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A mathematical analysis of carrier-facilitated diffusion
A Mathematical Analysis of Carrier-Facilitated ROBERT
W. KOLKKA*
Center for the Applicution Lehigh
University,
AND
ERIC P. SALATHe
of Muthematics.
Bethlehem,
Received 17 November
Diffusion
Pennsyh~ania
18015
1983; revised 10 Mu), 1984
ABSTRACT Carrier-facilitated matched asymptotic the small facilitation in the analysis,
1.
diffusion in a one dimensional slab is analyzed. The method of expansions is used to obtain solutions in both the near-equilibrium and limit. Consumption of the diffusing species within the slab is included
so that the solutions
are applicable
to situations
of physiological
interest.
INTRODUCTION
Carrier-facilitated, or mediated, diffusion is a reaction-diffusion phenomenon in which mass transport processes are enhanced by chemical reaction. The potential biological significance of facilitated diffusion was recognized by Wittenberg [16] and Scholander [14], as a result of experiments on the diffusion of oxygen through solutions of hemoglobin and myoglobin. There are also important applications in chemical engineering, including, for example, commercial membrane separation processes. In all cases a carrier molecule is contained within the medium in which transport occurs, and the substrate molecule combines reversibly with this carrier. Transport of the substrate is therefore a result of free diffusion and of diffusion by the complexed carrier molecule. A reversible chemical reaction with stoichiometry expressed by k+ nu+pw+
mv,
(1.1)
k-
in which fi moles of the substrate (I and p moles of the uncomplexed carrier W combine to form Fii moles of the complexed carrier I/, is typical of the reactions that occur in most applications. *Present address: Technical University. MATHEMATICAL
Department of Mathematics Houghton, MI 49981. BIOSCIENCES
71:147-179
TElsevier Science Publishing Co.. Inc., 19X4 52 Vanderbilt Ave., New York. NY 10017
and
Computer
Sciences,
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(1984)
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0025.5564/X4/$03.00
148
ROBERT W. KOLKKA
AND ERIC P. SALATHI?
Early work in chemical engineering by Olander [12] examined mediated diffusion in a membrane, under the assumption that the reaction (1.1) remained in chemical equilibrium. This assumption is applicable if the characteristic diffusion time for the system is much larger than the characteristic reaction time for the reaction (1.1). In chemical engineering terms, this is referred to as the large Damkbhler number limit. Since the diffusion time is proportional to the membrane thickness, the near-equilibrium limit is frequently referred to as “thick film theory.” Goddard, Schultz, and Basset (21 subsequently recognized that the near-equilibrium limit was a singular perturbation problem, in which nonequilibrium boundary layers exist near the surfaces of the membrane. They discussed systems corresponding to the reaction (1.1) in which only the substrate U was transferred, the carrier species W and P’ being confined to the membrane. The method of matched asymptotic expansions was used to determine the concentrations of the various species as a function of position within the membrane. A review of facilitated diffusion in membrane technology has been given by Goddard, Schultz, and Suchdeo [3,4]. The study of facilitated diffusion in biological systems developed independently of the work of chemical engineering. Wyman [17] examined the transport of oxygen by hemoglobin and myoglobin in order to explain the observations of Wittenberg and Scholander on facilitated diffusion, and derived a differential equation governing the concentration of oxygen within the solution. An analysis of this equation under the assumption of chemical equilibrium was given by Murray [8]. Because of the singular nature of the problem in the near-equilibrium limit, Murray was not able to satisfy the condition of zero flux for the carrier molecule at the surface of the hemoglobin or myoglobin solution. A singular perturbation solution to Wyman’s equation in the near-equilibrium limit has been given by Rubinow and Dembo [13]. Nedelman and Rubinow [ll] subsequently provided the analysis when the forward reaction rate is very fast compared to the reverse reaction, the “large affinity regime.” In this case the singular perturbation analysis of Rubinow and Dembo breaks down, but the solution can be obtained as a regular perturbation expansion. Numerical solutions to the problem have been given by Kutchai et al. [7] using a quasilinearization technique, and by Gonzalez-Femandez and Atta [5] using a piecewise linear technique. It has been conjectured that myoglobin present in skeletal muscle facilitates the diffusion of oxygen from the blood capillary into the tissue, where consumption of oxygen occurs. For this reason, an understanding of the mechanism of carrier-mediated diffusion is of considerable interest to physiologists. Murray [9] extended his analysis of Wyman’s equation for a one dimensional slab to diffusion into a cylinder of tissue. Again, he considered only the equilibrium case, and so did not satisfy the boundary conditions for the myoglobin concentration at the cylindrical surface. Murray included
ANALYSIS OF CARRIER-FACILITATED
DIFFUSION
149
oxygen consumption in his equations, and for sufficiently large values of the consumption rate found that regions of oxygen depletion occurred within the cylinder. The mathematical treatment of the problem consisted of increasing consumption until negative oxygen values were obtained in the core. Since negative concentrations cannot occur, this was interpreted as a region of zero concentration. Therefore, the concentration profiles obtained do not have zero slope at the border of the anoxic region, and so do not represent the correct solution. The problem must be solved so that when the concentration falls to zero it has zero derivative at that location, and therefore zero flux of oxygen into the anoxic region. An analysis using these “Stefan” boundary conditions was subsequently given by Murray [lo]. This study also was restricted to the equilibrium case, so the boundary conditions at the surface of the slab were not satisfied. In the present paper, one dimensional diffusion through a slab is analyzed. It is assumed that the substrate reacts reversibly with the carrier molecule according to Equation (l.l), and that the substrate may be consumed within the slab. The extent of the depletion zones that develop for large consumption rates is investigated. There are two parameters of fundamental importance in the analysis. The first, E, is the reciprocal of the Damkiihler number, and measures the departure from chemical equilibrium. The second, 6, is the ratio of the flux transported by the carrier to that transported by free diffusion. Two different solutions are obtained, one as an asymptotic expansion for small 6 and one as an asymptotic expansion for small E. Both expansions are singular, and the methods of matched asymptotic expansions are used. The small 6 expansion is applicable to situations in which large departures from equilibrium occur, and has not been considered before. The small E expansion extends previous analyses, so that the effects of small departures from equilibrium can be determined when consumption occurs. This is of particular importance in physiological applications.
2.
ANALYSIS
A one dimensional slab occupying the region 0 d .Cd I contains a diffusable species of molar concentration a( .?). This substrate is maintained at the surfaces 2 = 0 and 5~= I at the uniform concentrations uA and us, respectively, and within the slab is consumed (or produced) at a constant rate fi. Diffusion of the substrate is facilitated by the presence within the slab of a generally large carrier molecule. The molar concentration of the uncomplexed carrier molecule is denoted by iv(Z), and of the complexed carrier molecule by D( 2). We denote by P( t, 0, %) the time rate of production of i+i moles per unit volume of the complexed carrier molecules and the rate of consumption of ?i moles per unit volume of the substrate or of p moles per
150
ROBERT
W. KOLKKA
AND ERIC P. SALATHE
volume of uncomplexed carrier molecules. It is assumed that each of the three types of molecules is freely diffusable and that the carrier molecule is confined within the slab. It therefore follows that the concentrations satisfy the following differential equations and boundary conditions: unit
(2.1) (2.2) (2.3)
k(O) =
di2 0
-=
di
dti -= d2
ii(I) =
UA,
0
UB)
(2.4)
at
i = 0, I,
(2.5)
at
.%=O,I,
(2.6)
where D,,, D,,, D,. are the diffusivities of the three types of molecules. Since the carrier molecule is confined to the slab, the total amount, in both complexed and uncomplexed form, must remain constant. Therefore, (2.7) where u is a constant. the boundary condition
Note that integrating (2.5) yields
Equation
1.
J0 F(ic,D,i!)d,t=O,
(2.2) and employing
(2.8)
the condition of “global equilibrium.” We assume that the reaction rate is of the form $‘(6,6,fi)
=k+fj”@Ppk-[,“‘,
(2.9)
so that n = p = m = 1 corresponds to the physiologically important case of myoglobin-facilitated oxygen diffusion. Arbitrary values of m , n, p are included in the analysis, so that the results are applicable to other than bimolecular reactions. Although Equation (2.9) does not adequately describe the kinetics of hemoglobin-oxygen reactions, such reactions have been approximated by the bimolecular case m = n = p = 1. A thorough discussion of hemoglobin-oxygen reactions has been given in a recent review by Jacquez
[61.
ANALYSIS
OF CARRIER-FACILITATED
151
DIFFUSION
Introducing the normalization u = O/u,, v = C/a, w = iv/a, ?# = uB/ uA yields the following nondimensional equations: d2u Em = F(u,v,w)+~M,
x = .%/I,
(2.10)
2
(2.11)
~t$$=-F(u,v,w),
(2.12)
u(0) =l,
u(1) = @,
(2.13)
v’(0) = 0,
v’(1) = 0,
(2.14)
w’(0) = 0,
w’(1) = 0,
(2.15) (2.16)
jol(;+;)dx=l,
where
F(u,
iiD,,a/iiiD,uA,
v, w) = u”wp - Ku” and E = DU/ii12ktu~p’oP, a,, = a,,, = iiD,a/jiD,u,, M = A%12/uADU, K = k-cTP/k+u;.
This system can be simplified by observing that adding Equations (2.12) integrating twice, and employing the boundary conditions (2.15) yields an equation for w(x) in terms of v(x):
4, w(x) =A - ,v(x). From Equation
(2.11), (2.14),
(2.17)
(2.16), it follows that (2.18)
For small substrate molecules and large carrier molecules, D,, and Dw are approximately equal, and Equation (2.18) reduces to A = jj. Equation (2.17) would then have the symmetric form v/iii + w/p =l. Using Equation (2.17) to eliminate w(x) yields EU”+f(U,V)=EM,
(2.19)
E&V”-f(U,V)=o,
(2.20)
where (2.21)
f( u, v) = Ku”’ - u”(h - vu)” and
6 = I?,, v = S,/&.
The solution
to Equations
(2.19)-(2.21)
must be
ROBERT W. KOLKKA
152
AND ERIC I’. SALATHk
obtained in terms of X, and the results then substituted into Equation (2.18) to determine X. If the backward reaction rate is as fast as or faster than the forward reaction rate, it follows from Equation (2.9) that the characteristic reaction time is given by T, =(iikidmlo")'.Since the characteristic diffusion time for the substrate molecule is T, = 1*/D,, it is clear that F = T,/T,,the ratio of the diffusion to the reaction time scale. Also, the parameter 8 in Equation (2.20) is proportional to D,,u/D~,u,~, and so is a measure of the amount of substrate that can be transported by the carrier molecule to the amount that can be transported through simple diffusion. 2.1.
THE SMALL
FACILIT4
TION LIMIT.
