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Parameter dependence of myoglobin-facilitated transport of oxygen in the presence of membranes
Parameter Dependence of Myoglobin-Facilitated Transport of Oxygen in the Presence of Membranes
JOSfi M. GONZALEZ-FERNANDEZ Mathematical Research Branch, National Institute of Diabetes, Digestive and Kidney Diseases, National Institutes of Health, Bethesda, Maryland 20892 Received I2 January 1989; revised 30 October 1989
ABSTRACT Some physicochemical entities involved in the facilitated transport of oxygen along a transport path zt < z < z, with membranes impermeable to myoglobin at zjr i = 1,. . , n, were identified in an earlier paper [Math. Biosci. 95:209 (198911. These entities are the partition between the oxygen and oxymyoglobin flows, the flow transfers taking place near a membrane, and the membrane resistance. Expressions for these entities were obtained that explicitly involve the parameters of the system. In this paper, for the case of prescribed boundary oxygen concentrations x’ and x,, these expressions are incorporated into (i) an explicit representation for the facilitated transport value in terms of the difference, E(x’)E(x,), between the boundary oxymyoglobin equilibrium values and the sum, f, of the membrane resistances, and (ii) a representation for the distribution of the membrane oxygen concentrations x, at zi, i = 2,. . , n - 1. This makes it possible to analyze the manner in which the facilitated transport depends on the parameters. For a physiological range of parameter values, the facilitated transport was found to increase as either the oxygen-myoglobin association rate constant k’, the dissociation rate constant k, the oxygen diffusion coefficient, or the oxymyoglobin diffusion coefficient increases. Thus, the facilitated transport does not depend directly on ratios of chemical and diffusion coefficients. Although the increase in the oxygen diffusion coefficient does not per se affect the chemical conductance, it diminishes the membrane resistance through an interface feature, with a resulting increase in the facilitated transport. For a larger range of values of k’ and k, the dependences of the facilitated transport on k’ and on k are both biphasic and are similar in shape. However, the mechanisms involved are different: The associated changes in E(x’)E(n,) and in F that result from the increase in k’ are opposite to those that result from an increase in k. The use of(i) and (ii) permits, also, discrimination between the different roles of the physicochemical entities involved in a given facilitated transport change. In some cases (e.g., the decreasing phase of the facilitated transport as k’ increases), this change depends in an essential manner on a secondary modification of the profile x,, i = 1,. , n, along the transport path.
M4THEMATICAL
BIOSCIENCES
lOO:l-20
(1990)
Published by Elsevier Science Publishing Co., Inc., 1990 655 Avenue of the Americas, New York, NY 10010
1
0025-5564/90/$00.00
2
JO& M. GONZALEZ-FERNANDEZ
INTRODUCTION In a previous paper [l], an analysis of the myoglobin-facilitated transport of oxygen was carried out for the case that membranes impermeable to myoglobin are present along the transport path. It was assumed that a region of near chemical equilibrium exists between, but away from, the membranes. That analysis identified some physicochemical entities having a local character, namely, the flow partition between the free and combined oxygen flows and the membrane flow transfer resistance. The flow transfers are associated with jump discontinuities in the oxymyoglobin concentration at the membranes; their cumulative effect is one of the determinants of the extent of the facilitation of transport. Mathematical expressions for these entities were obtained that explicitly contain the parameters of the problem. In this paper, for the case that the oxygen concentrations at the boundaries of the transport path are prescribed, those results will be incorporated into a representation for facilitated transport. Also, a set of relations that determine the profile along the transport path of the oxygen concentration at the membranes will be obtained. With these expressions the dependence of the facilitated transport on the parameters will be analyzed. THE STRUCTURE
OF THE FACILITATED
TRANSPORT
On the space interval 0 ( z G z,, let the free oxygen concentration x(z), the oxymyoglobin concentration y(z), and the reduced myoglobin concentration c,, - y be defined. The constant c/, stands for the total concentration of reduced plus combined myoglobin. These species are able to diffuse and interact chemically in a reversible manner with each other. The diffusion coefficient for oxygen and that for oxymyoglobin are denoted by d, and d,, respectively. It is assumed that membranes impermeable to myoglobin exist at zi, i = l,..., II, where z, = 0, z, = z,, and zi < z~+~; let xi =x(2,), i = l,..., n. The rate constant for the association of oxygen with myoglobin and that for the dissociation of oxymyoglobin are denoted by k’ and k, respectively. The interface condition of zero myoglobin and oxymyoglobin flows is assumed to hold at the membranes. The boundary values x(0) and x(2,) are prescribed. The system is in the time-independent state. The facilitated transport ff is defined to equal f, - f,,f, where f, is the total flow of oxygen and f,,f = d,(x, - x,)/z, is the nonfacilitated flow. To avoid repetition, the reader is referred to [ll for the remaining terminology and results used below. TRANSFERS
OF FLOWS IN INTERVALS
NEXT TO THE MEMBRANES
The existence of nonzero facilitated transport implies the sharing of the total flow f, between the free oxygen flow f&) and the oxymyoglobin flow
PARAMETER
DEPENDENCE
OF FACILITATED
TRANSPORT
3
f,(z). It was shown in [I] that the sharing is effected through transfers between fd(z) and f,(z). The flow transfer that takes place in a right interval, (z;, z, + g[,>, [left interval, (z, - oI, zi)] of the membrane at z, results in a difference A,+ (A;> between the left (right) side limit of y(z) and the reference equilibrium oxymyoglobin concentration E(x) at zl; these differences have been called costs because they subtract from the possible maximum facilitated flow [2]. Let xi* = xj(zi _+ai). For reference purposes, explicit representations for the flow partition a,*, the eigenvalue A, associated with the interface condition at zi, and the flow transfer resistance pi obtained in [l] are collected here. Thus d,c,k’k
ay’ = d,c,k’k
+ dl( k’x’
+ k)* ’
&=hi/(k’x,+k).
The flow transfer on the interval (zi,zi + o,> is approximated by the and that on (zi - cr,, zj> by f,~;. It was shown that approxiproduct fin:, mately A: = The following representation
ff=4
f,a’&.
for the facilitated
E(x,)-E(x,)-(A;
flow
+El;-‘A: =1
Define the membrane
resistance
and
was obtained
in [2]:
+A;) (1)
yi* by
y;f = ar+&. Since A: = fry,’
ff
(2)
f, = f,,r + fr,
f =d E(x,)-E(x,)-d,[(x,-x,)/zllr f 2 zl + d2T
(3)
n-1
r= y: +
c yj++r,. 2
(4)
4
JO&
M. GONZALEZ-FERNANDEZ
Expression (3) gives the facilitated flow in terms of the difference E(x,)Hx,,) obtained by evaluating the equilibrium oxymyoglobin concentration for the prescribed boundary values, the nonfacilitated flow d,(x, - x,,)/z, and the sum I of the membrane resistances 7;‘. The term -yi* incorporates the flow partition CY;*,the chemical conductance k’x, + k, and the eigenvalue A,. Therefore, (3) can be used to analyze how the dependence of the facilitated transport on the parameters is mediated through changes in these physicochemical entities. The right-hand side of (3) contains the unknowns xi. Therefore it will be desirable to obtain another relationship that involves the xi. An expression that relates the profile of xi, i = 1,. . ., n, to the above entities for the prescribed boundary values X, and x,, will be developed next. THE DISTRIBUTION OF THE OXYGEN CONCENTRATION, xi, i = 1,. .,n, ALONG THE TRANSPORT PATH
By specializing
(1) to an interval (zi, z;+,), we can write the total flow f,
as
r,=d,-*‘;;
+d,
Introduce
+YLl)ft
E(Xi)-E(xi+l)-(Yi+
zi+l
I
-
zi
the notation
which represents the slope of the secant to the oxymyoglobin dissociation curve between the points that correspond to xi and xi+t, and solve for f, to obtain
h=(xi-xi+l)
d, + d2Si,i+ 1 Zi+l_Zi+d2(YT +YiGl) ’
The second factor on the right-hand side of this expression defines the interval conductance for (zi, zi+ ,). It can be shown that the terms yi* tend fixed; then the interval to zero as k’ and k tend to 00, with k/k’ conductance tends to
d, + dZSi,i+l zi+I-zi
.
