A computer simulation of the effect of heart rate on ion concentrations in the heart

J. theor. Biol. (1988) 132, 15-27

A Computer Simulation of the Effect of Heart Rate on Ion Concentrations in the Heart M. R. BOYE'I-r AND D. FEDIDAt

Department of Physiology, University of Leeds, Leeds LS2 9JT, England (Received 10 September 1987) At steady-state the passive fluxes of Na ÷ and K ÷ across the cell membrane of a heart cell are exactly matched by active fluxes of the two ions in the opposite direction via the Na-K pump, and the concentrations of Na ÷ and K÷ both within the cell and in the clefts between cells are steady. An alteration of the heart rate (or the rate of stimulation) disrupts this balance because the passive fluxes are affected, and there are changes in pump activity as well as the Na ÷ and K + concentrations. A computer model incorporating a cell separated from the bathing medium by a restricted extracellular cleft was devised to investigate these changes further. The model was able to simulate the changes observed with a variety of stimulation protocols as well as the effect of block of the Na-K pump. It is concluded that the changes in Na ÷ and K ÷ balance with heart rate can be explained in terms of the known properties of cardiac tissue incorporated into the model.

1. Introduction The heart rate of an animal varies continuously and these changes are known to have many varied and important effects on both the electrical and the mechanical activity o f the heart (reviews: Koch-Weser & Blinks, 1963; Boyett & Jewell, 1980). For example, an increase in heart rate shortens the action potential, increases the resting potential (or maximum diastolic potential), suppresses pacemaker activity and increases the force of contraction. Many of these effects are thought to result from an increase in the intracellular Na ÷ concentration ([Na]i) at high heart rates (Cohen et al., 1982; Lederer & Sheu, 1983; January and Fozzard, 1984; Ellis, 1985; Boyett et al., 1987a): the resultant increase in N a - K pump current has an important influence on electrical activity (see Boyett & Fedida, 1984, for an experimental study, and C h a p m a n et al., 1979, and Johnson et al., 1980, for theoretical studies), whereas the resultant rise in the intracellular Ca 2÷ concentration, [Ca]i, (via N a - C a exchange) has an important influence on mechanical activity (Eisner et al., 1981; Boyett et al., 1987b). In addition to these important changes in [Na]~ and N a - K pump activity, there are known to be heart rate dependent changes in K ÷ concentration, both within the cells, [K]~ (Langer, 1968; Cohen et al., 1982), and in the clefts between cells, [K]c (Kunze, 1977; Kline & Morad, 1978; Kline et al., 1980; Cohen & Kline, 1982; Kline & Kupersmith, 1982; Martin & Morad, 1982). The aim o f this study was to determine whether the experimentally observed changes in [Na]i, [K]~, [K]c and N a - K pump activity can be explained in terms of the known properties t Present address: University Laboratory of Physiology,Parks Road, Oxford OXI 3PT, England. 15

0022-5193/88/090015+13 $03.00/0

O c 1988 Academic Press Limited

16

M.

R.

BOYETT

AND

D.

FEDIDA

of cardiac muscle. To answer this question a computer model was devised. A preliminary account of this work has been presented to the Physiological Society (Boyett & Fedida, 1983).

2. Theory and Methods A list of the abbreviations used, together with brief definitions, and in the case of constants, their values, are given in Table 1. (A) THREE

COMPARTMENT

MODEL

Cardiac muscle is composed of tightly packed cells separated by narrow clefts, which greatly restrict the diffusion of ions between the cells and the bathing medium (Sommer & Johnson, 1968). To simulate this arrangement, the three compartment model (Fig. 1), which has been extensively studied by others (Daut, 1978; Attwell et al., 1979; DiFrancesco & Noble, 1980, 1985; Hart et aL, 1983), has been used. In this model, a standard cell is separated from the bathing medium by a standard cleft. This assumes that the K ÷ concentration is homogeneous in the clefts and it ignores the possibility of radial K ÷ gradients in the clefts (see DiFrancesco & Noble, 1985, for further discussion). The cleft space (c) was taken to be 20% of the total volume of the preparation; this is intermediate in the range (5-30%) studied by TABLE

1

M o d e l variables and constants !

