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Using V̇max to estimate changes in the sodium membrane conductance in cardiac cells
COMPUTERS
AND
BIOMEDICAL
RESEARCH
20, 351-365 (1987)
Using V,,, to Estimate Changes in the Sodium Membrane Conductance in Cardiac Cells FERNAND A. ROBERGEANDJEAN-PIERREDROUHARD Institute of Biomedical Engineering, Ecole Polytechnique and Faculty of Medicine, P.O. Box 6128, Station A, H3C 357, UniversitP de Mont&al, Qut!bec, Canada Received December 23. 1986
Relative changes in the sodium conductance of the resting cardiac cell membrane are often estimated from relative changes in the maximum rate of rise of the action potential (V,,). This approach has given rise to some controversy and it has not been possible so far to test it directly on an experimental basis. We have examined here the validity of this estimation using three different Hodgkin-Huxley representations of the cardiac membrane sodium current. The two basic requirements are a constant membrane capacitance and a negligible relative value of the nonsodium membrane currents at the time of V,,,. It is shown further that the approach leads to a satisfactory estimation only when the latency of V,,, is kept constant and a correction factor for the sodium driving force is applied to p,,,,, measurements. This conclusion applies either to a nonpropagated action or to an action potential propagated at constant velocity, provided that the membrane is not too strongly depolarized. It is valid for a wide range of sodium equilibrium potentials and a range of maximum sodium conductances limited to about 50% of the nominal value. 0 1987 Academic press, Inc.
INTRODUCTION
The partial blockade of the sodium membrane conductance (gNa) is probably a major mechanism of action of antiarrhythmic drugs in cardiac cells. The drug effects on gNa are currently determined either by direct measurements of the sodium current, INa, using voltage clamp techniques, or by indirect estimation from ti,,, of action potentials. As discussed recently by Grant et al. (I), both approaches have their respective advantages and limitations. Direct measurements provide insight into the precise mechanisms of drug actions but the difficulty of the techniques and the relatively short periods of time over which reliable records can be obtained make the direct approach unsuitable for the comparative study of many drugs with respect to their potency and speed of action. In contrast, V,,, measurements are convenient for the study of drug effects at the tissue level because they can be performed with relative ease, under a variety of controlled conditions, and in preparation with realistic geometry that are stable over several hours. But the reliability of e,,, as an estimate of g&, in cardiac tissue is uncertain. The subject has given rise to some controversy, focusing mainly on the linearity of the relationship between the two 351 OOlO-4809/87 $3.00 Copyright 0 1987 by Academic Press. Inc. All rights of reproduction in any form reserved.
352
ROBERGE AND DROUHARD
quantities (2-5). A recent claim of demonstrated nonlinearity by Cohen rt o/. (6, 7) has been disputed by Hondeghem (8) and Courtney (9). There are two basic requirements for the indirect approach which are well understood. These are the fixed value of the membrane capacitance. C,. during the action potential upstroke, and the small value of nonsodium currents compared to IN:, at the time of Q,,,. Since the maximum value of gN,, during the upstroke (peak g& occurs nearly at the time of i’,,;,.,, there is a very reasonable likelihood that relative changes in p,,,,, will reflect relative changes in gs,, at that moment. But the most critical underlying assumption of the method is that relative changes in Gus, at the time of e,,,. are strictly proportional to corresponding changes in gNa just before the action potential upstroke. at the resting potential level. The validity of these assumptions is examined in the present study by simulating the experimental technique. employing three different Hodgkin-Huxley representations of the sodium current passing through the myocardial cell membrane. It is shown that i’,,,,, measurements made at a constant latency can provide a satisfactory estimate of changes in
The sodium current is modeled by the equation II\;, = g,,( V - Vha) with ,qNa = m3h&,. The symbols frequently used in this discussion are defined in Table 1. For a membrane action potential (nonpropagated). the total ionic current (/,I is equal to the capacitive current C,v. In cardiac cells. the nonsodium currents at the time of ri,,, are much smaller than the sodium current and can be neglected. This condition is well reflected in the sodium current models used in the present study (see Results). Then we have TABLE LIST
Vha)
I
OF SYMHOLS
Membrane potential Resting membrane potential First derivative of potential with respect to time (dV/dr) Maximum value of V during action potential upstroke Membrane potential at which V,,, occurred Time of occurrence of V,;,, Sodium equilibrium potential Electrochemical sodium driving force acrobb membrane Sodium current crossing membrane per unit area Total ionic current crossing the membrane per unit area Maximal conductance of the sodium channel Sodium channel activation variable (voltage- and time-dependent) Sodium channel inactivation variable (voltage- and time-dependent) Actual conductance of the sodium channel defined as gNa = rn3hRy.
ESTIMATING
CHANGES
IN gNa IN CARDIAC
CELLS
- -t) = mw, i) . h(V, - t)- *& - (V- V&). C,V,,,(V,
353
HI
We assume throughout that the membrane capacitance, C,, is constant. Then peak ZNa (right-hand side of [l]) is strictly proportional to e,,,,, and the two quantities coincide in time. Note, however, that peak gNa does not necessarily coincide in time with peak ZN~. In fact, for a membrane action potential, peak gNa is slightly delayed relative to peak ZN, (see Results). Co_nsider a reference action potential characterized by vO, to, ~,,,(vO, to), gNa(VO, to), and a given reference resting potential. From [l], we can express the relative changes in &a as
MV, gNa( 6,
- _ t) = +~&x(v, f) (6 -
VNa)
to)
VNa)
%nax(~O,
iI> ’ (v
-
PI ’
where the primed quantities are functions of v and i, and correspond to the actual state of the system. Since vand f,,, can be obtained directly from the recorded action potential upstroke, [2] provides a convenient estimation of changes in &a, Ideally, one would like to make all e,,, measurements at the same values of v and t (i.e., v = v, and i = to> in order to minimize the variations in the voltage- and time-dependent gating variables. In practice, it is relatively easy to keep the latency constant (i.e., i = iO) by adjusting the amplitude of the fixed-duration stimulus. But the voltage v cannot be maintained constant over a wide range of conditions using simple procedures. Thus the best practical estimation of changes in &a at the time of ?,, is
- - gidv, h> = kx(v, to> gNa(6
, id
kd%
id
. (6
-
VNa)
(v -
VNa)
[31
which is more accurate if the range of variation of v is as restricted as possible. When an action potential propagates at constant velocity, the axial current is proportional to the second derivative of the membrane potential, d*V/dt*. At the time of emax, we have d2V/dt2 = 0 and the axial current is equal to zero. Therefore, the above analysis applies to this particular propagated case also provided that the nonsodium membrane currents are again negligible with respect to zNa. Our purpose is to examine the applicability of [3] in cardiac cells and to show that relative changes in gNa at the time of V,,, are equivalent to relative changes in gNa at the resting level. To do that, we have compared the results of simulated experimental techniques for three different Hodgkin-Huxley representations of the Na+ current. One is the original formulation of the BR model (10). Another is based on the synthesis of recent voltage clamp data which we have substituted into the BR model to constitute the BRDR model (11). A third one is the Ebihara and Johnson (12) representation substituted into the BR model to yield the BREJ model. While the BRDR, BR, and BREJ models differ by their ZNa representations, they have identical nonsodium current expressions .
