Theoretical interpretation of isotope labelling experiments in cells in which the label is chemically incorporated: The example of orthophosphate

J. theor. Biol. (1988) 134, 351-364

Theoretical Interpretation of Isotope Labelling Experiments in Cells in which the Label is Chemically Incorporated: The Example of Orthophosphate G. J. KEMP, A. BEVINGTONt AND R. G. G. RUSSELL

Department of Human Metabolism and Clinical Biochemistry, University of Sheffield Medical School, Beech Hill Road, Sheffield S10 2RX, U.K. (Received 10 September 1987, and in revised form 28 April 1988) Studies of transport across the plasma membrane in intact cells frequently involve measuring the incorporation of a labelled extracellular species into the cells. Unfortunately, if the labelled species is metabolized in the cell, the kinetics of labelling are made more complicated. Using the example of the incorporation of 32p-labelled orthophosphate into cells, we describe a mathematical model which allows for this complication, and show how this may alter the interpretation of experiments. The analysis is widely applicable to cellular labelling studies with any species that undergoes chemical exchange with a large cellular pool.

I. Introduction

A common method for studying transport across the plasma membrane in intact cells is to add to the extracellular medium a labelled species (normally a radioactively labelled molecule or ion) and to monitor incorporation of the label into the cells, or efflux of the label from prelabelled cells. If the species is metabolically inert (e.g. the chloride ion) mathematical analysis of the labelling kinetics is straightforward. Unfortunately, many membrane-permeant species of biological interest can be metabolized inside the cell, and this complicates the analysis. General mathematical treatments (Sheppard & Householder, 1950; Robertson, 1957) have often been neglected in the practical analysis of experimental results. For example, in the field of phosphate metabolism, studies of the transport of the 32p-labelled mineral precursor, orthophosphate (Pi), across the plasma membrane have frequently disregarded the fact that 32pi readily exchanges with unlabelled phosphate groups chemically incorporated into organic phosphates (Kemp et al., 1986). In view of the increasing interest in the role of Pi in the control of cell function (Bevington et al. 1986), we have developed a mathematical model which describes the effect on 32p-labelling kinetics of exchange between cell Pi and a single organic pool. This model is also applicable to similar studies using any labelled species that undergoes exchange with a large, metabolically transformed pool inside the cell. A preliminary account of this work has been presented (Kemp et al., 1987a). ? To whom correspondence should be addressed. 351

0022-5193/88/190351 + 13 $03.00/0

O 1988 Academic Press Limited

352

G.J.

KEMP

ET

AL.

2. A Model of Cellular Pi Metabolism

The simplest model of cellular Pi metabolism is a catenary model (Sheppard & Householder, 1950) consisting of three pools: extracellular Pi (1); cellular Pi (2); and a pool (3) representing all organic phosphates, which need not be further specified (see Fig. 1). Conventional analysis of 32P-labelling experiments neglects this third pool (the "two-pool model"): we are interested in the consequences of exchange between the cellular pools of Pi and organic phosphate (the "three-pool model"). Let I (mmol/1 cells/hour) be the rate of exchange of Pi between cellular and extracellular Pi, and G (mmol/l cells/hour) the rate of chemical incorporation of Pi into the organic pools (the rate of organic exchange). Let 3' be the ratio of G to L (Probable values of these parameters for the human erythrocyte are discussed in section 5.). We assume unless stated that all pools are at steady state. We represent concentration by C (mmol/l), 32p activity by A (cpm/l), and specific activity ( A / C ) by a (cpm/mmol). It is convenient to define a membrane permeability constant, k (1/hour), as the ratio of transmembrane Pi flux (I) to the cellular concentration of Pi (C2). (Note that this involves no assumptions about the mechanisms of membrane transport or other factors determining transmembrane Pi distribution.) We define a parameter 8 as ( C 2 / C 1 ) [ h / ( 1 - h ) ] , where C~ is extracellular Pi concentration, and h is the fractional cell volume of the incubated cell suspension. (For an incubation of whole human blood, 8 =0.5-0.7.) We define to as C2/C3, the ratio of the cellular concentrations of Pi and of all exchangeable organic phosphate groups: to will generally be small. In the human erythrocyte, C2 is 0.6 mmol/l cells and the total concentration of organic phosphate groups is about 14 mmol/l cells (Grimes, 1980), so to could be as low as 0.04. In sections 3a to 3c we have simplified the calculations by taking to as zero. Three differential equations describe the changes in the specific activities of the three pools following addition of 32pi to the extracellular medium ("influx experiments") or the addition of prelabelled cells to unlabelled medium ("efflux experiments"): d a l / dt = k S ( a 2 - al)

(la)

da2/dt = k[al - (1 + 'Y)a2 + a 3 ]

(lb)

da3/dt = k'yto(a2- a3)

(lc)

POOL 1

POOL2

],, (3

] POOL 3

F FIG. 1. Model of cellular Pi metabolism. Pool 1 is extracellular Pi; pool 2 is cellular Pi; pool 3 is cellular organic phosphate. I is the transmembrane Pi flux; G is the exchange flux between cellular Pi and organic phosphate.

I S O T O P I C L A B E L L I N G OF C E L L U L A R O R T H O P H O S P H A T E

353

TABLE 1

Labelling kinetics according to two-pool model (no organic exchange) during 32pi influx and efflux experiments influx experiments a 2 / a ° = [1/(1 +8)]{1 - e x p [ - k ( l +B)t]} a t / a ° = [1/(1+ 8)]{1 + 8 x exp [ - k ( l + 8)t]}

(al - a2)/a ° = exp [ - k ( 1 + 8)t] efliux experiments a2/a °=[l/(l+8)]{8+exp

[-k(l+8)t]}

a , / a ° = [B/(1 + 8 ) ] { 1 - e x p [-k(1 +8)t]} ( a 2 - a t ) / a ° = e x p [-k(1 + 8 ) t ]

8 ~ 0 in dilute suspensions.

