20 Electrochemical Model of Voltage-Gated Channels Martin Blank Department of Physiology and Cellular Biophysics, Columbia University, 630 West 168th Street, New York, NY 10032
An electrochemical model of protein aggregation has been developed based on changes in surface free energy that accompany changes in molecular surface area and charge density. The surface free energy model quantitatively predicts the disaggregation of hemoglobin tetramers into dimers at alkaline pHs, and also accounts for the energetics of the conformational changes in the reactions of hemoglobin with ligands (e.g., the Bohr effects, the Hill coefficient). Because membrane channels are oligomeric proteins, the model has been applied to channel opening and closing. Many physical properties appear to be direct consequences of this model, including the observed variation of the ion selectivity of voltage-gated channels with gating current and the cooling that accompanies channel opening.
Electrochemical Perspective T h e two m a i n threads o f recent research o n i o n channels—investigations o f channel structure a n d channel f u n c t i o n — a r e interrelated i n that research i n each area illuminates b o t h . Biologists realize that an understanding o f func tion must b e based o n structure, b u t it has b e e n difficult to extract clues relevant to f u n c t i o n f r o m the very detailed i n f o r m a t i o n about amino acid sequences a n d their arrangements i n the channel proteins. T h i s is where the concepts o f electrochemistry have p r o v e n useful. C o n s i d e r a t i o n o f average properties a n d large populations o f channels, rather than specific groups i n an individual channel (the focus o f m u c h current research), has made it possible 0065-2393/94/0235-0429 $08.00/0 © 1994 American Chemical Society
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BIOMEMBRANE ELECTROCHEMISTRY
to treat channels as charged surfaces o f proteins a n d to develop a macro scopic description o f channel function. T h i s approach has p r o v i d e d insights into a n u m b e r o f important problems. I n this chapter, w e w i l l summarize o u r research o n the energetics a n d kinetics o f c h a n n e l f u n c t i o n a n d show h o w the electrochemical m o d e l deals w i t h two specific problems: i o n specificity a n d t h e r m a l responses that accompany channel o p e n i n g a n d closing.
Aggregated Protein Structures M a n y m e m b r a n e processes i n cells (e.g., i o n p u m p i n g , sensory transduction, impulse conduction, energy transduction) involve charge movement
across
integral p r o t e i n structures c a l l e d channels. C h a n n e l s are c o m p o s e d o f subunits that have specialized h y d r o p h i l i c a n d h y d r o p h o b i c regions organized i n a cyhndrical geometry. T h e r e are many structural similarities i n groups o f protein assemblies w i t h different physiological properties ( I ) , so specific biological properties must b e achieved through relatively small variations o f physical properties such as the surface charge ( 2 ) . T h e o p e n i n g a n d closing o f all channels requires changes i n p r o t e i n - p r o t e i n contacts, m o r e accurately changes i n the area o f p r o t e i n - w a t e r interface, so the gating process depends u p o n the same physical principles that govern aggregation a n d disaggregation i n oligomeric proteins. These principles are w e l l k n o w n . I n fine w i t h o u r knowledge o f mass action, proteins o f many sizes, shapes,
a n d degrees
o f aggregation t e n d to aggregate as t h e m o n o m e r
concentration increases. Aggregation decreases as the p H is further away f r o m the isoelectric point a n d t h e charge o n t h e molecule increases. Surface charge plays a n important role because t h e surface free energy o f dissolved proteins is l o w a n d the electrostatic c o n t r i b u t i o n is relatively large ( 3 ) . P r o t e i n aggregation reactions are generally described as entropy-driven because they p r o c e e d spontaneously w i t h the evolution o f heat; that is, a positive change o f enthalpy ( 4 ) . Because the net free energy change is negative f o r a spontaneous process, a large positive entropy change must accompany this reaction, most likely f r o m the release o f adsorbed water molecules w h e n t h e oligomers associate. The accepted description is correct but incomplete. T h e description does not indicate w h y aggregation occurs only i n particular ranges o f p H . T o understand the p H dependence t h e aggregation process should b e considered i n terms o f surface free energy, w h i c h varies w i t h the surface charge. Because aggregation does not involve molecular rearrangements w i t h i n the subunits, b u t rather interactions b e tween the subunit surfaces, w e can approximate the total free energy change i n these processes b y the surface free energy. T h e loss o f adsorbed waters u p o n oligomer aggregation is equivalent to the loss o f interfacial area, a n d disaggregation involves the formation o f additional p r o t e i n - w a t e r interface.