354
ROBERGE
AND
DROUHARD
Details of the simulation technique have been given previously (If ). The membrane action potential was elicited by a constant current pulse, 2 msec in duration, from a stable resting potential (V,). The amplitude of the stimulation pulse was adjusted whenever necessary to keep the latency of V,,,, constant relative to the onset of the stimulus. Thus all measurements were made at the same time of occurrence of Q,,, (i.e., i = t(l). RESULTS
I. Nonsodium
Currents
The three membrane models (BRDR, BR, and BREJ) used in this study have similar nonsodium currents but quite different ZNaat the time of c,,, (see Fig. 6 in (II)). At the time of V,,,, in agreement with experimental measurements in cardiac cells, the size of the nonsodium currents in these models is quite small in comparison with the amplitude of the Na+ current. We see from [l] that the magnitude of the nonsodium currents at the time of c,,, corresponds to the difference between peak ZNa and C,e,,,. In Table 2, we have normalized this difference with respect to peak ZNRfor various values of ,&, and VN~. For nominal values of gNa and VNa (i.e.. lOO%), the difference between peak Z,, and peak capacitive current is less than 1% of peak ZXa in the BRDR model, and only slightly larger in the other two models. Increasing ,&, or V,, (e.g., 125%) reduces the relative importance of the nonsodium currents still further. Conversely, decreasing gNa or V,, (down to 25%) yields a small increase of less than 2% in the BRDR model. Therefore, as verified here in the case of a membrane action potential, ZNa is clearly the dominant membrane current component at the time of p,,,,, in the BRDR and BREJ models, while it is only slightly less so in the BR model (Table 2). 2. Sodium
Current Activation
and Inactivation
For membrane action potentials elicited from resting potentials more negative than about -70 mV, and provided that to is kept constant, the time course
TABLE RELATIVE
2
VALUE OF NONSODIUM CURRENTS FOR A MEMBKANE ACTION POTENTIAL. TIME OF V,,,, EXPRESSED BY THE RATIO (PEAK INa - C.,~,,,,)IPEAK Iv& Ru.,
Model -BRDR BR BREJ
125% 0.55% 1.66% 1.07%
100% 0.70% 2.09% 1.22%
AT THE
vv, 100%
50%
25%
125%
0.85% 2.83% 1.48%
1.15%
1.97% -
0.62%
0.70%
0.82%'
0.98%
1.209t
4.50%
1.77To
2.09Yc
2.55%
3.257~
4.42Tc
2.00%
3.61%
1.11%
1.22%
1.40%
1.64%
1.97%
75%
50%
25%
75%
ESTIMATING
CHANGES
IN g,, IN CARDIAC CELLS
355
./ .. / ‘\ amv ~--
Vo=-65rn”
0’
B
i
._ 1\ ) t (ms) A T- 5
1
1.00 h,-. m 0.75
,
- ---.- .__ _ _ _
BR -
i
-.
“.==-ssmv Vo--55m”
0.50
i’
I
,!’
0.25
,.*‘.,y
,
,/
-\
T.e:*:::
0
C
1
2
t Ons)
3
4
5
h, m
-
V,=-95mV
- -
V,=-75mv
~~~
V,=-65mV
V,--65mV
0
1
2
3
4
5
FIG. 1. Temporal variations in Na+ activation (WI) and inactivation (h) variables during the rising phase of the action potential, for action potentials initiated from different resting potentiallevels: BRDR model (panel A), BR model (panel B), BREJ model (panel C). The occurrence of V,,, (at time iO = 2.76 msec) is the same for all action potentials in all models, and is indicated by the vertical thin line. Note that the termination of the stimulus pulse, at 2 msec, causes slight undulations of the m(t) and h(t) curves.
of the Na+ activation variable, m(t), is practically independent of the resting potential (VO), and its magnitude, m(to>, reaches about 80% of full activation. This is illustrated in Fig. 1, for all three models, with V. values of -95, -85, and -75 mV. Under these specific conditions, the relative changes in &a at the time of ti’,, are independent of the Na+ current activation, since the m3 factor of the Na+ conductance cancels out when relative measurements are considered (see [2]). This observation is no longer valid for stronger depolarization, as illustrated for VO = -65 mV in Fig. 1. Note also that at -65 mV, in the BR model, it was no longer possible to trigger an action potential having the same i,, which was obtained with more negative resting potentials. Assuming that & has the same value in the actual and reference states, the above result on the invariance of m(&) allows [3] to be rewritten as
356
ROBERGE
AND
DROUHARD
h’(V, - lo) - = claxw, . -_‘_to) (c’n - Viva) M Vn,
to)
VmdVn.
h)
(V
-
141
VN;,)
where h(Vo. io) and h’(V, to) are the Na’ current inactivation values in the reference and actual states, respectively, and the other symbols are as previously defined. Equation [4] expresses the fact that the relative changes in h at the time of e,,,, as a function of Vn. can be estimated from measurements on cm and v, assuming that VNais known with sufficient accuracy. Although to is kept constant, there are nevertheless fairly substantial changes in the time course of the Na+ inactivation variable, h(t), and in its value at the time of C,,, , as V. is varied. This is illustrated in Fig. I for resting potentials in the range of -95 to -65 mV. But we note that the relative changes in Na’ inactivation at the time of V,,,, (h’(~o)/h(~o)) are comparable to those at time zero (h’(O)lh(O)), as shown by the calculations made in Table 3 with respect to a reference state of V. = -95 mV. The last column in Table 3 gives the percentage change of h’(&)lh(io) relative to h’(O)lh(O) for each value of V,, A deviation of less than 5% is observed in the BRDR model for V,, more negative than -75 mV, while it is a little larger in the other two models. Hence the right-hand side of [4] provides an acceptable estimation of the relative changes in Na + inactivation for the resting potential level (time zero in Fig. I and Table 3). Therefore. the steady-state characteristic h,(V) can be obtained in this way provided that the membrane is not too strongly depolarized. TABLE RELATIVE MEMBRANE
IN Na+
CHANGES POTENTIAL.
/I
Note.
The primed
INACTIVATION
V,, = -95
TAKING
Time
BRDR model V, = -95 mV -85 mV -75 mV -65 mV BR model V, = -95 mV -85 mV -75 mV -65 mV BREJ model V, = -95 mV -85 mV -75 mV -65 mV
CURRENT
3 WI I I-I THE
mV AS THE REFERENCE Time
zero
of V,,,<
RESTING STATE
Deviation (‘4
h’(O)/h(O)
I1
0.999 0.987 0.810 0.195
1.000 0.988 0.81 I 0.195
0.731 0.718 0.565 0.120
I.000 0.982 0.773 0.165
0.0 0.5 4.6 15.5
1.000 0.989 0.771 -
1.000 0.990 0.772
0.740 0.720 0.524 -
1.000 0.973 0.708 -.
0.0 I.6 x.2
0.998 0.985 0.872 0.415
1.000 0.986 0.873 0.415
0.568 0.557 0.466 0.201
l.UOO 11.980 0.821 0.355
0.0 0.6 6.0 14.5
quantities
correspond
to the actual
h’(i,,lih(i,,)
state
ESTIMATING
CHANGES
IN gNa IN CARDIAC CELLS
357
BRDR
V, (mv) 100
-60
-100 -60
-60
-60
0
-40
BREJ
1.0.
0.5
OJ
100 -60
-60
-40
FIG. 2. Relative changes in V,,,, peak gNa. and (v - V,,) with the resting potential level for the BRDR (panel A), BR (panel B), and BREJ (panel C) models. The ordinate values are all normalized relative to the reference state values given in Table 4. The steady-state inactivation characteristic, h,(V), is drawn as a continuous curve.