The numerator of this term expresses the additive nature of the free and combined oxygen flows and the role of the product of d, and the slope of the oxymyoglobin dissociation curve in the facilitation of the transport
PARAMETER
DEPENDENCE
OF FACILITATED
5
TRANSPORT
(Wittenberg [6]). The interval conductance generalizes this by taking into account the terms yi* that allow for the incorporation of departures from chemical equilibrium. Next, for (z;, zi+ ,), define the interval resistance Gi to be the inverse of the above conductance. Thus, +; =
*i+l
-Zi+d2(Yi+ dl
++Yi+l)
+
dZSi,i+*
’
and
The global relation xi-xi+l
=(x1-x
5
) n
i=l
cj”&
,...,a-1,
’
is then obtained. Therefore, the concentration drop x, - xi+t for the subinterval (zi, zi+il equals the prescribed boundary concentration difference x1 - x, times the fractional resistance for this interval. The greater the nonuniformity of J)~, i=l , . . . , n - 1, the greater the departure of xi, i = 1,. . ., n, from a straightline distribution. As an illustration consider the case, referred to above, where the products d2(yi+ + yi<,) are negligibly small compared with the terms zi; then zi+l‘i+l+i
On the and i=l
=
d,
+
‘i d*Si,i+l
.
the path 0 Q z d z, the xi decrease as i increases; therefore, owing to shape of the oxymyoglobin dissociation curve, the slopes si,i+l increase the ratios lcli/(zi+ 1- zi) decrease. As a consequence, the profile of xi, , . . . , n, as a function of zi, i = 1,. . . , n, is concave upwards.
THE DEPENDENCE THE PARAMETERS LOCAL AND GLOBAL
OF THE FACILITATED OF THE SYSTEM
TRANSPORT
ON
FEATURES
By using the above, the dependence of the facilitated transport on a given parameter can be analyzed. Assume that the solution has been obtained for a given set of boundary and parameter values. Increment the value p of a given parameter to p + Ap. Maintaining the xi and xi* fixed, approximate the new membrane resistances by evaluating the right-hand side of (2) for i = 1, . . . , n. Denote these approximations by partial yi*, and
JOSE M. GONZALEZ-FERNANDEZ TABLE 1 Basic Parameters CP
k' k
4 4 =1
x(O) x(2,)
7.50X lo-’ mol/cms 2.40 X 10’” cm3/(mol. s) 6.50~ 10 s-’ 1.50X 10m5 cm’/s 8.00~ 1OW’cm’/s 2.50 x 10V3 cm 1.35X 10-s mol/cm3 0.00” mol/cm3
For reference purposes this table lists the basic values of the physiological parameters that occur in red striated muscle; for a critical review and further references see [3], [4], and 161.The solubility coefficient for oxygen at 37°C is taken to be 1.35 x IO-” [5] mol/torr. “In studying the dependence of f/ on k’, the value of x(z,) was also set to 6.75 x lo-“’ mol/cm” (0.5 torr). 101
c
x 0
x’ c‘
7
-0
k’ X10-”
cm3 mol-’ s-’
FIG. 1. Dependence of the facilitated flow f, on the “on” rate constant k’. Solid line, fr; short-dashed line, E(x,)- E(x,); long-dashed line, F. A: _ff for k’= 2.40x 10”; B and C: partial fi and first iterate fr, respectively, for k’+ Ak’= 2.6OxlO’o. D: ff for k’ = 6.00 x lOlo; E and F: partial fr and first iterate j,, respectively, for k’ + A.k’= 6.50 x lot’. G: ff for k’ = 8.00~ 10”; H and I: partial f, and first iterate fr, respectively, for k’+ Ak’ = 9.00 x 10’~. The units of k’ are cm3/(mol. s). The values of the other parameters are listed in Table 1.
6.225 6.247 6.328 6.463 6.694 6.745 6.943 6.983 7.074 7.119 7.211
2.35
2.40 2.60
3.00
4.00
4.30 6.00 6.50
8.00 9.00
12.00
s)
2.554
2.339 2.398
2.068 2.205 2.241
2.041
1.939
The symbols
A- I correspond
to the stars plotted
1. in Figure
‘Original k’ = 8.0 X 10” cm3/(mol .s) The values of the other parameters are listed in Table
9.053
G 9.480 9.372
9.763 D 9.675 9.631
9.759
9.660
1.861 1.868 1.893
9.461
1.729 1.812
1.
H 9.567’
E 9.75gb
B 9.649a
x 10”
1.378 1.612
x10-3
A 9.482 9.556
8.811 9.269
6.005 7.932
ff [mol/(cm*~s)]
I-
Partial
and First Iterative
2
(cm/s)
Values for Partial
[mol/? cm*.s)] x 10”
k’= 2.4 X 10” cm3/(mol.s) k’ = 6.0 X 10” cm3/(mol.
5.678 6.045
1.50 2.00
aOriginal bOriginal
x 10’ 3.821 5.063
0.50 1.00
E(x 1
(mol/cm3j
E(x,)-
[cm3/Fmol. x lo-la
SI]
Computed
TABLE
First
2.337’
2.203b
I 9.418’
F 9.656b
c 9.559a
2.384’
2.233b
1.890a
x10-3
x 10”
x10-3
1.866a
r km/s>
ff tmol/km2~sll
iterate
lkm/s)
First iterate
Partial
Quantities
2 w
JO&
8
M. GONZALEZ-FERNANDEZ
their sum by partial I. Then approximate the new value of the facilitated flow by evaluating the right-hand side of (3) with I replaced by partial I, and denote this approximation by partial ff. The “partial” quantities give an estimate of the direct changes associated with the increment Ap of the parameter in question; they reflect local changes. Next, by using these partial yi* in expressions (5) and (61, approximate the new oxygen concentrations xi, i = 1,. . . , R - 1, at the membranes, and denote these approximations by first iterate xi, i = 2,. . . , II - 1. For these, solve the corresponding n - 1 boundary value problems on (zi, zi+i), i = 1,..*, n - 1, and by using (2) and (3) obtain further approximations to yi*, I, and fr, which will be denoted by first iterate yi* , first iterate I, and first iterate ff. These “first iterate” quantities reflect the secondary changes related to the modified values of xi, i = 2,. . . , n - 1. They express global properties associated with (61. The possibility of building up, with the above, an iterative algorithm to solve the general problem-thus the notation “first iterate” used here-will be taken up later in this paper. In what follows, the partial and first iterate quantities will be used to analyze the dependence of the facilitated transport on the parameters. Table 1 lists the basic parameter values used. For most of the considera-
x x10$
mol
cm-’
FIG. 2. Dependence of the approximate membrane resistance g(x) on the “on” rate constant k’. A: k’= 1.00~ 10”; B: k’= 2.40~ 10”; C: k’ = 5.00X 10”. The units of k’ are cm3/(mol.s). The values of the other parameters are listed in Table 1.
PARAMETER
DEPENDENCE
OF FACILITATED
TRANSPORT
9
tions that follow, the parameter values are taken from a physiological range. However, since the number of possible combinations of parameter values is very large, some selections were made for analyzing and illustrating the structure of the facilitated transport process. In what follows, each parameter value is changed over some range while the other parameters are kept constant. The numerical results analyzed were obtained by the algebraic method presented in [l]. They showed good agreement with the ones obtained by the continuum model, also described in [ll. For the discussion, it will be useful to estimate the behavior of the membrane resistance yi* by the associated function d’/2c g(x’
introduced
= d;“2(k’x
k’k
+ k)‘/‘(d;cp;‘k
+ d,(k’x
+ kj2] ‘I2 ’
in [l]. TABLE Dependence
3
of First Iterate
on k’
k’ [cm3/(mol
x lo4
2.4 x 10”
2.6 x 10’0
2.6 x lOlo
Solution
1st iterate
Solution
xi x 109 (mol/cm3)
(mol/cm3)
13.500
13.500
i
(cm)
xi x 109 (mol/cm3) 13.500
zi
.s)]
xi x 109
1
0.0000
2 3 4
4.1667 8.3333 12.5000
9.9823 6.9437 4.4621
9.9510 6.8859 4.3916
9.9330 6.8615 4.3729
5 6 7
16.6667 20.8333 25.0000
2.5357 1.0850 0.0000
2.4723 1.0470 0.0000
2.4647 1.0472 0.0000
k’ [cm3/(mol.s)]
2‘ x 104 i
(cm)
1 2 3 4 5 6 7
0.0000 4.1667 8.3333 12.5000 16.6667 20.8333 25.0000
6.0 x 10”
6.5 x 10”
6.5 x 10”
Solution
1st iterate
Solution
xi x 109
xi x 109 (mol/cm3)
xi x 109 (mol/cm3)
(mol/cm3) 13.500 9.492 6.073 3.497 1.784 0.712 0.000
13.500 9.487 6.050 3.454 1.741 0.676 0.000
13.500 9.461 6.010 3.422 1.727 0.674 0.000
(7)
10 DEPENDENCE ASSOCIATION
JO& M. GONZALEZ-FERNANDEZ OF THE FACILITATED RATE CONSTANT k’
FLOWff
ON THE OXYGEN-MYOGLOBIN
The solid line in Figure 1 shows the dependence of the facilitated flow on k’, for k’ between _5.00X109 and 1.20~ 10” cm3/(mol.s); the values of the other parameters listed in Table 1 are kept fixed. The transport path contains seven equally spaced membranes. As k’ increases, the facilitated flow increases for k’ < 4.30X lOlo cm3/(mol.s), and decreases for k’ larger. Both the boundary value difference E(x,)E(x,) and the sum of the membrane resistances T increase with k’. However, the increase of E(x,)E(x,) slows down as k’ grows. By using (3) it can be calculated that for k’ < 4.30 x 10” cm3/(mol* s) the increase in E(x,)E(x,) is large enough to balance the increase in r, so that the facilitated flow ff increases. For k’ > 4.30 X lOlo cm3/(mol.s), the increase in r dominates, so ff decreases. The dependence of E(x,)E(x,) on k’ is a consequence of the dependence of the oxymyoglobin dissociation curve on k’. To analyze the dependence of T on k’, the partial and first iterate quantities were computed for
I
I
10 k X10-l
20
s-’
FIG. 3. Dependence of the facilitated flow ff on the “off’ rate constant k. Solid line, f,-; short-dashed line, E(x,)E(x,); long-dashed line, r. A: fr for k = 6.50X 10; B and C: partial fr and first iterate fr, respectively, for k + Ak = 8.50 X 10. D: ff for k = 1.60 x 10’; E and F: partial f, and first iterate ff, respectively, for k + Ak = 1.80X 10’. The units of k are s- I. The values of the other parameters are listed in Table 1.
