Theory of the rotational contribution to facilitated diffusion

Z theo~ BioL (1986) 118, 231-246

Theory of the Rotational Contribution to Facilitated Diffusion J. D. MURRAYt AND D. A. SMITH~

"~Centre for Mathematical Biology, Mathematical Institute, Oxford O X 1 3 LB, England and 5; Department of Physics, Monash University, Clayton, Victoria 3168, Australia (Received 27 June 1985, and in revised form 15 August 1985) Gros and others have recently shown experimentally that the facilitated diffusion of protons carded by a form of haemoglobin is enhanced by rotational diffusion of the carrier, whereas facilitated diffusion of oxygen by the same carrier is not. In this paper the theory of facilitated transport by rotating carriers is developed from first principles. The theory confirms Gros's findings that (i) the rotational contribution appears only when the angle of rotational diffusion over the average time the proton remains bound is small and (ii) under these conditions rotation enhances the normal translational contribution by a factor½ at the lowest carder concentrations. The theory also shows that there must be a rotational boundary layer. The phenomenon of facilitated diffusion has been known for some time (Widdas, 1952 and references therein). Facilitated diffusion of oxygen by haemoglobin was first studied in depth by Wittenberg (1959, 1966) and Scholander (1960) for haemoglobin in solution, and by Scholander (1960), Hemmingsen (1965), Kutchai & Staub (1969) and Moll (1969) for in situ haemoglobin in erythrocytes. The main physiological importance of facilitated diffusion seems to be oxygen transport within muscle cells, where the cartier is not haemoglobin but myoglobin (Wittenberg, 1970; Murray, 1974; Wittenberg et al., 1975). The phenomenon is now well understood and requires a zone of reaction equilibrium within the cell (Wyman, 1966; Widdas, 1952) which implies that non-equilibrium boundary layers exist near the surfaces (Murray, 1971; Mitchell & Murray, 1971; Rubinow & Dembo, 1977; Nedelman & Rubinow, 1981). The conditions for existence of this equilibrium zone serve to explain why haemoglobin is a better facilitator of oxygen than myoglobin (Wyman, 1966; Wittenberg, 1970; Murray, 1974) and why oxygen is facilitated by myoglobin but carbon monoxide is not (Murray & Wyman, 1971; Nedelman & Rubinow, 1981). Pedagogical discussions have been given by Murray (1977) and Rubinow (1980). Gros and co-workers (1974, 1976, 1980) have established that proton transport is also facilitated in phosphate, myoglobin and 30% albumin solutions. In all the examples given here, the diffusion flux is less than or equal to the value specified by Wyman's formula (1966), which assumes carrier molecules are in reaction equilibrium with the small molecules and undergo translational diffusion only. However, more recent important work by Gros' group (1984) shows that proton transport by earthworm haemoglobin, which has a very high molecular weight, occurs with a flux in excess of Wyman's formula. In fact they observed not facilitated proton transport as such, but its effect on the ionic transport of CO2 in buffered 231

0022-5193/86/020231 + 16 $03.00/0

(~) 1986 Academic Press Inc. (London) Ltd

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A. S M I T H

solutions near neutrality where proton transport by haemoglobin is the rate-limiting process. They interpreted the excess flux as evidence for facilitation by rotational diffusion o f the haemoglobin molecules.

The Reaction-equilibrium Approximation Here we obtain the generalization of Wyman's formula to include rotational diffusion by a straightforward application of the assumption of reaction equilibrium. The derivation is quick but does not indicate when this assumption is valid. This deficiency is rectified later in the paper. Suppose a flux Jo o f protons is passing through a layer containing haemoglobin in solution. The haemoglobin cannot flow out of the layer (see Gros & Moll 1971 for experimental details). Let h(x) be the proton concentration at position x in the layer, pS(x) the fraction of haemoglobin binding sites (for protons) at x occupied by protons, and p(x, O) the bound fraction for haemoglobin molecules with centre of mass at x and binding site at orientation 0 to the x axis (Fig. 1). Rotational

.X

FIG. 1. Direction of the rotational fluxJo at the binding site of a haemoglobin molecule. diffusion about the y and z axes creates a flux jo of binding sites across the line o f latitude 0 = constant on the surface o f the unit sphere. This flux is given by the rotational equivalent o f Fick's law o f diffusion, viz.

jo(x, O)=-do-~(Np(x , 0))

(1)

where N is the total haemoglobin concentration, assumed independent of position, and d the rotational diffusion constant. This flux has the units o f [density] x [angular velocity] and is tangential to the unit sphere as shown in Fig. 1. The corresponding flux of bound protons in real space along the x-axis is -Rjo(x, 0) sin 0 where R is the rotation radius of the binding site from the centre of mass of the molecule. However, this is the rotational contribution to the flux of bound protons at position x + R cos 0 since x here is the centre-of-mass coordinate. The full rotational part