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Electrochemical Model of Voltage-Gated Channels
431
Surface Free Energy Changes in Protein Aggregation T h e surface free energy, F , o f the p r o t e i n can be calculated i n terms o f two s
parameters: A , the area o f the m o l e c u l e i n contact w i t h the solution, a n d σ , the surface charge density ( σ = charge p e r A ) : F
s
(1)
= ( / - T T ) A 0
w h e r e A is the m o l a r area o f the p r o t e i n surface, f
0
is the free energy p e r
unit area o f an u n c h a r g e d p r o t e i n surface, a n d IT is the surface pressure due to the charged surface layer, w h e r e IT < / . 0
(below 10 charges p e r 100 n m ) , π = kv , 2
2
A t l o w surface charge density w h e r e k = 0.016
if σ
is i n
charges p e r nanometer squared a n d ττ is i n ergs p e r centimeter squared. Therefore,
(2)
F = (f -ka )A s
2
0
a n d a small change i n F , s
A
F
s
= (/o " & σ ) Δ Α - 2 & σ Α Δ σ
(3)
2
F r o m the positive Δ A t e r m i n e q 3, it appears that increases i n area l e a d to a positive A F , so this process w i l l not o c c u r spontaneously. T h e unfavor S
able free energy that accompanies an increase i n area can be overcome b y an increase i n σ (i.e., Δ σ > 0), w h i c h contributes directly to a negative A F
S
and
also contributes indirectly b y decreasing the magnitude o f the Δ A t e r m . T h i s p r i n c i p l e is i n action i n the disaggregation o f h e m o g l o b i n , w h i c h involves an energetically unfavorable increase i n surface area. H o w e v e r , as the h e m o g l o b i n tetramer becomes positively or negatively charged, there is a c o m p e n satory decrease i n surface free energy w h e n the charge is spread over a greater area, a n d this decrease leads to disaggregation into dimers. I n general, processes involve a change o f b o t h charge density (due to a change o f p H or the b i n d i n g o f an ion) a n d area (due to changes i n aggregation). S u c h changes alter the free energy m i n i m u m a n d shift the disaggregation constant, K
D
:
Δ1ηΚ
0
= ^
(4)
w h e r e R is the universal gas constant a n d Γ is the absolute temperature. Substituting for AF
S
f r o m e q 3, w e can obtain an expression relating Δ In
to the change i n surface charge density.
Κ
Ό
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BO IMEMBRANE ELECTROCHEMS ITRY
Hemoglobin Disaggregation T o test the quantitative predictions o f the m o d e l , w e measured the disaggre gation o f the h e m o g l o b i n tetramer b y t h e increase i n osmotic pressure w i t h p H ( 5 ) . W h e n the pressure is c o m p a r e d to that observed i n solutions o f bovine serum a l b u m i n ( B S A ; a p r o t e i n o f comparable molecular weight that does not disaggregate), w e c a n ascribe the difference i n pressure to disaggre gation. I n the titration o f b o t h p r o t e i n solutions, the m i n i m u m pressure occurs at the isoelectric p o i n t ( I E P ) , w h i c h is 6.82 ± 0.06 f o r h e m o g l o b i n at 25.0 ° C . T h e osmotic pressures obtained d u r i n g alkaline titration are shown i n F i g u r e 1 f o r h e m o g l o b i n a n d B S A as functions o f the molecular charge. T h e B S A curve represents the osmotic pressure o f a charged b i o p o l y m e r i n a complex G i b b s - D o n n a n e q u i h b r i u m . Because the h e m o g l o b i n solution should give the same osmotic pressure as B S A at the same molar concentration a n d b i o p o l y m e r charge, the h i g h e r pressure is d u e to the increase i n concentra tion caused b y disaggregation. T h e difference b e t w e e n the t w o curves (the dashed curve) indicates that disaggregation occurs at a l l degrees o f charge, and that above 25 charges p e r molecule o r pH = 9.5, disaggregation goes b e y o n d the d i m e r to the m o n o m e r .
30 τ
Q (charges/molecule)
Figure 1. The colloid osmotic pressure (in millimeters of water) as a function of the charges per molecule in the alkaline region of hemoglobin (Hb) and BSA. The difference between the two protein solutions at comparable charges per molecule is the dashed line labeled difference. (Reproduced with permission from reference 5. Copyright 1987.)
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Electrochemical Model of Voltage-Gated Channels
T h e excess concentration o f osmotically active molecules, C, c a n b e calculated f r o m the dashed curve. T h e ratio o f C to the initial concentration, C , is a direct measure o f the degree o f disaggregation a n d c a n b e used to 0
calculate the e q u i l i b r i u m constant,
Κ: Ό
4M(C/C f 0
K
° ~
c/c )
(1
0
where M is the molar concentration o f dissolved h e m o g l o b i n tetramer. A plot of log Κ
Ό
versus the molecular charge, Q, is given i n F i g u r e 2. T h e slope
appears to b e constant b e l o w 10 charges p e r molecule, a n d extrapolation to the I E P determines
K
D
= 2.0 Χ 1 0 " m o l / L at zero charge. Ackers ( 6 ) 7
10"'
/
/
/ .