[ ],,[ ],. [Na]b

[K]~ JNa

Jr¢ ip

,S Km,Na n Km.K r T

C (3"

F

Time l n t r a c e l l u l a r a n d cleft ion c o n c e n t r a t i o n s B a t h i n g Na ÷ c o n c e n t r a t i o n Bathing K ÷ concentration Passive Na* influx: (i) at rest (ii) extra p e r a c t i o n p o t e n t i a l Passive K ÷ efflux Na-K pump current Maximum pump current [ N a ] , at w h i c h the p u m p is h a l f m a x i m a l l y a c t i v a t e d Hill coefficient [ K ] , at which the p u m p is half maximally activated R a t i o of the N a +: K ÷ p u m p fluxes T i m e c o n s t a n t o f diffusion b e t w e e n the clefts a n d the bathing solution Cleft v o l u m e (% o f the total v o l u m e of the p r e p a r a t i o n ) Cell surface : v o l u m e ratio Faraday's constant

--

S

--

mol 1- t

1 4 0 x 10 -3

mol 1-~ tool I -~

5 X 10 -3 5 x 10 -~2

3-1 x I0 -J2 T w o - t h i r d s of J~,~

m o l cm --~ s -j mol cm -2

--

A cm-2

5 X 10 - 6 2 0 X 10 - 3

A cm -2 mol I-~

3 1x

10 -3

m o I I -I

3:2 50

s

2O

%

4000 96,500

cmC mol -~

HEART

RATE AND

17

1ON CONCENTRATIONS

[K],,

"ffl. Bathing medium

FIG. 1. A schematic diagram of the three c o m p a r t m e n t model.

DiFrancesco & Noble (1982). (The effect of changing the cleft space volume was to alter the magnitude of the changes in the ion concentrations in the clefts--a reduction in the volume of the clefts results in larger changes in concentration. The value of the cleft space volume was chosen so that the computed changes matched those measured experimentally.) The surface:volume ratio (cr) of the cell was taken to be 4000 cm -' (this is a value for sheep Purkinje fibres, taken from Mobley & Page, 1972). (B) PASSIVE M E M B R A N E FLUXES

Models of the cardiac action potential and thus of membrane ion fluxes are available (Fedida, 1983; DiFrancesco & Noble, 1985), but the computation of action potentials for up to 40 min (Fig. 4B) would require a large amount of computation time. To simplify the calculation, therefore, the passive ion fluxes (i.e. fluxes not associated with the N a - K pump) at rest and during the action potential were given fixed values. The resting passive Na ÷ influx was taken to be 5 p mol cm -2 s-'. This is within the range of estimates reported for a number of cardiac tissues (Langer, 1968; Daut, 1978; Deitmer & Ellis, 1978; Eisner et aL, 1981). The passive membrane fluxes of both Na + and K + are greater during an action potential. The extra Na ÷ influx per action potential was assumed to be 3.1 p mol cm -2. This is less than the values that can be calculated from isotope flux data reviewed by Langer (1968), but comparable to that recently reported by Ellis (1985) for sheep Purkinje fibres. Passive K + efllux was taken to be two-thirds of the passive Na + influx for the following

18

M.R. BOYETT AND D. FEDIDA

reason: Na ÷ effiux and K ÷ influx are linked and occur via the N a - K p u m p in a ratio of 3 N a ÷ : 2 K ÷ (see below). It follows from this that the passive m e m b r a n e fluxes of Na + and K ÷ must be in the same ratio if a cell is to be in a state of ionic balance. (C) N a - K PUMP C U R R E N T

The passive m e m b r a n e fluxes of Na ÷ and K ÷ are countered by the N a - K pump: this removes the Na ÷ from the cell and p u m p s K ÷ back in. The N a - K p u m p is known to be electrogenic: the p u m p generates a net outward current because the stoichiometry of the p u m p is thought to be 3 N a ÷ : 2 K + (Eisner et al., 1981). The following equation was used to describe the p u m p current, ip: -= l/ [Na];' "~ ( [_K]<, i,, = ,p tK~,.N--~ [--Na]:/] \ K,,,,K + [K]
(1)

The m a x i m u m p u m p current, Tp, was set to 5 / z A cm -2 (of. Glitsch, 1979). Equation (1) is similar but not identical to the expression used by Hart et al. (1983), and shows that N a - K p u m p activity is thought to be primarily determined by [Na]i and [K]c. It was assumed that the p u m p is half-maximally activated when [Na]~ is 20 mM (KroNa ; Akera et al., 1976; Glitsch, 1979; Hart et al., 1983) and [K]c is 1 mM (K,,,,K" Gadsby, 1980), and the dependence of p u m p activity on [K]< is hyperbolic (Gadsby, 1980), whereas the dependence on [Na]~ is sigmoidal (Akera et al., 1976; Garay & Garrahan, 1973; the Hill coefficient, n, was set at 3).t (D) D I F F U S I O N BETWEEN THE CLEFTS A N D THE B A T H I N G M E D I U M

This was described by the Fick equation: d[ion]c 1 dt - ([ion]b -- [ion]c) -T

(2)

The time constant, ~-, for diffusion between the clefts and the bathing medium was set at 50 s (see the Results section for further discussion about the choice of r). (E) INTRACELLULAR AND CLEFT

ION C O N C E N T R A T I O N S

Changes in [Na]i and [K]i depend on the balance of the passive and active fluxes of the two ions across the cell membrane: d[Na]i

dt dt

( = O" JNa

-- Or --JK-'t

i,r F(r-1)]

1000

(3)

--

1000

(4)

F ( r - 1)

t All estimates of the extra Na + influx per action potential suggest that it is approximately the same as, or greater than, resting Na + influx per second. It follows that if the dependence of ip on [Na]i is hyperbolic, or linear (as has been suggested by Eisner e t a l . , 1981), [Na]~ should at least double at 60 beats/rain. This is not the case: Ellis (1985) observed [Na], to rise by only about 5-35% at 60 beats/min. However there is some evidence in heart and good evidence in other tissues that the dependence of i t, on [Na], is sigmoidal (see references above), and when this feature was introduced into our model, it greatly reduced the rise in [Na]i during stimulation.