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of the facilitated proton flux at x is given by replacing x by x - R cos 0 above and averaging over solid angles, i.e. integrate over ½sin 0 dO. The total proton flux Jo at x is therefore given by

Jo = - D , h x ( x ) - NDpS(x) - - N - ~ d R Io'~ sin20 po(x - R cos 0, 0) dO

(2)

where subscripts x and 0 imply partial differentiation and DH and D are diffusion constants. The first term represents diffusion of free protons by Fick's law and the remaining two terms are the fluxes of bound protons arising from translational and rotational diffusion respectively of the haemoglobin carrier. For simplicity, we assume only one binding site per haemoglobin molecule. This is not correct, but it turns out that the kinetics of the oxygenation reaction (Roughton, 1966) can be described phenomenologically to conform with our assumption provided N is interpreted as the density o f haem units (Wyman, 1966). The generalization of Wyman's formula follows by making an appropriate assumption of reaction equilibrium. We write the reaction for a single haem unit, which binds one proton, as k

A+ H.

"B

(3)

k'

where A and B are the two forms of the haemoglobin unit. If units of every orientation are in reaction equilibrium with protons then

kN[1 -p(x, O)]h(x + R cos 0) = k'Np(x, 0), where k and k' are the rate constants, giving

p(x, O)-

K h ( x + R cos 0) l + K h ( x + R cos 0)"

(4)

K = k/k' is the mass action constant. Inserting this into (2) and using the identity

-,Io

pS(x)-5

p ( x - R c o s O , O) sinOdO

which follows from the definitions, gives the required formula J0 = -

DH-~

NKDT "] (l+-~-~))2Jhx(x).

(5)

The transitional and rotational contributions combine to give Wyman's original formula with D, the transitional diffusion constant of haemoglobin, replaced by an effective diffusion constant D r = D + 2 d R 2.

(6)

It might be imagined that the haemoglobin molecule is small enough to render this second term negligible. This is not the case, because in the limit of very dilute

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A. SMITH

haemoglobin solutions D and d both obey the Einstein relations

kaT D = 6¢r~R'

knT d

- -

8,rr,r/R3,

where T is temperature, which relate them to the solvent viscosity 77 (Einstein, 1906), so that

dR2/D-*3/4

(dilute limit).

(7)

Hence Dr-* 3D/2 at infinite dilution. This elegant result is confirmed to an accuracy o f 8% by Gros' measurements. Gros et al. (1984) obtained equation (6) by a heuristic argument involving an analogy between rotational and translational diffusion for the mean-square displacement after time t. Their analogy is not quite exact, as it predicts an unbounded displacement along the x axis from rotations at large times. The present derivation avoids such analogies and uses only the assumption of reaction equilibrium. Rotational enhancement o f the bound proton flux can only occur when reaction equilibrium is achieved for each orientation of every haemoglobin unit. Gros et al. (1984) conjectured that this happens provided the root mean square diffusion angle during the dissociation time 1/k' of the proton is small. Fortunately, this is precisely the limit in which Gros' rotation-translation analogy holds good. Conversely, when this angle is large, i.e. many revolutions, the bound fraction is not able to equilibrate with the local proton concentration over distances of the order of 2R or less, and so the rotational contribution in equation (5) should vanish. Thus

d~ k' << 1 ¢:> (trans. + rot. equilibrium) ~

DT

=

D + 2dR 2

d/k'>> 1 ¢:> (trans. equilibrium only) ~ D T = D

(8a) (8b)

summarizes Gros' conjectures. They appear to fit the facts, since d/k'=O.O1 for protons and 2300 for oxygen in their experiments with earthworm haemoglobin (Table 1). In the last section we show that these conditions are not generally correct, as they hold only when k ' > kh, i.e. dissociation is faster than binding. This is not always so in Gros' experiments, as k h / k ' - 1 0 -4 for protons but - 1 0 for oxygen. When k ' < kh, formulae (8) hold with k' replaced by kh.

The Reaction-diffusion Equations As a preliminary to setting up perturbation expansions about the two limiting equilibrium situations, the form of the continuity equations for each chemical component in the presence of reaction and diffusion must be written down. Consider a one-dimensional stationary flow of protons H and haemoglobin in the unprotonated and protonated forms A and B respectively. Let h(x) be the proton concentration, and let a(x, u) du and b(x, u) du be the concentrations of A and B with centres-of-mass at x and binding sites in the orientation range u to u + du, where u = cos 0. Both A and B are assumed to have the same translational diffusion constant D and the same orientational diffusion constant d (in practice,

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TABLE 1

Parameter values for Gros' ( 1 9 8 4 ) and Wittenberg's ( 1 9 7 0 ) experiments H-EWHb (Gros, 1984)

O2-EWHb (Gros, 1984)