The equations in Table 1 describe the kinetics of a2 and a~ during experiments with cells conforming to the two-pool model. These are derived from eqns (la) and (lb) with y = 0, and are appropriate to the simple case of labelling experiments with non-metabolizable substances. We shall derive the corresponding equations for the three-pool model, which are appropriate to experiments with 32pi, and show how inaccurate estimates o f model parameters may result from a " t w o - p o o l " analysis.

3. Influx Experiments

( a ) Initial rates In 32pi influx experiments, the initial rate of cellular labelling (the initial rate of increase of a2/a~), is k, and the initial rate of loss of extracellular label (the initial rate of decrease of a~/a ° or A~/A °, where the superscript refers to the value at t = 0), is k& These estimates are not affected by organic exchange. We shall assume in what follows that the concentrations of all Pi and organic pools are at steady state. However, even if they are not, the rate of Pi influx, I~2, is always equal to the initial rate of increase of A2/a ° (e.g. Tenenhouse & S c r i v e r , 1975; Johnson & Freinkel, 1985) and to the initial rate of decrease of ( A J a ° ) / [ ( 1 - h)/h].

( b ) Short incubations in dilute suspensions Firstly, we assume that h is small (8-~0), so that a~ can be taken as constant. The solution for a2/a~ is the first equation in Table 2. Instead of increasing to 1 as in the two-pool model (Table 1; see Fig. 2(a), curve 1), it reaches a " p l a t e a u " value of E = 1/(1 + y), which is less than 1 (see Fig. 2(a), curve 2). This arises from the removal of labelled Pi and release of unlabelled Pi by the large, incompletely-labelled organic pool. Furthermore, the exponential rate constant describing this increase is K = k(1 + y), not k as in the two-pool case. Therefore, since k = KE, k is correctly

354

G.J.

KEMP

ET AL.

TABLE 2

Labelling kinetics according to three-pool model during short 32p/influx experiments Dilute cell suspensions (where 8 = 0) a2/a~ = [1/(1 + y)]{l - e x p [ - k ( l + 3,)t]} A3/al = [G/(1 + y)]{t-[(a2/al)/k]} Concentrated cell suspensions (where 8 > 0) ~ [ 2 / ( a 2 / a t ) ] - (1 + 3, - 8) = ( h " - h '){1 + exp [ - k ( a " -

h ')t]}/{1 - e x p [ - k ( a " - h ')t]}

a2/a° = [exp (-ka't) - exp (-k,V't)]/(,~"- ,~') al/a ° = {[(8 - ;t') x exp (-kX"t)] - [(6 - ,V')x exp (-kX't)]}/(,~"- X') ~,V and A" are roots of A 2 - ( I + 8 + y ) A + S y = 0 .

A useful approximation for fairly small 8 is

~'=(1 + 8+ y ) - x " ~ 8y/(l+ 8 + ~). estimated as k ' = K ' E ' , where E ' is the observed plateau and K ' is the observed slope of the theoretically linear plot of - I n [ E ' - ( a 2 / a l ) ] against t. An equivalent plot of - I n {1 - [ ( a 2 / a l ) / E ' ] } has been used to estimate k (Levinson, 1966), but this too should have slope K, and therefore overestimates k by a factor (1 + 3')- Obviously this overestimation becomes greater as 3' increases, although it has been wrongly assumed elsewhere that when y becomes large this makes the error negligible (Levinson, 1966). If the plateau value cannot be established because the experiment is too short, a simple plot of - I n [ 1 - (a2/al)] against t, though non-linear, has a slope of k near t=0. In the two-pool model, since a2 = a~ at isotopic equilibrium, cellular Pi concentration can be estimated from cellular 32p activity as C 2' - A2/a~ However, as we have seen, in the presence of a large exchangeable organic pool, a2/a~ attains a plateau less than 1, so A2/a~ will underestimate C2 by a factor E = 1/(1 + 3') (see Section 3, below). Other methods relying on the assumption that a~ = a2 (e.g. Prins et al., 1986) may also be subject to a similar error. Although the mean specific activity of the organic pool is, by hypothesis, negligible, its total activity, A 3 , c a n easily be measured as the cellular 32p activity remaining when the 32pi has been extracted (Kemp et ai. 1986). The kinetics of A3/a~ are as given in Table 2. After a2/a~ has reached a plateau, A3/al is linear with slope 1 / [ ( 1 / I ) + ( 1 / G ) ] . If 3/ is large, a2 is negligible and A 3 / a ~ = l x t. If 3' is not too large (see Section 3(g) below) the rate of organic incorporation can be estimated from the mean value of a2 and the mean rate of increase of A 3 a s

G'= (dA3/dt) . . . .

/(a2) . . . .

(2)

For Pi, this yields two tests of the three-pool model. Firstly, for a cell whose energy metabolism relies entirely on anaerobic glycolysis (e.g. the human erythrocyte) G' should be at least equal to the rate of cellular generation of lactate since, in such a cell, Pi incorporation proceeds solely by means of the reaction catalysed by

ISOTOPIC LABELLING OF CELLULAR ORTHOPHOSPHATE

I

-

(a)

355

I

?

0

¢ (hours)

f /

/ / / / / / ........

I

I

, I

I

I

I

I

I

I....