HEART

RATE

AND

ION

19

CONCENTRATIONS

Changes in [Na]c and [K]c depend on the balance of the passive and active m e m b r a n e fluxes as well as diffusion between the clefts and the bathing medium:

d[Na]~ o.(l-c)(_jN.+ ipr dt c F(r-1) d[K]c

dt

o'(1-c)(jK_

-

c

\

) 1000+([Na]b-[Na]c)lr

(5)

ip ~ 1 0 0 0 + ( [ K ] b - [ K ] c ) l F(r-1)] -~

(6)

Differential equations (3) to (6) were solved numerically on an Amdahl computer. The integration time step was set to the cycle length (the interval between "action potentials") or 1 s, whichever was shorter.

3. Results (A) THE

RATE-DEPENDENT

CHANGES

IN

THE

Na + AND

K + CONCENTRATIONS

Figure 2 illustrates the changes predicted by the model in the N a ÷ and K ÷ concentrations as well as N a - K p u m p current (measure of p u m p activity) during a period of stimulation at 150 min -~ after a rest. The top left hand panel illustrates the stimulation protocol. At rest, the c o m p u t e d values o f [Na]i and [K]~ are typical of cardiac tissue (eg. Boyett et al., 1987a,b; Reverdin et al., 1986). In the model, the effect of stimulation is mimicked by increasing the passive fluxes of Na ÷ and K+; as expected, this leads to a rise in [Na]~ and a fall in [K]~. These changes do not continue unabated because the rise in [Na]~ is a potent stimulus to the N a - K p u m p [Equation (1)] and p u m p activity rises. Eventually a new equilibrium is reached and this occurs when [Na]~ has risen sufficiently in order that p u m p activity is once again able to match the larger passive fluxes o f the two ions. [No]/ (mM)

Roie (min -I) 150

o_J--

o .............

'

'

'

'

'

'

[K]i (raM)

'

'

'

'

'

I

I

128 I . . . . . . . .

, ....

I

/pump (#A/cm 2)

[No]c (mM)

• [K]c ( m M )

,42

-

'

0

1300

0

1300

0

I

I

I

I

I

i 300

Time (s) FIG. 2. Computed changes in Na-K pump activity and ion concentrations during a period of stimulation at 150/min. The stimulation protocol is shown in the top left-hand panel.

20

M.

R. B O Y E T T

AND

D. F E D I D A

At rest, the values o f [ Na]+ a n d [ K]~ a r e the s a m e as t h o s e o f t h e b a t h i n g m e d i u m . At the start o f s t i m u l a t i o n , the i n c r e a s e o f the p a s s i v e fluxes l e a d s to a fall in [ N a ] c a n d a rise in [K],. as e x p e c t e d . T h e s e c h a n g e s are n o t m a i n t a i n e d , h o w e v e r , b e c a u s e as [Na]+ rises, the activity o f the N a - K p u m p rises also. W h e n a s t e a d y - s t a t e is r e a c h e d , the larger p a s s i v e fluxes o f the two ions are e x a c t l y m a t c h e d b y the active fluxes via the p u m p . T h e r e is t h e r e f o r e n o net m o v e m e n t o f the i o n s b e t w e e n the cells a n d the clefts a n d thus, o n c e a g a i n , t h e ion c o n c e n t r a t i o n s in the clefts are the s a m e as t h o s e in the b a t h i n g m e d i u m . W h e n s t i m u l a t i o n is s t o p p e d , the p a s s i v e fluxes are r e t u r n e d to t h e i r resting values a n d all the c h a n g e s o c c u r in r e v e r s e . t The p r e d i c t e d c h a n g e s in [ N a ] i , [K]+, [K]c a n d p u m p c u r r e n t in Fig. 2 a r e s i m i l a r to c h a n g e s o b s e r v e d e x p e r i m e n t a l l y (this p o i n t is d e a l t with in d e t a i l in the Discussion) a n d t h e r e f o r e the k n o w n p r o p e r t i e s o f c a r d i a c m u s c l e that a r e b r o u g h t t o g e t h e r in this m o d e l a r e a b l e to a c c o u n t for the v a r i o u s r a t e - d e p e n d e n t changes. This c o n c l u s i o n is r e i n f o r c e d by the d i r e c t c o m p a r i s o n o f e x p e r i m e n t a l a n d p r e d i c t e d results in the next two sections.