O2-human Hb (Wittenberg, 1966)

1.48×10 -7 6.9x 104 1.38x 10-6 9"2 x 10-s 3-1 x 10-6

1.48×10 -7 6-9x 104 1.38x 10-6 1-6 × 10-5 4.2 × 10 -6

6.4x 10-7

h(0) ( m o l e s . c m -3) k (sec -1 m o l e - l c m 3)

10 -1° 1013

1.3 x 10 -7 2.3 × 10 9

1 '3 x 10 -7 2.3 x 109

k' (sec -t) -Io ( m°le-cm-2 sec-l) L (cm) av. mol. wt. Hb

7 x 106 -0.015 3.7 X 10 6

30 -0.015 3.7 x 106

D (cm2 sec -1 ) d (sec -1) R(cm) D H ( c m 2 sec - a )

N (haem.moles.cm -3)

l ' 0 x 10 6

3-6×<10 -7 1.2 x 10-5 1"1 x 10-5 40 9 x 10-II 0.022 64 450

All values are from the two original papers unless otherwise stated. The molecular radius for human haemoglobin (Hb) is scaled from the earthworm haemoglobin value using the molecular weights given, and the orientational diffusion constant d for human Hb likewise using d oc R -2. Haemoglobin conc~ntrations are based on a haem unit of 16 100, so there are 4 haem units in human Hb and about 230 in EWHb. Where a range of concentrations was used, lower values were preferred, h(0) for protons is for a pH of 7-0. h(0) for oxygen corresponds to a partial pressure of 70 torr, using a Henry's law constant of 5.6x 105 torr/M (Sendroy et al., 1932). k and k' for protons are as estimated in Gros et al. (1984). Jo is well defined only for Wittenberg's experiment, and is based on a flux of 1.3 p.litres/minute, but the same notional value is used for Gros' work. The value of the diffusion constant for protons is that given by Kortum (1972). a very good approximation continuity equations are

-Daxx(X, u ) - d ~ u

i n d e e d ) . I n t h e p r e s e n c e o f r e a c t i o n (3), t h e s t e a d y s t a t e

[(1-uE)a,(x,

u)]=-ka(x,

u ) h ( x + R u ) + k ' b ( x , u)

(9a)

,q

u ) - d ~ u [ ( 1 - u 2 ) b ~ ( x , u ) ] = k a ( x , u ) h ( x + R u ) - k ' b ( x , u) - D h ~ ( x ) = k a S ( x ) h ( x ) + k'bS(x).

(9b) (10)

Here

aS(x) =

11 a(x - Ru, u) du, -1

bS(x) =

I l b(x - Ru, u) du

(11)

-1

m u s t b e i n t e r p r e t e d a s t h e d e n s i t i e s o f b i n d i n g sites, u n o c c u p i e d a n d o c c u p i e d r e s p e c t i v e l y b y p r o t o n s , a t x. O t h e r s y m b o l s w e r e d e f i n e d e a r l i e r . T h e d i f f u s i o n t e r m s a r e d i v e r g e n c e s ( d e r i v a t i v e s w i t h r e s p e c t t o x, u ) o f t h e a p p r o p r i a t e f l u x e s . T h u s t h e r o t a t i o n a l t e r m i n ( 9 a ) c a n b e w r i t t e n a s ajx/au, w h e r e j~ = - j o s i n 0 a n d jo = -d(Oa/O0) is F i c k ' s l a w f o r r o t a t i o n a l d i f f u s i o n . O n l y a x i a l l y

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symmetric rotational diffusion need be considered here. Thus the forms written down here can be seen to be equivalent to those in the previous section. Two kinds o f conservation law follow from equations (9)-(11). The first is somewhat accidental as it requires equal diffusion constants for A and B. Adding (9a) and (9b) gives

+dO n~(x, u) - ~ u [ ( 1 - u 2 ) n , ( x , u)] = 0

(12)

for the total density n(x, u)= a(x, u)+ b(x, u) of haemoglobin units of fixed x, u. We assume that the only physically reasonable solution of this equation is the trivial one

n(x, u)=½N

(13)

where N is the haemoglobin density integrated over u. We may now introduce the bound fraction at fixed orientation p(x, u) and the bound fraction pS(x) for all orientations

p(x, u)

2b(x, u) N

pS(x) = bS(x) N

(14) (15)

so that

f

l

pS(x) = ½

p(x - Ru, u) du

(16)

--I

as before. Then (9a), (9b) and (10) are replaced by two equations,

-Dp,~(x, u) - d °au [(1 -u2)pu(x, u ) ] = k ( 1 - p ( x , u ) ) h ( x + R u ) - k ' p ( x , u) - N - I D , h ~ ( x ) = - k [ 1 -pS(x)]h(x) + k'pS(x).