4.309 4.233
24.0 25.0
‘Original k = 1.60~10~ s-l.
“Original k = 6.50 x 10 s- I.
4.386
5.021 4.821
5.237
5.535 5.424
9.386 9.300
9.858 9.471
D 9.978
10.087 10.061
10.079
10.018
6.731
0.786 0.763
0.812
1.052 0.969
1.249 1.152
1.312
1.435
1.868 1.589
18.0 23.0
16.0
12.4 14.0
11.5
8.5 10.0
2.173
3.145 2.459
x 1o-3
A 9.482 9.874
5.0 6.5
7.042
x 10”
l(cm/s)
6.247 5.941
6.497
2.5 4.0
ff [mol/(cm2.s)]
8.388 8.942
x 10’ 6.963 6.676
x lo-’
%,IE(x,) (mol/cm3)
(cm/s)
[mol/(cm2.s)l
E 9.819b
B 9.727”
x 10”
0.978 b
1.62Sa
x10-3
Partial r
fJ
Partial
TABLE 4
F 9.845b
C 9.850a
x 10”
[mol/(cm2.s)l
fJ
1st iterate
0.972b
1.595”
x 10-3
(cm/d
r
1st iterate
12
JO& M. GONZALEZ-FERNANDEZ
several pairs k’, k’+ Ak’. The results are listed in Table 2 and plotted by stars in Figure 1. In each instance, partial I is almost equal to the original I, and therefore the increase in E(x,) - E(x,) results in partial ff (stars B, E, and H in Figure 1) larger than the original ff (stars A, D, and G, respectively). For k’ > 4.30~ 10” cm3/(mol.s), partial fZ fails to detect the decreasing phase of the facilitated transport. In each instance, first iterate I? is significantly larger than partial I, Table 2, so that from (3) the change in fZ, estimated by partial fZ, is attenuated or reversed. The approximation given by first iterate ff (stars C, F, and Z in Figure 1) is very close to the correct value of the facilitated flow for k’+ Ak’. The mechanism governing the above can be analyzed as follows. As k’ increases, the approximating membrane resistance, g(x), increases for small x and decreases for large x (see Figure 21, so that the sum partial I is about equal to the original sum I. On the other hand, the increase in k’ is accompanied by a decrease in the slope si of the associated oxymyoglobin dissociation curve for i small, and by an increase in si for i large. Then from (51, as k’ increases, tji increases for i small and decreases for i large. This leads to a larger nonuniformity of the interval resistance tji, i = 1,. . . ,
A
FIG. 4. Dependence of the approximate membrane resistance g(x) on the “off’ rate constant k. A: k = 4.00~10;B: k = 6.50X10; C: k = 1.50~10’. The units of k are s-l. The values of the other parameters are listed in Table 1.
PARAMETER DEPENDENCE OF FACILITATED TRANSPORT
13
n - 1, and hence, by (6), to a first iterate xi smaller than the original xi, for i=2 , . . . , n - 1, as is shown in Table 3. Since g(x), which approximates the behavior of yik, is a decreasing function of x (Figure 21, it can be expected that the decrease in the xi is associated with the increase of the yi* and thus of their sum, first iterate r. Therefore, a distinctive feature of the dependence of r on k’, which accounts for the decreasing phase of the facilitated flow ff observed above, results from the secondary change in the profile of xi, i = 1,. . . , n. DEPENDENCE DISSOCUTION
OF THE FACILITATED RATE CONSTANT k
FLOWff
ON THE OmOGLOBIN
For the values of the parameters listed in Table 1, the solid line in Figure 3 shows the dependence of ff on k when k varies between 2.5 x 10 and 2.5 X lo2 s-l. As k increases, fr increases for k < 1.2X 10’ s-l and decreases for k larger. Both E(x,)-- E(x,) and r decrease as k increases; by using (3) one finds that for k small the decrease of r is relatively more
I 3
d,
I 6
Xl O5
cm2
s-’
FIG. 5. Dependence of the facilitated flow fr on the oxygen diffusion coefficient d,. Solid line, fr. A: f, for d, = 1.50~ lo- 5; B and C: partial f, and first iterate f,, respectively, for d, + Ad, = 2.50X 10w5. The units of d, are cmz/s. The values of the other parameters are listed in Table 1.
14
JO& M. GONZALEZ-FERNANDEZ
pronounced than that of E(x,)E(x,), so that ff increases, and that for k large the reverse takes place. The mechanism that governs the dependence of the facilitated flow ff on k can be analyzed in a manner similar to that for the dependence of ff on k’. For two pairs k, k + Ak, the associated partial and first iterate quantities were computed; the results are listed in Table 4. The values of partial fr and first iterate ff are represented by stars in Figure 3. The associated g(x) functions are plotted in Figure 4. DEPENDENCE OF THE FACILITATED FLOWff CONSTANT k’ AND k, FOR FIXED RATIO k/k’
Let
K =
k/k’;
ON THE RATE
then c~J/~C~K[
‘1’2 d;‘2(
Xi +
d2Cp~ + d,( xi + K)3’2[
d2CpK
+
d,(
1’2
K)‘] Xi’
+
K)“]
.
For K constant, the value of k’ can be used as the variable parameter to examine the changes in fr. Here, the dependence of -yi* on k’ and on the
1001
‘0
-x
50
A .z m
FIG. 6. Dependence of the approximate membrane resistance g(x) on the oxygen diffusion coefficient d,. A: d, = 5.00~ 10W6; B: d, = 1.50X 10e5; C: d, = 7.00X 10-5. The units of d, are cm’/s. The values of the other parameters are listed in Table 1.
PARAMETER
DEPENDENCE
OF FACILITATED
1.5
TRANSPORT
xi is expressed by separate factors; hence, for each xi, as k’ increases yik decreases toward zero, and with it r. Also the term E(x,)E(x,) in (3) is constant. Therefore, as k’ increases, ff tends monotonically to the maximum possible value, d,[E(x,)E(x,)]/ zI. This corresponds to the attainment of chemical equilibrium everywhere. DEPENDENCE DIFFUSION
OF THE FACILITATED
COEFFICIENT
FLOWf/
ON THE OXYGEN
d,
For the values of the parameters listed in Table 1, the facilitated flow was found to increase monotonically with d, (Figure 5). The magnitude of the jump discontinuities of the oxymyoglobin concentration Y(Z) at the membranes (not shown) decreases as d, increases. This leads to the decrease in the A: in expression (1) with a resulting increase in the facilitated flow. The mechanism responsible for this will be examined next. For d, small, from (7), g(x) is 0(d;‘/2); therefore, the product of d, and r in the numerator of (3) is 0(di12>, and the denominator of (3) is O(d; ‘I*). Thus, partial ff is expected to be O(dl12> for d, small; this was
0
I
I
50
0 d2 X10’
100
cm2
s-’
FIG. 7. Dependence of the facilitated flow ff on the oxymyuglobin diffusion coefficient d,. Solid line, f,. A: frford, = 1.00X lo-‘; B and C: partial fJ and first iterate fr, respectively, for d, + Ad, = 8.00~ lo-‘. D: fi for d, = 8.00x lo-‘; E and F: partial fr and first iterate f,, respectively, for d, + Ad, = 3.00~ 10-6. The units of d, are cm’/s. The values of the other parameters are listed in Table 1.