(B) T H E E F F E C T

OF STIMULATION

FOR

DIFFERENT

TIMES

A u t h o r s i n t e r e s t e d in cleft K + have c a r r i e d o u t a n u m b e r o f o t h e r e x p e r i m e n t s in a d d i t i o n to the one r e p r o d u c e d in Fig. 2. O n e o f these a d d i t i o n a l e x p e r i m e n t s will be c o n s i d e r e d in full (Fig. 3), w h e r e a s the r e m a i n d e r will be c o n s i d e r e d m o r e briefly in the next section. F i g u r e 3 illustrates the effect o f s t i m u l a t i o n for different times. Panel A c o m p a r e s the e x p e r i m e n t a l a n d c o m p u t e d c h a n g e s in [ K ] c - - t h e r e c o r d s o b t a i n e d d u r i n g the different runs h a v e b e e n s u p e r i m p o s e d . T h e l o w e r trace o f each p a i r in p a n e l A s h o w s the s t i m u l a t i o n p r o t o c o l . It can b e seen in the e x p e r i m e n t a l trace ( u p p e r one; f r o m K u n z e , 1977) t h a t with s t i m u l a t i o n for a s h o r t p e r i o d , t h e r e is a s i m p l e rise in [K]c d u r i n g s t i m u l a t i o n a n d a fall after; t h e r e is little u n d e r s h o o t o f [K]c after such a s h o r t p e r i o d o f s t i m u l a t i o n . H o w e v e r , with l o n g e r p e r i o d s o f s t i m u l a t i o n , [K]c falls b a c k to its o r i g i n a l level d u r i n g s t i m u l a t i o n a n d after s t i m u l a t i o n t h e r e is a t r a n s i e n t d e p l e t i o n o f K + f r o m the clefts. All o f these f e a t u r e s are r e p r o d u c e d b y the m o d e l (Fig. 3A, l o w e r p a i r o f traces). T h e t r a n s i e n t d e p l e t i o n o f cleft K + after a l o n g p e r i o d o f s t i m u l a t i o n has b e e n a s c r i b e d to e n h a n c e d activity o f the N a - K p u m p at high rates (e.g. K u n z e , 1977) a n d this is c o n f i r m e d by the m o d e l (Fig. 3B). D u r i n g a s h o r t p e r i o d o f s t i m u l a t i o n , the i n c r e a s e o f the p a s s i v e N a + a n d K + fluxes l e a d s to a rise in [K]c o v e r a b o u t 60 s t Kinetics: It was found that in our model, the time course of the initial fall in [Na], and rise in [K], on start of stimulation is primarily determined by the time constant for diffusion between the clefts and the bathing medium [Eq. (2)]. Initially the value was taken to be 2 s (DiFrancesco & Noble, 1985), but the initial changes were then complete in about 6 s, whereas Kunze (1977), Kline & Morad (1978), Kline & Kupersmith (1982) and Martin & Morad (1982) found the changes to be complete afler about 60 s; this could be obtained in our model by increasing the time constant to 50 s. The only experimental estimate of the speed of diffusion between the clefts and the bathing medium known to the authors is that of Martin & Morad (1982) who measured the half-time to be 60-90 s in frog ventricular muscle. As an aside, an increase in the time constant for diffusion between the clefts and the bathing medium in our model also resulted in larger changes in ion concentrations within the clefts after a change in the stimulation rate. The computed changes in [Na]i, [K],, pump current and the slower changes in [Na], and [K], are monotonic and they reach completion in about l0 rain.

H E A R T

RATE

A

AND

ION

21

C O N C E N T R A T I O N S

Experimentol 2-8

I

120

l I I I I

Ro,e(min-'): 0_.I

i min

]

o

Computed

120

Rote (min-'):

OI

[

,

1

I

l

,rain

0

Computed

B

Rote (min-i)

II

[K]i (mM)

[No],. (mM) 130

120

Jlllll t o I

I

l

I

I

I

i

'

,/'pump (~,'m/cruz )

l

5

144

f

i

~

l

l

I I I

[K]c (mM)

[No]c (mM)

140

/

i

0

/,"

128 i

0.5 0'25

j/ /r',/':/~

•

7 ~ ~ . . _,,,.~-\ i_[.9 \.