I

50

0 f (hours)

FIG. 2. Example of the theoretical kinetics of relative labelling of cellular Pi (az/al) in an influx experiment. (Assume k = 1, which is comparable with the value observed in some experiments with human erythrocytes (see section 2).) (a) Short incubations (7 hours). Curve 1: Cell with no organic incorporation of Pi (3, =0), incubated at low haematocrit (B=0). See Table 1. Note that the final value is unity; Curve 2: Cell with a very large organic phosphate pool (o~ =0) which exchanges with cellular Pi (assume 7 = 1), incubated at low haematocrit (B -~0). See Table 2 and section 3b. Note that the stable "plateau" value is less than unity; Curve 3: Cell with a very large organic pool (co = 0) which exchanges with cellular Pi (assume y = 1), incubated at high haematocrit (assume 8 = 0-5). See Table 2 and section 3c. Note that the stable plateau is slightly higher than at low haematocrit (curve 2); Curve 4: Cell with a moderately large organic pool (assume o~= 0-1 ) which exchanges with cellular Pi (assume 3' = 1), at low haematocrit (B = 0). See Table 3 and section 3d. Note that the plateau value increases slowly as the organic pool labels. The broken line shows the corresponding organic phosphate labelling, aa/a t . (b) Long incubation (50 hours). Conditions as for curve 4 in Fig. 2(a), Note how slowly a2/a~ approaches unity. The broken line shows the corresponding organic phosphate labelling, a3/a I .

g l y c e r a l d e h y d e - 3 - p h o s p h a t e d e h y d r o g e n a s e ( G 3 P D H ) [E.C. 1.2.1.12], so at least 1 m o l e o f l a b e l l e d Pi is i n c o r p o r a t e d for each m o l e o f lactate g e n e r a t e d . If G 3 P D H is n e a r to e q u i l i b r i u m ( N e w s h o l m e & Start, 1973) G ' will be c o n s i d e r a b l y greater t h a n the rate o f lactate g e n e r a t i o n , since i n c o r p o r a t i o n by e x c h a n g e b e t w e e n reactants a n d p r o d u c t s c a n o c c u r in a n e a r - e q u i l i b r i u m r e a c t i o n even if the net flux t h r o u g h the system is negligible. S e c o n d l y , even i n a m o r e c o m p l e x cell in which Pi is i n c o r p o r a t e d at several o t h e r steps the f o l l o w i n g test o f the self-consistency o f the t h r e e - p o o l m o d e l can be used: from their definitions, G ' , m e a s u r e d b y m e a n s o f e q n (2), s h o u l d be e q u a l to y ' k ' C 2 , where k' is the initial rate o f a2/a~ a n d 7' is e s t i m a t e d f r o m the p l a t e a u v a l u e o f a j a r as ( 1 - E ' ) / E ' .

356

G . J.

KEMP

ET

AL.

( c) Short incubations in concentrated cell suspensions It is often convenient to conduct labelling experiments at high h, either for technical reasons or to allow measurement of changes in extracellular 32p activity. Table 2 gives solutions of equations (la) and (lb) for incubations at high h, and an approximation valid when 8 is fairly small. At high h, a2/aj increases to a plateau E* =2/[1 + 3 ' - 8 + A " - A ' ] - - ~ 1/{1 + y - [ S y / ( 1 + 8 + y)]} (see Fig. 2(a), curve 3), which is greater than E (see section 3b), although close to it when 3' is large. It follows that ( 1 - E ' ) / E ' = y(1 + 8)/(1 + 8 + 3'), which slightly underestimates 3'. (In the example of Fig. 2(a), curve 3, the underestimate is about 40%.) In the two-pool model (Table 1) the non-linear plot of - I n [ 1 - (a2/a~)] against t has slope k near t = 0, and the same is true in the present case. In the three-pool model the similar (and plausible) plot of - I n [ E ' - ( a f f a ~ ) ] against t has slope near t = 0 of K*=(k/2)[I+3"-8+A"-A']~-k{1+3"-[83"/(1+8+3")]}, which is between k and K. As in experiments in dilute suspension (see section 3b), k is correctly estimated as K'E'. In experiments with concentrated suspensions, a~ must eventually decrease because of 32p consumption by the cells; and because of organic incorporation of 32pi so eventually must a2. This is shown by the third and fourth equations in Table 2, which describe how affa ° decreases, while a2/a ° rises to a peak and then declines slowly. Measurements of A f f A ° (=affa °) have often been used to estimate membrane permeability. In the two-pool model (Table 1) the plot o f - l n [(A~ - AIoo)/(A~-A~ 0 )] against t (that is, a semilogarithmic plot of the fraction of transported 32p remaining outside the cell) is linear with slope k ( 1 + 8 ) . However, in the presence of an exchangeable organic pool there is no clear "plateau" value, A~. The analogous plot of - I n ( A f f A °) against t (e.g. Prankerd & Altman, 1954; Vestergaard-Bogind, 1963; Chedru & Cartier, 1966) is non-linear, with slope near t = 0 of k8 (just as in the two-pool model, although the subsequent kinetics are different): if 3' is very large, this plot is linear with slope k8 throughout. Alternatively we might take the final value at the end of the incubation, say F, as A~° (e.g. Ho & Guidotti, 1975; Craik et al. 1986). In the two-pool model F / A ° is 1/(1+ 8), but it is less than this in the presence of organic exchange because of net loss of 32p into the organic pools (see section 3d below); and so the plot is non-linear with slope near t = 0 of kS~[ 1 - (F/A°)], which is between k8 and k( 1 + 8). Thus k might be underestimated, in the worst case when F is very small, by a factor 8/(1+8), which is typically about 0-3 for incubations of whole blood (see section 2). ( d) Longer incubations We have so far assumed to to be zero, so that a3 can be ignored. The resulting equations apply to incubations where t is not much larger than 1 / [ k ( 1 + 3')]. In longer incubations, however, we must take account of back-flux of 32pi from the organic pool. We will assume now that to is small, though non-zero, and consider only incubations in dilute suspension. As shown by the first expression in Table 3, affal rises to an early "plateau value" of approximately E, with rate constant K, as in section 3. It then rises slowly to one with rate constant kto3'/(1 + 3') (see Fig.