MODEL
I
In this subsection the solution to Equations (2.19)-(2.21) will be obtained for the case in which the total quantity of solute transported is increased by only a small amount as a result of facilitation. This corresponds to the limit S -+ 0, and the solution can be obtained as an asymptotic expansion in 6. It turns out that the expansion is singular, and that there are boundary layers at x = 0 and at x = 1. In the core region, bounded away from x = 0, the expansion has the form
u(x)-
u~~(.~)+s1”u,(x)+su*(x)+o(s3”), (2.22)
D(X) - L~~~(x)+s’~“~,(x)+sli~(x)+o(8~‘~). Substituting this expansion into Equations (2.19), (2.20) and equating like powers in 8l/* yields a sequence of equations from which the various terms in this expansion can be determined. The leading terms zlO(x), ~a( x) satisfy u;; = M and f( uO, ~1~)= 0. Therefore, u0 = M,x’/~+ A,x + B, and Q = P’(uo), where V( N,,) is the inverse of (2.23)
Note that the solution for uU represents simple diffusion, with consumption, through the slab, and that to this order the substrate is in equilibrium with the carrier molecule. The terms of order B1/* give Ul
-
0,
h(X) = where
f”(U”, %I f,,(uo,uo)
(2.24)
h(X),
f, = df/au, f,, = df/dv. Therefore, pi = A,x + B,. The constants A,,,
ANALYSIS OF CARRIER-FACILITATED
153
DIFFUSION
A,, B,,, and B, cannot be determined from the boundary conditions on u, since the expansion assumed here is not valid in the neighborhood of x = 0 and x = 1. In fact, ua and vi do not satisfy the no-flux condition cl’= 0 at the boundaries of the slab, for any choice of A,, A,, B,, and B,.It is necessary to construct different expansions for the regions near x = 0 and x = 1. Matching them to the above expansion will determine the constants A,, A,, B,, and B,.The nature of the expansions near x = 0,l (the inner expansions or boundary layers) will be briefly described here. Higher order terms in the expansion bounded away from x = 0,l (the outer expansion) and in the boundary layers at x = 0 and x = 1 are presented in the appendix. In order to analyze the solution near x = 0, it is necessary to introduce the boundary layer variable z = .x/#‘~, which has the property that x -+ 0 as 6 + 0 with z fixed. In terms of this variable, Equations (2.19), (2.20) become
&iiz;+8f(ii,6)=&SM,
(2.25)
&fizz-f(C,G)=O,
(2.26)
where fi( z) = u( 6112z), 6 (z) = v ( a’/2z). Assuming form
a(z)-
an inner expansion
n,(Z)+s”‘ii,(Z)+0(s),
of the
(2.27)
a(Z)-fi,(z)+C%i(Z)+0(6), substituting this into Equations (2.25), (2.26), and equating order in #I2 yields the equations Goz_= 0,
terms of similar
(2.28)
&fiOzz-f(&,QJ)=O
(2.29)
and iilr; = 0,
&Dlrl-fU(~O,~O)~l-fr,(~O,~O)~l=O. Equations
(2.30) (2.31)
(2.28) and (2.30) have the solutions
ii,=&,+ Ii,=A,z+B,.
B,,
(2.32) (2.33)
The boundary conditions give G,(O) = 1, C,(O) = 0, C&O) = 0, C13(0) = 0. Therefore, &, = 1 and El = 0. Matching of the boundary layer solution to the outer solution is achieved by substituting x = #I2 z into the Equations (2.22) and expanding in powers
154
ROBERT
W. KOLKKA
AND ERIC P. SALATHk
of f3”z: u - us(O) + IV2 { =;,(O)+tl,(O)}+O(6), (2.34) c’- U,](O)+ W{ Comparing Z’DO
this with the inner
zU;,(o)+U,(o)}+o(6). expansion,
Equation
(2.27), shows that as
n,(z)
+ U”(0) = B”,
(2.35)
b,(z)
+ U,(O),
(2.36)
n,(z)
+ zu;j(o)+
U,(O) = A,z + B,,
~,(=)-z/~;,(o)+u,(o).
(2.37) (2.38)
It therefore follows that A,, = 0, B,, = 1, A, = A,, and B, = 0. The solution to Equations (2.29) and (2.31) subject to the boundary and matching conditions is fiO= V(l), and
fi1(Z)=-&“2 _
fu(l>V(l))
A,crp[ _{T;]
cm~ w>)13’2
fu(l, V(l))
A
z,
(2.39)
f,.(l? V(l)) (I
Two constants, A, and A,, remain unknown. These are determined when the boundary layer solution at x =l is analyzed. Introducing the boundary layer variable .$= (1 - .~)/8”‘~, which has the property x + 1 as 6 + 0. [ fixed, the boundary layer expansion at x = 1 is assumed to have the form u(~)=uo(~)+s”‘ui(~)+o(s), L’(~)=L$)(‘$)+W%i([)+o(8),
(2.40)
where U(t) = u(l- S’/‘[), G(t) = z~(l- S’/‘,$). Substituting this expansion into Equations (2.19), (2.20), rewritten in terms of the boundary layer variable 6, yields a sequence of equations from which u,,, Q, pi, c~i, etc. can be determined. The procedure is analogous to the analysis of the boundary layer at x = 0. The equations for ‘A”,Ui have the solutions a0 = A06 + B,, U, = x1.$ + B,. The boundary conditions at x = 1 give B, = %, 3, = 0, while the matching conditions give (2.41)
uo + U”(l), Ui + - ‘$ub(l)+
U,(l)
(2.42)
asE+so.Fromtheseitfollowsthat&=O, A,,=%-l-M/2, A,=%-1 + M/2, A, = 0. The solutions for co, 8, are obtained in a manner analogous
ANALYSIS OF CARRIER-FACILITATED
to the determinations
DIFFUSION
155
of Co, fii, and are found to be
Uo=V(@),
(2.43)
(2.44)
This completes the solution to order 6 1/2 in the central core region and in both boundary layers. Higher order terms can readily be obtained by continuing the analysis outlined here. The details are provided in the appendix. 2.2.
THE NEAR-EQUILIBRIUM
LIMIT,
MODEL
II
In this section the solution to Equations (2.19)-(2.21) will be obtained in the limit e -+ 0, corresponding to the case where the reaction between substrate and carrier remains near equilibrium. Since E is inversely proportional to I*, small E is frequently referred to as “ thick film theory” and large E as “ thin film theory.” As in the previous case, the expansion in terms of e is singular, and boundary layers must be constructed at x = 0 and x = 1 to complete the solution. In the region bounded away from x = 0 and x = 1, the solution can be expanded in the form
u(x)- Uo(x)+d’2U1(X)+O(&), 4x) - uo(x) + Pq( x) + O( E).
(2.45)
Substituting this expansion into Equation (2.20) shows that the leading terms satisfy f( aa, ~a) = 0, so that u,, = V( uo), as before. Adding Equations (2.19) and (2.20) and substituting the expansion (2.45) into the resulting equation gives [u. + 6V( uo)]“=
M.
(2.46)
Since the inner expansions are not valid at x = 0, L, the boundary conditions cannot be applied to the terms of this expansion. Boundary layer solutions must be constructed at x = 0 and x = L that satisfy the boundary conditions, and the inner expansions must be matched to these solutions. However, it was found in the last section that the leading term in the inner expansion for u(x), u,(x), did satisfy the boundary conditions at x = 0 and L, although this was not the case for U(X). The analysis in this section is considerably simplified by trying a solution for us(x) that satisfies the
156
ROBERT W. KOLKKA AND ERIC I’. SALATHk
boundary
conditions uo(O) =l,
uo(l)
= q.
(2.47)
Justification for this assumption follows from the ability to construct innner and outer solutions that can be matched, and that satisfy the appropriate equations and boundary conditions. It therefore follows that
U() +
SV( zq))=
F
+(4-1+s[v(“z/)-li(l)]-~).~+I+sv(1). (2.48)
which can readily be inverted satisfy the equations
to yield ug( x). The higher order terms,
U, , ~1, ,
(2.49)
(2.50)
Therefore,
(2.51)
Boundary conditions at x = 0 and x = 1 cannot be applied to determine the constants A, and B,, since the expansion (2.45) is not applicable in the neighborhood of the boundaries. In addition, L’,) and ~1~ do not satisfy the boundary condition of zero flux at x = 0,l for any choice of the constants A,, B,. It is necessary, as before, to construct different expansions in those regions and to join them to the expansion developed here for the core region. The boundary layer solution near x = 0 is obtained by defining the boundary layer variables z = x/E’/~, ii(z) = U( F’/~z), r?(z) = u( F’/‘z). In terms of these variables, Equations (2.19), (2.20) become
(2.52)
Clearly,
the boundary
layer solutions
are unaffected
by M until order
F.
ANALYSIS
Introducing
OF CARRIER-FACILITATED
the boundary
DIFFUSION
157
layer expansions
ii(Z) -1-r &"'k,(Z)+o(E), (2.53) a(z)
- V(l)+&“%,(Z)+O(&),
and substituting these into the boundary equations for 1, and r?,:
where f,,, = f,[l,
V(l)], f,,O =f,,[l,
V(l)]. The boundary
8, (0) = 0, and the matching
requirement
as z + 00. Equations
layer equations
61: (0)
= 0,
yields the following
conditions
are (2.56)
gives
a,(z)
- zr&(O)+ Ui(O),
a,(z)
- z,;,(O)+ u,(O),
(2.54)-(2.57)
(2.57)
yield the solution
(2.58)
(2.59)
and the result that (2.60)
where7 = KfdWl
- ~fuo/f,d1'~2. The boundary layer analysis at z = 1 proceeds in exactly the same manner. Introducing the boundary layer variables 5 = (1- X)/E’/‘, ii([) = U(1 - P[), U(5) = u(l- P2 I), the expansion in the region near z = 1 has the form ti([)
-
q-t
E1’2ii,([)+@E)!
(2.61) u(~)-V(q+&“2u,(~)+O(&).
ROBERT W. KOLKKA AND ERIC P. SALATHk
158
It can be shown that pi and ui are given by (2.62)
(2.63)
where ful = f,[udl), udl)l~ .f,.l= .fJud), ~+AlN, and SfUl/f,,i)]“‘. the result
The matching
7 = [G/6)(1 of this solution with the outer solution also yields
Higher order terms in both the outer and the two inner expansions obtained in the appendix.
2.3.
DEPLETION
arc
ZONES
For sufficiently large values of the consumption rate, there is a region within the slab in which the substrate concentration is zero. When such a depletion zone is present, the solutions obtained above are not valid. The governing equations must be solved subject to the condition that at the boundaries of the depletion zone both the value and the derivative of the substrate and the carrier concentration vanish. If the depletion zone is located between the values x = L and x =lR, then the substrate and carrier concentrations must satisfy the Stefan boundary conditions
u(L) =u(L)=O,
(2.65)
u'(L)=v'(L)=O and u(l-R)=v(l-R)=O, u’(l-
R)=v’(l-
(2.66) R)=O.
The additional boundary conditions do not over determine the system, because now the unknowns L and R must be found as part of the solution. The governing equations are Equations (2.19)-(2.21), and the boundary conditions at x = 0 and x =l are given by Equations (2.13) and (2.14). The
ANALYSIS
OF CARRIER-FACILITATED
DIFFUSION
159
consumption rate is assumed to be a constant, M, when the substrate concentration is nonzero, and to drop discontinuously to zero when u = 0. Different singular perturbation expansions must be developed for the region to the left and to the right of the depletion zone. The analysis will be presented for the left side, and only the results will be given for the right side. The outer expansion, valid away from the boundary layer and up to the boundary of the depletion zone, has the form of Equation (2.22) for model I and of Equation (2.45) for model II. It is not necessary to construct a boundary layer at the boundary of the depletion zone, since the vanishing of u and U’ assures that both u and u’ will also vanish, as can be seen from the outer solutions obtained in Sections 2.1 and 2.2. The leading order equilibrium solutions for model II are
u,(x)+sv(u,(x))=~ +Ax+l+W(l),
(2.67) (2.68)
00(x) = l+%(X))~ where the constant A is to be determined (2.65). The first order corrections are
by the Stefan boundary
conditions
1’ q(x) =-;;;oj;; 07 0u,(x>, u,(x)=(Bx+C)
fll(Uo,uo) -l
l-6
L~(U,, 00)
[
where the constants B and C are to be determined by the Stefan conditions, and by the analysis of the boundary layer. The inner expansions have the form (2.53), and the solutions are given by (2.58), (2.59), with A
u;l(O) =
1+ 6V’(l) The inner solutions
show that the constant
C is related to the constant
(2.71)
A by (2.72)
At this point all that remains to complete the solution is to determine the values of the unknown constants A and B, and the location of the depletion zone boundary point L. The value of L can be expressed as a perturbation expansion : L-
Lo+&1’2L1+O(&),
(2.73)
160 so
ROBERT W. KOLKKA AND ERIC P. SALATHk
that the Equations
(2.65) can be expanded,
to order
F"',
in the form
Uo(L,,)+&“?[L,U;)(LO)+UI(LO)l
=o,
(2.74)
u;,(L,,)+E’,‘?[LIU;;(LO)+U;(LO)]
=o.
