF HEMOGLOBIN
THE COOPERATIVE
A more precise representation than the one used above for the chemical reaction between hemoglobin and oxygen would involve a four step binding model (Adair [ 1I), px,_,+xZpx,, k:,
i=1,...,4,
where PX, stands for the compound formed by one molecule of hemoglobin and i molecules of oxygen. An analysis of the facilitated transport will be carried out to include the influence of the different constants kk, kb. The resulting model will retain the simple description in terms of the two variables x, and x2. Let J, be the increment of the molar concentration of PX, per unit of time contributed by the chemical reactions. Then the second equation of (1) generalizes to d*PX D,-z dz
+J,=o,
(21)
i=1,...,4,
where J,=kXPX,_,X-k’,PX,-kh+‘PX,X+k’,i’PX,+,,
i=1,2,3,
(22)
and J,=k; Multiply
PX,X-k;PX,.
(21) by i and sum over i= I,...,4 4
to obtain
d2PX.
4X+-&
(23) i=l
i=l
Let 4 x2=
iPX,,
x i=l
and as customary
define the intrinsic kk=(4-i-t
l)ki,
rate constants k,=iki,
ki and ki by i=1,...,4.
282
JOSE M. GONZALEZ-FERNANDEZ
AND
SUSIE E. ATTA
Then from (23) and (22) d2x2 D2-
+X
dz2
ki+‘(4-ii)
i
PX, -
i
r=O
k;iPX,
1=l
=O.
(24)
Since 2:=,(4-i) PX, equals the concentration of unoccupied sites (reduced hemes) and Cp=, i PX, equals the concentration of occupied sites (oxygenated hemes), it is natural to define an average recombination coefficient by
; k:+‘(i-i) r=a
k;,=
j.
and an average dissociation
(4-i)
coefficient
i k,,=
(25) PX,
by
k;iPX,
‘=’
(26)
,i,iPX,
Since C, =4X:=,
PX,
.
PX,, then from (24) d’x, D2-
In a similar manner
dz2
+k:&-x,)x,
-Lx2
=O.
(27)
one obtains d2x, -k:,(C, D,-dz2
-x2)x,
+k,,x,=O.
Equations (27) and (28) are formally similar to (1) plus (3). However, the chemical kinetic coefficients kiv and k,, are not constants. Their values depend on the prevailing families PXi, i=O, I,. . ,4, which themselves are part of the solution to the problem in question. As a first approximation, the families (PX;) associated with the hemoglobin oxygen dissociation curve will be used in (25), (26). For this choice one defines k,(x2)=kLv and k2(x2)=kav to be used in (27) and (28). Equation (3) is then replaced by +(x17x2)=
-k,(x,)(Cp
-x2)x,
+&(x2)x2.
(29)
TRANSPORT
OF OXYGEN
283
The difference between the linear x7, XT and $(x,, x2) is given by
approximation
4(x:,
xz; x,, x2) about
stand for the second-order partial derivatives of $J where%,x2ad +.x2.x, evaluated between xi,xz and x7,x;. Observe that &_+,=O. The error between the correct and the approximate second derivatives of solutions to (1) at x,, x2 is then 0(x, -xz) uniformly in x,, x2. Because of the form of this error bound one needs to change the grid used in the preceding numerical approximation. Instead of the set (10) one may consider the set x:‘, i= 1,. . , I, and may associate to the interval x:’
TABLE 2 I
k: (Mm’secm’)
k; (set _ ‘)
1 2
9.00x IO6 4.0x IOh
1.080X IO3 2.44x 102
3 4
3.5 x IO6 4.0x IO’
2.8X IO 4.80X IO
284
JOSE M. GONZALEZ-FERNANDEZ
AND
SUSIE E. AT-I-A
The functions k,( x2) and k2( x2) were calculated by using the values of ki and k; obtained by Ilgenfritz et al. (Table 2). To describe them numerically it is convenient to consider them to be functions of f, where f =x2/CP. The value of k, is approximately equal to 8.5X IO6 M-‘see-’ for O
Parameter
C’=8.93X
10e3
TABLE 3 Values Used for the Computations
M heme;
D, =3.0X
IO-’
cm’/sec;
Pertaining
to Fig. 7
D, = 1.5X 10 -5 cm2/sec;
xp=
190
Torr; x( =O Torr
A.
Model with cooperative properties The values for k,( x2) and k2(x2) are derived from Ilgenfritz (Table 2) and PX, in equilibrium; P5c = 10 Torr; O2 sol. coeff.=1.7X10-6
et al.% values for k;, k:,
M Torr-‘. Single step model
B. C. D.
k,=3.00X106 M-‘set-‘; k,=42.5 set-‘; P5n=8.9Torr. k,=7.00X106 M-‘set-‘; kzz99.16 set-‘; Ps,,=8.9Torr; O2 sol. coeff.= 1.6X 10e6 M Torr-‘. Jacquez et al. 1141: C,=9,375XlO~-‘Mheme; D2=3.6X10-‘cm*/sec; D,=l.4X10~5cm2/sec; xy =200 Torr; xi =0.05 Torr; k,=3.00X106 M-‘set-‘; k2=40.0 O2 sol. coeff.=l.8X10-6 M Torr-‘.
set-‘;
P,,=7.4Torr;
TRANSPORT
285
OF OXYGEN
2
6
10
20
30
40
60
70
80
FIG. 7. Dependence of F,/FfmaX on the length of the diffusion path for hemoglobin solution. Curve A represents results from the cooperative model; curves B, C, and represent
results from
the single step model. The values used for the parameters
in D
are listed in
Table 3.