ISOTOPIC LABELLING OF CELLULAR ORTHOPHOSPHATE

357

TABLE 3

Labelling kinetics according to three-pool model during long 32Pi experiments in dilute cell suspensions I a2/a ' ~ [ 1 / ( 1 + 3')]{1 - e x p I - k ( 1 + 7)t]}+['y/(1 + 30]{1 - e x p [-kto'yt/(1 + 7)]} a3/a I = 1 - e x p [ - k ~ y t / ( l + 7)] Assume that ~5=0, and that to is small, but non-zero.

2(b)). In its early stages this slow phase will appear to be approximately linear with slope k t o [ 3 , / ( l + 3,)] 2 (see Fig. 2(a), curve 4). We could use this slow labelling to estimate to. In 48-hour experiments with h u m a n erythrocytes ( K e m p et al. 1988) we have found that a2/al has a time course resembling that in Fig. 2(b), and whose half-time is consistent with to < 0.1. For how long will the plateau appear flat? It is convenient to answer this in terms of the time taken to reach the plateau. We define this as ta, at which point ( a 2 / a l ) / E - - 1 - 0; that is, ta is the time at which a 2 / a 1 is within a small fraction, 0, of the plateau, E: then t ~ - - I n O/[k(1 + 3,)]. Similarly, we define tb as the time at which a2/al exceeds E by the same fraction: then if tb and to are fairly small, tb~--(l+3,)O/(3,2tok), which is if anything a slight underestimate. So tb/ta~[(1 + 3,)/3,]2(1/oo)[0/-In 0]. If, for example, to = 0 - 0 4 and 3,>3 (see section 2), it will take up to twice as long for a 2 / a I tO exceed the initial plateau by 10% as to approach within 10% of it. This factor is directly proportional to the size of the organic pool, and decreases as 3, increases. Table 3 also shows that the mean relative specific activity of the organic pool (aa/al) increases slowly to 1 with rate constant kw3,/(1 + 3,) (see Figs 2(a) and 2(b), broken lines). In the early stages when this is approximately linear, eqn (2) can still be used to estimate G. However, if the incubation is long enough for a3 to become significant, the back-flux of 32Pi from the organic pool causes eqn (2) to underestimate G by a factor f = (a2-a3) . . . . /(a2) . . . . . It can be shown that, if t is fairly large, f is given approximately by [1/(3,f)] + 1 = k t o t / { 1 - e x p [-koJ3,t/(1 + 3,)]}. Therefore G ' becomes a worse underestimate of G as k, to, 3, and t are made larger (i.e. in an experiment with a highly-permeable membrane, a small organic pool exchanging rapidly with cellular Pi, and a long incubation time).

( e) Total cellular 32p activity The simplest measure of cellular 32p uptake is the total cellular activity AT--

A2+A3, conveniently expressed as A r / a ~ , which can be calculated from the definitions of a2, a3 and to as A T / a l = C2{(a2/al)+[(1/to)(aa/a~)]}. Influx can be estimated simply as the intial rate of A-r/a~ (e.g. Deuticke, 1967; Bowen & Levinson, 1982; Biber et al., 1983). In a fairly short incubation at low h, after cell Pi has labelled to the plateau but before the plateau becomes significantly "sloped", the expression above becomes AT/a~ ~--[C2/(1 + 3,)]{1 - e x p I - k ( 1 + 3,)t] + k3,t}. The mean slope of AT/a~ can then be taken as a minimum estimate of (3.

358

G.

J.

KEMP

ET

AL.

We have seen (section 3(b), above) that A2/a~ may underestimate C2 in short incubations, even when it appears to have reached steady state. The use of A r instead of A2 (e.g. Schnell et al., 1981) introduces an opposing error. These two errors can be shown to cancel at approximately t = 1/k, if 7 is fairly large. For the human erythrocyte, this represents about 1-2 hours (see section 2). (If 7 is small this is a slight underestimate.) At this point incomplete inorganic labelling is compensated by organic labelling, so C~ calculated as Ar/a~ is fortuitously equal to the true value. Beyond this point, it becomes an overestimate.

(f) Very long incubations Eventually, al = a2 = a3 (see Fig. 2(b)), so that total organic 32p activity can be used to estimate the size of the exchangeable organic pool as C~= A3/a~; and cellular 32Pi activity can now reliably be used to estimate cellular Pi concentration since C~= A2/a,. It follows that to can be estimated as to'= A2/A 3. What is the time needed to attain this steady state i.e. for a2/a~ to reach 1 - 0, where 0 is a small fraction? At low h this will occur at tc = [(1 + 7)/(kxoT)]{-ln [0(1 + ~,)/'y]}. If 7 is fairly large, tc is approximately ( - I n 0)/(kxo), which is roughly ( 1 + 7 ) / t o times longer than a2/a~ takes to approach within the same fraction of the initial plateau E (section 3). This ratio is proportional to the size of the organic pool and the rate of organic incorporation, and for the human erythrocyte, it is at least 25 (section 2). Our observation that a2/a 1 is about 0.9 at 48 hours in the human erythrocyte (Kemp et al., 1988) is therefore consistent with to <0.1.