(2.75)
It then follows that
(2.76) A = - \,‘2M[l+
SV(I)]
,
fj-F,
(2.77) (2.78)
,’
It can be shown in a similar manner
that for the right hand side,
The expressions (2.76) and (2.79) are asymptotically valid approximations only if L,,, R,, # O(E”‘). When L,,, R,, = O(F’/~). the depletion zone has “invaded” the boundary layer, and the previous analysis is not appropriate. The critical value for the onset of depletion occurs when L = 1~ R. This condition gives A4, =2(1+%)+4[1+6V(l)]‘~Q?+SV(~)] +2G[V(I)+Y(*)]-?~~~-~i+O(r).
(2.80)
where A4, is the consumption rate for which the substrate concentration falls to zero at some point in the interior of the slab. For M < M, the solution obtained in the previous section is applicable. For M > M, , finite regions of zero concentration occur and the solutions obtained in this section must be used. The otcurrence of depletion zones is a consequence of the assumption that consumption rate is constant for nonzero concentration, and falls discontinuously to zero when concentration is zero. It is known, however, that although oxygen consumption in tissue is uniform for normal values of concentration, for very low values the consumption rate drops rapidly, falling continuously to zero. For such a continuous drop to zero in consumption rate there is no sharp border separating oxygenated and anoxic regions. However. an analysis by Berger et al. [l] using Michaelis-Menton kinetics showed that in the
ANALYSIS
OF CARRIER-FACILITATED
DIFFUSION
161
limiting case in which consumption remained nearly constant until very low concentrations were reached, the solution was very close to the solution corresponding to the step function consumption rate assumed here. In the absence of facilitation, Rogers (see Berger et al. [l]) has proved that the solution with step function consumption can be obtained as a limit from the solution with continuously varying consumption rate. Therefore, the step function variation assumed for the consumption rate should result in an accurate description of the concentration profiles for many situations where the consumption rate varies continuously. Similar conclusions were reached by Taylor and Murray [15] after examining the case where the consumption rate is of the Michaelis-Menton form.
3.
RESULTS
AND DISCUSSIONS
A composite expansion, uniformly valid throughout the slab, can be obtained from the perturbation solutions for the central core and for each of the boundary layers. This is done by adding the three separate expansions together and subtracting any part that is common to more than one region. For the small facilitation model, the composite expansion for the substrate concentration u(x) is given, to order 6, by
u,(x)=u,+8
1’2u, + 62.4,+1+
8wi, + 6B2
- {1+ 81’2[ 24i(o)+ z~;,(0)]+6[u*(0)+zu;(0)+~z2u;;(0)]} +%+$1’2u
+6u
-~“+s’/‘;ul(l~-~~;(I~]+s[u,(I)-bu;o+i~’u;~l)]}, (3.1) with a similar expression for the carrier concentration v(x). Analogous results can be written for the near-equilibrium model. Clearly, these formulas can be considerably reduced when the appropriate solution is substituted for each of the terms appearing in them. The total flux through the slab is given in nondimensional terms by J=
u’(x)+sL~‘(x).
(3.2)
If there is no consumption, J must be constant. This can easily be proved by adding Equations (2.19) and (2.20). Therefore, since u’ = 0 at x = 0 or x = 1, the total normalized flux for M = 0 is J = u’(O) = u’(l). The facilitated flux is by definition the difference between the flux when 6 = 0 and the flux when 6 # 0.
162
ROBERT
W. KOLKKA
AND ERIC I’. SALATHI?
The derivative of v at x = 0 and x = 1 vanishes in the boundary layer solutions. However, this no flux condition is only approximately satisfied by the composite solution. When the solution is obtained to order 6 for model I, it can be seen that v:(O) = i3u;(O), with a similar result at x =l. The contribution of this inaccuracy to the computation of the flux is an indication of the error involved in the perturbation solution. Defining e,. = S’P;/( u: + 8~:) for model I, and analogously e,, = E~u;( ui. + 6~:) for model II, the values e,.(O) and e,.(l) provide an effective error measure. For model I. the composite expansion and the outer expansion for N are identical However, for model II, it is desirable to introduce an additional error measure, analogous to that for 11above, that determines the extent to which the slopes of the composite and inner solutions differ at .Y= 0 and x = 1. Clearly, this can be done by defining eU = &US/( u: + 6~~‘). and examining the values e,,(O) and e,,(l).
0.8
0.6
5 3 0.4
0.2
0 0
0.2
0.4
0.6
0.8
1.0
x FIG. 1.
Substrate
and S = 0.0.1.
concentration
variation
for model
I, with
F = 1.0. K = 1.0. @ = 0.
ANALYSIS
OF CARRIER-FACILITATED
163
DIFFUSION
Figures 1 and 2 show the effect of varying 6 from 0 to 0.1 in model I, for F = 1.0, K = 1.0, and % = 0, when there is no consumption (M = 0).In these and subsequent figures the broken line represents the leading order solution. All numerical results shown include the higher order terms obtained in the appendix, and so correspond to m = n = p = 1. The concentration u(x) is independent of 6 to order 6, and Figure 1 shows the small influence of 6 on the u(x) profile. However, the profiles for u(x) are strongly dependent on 6, as illustrated in Figure 2. An alternate expression for the total flux can be obtained by integrating Equation (3.2) from x = 0 to x =l. Subtracting the portion of the flux that would exist in the absence of facilitation gives the normalized facilitated flux,
.$=S[rJ(l)-U(O)].
(3.3)
1. .O.-
0. 8-
0. 65 > 0. 4-
0. 2’
00
0.2
0.4
0.6
0.8
1.0
x FIG. 2. Complexed carrier concentration S = 0.01, (3) 6 = 0.001. and (4) 6 = 0.
variation
for model
I. with (1) 6 = 0.1, (2)
164
ROBERT
W. KOLKKA
AND
ERIC P. SALATHE
This flux was computed both using values for r,(O), cl(l) from the inner expansions and using the composite expansion. The two methods give virtually identical results. For q = 0, IJ,I varies nearly linearly from 0 to about 5% as 6 varies from 0 to 0.12. The normalized facilitated flux increases with 6, even though the difference v(l)- v(O) decreases with increasing 8. For these results the error remained well below 1%. as determined by e, , except that for 9 = 0 it increased to 7% at Y = 1 when 6 = 0.1. Figure 3 illustrates the effect of varying F on P(X). The solution for U(X) does not depend on E and very nearly approximates a straight line betwen zero and one. As e increases, v(O) decreases and ~(1) incrcascs, so that the normalized facilitated flux decreases. For the near-equilibrium model, the effect of varying F from 0 to 0.1 is illustrated in Figures 4 and 5 when 6 = 1.0, K =l.O,and 9 = 0. As F
1.
0. 8-
0. 6-
5 >
a
0. 4-
0. 2’
O0.2
0.4
0.6
0.8
FIG. 3. The effect of var)ilng F on the complexed carrier concentration wth I(=l.O. Dz/=O fixed. and(4) r={),(3) ~=1.0,(2) ~=5.().clnd(l) ~=l().().
1. 0
8 = ().()()l.
ANALYSIS
OF CARRIER-FACILITATED
165
DIFFUSION
0.6
5 3 0.4
0.2
0 0
0.2
0.4
0.6
0.8
1.0
x FIG. 4.
The effect
6=1.0, K=l.O,
Q=O,
of varying
e on the substrate
and e=O,O.Ol,
concentration
for
model II. with
andO.l.
increases, u(0) decreases and u(1) increases, giving the dependence of the normalized facilitated flux on E shown in Figure 6. The calculation using the inner solutions and the calculation using the composite expansion give very nearly the same results. The error, as determined from e,, and e,,, remained small for all cases, except at z =1 when Q = 0. For +Y= 0.5, the error was always below 5%, even for E = 0.2, but it increased to more than 10% at z = 1 for 4? = 0. Figures 7, 8 illustrate the effect on the substrate and complexed carrier concentrations of variation in 6. The results correspond to E= 0.001, K = 1, @ = 0 and to various 6 from 0 to 10. The normalized facilitated flux is an almost linear function of 6 and varies from zero to about 3.7 as 6 varies from zero to 10. In physiological applications K is small, E very small, and usually a = 0 for in vitro experiments with slabs. Figures 9, 10 illustrate typical concentra-
166
ROBERT
W. KOLKKA
AND ERIC P. SALATHk
1.
0. 8-
0. 6-
5 i 0. 4-
0. 2-
I
a-
0. 2
FIG. 5. withS=l.O.
The cffcct K=l.O,
of varying q=O,and
I
I
0. 4
0. b
F on the complcxcd
I
carrier
0. 8
conccntraticln
1 0
for model
II.
~=O.O.Ol,andO.1.
tion profiles for representative values of the parameters. It appears that the derivative of L’ is not zero at x = 0 and x = 1, because the boundary layer thickness is very smaI1 and on the scale of the figures the vanishing slopes at the ends cannot be seen. The boundary layers play a crucial role in determining the concentration profiles, since the higher order solutions could not be determined without a simultaneous analysis of the outer and inner solutions. The higher order terms contribute significantly to the concentration profiles, as is clearly illustrated in Figures 11 and 12. When 6 and E are both small, either model I or model II can be used to determine the profiles. Results for nonzero consumption rate are shown in Figures 13 and 14 for both models I and II, corresponding to E= 0.1, 6 = 0.1. K = 1.0, % = I, and M = 8. The zero order solutions for model I and for model II are different, but since the data used in both cases are the same,
ANALYSIS OF CARRIER-FACILITATED
167
DIFFUSION
0.50
0.46
0.42
7
r, 0.38
0.34
0.30 0
0.02
0.04
0.06
0.08
0.1
FIG. 6. The facilitated flux as a function of E for model II, with 6 = 1.0, K = 1.0, and Q = 0. Curve (1) is computed from the inner expansions, and curve (2) from the composite expansion.
both models should yield the same perturbation the figures that this is approximately true.
profiles. It can be seen from
APPENDIX The asymptotic expansions described in Sections 2.1 and 2.2 are given to order 6 and to order e, respectively, in this appendix, for the special case of bimolecular reaction, i.e., m = n = p = 1, and for SC,/&,,= 1. For the outer expansion in the small facilitation limit [cf, Equation (2.22)], it was found that
%-I-$
1
x+1
(Al)
268
ROBERT
W. KOLKKA
AND ERIC P. SALATHI?