i

[

i

i

3

136 900 0

I

I

900

0

1

|

I

D

I

I

I

9O0

Time (s) FIG. 3. The effects of stimulation at 120/min for different periods of time. A, Companson of experimental and computed changes in the cleft K+ concentration during stimulation for different times. The upper record is from the work of Kunze (1977) and shows changes in cleft K÷ activity, a"K, (0"74 X [K],.) of rabbit atrium at 36-37 °C. The lower record shows the computed changes in cleft K +. The lower trace of each pair shows the stimulation protocol. B, Computed changes in Na-K pump activity and ion concentrations during stimulation for different times. The stimulation protocol is shown in the top left hand corner of B. In both A and B, records obtained during different runs have been superimposed. b u t little c h a n g e i n p u m p a c t i v i t y b e c a u s e p u m p a c t i v i t y is p r i m a r i l y g o v e r n e d b y [ N a ] i a n d this i n c r e a s e s r e l a t i v e l y s l o w l y o v e r a b o u t 600 s. A f t e r s u c h a s h o r t p e r i o d o f s t i m u l a t i o n , t h e r e is a r a p i d fall in [K],. o v e r a b o u t 60 s as a r e s u l t o f d i f f u s i o n o f K ÷ o u t o f t h e clefts i n t o the b a t h i n g m e d i u m . D u r i n g a long p e r i o d o f s t i m u l a t i o n , t h e r e is a large i n c r e a s e in [ N a ] i a n d t h e r e f o r e in N a - K p u m p activity, a n d [K],. is

22

M. R. B O Y E T T A N D

D. F E D I D A

returned to that of the bathing medium. When such a long period of stimulation is stopped there is an immediate decrease in passive K + efflux, but as pump activity declines only very slowly (as a result of the slow decline in [Na]~), K ÷ uptake out of the clefts into the cells remains high and therefore there is a large and rapid decline in [K],. to below its resting value. [K]L. falls to a value such that diffusion from the bathing medium into the clefts down the concentration gradient equals K + uptake into the cells via the pump. In Fig. 3B, it can be seen that the extent of K + depletion after stimulation is indeed a rough measure of the activation of the N a - K pump during the previous period of stimulation.

(C) THE EFFECT OF RATE, HISTORY OF S T I M U L A T I O N , A N D BLOCK OF THE N a - K PUMP ON THE R A T E - D E P E N D E N T C H A N G E S IN [K],

In this section, a number of other experiments will be briefly considered. Stimulation at higher rates has been shown to lead to larger changes in [Na]i (Ellis, 1985) and pump current (Boyett & Fedida, 1984), as well as [K],. (upper panel of Fig. 4A--from Kunze, 1977). This behaviour is also predicted by the model and is the result of the fact that the passive fluxes are greater the higher the stimulus rate. The lower panel of Fig. 4A shows the computed changes in [K]c at different rates (the changes in the other variables are not illustrated).? The interesting experimental result in Fig. 4B from Martin & Morad (1982) shows the dependence of the changes in [K]c on the history of stimulation. It shows that when a preparation is stimulated at a high rate (in the experimental case, 60/min), the initial rise in [K],. is greater when the preparation has been previously resting rather than stimulated at a low r a t e - - c o m p a r e the response of [K],. to the first and second periods of stimulation at 60/min. Once again this result is qualitatively reproduced by the model (lower panel of Fig. 4B). In the model the explanation for this behaviour is that the transient rise of [K],. after an increase in rate is dependent on the change in passive K + efflux rather than its new value. On the other hand, [Na]i (not illustrated), under steady-state conditions, is independent of the history of stimulation and is dependent on the absolute value of the passive Na + influx. When a preparation is stimulated rapidly, there is an initial rise of [K],. but then [K]c returns to its original value, as already described. Various authors (Kunze, 1977; Martin & Morad, 1982) have attributed the slow decline to enhanced activity of the N a - K pump. Their evidence for this is that the slow decline is abolished after block of the N a - K pump. The experimental result obtained by Kunze (1977) is illustrated in Fig. 4C; after block of the pump there is a maintained rise in [K],. during stimulation. A similar effect is observed in the model after N a - K pump "block" (the maximal pump current ip was set to zero). The maintained rise in [K]c ? T h e model predicts that [Na]~ is roughly linearly dependent upon the stimulus rate, whereas experimental studies have shown that this is not the case: Ellis (1985) has shown that the relationship between [Na], and rate becomes less steep at higher rates. In the model this can be reproduced by reducing the increment in passive Na + influx per action potential at higher rates. This is not unrealistic because at higher rates the action potential upstroke can be slowed and the action potential is dramatically shortened (BoyetI & Jewell, 1980).

HEART

/~, ~::~

RATE AND

E_._xperimeni'ol

~

4.5

60

l

3.6 ~ 2.9E (90

~

B E×perimenfol R a t e ol 60 7,4. 60 ;)4 (min-):" I o i J ' tO-..-~-5 , 0 ~

-~

~

, ~

23

ION CONCENTRATIONS

4.0 3.0

m

t min

;)-O

Computed

~'~

;)L t g o

-

1.......