In a membrane model, the peak value of gNa does not coincide exactly in time with the occurrence of ri max. Under the conditions of the present simulation, in all three models, peak gNa is delayed slightly (about 0.1 msec) with respect to to, so that the estimated value of g &, at to is a little smaller than peak &a. When relative changes are considered, however, the difference is negligible and peak gNa (which is more conveniently measured in the models) may be taken to be equivalent to &$a at the time of Vi,,,. Hence the ratio peak gh,/(peak g& used in Figs. 2-4 is essentially - - equal to the ratio gha(V, tO)/gNJVO, to) of 133, or the equivalent ratio h’(V, to)lh(vO, ta> of [4]. In view of the result of Table 3, the ratio peak gh,/(peak g&O is also equivalent to h,(V). The use of [4] to estimate h,(V) is illustrated in Fig. 2 where relative changes in timax, peak gNa, and (v - VN,) are plotted as a function of the resting potential for all three models. For the BRDR and BREJ models, the data points for peak gNa follow very closely the expected sigmoidal steady-state inactivation curve, h,(V) (see [12] and Fig. 5 in (II)). But, for these two models, the relative
358
ROBERGE
AND DROUHARD
changes in 6’ maxdeviate slightly from the h,(V) curve and the correction by the ratio (7, - V&V - V,,) indicated by [4] is necessary. This correction factor becomes important as the resting potential reaches values more depolarized than about -80 mV (Fig. 2). The situation is different for the BR model which shows a shift of the c,,, and peak gNa data points toward more negative potentials. As discussed by Cohen ef ul. (51. this shift is due to the fact that the nonsodium currents are relatively more important in the BR model (see Table 2). The data of Fig. 2 have been replotted in Fig. 3 in order to display more clearly the relationship between V,,, and peak gNa. All three models show a slightly nonlinear relationship which is a little less marked in the case of the BR model. When applying the correction factor (v - VN&/(V~I - VN,), all the data points of Fig. 3 fall on the 45” line so that the estimation is strictly proportional to relative changes in peak gNa. We have also examined the influence of changes in I/‘&,.,and &, on the time course of m and h during the action potential upstroke for the nominal resting potential of VO = -85 mV. Even substantial changes in VNa (from 25 to 125%) affect the Na’ gating variables very little. Similar changes in gilrja are slightly more potent but, in comparison with the effects resulting from changes in Vet. the variations observed on m(t) and h(t) are quite insignificant. It was also verified that the h,(V) curve estimated via [4] is practically independent of 1’,, and gNa, except when j& is reduced below 50% of its nominal value. 3. Changes in km,, Due to Variations
in VNa and &;a
The relative changes in p,,,, peak &a, and (v - Vy,) due to variations in VN~ are illustrated in Fig. 4 for all three models. The reference state values are as given in Table 4. As VN~ was changed by discrete amounts, action potentials 1.00
-
\i’ max 01, to) . vrn,, 010, to)
0.75 ,,I’
,’
.,,.
0.50 -
A
,/.
. I+.,.’ /‘
.,I.
.,,’
l
o,/’ .
0.25 1
1
,,.’ _.,’
0
,.:
,/’
,/’
BRDR
0 BR
/
* BREJ peak
0.25
0.50
0.75
7 1 .oo
beak
dNa g,,),
FIG. 3. Relationship between relative changes in c,,, and relative changes in peak gNa for the BRDR, BR, and BREJ models. The broken line at 45” represents a linear relationship between relative changes in V,,, and peak gNa.
ESTIMATING
CHANGES
IN gNa IN CARDIAC
CELLS
359
A
0.25
0
B
0.50
0.75
1.00
1.25
,..‘.
1.25
...‘.
Peak g’,, (peak t&,J,, 1.oo
I
0
& ,.,“b’ -..__....u----“a
,/.,.../.
-
0.25
0.50
0.75
1.00
1.25
C
-
0
0.25
0.50
0.75
BRDR
1 .oo
1.25
FIG. 4. Changes in V,,, (panel A), peak &,a (panel B), and (V - VNJ (panel C) due to variations in VNa. All quantities are normalized relative to the reference state values given in Table 4. The relationships are linear for all three models.
were elicited from a resting level of -85 mV using a 2-msec current pulse whose amplitude was adjusted in order to have a constant &,. The three models display similar relative changes in qi,,,, peak &a, and (v - VN,) with respect to the reference state of V,,,(~O, $, (peak g&, and (vO - VNJ, and these changes are linearly related to variations in V&(VN&. The most remarkable feature of these results is the insensitivity of the relative changes in peak g& to variations in VN~, especially for the BRDR and BREJ models. Consequently, for fixed V. and &a, the changes in V,,, are paralleled by changes in the sodium driving force, as indicated in Figs. 4A and 4C.
360
ROBERGEANDDROUHARD TABLE REFERENCE
4
STATE VALUES FOR THE UK. BRDR, MODELS, AS USED IN FIGS. 3 .AND 4
BRDR V. (mV) (&:,,), (mS/cm’) ( VNah WV) fe (msec) To (mV) V,,,CV,. id (Visec) (peak gNa),, (mS/cm’) ( F,, ~ VNs) (mV)
model
-8s IS.00 40.00 3.76 -6.19 423. IX IO.91 -46.19
BR model x5 3.00 50.00 1.76 16.60 131.69 2.09 --66.60
AND
BREJ
BREJ
model
x5 23.00 29 .oo 2.76 -x.33 234.80 7.09 37.33
Changes caused by variations in gNa are illustrated in Fig. 5 for all three models. The conditions were similar to those used to study changes in VNd. I;Iere also the models display a similar behavior and the relative changes in V max3 peak &a, and (v - V& are linearly related to &,&&, provided that &+ is not reduced more than about 50%. The slope of the relationship between peak &a and &a is close to unity, particularly for the BRDR and BREJ models (Fig. 5B). On the other hand, variations in &a have only a small influence on the sodium driving force (Fig. SC). These measurements were repeated for different resting potential values in the range of -65 to -95 mV. For potentials more negative than about -70 mV. there is very little variability in the results and changes due to variations in VNa and &a are well represented by the curves of Figs. 4 and 5. DISCUSSION
The above calculations apply to a nonpropagated action potential or to an action potential propagated at constant velocity. Their validity depends on the assumption that the Hodgkin-Huxley representation of the sodium current used here does reflect the behavior of the myocardial cell membrane. If this assumption is accepted, then it can be concluded from our simulation results that the experimental method of using e,,,,, to provide a measure of relative changes in &a (see [3]) is applicable in the following circumstances: 1. The membrane capacitance, C,, does not change during the action potential upstroke. 2. At the time of v max, the sodium current, fNa, is so much greater than the other currents that the difference between the total ionic current, Ii, and INa is not significant. 3. The time of occurrence of c,,, is . kept constant so that the time-dependent sodium gating variables have an opportunity to change by similar amounts.
ESTIMATING
CHANGES
IN gNa IN CARDIAC CELLS
4. Relative c,,,,, measurements must be corrected for variations dium driving force at the time of e,,,.
361 in the so-
Condition 3 refers to the relationship between relative changes in &a at the time of G,,,, as measured by the method, and corresponding changes in &a at the resting level, prior to the stimulation. We have examined this question in detail here for a membrane action potential. First we have noted that relative changes in peak gNa correspond well to relative changes in gNa at the time of
B Peak g’,, beak g,&,
1.25-
1.000.75. 0.50 ,..’
0.25 \,,,,.......... ...i., , a;:,,;5 & C - P-U (vo - VNJO
0
0.26
1.25-
1
0.50
0.75
&G 1 .oo -
0.75.
,..’ ...’ ...’ ,..’ ,..’ ,..’ ,...” . ..’
FIG. 5. Changes in G,,,,, (panel A), peak gNa (panel B), and (v - VN,) (panel C) due to variations in RNa.As in Fig. 4, all quantities are normalized relative to the reference state values given in Table 4. The relationships are linear for all three models as long as & is not reduced by more than about 50%.