10"* Η
log Kp
/
10
/
10
H
2.0 χ 10"
7
10"
10
15
20
25
30
Q (charges/molecule) Figure 2. A semilog plot of the disaggregation equilibrium constant, K , as a function of the charge on the molecule, Q . The slope of the initial part of the curve is 0.21 per charges per molecule and the intercept is 2.0 X 10 ~ (Reproduced with permission from reference 5. Copyright 1987.) D
7
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BIOMEMBRANE ELECTROCHEMISTRY
used somewhat different conditions to find a value o f 1.08 X 10 p H 7.4. A t p H 7.4, o u r value is K
D
= 3.0 X 1 0 ~
6
6
m o l / L at
m o l / L . I n addition to
d e t e r m i n i n g the p H s at w h i c h tetramers disaggregate into dimers, the calcu lations agree w i t h observations i n regard to the effects o f oxygenation a n d ionic strength (3). The
aggregation
o f sickle h e m o g l o b i n , w h i c h
differs
from
normal
h e m o g l o b i n by only one amino acid o n the beta chain, proceeds markedly u n d e r certain conditions. I f w e assume that aggregation b e y o n d the tetramer stage can also o c c u r i n n o r m a l h e m o g l o b i n , it is possible to explain the unusually large increase i n viscosity that occurs i n h e m o g l o b i n solutions at h i g h concentration ( 7 ) . A s s u m i n g that h e m o g l o b i n associates isodesmically, w e can calculate the concentrations o f the different oligomeric species a n d their contributions to the viscosity. T h e calculations show an increasing viscosity w i t h concentration a n d also suggest a change f r o m flexible chain to rigid
r o d as a result o f head-to-tail interactions i n the same chain w h e n it
becomes longer.
Conformational Changes in Hemoglobin Reactions C o n f o r m a t i o n a l changes are
frequently i n v o k e d to describe
biochemical
mechanisms qualitatively. W i t h the surface free energy m o d e l , it is possible to evaluate the energetic consequences o f conformational changes i n proteins. W e have done this for the l i g a n d - b i n d i n g reactions o f h e m o g l o b i n (8). T h e oxygen b i n d i n g curve o f h e m o g l o b i n is sigmoidal i n shape, w h i c h indicates a change i n the affinity w i t h degree o f oxygenation. T h e curve also shifts w i t h p H (the B o h r effect). B o t h properties are due to the conforma tional changes i n the
h e m o g l o b i n molecule u p o n oxygenation. W e
can
estimate the free energy o f these changes along the fines indicated i n F i g u r e 3, w h i c h is a diagram o f the changes that o c c u r i n h e m o g l o b i n as a result o f combination w i t h oxygen. T h e results o f these calculations, shown i n Figures 4 a n d 5, indicate that the oxygen-binding constant o f h e m o g l o b i n varies w i t h p H , as i n the a c i d a n d alkaline (physiological) B o h r effects, a n d also w i t h the ionic strength. I n F i g u r e 4, the b i n d i n g constant
is calculated to be
a
m a x i m u m b e l o w the I E P , as has b e e n observed. Because the free energy o f the h e m o g l o b i n system varies as a result o f b o t h the subunit aggregation reactions and the combination w i t h ligands, the two processes s h o u l d be finked; that is, oxygenation s h o u l d affect aggregation a n d vice versa. T h i s l i n k i n g is one o f the characteristics
of hemoglobin
reactions, and it is possible to understand it quantitatively w i t h the a i d o f the surface free energy m o d e l (3). T h e surface free energy m o d e l also suggests a simple physical m e a n i n g for the H i l l coefficient, w h i c h is the e m p i r i c a l constant used to
describe
20.
435
Electrochemical Model of Voltage-Gated Channeb
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Figure 3. A model for calculating the energetics of the conformational change of hemoglobin when it reacts with oxygen or another ligand. The decrease in area (ΔΑ) is due to the relative movement of the chains of the hemoglobin tetramer toward each other. The change in surface charge density (Δσ) is due to the changes in ionization of the histidine groups that are affected by the movements of the chains. The magnitudes of these two changes have been estimated from pub lished X-ray and titration data. (Repro duced with permission from reference 8. Copyright 1975.)