(17) (18)

A continuity equation for pS(x) can be derived from (17). The rotational term is processed by integrating by parts over u, using the identity

d

±

d-u [(1 - u2)p,(x - Ru, u)] = au [(1 - u2)p~(y, u)]y=x-e, - R(1 - u2)px~(x - Ru, u). Hence

-DpS(x) -

(1 - u2)px,(x - Ru, u) du = k[1 -pS(x)]h(x) - k'pS(x).

(19)

--1

In this equation the effect of rotations on the transport of binding sites is clear. However, (18) and (19) are not closed equations for h(x) and pS(x) in the presence o f rotational diffusion. The main importance of (19) is that it leads to a second conservation law on combining with (18). Adding the two equations and integrating

ROTATIONAL

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237

once with respect to x gives

_ N _ l Dnhx(x) _ DpS(x) _ d R f l (1 - u2)pu(x- Ru, u) du = constant. 2 J-I

(20)

To determine the constant in this conservation law, we require boundary conditions at the surfaces of the membrane. These conditions are complicated by the rotational degrees of freedom. In fact, there are three independent flux conditions at each surface, say x = 0 and x = L, viz.

hx(x) = - J o / D , px(x, u) = 0 p~x(x) = o

x = O, L

(21a)

x = R, L - R

(21b)

x = o, L

(21c)

since the surfaces are supposed to be permeable to protons but impermeable to haemoglobin. Jo is the external proton flux. There can be no flux o f the total haemoglobin concentration N, which is constant throughout the membrane, so (21b) and (21c) involve only the bound fraction. Equation (21b) implies no translational flux of the centre-of-mass of protonated haemoglobin, and (21c) no flux of bound protons. In the case of point molecules (21b) and (21c) are replaced by a single boundary condition; here we must establish that they are independent conditions. Suppose (21b) only is valid. Then the behaviour of p(x, u) near x = R must be o f the form p ( x , u) = o

(x < R)

=po(u)+½p2(u)(x-R)2+...

(x>-R)

(22)

as (21b) removes the term linear in x - R. Hence p(x - Ru, u) > 0 only for x - Ru < R, i.e. u < - 1 + x~ R, and so

f -l+x/R pS(x) = ½

d--I

=½po(-1) R

p(x - Ru, u) du O ~

.

(23)

So pS(x) ~ Po(- 1)/R as x ~ 0, and (21c) is indeed an independent boundary condition, equivalent to requiring that P o ( - 1 ) = 0, i.e. the bound fraction must be zero at the walls. The physical reason for this extra condition can be uncovered by noticing that a gradient in pS(x) can occur even when the distribution p(x, u) is constant in x and u; the gradient arises purely from the assumed geometrical constraint that haemoglobin molecules lie entirely within the membrane. Equation (21b) also leads to the estimate -1

I , x ( l - u 2 ) p ~ ( x - R u , u) du=~po(-1) -R

+ 0 -R

(x<< R)

for the last term in (20). On using all these conditions in (20), we have an expression

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A. S M I T H

for the proton flux

Jo = -DHhx(x) - NDpS(x) - N 2dR fJ l (1 - u2)pu(x - Ru, u) du

(24)

-1

which agrees with that in equation (2), written down by inspection. This result can be regarded as a connecting formula, since it relates the external proton flux Jo to the concentrations of reactants in the interior o f the membrane, e.g. in some inner reaction equilibrium zone if such exists. Alternatively, (24) can be formally integrated to yield an expression for the proton concentration h(x), viz.

h(x)=h(O)

Jox DH

N dR 2Dn

ND s s ~ [p (x)--p (0)] dx'

f o l -1'

(1 - u2)p~(x ' - Ru, u) du.

(25)

"Inner" Perturbation Theory At the beginning of this paper, the assumption of reaction equilibrium was applied in a naive way to obtain an expression for the rotational contribution to facilitated diffusion. We now construct explicit perturbation expansions to obtain conditions of validity of the two equilibrium regimes, namely translation + rotation equilibrium where a rotational contribution to the flux is present, and the translational equilibrium only where it is absent. To do this it is sufficient to obtain perturbation expansions for the inner or reaction-equilibrium zone. It is well known that this zone cannot extend to the membrane surfaces because the extra reaction condition makes it impossible to satisfy all the boundary conditions (21). No attempt will be made here to complete the mathematical solution of the problem by finding concentration distributions within the boundary layers, since one object of this paper is to identify the correct perturbation parameters. The equations to be solved within the membrane for p(x, u) and h(x) are (17) and (25). The parameters involved are N, h(0), J0, D, DH, d, k, k', R and the membrane thickness L. The order of magnitude of d is assumed to be given by (7) even though the solution might not be dilute; this is indicated by writing

d = s D / R 2,

s = O(1)

(26)

so s ~ 3/4 as n -~ 0. Apart from L and R, there are four characteristic lengths

( D

l=\\]

,

l'=~-~)

,

l,=\-~n ]

,

l,,,-

o.h(o) Jo

(27)

l is the " o n " reaction length for haemoglobin, i.e. the distance it diffuses before binding a proton, l' is the "off" reaction length for protonated haemoglobin and IH the " o n " length for protons, li,j is the injection length for protons injected at flux Jo and concentration h(0), over which diffusion reduces the proton concentration to zero.