362
ROBERGEANDDROUHARD
V,,, so that we can discuss the problem in terms of the former, more easily measured, parameter. If the latency of V,,, is kept constant, the Nat activation variable reaches roughly the same value at the time of c,,,,. even though the resting potential is varied over a fairly wide range (Fig. I ). Then, with a fixed gNa, we see that relative changes in peak gNa and Na- inactivation are equivalent at the time of V,,,. We have shown here that relative changes in peak gNa are adequately estimated by relative values of c,,,, . when a correction factor for the sodium driving force is applied (Fig. 2). In addition, we have established that relative changes in Na’ inactivation at the resting potential level (i.e., in the steady state) are equivalent to relative changes in Na’ inactivation at the time of c,,,, provided that the membrane is not too strongly depolarized. Consequently, we find that the pattern of relative changes in peak &+,,ais equivalent to the steady-state Na+ inactivation curve, hJ E’). Therefore. the present simulation study demonstrates that relative G’,,,;,, measurements. with a correction for the sodium driving force, give a good estimation of relative changes in KNa at the resting potential level. This result is also valid in the presence of variations in CNi, since such variations do not alter the behavior of the Nat activation variable at the time of elEix. In fact, relative changes in peak Key, or its equivalent h,(v). are practically unaffected by variations in Vpqa(Fig. 4B). As illustrated in Fig. 4A. our simulation results differ from the experimental observations of Kohlhardt (13 ) who found a nonlinear relationship between p,,,,,, and the extracellular Na’ concentration. This discrepancy is probably due to the fact that the time ot occurrence of G,,, was allowed to vary in his experiments, thus bringing into play differences in the Nat activation and inactivation processes, Variations in KNa produced linear relative changes in c’,;,,, peak #ha. and (v - V,,), as long as the maximum range of variations is not much more than about 50% (Fig. 5). Otherwise, the relationship becomes nonlinear. This result concurs with a similar simulation study by Walton and Fozzard (4) using the Purkinje fiber model of McAllister rt ~1. (/4 ). The influence of van the accuracy of the indirect approach has been recognized (I, 4). Since v cannot be usually maintained constant. the resulting variations in the sodium driving force must be taken into account. The present simulation study has shown that the use of a correction factor for V(see [3] and [4]) allows an accurate estimation of gha at the resting potential level. This correction factor also removes the nonlinearity seen in the relationship between relative changes in v,,, and peak ~~~ (Fig. 3). Therefore, from the experimental point of view, the proper use of the indirect approach requires, in addition to maintaining i constant, the accurate monitoring of v and a correct estimation of VNil. The results of the present study differ from those of Cohen rt ~1. (6) who reported a strongly nonlinear relationship between gNa and $‘,,,,. In order to better understand the major discrepancies between their results and ours, we have compared the membrane current characteristics of the model of Cohen et al. (6) (referred to here as the CBT model) with those of the models used here.
ESTIMATING
CHANGES
363
IN gNa IN CARDIAC CELLS
The CBT was implemented using their Na+ current activation and inactivation equations ([5]-[ll] in Cohen et al. (6)), the expression for the leak current given in the legend to their Fig. 7, and VNa = 50 mV. The action potential was triggered by a 2-msec pulse of current from a resting potential of - 104 mV, and it was necessary to use & = 85 mS/cm2 in order to obtain a p,,,,, of about 230 V/set. The stimulus intensity required to have t = 2.76 msec was 27.25 pA/cm2. At the time of p max, we have in the CBT model fT = 17.11 mV, peak ZNa = 274.3 pAlcm2, and a leak current equal to 16.2% of ZN~. There are, in addition, very important differences between the Na+ current characteristics of the CBT model and those of the models used in the present study, as shown in Fig. 6. We note, first of all, that the m,(V) and h,(V) curves of the CBT model are shifted some 8 mV toward more positive and more negative potentials, respectively, compared to the BRDR model (Fig. 6B). As a result, these characteristics of the CBT model are at the limit, and even outside in the case of h,(V), of the range of usual experimental measurements. This can be readily appreciated by comparing Fig. 6B with Figs. 1 and 3 of Drouhard and Roberge (II). Similarly, the time constants of activation (r,,J and inactivation (7h) of the CBT model differ widely from those of the BRDR and BR models (Figs. 6C and 6D). Finally, because of the very large time constant of activation and inactivation, and the large value of RNar the 8Na pulse during the upstroke is considerably larger and longer lasting in the CBT model (Fig. 6A). C
rm0-d
1.00 1 0.7%
0.50-
-
-‘XT .~ BRDR --- BR
0.25-
B
h,,m, I.007
D
200
5 (ms) -CBT -BRDR --- BR
FIG. 6. Comparison of the main characteristics of the CBT, BRDR, and BR models. Panel A shows the gNa pulse during the action potential upstroke, panel B the steady-state inactivation (h,) and activation (m,) curves, panel C the time constant of activation (T,), and panel D the time constant of inactivation (Q).
364
ROBERGEANDDROUHARD
Obviously, the presence of very large nonsodium currents in the CBT model violates condition 2 above and invalidates the use of V,,, as an estimate of gNe. It is not surprising then that these authors have found a nonlinear relationship between these two quantities. Moreover, the CBT model has very peculiar Na+ current characteristics which deviate importantly from the range of usually observed experimental values. These features make it difficult to accept the CBT model as a valid representation of the cardiac cell membrane. Previous discussions of the relationship between V,;,, and ,& have focused the attention on the ratio r/, : T,, stressing the fact that a small value of this ratio (e.g., 6: 1) could contribute to the nonlinearity of the relationship (1, 5). As the present simulation study shows, however, more or less rapid changes in m(t) and h(t) are not determinant factors in this regard. Since only relative changes from a reference state are considered, the requirement for applicability of the theory is to have roughly equal percentage changes in m(t). or /z(t). both at the resting potential level and at the time of k,,,,. In other words, incomplete activation at & and inactivation occurring during the stimulus and latency period do not affect the accuracy of the method, provided that the drug effects produce roughly equal relative changes in the gating variables at both time zero and &.
ACKNOWLEDGMENTS This work was supported by the Medical Research Council and the Natural Sciences and Engineering Research Council of Canada, the Quebec Heart Foundation. and the Departement d’Enseignement Superieur et des Sciences du Quebec.
REFERENCES 1. GRANT, A. 0.. STARMER. C. F., AND STRAUSS, H. C. Antiarrhythmic drug action: Blockade of the inward sodium current. Circ. Res. 55, 427 (1984). 2. STRICHARTZ, G. R.. AND COHEN. I. S. c ,,,‘,j as a meabu~-c of ,gv,, 111net-vc and cardiac mcmbranes. Biopll~s. .I. 23, 153 (1978). .i. HONDEC~HEM, 1.. M. Validity of G,,.,, a\ II measure of the sodium current in cardiac and nervous tissues. Riop1ry.c. .I. 23, 147 (1978). 4. WAI.TON. M.. AND FOZARD. H. A. The relation of i’,,,,,, and /I in a model of the cardiac Purkinje fiber. Rioply.c. ./. 25, 407 ( 197’)). 5. COHEN, I.. ATWELL, D.. AND STRICHARTZ, G. The dependence of the maximum rate of rise of the action potential upstroke on membrane properties. Proc~. R. Sot. London B 214, 85 (1981). 6. COHEN, C. J., BEAN, B. P., AND TSIEN, R. W. Maximal upstroke velocity as an index of available sodium conductance-Comparison of maximal upstroke velocity and voltage clamp measurements of sodium current in rabbit Purkinje fibers. Circ. Res. 57, 636 (1984). 7. COHEN, C. J.. BEAN, B. P., AND TSIEN. R. W. Strengths and weaknesses of 6’,,,,, as a measure of available gN,,. Circ. Res. 57, 195 (1985). 8. HONDEGHEM. L. M. Comments on Cohen rt N/. (19X4). Circ,. Rex. 57. iY2 (IYXS). V. COURTNEY, K. R. Comments on Cohen (‘I cl/. (1984). Cirr,. Re.s. 57, 194 (1985). IO. BEELER, G. W., AND REUTER, H. Reconstruction of the action potential of ventricular myocardial fibres. J. Physiol. (London) 268, 177 (1977).
ESTIMATING
CHANGES
IN gNa IN CARDIAC CELLS
365
DROUHARD, .I. P., AND ROBERGE, F. A. Revised formulation of the Hodgkin-Huxley representation of the sodium current in cardiac cells. C’omp~f. Biomed. Res. 20, 333 (1987). 12. EBIHARA, L., AND JOHNSON, E. A. Fast sodium current in cardiac muscle. A quantitative description. Biophys. J. 32, 779 (1980). 13. KOHLHARDT. M. A quantitative analysis of the Na-dependence of ?,,, of the fast action potential in mammalian ventricular myocardium: Saturation characteristics and the modulation of a drug induced INa blockade by [Na+],,. efl II<‘x em Arch. 392, 379 (1982). 14. MCALLISTER, R. E., NOBLE, D., AND TSIEN, R. W. Reconstruction of the electrical activity of cardiac Purkinje fibres. J. Physiol. (London) 251, 1 (1975). If.