(g) Problems arising in cells with rapid organic exchange If organic exchange is much more rapid than transmembrane flux, then y is very large. It follows that in a short incubation (Table 2) with a concentrated cell suspension, A i / A ° ( = a l / a °) ~-exp [-3kt] (see section 3c). However, a2/al will be too low to measure so eqn (2) cannot be used to estimate G. In a longer incubation at low PCV (Table 3), a2/al -~ 1 - e x p [-ktot]. At first this will appear linear with slope kto, but this is not useful for estimating k since to is not precisely known.

4. Etttux Experiments 32p-efflux experiments, in which prelabelled cells are added to unlabelled medium, are not simply the reverse of an influx experiment. In an influx experiment 32p is initially in only one pool (extracellular Pi), whereas in an eittux experiment there is the added complication that ~2p is present in both cellular Pi and organic phosphate pools.

( a ) Initial rates The results of these experiments (e.g. Tenenhouse & Scriver, 1975; Schnell et al., 1981) depend on the degree to which the organic pool has been labelled by preincubation with 32Pi before the start of the efflux experiment. This can be

ISOTOPIC

LABELLING

OF

CELLULAR

ORTHOPHOSPHATE

359

c h a r a c t e i r z e d b y / 3 0 , d e f i n e d as the value o f a3/a2 at t = 0 , the start o f the efflux e x p e r i m e n t : after a s h o r t p r e i n c u b a t i o n /3o is negligible, a n d after a very l o n g p r e i n c u b a t i o n it is a p p r o x i m a t e l y one. F r o m eqns ( l a ) a n d ( l b ) , the initial rate o f rise o f a f f a ° (the specific activity o f e x t r a c e l l u l a r Pi e x p r e s s e d relative to that o f c e l l u l a r Pi at t = 0) is 8k if all c o n c e n t r a t i o n s are at s t e a d y state; if not, the rate o f efflux, I2~, c a n still be e s t i m a t e d as the initial rate o f rise o f ( A f f a ° ) [ ( 1 - h)/h]. As we have seen (see s e c t i o n 3 ( f ) ) , a ° (at the start o f the efflux e x p e r i m e n t ) c a n n o t be a s s u m e d to be equal to a~ (at the e n d o f the p r e i n c u b a t i o n ) unless the p r e i n c u b a t i o n is very long. I f it is n o t very long a ° m u s t be m e a s u r e d o t h e r w i s e s u b s e q u e n t m e a s u r e m e n t s o f e x t r a c e l l u l a r activity c a n n o t b e i n t e r p r e t e d . T h e initial rate o f fall o f c e l l u l a r labelling, e x p r e s s e d as A f f A ° ( = a f f a °) is k [ l + T ( 1 - / 3 ° ) ] , which is g r e a t e r t h a n k s i n c e / 3 o -< 1. This o v e r e s t i m a t e o f k arises b e c a u s e the i n c o m p l e t e l y l a b e l l e d o r g a n i c p o o l c o n t i n u e s to c o n s u m e 32p d u r i n g the efflux e x p e r i m e n t . So u n l i k e the s i t u a t i o n in a n influx e x p e r i m e n t (section 3(a)), the initial rate o f c h a n g e o f c e l l u l a r l a b e l l i n g is affected b y o r g a n i c e x c h a n g e .

, (b) Incubations in dilute cell suspensions In i n c u b a t i o n s at low h, a~ = 0 t h r o u g h o u t . The eqns in T a b l e 4 s h o w that affa ° (see Fig. 3 (a), w h e r e flo is t a k e n as zero) falls t o w a r d s a p l a t e a u value o f / 3 ° y / ( 1 + y) with rate c o n s t a n t K = k(1 + y ) , not k as in the t w o - p o o l m o d e l ( T a b l e 1). M o r e g e n e r a l l y , i f / 3 ° ~ 0 , a2/a ° t h e n falls m o r e slowly with rate c o n s t a n t k~oy/(1 + y), a n d this s l o w p h a s e will a p p e a r to be l i n e a r at first with s l o p e - k x o y / ( 1 + y). T h e o r g a n i c l a b e l l i n g , a3/a °, d e c r e a s e s s l o w l y with the s a m e rate c o n s t a n t , a n d a g a i n this is a p p r o x i m a t e l y l i n e a r at first.

( c) Incubations in concentrated cell suspensions D u r i n g an efflux e x p e r i m e n t at high h, affa ° increases as 32pi e s c a p e s from the cells. In the t w o - p o o l m o d e l ( T a b l e 1) it i n c r e a s e s with rate c o n s t a n t k(1 + 8) to an

TABLE 4

Delabelling kinetics according to three-pool model during 32p/efflux experiments Dilute suspensions (where 8 = 0) 1

az/a ° = {[1/(1 + 3,)][1 + y(l -/3°)] x exp [-k(1 + y)t]} + {[3,/(1 + 3,)](affa°)} a3 / a 0 ~ flo × e x p [ - k w 3 , t / ( 1 + 3,)]

Concentrated cell suspensions (where 8 > 0) 2

(a2 - a, )/a ° = {[1 + 6 + y(1 -13 °) - ~b']x exp [-k~b"t]/( (a"- ~')} - {[1 + ~5+ 3,(1 - flo) _ 4~"] x exp [-kr't]/(6"- 6')} t We also assume here that w is small but non-zero. 2This is a general expression: d~' and ~b" are roots of f f 2 - [ l + 8 + y ( 1 +to)]~b+y[&+to(l+&)]=0A useful approximation for fairly small 8 is th' -- ( t + 8 + 3') - ~b"= 3,[8+ w( 1+ 8)]/[ 1 + 8 + y( 1+ to)].