ROTATIONAL

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The q u a n t i t y of most relevance is actually the e q u i l i b r i u m length for h a e m o g l o b i n , b e i n g the d i s t a n c e a stream of h a e m o g l o b i n diffuses before c o m i n g to reaction e q u i l i b r i u m with protons. This q u a n t i t y c a n n o t be defined precisely for arbitrary d e v i a t i o n s from e q u i l i b r i u m , b u t a good a p p r o x i m a t i o n to it is to take the smaller o f I a n d 1'. This is a good estimate for the width o f the t r a n s l a t i o n a l b o u n d a r y layer. So to establish a n i n n e r z o n e o f t r a n s l a t i o n a l r e a c t i o n e q u i l i b r i u m we require

min(l,l')<
/n<
(28)

We also require l~,j -> L in order to have s o m e residual p r o t o n c o n c e n t r a t i o n at the back o f the m e m b r a n e . I n r e d u c i n g the p r o b l e m to d i m e n s i o n l e s s form, it is c o n v e n i e n t to choose the scale length to be not L b u t l~,~, since the i n j e c t i o n length d e t e r m i n e s the spread of the p r o t o n d i s t r i b u t i o n . H e n c e define d i m e n s i o n l e s s variables

£ = x/l,.,

/~(£) = h ( x ) / h ( O )

and dimensionless parameters

e = (l'/l~.j) 2 = D/k'l~nj, r = R/l~.j,

ct = k h ( O ) / k ' ,

~z = d / k '

fl = N D / h ( O ) D n ,

[3'= N d R 2 / h ( O ) D n .

(29a) (29b)

F o r definiteness, we a s s u m e that

l'
(i.e.a
(30)

so that l' m a y be t a k e n as the e q u i l i b r i u m length for h a e m o g l o b i n . The changes required w h e n l ' > l are discussed later. By ( 2 6 ) , / 3 ' = s/3 a n d ~ = s e / r 2. Since r<< 1 always ( T a b l e 1), we see t h a t / z >> e a n d so tz <> 1 are both possible while k e e p i n g TABLE 2

Derived parameters

in the Gros and experiments

Wittenberg

H-EWHb O 2 - E W H b O2-humanHb (Gros, 1984) (Gros, 1984) (Wittenberg, 1966) 1 (cm) I' ( c m )

1.2 x 10 -5 1 . 5 x 1 0 -7

ln (cm)

l'7x 10-6 1"0 x 10-4 2.0x 10-6 -0.017 --

li. j ( c m )

e e'= e/a /z' =/z/a a /3

l ' 4 x 10 -4

50

2-2

×

10 - s

7"0x 10-s 4"1 x 10-s 0"023 -9-3x 10-7 -230 10 0"30

4"6 x 10 -3 l ' 3 x l 0 -4 22"2x 10-~ 0-017 7.1 X 10 -6 -3300 7"5 4"5

All values are derived from Table 1. Where I'< I(a < 1), as for H-EWHb, the values of e and p. are given, but when l'> l(a > 1), as for the two oxygen experiments, e' and p~' are given, p. < 1 for H-EWHb implies that there is a rotational contribution to facilitated diffusion, but p.'> 1 for O2-EWHb and O2-human Hb implies there is not.

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e<< 1. Then the governing equations (17) and (25) take the form

ep,~(x, u)+l~-~u [(1 - u2)p~(x, u)] = ~b[p, h] 4~[P, h] =p(x, u ) - a ( 1

h(x)=l-x-/3[pS(x)-pS(O)] where

-p(x, u))h(x+Ru)

I0f -'I_ dx'

pS(x)-5

(1-u2)p',,(x-ru, u) du

(31a) (31b) (31c)

!

p(x-ru, u) du

(31d)

I

and tilde symbols have been dropped. 4~ is a scaled reaction rate. Now 4~ =0(1) if

p(x,u)=O(1), because o~<1 and because we expect 0-
This is expected when e, /x are both small quantities. To describe the purely rotational part of the problem, set e = 0,

/z << 1

(32)

and construct an expansion in positive powers o f / ~ to show that is the correct convergence parameter. The details of this procedure are given in Appendix A. Here we quote the solution to zeroth order in /z, which is of course the full reactionequilibrium solution discussed earlier

oth(°)(x + ru) p~°)(x, u) = 1 +oth(°)(x+ ru) {

1

h(°)(x)=l-x+flr l + a h ( ° ) ( x )

(33) 1 } l+~t

(34)

where/3T =/3 +]/3'. (34) is the integrated version of Wyman's formula (5) with the rotational contribution included, and written in dimensionless units. It is valid only in the limit of very rapid translational and rotational reaction equilibrium. The solution to first order in/.t is given in Appendix A. (B) T R A N S L A T I O N A L E Q U I L I B R I U M O N L Y

This is expected when e is small but/~ is large. Put e=0,

/~>>1

(35)

so that (31a) becomes / z - ' 6 [ p , h] =

[(1 -

u2)p,,(x, u)].