AND
BIOMEDICAL
RESEARCH
20, 351-365 (1987)
Using V,,, to Estimate Changes in the Sodium Membrane Conductance in Cardiac Cells FERNAND A. ROBERGEANDJEAN-PIERREDROUHARD Institute of Biomedical Engineering, Ecole Polytechnique and Faculty of Medicine, P.O. Box 6128, Station A, H3C 357, UniversitP de Mont&al, Qut!bec, Canada Received December 23. 1986
Relative changes in the sodium conductance of the resting cardiac cell membrane are often estimated from relative changes in the maximum rate of rise of the action potential (V,,). This approach has given rise to some controversy and it has not been possible so far to test it directly on an experimental basis. We have examined here the validity of this estimation using three different Hodgkin-Huxley representations of the cardiac membrane sodium current. The two basic requirements are a constant membrane capacitance and a negligible relative value of the nonsodium membrane currents at the time of V,,,. It is shown further that the approach leads to a satisfactory estimation only when the latency of V,,, is kept constant and a correction factor for the sodium driving force is applied to p,,,,, measurements. This conclusion applies either to a nonpropagated action or to an action potential propagated at constant velocity, provided that the membrane is not too strongly depolarized. It is valid for a wide range of sodium equilibrium potentials and a range of maximum sodium conductances limited to about 50% of the nominal value. 0 1987 Academic press, Inc.
INTRODUCTION
The partial blockade of the sodium membrane conductance (gNa) is probably a major mechanism of action of antiarrhythmic drugs in cardiac cells. The drug effects on gNa are currently determined either by direct measurements of the sodium current, INa, using voltage clamp techniques, or by indirect estimation from ti,,, of action potentials. As discussed recently by Grant et al. (I), both approaches have their respective advantages and limitations. Direct measurements provide insight into the precise mechanisms of drug actions but the difficulty of the techniques and the relatively short periods of time over which reliable records can be obtained make the direct approach unsuitable for the comparative study of many drugs with respect to their potency and speed of action. In contrast, V,,, measurements are convenient for the study of drug effects at the tissue level because they can be performed with relative ease, under a variety of controlled conditions, and in preparation with realistic geometry that are stable over several hours. But the reliability of e,,, as an estimate of g&, in cardiac tissue is uncertain. The subject has given rise to some controversy, focusing mainly on the linearity of the relationship between the two 351 OOlO-4809/87 $3.00 Copyright 0 1987 by Academic Press. Inc. All rights of reproduction in any form reserved.
352
ROBERGE AND DROUHARD
quantities (2-5). A recent claim of demonstrated nonlinearity by Cohen rt o/. (6, 7) has been disputed by Hondeghem (8) and Courtney (9). There are two basic requirements for the indirect approach which are well understood. These are the fixed value of the membrane capacitance. C,. during the action potential upstroke, and the small value of nonsodium currents compared to IN:, at the time of Q,,,. Since the maximum value of gN,, during the upstroke (peak g& occurs nearly at the time of i’,,;,.,, there is a very reasonable likelihood that relative changes in p,,,,, will reflect relative changes in gs,, at that moment. But the most critical underlying assumption of the method is that relative changes in Gus, at the time of e,,,. are strictly proportional to corresponding changes in gNa just before the action potential upstroke. at the resting potential level. The validity of these assumptions is examined in the present study by simulating the experimental technique. employing three different Hodgkin-Huxley representations of the sodium current passing through the myocardial cell membrane. It is shown that i’,,,,, measurements made at a constant latency can provide a satisfactory estimate of changes in
The sodium current is modeled by the equation II\;, = g,,( V - Vha) with ,qNa = m3h&,. The symbols frequently used in this discussion are defined in Table 1. For a membrane action potential (nonpropagated). the total ionic current (/,I is equal to the capacitive current C,v. In cardiac cells. the nonsodium currents at the time of ri,,, are much smaller than the sodium current and can be neglected. This condition is well reflected in the sodium current models used in the present study (see Results). Then we have TABLE LIST
Vha)
I
OF SYMHOLS
Membrane potential Resting membrane potential First derivative of potential with respect to time (dV/dr) Maximum value of V during action potential upstroke Membrane potential at which V,,, occurred Time of occurrence of V,;,, Sodium equilibrium potential Electrochemical sodium driving force acrobb membrane Sodium current crossing membrane per unit area Total ionic current crossing the membrane per unit area Maximal conductance of the sodium channel Sodium channel activation variable (voltage- and time-dependent) Sodium channel inactivation variable (voltage- and time-dependent) Actual conductance of the sodium channel defined as gNa = rn3hRy.
ESTIMATING
CHANGES
IN gNa IN CARDIAC
CELLS
- -t) = mw, i) . h(V, - t)- *& - (V- V&). C,V,,,(V,
353
HI
We assume throughout that the membrane capacitance, C,, is constant. Then peak ZNa (right-hand side of [l]) is strictly proportional to e,,,,, and the two quantities coincide in time. Note, however, that peak gNa does not necessarily coincide in time with peak ZN~. In fact, for a membrane action potential, peak gNa is slightly delayed relative to peak ZN, (see Results). Co_nsider a reference action potential characterized by vO, to, ~,,,(vO, to), gNa(VO, to), and a given reference resting potential. From [l], we can express the relative changes in &a as
MV, gNa( 6,
- _ t) = +~&x(v, f) (6 -
VNa)
to)
VNa)
%nax(~O,
iI> ’ (v
-
PI ’
where the primed quantities are functions of v and i, and correspond to the actual state of the system. Since vand f,,, can be obtained directly from the recorded action potential upstroke, [2] provides a convenient estimation of changes in &a, Ideally, one would like to make all e,,, measurements at the same values of v and t (i.e., v = v, and i = to> in order to minimize the variations in the voltage- and time-dependent gating variables. In practice, it is relatively easy to keep the latency constant (i.e., i = iO) by adjusting the amplitude of the fixed-duration stimulus. But the voltage v cannot be maintained constant over a wide range of conditions using simple procedures. Thus the best practical estimation of changes in &a at the time of ?,, is
- - gidv, h> = kx(v, to> gNa(6
, id
kd%
id
. (6
-
VNa)
(v -
VNa)
[31
which is more accurate if the range of variation of v is as restricted as possible. When an action potential propagates at constant velocity, the axial current is proportional to the second derivative of the membrane potential, d*V/dt*. At the time of emax, we have d2V/dt2 = 0 and the axial current is equal to zero. Therefore, the above analysis applies to this particular propagated case also provided that the nonsodium membrane currents are again negligible with respect to zNa. Our purpose is to examine the applicability of [3] in cardiac cells and to show that relative changes in gNa at the time of V,,, are equivalent to relative changes in gNa at the resting level. To do that, we have compared the results of simulated experimental techniques for three different Hodgkin-Huxley representations of the Na+ current. One is the original formulation of the BR model (10). Another is based on the synthesis of recent voltage clamp data which we have substituted into the BR model to constitute the BRDR model (11). A third one is the Ebihara and Johnson (12) representation substituted into the BR model to yield the BREJ model. While the BRDR, BR, and BREJ models differ by their ZNa representations, they have identical nonsodium current expressions .