1.0
0.8
0.6
5 r! 0.4
0.2
0 0
0.2
0.4
0.6
0.8
1.0
x FIG. 7. The effect of varying 6 on the substrate Du = 0, and (3) 6 = 0, (2) 6 = 5.0. and (1) 6 = 10.0.
concentration
with F = 0.001
1 K = 1.0.
and that u1 = uI = 0. For a bimolecular reaction, it follows from Equation (2.23) that u0 = uo/( K + ug). The order 6 terms are given by
U2(X)=-U()(X)+ u*(x) =
[&-+K
&lJ;I(x)
1 x+
K
K+ %(X) +[
K-t
1 ~ l+K
u7_(x).
uo(x)12
The solution obtained in Section 2.1 for the boundary reduces, for bimolecular reactions, to n,(z)
=hz, q(l+
(A3) layer at x = 0
(A4)
Kh
fit(z)=
(A21
Kf
-?1:
+
Kh
(l+ K)+
(A5)
ANALYSIS OF CARRIER-FACILITATED
169
DIFFUSION
1.0
0.8
0.6
0 0
0.2
0.4
0.6 x
0.8
1.0
FIG. 8 The effect of varying 6 on the complexed carrier concentration, with F = 0.001, K = 1.0, Q = 0, and (3) 6 = 0, (2) 6 = 5.0, and (1) 6 =lO.O.
where order
77=/m, 6 terms
and
h = e-l-
M/2. It can be shown
that
the
are given by
M fi,=_-z2
(A61
2
_ q4(;:K) [A_
Kh2 4eq2(1+
K)2
*2e-v .
$I-
4Eq~j(l;K)1iemT'
(A7)
170
ROBERT W. KOLKKA AND ERIC P. SALATHk 1
0
0. x 3 0.
0.
0
0. 2
0.4
0.6
0.8
x
1.0
FIG. 9. The variation in the substrate concentration for rn PIG-Ophysiological cxperimerits. The values of the parameters are F = lo-‘, K=lO ‘, Q=O.and 6=2.5. 1.0,O.l.
The solutions
for the boundary
layer at x = 1 are given by
(AlO)
_
Kp? 4&$(K$
t2e qt, ?P)’ and p=q--l+M/2.
(All)
ANALYSIS OF CARRIER-FACILITATED
DIFFUSION
0.8
0.6
X > 0.4
0.2
0 0.2
0
0.4
0.6
0.8
1.0
x FIG. 10. The variation in the complexed carrier concentration for in oitro physiological experiments. The values of the parameters are E= 10m6, K = lo-‘, “z/= 0, and 6 = 2.5, l.O,.l.
The dominant term in the outer expansion of u for the near-equilibrium limit, given by Equation (2.48), can be solved explicitly when m = n = p = 1, i.e., for bimolecular reactions. The result is
4KF(x) %(X1
=
[F(X)-~-81~
,
(A121
where F(x)=r~+l+&+~,
Mx2 (A131
and (Al4)
172
ROBERT W. KOLKKA AND ERIC F SALATHE l.C
0.6 x 3 0.4
0.2
a
The leading order term in the expansion of ZJis given by
The next terms in the outer expansions arc u*(x)
K + 110 = (A,.x -t B,)~--_ K+u,~+S{l-q,)’
(A16)
(A171 where A, and I?,, determined as a result of the boundary layer analysis at
ANALYSIS OF CARRIER-FACILITATED
DIFFUSION
1.0
0.8
0.6
5 > 0.4
0.2
0 0
0.2
0.4
0.6
0.8
6
1.0
FIG. 12. The effect of the perturbation on the complexed carier concentration for ut oirro physiological experiments. The values of the parameters are E= 10m6. S = 2.5, K=W2, and %=O.
x=Oand
x=l,aregivenby
B
1
=
K~4(0)~, 7(1+K)*
(A18)
’
‘4, = _ B, _ KSull(l)h, T(KS
q’
’
(A19)
with
x,=1+
6K (1 + K)* ’
h,=1+
6K (K+@)’
(A20)
ROBERT
0
0.2
W. KOLKKA
0.4
0.6
AND ERIC P. SALATHE
0.8
x FIG. 13. consumption,
The substrate concentration variation for models M = 8, and e = 0.1, 6 = 0.1, and K = 1.0.
I and
II with
nonzero
and
The boundary layer solutions, obtained in Section 2.2 to O(E’/~), reduce for the special case of bimolecular reactions to
ii,(Z)
=
2 +u’(O)z - ;P, I_
a,(z)=
Bl +ve-+ K12
KBl X,,(1+
Ku;,(0) (l+K)‘z
6423)
ANALYSIS OF CARRIER-FACILITATED 1.
.
0.
.
175
DIFFUSION
0.
x > 0.
0.
0.2
0.4
0.6
1.0
0.8
x FIG. 14 The complexed carrier concentration for models I and II with nonzero consumption, M = 8, and F = 0.1, 6 = 0.1. and K = 1.0
for the boundary
layer at x = 0, and to
K(A, c’,(5)
for the boundary The equations
=
+
4)
hR(K+@)*
WI
(1)
- (KS%)’
[
+
(A,
+
4)
,-;i[
AR6
layer at x =l. for the O(E) terms in the outer expansions
are
ROBERT W. KOLKKA
176
Their solutions
AND ERIC I’. SALATHE
are given by
(A28)
(A29) where again the constants A2 and B, can only be found through an analysis of the O(E) terms of the boundary layer expansions at x = 0.1. The equations for the O(E) terms in the boundary layer at x = 0 are
(l+K)B,-f$
ii2._;+
1
[
8&-
[
(l+K)+f$
1
They must be solved subject to the boundary
Equations
= u1u1. - -
(A31)
conditions (~32)
lim ii2(z) ; + cc
- ;U;;(O)+zN;(O)+
lim k(z) Z’CI)
- ~L~;I(o)+zo;(o)+
u,(O), (A33) 02(0).
yield the solutions
= (It + u2z + p2z2 +[a~+(r,+m,)~+m~z~]e~~‘+Pe
fi,(Z)=
(A30)
conditions
(A30)-(A33) n,(z)
1 1'
/L._(0) = 0,
&(O) = 0, and the matching
=-M-iir;
Kul (1 + K)’
“‘,
Ku2 41+ (I+ K)Z
+40+ i
-~+(sl+n,)z+nLZ:
(A34)
z+923? i
+Qe- lr’ 1 Tz e
(A35)
ANALYSIS
OF CARRIER-FACILITATED
where the coefficients
are given by
P2=2_a
1
6K[ U;(o)]?
2K44,(0) =
-
r = 1
3,=-p,
P=
KPZ
+
(l+
K)” ’ KBf
2P2
A’,(1 + K)’
l+K’
M
qo = 1 + K
K)3 ’
X,.(1+
+
(1-t K)’
q_ = _ Kb;,(0)12 7 (1-r K)”
41
177
DIFFUSION
aK2[ 4U912 K)4
2~~(1+
44,(O) [l-(l:;)2]ml=-,;((lpK)‘-;j. ml ( 6
r1
n, = - -
6 Bf
nz, =
-
Trnl,
?12 =
Q-f,
3X:-(2-
(Y=
a) ’
711 I’
1-6Kl+K
K2 ’
26TKB, al= A;_(2- (~)(l+
K)’
27Q - q1 - sI - n, -
’
KU2 (l+K)’
u1 =
The constant
-
a3
-
P,
1 ’
B2 is found to be
B, = ulh,
-
SBf
h’,(l+
+ S%;;(O) K)j
6436)
l+K
In a completely analogous manner it is found that the O(e) terms in the boundary layer at x = 1 are given by
a,(t)
=
KiT, (K+‘i’?)’
+
[
6 + q2‘F
-~+(s,+n,)[+A2<2 e-‘6 + Qe-w, 1
(A38)
ROBERT
178
where
the coefficients
M( K+
PI=
AND ERIC P. SALATHE
are given by
9)
6K[ u;)(l)]’
+
2(2-a)
z)( K + Qq’ ’
(2-
4_ = - Kb;(1)12
+
,
w[
MK 2(2-a)(K+Q)
(K-t@ -26K?[
W. KOLKKA
24gl)]?
+ (2-ii)(KIqq4.
U;,(l)]2
41 = 7(K+9)5
’
WK-t@-_(2-a)l
40=_
6’K3[
_
(2-a)(K+%)
r =
?(
26K[
u6(1)]’
K + @)’
u;,(l)]’
(2M)(K+‘&’
SK2[ 4(1)12
I
2?( m
= _ I
K + %)” [l-
(Kr”,,l]’
Kb;(l)l’ ni.b(K+&
(K:&]’
cy= %(2-@)+2K(l-‘P-6K-
K’
K+‘% _ 26Ku;,(l)B, U? &(K+%)3
~ A
1 ’ i ‘R KZ,
2?Q - q1 ~ s, - n, -
(K+&)”
The constant
A,
is found
A, =5,X,
1’
to be
- B2 -
6( A, + B,)‘K A’R( K + 9)’
This work wus supported Progrum Project Grunt POI-I
by NSF 7421.
Grant
+ S’v;;( 1) (A391
K+‘%
CME-8006782
und
Lyv NIH
ANALYSIS
OF CARRIER-FACILITATED
DIFFUSION
179
REFERENCES 1
2 3
4
5 6 I
A. E. Berger, M. Ciment. and J. C. W. Rogers, Numerical solution of a diffusion consumption problem with a free boundary, SIAM J. Numer. And. 12:646-672 (1975). J. D. Goddard, J. S. Schultz. and R. J. Bassett, On membrane diffusion with near-equilibrium reaction, Chenl. Engrg. Ser. 25:665-6X3 (1970). J. D. Goddard, J. S. Schultz. and S. R. Suchdeo. Facilitated transport via carrier mediated diffusion in membranes; Part II. Mathematical aspects and analyses. AIChE J. 201625~645 (1974). J. D. Goddard. J. S. Schultz, and S. R. Suchdeo, Facilitated transport via carrier mediated diffusion in membranes: Part I. Mechanistic aspects. experimental systems, and characteristic regimes, .4IChE J. 20:417-445 (1974). J. M. Gonzalez-Fernandez and S. E. Atta, Transport of oxygen in solutions of hemoglobin and myoglobin, Math. Biosci. 54:265-290 (1981). J. A. Jacquez, The physiological role of myoglobin: More than a problem in reactiondiffusion kinetics, M&h. Brow. 68:57-97 (1984). H. Kutchai. J. A. Jacquez, and F. J. Mather. Nonequilibrium facilitated oxygen transport in hemoglobin solution, BiophJ,s. J. 10:3X-54 (1970)
8
J. D. Murray. On the molecular mechanism of facilitated oxygen diffusion globin and myoglobin, Proc. Rq,,. Sot. London Ser. B 17X:95-110 (1971).
9
J. D. Murray, On the role of myoglobin in muscle respiration, J. Throret. B/o/. 47:115-126 (1974). J. D. Murray, Lecfwes on Nonlineur Differentml Equution Models in Bdog,. Oxford U.P., 197X J. Nedelman and S. I Rubinow. Facilitated diffusion of oxygen and carbon monoxide in the large affinity regime, J. Math. Biol. 12:73-90 (1981). D. R. Olander, Simultaneous mass transfer and equilibrium chemical reaction, AIChE J. 6:233-239 (1960). S. I. Rubinow and M. Dembo, The facilitated diffusion of oxygen by hemoglobin and myoglobin, Biophys. J. 18:29-42 (1977). P. F. Scholander, Oxygen transport through hemoglobin solutions, Science 131:5X5-590 (1960). B. A. Taylor and J. D. Murray, Effect of the rate of oxygen consumption on muscle respiration, J. Muth. Biol. 4:1-20 (1970). J. B. Wittenberg, Myoglobin facilitated oxygen diffusion: Role of myoglobin in oxygen entry into muscle, Phpiol. Rev. 50(4):559-635 (1970). J. Wyman, Facilitated diffusion and the possible role of myoglobin as a transport mechanism, J. Biol. Chem. 241:115-121 (1966)
10 11 12 13 14 15 16 17
of hemo-
W. KOLKKA*
Center for the Applicution Lehigh
University,
AND
ERIC P. SALATHe
of Muthematics.