360

G.J.

KEMP

AL.

ET

o)

\

,-----...~, f (hours)

( ) oN

I

o I

I

I

r

0 t (hours)

FIG. 3. Example of the theoretical kinetics of delabelling during efflux experiments. (Assume k = 1). (a) Delabelling of cellular I~" (a2/a °) in incubations at low haematocrit (8 =0). Curve 1: Cell with no organic incorporation of Pi (y = 0). See Table 1; Curve 2: Cell with organic incorporation of Pi. (Assume = 1 and to = 0). See Table 4 and section 4b. Note that delabelling is more rapid than in curve 1. ( b ) The difference between the specific actioities of cellular Pi and extracellular medium, (a 2- al)/ a~, in incubations at high haematocrit (assume t5 = 0-5). See Table 4 and section 4d. Curve 1: After a very long preincubation which completely labels the organic phosphate pool. (Assume /3o= 1); Curve 2: after a very short preincubation, during which organic labelling is negligible. (Assume /30=0). Note that a~ exceeds a 2 (the "'crossover" effect) after about 1 hour, as a result of continued organic incorporation of cellular 32pi.

e q u i l i b r i u m v a l u e 8 / ( 1 + 8). In the t h r e e - p o o l m o d e l , the e x p r e s s i o n s are c o m p l i cated, b u t if/30 is n e g l i g i b l e t h e n a ~ / a ° = 8 [ e x p ( - k q b ' t ) - e x p (-kqb"t)]/(~"-¢b'), w h e r e ~b" a n d ~b' are d e f i n e d in T a b l e 4. P l a s m a activity, a ~ / a °, rises to a m a x i m u m value, say M, a n d t h e n falls slowly: if/30 is n e g l i g i b l e , t h e n M = (8/4~")a t~/cl-~l, w h e r e a = qb"/qb'; i f / 3 ° > 0 t h e n M is l a r g e r t h a n this. In g e n e r a l , the s l o p e n e a r t =0 of a plot of-In {1-[(aJa°)/M]} a g a i n s t t ( u s e d in e q u i v a l e n t f o r m b y e.g. Schnell et al. 1981; B e r g h o u t et al. 1985) is k S ~ M , w h i c h o v e r e s t i m a t e s k if there is significant o r g a n i c e x c h a n g e . In the e x a m p l e o f Fig. 3(b) (curve 2), M = 0.3, a n d k S / M = 1.7; so k w o u l d b e o v e r e s t i m a t e d b y a b o u t 7 0 % . T h e kinetics o f c e l l u l a r "Pl activity, a 2 / a °, d e p e n d on to a n d / 3 since 32p c o n t i n u e s to e n t e r the o r g a n i c p o o l d u r i n g the efflux e x p e r i m e n t . F o r e x a m p l e , if to a n d

ISOTOPIC

LABELLING

OF

CELLULAR

ORTHOPHOSPHATE

361

/3 are negligible, then a2/a ° ~- {[(1 + y - A') x exp (-kA"t)] - [(1 + y - A") x exp ( - k A ' t ) ] } / ( A " - A ' ) , where A' and A" are defined in Table 2. In general the slope near t = 0 of a plot o f - I n (a2/a °) against t will be k [ l + y ( 1 _/30)].

( d) The "crossover" effect One striking prediction of the model, for concentrated cell suspensions, which follows from the equation for ( a 2 - a l ) / a ° in Table 4, is that if the initial extent of organic labelling, fl0, is less than a critical value flo, then both a2/a ° and al/a ° become equal to, say (a/a°)d, after a time ta after which, surprisingly, a2 falls below al (Fig. 3(b)). As with the plateau in a2/al during influx experiments (section 3), this paradoxical "crossover" between az and al arises from the continued release of "unlabelled" Pi and consumption of 32Pi by the large, incompletely-labelled organic pool. If the extent of prelabelling is too high (/3°> flo), the 32p activity gradient between cellular Pi and organic phosphate pools is insufficient to suppress a2 in this way. The time at which crossover occurs in an incubation at high h is given by td = { 1 / [ k ( d p " - q b ' ) ] } l n { [ l + y ( 1 - f l ° ) + 6 - d / ] / [ l + y ( 1 - f l ° ) + 6 - q b " ] } . This "crossover time" increases with /3o and to, but if these are fairly small they have little influence on it. We can therefore simplify the expression for ta to {1/[k(1 + 6 + 3')]} In [(1 + 6 + y)2/(yr)], which is a lower limit. In the example of Fig. 3(b) (curve 2), td = 1"1 hours. The highest value of/3 o at which a crossover is theoretically possible is that at which td becomes very large. This is / 3 ° = ( ~ b ' / y ) - t o which, when to is small, approximates to /3o ~_ ( 6 - yto)/(1 + 8 + y). A question of practical significance is: for how long must cells be preincubated with 3Zpi in dilute suspension for/3o to attain the critical value/3o? For a given preincubation time, t,/3 can be calculated as (a3/aO/(a2/aO from the expressions in Table 3. If to is small, then after a fairly short preincubation, /3 is given approximately by 1~fly~(1 + 3')] + [1/(kytot)]}, or more approximately by kytot. Therefore the preincubation time, t,,, required to give the critical degree of prelabelling, can be calculated from this approximate expression for /3, and the expression for /3o, above, to be roughly t , . = {1/[k(1 + 6 + y)]}{[6/(yto)]- 1}, which is a lower limit. If the duration of the preincubation is less than t,,, a crossover will eventually be observed in the subsequent efflux experiment in a concentrated cell suspension. In the example of Fig. 3(b),/3o = 0.2 and t,, = 2 hours. The expression derived from Table 4 for the value of a~/a °= az/a °= (a/a°)d at which crossover occurs is complicated but can be simplified by assuming that to and/3 are very small. This gives (a/a°)d = {1 + ( 1 / o r ) - [(1 + y)/qb"]}a EuC~-~)J. This is increased if/3o>0. In the example of Fig. 3(b) (curve 2), (a/a°)d ~--0.1. 5. Discussion