354
ROBERGE
AND
DROUHARD
Details of the simulation technique have been given previously (If ). The membrane action potential was elicited by a constant current pulse, 2 msec in duration, from a stable resting potential (V,). The amplitude of the stimulation pulse was adjusted whenever necessary to keep the latency of V,,,, constant relative to the onset of the stimulus. Thus all measurements were made at the same time of occurrence of Q,,, (i.e., i = t(l). RESULTS
I. Nonsodium
Currents
The three membrane models (BRDR, BR, and BREJ) used in this study have similar nonsodium currents but quite different ZNaat the time of c,,, (see Fig. 6 in (II)). At the time of V,,,, in agreement with experimental measurements in cardiac cells, the size of the nonsodium currents in these models is quite small in comparison with the amplitude of the Na+ current. We see from [l] that the magnitude of the nonsodium currents at the time of c,,, corresponds to the difference between peak ZNa and C,e,,,. In Table 2, we have normalized this difference with respect to peak ZNRfor various values of ,&, and VN~. For nominal values of gNa and VNa (i.e.. lOO%), the difference between peak Z,, and peak capacitive current is less than 1% of peak ZXa in the BRDR model, and only slightly larger in the other two models. Increasing ,&, or V,, (e.g., 125%) reduces the relative importance of the nonsodium currents still further. Conversely, decreasing gNa or V,, (down to 25%) yields a small increase of less than 2% in the BRDR model. Therefore, as verified here in the case of a membrane action potential, ZNa is clearly the dominant membrane current component at the time of p,,,,, in the BRDR and BREJ models, while it is only slightly less so in the BR model (Table 2). 2. Sodium
Current Activation
and Inactivation
For membrane action potentials elicited from resting potentials more negative than about -70 mV, and provided that to is kept constant, the time course
TABLE RELATIVE
2
VALUE OF NONSODIUM CURRENTS FOR A MEMBKANE ACTION POTENTIAL. TIME OF V,,,, EXPRESSED BY THE RATIO (PEAK INa - C.,~,,,,)IPEAK Iv& Ru.,
Model -BRDR BR BREJ
125% 0.55% 1.66% 1.07%
100% 0.70% 2.09% 1.22%
AT THE
vv, 100%
50%
25%
125%
0.85% 2.83% 1.48%
1.15%
1.97% -
0.62%
0.70%
0.82%'
0.98%
1.209t
4.50%
1.77To
2.09Yc
2.55%
3.257~
4.42Tc
2.00%
3.61%
1.11%
1.22%
1.40%
1.64%
1.97%
75%
50%
25%
75%
ESTIMATING
CHANGES
IN g,, IN CARDIAC CELLS
355
./ .. / ‘\ amv ~--
Vo=-65rn”
0’
B
i
._ 1\ ) t (ms) A T- 5
1
1.00 h,-. m 0.75
,
- ---.- .__ _ _ _
BR -
i
-.
“.==-ssmv Vo--55m”
0.50
i’
I
,!’
0.25
,.*‘.,y
,
,/
-\
T.e:*:::
0
C
1
2
t Ons)
3
4
5
h, m
-
V,=-95mV
- -
V,=-75mv
~~~
V,=-65mV
V,--65mV
0
1
2
3
4
5
FIG. 1. Temporal variations in Na+ activation (WI) and inactivation (h) variables during the rising phase of the action potential, for action potentials initiated from different resting potentiallevels: BRDR model (panel A), BR model (panel B), BREJ model (panel C). The occurrence of V,,, (at time iO = 2.76 msec) is the same for all action potentials in all models, and is indicated by the vertical thin line. Note that the termination of the stimulus pulse, at 2 msec, causes slight undulations of the m(t) and h(t) curves.
of the Na+ activation variable, m(t), is practically independent of the resting potential (VO), and its magnitude, m(to>, reaches about 80% of full activation. This is illustrated in Fig. 1, for all three models, with V. values of -95, -85, and -75 mV. Under these specific conditions, the relative changes in &a at the time of ti’,, are independent of the Na+ current activation, since the m3 factor of the Na+ conductance cancels out when relative measurements are considered (see [2]). This observation is no longer valid for stronger depolarization, as illustrated for VO = -65 mV in Fig. 1. Note also that at -65 mV, in the BR model, it was no longer possible to trigger an action potential having the same i,, which was obtained with more negative resting potentials. Assuming that & has the same value in the actual and reference states, the above result on the invariance of m(&) allows [3] to be rewritten as
356
ROBERGE
AND
DROUHARD
h’(V, - lo) - = claxw, . -_‘_to) (c’n - Viva) M Vn,
to)
VmdVn.
h)
(V
-
141
VN;,)
where h(Vo. io) and h’(V, to) are the Na’ current inactivation values in the reference and actual states, respectively, and the other symbols are as previously defined. Equation [4] expresses the fact that the relative changes in h at the time of e,,,, as a function of Vn. can be estimated from measurements on cm and v, assuming that VNais known with sufficient accuracy. Although to is kept constant, there are nevertheless fairly substantial changes in the time course of the Na+ inactivation variable, h(t), and in its value at the time of C,,, , as V. is varied. This is illustrated in Fig. I for resting potentials in the range of -95 to -65 mV. But we note that the relative changes in Na’ inactivation at the time of V,,,, (h’(~o)/h(~o)) are comparable to those at time zero (h’(O)lh(O)), as shown by the calculations made in Table 3 with respect to a reference state of V. = -95 mV. The last column in Table 3 gives the percentage change of h’(&)lh(io) relative to h’(O)lh(O) for each value of V,, A deviation of less than 5% is observed in the BRDR model for V,, more negative than -75 mV, while it is a little larger in the other two models. Hence the right-hand side of [4] provides an acceptable estimation of the relative changes in Na + inactivation for the resting potential level (time zero in Fig. I and Table 3). Therefore. the steady-state characteristic h,(V) can be obtained in this way provided that the membrane is not too strongly depolarized. TABLE RELATIVE MEMBRANE
IN Na+
CHANGES POTENTIAL.
/I
Note.
The primed
INACTIVATION
V,, = -95
TAKING
Time
BRDR model V, = -95 mV -85 mV -75 mV -65 mV BR model V, = -95 mV -85 mV -75 mV -65 mV BREJ model V, = -95 mV -85 mV -75 mV -65 mV
CURRENT
3 WI I I-I THE
mV AS THE REFERENCE Time
zero
of V,,,<
RESTING STATE
Deviation (‘4
h’(O)/h(O)
I1
0.999 0.987 0.810 0.195
1.000 0.988 0.81 I 0.195
0.731 0.718 0.565 0.120
I.000 0.982 0.773 0.165
0.0 0.5 4.6 15.5
1.000 0.989 0.771 -
1.000 0.990 0.772
0.740 0.720 0.524 -
1.000 0.973 0.708 -.
0.0 I.6 x.2
0.998 0.985 0.872 0.415
1.000 0.986 0.873 0.415
0.568 0.557 0.466 0.201
l.UOO 11.980 0.821 0.355
0.0 0.6 6.0 14.5
quantities
correspond
to the actual
h’(i,,lih(i,,)
state
ESTIMATING
CHANGES
IN gNa IN CARDIAC CELLS
357
BRDR
V, (mv) 100
-60
-100 -60
-60
-60
0
-40
BREJ
1.0.
0.5
OJ
100 -60
-60
-40
FIG. 2. Relative changes in V,,,, peak gNa. and (v - V,,) with the resting potential level for the BRDR (panel A), BR (panel B), and BREJ (panel C) models. The ordinate values are all normalized relative to the reference state values given in Table 4. The steady-state inactivation characteristic, h,(V), is drawn as a continuous curve.