16
JO&
M. GONZALEZ-FERNANDEZ
confirmed computationally for values of d, ranging from lo-l2 to 5.0~ lop8 cm*/s. For d, large, the product of d, and P in the numerator of (3) is O(l), and the denominator of (3) is 1+ O(d;‘). Thus for large d,, ff behaves as
d,(a - b)/(d, + c>> where a, b, and c are appropriate constants. This describes qualitatively the dependence of ff on d, shown in Figure 5. To analyze the dependence on d, for intermediate values of d,, partial fr and first iterate ff were computed for the pair d, = 1.5 x 10W5, d, + Ad, = 2.5 X lop5 cm2/s and represented in Figure 5 by stars B and C, respectively. Partial ff (star B) is larger than the original ff (A). As d, increases, the nonuniformity of the term d2si,i+l in (5) becomes relatively less important. Also, as Figure 6 indicates, the yi* are more uniform and so are (I~, i=l,..., IE. Therefore, from (6) the ni, i = 2,. . . , n - 1, increase. Since g(x) is monotonically decreasing, then, from (4), P decreases further, and from (3) first iterate ff is larger than partial ff; fr is larger at C than at B
x xlog FIG. 8. Dependence globin diffusion coefficient lo-‘; D: d, = 5.00~ lo-‘; the other
parameters
mol
cms3
of the approximate membrane resistance g(x) on the oxymyoC: d, = 8.00x d,. A: d, = 1.00x 10-9; B: d, = 5.00x lo-‘; E: d, = 5.00~ 10m4. The units of d, are cm*/s. The values of
are listed in Table
1.
PARAMETER
DEPENDENCE OF FACILITATED TRANSPORT
in Figure 5. Observe that first iterate fr essentially of the facilitated flow for d, = 2.5x 1O-5 cm’/s. DEPENDENCE OF THE FACILITATED FLOWff ON THE OXKMYOGLOBIN DIFFUSION COEFFICIENT
17
equals the correct value
d,
The facilitated flow as a function of d, is represented in Figure 7; ff increases monotonically with d,, and its rate of growth decreases as d, increases. The values assigned to the other parameters are listed in Table 1. For two pairs k, k + Ak, the associated partial fr and first iterate fr were computed and plotted in Figure 7. For different values of k, the g(x) functions were computed and plotted in Figure 8. By using (3)-(71, the dependence of fr on d, can be analyzed in a manner similar to that used above. The details are omitted. DISCUSSION It was shown in [l] that the myoglobin-facilitated transport with membranes in the transport path involves 1) homogeneous neighborhoods for the reaction diffusion equations with the associated chemical conductance k’x + k and the reference chemical equilibrium relation E(x), and 2) the interface condition of zero oxymyoglobin flow at each membrane with the associated eigenvalue Ai. For the case of prescribed oxygen concentrations at the boundaries, the description is completed by incorporating 3) a global relation governed by Equation (6) that determines the profile of the oxygen concentrations xi, i = 2,. . . , n - 1, along the transport path. In assessing the dependence of the facilitated transport on the parameters, the “partial” quantities incorporate 1) and 2); the “first iterate” quantities incorporate 3). The relations (31 and (6) obtained above can be used to analyze the interplay of the physicochemical entities involved in H-3) that result in the value of the facilitated transport. For the parameter values of Table 1, as either the association rate constant k’ or the dissociation rate constant k increases, the facilitated flow changes in a similar manner (solid lines in Figures 1 and 3). However, the interplays of the physicochemical entities involved are different: The changes both in the boundary oxygen concentration difference E(x,)E(x,) and in the sum of the membrane resistances r that result from the increase in k’ are opposite to those that result from the increase in k, dashed lines in Figures 1 and 3. Moreover, the dependencies of E(x,)- E(x,) and r on k’ are influenced by the values of xi and x,. The boundary value difference E(x,)E(x,) increases monotonically with the association rate constant k’ for x, = 0, but not necessarily for x, > 0; for x, > 0, E(x,)E(x,) eventually
18
JO&
M. GONZALEZ-FERNANDEZ
TABLE 5 k’ [cm3/(mol. x 10-10
E(q)91
(mol/cm3) x 107
0.50 1.00 1.50 2.00 2.35 2.40 2.60 3.00 4.00 5.00 6.00 8.00 10.00 12.00 X, = 6.75 X 10-‘”
Eb”)
mol/cm3;
ff [mol/kmZ~s)l x 10L’
3.451 4.357
5.517 7.096
4.667 4.755 4.753 4.750 4.734 4.681 4.493 4.278 4.064 3.671 3.335 3.050
7.699 7.928 7.978 7.981 7.982 7.950 7.760 7.508 7.244 6.736 6.282 5.885
r km/s) x 10-3 1.307 1.446 1.469 1.458 1.442 1.439 1.428 1.404 1.338 1.270 1.205 1.085 0.981 0.892
the values of the other parameters are listed in
Table 1.
decreases to zero. Computations were performed for x, = 6.75 x 10P1’ mol/cm3 (0.5 torr) (see Table 5). The dependence of the facilitated flow ff on k’ is biphasic, like that for x, = 0; however, the behaviors of ,5(x,)I&X,,) and of r are, for the decreasing phase of ff, opposite the ones found for X, = 0. The decreasing phase of ff is due, for x, = 0 mol/cm3, to the preponderance of the increase in I’ over the increase in E(x,)E(x,), but for x, = 6.75 x lo-” mol/cm3 it is due to the preponderance of the decrease in E(x,) - E(x,) over the decrease in r. This illustrates that, for different sets of parameter values, the resulting dependences of the facilitated flow ff on k’ may be qualitatively similar, but the interplay of the physicochemical entities involved may be different. Expression (3) helps to discriminate between those differences. When the oxygen diffusion coefficient d, increases, the relative availability of the oxymyoglobin channel, expressed by rui*, diminishes. Since the existence of facilitated flow is predicated on the availability of the combined oxygen channel, it is prima facie unexpected that the decrease in the partition coefficient ai* is associated with the increase in facilitated flow fr. The antinomy is resolved by observing that here the decrease in ai* tends to limit the increase in the flow transfers and thus in the costs A: at the membranes. The increase of d, also decreases the eigenvalue Ai associated with the ith interface. Thus the numerator of the flow transfer resistance pi decreases. This results in the decrease in the cost A: -the
PARAMETER
DEPENDENCE
OF FACILITATED
TRANSPORT
19
oxymyoglobin concentration becomes closer to the equilibrium value at the membranes-and from (1) the facilitated transport increases. The transfers between the oxygen and the oxymyoglobin flows depend pointwise on the values of the chemical conductance, k’x + k; yet the increase in the oxygen diffusion coefficient results in a decrease in the flow transfer resistance. The geometrical character of this interface feature is related to the increase in the characteristic length l/A, of the boundary layers next to the ith membrane, so that more space is available for the corresponding flow transfers. With the increase in the myoglobin diffusion coefficient d2, the membrane resistance yi* increases owing to the increase in the flow partition cyi*, which predominates over the decrease in the transfer flow resistance pi Thus, the jump discontinuities in the oxymyoglobin concentration at the membranes increase; the oxymyoglobin becomes farther away from equilibrium in the intervals next to the membranes. This leads to a decrease in the second factor on the right-hand side of (1). This effect, however, is overridden by an increase in the first factor, d,, with a resulting increase in the facilitated flow ff. Therefore, the increases in the facilitated flow brought about by the increase in either the oxygen diffusion or the myoglobin diffusion coefficient (compare Figures 5 and 7) result from almost opposite changes in the intervening physicochemical entities. A NUMERICAL
APPLICATION
In this report expressions (3) and (6) have been used to analyze the numerical results obtained with the algebraic or continuum methods presented in [l]. Motivated by the satisfactory approximations obtained with the use of first iterate ff, as shown in Figure 1, 3, 5, and 7, an m-iteration approach was tried for a number of combinations of parameter values. In general, three to five iterations provided a satisfactory approximation. This reduces the computation times compared with the methods used in [l], because finding the mth-iteration solution involves solving IZ- 1 algebraic subproblems with only two unknowns each, and iterations replace the computation of Jacobians and the solving of large linear problems.
REFERENCES 1
2
J. M. Gonzalez-Fernandez, The transfers between the free and combined oxygen flows in determining the facilitated transport with membranes on the transport path, Math. Biosci. 95:209-231 (1989). J. M. Gonzalez-Fernandez and S. E. Atta, Facilitated transport of oxygen in the presence of membranes in the diffusion path, Biqhys. J. 38:133-141 (1982).
20 3 4 5
6
JOSE M. GONZALEZ-FERNANDEZ J. A. Jacquez, The physiological role of myoglobin: more than a problem in reaction-diffusion kinetics, Math. Biosci. 68:57-97 (1984). F. Kreuzer and L. J. C. Hoofd, Facilitated diffusion of oxygen in the presence of hemoglobin, Respir. Physiol. 8:280-302 (1970). J. Sendroy, Jr., R. T. Dillon, and D. D. VanSlyke, Studies of gas and electrolyte equilibria in blood. XIX. The solubility and physical state of combined oxygen in blood, J. Biol. Chem. 105:597-632 (1934). J. B. Wittenberg, Myoglobin-facilitated oxygen diffusion: role of myoglobin in oxygen entry into muscle, Physiol. Rev. 50:559-636 (1970).