The three-pool model shows how exchange between cellular Pi and organic phosphate may affect the interpretation of kinetic experiments with 32pi. Comparison

362

G.

J.

KEMP

ET

AL.

with the two-pool model, which neglects this, allows the errors resulting from the two-pool simplification to be summarized as follows. Analysis of initial rates of change of cellular and extracellular Pi labelling, to obtain estimates of transmembrane flux, is independent of organic exchange in influx experiments (section 3a). However, interpretation of initial rates in efflux experiments (section 4a) must take account of the degree of organic labelling (/3 °) at the start of the experiment; and analysis of the subsequent kinetics of cellular and extracellular 32p is affected by organic exchange in both influx (sections 3b, 3c) and efflux experiments (sections 4b, 4c). Estimation of the cellular Pi concentration from the specific activity of extracellular Pi and the total 32p labelling of the cell during influx experiments is rendered inaccurate by organic exchange, since the specific activity of cellular Pi may initially stabilize at a value less than that of extracellular Pi (sections 3b, 3c) and a significant fraction of cellular 32p may be in the form of organic phosphates (section 3e). If the organic phosphate pool is large compared with cellular Pi, the errors arising from organic phosphate labelling are not negligible even if the absolute or specific activity of the organic phosphate pool is low (contrary to what has been assumed in earlier reports e.g. Schnell et al. (1981)), and they are generally worse when organic exchange is rapid (contrary to what has been assumed in earlier reports e.g. Levinson (1966)). In principle, these errors can be minimized by careful choice of incubation time (sections 3b, 3e, 3f), but in practice such conditions are often ignored. Less obvious consequences of organic exchange are the predictions that the specific activity of cellular Pi will stabilize temporarily at a plateau value less than that of extracellular Pi during influx experiments (sections 3b, 3c), and that it may fall below that of extracellular Pi (the crossover effect) during etflux experiments in concentrated cell suspensions (section 4d). Both the plateau (Latzkovits et al. 1966; Till et al. 1973; Tenenhouse & Scriver, 1975; Niehaus & Hammerstedt, 1976) and crossover (Till et al. 1973) have been observed experimentally in human erythrocytes, and have sometimes been taken as evidence for a slow-exchanging pool of cellular Pi (Till et al. 1973; Niehaus & Hammerstedt, 1976). However, although there may indeed be functional heterogeneity of cellular Pi in erythrocytes (see below), these observations cannot necessarily be used as evidence for it, since the theoretical results presented here show that they could arise under attainable conditions with only one homogeneous cellular Pi pool One consequence of organic exchange is that mathematical description becomes complicated, especially in efflux experiments (section 4). Neglect of these complications may make detailed quantitative interpretation of much published data, even of straightforward measurements such as plasma activity, almost impossible (sections 3c, 4c). From published data for the human erythrocyte we can use initial rate analysis (section 3a) to obtain estimates of •=0.2-0.9 mmol/l cells/hour and k=0-5-1.0 (Prankerd & Altman, 1954; Vestergaard-Bogind, 1963; Chedru & Cartier, 1966; Tenenhouse & Scriver, 1975; Niehaus & Hammerstedt, 1976). G must be at least equal to the rate of lactate generation (section 3b), which is about 3 mmol/l cells/hour (Kemp et al. 1986), so 3' is probably greater than 3. This is reasonably