In a membrane model, the peak value of gNa does not coincide exactly in time with the occurrence of ri max. Under the conditions of the present simulation, in all three models, peak gNa is delayed slightly (about 0.1 msec) with respect to to, so that the estimated value of g &, at to is a little smaller than peak &a. When relative changes are considered, however, the difference is negligible and peak gNa (which is more conveniently measured in the models) may be taken to be equivalent to &$a at the time of Vi,,,. Hence the ratio peak gh,/(peak g& used in Figs. 2-4 is essentially - - equal to the ratio gha(V, tO)/gNJVO, to) of 133, or the equivalent ratio h’(V, to)lh(vO, ta> of [4]. In view of the result of Table 3, the ratio peak gh,/(peak g&O is also equivalent to h,(V). The use of [4] to estimate h,(V) is illustrated in Fig. 2 where relative changes in timax, peak gNa, and (v - VN,) are plotted as a function of the resting potential for all three models. For the BRDR and BREJ models, the data points for peak gNa follow very closely the expected sigmoidal steady-state inactivation curve, h,(V) (see [12] and Fig. 5 in (II)). But, for these two models, the relative
358
ROBERGE
AND DROUHARD
changes in 6’ maxdeviate slightly from the h,(V) curve and the correction by the ratio (7, - V&V - V,,) indicated by [4] is necessary. This correction factor becomes important as the resting potential reaches values more depolarized than about -80 mV (Fig. 2). The situation is different for the BR model which shows a shift of the c,,, and peak gNa data points toward more negative potentials. As discussed by Cohen ef ul. (51. this shift is due to the fact that the nonsodium currents are relatively more important in the BR model (see Table 2). The data of Fig. 2 have been replotted in Fig. 3 in order to display more clearly the relationship between V,,, and peak gNa. All three models show a slightly nonlinear relationship which is a little less marked in the case of the BR model. When applying the correction factor (v - VN&/(V~I - VN,), all the data points of Fig. 3 fall on the 45” line so that the estimation is strictly proportional to relative changes in peak gNa. We have also examined the influence of changes in I/‘&,.,and &, on the time course of m and h during the action potential upstroke for the nominal resting potential of VO = -85 mV. Even substantial changes in VNa (from 25 to 125%) affect the Na’ gating variables very little. Similar changes in gilrja are slightly more potent but, in comparison with the effects resulting from changes in Vet. the variations observed on m(t) and h(t) are quite insignificant. It was also verified that the h,(V) curve estimated via [4] is practically independent of 1’,, and gNa, except when j& is reduced below 50% of its nominal value. 3. Changes in km,, Due to Variations
in VNa and &;a
The relative changes in p,,,, peak &a, and (v - Vy,) due to variations in VN~ are illustrated in Fig. 4 for all three models. The reference state values are as given in Table 4. As VN~ was changed by discrete amounts, action potentials 1.00
-
\i’ max 01, to) . vrn,, 010, to)
0.75 ,,I’
,’
.,,.
0.50 -
A
,/.
. I+.,.’ /‘
.,I.
.,,’
l
o,/’ .
0.25 1
1
,,.’ _.,’
0
,.:
,/’
,/’
BRDR
0 BR
/
* BREJ peak
0.25
0.50
0.75
7 1 .oo
beak
dNa g,,),
FIG. 3. Relationship between relative changes in c,,, and relative changes in peak gNa for the BRDR, BR, and BREJ models. The broken line at 45” represents a linear relationship between relative changes in V,,, and peak gNa.
ESTIMATING
CHANGES
IN gNa IN CARDIAC
CELLS
359
A
0.25
0
B
0.50
0.75
1.00
1.25
,..‘.
1.25
...‘.
Peak g’,, (peak t&,J,, 1.oo
I
0
& ,.,“b’ -..__....u----“a
,/.,.../.
-
0.25
0.50
0.75
1.00
1.25
C
-
0
0.25
0.50
0.75
BRDR
1 .oo
1.25
FIG. 4. Changes in V,,, (panel A), peak &,a (panel B), and (V - VNJ (panel C) due to variations in VNa. All quantities are normalized relative to the reference state values given in Table 4. The relationships are linear for all three models.
were elicited from a resting level of -85 mV using a 2-msec current pulse whose amplitude was adjusted in order to have a constant &,. The three models display similar relative changes in qi,,,, peak &a, and (v - VN,) with respect to the reference state of V,,,(~O, $, (peak g&, and (vO - VNJ, and these changes are linearly related to variations in V&(VN&. The most remarkable feature of these results is the insensitivity of the relative changes in peak g& to variations in VN~, especially for the BRDR and BREJ models. Consequently, for fixed V. and &a, the changes in V,,, are paralleled by changes in the sodium driving force, as indicated in Figs. 4A and 4C.
360
ROBERGEANDDROUHARD TABLE REFERENCE
4
STATE VALUES FOR THE UK. BRDR, MODELS, AS USED IN FIGS. 3 .AND 4
BRDR V. (mV) (&:,,), (mS/cm’) ( VNah WV) fe (msec) To (mV) V,,,CV,. id (Visec) (peak gNa),, (mS/cm’) ( F,, ~ VNs) (mV)
model
-8s IS.00 40.00 3.76 -6.19 423. IX IO.91 -46.19
BR model x5 3.00 50.00 1.76 16.60 131.69 2.09 --66.60
AND
BREJ
BREJ
model
x5 23.00 29 .oo 2.76 -x.33 234.80 7.09 37.33
Changes caused by variations in gNa are illustrated in Fig. 5 for all three models. The conditions were similar to those used to study changes in VNd. I;Iere also the models display a similar behavior and the relative changes in V max3 peak &a, and (v - V& are linearly related to &,&&, provided that &+ is not reduced more than about 50%. The slope of the relationship between peak &a and &a is close to unity, particularly for the BRDR and BREJ models (Fig. 5B). On the other hand, variations in &a have only a small influence on the sodium driving force (Fig. SC). These measurements were repeated for different resting potential values in the range of -65 to -95 mV. For potentials more negative than about -70 mV. there is very little variability in the results and changes due to variations in VNa and &a are well represented by the curves of Figs. 4 and 5. DISCUSSION
The above calculations apply to a nonpropagated action potential or to an action potential propagated at constant velocity. Their validity depends on the assumption that the Hodgkin-Huxley representation of the sodium current used here does reflect the behavior of the myocardial cell membrane. If this assumption is accepted, then it can be concluded from our simulation results that the experimental method of using e,,,,, to provide a measure of relative changes in &a (see [3]) is applicable in the following circumstances: 1. The membrane capacitance, C,, does not change during the action potential upstroke. 2. At the time of v max, the sodium current, fNa, is so much greater than the other currents that the difference between the total ionic current, Ii, and INa is not significant. 3. The time of occurrence of c,,, is . kept constant so that the time-dependent sodium gating variables have an opportunity to change by similar amounts.
ESTIMATING
CHANGES
IN gNa IN CARDIAC CELLS
4. Relative c,,,,, measurements must be corrected for variations dium driving force at the time of e,,,.
361 in the so-
Condition 3 refers to the relationship between relative changes in &a at the time of G,,,, as measured by the method, and corresponding changes in &a at the resting level, prior to the stimulation. We have examined this question in detail here for a membrane action potential. First we have noted that relative changes in peak gNa correspond well to relative changes in gNa at the time of
B Peak g’,, beak g,&,
1.25-
1.000.75. 0.50 ,..’
0.25 \,,,,.......... ...i., , a;:,,;5 & C - P-U (vo - VNJO
0
0.26
1.25-
1
0.50
0.75
&G 1 .oo -
0.75.
,..’ ...’ ...’ ,..’ ,..’ ,..’ ,...” . ..’
FIG. 5. Changes in G,,,,, (panel A), peak gNa (panel B), and (v - VN,) (panel C) due to variations in RNa.As in Fig. 4, all quantities are normalized relative to the reference state values given in Table 4. The relationships are linear for all three models as long as & is not reduced by more than about 50%.