JOSfi M. GONZALEZ-FERNANDEZ Mathematical Research Branch, National Institute of Diabetes, Digestive and Kidney Diseases, National Institutes of Health, Bethesda, Maryland 20892 Received I2 January 1989; revised 30 October 1989
ABSTRACT Some physicochemical entities involved in the facilitated transport of oxygen along a transport path zt < z < z, with membranes impermeable to myoglobin at zjr i = 1,. . , n, were identified in an earlier paper [Math. Biosci. 95:209 (198911. These entities are the partition between the oxygen and oxymyoglobin flows, the flow transfers taking place near a membrane, and the membrane resistance. Expressions for these entities were obtained that explicitly involve the parameters of the system. In this paper, for the case of prescribed boundary oxygen concentrations x’ and x,, these expressions are incorporated into (i) an explicit representation for the facilitated transport value in terms of the difference, E(x’)E(x,), between the boundary oxymyoglobin equilibrium values and the sum, f, of the membrane resistances, and (ii) a representation for the distribution of the membrane oxygen concentrations x, at zi, i = 2,. . , n - 1. This makes it possible to analyze the manner in which the facilitated transport depends on the parameters. For a physiological range of parameter values, the facilitated transport was found to increase as either the oxygen-myoglobin association rate constant k’, the dissociation rate constant k, the oxygen diffusion coefficient, or the oxymyoglobin diffusion coefficient increases. Thus, the facilitated transport does not depend directly on ratios of chemical and diffusion coefficients. Although the increase in the oxygen diffusion coefficient does not per se affect the chemical conductance, it diminishes the membrane resistance through an interface feature, with a resulting increase in the facilitated transport. For a larger range of values of k’ and k, the dependences of the facilitated transport on k’ and on k are both biphasic and are similar in shape. However, the mechanisms involved are different: The associated changes in E(x’)E(n,) and in F that result from the increase in k’ are opposite to those that result from an increase in k. The use of(i) and (ii) permits, also, discrimination between the different roles of the physicochemical entities involved in a given facilitated transport change. In some cases (e.g., the decreasing phase of the facilitated transport as k’ increases), this change depends in an essential manner on a secondary modification of the profile x,, i = 1,. , n, along the transport path.
M4THEMATICAL
BIOSCIENCES
lOO:l-20
(1990)
Published by Elsevier Science Publishing Co., Inc., 1990 655 Avenue of the Americas, New York, NY 10010
1
0025-5564/90/$00.00
2
JO& M. GONZALEZ-FERNANDEZ
INTRODUCTION In a previous paper [l], an analysis of the myoglobin-facilitated transport of oxygen was carried out for the case that membranes impermeable to myoglobin are present along the transport path. It was assumed that a region of near chemical equilibrium exists between, but away from, the membranes. That analysis identified some physicochemical entities having a local character, namely, the flow partition between the free and combined oxygen flows and the membrane flow transfer resistance. The flow transfers are associated with jump discontinuities in the oxymyoglobin concentration at the membranes; their cumulative effect is one of the determinants of the extent of the facilitation of transport. Mathematical expressions for these entities were obtained that explicitly contain the parameters of the problem. In this paper, for the case that the oxygen concentrations at the boundaries of the transport path are prescribed, those results will be incorporated into a representation for facilitated transport. Also, a set of relations that determine the profile along the transport path of the oxygen concentration at the membranes will be obtained. With these expressions the dependence of the facilitated transport on the parameters will be analyzed. THE STRUCTURE
OF THE FACILITATED
TRANSPORT
On the space interval 0 ( z G z,, let the free oxygen concentration x(z), the oxymyoglobin concentration y(z), and the reduced myoglobin concentration c,, - y be defined. The constant c/, stands for the total concentration of reduced plus combined myoglobin. These species are able to diffuse and interact chemically in a reversible manner with each other. The diffusion coefficient for oxygen and that for oxymyoglobin are denoted by d, and d,, respectively. It is assumed that membranes impermeable to myoglobin exist at zi, i = l,..., II, where z, = 0, z, = z,, and zi < z~+~; let xi =x(2,), i = l,..., n. The rate constant for the association of oxygen with myoglobin and that for the dissociation of oxymyoglobin are denoted by k’ and k, respectively. The interface condition of zero myoglobin and oxymyoglobin flows is assumed to hold at the membranes. The boundary values x(0) and x(2,) are prescribed. The system is in the time-independent state. The facilitated transport ff is defined to equal f, - f,,f, where f, is the total flow of oxygen and f,,f = d,(x, - x,)/z, is the nonfacilitated flow. To avoid repetition, the reader is referred to [ll for the remaining terminology and results used below. TRANSFERS
OF FLOWS IN INTERVALS
NEXT TO THE MEMBRANES
The existence of nonzero facilitated transport implies the sharing of the total flow f, between the free oxygen flow f&) and the oxymyoglobin flow
PARAMETER
DEPENDENCE
OF FACILITATED
TRANSPORT
3
f,(z). It was shown in [I] that the sharing is effected through transfers between fd(z) and f,(z). The flow transfer that takes place in a right interval, (z;, z, + g[,>, [left interval, (z, - oI, zi)] of the membrane at z, results in a difference A,+ (A;> between the left (right) side limit of y(z) and the reference equilibrium oxymyoglobin concentration E(x) at zl; these differences have been called costs because they subtract from the possible maximum facilitated flow [2]. Let xi* = xj(zi _+ai). For reference purposes, explicit representations for the flow partition a,*, the eigenvalue A, associated with the interface condition at zi, and the flow transfer resistance pi obtained in [l] are collected here. Thus d,c,k’k
ay’ = d,c,k’k
+ dl( k’x’
+ k)* ’
&=hi/(k’x,+k).
The flow transfer on the interval (zi,zi + o,> is approximated by the and that on (zi - cr,, zj> by f,~;. It was shown that approxiproduct fin:, mately A: = The following representation
ff=4
f,a’&.
for the facilitated
E(x,)-E(x,)-(A;
flow
+El;-‘A: =1
Define the membrane
resistance
and
was obtained
in [2]:
+A;) (1)
yi* by
y;f = ar+&. Since A: = fry,’
ff
(2)
f, = f,,r + fr,
f =d E(x,)-E(x,)-d,[(x,-x,)/zllr f 2 zl + d2T
(3)
n-1
r= y: +
c yj++r,. 2
(4)
4
JO&
M. GONZALEZ-FERNANDEZ
Expression (3) gives the facilitated flow in terms of the difference E(x,)Hx,,) obtained by evaluating the equilibrium oxymyoglobin concentration for the prescribed boundary values, the nonfacilitated flow d,(x, - x,,)/z, and the sum I of the membrane resistances 7;‘. The term -yi* incorporates the flow partition CY;*,the chemical conductance k’x, + k, and the eigenvalue A,. Therefore, (3) can be used to analyze how the dependence of the facilitated transport on the parameters is mediated through changes in these physicochemical entities. The right-hand side of (3) contains the unknowns xi. Therefore it will be desirable to obtain another relationship that involves the xi. An expression that relates the profile of xi, i = 1,. . ., n, to the above entities for the prescribed boundary values X, and x,, will be developed next. THE DISTRIBUTION OF THE OXYGEN CONCENTRATION, xi, i = 1,. .,n, ALONG THE TRANSPORT PATH
By specializing
(1) to an interval (zi, z;+,), we can write the total flow f,
as
r,=d,-*‘;;
+d,
Introduce
+YLl)ft
E(Xi)-E(xi+l)-(Yi+
zi+l
I
-
zi
the notation
which represents the slope of the secant to the oxymyoglobin dissociation curve between the points that correspond to xi and xi+t, and solve for f, to obtain
h=(xi-xi+l)
d, + d2Si,i+ 1 Zi+l_Zi+d2(YT +YiGl) ’
The second factor on the right-hand side of this expression defines the interval conductance for (zi, zi+ ,). It can be shown that the terms yi* tend fixed; then the interval to zero as k’ and k tend to 00, with k/k’ conductance tends to
d, + dZSi,i+l zi+I-zi
.