Bethlehem,
Received 17 November
Diffusion
Pennsyh~ania
18015
1983; revised 10 Mu), 1984
ABSTRACT Carrier-facilitated matched asymptotic the small facilitation in the analysis,
1.
diffusion in a one dimensional slab is analyzed. The method of expansions is used to obtain solutions in both the near-equilibrium and limit. Consumption of the diffusing species within the slab is included
so that the solutions
are applicable
to situations
of physiological
interest.
INTRODUCTION
Carrier-facilitated, or mediated, diffusion is a reaction-diffusion phenomenon in which mass transport processes are enhanced by chemical reaction. The potential biological significance of facilitated diffusion was recognized by Wittenberg [16] and Scholander [14], as a result of experiments on the diffusion of oxygen through solutions of hemoglobin and myoglobin. There are also important applications in chemical engineering, including, for example, commercial membrane separation processes. In all cases a carrier molecule is contained within the medium in which transport occurs, and the substrate molecule combines reversibly with this carrier. Transport of the substrate is therefore a result of free diffusion and of diffusion by the complexed carrier molecule. A reversible chemical reaction with stoichiometry expressed by k+ nu+pw+
mv,
(1.1)
k-
in which fi moles of the substrate (I and p moles of the uncomplexed carrier W combine to form Fii moles of the complexed carrier I/, is typical of the reactions that occur in most applications. *Present address: Technical University. MATHEMATICAL
Department of Mathematics Houghton, MI 49981. BIOSCIENCES
71:147-179
TElsevier Science Publishing Co.. Inc., 19X4 52 Vanderbilt Ave., New York. NY 10017
and
Computer
Sciences,
Michigan
(1984)
147
0025.5564/X4/$03.00
148
ROBERT W. KOLKKA
AND ERIC P. SALATHI?
Early work in chemical engineering by Olander [12] examined mediated diffusion in a membrane, under the assumption that the reaction (1.1) remained in chemical equilibrium. This assumption is applicable if the characteristic diffusion time for the system is much larger than the characteristic reaction time for the reaction (1.1). In chemical engineering terms, this is referred to as the large Damkbhler number limit. Since the diffusion time is proportional to the membrane thickness, the near-equilibrium limit is frequently referred to as “thick film theory.” Goddard, Schultz, and Basset (21 subsequently recognized that the near-equilibrium limit was a singular perturbation problem, in which nonequilibrium boundary layers exist near the surfaces of the membrane. They discussed systems corresponding to the reaction (1.1) in which only the substrate U was transferred, the carrier species W and P’ being confined to the membrane. The method of matched asymptotic expansions was used to determine the concentrations of the various species as a function of position within the membrane. A review of facilitated diffusion in membrane technology has been given by Goddard, Schultz, and Suchdeo [3,4]. The study of facilitated diffusion in biological systems developed independently of the work of chemical engineering. Wyman [17] examined the transport of oxygen by hemoglobin and myoglobin in order to explain the observations of Wittenberg and Scholander on facilitated diffusion, and derived a differential equation governing the concentration of oxygen within the solution. An analysis of this equation under the assumption of chemical equilibrium was given by Murray [8]. Because of the singular nature of the problem in the near-equilibrium limit, Murray was not able to satisfy the condition of zero flux for the carrier molecule at the surface of the hemoglobin or myoglobin solution. A singular perturbation solution to Wyman’s equation in the near-equilibrium limit has been given by Rubinow and Dembo [13]. Nedelman and Rubinow [ll] subsequently provided the analysis when the forward reaction rate is very fast compared to the reverse reaction, the “large affinity regime.” In this case the singular perturbation analysis of Rubinow and Dembo breaks down, but the solution can be obtained as a regular perturbation expansion. Numerical solutions to the problem have been given by Kutchai et al. [7] using a quasilinearization technique, and by Gonzalez-Femandez and Atta [5] using a piecewise linear technique. It has been conjectured that myoglobin present in skeletal muscle facilitates the diffusion of oxygen from the blood capillary into the tissue, where consumption of oxygen occurs. For this reason, an understanding of the mechanism of carrier-mediated diffusion is of considerable interest to physiologists. Murray [9] extended his analysis of Wyman’s equation for a one dimensional slab to diffusion into a cylinder of tissue. Again, he considered only the equilibrium case, and so did not satisfy the boundary conditions for the myoglobin concentration at the cylindrical surface. Murray included
ANALYSIS OF CARRIER-FACILITATED
DIFFUSION
149
oxygen consumption in his equations, and for sufficiently large values of the consumption rate found that regions of oxygen depletion occurred within the cylinder. The mathematical treatment of the problem consisted of increasing consumption until negative oxygen values were obtained in the core. Since negative concentrations cannot occur, this was interpreted as a region of zero concentration. Therefore, the concentration profiles obtained do not have zero slope at the border of the anoxic region, and so do not represent the correct solution. The problem must be solved so that when the concentration falls to zero it has zero derivative at that location, and therefore zero flux of oxygen into the anoxic region. An analysis using these “Stefan” boundary conditions was subsequently given by Murray [lo]. This study also was restricted to the equilibrium case, so the boundary conditions at the surface of the slab were not satisfied. In the present paper, one dimensional diffusion through a slab is analyzed. It is assumed that the substrate reacts reversibly with the carrier molecule according to Equation (l.l), and that the substrate may be consumed within the slab. The extent of the depletion zones that develop for large consumption rates is investigated. There are two parameters of fundamental importance in the analysis. The first, E, is the reciprocal of the Damkiihler number, and measures the departure from chemical equilibrium. The second, 6, is the ratio of the flux transported by the carrier to that transported by free diffusion. Two different solutions are obtained, one as an asymptotic expansion for small 6 and one as an asymptotic expansion for small E. Both expansions are singular, and the methods of matched asymptotic expansions are used. The small 6 expansion is applicable to situations in which large departures from equilibrium occur, and has not been considered before. The small E expansion extends previous analyses, so that the effects of small departures from equilibrium can be determined when consumption occurs. This is of particular importance in physiological applications.
2.
ANALYSIS
A one dimensional slab occupying the region 0 d .Cd I contains a diffusable species of molar concentration a( .?). This substrate is maintained at the surfaces 2 = 0 and 5~= I at the uniform concentrations uA and us, respectively, and within the slab is consumed (or produced) at a constant rate fi. Diffusion of the substrate is facilitated by the presence within the slab of a generally large carrier molecule. The molar concentration of the uncomplexed carrier molecule is denoted by iv(Z), and of the complexed carrier molecule by D( 2). We denote by P( t, 0, %) the time rate of production of i+i moles per unit volume of the complexed carrier molecules and the rate of consumption of ?i moles per unit volume of the substrate or of p moles per
150
ROBERT
W. KOLKKA
AND ERIC P. SALATHE
volume of uncomplexed carrier molecules. It is assumed that each of the three types of molecules is freely diffusable and that the carrier molecule is confined within the slab. It therefore follows that the concentrations satisfy the following differential equations and boundary conditions: unit
(2.1) (2.2) (2.3)
k(O) =
di2 0
-=
di
dti -= d2
ii(I) =
UA,
0
UB)
(2.4)
at
i = 0, I,
(2.5)
at
.%=O,I,
(2.6)
where D,,, D,,, D,. are the diffusivities of the three types of molecules. Since the carrier molecule is confined to the slab, the total amount, in both complexed and uncomplexed form, must remain constant. Therefore, (2.7) where u is a constant. the boundary condition
Note that integrating (2.5) yields
Equation
1.
J0 F(ic,D,i!)d,t=O,
(2.2) and employing
(2.8)
the condition of “global equilibrium.” We assume that the reaction rate is of the form $‘(6,6,fi)
=k+fj”@Ppk-[,“‘,
(2.9)
so that n = p = m = 1 corresponds to the physiologically important case of myoglobin-facilitated oxygen diffusion. Arbitrary values of m , n, p are included in the analysis, so that the results are applicable to other than bimolecular reactions. Although Equation (2.9) does not adequately describe the kinetics of hemoglobin-oxygen reactions, such reactions have been approximated by the bimolecular case m = n = p = 1. A thorough discussion of hemoglobin-oxygen reactions has been given in a recent review by Jacquez
[61.
ANALYSIS
OF CARRIER-FACILITATED
151
DIFFUSION
Introducing the normalization u = O/u,, v = C/a, w = iv/a, ?# = uB/ uA yields the following nondimensional equations: d2u Em = F(u,v,w)+~M,
x = .%/I,
(2.10)
2
(2.11)
~t$$=-F(u,v,w),
(2.12)
u(0) =l,
u(1) = @,
(2.13)
v’(0) = 0,
v’(1) = 0,
(2.14)
w’(0) = 0,
w’(1) = 0,
(2.15) (2.16)
jol(;+;)dx=l,
where
F(u,
iiD,,a/iiiD,uA,
v, w) = u”wp - Ku” and E = DU/ii12ktu~p’oP, a,, = a,,, = iiD,a/jiD,u,, M = A%12/uADU, K = k-cTP/k+u;.
This system can be simplified by observing that adding Equations (2.12) integrating twice, and employing the boundary conditions (2.15) yields an equation for w(x) in terms of v(x):
4, w(x) =A - ,v(x). From Equation
(2.11), (2.14),
(2.17)
(2.16), it follows that (2.18)
For small substrate molecules and large carrier molecules, D,, and Dw are approximately equal, and Equation (2.18) reduces to A = jj. Equation (2.17) would then have the symmetric form v/iii + w/p =l. Using Equation (2.17) to eliminate w(x) yields EU”+f(U,V)=EM,
(2.19)
E&V”-f(U,V)=o,
(2.20)
where (2.21)
f( u, v) = Ku”’ - u”(h - vu)” and
6 = I?,, v = S,/&.
The solution
to Equations
(2.19)-(2.21)
must be
ROBERT W. KOLKKA
152
AND ERIC I’. SALATHk
obtained in terms of X, and the results then substituted into Equation (2.18) to determine X. If the backward reaction rate is as fast as or faster than the forward reaction rate, it follows from Equation (2.9) that the characteristic reaction time is given by T, =(iikidmlo")'.Since the characteristic diffusion time for the substrate molecule is T, = 1*/D,, it is clear that F = T,/T,,the ratio of the diffusion to the reaction time scale. Also, the parameter 8 in Equation (2.20) is proportional to D,,u/D~,u,~, and so is a measure of the amount of substrate that can be transported by the carrier molecule to the amount that can be transported through simple diffusion. 2.1.
THE SMALL
FACILIT4
TION LIMIT.