20 Min

30

40

3 x I0 -6 M Ouaboin

6o

j

~5o

6o

, o

~. 0

Confrol

E

.~. I

E__xperimental 4,3 3.6

to

Rote ~5o (min-I):ol I o ~ 7.o

Imin C

0

Computed

I

~20

. . . . . . . . . . .

I

I0

I

20 Min

I

30

I

40

I min

180

Rate o_J (min-~):

II

o

Computed Control

6.o~ 9.4 ~

Na-K pump "blocked"

~

.............

,---,

I min

60

Rate o I (min-I): -

I

FIG. 4. A comparison o f e x p e r i m e n t a l and c o m p u t e d rate-dependent changes in the cleft K + concentration. A, The effect of stimulation at different rates ranging from 6 0 / m i n to 200/rain. The rate is s h o w n above each of the traces. The experimental traces are from the work of Kunze (1977) a n d show changes in cleft K + activity (0.74 x [ K ] , ) (rabbit atrial muscle, 36-37°C). B, The effect of the history of stimulation on the rate-dependent changes in cleft K +, The upper trace of each pair shows the stimulation protocol. The experimental record is from the work o f Martin & Morad (1982) and shows changes in the cleft K + activity of frog ventricular muscle at 22°C. C, The effect of block of the N a - K p u m p on the rate-dependent changes in cleft K +. The upper set of traces is taken from the work of Kunze (1977) and shows changes in the cleft K + activity under control conditions and in the presence of 3 x 1 0 - 6 M ouabain to block the N a - K p u m p (rabbit atrial muscle, 36-37°C). The lower set of traces are the equivalent computed results. In the model the N a - K p u m p was blocked by reducing ~ to zero. The lower trace o f each set shows the stimulation protocol.

24

M. R. B O Y E T T A N D

D. F E D I D A

during stimulation occurs because the increased passive loss of K ÷ from the cells is not matched by a large uptake of K ÷ out of the clefts and back into the cells via the p u m p and therefore this confirms the original conclusions of the authors referred to above.¢ (D) CAN THERE BE A R A T E - D E P E N D E N T RISE IN [K],

UNDER

STEADY-STATE C O N D I T I O N S ' ?

For more than 20 years, a possible accumulation of K ÷ in the clefts at high stimulus rates has been considered in the literature (Carmeliet & Lacquet, 1958; Boyett & Jewell, 1980). This has been an attractive explanation for the shortening of the cardiac action potential at high heart rates (e.g. Kline & Morad, 1978). Figure 1 shows, however, that with the model, during prolonged stimulation, a rise in [K],. is not maintained and [K],. returns to its original value. Nevertheless, it is easy to obtain a maintained rise in [K]~ during stimulation with the model. In Fig. 5 it has

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FIG. 5. Can there be a rate-dependent rise in [K], under steady-state conditions? Computed ratedependent changes in N a - K p u m p current and ion concentrations when the ratio of the passive Na ÷ and K ÷ fluxes was changed from 3 : 2 to 3 : 2.3. The active fluxes of Na + and K + were maintained at the original ratio of 3:2. The stimulation protocol is shown in the top left hand panel. The dashed line in the bottom right hand panel marks the resting level of [K], and it shows that there was a steady rise in [K], during stimulation.

been achieved by changing the ratio of passive Na ÷ and K ÷ fluxes from 3:2 to 3:2-3. The rise of [K], under apparent "steady-state" conditions is only modest. However, Fig. 5 shows that the system is not in a true steady-state. If [K],. is raised, there is of course a continuous diffusion of K ÷ out of the clefts into the bathing ¢ In the model, after block of the N a - K pump, the steady level of [K],., both at rest and during stimulation, is higher than [K]h and it represents an equilibrium between the steady ioss of K ÷ from the cells and diffusion out of the clefts into the bathing medium along the generated concentration gradient. As would be expected, in the model, [K], (not illustrated) declines continuously after pump block and the decline is more rapid during stimulation.