Equations (27) and (28) with the boundary conditions (2) were solved for different lengths of the diffusion path 1. The values used for the parameters are listed in Table 3. To express the magnitude of the facilitated transport the ratios F,/F,““, where Ffmax= D2[x~4(x~)-x~4(x~)J, are graphed in Fig. 7. For diffusion lengths larger than 35.75 microns this ratio is nearly constant. By using the tortuosity factor 1.43 these lengths correspond to values of slab thickness larger than 25 microns. Therefore, the values of facilitated flow are approximately of the form constant/l, as was observed experimentally in this range by Wittenberg [30]. For comparison purposes the values of F,/F;“” for several single step chemical representations are included. The profile of for the cooperative model contrasts with the ones for the single step F, /F,“” models. For the former the value of this ratio is larger than for the latter ones for values of I down to 10 microns. However, for values of I smaller than 10 microns it decreases rapidly with decreasing I, to eventually become smaller than the values associated with the single step model. One may note that the overall apparent second order rate constant k’ obtained in conditions of excess ligand by Ilgenfritz et al. was 5X lo6 M-‘set-’ [13]. This is significantly smaller than the least value 8.5X IO6 M - ’ set - ’ taken by k, in the model as above. order to this discrepancy the results by Gibson were analyzed in terms the Adair using the values the constants obtained by Ilgenfritz et Gibson the course of of deoxyhemoglobin 10e5 M oxygen,
286
JOSE M. GONZALEZFERNANDEZ
AND
SUSIE E. A’ITA
and (D) 1.55X 10 p4 M oxygen. It was verified that the numerical solutions agree closely with the experimental results. The values of kX and k,, were computed and their relationships with the values of j were studied. It was found that at each j the values of kiv are smaller (k,, are larger) for faster transients. For. instance, at j=0.3 kj, =7.7X lo6 M p’sec-’ for curve C (at equilibrium k iv =8.2X10h M-‘see-‘), kh,=6.2X10h M-‘set-’ for curve B, and ki,=5.6X106 Mp’secm-’ for curve A. The latter is very close to the overall k’ quoted above. It was found that these different values reflect the different populations PX,, i=O, 1,. . , 4. The relative amounts of the intermediate species PX, and PX, are larger for faster transients. One may examine the meaning of the values of k’= 3 X 10” M ~ ’ set- ’ and k=42.5sec-’ that Hartridge and Roughton [ 10, 1 I] proposed in 1923 for the single step model. From the above analysis the value of k’ appears to reflect the values of kf and k:, and the value of k appears to reflect the value of k:. These values embody the low range values of the recombination and of the dissociation constants respectively. Thus, in terms of kinetics the values of Hartridge and Roughton yield a lower bound for the speed of the chemical kinetics of hemoglobin with oxygen. Those values would be too small when the process studied corresponded to either a slow transient or a situation where the species were close to chemical equilibrium and their use resulted in a solution away from equilibrium. Therefore, the results for hemoglobin reported in the previous section are still correct. On the other hand, one could expect the values of curve A in Fig, 7 to be correct for values of F, / Ffmax larger than 0.85. For smaller values of this ratio one could expect, because of the departure from equilibrium, that the correct solution will decrease with decreasing 1 faster than the computed solution of Fig. 7. The results of Moll [20] and Kutchai et al. [19] provided experimental evidence of facilitated transport of oxygen through 300 and 165 micron layers of packed erythrocytes. Since the dimensions of a single red cell are much smaller, Kutchai et al. [ 181 studied theoretically the transport in layers of thickness commensurate with that of the red cell; a single step chemical kinetics model was used. The problem will be reexamined here by using a fuller description of the chemical kinetics as it pertains to hemoglobin in the red cell. The half saturation point psO for whole blood, 37°C pH 7.4, is approximately 24 Torr [23]. The effect of 2,3-diphosphoglycerate on the oxygen affinity of hemoglobin has been elucidated by Benesch et al. [4, 51. Its chemical kinetic interpretation has been obtained by Gibson [7]. This author showed that the change in the Adair rate constants of hemoglobin in solution from no phosphates ( psO = 2 Torr) to about 0.5 mole DPG per mole Hb ( pso = 10 Torr) was effected fundamentally by increments in the dissociation constants. One may extrapolate to the situation in the erythrocite in which one mole