362
ROBERGEANDDROUHARD
V,,, so that we can discuss the problem in terms of the former, more easily measured, parameter. If the latency of V,,, is kept constant, the Nat activation variable reaches roughly the same value at the time of c,,,,. even though the resting potential is varied over a fairly wide range (Fig. I ). Then, with a fixed gNa, we see that relative changes in peak gNa and Na- inactivation are equivalent at the time of V,,,. We have shown here that relative changes in peak gNa are adequately estimated by relative values of c,,,, . when a correction factor for the sodium driving force is applied (Fig. 2). In addition, we have established that relative changes in Na’ inactivation at the resting potential level (i.e., in the steady state) are equivalent to relative changes in Na’ inactivation at the time of c,,,, provided that the membrane is not too strongly depolarized. Consequently, we find that the pattern of relative changes in peak &+,,ais equivalent to the steady-state Na+ inactivation curve, hJ E’). Therefore. the present simulation study demonstrates that relative G’,,,;,, measurements. with a correction for the sodium driving force, give a good estimation of relative changes in KNa at the resting potential level. This result is also valid in the presence of variations in CNi, since such variations do not alter the behavior of the Nat activation variable at the time of elEix. In fact, relative changes in peak Key, or its equivalent h,(v). are practically unaffected by variations in Vpqa(Fig. 4B). As illustrated in Fig. 4A. our simulation results differ from the experimental observations of Kohlhardt (13 ) who found a nonlinear relationship between p,,,,,, and the extracellular Na’ concentration. This discrepancy is probably due to the fact that the time ot occurrence of G,,, was allowed to vary in his experiments, thus bringing into play differences in the Nat activation and inactivation processes, Variations in KNa produced linear relative changes in c’,;,,, peak #ha. and (v - V,,), as long as the maximum range of variations is not much more than about 50% (Fig. 5). Otherwise, the relationship becomes nonlinear. This result concurs with a similar simulation study by Walton and Fozzard (4) using the Purkinje fiber model of McAllister rt ~1. (/4 ). The influence of van the accuracy of the indirect approach has been recognized (I, 4). Since v cannot be usually maintained constant. the resulting variations in the sodium driving force must be taken into account. The present simulation study has shown that the use of a correction factor for V(see [3] and [4]) allows an accurate estimation of gha at the resting potential level. This correction factor also removes the nonlinearity seen in the relationship between relative changes in v,,, and peak ~~~ (Fig. 3). Therefore, from the experimental point of view, the proper use of the indirect approach requires, in addition to maintaining i constant, the accurate monitoring of v and a correct estimation of VNil. The results of the present study differ from those of Cohen rt ~1. (6) who reported a strongly nonlinear relationship between gNa and $‘,,,,. In order to better understand the major discrepancies between their results and ours, we have compared the membrane current characteristics of the model of Cohen et al. (6) (referred to here as the CBT model) with those of the models used here.
ESTIMATING
CHANGES
363
IN gNa IN CARDIAC CELLS
The CBT was implemented using their Na+ current activation and inactivation equations ([5]-[ll] in Cohen et al. (6)), the expression for the leak current given in the legend to their Fig. 7, and VNa = 50 mV. The action potential was triggered by a 2-msec pulse of current from a resting potential of - 104 mV, and it was necessary to use & = 85 mS/cm2 in order to obtain a p,,,,, of about 230 V/set. The stimulus intensity required to have t = 2.76 msec was 27.25 pA/cm2. At the time of p max, we have in the CBT model fT = 17.11 mV, peak ZNa = 274.3 pAlcm2, and a leak current equal to 16.2% of ZN~. There are, in addition, very important differences between the Na+ current characteristics of the CBT model and those of the models used in the present study, as shown in Fig. 6. We note, first of all, that the m,(V) and h,(V) curves of the CBT model are shifted some 8 mV toward more positive and more negative potentials, respectively, compared to the BRDR model (Fig. 6B). As a result, these characteristics of the CBT model are at the limit, and even outside in the case of h,(V), of the range of usual experimental measurements. This can be readily appreciated by comparing Fig. 6B with Figs. 1 and 3 of Drouhard and Roberge (II). Similarly, the time constants of activation (r,,J and inactivation (7h) of the CBT model differ widely from those of the BRDR and BR models (Figs. 6C and 6D). Finally, because of the very large time constant of activation and inactivation, and the large value of RNar the 8Na pulse during the upstroke is considerably larger and longer lasting in the CBT model (Fig. 6A). C
rm0-d
1.00 1 0.7%
0.50-
-
-‘XT .~ BRDR --- BR
0.25-
B
h,,m, I.007
D
200
5 (ms) -CBT -BRDR --- BR
FIG. 6. Comparison of the main characteristics of the CBT, BRDR, and BR models. Panel A shows the gNa pulse during the action potential upstroke, panel B the steady-state inactivation (h,) and activation (m,) curves, panel C the time constant of activation (T,), and panel D the time constant of inactivation (Q).
364
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Obviously, the presence of very large nonsodium currents in the CBT model violates condition 2 above and invalidates the use of V,,, as an estimate of gNe. It is not surprising then that these authors have found a nonlinear relationship between these two quantities. Moreover, the CBT model has very peculiar Na+ current characteristics which deviate importantly from the range of usually observed experimental values. These features make it difficult to accept the CBT model as a valid representation of the cardiac cell membrane. Previous discussions of the relationship between V,;,, and ,& have focused the attention on the ratio r/, : T,, stressing the fact that a small value of this ratio (e.g., 6: 1) could contribute to the nonlinearity of the relationship (1, 5). As the present simulation study shows, however, more or less rapid changes in m(t) and h(t) are not determinant factors in this regard. Since only relative changes from a reference state are considered, the requirement for applicability of the theory is to have roughly equal percentage changes in m(t). or /z(t). both at the resting potential level and at the time of k,,,,. In other words, incomplete activation at & and inactivation occurring during the stimulus and latency period do not affect the accuracy of the method, provided that the drug effects produce roughly equal relative changes in the gating variables at both time zero and &.
ACKNOWLEDGMENTS This work was supported by the Medical Research Council and the Natural Sciences and Engineering Research Council of Canada, the Quebec Heart Foundation. and the Departement d’Enseignement Superieur et des Sciences du Quebec.
REFERENCES 1. GRANT, A. 0.. STARMER. C. F., AND STRAUSS, H. C. Antiarrhythmic drug action: Blockade of the inward sodium current. Circ. Res. 55, 427 (1984). 2. STRICHARTZ, G. R.. AND COHEN. I. S. c ,,,‘,j as a meabu~-c of ,gv,, 111net-vc and cardiac mcmbranes. Biopll~s. .I. 23, 153 (1978). .i. HONDEC~HEM, 1.. M. Validity of G,,.,, a\ II measure of the sodium current in cardiac and nervous tissues. Riop1ry.c. .I. 23, 147 (1978). 4. WAI.TON. M.. AND FOZARD. H. A. The relation of i’,,,,,, and /I in a model of the cardiac Purkinje fiber. Rioply.c. ./. 25, 407 ( 197’)). 5. COHEN, I.. ATWELL, D.. AND STRICHARTZ, G. The dependence of the maximum rate of rise of the action potential upstroke on membrane properties. Proc~. R. Sot. London B 214, 85 (1981). 6. COHEN, C. J., BEAN, B. P., AND TSIEN, R. W. Maximal upstroke velocity as an index of available sodium conductance-Comparison of maximal upstroke velocity and voltage clamp measurements of sodium current in rabbit Purkinje fibers. Circ. Res. 57, 636 (1984). 7. COHEN, C. J.. BEAN, B. P., AND TSIEN. R. W. Strengths and weaknesses of 6’,,,,, as a measure of available gN,,. Circ. Res. 57, 195 (1985). 8. HONDEGHEM. L. M. Comments on Cohen rt N/. (19X4). Circ,. Rex. 57. iY2 (IYXS). V. COURTNEY, K. R. Comments on Cohen (‘I cl/. (1984). Cirr,. Re.s. 57, 194 (1985). IO. BEELER, G. W., AND REUTER, H. Reconstruction of the action potential of ventricular myocardial fibres. J. Physiol. (London) 268, 177 (1977).
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DROUHARD, .I. P., AND ROBERGE, F. A. Revised formulation of the Hodgkin-Huxley representation of the sodium current in cardiac cells. C’omp~f. Biomed. Res. 20, 333 (1987). 12. EBIHARA, L., AND JOHNSON, E. A. Fast sodium current in cardiac muscle. A quantitative description. Biophys. J. 32, 779 (1980). 13. KOHLHARDT. M. A quantitative analysis of the Na-dependence of ?,,, of the fast action potential in mammalian ventricular myocardium: Saturation characteristics and the modulation of a drug induced INa blockade by [Na+],,. efl II<‘x em Arch. 392, 379 (1982). 14. MCALLISTER, R. E., NOBLE, D., AND TSIEN, R. W. Reconstruction of the electrical activity of cardiac Purkinje fibres. J. Physiol. (London) 251, 1 (1975). If.