15
log P,
ο ι-
0.5
1.0
2\
3-
10.5
4/ 5-
ι
ι
ι
ι
ι
ι
4
5
6
7
8
9
I
10
PH
Figure 4. The calculated values of the equilibrium constant (Φ) in units of k' = ^ 1 F / R T , the surface free energy change due to the conformational change, superimposed upon the dashed curve of the Bohr effect in units of log P , the oxygen pressure for half saturation of hemoglobin. The sets of values are made to coincide at pH 6.3 and are in the same units over the same pH range. (Reproduced with permission from reference 8. Copyright 1975.) S
0 5
cooperative oxygen b i n d i n g . O r d i n a r y b i n d i n g is usually described b y a simple equation that is based o n a balance between the rates o f b i n d i n g a n d release. F o r a ligand at concentration C that reacts w i t h a fraction y o f available b i n d i n g sites, the e q u i l i b r i u m b i n d i n g constant
y
1
(l-y) C
(5)
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BIOMEMBRANE ELECTROCHEMISTRY
1.5 ^
log
PH6
Ρ 0.5
10 Q5
-3
-2
-1
Ο
logq
Figure 5. The calculated values of the equilibrium constant at pH 6.3 (Ο) and at pH 7.3 (Φ) in units of k', the surface free energy change due to the conformational change, as a function of the logarithm of the ionic strength. The measured values at pH 6 and pH 7 are shown as dashed lines in units of log P . The calculated and observed values are made to coincide at pH 6.3 and logon ~ 1- (Reproduced with permission from reference 8. Copyright 1975.) 0 5
=
where y/iX — y) is the ratio o f reacted to unreacted sites. T h e affinity is related to A G , the free energy change p e r mole due to the b i n d i n g reaction: 0
AG —
0
* = exp -
(6)
A d d i t i o n a l changes i n free energy ( A G ) that o c c u r as a result o f the conformational changes o f h e m o g l o b i n cause a displacement o f the affinity to a n e w value: S
A G + AG (y) 0
Κ
= exp
s
—
RT
r
K' = Kexp
1^
(7)
RT W h e n AG is negative, the affinity increases, a n d because AG varies w i t h y, there is a gradual change o f affinity d u r i n g a reaction. I n the H i l l equation for the combination o f oxygen w i t h h e m o g l o b i n , the affinity is assumed to r e m a i n constant at a n e w value, K , and the increase i n b i n d i n g is accom plished b y raising the concentration to the p o w e r n, w h e r e η > 1 for S
S
e
W h e n η > 1 the interaction is t e r m e d cooperative.
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Electrochemical Model of Voltage-Gated Channels
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Equations 5, 7, a n d 8 provide t w o equivalent expressions f o r the effect o f a cooperative process o n the fraction o f sites b o u n d . I n e q 9, the extra free energy change is h a n d l e d empirically w i t h the exponent n , a n d i n e q 10 the extra free energy change is i n c l u d e d explicitly:
AG
0
AG
0
4- AG ( S
+ RT I n — ^ — - nRT In C = 0 (! -y) y) + RT In
V
- RT In C = 0
(9)
(10)
\ -y) l
I f w e differentiate the t w o equations w i t h respect to In C a n d subtract one f r o m the other,
( η — 1) has the f o r m o f a surface excess i n the G i b b s equation a n d c a n b e interpreted simply as a higher concentration o f ligand that results f r o m the extra free energy change. A plot o f ( η — 1) versus C has the same f o r m as graphs o f surface excess i n a classical interfacial system ( 9 ) . N o t e that η is not constant b u t varies w i t h y a n d is equal to unity at the extremes o f the range, y — 0 a n d y = 1, as has b e e n observed f o r h e m o g l o b i n oxygenation. T h e agreement b e t w e e n o u r predictions a n d the measurements shows that surface free energy changes are a good measure o f the total free energy change that occurs i n h e m o g l o b i n reactions. T h e surface free energy m o d e l has reasonable predictive value a n d appears to b e particularly u s e f u l to describe the energetics o f aggregation e q u i l i b r i a a n d conformational changes.
Channel-Gating Processes M e m b r a n e channels are made u p o f p r o t e i n subunits i n a cylindrical array, so the o p e n i n g o f voltage-gated channels c a n b e thought o f as the partial disaggregation o f oligomeric proteins triggered b y changes i n the surface charge density. Because the interactions between the parts o f the channel proteins exposed to the aqueous phase are similar to those o f the subunits o f globular proteins i n solution, w e can assume that changes i n the e q u i l i b r i u m constant can b e estimated f r o m the disaggregation o f h e m o g l o b i n tetramers into d i m e r s . T h e fraction o f channels that are o p e n , a , w o u l d b e similar to the fraction o f tetramers disaggregated based o n the surface free energy change given i n e q 3, except that the e q u i l i b r i u m w o u l d not involve an increase i n the n u m b e r o f species.
438
BO IMEMBRANE ELECTROCHEMS ITRY T h e e q u i l i b r i u m constant f o r the channel-opening reaction is i n d e p e n
dent o f the n u m b e r o f channels a n d can b e w r i t t e n as
Ko =
~7~
( ) 12
Solving f o r α , a
Ko 1 +
(13)
K
0
F r o m F i g u r e 2 w e see that at zero charge Κ