TRANSPORT
287
OF OXYGEN TABLE 4 Parameters for the Cooperative Model for Hemoglobin in the Red Cell C,=2.20XlO-‘Mheme &=6.80X IO-‘cm2sec D,=7.00XIO~-hcm2sec~’
-’
Ps,,= 23.45 Torr 02sol.cocff.=1.7X10~6MTorr~’
DPG is present per mole Hb and the pSO=24 Torr. As a first approximation the values of k,(x,) were kept as the ones described above. Then, the values of k,(x,) were calculated so that the associated oxygen dissociation curve, pSO=23.45 Torr, was a good fit to the one for whole blood. The calculated values of k, decrease from 2.28X103 set-’ at j=O to 1.05X103 sect’ at f= 0.15; then they decrease more gradually to 4.1 X 10 2 set - ’ at f= 0.4; and for larger values off they decrease at about a constant rate to 9.6 X 10 set - ’ at f= 1.0. For these functions k, and k, one may calculate approximately the associated intrinsic chemical constants. The values of ki, i= 1,. . . ,4, are essentially the ones listed in Table 2. The values of k; are as follows: k: =2.28X lo3 set-‘, k; =2.90X IO2 set-‘, k: = 1.71 X IO2 set-‘, and kj = 9.6X 10 set-‘. Again, Eqs. (27) and (28) with the boundary conditions (2) were solved for different lengths of the diffusion path, [. The values of the other parameters are listed in Table 4. The ratios Ff/Fnnf and F,/F,““” for the boundary values xp = 100 Torr and xi =O Torr are listed in Table 5. The magnitude of the facilitated flow, F,, is about the same as the one of the nonfacilitated flow, Fnr, for I= 1 micron. For comparison purposes the results for the single step
TABLE x1’= 100 Torr
5
Torr 1x’=O I F,/FfmaX
F,/F,r I (microns) 0.50 0.75 1.00 2.00 5.00 15.00 25.00
Single step
Cooperative
model
model
0.3609 0.4769 0.5566 0.7278 0.8848 0.9774 0.9982
0.591 0.845
Single step model
Cooperative model
1.206
0.3499 0.4620 0.5396 0.7053 0.8577
0.485 0.693 0.845 0.960 0.990
1.217
0.9475 0.9677
0.996
1.03I 1.160
JOSE M. GONZALEZ-FERNANDEZ
AND
SUSIE E. A’ITA
2 MICRONS
FIG. 8. Details of the unloading of oxygen in the cooperative model for hemoglobin the red cell. Boundary values: x,” - I00 Torr ( X{ =O Torr; I= IO microns.
in
model with k, =5.4X IO6 M-‘see-’ and k, =202 set-‘, PSO=22 Torr, are also listed. Some details of the numerical solution for the cooperative model for f= 10 microns are graphed in Fig. 8. It can be observed that the unloading takes place in the interval 7.83- 10 microns. The steep phase occurs in the boundary layer, 9.85 10 microns; its width agrees with the a priori calculated u. The slow phase occurs in the interval 7.83-9.85 microns, and it is attended by
TABLE 6 xj’=lOOTorr x’= 40Torr I (microns) 0.50 0.75 l.O4l 2.00 4.00 5.00 10.00 20.00 25.00
F,/F,r
F, / Ffmu
x:’ =40 Torr x{=l5Torr G/F,,
F,/Ffm””
0.23 I 0.280 0.319 0.379
0.508 0.595 0.702 0.834
0.938 I.296 I.514 I .937
0.360 0.496 0.580 0.741
0.422
0.929
2.321 2.462
0.889 0.943
0.448
0.985
XI’ = 25 Torr xl= 5 Torr h/F,,
F,/ Ffmax
I.500 1.971 2.238 2.670 2.900
0.477 0.627 0.712 0.849 0.922
3.046 3.095
0.969 0.984
TRANSPORT
289
OF OXYGEN
near chemical equilibrium there. Its beginning corresponds to x, = 15.8 Torr, which is the inflection point of the oxygen dissociation curve used. Compare with the single step model profiles of Fig. 5. To take into account the ranges of partial pressures of oxygen in the pulmonary circuit and in the systemic capillaries [6], further computations were carried out. The ratios F,/F,, and F,/F;““” are listed in Table 6. One can observe that for diffusion paths of 0.75 to 1.00 micron the facilitated flow contributes 56 to 60% of the total flow for xp ~40 Torr, XI = 15 Torr, and 66 to 69% for xp = 25 Torr, xi =5 Torr. Since the length of the diffusion path in an erythrocite inside a capillary is likely to be of that magnitude, these results make it plausible that facilitated flow by hemoglobin may contribute to the transport of oxygen in the systemic capillaries. To further elucidate this question it will be pertinent to study models that include other features of the capillary transport, especially the transit time.
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4
G. S. Adair. The hemoglobin system. VI. The oxygen dissociation curve of hemoglobin. J. Biol. Chem. 63~529-545 (1925). E. Antonini and M. Brunori. Hemoglobin and Mvoglohin in Therr Reactions with Ligunds, North-Holland, Amsterdam, I97 I. R. J. Basset and J. S. Schultz, Nonequilibrium facilitated diffusion of oxygen through membranes of aqueous cobaltohistidine, Biochim. Bioph_vs. Actu. 2 I I : 194-2 15 ( 1970). R. Benesch, R. E. Benesch, and C. I. Yu, Reciprocal binding of oxygen and diphos-
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