The numerator of this term expresses the additive nature of the free and combined oxygen flows and the role of the product of d, and the slope of the oxymyoglobin dissociation curve in the facilitation of the transport
PARAMETER
DEPENDENCE
OF FACILITATED
5
TRANSPORT
(Wittenberg [6]). The interval conductance generalizes this by taking into account the terms yi* that allow for the incorporation of departures from chemical equilibrium. Next, for (z;, zi+ ,), define the interval resistance Gi to be the inverse of the above conductance. Thus, +; =
*i+l
-Zi+d2(Yi+ dl
++Yi+l)
+
dZSi,i+*
’
and
The global relation xi-xi+l
=(x1-x
5
) n
i=l
cj”&
,...,a-1,
’
is then obtained. Therefore, the concentration drop x, - xi+t for the subinterval (zi, zi+il equals the prescribed boundary concentration difference x1 - x, times the fractional resistance for this interval. The greater the nonuniformity of J)~, i=l , . . . , n - 1, the greater the departure of xi, i = 1,. . ., n, from a straightline distribution. As an illustration consider the case, referred to above, where the products d2(yi+ + yi<,) are negligibly small compared with the terms zi; then zi+l‘i+l+i
On the and i=l
=
d,
+
‘i d*Si,i+l
.
the path 0 Q z d z, the xi decrease as i increases; therefore, owing to shape of the oxymyoglobin dissociation curve, the slopes si,i+l increase the ratios lcli/(zi+ 1- zi) decrease. As a consequence, the profile of xi, , . . . , n, as a function of zi, i = 1,. . . , n, is concave upwards.
THE DEPENDENCE THE PARAMETERS LOCAL AND GLOBAL
OF THE FACILITATED OF THE SYSTEM
TRANSPORT
ON
FEATURES
By using the above, the dependence of the facilitated transport on a given parameter can be analyzed. Assume that the solution has been obtained for a given set of boundary and parameter values. Increment the value p of a given parameter to p + Ap. Maintaining the xi and xi* fixed, approximate the new membrane resistances by evaluating the right-hand side of (2) for i = 1, . . . , n. Denote these approximations by partial yi*, and
JOSE M. GONZALEZ-FERNANDEZ TABLE 1 Basic Parameters CP
k' k
4 4 =1
x(O) x(2,)
7.50X lo-’ mol/cms 2.40 X 10’” cm3/(mol. s) 6.50~ 10 s-’ 1.50X 10m5 cm’/s 8.00~ 1OW’cm’/s 2.50 x 10V3 cm 1.35X 10-s mol/cm3 0.00” mol/cm3
For reference purposes this table lists the basic values of the physiological parameters that occur in red striated muscle; for a critical review and further references see [3], [4], and 161.The solubility coefficient for oxygen at 37°C is taken to be 1.35 x IO-” [5] mol/torr. “In studying the dependence of f/ on k’, the value of x(z,) was also set to 6.75 x lo-“’ mol/cm” (0.5 torr). 101
c
x 0
x’ c‘
7
-0
k’ X10-”
cm3 mol-’ s-’
FIG. 1. Dependence of the facilitated flow f, on the “on” rate constant k’. Solid line, fr; short-dashed line, E(x,)- E(x,); long-dashed line, F. A: _ff for k’= 2.40x 10”; B and C: partial fi and first iterate fr, respectively, for k’+ Ak’= 2.6OxlO’o. D: ff for k’ = 6.00 x lOlo; E and F: partial fr and first iterate j,, respectively, for k’ + A.k’= 6.50 x lot’. G: ff for k’ = 8.00~ 10”; H and I: partial f, and first iterate fr, respectively, for k’+ Ak’ = 9.00 x 10’~. The units of k’ are cm3/(mol. s). The values of the other parameters are listed in Table 1.
6.225 6.247 6.328 6.463 6.694 6.745 6.943 6.983 7.074 7.119 7.211
2.35
2.40 2.60
3.00
4.00
4.30 6.00 6.50
8.00 9.00
12.00
s)
2.554
2.339 2.398
2.068 2.205 2.241
2.041
1.939
The symbols
A- I correspond
to the stars plotted
1. in Figure
‘Original k’ = 8.0 X 10” cm3/(mol .s) The values of the other parameters are listed in Table
9.053
G 9.480 9.372
9.763 D 9.675 9.631
9.759
9.660
1.861 1.868 1.893
9.461
1.729 1.812
1.
H 9.567’
E 9.75gb
B 9.649a
x 10”
1.378 1.612
x10-3
A 9.482 9.556
8.811 9.269
6.005 7.932
ff [mol/(cm*~s)]
I-
Partial
and First Iterative
2
(cm/s)
Values for Partial
[mol/? cm*.s)] x 10”
k’= 2.4 X 10” cm3/(mol.s) k’ = 6.0 X 10” cm3/(mol.
5.678 6.045
1.50 2.00
aOriginal bOriginal
x 10’ 3.821 5.063
0.50 1.00
E(x 1
(mol/cm3j
E(x,)-
[cm3/Fmol. x lo-la
SI]
Computed
TABLE
First
2.337’
2.203b
I 9.418’
F 9.656b
c 9.559a
2.384’
2.233b
1.890a
x10-3
x 10”
x10-3
1.866a
r km/s>
ff tmol/km2~sll
iterate
lkm/s)
First iterate
Partial
Quantities
2 w
JO&
8
M. GONZALEZ-FERNANDEZ
their sum by partial I. Then approximate the new value of the facilitated flow by evaluating the right-hand side of (3) with I replaced by partial I, and denote this approximation by partial ff. The “partial” quantities give an estimate of the direct changes associated with the increment Ap of the parameter in question; they reflect local changes. Next, by using these partial yi* in expressions (5) and (61, approximate the new oxygen concentrations xi, i = 1,. . . , R - 1, at the membranes, and denote these approximations by first iterate xi, i = 2,. . . , II - 1. For these, solve the corresponding n - 1 boundary value problems on (zi, zi+i), i = 1,..*, n - 1, and by using (2) and (3) obtain further approximations to yi*, I, and fr, which will be denoted by first iterate yi* , first iterate I, and first iterate ff. These “first iterate” quantities reflect the secondary changes related to the modified values of xi, i = 2,. . . , n - 1. They express global properties associated with (61. The possibility of building up, with the above, an iterative algorithm to solve the general problem-thus the notation “first iterate” used here-will be taken up later in this paper. In what follows, the partial and first iterate quantities will be used to analyze the dependence of the facilitated transport on the parameters. Table 1 lists the basic parameter values used. For most of the considera-
x x10$
mol
cm-’
FIG. 2. Dependence of the approximate membrane resistance g(x) on the “on” rate constant k’. A: k’= 1.00~ 10”; B: k’= 2.40~ 10”; C: k’ = 5.00X 10”. The units of k’ are cm3/(mol.s). The values of the other parameters are listed in Table 1.
PARAMETER
DEPENDENCE
OF FACILITATED
TRANSPORT
9
tions that follow, the parameter values are taken from a physiological range. However, since the number of possible combinations of parameter values is very large, some selections were made for analyzing and illustrating the structure of the facilitated transport process. In what follows, each parameter value is changed over some range while the other parameters are kept constant. The numerical results analyzed were obtained by the algebraic method presented in [l]. They showed good agreement with the ones obtained by the continuum model, also described in [ll. For the discussion, it will be useful to estimate the behavior of the membrane resistance yi* by the associated function d’/2c g(x’
introduced
= d;“2(k’x
k’k
+ k)‘/‘(d;cp;‘k
+ d,(k’x
+ kj2] ‘I2 ’
in [l]. TABLE Dependence
3
of First Iterate
on k’
k’ [cm3/(mol
x lo4
2.4 x 10”
2.6 x 10’0
2.6 x lOlo
Solution
1st iterate
Solution
xi x 109 (mol/cm3)
(mol/cm3)
13.500
13.500
i
(cm)
xi x 109 (mol/cm3) 13.500
zi
.s)]
xi x 109
1
0.0000
2 3 4
4.1667 8.3333 12.5000
9.9823 6.9437 4.4621
9.9510 6.8859 4.3916
9.9330 6.8615 4.3729
5 6 7
16.6667 20.8333 25.0000
2.5357 1.0850 0.0000
2.4723 1.0470 0.0000
2.4647 1.0472 0.0000
k’ [cm3/(mol.s)]
2‘ x 104 i
(cm)
1 2 3 4 5 6 7
0.0000 4.1667 8.3333 12.5000 16.6667 20.8333 25.0000
6.0 x 10”
6.5 x 10”
6.5 x 10”
Solution
1st iterate
Solution
xi x 109
xi x 109 (mol/cm3)
xi x 109 (mol/cm3)
(mol/cm3) 13.500 9.492 6.073 3.497 1.784 0.712 0.000
13.500 9.487 6.050 3.454 1.741 0.676 0.000
13.500 9.461 6.010 3.422 1.727 0.674 0.000
(7)
10 DEPENDENCE ASSOCIATION
JO& M. GONZALEZ-FERNANDEZ OF THE FACILITATED RATE CONSTANT k’
FLOWff
ON THE OXYGEN-MYOGLOBIN
The solid line in Figure 1 shows the dependence of the facilitated flow on k’, for k’ between _5.00X109 and 1.20~ 10” cm3/(mol.s); the values of the other parameters listed in Table 1 are kept fixed. The transport path contains seven equally spaced membranes. As k’ increases, the facilitated flow increases for k’ < 4.30X lOlo cm3/(mol.s), and decreases for k’ larger. Both the boundary value difference E(x,)E(x,) and the sum of the membrane resistances T increase with k’. However, the increase of E(x,)E(x,) slows down as k’ grows. By using (3) it can be calculated that for k’ < 4.30 x 10” cm3/(mol* s) the increase in E(x,)E(x,) is large enough to balance the increase in r, so that the facilitated flow ff increases. For k’ > 4.30 X lOlo cm3/(mol.s), the increase in r dominates, so ff decreases. The dependence of E(x,)E(x,) on k’ is a consequence of the dependence of the oxymyoglobin dissociation curve on k’. To analyze the dependence of T on k’, the partial and first iterate quantities were computed for
I
I
10 k X10-l
20
s-’
FIG. 3. Dependence of the facilitated flow ff on the “off’ rate constant k. Solid line, f,-; short-dashed line, E(x,)E(x,); long-dashed line, r. A: fr for k = 6.50X 10; B and C: partial fr and first iterate fr, respectively, for k + Ak = 8.50 X 10. D: ff for k = 1.60 x 10’; E and F: partial f, and first iterate ff, respectively, for k + Ak = 1.80X 10’. The units of k are s- I. The values of the other parameters are listed in Table 1.