(36)

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241

Now seek an expansion of p, ~b and h in positive powers o f / z -1. Results to O0z -1) are given in Appendix B. Here we give only the solution in the limit/z ~ oo, which is of some interest for it has not yet been shown that this limit is identical to the case e = 0 for point carders. In fact, the point carrier limit (/x = r = fl' = 0) cannot be achieved simply by letting r ~ 0 because of the Einstein law (26) which implies t h a t / z oc r -2. Instead, r ~ 0 implies/z ~ oo, which is the limit considered here. W h e n / z ~ oo, the solution p(O) to (36) satisfies (1 where f(x) p<°)(x, u) is

u2)p~)(x, u) = f ( x )

is an arbitrary real function. On physical grounds we assume that regular at u = +1. Hence f(x)=0 and we may write

pC°)(x, u) = p(°)(x).

(37)

To determine this function of x, we require a mathematical form of the idea that rotational diffusion averages out the proton concentration seen by a haemoglobin binding site. At a particular orientation u, the reaction rate fb[p(x, u), h(x)] will in general be non-zero. However, the orientationally averaged rate is

~=½

I' 4~[p,h]du --1

_- ~1 f l

- 0- [ ( 1 - u 2 ) p . ( x , _IOu

=½[(1 -

u)] du

u2)p.(x, u)]'_,.

(38)

Hence ~=0

if[ lim (1--u2)pu(X,

u)=0.

(39)

M~±I

This result holds to all orders in/~-1. This restriction is satisfied if p(x, u) is regular in u at u = + 1, which restricts p(x, u) to linear combinations of Legendre polynomials of the first kind and degree zero. In particular, q~(o) = 0

(40)

which determines p(O). We find

P(°)(x) -

ah(°)(x)

1 + ah(°)(x)

(41)

where l

h(°)(x) =½

ff ht°)(x + ru) du --I

2

r (o) =h (o) (xl+-~h~(x)+...

(42)

is the expected average of the proton concentration over the haemoglobin molecule.

242

J.D.

M U R R A Y A N D D. A. S M I T H

Equation (31c) gives h(°)(x) = 1 - x - fl[pS°(x) - pS°(0)]

(43)

where pS°(x) is the centre-of-mass average of p(°)(x, u) and given by the usual formula (16). To zeroth order in r, /~co)= h(0) and /5c°) =pSO. Then we recover the integrated version of Wyman's formula, viz.

h(°)(x)=l-x+~

lla

+O(r2).

The corrections to first order in - 1 given in Appendix B confirm that - 1 correct perturbation parameter in this case.

(44) is the

When l ' > l, i.e. c~> 1, then l is a better approximation to the equilibrium length than l' as binding is fast and dissociation slow. The perturbation expansions must be rearranged accordingly. Specifically, (31a) must be divided by a, giving an equation of the same form but with e,/x, ~b replaced by

e'= e/s,

~'= ~1~,

~'= 4~la.

Now ~b'=0(1) if p(x, u ) = 0 ( 1 ) (see (31b)) so that e' a n d / x ' are the new expansion parameters. When e ' = 0 , the condition for a rotational contribution is /~'<< 1, not /x << 1. Table 3 gives the full set of conditions for the two regimes. It is seen to include Gros' conjectures (8) as a special case. TABLE 3

Conditions of validity for the two reaction-equilibrium regimes I"<1 [ k ' > kh(0)] d

I'>1 [ k ' < kh(0)] d

Translational + rotational equilibrium

-~ 1

kh~O-----S)<,1

Translational equilibrium only

d -->>1 k'

d >>1 kh(O)

Concluding Discussion Gros' (1984) expression for the rotational contribution to the facilitated diffusion of protons by haemoglobin has been derived from first principles, and Gros' conjectured condition for the existence of this effect is shown to be included in a more general condition, viz. that reaction equilibrium be rapid over the orientational diffusion time of the haemoglobin molecule. The theory presented shows that rotational diffusion alone can facilitate the transport of small molecules: this situation may be approached as the solution viscosity is increased if, z.~ Lindstrom et al. (1976) suggest, D vanishes faster than d.