MODEL
I
In this subsection the solution to Equations (2.19)-(2.21) will be obtained for the case in which the total quantity of solute transported is increased by only a small amount as a result of facilitation. This corresponds to the limit S -+ 0, and the solution can be obtained as an asymptotic expansion in 6. It turns out that the expansion is singular, and that there are boundary layers at x = 0 and at x = 1. In the core region, bounded away from x = 0, the expansion has the form
u(x)-
u~~(.~)+s1”u,(x)+su*(x)+o(s3”), (2.22)
D(X) - L~~~(x)+s’~“~,(x)+sli~(x)+o(8~‘~). Substituting this expansion into Equations (2.19), (2.20) and equating like powers in 8l/* yields a sequence of equations from which the various terms in this expansion can be determined. The leading terms zlO(x), ~a( x) satisfy u;; = M and f( uO, ~1~)= 0. Therefore, u0 = M,x’/~+ A,x + B, and Q = P’(uo), where V( N,,) is the inverse of (2.23)
Note that the solution for uU represents simple diffusion, with consumption, through the slab, and that to this order the substrate is in equilibrium with the carrier molecule. The terms of order B1/* give Ul
-
0,
h(X) = where
f”(U”, %I f,,(uo,uo)
(2.24)
h(X),
f, = df/au, f,, = df/dv. Therefore, pi = A,x + B,. The constants A,,,
ANALYSIS OF CARRIER-FACILITATED
153
DIFFUSION
A,, B,,, and B, cannot be determined from the boundary conditions on u, since the expansion assumed here is not valid in the neighborhood of x = 0 and x = 1. In fact, ua and vi do not satisfy the no-flux condition cl’= 0 at the boundaries of the slab, for any choice of A,, A,, B,, and B,.It is necessary to construct different expansions for the regions near x = 0 and x = 1. Matching them to the above expansion will determine the constants A,, A,, B,, and B,.The nature of the expansions near x = 0,l (the inner expansions or boundary layers) will be briefly described here. Higher order terms in the expansion bounded away from x = 0,l (the outer expansion) and in the boundary layers at x = 0 and x = 1 are presented in the appendix. In order to analyze the solution near x = 0, it is necessary to introduce the boundary layer variable z = .x/#‘~, which has the property that x -+ 0 as 6 + 0 with z fixed. In terms of this variable, Equations (2.19), (2.20) become
&iiz;+8f(ii,6)=&SM,
(2.25)
&fizz-f(C,G)=O,
(2.26)
where fi( z) = u( 6112z), 6 (z) = v ( a’/2z). Assuming form
a(z)-
an inner expansion
n,(Z)+s”‘ii,(Z)+0(s),
of the
(2.27)
a(Z)-fi,(z)+C%i(Z)+0(6), substituting this into Equations (2.25), (2.26), and equating order in #I2 yields the equations Goz_= 0,
terms of similar
(2.28)
&fiOzz-f(&,QJ)=O
(2.29)
and iilr; = 0,
&Dlrl-fU(~O,~O)~l-fr,(~O,~O)~l=O. Equations
(2.30) (2.31)
(2.28) and (2.30) have the solutions
ii,=&,+ Ii,=A,z+B,.
B,,
(2.32) (2.33)
The boundary conditions give G,(O) = 1, C,(O) = 0, C&O) = 0, C13(0) = 0. Therefore, &, = 1 and El = 0. Matching of the boundary layer solution to the outer solution is achieved by substituting x = #I2 z into the Equations (2.22) and expanding in powers
154
ROBERT
W. KOLKKA
AND ERIC P. SALATHk
of f3”z: u - us(O) + IV2 { =;,(O)+tl,(O)}+O(6), (2.34) c’- U,](O)+ W{ Comparing Z’DO
this with the inner
zU;,(o)+U,(o)}+o(6). expansion,
Equation
(2.27), shows that as
n,(z)
+ U”(0) = B”,
(2.35)
b,(z)
+ U,(O),
(2.36)
n,(z)
+ zu;j(o)+
U,(O) = A,z + B,,
~,(=)-z/~;,(o)+u,(o).
(2.37) (2.38)
It therefore follows that A,, = 0, B,, = 1, A, = A,, and B, = 0. The solution to Equations (2.29) and (2.31) subject to the boundary and matching conditions is fiO= V(l), and
fi1(Z)=-&“2 _
fu(l>V(l))
A,crp[ _{T;]
cm~ w>)13’2
fu(l, V(l))
A
z,
(2.39)
f,.(l? V(l)) (I
Two constants, A, and A,, remain unknown. These are determined when the boundary layer solution at x =l is analyzed. Introducing the boundary layer variable .$= (1 - .~)/8”‘~, which has the property x + 1 as 6 + 0. [ fixed, the boundary layer expansion at x = 1 is assumed to have the form u(~)=uo(~)+s”‘ui(~)+o(s), L’(~)=L$)(‘$)+W%i([)+o(8),
(2.40)
where U(t) = u(l- S’/‘[), G(t) = z~(l- S’/‘,$). Substituting this expansion into Equations (2.19), (2.20), rewritten in terms of the boundary layer variable 6, yields a sequence of equations from which u,,, Q, pi, c~i, etc. can be determined. The procedure is analogous to the analysis of the boundary layer at x = 0. The equations for ‘A”,Ui have the solutions a0 = A06 + B,, U, = x1.$ + B,. The boundary conditions at x = 1 give B, = %, 3, = 0, while the matching conditions give (2.41)
uo + U”(l), Ui + - ‘$ub(l)+
U,(l)
(2.42)
asE+so.Fromtheseitfollowsthat&=O, A,,=%-l-M/2, A,=%-1 + M/2, A, = 0. The solutions for co, 8, are obtained in a manner analogous
ANALYSIS OF CARRIER-FACILITATED
to the determinations
DIFFUSION
155
of Co, fii, and are found to be
Uo=V(@),
(2.43)
(2.44)
This completes the solution to order 6 1/2 in the central core region and in both boundary layers. Higher order terms can readily be obtained by continuing the analysis outlined here. The details are provided in the appendix. 2.2.
THE NEAR-EQUILIBRIUM
LIMIT,
MODEL
II
In this section the solution to Equations (2.19)-(2.21) will be obtained in the limit e -+ 0, corresponding to the case where the reaction between substrate and carrier remains near equilibrium. Since E is inversely proportional to I*, small E is frequently referred to as “ thick film theory” and large E as “ thin film theory.” As in the previous case, the expansion in terms of e is singular, and boundary layers must be constructed at x = 0 and x = 1 to complete the solution. In the region bounded away from x = 0 and x = 1, the solution can be expanded in the form
u(x)- Uo(x)+d’2U1(X)+O(&), 4x) - uo(x) + Pq( x) + O( E).
(2.45)
Substituting this expansion into Equation (2.20) shows that the leading terms satisfy f( aa, ~a) = 0, so that u,, = V( uo), as before. Adding Equations (2.19) and (2.20) and substituting the expansion (2.45) into the resulting equation gives [u. + 6V( uo)]“=
M.
(2.46)
Since the inner expansions are not valid at x = 0, L, the boundary conditions cannot be applied to the terms of this expansion. Boundary layer solutions must be constructed at x = 0 and x = L that satisfy the boundary conditions, and the inner expansions must be matched to these solutions. However, it was found in the last section that the leading term in the inner expansion for u(x), u,(x), did satisfy the boundary conditions at x = 0 and L, although this was not the case for U(X). The analysis in this section is considerably simplified by trying a solution for us(x) that satisfies the
156
ROBERT W. KOLKKA AND ERIC I’. SALATHk
boundary
conditions uo(O) =l,
uo(l)
= q.
(2.47)
Justification for this assumption follows from the ability to construct innner and outer solutions that can be matched, and that satisfy the appropriate equations and boundary conditions. It therefore follows that
U() +
SV( zq))=
F
+(4-1+s[v(“z/)-li(l)]-~).~+I+sv(1). (2.48)
which can readily be inverted satisfy the equations
to yield ug( x). The higher order terms,
U, , ~1, ,
(2.49)
(2.50)
Therefore,
(2.51)
Boundary conditions at x = 0 and x = 1 cannot be applied to determine the constants A, and B,, since the expansion (2.45) is not applicable in the neighborhood of the boundaries. In addition, L’,) and ~1~ do not satisfy the boundary condition of zero flux at x = 0,l for any choice of the constants A,, B,. It is necessary, as before, to construct different expansions in those regions and to join them to the expansion developed here for the core region. The boundary layer solution near x = 0 is obtained by defining the boundary layer variables z = x/E’/~, ii(z) = U( F’/~z), r?(z) = u( F’/‘z). In terms of these variables, Equations (2.19), (2.20) become
(2.52)
Clearly,
the boundary
layer solutions
are unaffected
by M until order
F.
ANALYSIS
Introducing
OF CARRIER-FACILITATED
the boundary
DIFFUSION
157
layer expansions
ii(Z) -1-r &"'k,(Z)+o(E), (2.53) a(z)
- V(l)+&“%,(Z)+O(&),
and substituting these into the boundary equations for 1, and r?,:
where f,,, = f,[l,
V(l)], f,,O =f,,[l,
V(l)]. The boundary
8, (0) = 0, and the matching
requirement
as z + 00. Equations
layer equations
61: (0)
= 0,
yields the following
conditions
are (2.56)
gives
a,(z)
- zr&(O)+ Ui(O),
a,(z)
- z,;,(O)+ u,(O),
(2.54)-(2.57)
(2.57)
yield the solution
(2.58)
(2.59)
and the result that (2.60)
where7 = KfdWl
- ~fuo/f,d1'~2. The boundary layer analysis at z = 1 proceeds in exactly the same manner. Introducing the boundary layer variables 5 = (1- X)/E’/‘, ii([) = U(1 - P[), U(5) = u(l- P2 I), the expansion in the region near z = 1 has the form ti([)
-
q-t
E1’2ii,([)+@E)!
(2.61) u(~)-V(q+&“2u,(~)+O(&).
ROBERT W. KOLKKA AND ERIC P. SALATHk
158
It can be shown that pi and ui are given by (2.62)
(2.63)
where ful = f,[udl), udl)l~ .f,.l= .fJud), ~+AlN, and SfUl/f,,i)]“‘. the result
The matching
7 = [G/6)(1 of this solution with the outer solution also yields
Higher order terms in both the outer and the two inner expansions obtained in the appendix.
2.3.
DEPLETION
arc
ZONES
For sufficiently large values of the consumption rate, there is a region within the slab in which the substrate concentration is zero. When such a depletion zone is present, the solutions obtained above are not valid. The governing equations must be solved subject to the condition that at the boundaries of the depletion zone both the value and the derivative of the substrate and the carrier concentration vanish. If the depletion zone is located between the values x = L and x =lR, then the substrate and carrier concentrations must satisfy the Stefan boundary conditions
u(L) =u(L)=O,
(2.65)
u'(L)=v'(L)=O and u(l-R)=v(l-R)=O, u’(l-
R)=v’(l-
(2.66) R)=O.
The additional boundary conditions do not over determine the system, because now the unknowns L and R must be found as part of the solution. The governing equations are Equations (2.19)-(2.21), and the boundary conditions at x = 0 and x =l are given by Equations (2.13) and (2.14). The
ANALYSIS
OF CARRIER-FACILITATED
DIFFUSION
159
consumption rate is assumed to be a constant, M, when the substrate concentration is nonzero, and to drop discontinuously to zero when u = 0. Different singular perturbation expansions must be developed for the region to the left and to the right of the depletion zone. The analysis will be presented for the left side, and only the results will be given for the right side. The outer expansion, valid away from the boundary layer and up to the boundary of the depletion zone, has the form of Equation (2.22) for model I and of Equation (2.45) for model II. It is not necessary to construct a boundary layer at the boundary of the depletion zone, since the vanishing of u and U’ assures that both u and u’ will also vanish, as can be seen from the outer solutions obtained in Sections 2.1 and 2.2. The leading order equilibrium solutions for model II are
u,(x)+sv(u,(x))=~ +Ax+l+W(l),
(2.67) (2.68)
00(x) = l+%(X))~ where the constant A is to be determined (2.65). The first order corrections are
by the Stefan boundary
conditions
1’ q(x) =-;;;oj;; 07 0u,(x>, u,(x)=(Bx+C)
fll(Uo,uo) -l
l-6
L~(U,, 00)
[
where the constants B and C are to be determined by the Stefan conditions, and by the analysis of the boundary layer. The inner expansions have the form (2.53), and the solutions are given by (2.58), (2.59), with A
u;l(O) =
1+ 6V’(l) The inner solutions
show that the constant
C is related to the constant
(2.71)
A by (2.72)
At this point all that remains to complete the solution is to determine the values of the unknown constants A and B, and the location of the depletion zone boundary point L. The value of L can be expressed as a perturbation expansion : L-
Lo+&1’2L1+O(&),
(2.73)
160 so
ROBERT W. KOLKKA AND ERIC P. SALATHk
that the Equations
(2.65) can be expanded,
to order
F"',
in the form
Uo(L,,)+&“?[L,U;)(LO)+UI(LO)l
=o,
(2.74)
u;,(L,,)+E’,‘?[LIU;;(LO)+U;(LO)]
=o.