ISOTOPIC

LABELLING

OF CELLULAR

ORTHOPHOSPHATE

363

c o n s i s t e n t with p u b l i s h e d v a l u e s o f the p l a t e a u in a 2 / a l , w h i c h lie b e t w e e n 0.2 ( L a t z k o v i t s et al., 1966; Till et al., 1973) a n d 0-6 ( N i e h a u s & H a m m e r s t e d t , 1976). It seems likely, t h e r e f o r e , that the errors resulting f r o m the a s s u m p t i o n that 3' = 0 are q u a n t i t a t i v e l y significant. T h e p r e s e n t m o d e l m a y be o v e r s i m p l i f i e d in at least two respects. Firstly, it ignores the fact that the o r g a n i c p h o s p h a t e s are h e t e r o g e n e o u s , a n d have different l a b e l l i n g kinetics (e.g. P r a n k e r d & A l t m a n , 1954; Till et al., 1973). H o w e v e r , in a g r e e m e n t with the p r e s e n t m o d e l , overall o r g a n i c l a b e l l i n g , A 3 / a l , is l i n e a r for the first few h o u r s in the h u m a n e r y t h r o c y t e ( K e m p et al., 1986 a n d 1987b). I r r e s p e c t i v e o f the p r e c i s e c o m p o s i t i o n o f the o r g a n i c p h o s p h a t e p o o l s , t h e i r low specific activity in relatively s h o r t e x p e r i m e n t s is likely to influence the l a b e l l i n g kinetics o f c e l l u l a r Pi in a w a y that is at least q u a l i t a t i v e l y s i m i l a r to t h a t d e s c r i b e d b y the p r e s e n t m o d e l . T h e s e c o n d o v e r s i m p l i f i c a t i o n is t h a t this m o d e l ignores the p o s s i b i l i t y that c e l l u l a r Pi m a y c o m p r i s e m o r e t h a n o n e p o o l , even in such a s i m p l e cell as the e r y t h r o c y t e ( K e m p et al., 1986, 1987b a n d 1988). N e v e r t h e l e s s , the m o d e l p r o v i d e s a useful basis for the i n t e r p r e t a t i o n o f e x p e r i m e n t a l results, can easily be m o d i f i e d to a c c o m m o d a t e a s e c o n d f a s t - l a b e l l i n g Pi p o o l , a n d p r o v i d e s w a y s o f d e t e c t i n g f u r t h e r f u n c t i o n a l h e t e r o g e n e i t y in cell Pi ( K e m p et al. 1986, 1987b a n d 1988). G.J.K. was supported by the Special Trustees of the Former United Sheffield Hospitals, and A.B. by the Wellcome Trust. REFERENCES BERGHOUT, A., RAIDA, M., ROMANO, L. & PASSOW, H. (1985). Biochim. Biophys. Acta 815, 281-286. BEVINGTON, A., KEMP, G. J. & RUSSELL, R. G. G. (1986). Adv. Exp. Med. Biol. 208, 469-478 (eds S. G. Massry, M. Olmer & E. Ritz). BIBER, J., BROWN, C. D. A. & MURER, H. (1983). Biochim. Biophys. Acta 735, 325-330. BOWEN, J. M. & LEVINSON, C. (1982) J. Cell. Physiol. 110, 149-154. CHEDRU, J. & CARTIER, P. (1966) Biochim. Biophys. Acta, 126, 500-512. CRAJK, J. D., GOUNDEN, K. & REITHMEIER, R. A. F. (1986) Biochim. Biophys. Acta 856, 602-609. DEUTICKE, B. (1967) Pfliig. Arch. 296, 21-38. GRIMES, A. J. (1980) Human Red Cell Metabolism, p. 92, Oxford: Blaekwells. Ho, M. K. & GUIDOTTI, G. (1975) J. biol. Chem. 250, 675-683. JOHNSON, R. C. & FREINKEL, N. (1985). Biochem. Biophys. Res. Comm. 129, 862-867. KEMP, G. J., KHODJA, D., CHALLA, A., BEVINGTON, A. & RUSSELL, R. G. G. (1986). Biochem. Soc. Trans. 14, 1194-1195. KEMP, G. J., BEVlNGTON, A. & RUSSELL, R. G. G. (1987a). Biochem. Soc. Trans. 15, 381-382. KEMP, G. J., KHODJA, D., BEVINGTON, A. & RUSSELL, R. G. G. (1987b). Biochem. Soc. Trans. 15, 1141-1142.

KEMP, G. J., KHODJA, D., BEVINGTON,A. & RUSSELL, R. G. G. (1988). Biochem. Soc. Trans. 16, 781-782. LATZKOVITS, L., HUSZAK, I. & SZECHENYI, F. (1966). Brain 89, 831-836. LEVlNSON, C. (1966). Biochim. Biophys. Acta 120, 292-298. NEWSHOLME, E. A. & START, C. (1973). Regulation in Metabolism, p. 97, London: John Wiley. NIEHAUS, W. G. & HAMMERSTEDT, R. H. (1976). Biochim. Biophys. Acta 443, 515-524. PRANKERD, T. A. J. & ALTMAN, K. 1. (1954). Biochem. J. 58, 622-633. PRI NS, A. P. A., KI t.JAN, E., STADT, R. J. v. d. & KORST,J. K. v. d. (1986). Analyt. Biochem. 152, 370-375. ROBERTSON, J. S. (1957). Physiol. Reo. 37, 133-154. SCHNELL, K. F., BESL, E. & MOSEL, R. v. d. (1981). J. Membrane Biol. 61, 173-192. SHEPPARD, C. W. & HOUSEHOLDER, A. S. (1950). J. Appl. Phys. 22, 510-520. TENENHOUSE, H. S. & SCRtVER, C. R. (1975). J. Clin. Invest. 55, 644-654. TILL, U., KOHLER, W., RUSCHKE, I., KOHLER, A. & LOSCHE, W. (1973). Eur. J. Biochem. 35, 167-178. VESTERGAARD-BOGIND, H. (1963). Biochim. Biophys. Acta 66, 93-109.

364

G.

J.

KEMP

ET

AL.

APPENDIX

Symbols used in this Paper t 1; 2; 3; T

Time (for subscripts see text) Subscripts denoting extracellular Pi; cellular Pi; organic P; total cellular Pi Estimated value of x; x at t = 0 Fractional cell volume of incubation Activity (cpm/l); concentration (mmol/l); specific activity (cpm/mmol) Transmembrane flux; organic incorporation rate (mmol/l cells/hour) See Tables 2 & 4 See text

x'; x ° h A; C; a /, G A, ~b 0, f, F, M

Derived dimensionless quantities: 3,=G/I;

to=C2/C3;

E = 1/(1+3,);

8 = (C21C,)[hl(1

-h)];

E* = 2 / ( l + 3 , - 8 + A " - A ' ) ;

= a3/a2 ; <~ = , ~ " 1 @ '

Rate constants (1/hour): k=IIC2;

K = k ( l + y);

K*=(k/2)[l+y-g+h"-h']