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T h e r e will also b e d i s t i n c t b o u n d a r y layers for t h e e s t a b l i s h m e n t o f t r a n s l a t i o n a l a n d r o t a t i o n a l e q u i l i b r i u m . T h e s t r u c t u r e o f these layers is n o t c o n s i d e r e d in this p a p e r , b u t t h e i r w i d t h s are easily e s t i m a t e d , e.g. f r o m (31a). T h e t r a n s l a t i o n a l l a y e r h a s a w i d t h AxT -- e~/Zli,j = l' = ( D / k ' ) ~/2 a n d t h e r o t a t i o n a l l a y e r a w i d t h AxR - - / ~ R = d R / k ' , a s s u m i n g l ' < I. U s i n g (26) gives A x r = ( t z / s ) ~ / 2 R w h e r e s = 0 ( 1 ) . H e n c e A X R << SI/2AXT << R

(/z << 1, i.e. r o t a t i o n a l e q u i l i b r i u m )

AXR >> St/~AxT >> R

(/~ >> t, i.e. n o r o t a t i o n a l e q u i l i b r i u m ) .

(43)

One of us (D.A.S.) wishes to thank the Centre for Mathematical Biology, University of Oxford, where most of this work was done, for its hospitality during 1983. REFERENCES EINSTEIN, A. (1906). Annalen der Physik 19, 371. (Reprinted in Investigations on the Theory of Brownian Motion (New York: Dover) 1956.) GROS, G. & MOLL, W. (1971). Pfliigers Arch. 324, 249. GROS, G. & MOLL, W. (1974). J. gen. PhysioL 64, 356. GROS, G., MOLL, W., HOPPE, H. & GROS, H. (1976). J. gen. Physiol. 67, 773. GROS, G., GROS, H., LAVALE'r'rE,D., AMAND, B. & POCHON, F. (1980). In Biophysics and Physiology of Carbon Dioxide. (Bauer, C., Gros, G. & Bartels, H. eds). Berlin: Springer. GRos, G., LAVALETTE,D., MOLL, W., GROS, H., AMAND, B. & POCHON, F. (1984). Proc. natl. Acad. Sci. U.S.A. 81, 1710. HEMMINGSEN, E. A. (1965). Acta Physiol. Scand. 64, suppl. 246, 1. KORTUM, G. (1972). Lehrbuch der Elektrochemie, 5th edn. Verlag Chemic. (Weinheim Bergstra). KUTCHAI, H. & STAUB, N. C. (1969). J. gen. Physiol. 53, 576. LINDSTROM, T. H., KOENIG, S. H., BOUSSIOS, T. & BERTLES, J. F. (1976). Biophys. J. 16, 679. MITCHELL, P. J. & MURRAY, J. D. (1971). Biophysik 9, 177. MOLL, W. (1969). Arch. gen. Physiol. 305, 269. MURRAY, J. D. (1971). Proc. Roy. So~ B178, 95. MURRAY, J. D. (1974). Z theor, biol. 57, 115. MURRAY, J. D. (1977). Lectures on Non-Linear Differential Equation Models in Biology. Oxford: Clarendon Press. MURRAY, J. D. & WYMAN, J. (1971). J. biol. Chem. 246, 5903. MURRAY, J. D. (1984). Asymptotic Analysis. New York: Springer-Verlag. NEDELMAN, J. & RUBINOW, S. I. (1981). J. math. Biol. 12, 73. ROUGHTON, F. J. W. (1966). J. gen. Physiol. 49, suppl. 1, 105. RUBINOW, S. I. (1980). In: Mathematical Models in Cellular and Molecular Biology (Segel, L.A. ed.). Cambridge: Cambridge University Press. RUBINOW, S. I. & DEMBO, M. (1977). Biophys. J. 18, 29. SENDROY, J., JR., DILLON, R. T. & VAN SLYKE, D. D. (1934). J. biol. Chem. 105, 597. SCHOLANDER, P. F. (1960). Science 131, 585. WIDDAS, W. F. (1952). 3". Physiol. 118, 23. WITTENBERG, J. B. (1959). Biol. Bull. 117, 402. WITTENBERG, J. B. (1966). J. biol. Chem. 241, 104. WITTENBERG, J. B. (1970). Physiol. Rev. 50, 559. WITTENBERG, B. A., WITTENBERG, J. B. & CALDWELL, P. R. B. (1975). J. biol. Chem. 250, 9038. WYMAN, J. (1966). J. biol. Chem. 241, 115. APPENDIX A P e r t u r b a t i o n t h e o r y for e = 0 , / ~ << 1 p ( x , u ) = p~°)(x, u) + IxpCl)(x, u) + . . .

(A1)

244

J.

D.