(2.75)
It then follows that
(2.76) A = - \,‘2M[l+
SV(I)]
,
fj-F,
(2.77) (2.78)
,’
It can be shown in a similar manner
that for the right hand side,
The expressions (2.76) and (2.79) are asymptotically valid approximations only if L,,, R,, # O(E”‘). When L,,, R,, = O(F’/~). the depletion zone has “invaded” the boundary layer, and the previous analysis is not appropriate. The critical value for the onset of depletion occurs when L = 1~ R. This condition gives A4, =2(1+%)+4[1+6V(l)]‘~Q?+SV(~)] +2G[V(I)+Y(*)]-?~~~-~i+O(r).
(2.80)
where A4, is the consumption rate for which the substrate concentration falls to zero at some point in the interior of the slab. For M < M, the solution obtained in the previous section is applicable. For M > M, , finite regions of zero concentration occur and the solutions obtained in this section must be used. The otcurrence of depletion zones is a consequence of the assumption that consumption rate is constant for nonzero concentration, and falls discontinuously to zero when concentration is zero. It is known, however, that although oxygen consumption in tissue is uniform for normal values of concentration, for very low values the consumption rate drops rapidly, falling continuously to zero. For such a continuous drop to zero in consumption rate there is no sharp border separating oxygenated and anoxic regions. However. an analysis by Berger et al. [l] using Michaelis-Menton kinetics showed that in the
ANALYSIS
OF CARRIER-FACILITATED
DIFFUSION
161
limiting case in which consumption remained nearly constant until very low concentrations were reached, the solution was very close to the solution corresponding to the step function consumption rate assumed here. In the absence of facilitation, Rogers (see Berger et al. [l]) has proved that the solution with step function consumption can be obtained as a limit from the solution with continuously varying consumption rate. Therefore, the step function variation assumed for the consumption rate should result in an accurate description of the concentration profiles for many situations where the consumption rate varies continuously. Similar conclusions were reached by Taylor and Murray [15] after examining the case where the consumption rate is of the Michaelis-Menton form.
3.
RESULTS
AND DISCUSSIONS
A composite expansion, uniformly valid throughout the slab, can be obtained from the perturbation solutions for the central core and for each of the boundary layers. This is done by adding the three separate expansions together and subtracting any part that is common to more than one region. For the small facilitation model, the composite expansion for the substrate concentration u(x) is given, to order 6, by
u,(x)=u,+8
1’2u, + 62.4,+1+
8wi, + 6B2
- {1+ 81’2[ 24i(o)+ z~;,(0)]+6[u*(0)+zu;(0)+~z2u;;(0)]} +%+$1’2u
+6u
-~“+s’/‘;ul(l~-~~;(I~]+s[u,(I)-bu;o+i~’u;~l)]}, (3.1) with a similar expression for the carrier concentration v(x). Analogous results can be written for the near-equilibrium model. Clearly, these formulas can be considerably reduced when the appropriate solution is substituted for each of the terms appearing in them. The total flux through the slab is given in nondimensional terms by J=
u’(x)+sL~‘(x).
(3.2)
If there is no consumption, J must be constant. This can easily be proved by adding Equations (2.19) and (2.20). Therefore, since u’ = 0 at x = 0 or x = 1, the total normalized flux for M = 0 is J = u’(O) = u’(l). The facilitated flux is by definition the difference between the flux when 6 = 0 and the flux when 6 # 0.
162
ROBERT
W. KOLKKA
AND ERIC I’. SALATHI?
The derivative of v at x = 0 and x = 1 vanishes in the boundary layer solutions. However, this no flux condition is only approximately satisfied by the composite solution. When the solution is obtained to order 6 for model I, it can be seen that v:(O) = i3u;(O), with a similar result at x =l. The contribution of this inaccuracy to the computation of the flux is an indication of the error involved in the perturbation solution. Defining e,. = S’P;/( u: + 8~:) for model I, and analogously e,, = E~u;( ui. + 6~:) for model II, the values e,.(O) and e,.(l) provide an effective error measure. For model I. the composite expansion and the outer expansion for N are identical However, for model II, it is desirable to introduce an additional error measure, analogous to that for 11above, that determines the extent to which the slopes of the composite and inner solutions differ at .Y= 0 and x = 1. Clearly, this can be done by defining eU = &US/( u: + 6~~‘). and examining the values e,,(O) and e,,(l).
0.8
0.6
5 3 0.4
0.2
0 0
0.2
0.4
0.6
0.8
1.0
x FIG. 1.
Substrate
and S = 0.0.1.
concentration
variation
for model
I, with
F = 1.0. K = 1.0. @ = 0.
ANALYSIS
OF CARRIER-FACILITATED
163
DIFFUSION
Figures 1 and 2 show the effect of varying 6 from 0 to 0.1 in model I, for F = 1.0, K = 1.0, and % = 0, when there is no consumption (M = 0).In these and subsequent figures the broken line represents the leading order solution. All numerical results shown include the higher order terms obtained in the appendix, and so correspond to m = n = p = 1. The concentration u(x) is independent of 6 to order 6, and Figure 1 shows the small influence of 6 on the u(x) profile. However, the profiles for u(x) are strongly dependent on 6, as illustrated in Figure 2. An alternate expression for the total flux can be obtained by integrating Equation (3.2) from x = 0 to x =l. Subtracting the portion of the flux that would exist in the absence of facilitation gives the normalized facilitated flux,
.$=S[rJ(l)-U(O)].
(3.3)
1. .O.-
0. 8-
0. 65 > 0. 4-
0. 2’
00
0.2
0.4
0.6
0.8
1.0
x FIG. 2. Complexed carrier concentration S = 0.01, (3) 6 = 0.001. and (4) 6 = 0.
variation
for model
I. with (1) 6 = 0.1, (2)
164
ROBERT
W. KOLKKA
AND
ERIC P. SALATHE
This flux was computed both using values for r,(O), cl(l) from the inner expansions and using the composite expansion. The two methods give virtually identical results. For q = 0, IJ,I varies nearly linearly from 0 to about 5% as 6 varies from 0 to 0.12. The normalized facilitated flux increases with 6, even though the difference v(l)- v(O) decreases with increasing 8. For these results the error remained well below 1%. as determined by e, , except that for 9 = 0 it increased to 7% at Y = 1 when 6 = 0.1. Figure 3 illustrates the effect of varying F on P(X). The solution for U(X) does not depend on E and very nearly approximates a straight line betwen zero and one. As e increases, v(O) decreases and ~(1) incrcascs, so that the normalized facilitated flux decreases. For the near-equilibrium model, the effect of varying F from 0 to 0.1 is illustrated in Figures 4 and 5 when 6 = 1.0, K =l.O,and 9 = 0. As F
1.
0. 8-
0. 6-
5 >
a
0. 4-
0. 2’
O0.2
0.4
0.6
0.8
FIG. 3. The effect of var)ilng F on the complexed carrier concentration wth I(=l.O. Dz/=O fixed. and(4) r={),(3) ~=1.0,(2) ~=5.().clnd(l) ~=l().().
1. 0
8 = ().()()l.
ANALYSIS
OF CARRIER-FACILITATED
165
DIFFUSION
0.6
5 3 0.4
0.2
0 0
0.2
0.4
0.6
0.8
1.0
x FIG. 4.
The effect
6=1.0, K=l.O,
Q=O,
of varying
e on the substrate
and e=O,O.Ol,
concentration
for
model II. with
andO.l.
increases, u(0) decreases and u(1) increases, giving the dependence of the normalized facilitated flux on E shown in Figure 6. The calculation using the inner solutions and the calculation using the composite expansion give very nearly the same results. The error, as determined from e,, and e,,, remained small for all cases, except at z =1 when Q = 0. For +Y= 0.5, the error was always below 5%, even for E = 0.2, but it increased to more than 10% at z = 1 for 4? = 0. Figures 7, 8 illustrate the effect on the substrate and complexed carrier concentrations of variation in 6. The results correspond to E= 0.001, K = 1, @ = 0 and to various 6 from 0 to 10. The normalized facilitated flux is an almost linear function of 6 and varies from zero to about 3.7 as 6 varies from zero to 10. In physiological applications K is small, E very small, and usually a = 0 for in vitro experiments with slabs. Figures 9, 10 illustrate typical concentra-
166
ROBERT
W. KOLKKA
AND ERIC P. SALATHk
1.
0. 8-
0. 6-
5 i 0. 4-
0. 2-
I
a-
0. 2
FIG. 5. withS=l.O.
The cffcct K=l.O,
of varying q=O,and
I
I
0. 4
0. b
F on the complcxcd
I
carrier
0. 8
conccntraticln
1 0
for model
II.
~=O.O.Ol,andO.1.
tion profiles for representative values of the parameters. It appears that the derivative of L’ is not zero at x = 0 and x = 1, because the boundary layer thickness is very smaI1 and on the scale of the figures the vanishing slopes at the ends cannot be seen. The boundary layers play a crucial role in determining the concentration profiles, since the higher order solutions could not be determined without a simultaneous analysis of the outer and inner solutions. The higher order terms contribute significantly to the concentration profiles, as is clearly illustrated in Figures 11 and 12. When 6 and E are both small, either model I or model II can be used to determine the profiles. Results for nonzero consumption rate are shown in Figures 13 and 14 for both models I and II, corresponding to E= 0.1, 6 = 0.1. K = 1.0, % = I, and M = 8. The zero order solutions for model I and for model II are different, but since the data used in both cases are the same,
ANALYSIS OF CARRIER-FACILITATED
167
DIFFUSION
0.50
0.46
0.42
7
r, 0.38
0.34
0.30 0
0.02
0.04
0.06
0.08
0.1
FIG. 6. The facilitated flux as a function of E for model II, with 6 = 1.0, K = 1.0, and Q = 0. Curve (1) is computed from the inner expansions, and curve (2) from the composite expansion.
both models should yield the same perturbation the figures that this is approximately true.
profiles. It can be seen from
APPENDIX The asymptotic expansions described in Sections 2.1 and 2.2 are given to order 6 and to order e, respectively, in this appendix, for the special case of bimolecular reaction, i.e., m = n = p = 1, and for SC,/&,,= 1. For the outer expansion in the small facilitation limit [cf, Equation (2.22)], it was found that
%-I-$
1
x+1
(Al)
268
ROBERT
W. KOLKKA
AND ERIC P. SALATHI?