HEART RATE AND ION CONCENTRATIONS

25

medium along the concentration gradient. For [K]c to remain high there must be a continuous net loss of K ÷ from the cells; the result of this is a steady decline in [K],. It may be concluded that within the framework of the model an accumulation of K ÷ in the clefts at high rates under true steady-state conditions cannot occur (this is considered further below). 4. Discussion Figures 3 and 4 show that the computed changes in [K]~ are similar in time course and magnitude to the changes in [K]c observed experimentally. Rate-dependent changes in [Na]~ (Cohen et al., 1982; Lederer & Sheu, 1983; January & Fozzard, 1984; Ellis, 1985; Boyett et al., 1987a,b), [K]~ (Langer, 1968; Cohen et aL, 1982) and N a - K pump current (Boyett & Fedida, 1984; Falk & Cohen, 1984) have also been measured in the heart and they too are similar to the computed changes. Ellis (1985) reported that in sheep Purkinje fibres [Na]~ gradually rises during stimulation over about 15 rain and, although the rise varies from one preparation to another, it is similar to that predicted by the model (for example, when a preparation is stimulated at 120/min after a rest [Na]i rises by 15-55%; in Fig. 2 the model predicted a rise in [Na]; by about 40% at a rate of 150/min). Langer & Brady (1966) found that in dog ventricular muscle [K]~ falls by about 1 mM (cf. Fig. 2) when the stimulation rate is increased by about 30 beats/min. Boyett & Fedida (1984) measured rate-dependent changes in pump current in voltage clamp experiments on dog Purkinje fibres, and found a slow build-up of pump current over about 7 min after an increase in rate (cf. Fig. 1). The cleft Na ÷ concentration has not been measured and evidence of the changes in [Na]c predicted by the model in Fig. 2 has yet to be sought. Because of the broad agreement between the experimental and computed results, the important conclusion from this study is that the experimentally observed changes c a n be explained within the framework of the model pictured in Fig. 1.t Figure 5 shows that a steady accumulation of K ÷ within the clefts is unlikely to occur at high rates under steady-state conditions. There is, however, one condition under which such a rise could occur, i.e. when the K + channels in the membrane are separated from the N a - K pump sites by a substantial diffusion barrier. In this circumstance, there would be an accumulation of K ÷ in the parts of the clefts where N a - K p u m p sites were absent, but where present, the enhanced activity of the N a - K pump would deplete cleft K ÷. This would seem to be an unlikely arrangement, but it cannot be ruled out. We wish to thank Drs Kunze and Morad for permission to reproduce some of their results. t However, it should be borne in mind that the model has its limitations. In the absence of data, it assumes step changes in the passive fluxes after changes in rate, but this may not be the case. At least it can be concluded that it is not essential to postulate slow changes in the passive fluxes after a change in rate in order to explain the events. The model predicts monotonic changes in intracellular Na+ after a change in rate (e.g. Fig. 2), and although monotonic changes in intracellular Na+ have been reported by mostauthors (Cohen el al., 1982;January & Fozzard, 1984;Ellis, 1985),biphasic changes in intracellular Na+ have been observed in certain circumstances (Boyett et al., 1987a,b). The cause of the biphasic changes in Na+ is not known and the model in its present form cannot account for them.