MURRAY

AND

D. A. SMITH

and similarly for 4' and h. On equating powers o f / z in equations (31) 4~o) = 0

(A2a)

~b(l) = au 0 [(1 - u2)p~)(x, u)]

(A2b)

etc. where 4, C°)= pC°)(x, u) - ceil -p~°)(x, u)]hC°)(x + ru)

(A3a)

#b~l)=[l+a'h~°)(x+ru)]p")(x, u)-a[1-pC°)(x, u)]h~l)(x+ru)

(A3b)

ht°)(x) = 1 - x - fl (pS°(x) - pS°(0)) ---

(1 - u2)p~)(x ' - ru, u) du

dx'

2r

(A4a)

-1

h
(1 - u2)p~X)(x ' - ru, u) du.

dx'

(A4b)

-1

The zeroth order solutions are given in (33), (34). A cursory look at (A2b) and (33) suggests that perhaps I~/r and not /z is the correct expansion parameter: the function pt°)(x, u) is a function of x + ru only, so the leading term in the right-hand side of (A2b) is O(r), taking p ~ ) = 0 ( 1 ) . In this case, first impressions are misleading. 0(r) terms must be retained to recover the correct solutions p~X)and h ~1)to zeroth order in r, because of the factor r -~ appearing in the last term of (A4b). After some manipulations, we find

h~,)(x)

afl' {

=--~-

otflr

~-' ~

l+[l+ah~O)(x)]2j

°th:°)(x')

[l~,)]3dx

,

2 +O(r )

(as) a p°)(x, u) = [1 + ah(°)(x)] 2 hO)(x)+O(r). Certain "internal" boundary conditions have been applied to get these results. They are not the true boundary conditions (21), but arise at the inner edge of the boundary layer as a result of choosing h(0) = 1 there. By equation (34), h<°)(0) = 1 also, and so h " ) ( 0 ) = 0 . Existing zeroth order results then enable zeroth order derivatives at x = 0 to be obtained, giving h~)(0) =

(l+a)2

#-1

(1 +a)2+a/3r PS°(0) =

(A6)

(1 + a ) 2 + a f l r ~ 0 .

The values on the extreme fight-hand side represent the true boundary conditions on the outer side of the boundary layer, whose width is neglected in this analysis. Hence we see that if the boundary layer is small (i.e. e,/~ << 1) then changes in the

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values o f h(x) and pS(x) across the layer are small and can perhaps be neglected, whereas changes in their derivatives are finite. This "matching" technique is standard singular perturbation technique; see, e.g., the book by Murray (1984). The analysis on which this Appendix is based assumes l' < 1 (30). If l' > l, a similar analysis based on the parameters e', /z' (45) shows that h~)(x) is increased by a factor a. APPENDIX B Perturbation theory for e = 0,/x >>1

p(x, u)= p{°)(x, u)+ tx-~p{1)(x, u ) + . . .

(B1)

and so on for ~b, h. Now q5tin, thm and h (°), h C~ are correctly given by the previous formulae (A3), (A4). Equating powers of ;z -~ gives

±

Ou [ ( 1 - uZ)p~)(x, u)] = 0

(B2a)

2_ OU [(1 -- u2)p~l)(X, U)] =

(B2b)

q~(Ot[p, h i .

Also, formula (38) holds to all orders in p - i provided (39) does, i.e. 4~(o~= q~l~ = . . . = 0.

(B3)

The zeroth-order solutions (45), (43) are derived in the main text. As explained in Appendix A, pCt)(x, u) must be evaluated at least to 0(r) to obtain h(~)(x) to zeroth order in r. The first integral of (B2b) is

1 p~)(x, u) = _--~2 1 _ r

~b(°)[

o~ht~°)(x)

ptO~, h~O~] d u + C,(x)

+ O ( r 2)

(B4)

2 1 + ah{°)(x) on choosing C~(x) to make p(O} regular at u = :t:1. Hence

r ah~)(x) p{')(x, u) - 2 1 + cth{°}(x) u + C2(x).

(B5)

C2(x) is determined by using the condition ~b(~)= 0. To evaluate th C~ use h{l)(x)=_fl[C2(x)_C2(O)]_a~ 3 fo ~ 1 +h!:)(x') otht°)(x ') d x ' + O(r2).

(B6)

Hence q~o) = [1 + cth(m(x)]C2(x)

ah(l)(x)

1 + aht°)(x) t- O(r") = 0,

(B7)

which in conjunction with (B6) fixes C2(x). On setting x = 0 in the resulting equation for C2(x) we find that C : ( 0 ) = 0.

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J.D.

MURRAY

AND

D. A. S M I T H

The final form for the proton distribution to 0(7/-~) is h(~)(x) = - - - ~ -

1 + [1 +h(O)(x)]2 ]

l,

l+~h(O)(x,)dx'+O(r2).

(B8)

The above hold only for l ' < I. When l ' > similar analysis with 1 / ~ ' as expansion parameter shows that the right-hand side of (B8) should be divided by ~ to get the right result.