1.0
0.8
0.6
5 r! 0.4
0.2
0 0
0.2
0.4
0.6
0.8
1.0
x FIG. 7. The effect of varying 6 on the substrate Du = 0, and (3) 6 = 0, (2) 6 = 5.0. and (1) 6 = 10.0.
concentration
with F = 0.001
1 K = 1.0.
and that u1 = uI = 0. For a bimolecular reaction, it follows from Equation (2.23) that u0 = uo/( K + ug). The order 6 terms are given by
U2(X)=-U()(X)+ u*(x) =
[&-+K
&lJ;I(x)
1 x+
K
K+ %(X) +[
K-t
1 ~ l+K
u7_(x).
uo(x)12
The solution obtained in Section 2.1 for the boundary reduces, for bimolecular reactions, to n,(z)
=hz, q(l+
(A3) layer at x = 0
(A4)
Kh
fit(z)=
(A21
Kf
-?1:
+
Kh
(l+ K)+
(A5)
ANALYSIS OF CARRIER-FACILITATED
169
DIFFUSION
1.0
0.8
0.6
0 0
0.2
0.4
0.6 x
0.8
1.0
FIG. 8 The effect of varying 6 on the complexed carrier concentration, with F = 0.001, K = 1.0, Q = 0, and (3) 6 = 0, (2) 6 = 5.0, and (1) 6 =lO.O.
where order
77=/m, 6 terms
and
h = e-l-
M/2. It can be shown
that
the
are given by
M fi,=_-z2
(A61
2
_ q4(;:K) [A_
Kh2 4eq2(1+
K)2
*2e-v .
$I-
4Eq~j(l;K)1iemT'
(A7)
170
ROBERT W. KOLKKA AND ERIC P. SALATHk 1
0
0. x 3 0.
0.
0
0. 2
0.4
0.6
0.8
x
1.0
FIG. 9. The variation in the substrate concentration for rn PIG-Ophysiological cxperimerits. The values of the parameters are F = lo-‘, K=lO ‘, Q=O.and 6=2.5. 1.0,O.l.
The solutions
for the boundary
layer at x = 1 are given by
(AlO)
_
Kp? 4&$(K$
t2e qt, ?P)’ and p=q--l+M/2.
(All)
ANALYSIS OF CARRIER-FACILITATED
DIFFUSION
0.8
0.6
X > 0.4
0.2
0 0.2
0
0.4
0.6
0.8
1.0
x FIG. 10. The variation in the complexed carrier concentration for in oitro physiological experiments. The values of the parameters are E= 10m6, K = lo-‘, “z/= 0, and 6 = 2.5, l.O,.l.
The dominant term in the outer expansion of u for the near-equilibrium limit, given by Equation (2.48), can be solved explicitly when m = n = p = 1, i.e., for bimolecular reactions. The result is
4KF(x) %(X1
=
[F(X)-~-81~
,
(A121
where F(x)=r~+l+&+~,
Mx2 (A131
and (Al4)
172
ROBERT W. KOLKKA AND ERIC F SALATHE l.C
0.6 x 3 0.4
0.2
a
The leading order term in the expansion of ZJis given by
The next terms in the outer expansions arc u*(x)
K + 110 = (A,.x -t B,)~--_ K+u,~+S{l-q,)’
(A16)
(A171 where A, and I?,, determined as a result of the boundary layer analysis at
ANALYSIS OF CARRIER-FACILITATED
DIFFUSION
1.0
0.8
0.6
5 > 0.4
0.2
0 0
0.2
0.4
0.6
0.8
6
1.0
FIG. 12. The effect of the perturbation on the complexed carier concentration for ut oirro physiological experiments. The values of the parameters are E= 10m6. S = 2.5, K=W2, and %=O.
x=Oand
x=l,aregivenby
B
1
=
K~4(0)~, 7(1+K)*
(A18)
’
‘4, = _ B, _ KSull(l)h, T(KS
q’
’
(A19)
with
x,=1+
6K (1 + K)* ’
h,=1+
6K (K+@)’
(A20)
ROBERT
0
0.2
W. KOLKKA
0.4
0.6
AND ERIC P. SALATHE
0.8
x FIG. 13. consumption,
The substrate concentration variation for models M = 8, and e = 0.1, 6 = 0.1, and K = 1.0.
I and
II with
nonzero
and
The boundary layer solutions, obtained in Section 2.2 to O(E’/~), reduce for the special case of bimolecular reactions to
ii,(Z)
=
2 +u’(O)z - ;P, I_
a,(z)=
Bl +ve-+ K12
KBl X,,(1+
Ku;,(0) (l+K)‘z
6423)
ANALYSIS OF CARRIER-FACILITATED 1.
.
0.
.
175
DIFFUSION
0.
x > 0.
0.
0.2
0.4
0.6
1.0
0.8
x FIG. 14 The complexed carrier concentration for models I and II with nonzero consumption, M = 8, and F = 0.1, 6 = 0.1. and K = 1.0
for the boundary
layer at x = 0, and to
K(A, c’,(5)
for the boundary The equations
=
+
4)
hR(K+@)*
WI
(1)
- (KS%)’
[
+
(A,
+
4)
,-;i[
AR6
layer at x =l. for the O(E) terms in the outer expansions
are
ROBERT W. KOLKKA
176
Their solutions
AND ERIC I’. SALATHE
are given by
(A28)
(A29) where again the constants A2 and B, can only be found through an analysis of the O(E) terms of the boundary layer expansions at x = 0.1. The equations for the O(E) terms in the boundary layer at x = 0 are
(l+K)B,-f$
ii2._;+
1
[
8&-
[
(l+K)+f$
1
They must be solved subject to the boundary
Equations
= u1u1. - -
(A31)
conditions (~32)
lim ii2(z) ; + cc
- ;U;;(O)+zN;(O)+
lim k(z) Z’CI)
- ~L~;I(o)+zo;(o)+
u,(O), (A33) 02(0).
yield the solutions
= (It + u2z + p2z2 +[a~+(r,+m,)~+m~z~]e~~‘+Pe
fi,(Z)=
(A30)
conditions
(A30)-(A33) n,(z)
1 1'
/L._(0) = 0,
&(O) = 0, and the matching
=-M-iir;
Kul (1 + K)’
“‘,
Ku2 41+ (I+ K)Z
+40+ i
-~+(sl+n,)z+nLZ:
(A34)
z+923? i
+Qe- lr’ 1 Tz e
(A35)
ANALYSIS
OF CARRIER-FACILITATED
where the coefficients
are given by
P2=2_a
1
6K[ U;(o)]?
2K44,(0) =
-
r = 1
3,=-p,
P=
KPZ
+
(l+
K)” ’ KBf
2P2
A’,(1 + K)’
l+K’
M
qo = 1 + K
K)3 ’
X,.(1+
+
(1-t K)’
q_ = _ Kb;,(0)12 7 (1-r K)”
41
177
DIFFUSION
aK2[ 4U912 K)4
2~~(1+
44,(O) [l-(l:;)2]ml=-,;((lpK)‘-;j. ml ( 6
r1
n, = - -
6 Bf
nz, =
-
Trnl,
?12 =
Q-f,
3X:-(2-
(Y=
a) ’
711 I’
1-6Kl+K
K2 ’
26TKB, al= A;_(2- (~)(l+
K)’
27Q - q1 - sI - n, -
’
KU2 (l+K)’
u1 =
The constant
-
a3
-
P,
1 ’
B2 is found to be
B, = ulh,
-
SBf
h’,(l+
+ S%;;(O) K)j
6436)
l+K
In a completely analogous manner it is found that the O(e) terms in the boundary layer at x = 1 are given by
a,(t)
=
KiT, (K+‘i’?)’
+
[
6 + q2‘F
-~+(s,+n,)[+A2<2 e-‘6 + Qe-w, 1
(A38)
ROBERT
178
where
the coefficients
M( K+
PI=
AND ERIC P. SALATHE
are given by
9)
6K[ u;)(l)]’
+
2(2-a)
z)( K + Qq’ ’
(2-
4_ = - Kb;(1)12
+
,
w[
MK 2(2-a)(K+Q)
(K-t@ -26K?[
W. KOLKKA
24gl)]?
+ (2-ii)(KIqq4.
U;,(l)]2
41 = 7(K+9)5
’
WK-t@-_(2-a)l
40=_
6’K3[
_
(2-a)(K+%)
r =
?(
26K[
u6(1)]’
K + @)’
u;,(l)]’
(2M)(K+‘&’
SK2[ 4(1)12
I
2?( m
= _ I
K + %)” [l-
(Kr”,,l]’
Kb;(l)l’ ni.b(K+&
(K:&]’
cy= %(2-@)+2K(l-‘P-6K-
K’
K+‘% _ 26Ku;,(l)B, U? &(K+%)3
~ A
1 ’ i ‘R KZ,
2?Q - q1 ~ s, - n, -
(K+&)”
The constant
A,
is found
A, =5,X,
1’
to be
- B2 -
6( A, + B,)‘K A’R( K + 9)’
This work wus supported Progrum Project Grunt POI-I
by NSF 7421.
Grant
+ S’v;;( 1) (A391
K+‘%
CME-8006782
und
Lyv NIH
ANALYSIS
OF CARRIER-FACILITATED
DIFFUSION
179
REFERENCES 1
2 3
4
5 6 I
A. E. Berger, M. Ciment. and J. C. W. Rogers, Numerical solution of a diffusion consumption problem with a free boundary, SIAM J. Numer. And. 12:646-672 (1975). J. D. Goddard, J. S. Schultz. and R. J. Bassett, On membrane diffusion with near-equilibrium reaction, Chenl. Engrg. Ser. 25:665-6X3 (1970). J. D. Goddard, J. S. Schultz. and S. R. Suchdeo. Facilitated transport via carrier mediated diffusion in membranes; Part II. Mathematical aspects and analyses. AIChE J. 201625~645 (1974). J. D. Goddard. J. S. Schultz, and S. R. Suchdeo, Facilitated transport via carrier mediated diffusion in membranes: Part I. Mechanistic aspects. experimental systems, and characteristic regimes, .4IChE J. 20:417-445 (1974). J. M. Gonzalez-Fernandez and S. E. Atta, Transport of oxygen in solutions of hemoglobin and myoglobin, Math. Biosci. 54:265-290 (1981). J. A. Jacquez, The physiological role of myoglobin: More than a problem in reactiondiffusion kinetics, M&h. Brow. 68:57-97 (1984). H. Kutchai. J. A. Jacquez, and F. J. Mather. Nonequilibrium facilitated oxygen transport in hemoglobin solution, BiophJ,s. J. 10:3X-54 (1970)
8
J. D. Murray. On the molecular mechanism of facilitated oxygen diffusion globin and myoglobin, Proc. Rq,,. Sot. London Ser. B 17X:95-110 (1971).
9
J. D. Murray, On the role of myoglobin in muscle respiration, J. Throret. B/o/. 47:115-126 (1974). J. D. Murray, Lecfwes on Nonlineur Differentml Equution Models in Bdog,. Oxford U.P., 197X J. Nedelman and S. I Rubinow. Facilitated diffusion of oxygen and carbon monoxide in the large affinity regime, J. Math. Biol. 12:73-90 (1981). D. R. Olander, Simultaneous mass transfer and equilibrium chemical reaction, AIChE J. 6:233-239 (1960). S. I. Rubinow and M. Dembo, The facilitated diffusion of oxygen by hemoglobin and myoglobin, Biophys. J. 18:29-42 (1977). P. F. Scholander, Oxygen transport through hemoglobin solutions, Science 131:5X5-590 (1960). B. A. Taylor and J. D. Murray, Effect of the rate of oxygen consumption on muscle respiration, J. Muth. Biol. 4:1-20 (1970). J. B. Wittenberg, Myoglobin facilitated oxygen diffusion: Role of myoglobin in oxygen entry into muscle, Phpiol. Rev. 50(4):559-635 (1970). J. Wyman, Facilitated diffusion and the possible role of myoglobin as a transport mechanism, J. Biol. Chem. 241:115-121 (1966)
10 11 12 13 14 15 16 17
of hemo-