4.309 4.233
24.0 25.0
‘Original k = 1.60~10~ s-l.
“Original k = 6.50 x 10 s- I.
4.386
5.021 4.821
5.237
5.535 5.424
9.386 9.300
9.858 9.471
D 9.978
10.087 10.061
10.079
10.018
6.731
0.786 0.763
0.812
1.052 0.969
1.249 1.152
1.312
1.435
1.868 1.589
18.0 23.0
16.0
12.4 14.0
11.5
8.5 10.0
2.173
3.145 2.459
x 1o-3
A 9.482 9.874
5.0 6.5
7.042
x 10”
l(cm/s)
6.247 5.941
6.497
2.5 4.0
ff [mol/(cm2.s)]
8.388 8.942
x 10’ 6.963 6.676
x lo-’
%,IE(x,) (mol/cm3)
(cm/s)
[mol/(cm2.s)l
E 9.819b
B 9.727”
x 10”
0.978 b
1.62Sa
x10-3
Partial r
fJ
Partial
TABLE 4
F 9.845b
C 9.850a
x 10”
[mol/(cm2.s)l
fJ
1st iterate
0.972b
1.595”
x 10-3
(cm/d
r
1st iterate
12
JO& M. GONZALEZ-FERNANDEZ
several pairs k’, k’+ Ak’. The results are listed in Table 2 and plotted by stars in Figure 1. In each instance, partial I is almost equal to the original I, and therefore the increase in E(x,) - E(x,) results in partial ff (stars B, E, and H in Figure 1) larger than the original ff (stars A, D, and G, respectively). For k’ > 4.30~ 10” cm3/(mol.s), partial fZ fails to detect the decreasing phase of the facilitated transport. In each instance, first iterate I? is significantly larger than partial I, Table 2, so that from (3) the change in fZ, estimated by partial fZ, is attenuated or reversed. The approximation given by first iterate ff (stars C, F, and Z in Figure 1) is very close to the correct value of the facilitated flow for k’+ Ak’. The mechanism governing the above can be analyzed as follows. As k’ increases, the approximating membrane resistance, g(x), increases for small x and decreases for large x (see Figure 21, so that the sum partial I is about equal to the original sum I. On the other hand, the increase in k’ is accompanied by a decrease in the slope si of the associated oxymyoglobin dissociation curve for i small, and by an increase in si for i large. Then from (51, as k’ increases, tji increases for i small and decreases for i large. This leads to a larger nonuniformity of the interval resistance tji, i = 1,. . . ,
A
FIG. 4. Dependence of the approximate membrane resistance g(x) on the “off’ rate constant k. A: k = 4.00~10;B: k = 6.50X10; C: k = 1.50~10’. The units of k are s-l. The values of the other parameters are listed in Table 1.
PARAMETER DEPENDENCE OF FACILITATED TRANSPORT
13
n - 1, and hence, by (6), to a first iterate xi smaller than the original xi, for i=2 , . . . , n - 1, as is shown in Table 3. Since g(x), which approximates the behavior of yik, is a decreasing function of x (Figure 21, it can be expected that the decrease in the xi is associated with the increase of the yi* and thus of their sum, first iterate r. Therefore, a distinctive feature of the dependence of r on k’, which accounts for the decreasing phase of the facilitated flow ff observed above, results from the secondary change in the profile of xi, i = 1,. . . , n. DEPENDENCE DISSOCUTION
OF THE FACILITATED RATE CONSTANT k
FLOWff
ON THE OmOGLOBIN
For the values of the parameters listed in Table 1, the solid line in Figure 3 shows the dependence of ff on k when k varies between 2.5 x 10 and 2.5 X lo2 s-l. As k increases, fr increases for k < 1.2X 10’ s-l and decreases for k larger. Both E(x,)-- E(x,) and r decrease as k increases; by using (3) one finds that for k small the decrease of r is relatively more
I 3
d,
I 6
Xl O5
cm2
s-’
FIG. 5. Dependence of the facilitated flow fr on the oxygen diffusion coefficient d,. Solid line, fr. A: f, for d, = 1.50~ lo- 5; B and C: partial f, and first iterate f,, respectively, for d, + Ad, = 2.50X 10w5. The units of d, are cmz/s. The values of the other parameters are listed in Table 1.
14
JO& M. GONZALEZ-FERNANDEZ
pronounced than that of E(x,)E(x,), so that ff increases, and that for k large the reverse takes place. The mechanism that governs the dependence of the facilitated flow ff on k can be analyzed in a manner similar to that for the dependence of ff on k’. For two pairs k, k + Ak, the associated partial and first iterate quantities were computed; the results are listed in Table 4. The values of partial fr and first iterate ff are represented by stars in Figure 3. The associated g(x) functions are plotted in Figure 4. DEPENDENCE OF THE FACILITATED FLOWff CONSTANT k’ AND k, FOR FIXED RATIO k/k’
Let
K =
k/k’;
ON THE RATE
then c~J/~C~K[
‘1’2 d;‘2(
Xi +
d2Cp~ + d,( xi + K)3’2[
d2CpK
+
d,(
1’2
K)‘] Xi’
+
K)“]
.
For K constant, the value of k’ can be used as the variable parameter to examine the changes in fr. Here, the dependence of -yi* on k’ and on the
1001
‘0
-x
50
A .z m
FIG. 6. Dependence of the approximate membrane resistance g(x) on the oxygen diffusion coefficient d,. A: d, = 5.00~ 10W6; B: d, = 1.50X 10e5; C: d, = 7.00X 10-5. The units of d, are cm’/s. The values of the other parameters are listed in Table 1.
PARAMETER
DEPENDENCE
OF FACILITATED
1.5
TRANSPORT
xi is expressed by separate factors; hence, for each xi, as k’ increases yik decreases toward zero, and with it r. Also the term E(x,)E(x,) in (3) is constant. Therefore, as k’ increases, ff tends monotonically to the maximum possible value, d,[E(x,)E(x,)]/ zI. This corresponds to the attainment of chemical equilibrium everywhere. DEPENDENCE DIFFUSION
OF THE FACILITATED
COEFFICIENT
FLOWf/
ON THE OXYGEN
d,
For the values of the parameters listed in Table 1, the facilitated flow was found to increase monotonically with d, (Figure 5). The magnitude of the jump discontinuities of the oxymyoglobin concentration Y(Z) at the membranes (not shown) decreases as d, increases. This leads to the decrease in the A: in expression (1) with a resulting increase in the facilitated flow. The mechanism responsible for this will be examined next. For d, small, from (7), g(x) is 0(d;‘/2); therefore, the product of d, and r in the numerator of (3) is 0(di12>, and the denominator of (3) is O(d; ‘I*). Thus, partial ff is expected to be O(dl12> for d, small; this was
0
I
I
50
0 d2 X10’
100
cm2
s-’
FIG. 7. Dependence of the facilitated flow ff on the oxymyuglobin diffusion coefficient d,. Solid line, f,. A: frford, = 1.00X lo-‘; B and C: partial fJ and first iterate fr, respectively, for d, + Ad, = 8.00~ lo-‘. D: fi for d, = 8.00x lo-‘; E and F: partial fr and first iterate f,, respectively, for d, + Ad, = 3.00~ 10-6. The units of d, are cm’/s. The values of the other parameters are listed in Table 1.