26

M. R. B O Y E T T A N D D. F E D 1 D A REFERENCES

AKERA, T., BENNE'VT, R. T., OLGAARD, M. K. & BRODY, T. M. (1976). Cardiac Na +, K+-adenosine triphosphatase inhibition by ouabain and myocardial sodium: a computer simulation. J. Pharmac. exp. Ther. 199, 287. ATTWELL, D., EISNER, D. & COHEN, 1. (1979). Voltage clamp and tracer flux data: effects of a restricted extra-cellular space. Q. Rev. Biophys. 12, 213. BOYEqrq', M. R., HART, G. & LEVI, A. J. (1987a). Factors affecting intracellular sodium during repetitive activity in isolated sheep Purkinje fibres. J. Physiol. 384, 405. BoYEl-r, M. R., HART, G., LEVI, A. J. & ROBERTS, A. (1987b). Effects of repetitive activity on developed force and intracellular sodium in isolated sheep and dog Purkinje fibres. J. Physiol. 388, 295. BOYETT, M. R. & FEDIDA, D. (1983). A simple model of ion concentration changes in the heart during repetitive activity. Z Physiol. 342, 52P. BoYEr-r, M. R. & FEDIDA, D. (1984). Changes in the electrical activity of dog cardiac Purkinje fibres at high heart rates. J. Physiol. 350, 361. BOYEVr, M. R. & JEWELL, B. R. (1980). Analysis of the effects of changes in rate and rhythm upon electrical activity in the heart. Prog. Biophys. molec. Biol. 36, 1. CARMELIET, E. & LACQUET, L. (1958). Duree du potentiel d'action ventriculaire de grenouille en fonction de la frequence influence des variations ioniques de potassium et sodium. Archs int. Physiol. Biochim. 66, 1. CHAPMAN, J. B., KOOTSEY, J. M. & JOHNSON, E. A. (1979). A kinetic model for determining the consequences of electrogenic active transport in cardiac muscle. J. theor. Biol. 80, 405. COHEN, C. J., FOZZARD, H. A. & SHEU, S.-S. (1982). Increase in intracellular sodium ion activity during stimulation in mammalian cardiac muscle. Circ. Res..~0, 651. COHEN, I. & KLINE, R. (1982). K + fluctuations in the extracellular spaces of cardiac muscle. Evidence from the voltage clamp and extracellular K+-selective microelectrodes. Circ. Res. 50, 1. DAUT, J. (1978). The Effects of Dihydro-ouabain on the Action Potential of Cardiac Purkinje Fibres. PhD. thesis, University of Oxford. DEITMER, J. W. & ELLIS, D. (1978). The intracellular sodium activity of cardiac Purkinje fibres during inhibition and re-activation of the Na-K pump. J. Physiol. 284, 241. DIFRANCESCO, D. & NOBLE, D. (1980). The time course of potassium current following potassium accumulation in frog atrium: analytical solutions using a linear approximation. J. Physiol. 306, 151. DIFRANCESCO, D. & NOBLE, D. (1982). Implications of the re-interpretation of iK.2 for the modelling of the electrical activity of pacemaker tissues in the heart. In: Cardiac Rate and Rhythm, (Bouman, L. N. & Jongsma, H. J., eds). The Hague: Martinus Nijhoff. DIFRANCESCO, D. & NOBLE, D. (1985). A model of cardiac electrical activity incorporating ionic pumps and concentration changes. Phil. Trans. Roy. Soc. B307, 353. EISNER, D. A., LEDERER, W. J. & VAUGHAN-JONES, R. O. (1981). The dependence of sodium pumping and tension on intracellular sodium activity in voltage-clamped sheep Purkinje fibres. J. Physiol, 317, 163. ELLIS, D. (1985). Effects of stimulation and diphenylhydantoin on the intracellular sodium activity in Purkinje fibres of sheep hearts. J. Physiol. 362, 331. FALK, R. T. & COHEN, I. S. (1984). Membrane current following activity in canine cardiac Purkinje fibers. J. Gen. Physiol. 83, 771. FEDIDA, D. (1985). A computer model of the ionic currents and the electrogenic Na pump in cardiac Purkinje strands predicts action potential changes during repetitive excitation. J. Physiol. 341, 27P. GAOSBY, D. C. (1980). Activation of electrogenic Na+/K + exchange by extracellular K + in canine cardiac Purkinje fibers. Proc. natn. Acad. Sci. U.S.A. 77, 4035. GARAY, R. P. & GARRAHAN, P. J. (1973). The interaction of sodium and potassium with the sodium pump in red cells. J. Physiol. 231, 297. GLITSCH, H. G. (1979). Characteristics of active Na transport in intact cardiac cells. Am. Z Physiol. 236, H189. HART, G., NOBLE, D. & SHI/VlONI, Y. (1983). The effects of low concentrations of cardiotonic steroids on membrane currents and tension in sheep Purkinje fibres. J. Physiol. 334, 103. JANUARY, C. T. & FOZZARD, H. A. (1984). The effects of membrane potential, extracellular potassium, and tetrodotoxin on the intracellular sodium ion activity of sheep cardiac muscle. Circ. Res. 54, 652. JOHNSON, E. A., CHAPMAN, J. }3. & KOOTSEY, J. M. (1980). Some electrophysiological consequences of electrogenic sodium and potassium transport in cardiac muscle: a theoretical study. J. theor. Biol. 87, 737.

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KLINE, R. P., COHEN, I., FALK, R. & KUPERSMITH, J. (1980). Activity-dependent extracellular K + fluctuations in canine Purkinje fibres. Nature 286, 68. KLINE, R. P. & KUPERSMITH, J. (1982). Effects of extracellular potassium accumulation and sodium pump activation on automatic canine Purkinje fibres. J. Physiol. 324, 507. KLINE, R. P. & MORAD, M. (1978). Potassium efflux in heart muscle during activity: extracellular accumulation and its implications. J. Physiol. 2110, 537. KOCH-WESER, J. & BLINKS, J. R. (1963). The influence of the interval between beats on myocardial contractility. Pharmac. Reo. 15, 601. KUNZE, D. L. (1977). Rate-dependent changes in extraeellular potassium in the rabbit atrium. Circ. Res. 41, 122. LANGER, G. A. (1968). Ion fluxes in cardiac excitation and contraction and their relation to myocardial contractility. Physiol. Rev. 48, 708. LANGER, G. A. & BRADY, A. J. (1966). Potassium in dog ventricular muscle: kinetic studies of distribution and effects of varying frequency of contraction and potassium concentration of perfusate. Circ. Res. 18, 164. LEOERER, W. J. & SHEU, S.-S. (1983). Heart rate-dependent changes in intracellular sodium activity and twitch tension in sheep cardiac Purkinje fibres. J. PhysioL 345, 44P. MARTIN, G. & MORAD, M. (1982). Activity-induced potassium accumulation and its uptake in frog ventricular muscle. J. Physiol. 328, 205. MOBLEY, B. A. & PAGE, E. (1972). The surface area of sheep cardiac Purkinje fibres. J. Physiol. 220, 547. REVERD1N, E. C., ILLANES, A. & McGUIGAN, J. A. S. (1986). Internal potassiuni activity in ferret ventricular muscle. Q. J. exp. Physiol. 71, 451. SOMMER, J. R. & JOHNSON, E. A. (1968). Cardiac muscle. A comparative study of Purkinje fibers and ventricular fibers. J. cell Biol. 36, 497.