16
JO&
M. GONZALEZ-FERNANDEZ
confirmed computationally for values of d, ranging from lo-l2 to 5.0~ lop8 cm*/s. For d, large, the product of d, and P in the numerator of (3) is O(l), and the denominator of (3) is 1+ O(d;‘). Thus for large d,, ff behaves as
d,(a - b)/(d, + c>> where a, b, and c are appropriate constants. This describes qualitatively the dependence of ff on d, shown in Figure 5. To analyze the dependence on d, for intermediate values of d,, partial fr and first iterate ff were computed for the pair d, = 1.5 x 10W5, d, + Ad, = 2.5 X lop5 cm2/s and represented in Figure 5 by stars B and C, respectively. Partial ff (star B) is larger than the original ff (A). As d, increases, the nonuniformity of the term d2si,i+l in (5) becomes relatively less important. Also, as Figure 6 indicates, the yi* are more uniform and so are (I~, i=l,..., IE. Therefore, from (6) the ni, i = 2,. . . , n - 1, increase. Since g(x) is monotonically decreasing, then, from (4), P decreases further, and from (3) first iterate ff is larger than partial ff; fr is larger at C than at B
x xlog FIG. 8. Dependence globin diffusion coefficient lo-‘; D: d, = 5.00~ lo-‘; the other
parameters
mol
cms3
of the approximate membrane resistance g(x) on the oxymyoC: d, = 8.00x d,. A: d, = 1.00x 10-9; B: d, = 5.00x lo-‘; E: d, = 5.00~ 10m4. The units of d, are cm*/s. The values of
are listed in Table
1.
PARAMETER
DEPENDENCE OF FACILITATED TRANSPORT
in Figure 5. Observe that first iterate fr essentially of the facilitated flow for d, = 2.5x 1O-5 cm’/s. DEPENDENCE OF THE FACILITATED FLOWff ON THE OXKMYOGLOBIN DIFFUSION COEFFICIENT
17
equals the correct value
d,
The facilitated flow as a function of d, is represented in Figure 7; ff increases monotonically with d,, and its rate of growth decreases as d, increases. The values assigned to the other parameters are listed in Table 1. For two pairs k, k + Ak, the associated partial fr and first iterate fr were computed and plotted in Figure 7. For different values of k, the g(x) functions were computed and plotted in Figure 8. By using (3)-(71, the dependence of fr on d, can be analyzed in a manner similar to that used above. The details are omitted. DISCUSSION It was shown in [l] that the myoglobin-facilitated transport with membranes in the transport path involves 1) homogeneous neighborhoods for the reaction diffusion equations with the associated chemical conductance k’x + k and the reference chemical equilibrium relation E(x), and 2) the interface condition of zero oxymyoglobin flow at each membrane with the associated eigenvalue Ai. For the case of prescribed oxygen concentrations at the boundaries, the description is completed by incorporating 3) a global relation governed by Equation (6) that determines the profile of the oxygen concentrations xi, i = 2,. . . , n - 1, along the transport path. In assessing the dependence of the facilitated transport on the parameters, the “partial” quantities incorporate 1) and 2); the “first iterate” quantities incorporate 3). The relations (31 and (6) obtained above can be used to analyze the interplay of the physicochemical entities involved in H-3) that result in the value of the facilitated transport. For the parameter values of Table 1, as either the association rate constant k’ or the dissociation rate constant k increases, the facilitated flow changes in a similar manner (solid lines in Figures 1 and 3). However, the interplays of the physicochemical entities involved are different: The changes both in the boundary oxygen concentration difference E(x,)E(x,) and in the sum of the membrane resistances r that result from the increase in k’ are opposite to those that result from the increase in k, dashed lines in Figures 1 and 3. Moreover, the dependencies of E(x,)- E(x,) and r on k’ are influenced by the values of xi and x,. The boundary value difference E(x,)E(x,) increases monotonically with the association rate constant k’ for x, = 0, but not necessarily for x, > 0; for x, > 0, E(x,)E(x,) eventually
18
JO&
M. GONZALEZ-FERNANDEZ
TABLE 5 k’ [cm3/(mol. x 10-10
E(q)91
(mol/cm3) x 107
0.50 1.00 1.50 2.00 2.35 2.40 2.60 3.00 4.00 5.00 6.00 8.00 10.00 12.00 X, = 6.75 X 10-‘”
Eb”)
mol/cm3;
ff [mol/kmZ~s)l x 10L’
3.451 4.357
5.517 7.096
4.667 4.755 4.753 4.750 4.734 4.681 4.493 4.278 4.064 3.671 3.335 3.050
7.699 7.928 7.978 7.981 7.982 7.950 7.760 7.508 7.244 6.736 6.282 5.885
r km/s) x 10-3 1.307 1.446 1.469 1.458 1.442 1.439 1.428 1.404 1.338 1.270 1.205 1.085 0.981 0.892
the values of the other parameters are listed in
Table 1.
decreases to zero. Computations were performed for x, = 6.75 x 10P1’ mol/cm3 (0.5 torr) (see Table 5). The dependence of the facilitated flow ff on k’ is biphasic, like that for x, = 0; however, the behaviors of ,5(x,)I&X,,) and of r are, for the decreasing phase of ff, opposite the ones found for X, = 0. The decreasing phase of ff is due, for x, = 0 mol/cm3, to the preponderance of the increase in I’ over the increase in E(x,)E(x,), but for x, = 6.75 x lo-” mol/cm3 it is due to the preponderance of the decrease in E(x,) - E(x,) over the decrease in r. This illustrates that, for different sets of parameter values, the resulting dependences of the facilitated flow ff on k’ may be qualitatively similar, but the interplay of the physicochemical entities involved may be different. Expression (3) helps to discriminate between those differences. When the oxygen diffusion coefficient d, increases, the relative availability of the oxymyoglobin channel, expressed by rui*, diminishes. Since the existence of facilitated flow is predicated on the availability of the combined oxygen channel, it is prima facie unexpected that the decrease in the partition coefficient ai* is associated with the increase in facilitated flow fr. The antinomy is resolved by observing that here the decrease in ai* tends to limit the increase in the flow transfers and thus in the costs A: at the membranes. The increase of d, also decreases the eigenvalue Ai associated with the ith interface. Thus the numerator of the flow transfer resistance pi decreases. This results in the decrease in the cost A: -the
PARAMETER
DEPENDENCE
OF FACILITATED
TRANSPORT
19
oxymyoglobin concentration becomes closer to the equilibrium value at the membranes-and from (1) the facilitated transport increases. The transfers between the oxygen and the oxymyoglobin flows depend pointwise on the values of the chemical conductance, k’x + k; yet the increase in the oxygen diffusion coefficient results in a decrease in the flow transfer resistance. The geometrical character of this interface feature is related to the increase in the characteristic length l/A, of the boundary layers next to the ith membrane, so that more space is available for the corresponding flow transfers. With the increase in the myoglobin diffusion coefficient d2, the membrane resistance yi* increases owing to the increase in the flow partition cyi*, which predominates over the decrease in the transfer flow resistance pi Thus, the jump discontinuities in the oxymyoglobin concentration at the membranes increase; the oxymyoglobin becomes farther away from equilibrium in the intervals next to the membranes. This leads to a decrease in the second factor on the right-hand side of (1). This effect, however, is overridden by an increase in the first factor, d,, with a resulting increase in the facilitated flow ff. Therefore, the increases in the facilitated flow brought about by the increase in either the oxygen diffusion or the myoglobin diffusion coefficient (compare Figures 5 and 7) result from almost opposite changes in the intervening physicochemical entities. A NUMERICAL
APPLICATION
In this report expressions (3) and (6) have been used to analyze the numerical results obtained with the algebraic or continuum methods presented in [l]. Motivated by the satisfactory approximations obtained with the use of first iterate ff, as shown in Figure 1, 3, 5, and 7, an m-iteration approach was tried for a number of combinations of parameter values. In general, three to five iterations provided a satisfactory approximation. This reduces the computation times compared with the methods used in [l], because finding the mth-iteration solution involves solving IZ- 1 algebraic subproblems with only two unknowns each, and iterations replace the computation of Jacobians and the solving of large linear problems.
REFERENCES 1
2
J. M. Gonzalez-Fernandez, The transfers between the free and combined oxygen flows in determining the facilitated transport with membranes on the transport path, Math. Biosci. 95:209-231 (1989). J. M. Gonzalez-Fernandez and S. E. Atta, Facilitated transport of oxygen in the presence of membranes in the diffusion path, Biqhys. J. 38:133-141 (1982).
20 3 4 5
6
JOSE M. GONZALEZ-FERNANDEZ J. A. Jacquez, The physiological role of myoglobin: more than a problem in reaction-diffusion kinetics, Math. Biosci. 68:57-97 (1984). F. Kreuzer and L. J. C. Hoofd, Facilitated diffusion of oxygen in the presence of hemoglobin, Respir. Physiol. 8:280-302 (1970). J. Sendroy, Jr., R. T. Dillon, and D. D. VanSlyke, Studies of gas and electrolyte equilibria in blood. XIX. The solubility and physical state of combined oxygen in blood, J. Biol. Chem. 105:597-632 (1934). J. B. Wittenberg, Myoglobin-facilitated oxygen diffusion: role of myoglobin in oxygen entry into muscle, Physiol. Rev. 50:559-636 (1970).