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Fractal Analysis of Channel Mechanisms Larry S. Liebovitch Department of Ophthalmology, Columbia College of Physicians and Surgeons, Columbia University, 630 West 168th Street, New York, NY 10032

The sequence of open and closed states measured from individual ion channels is fractal; namely, the pattern of openings and closings at one temporal resolution is similar to that viewed at other temporal resolutions. This phenomenon suggests that ion channel proteins have (1) many states that are shallow energy minima, (2) different physical processes that cooperate to open and close the channel, (3) activation energy barriers between states that vary in time, and (4) a set of processes that have different widths in their distribution of activation energy barriers. If the switching between different conformational states is deterministic chaos rather than a stochastic process, then channels may be able to organize structure fluctuations into coherent patterns of motion.

Ion Channels As already described i n detail i n the previous chapters, i o n channels

are

proteins that consist of a few thousand amino acid residues a n d a few h u n d r e d carbohydrate residues that span the l i p i d c e l l membrane ( I ,

2).

These channel proteins can have several different three-dimensional struc tures called conformational states. Some conformational states have a central hole, so that the channel is o p e n to the flow o f ions into or out o f the cell. O t h e r conformational states are closed to the flow o f ions. Because these conformational states differ b y energies fluctuations,

that are less than the

thermal

a channel is always spontaneously switching between different

o p e n a n d closed states. 0065-2393/94/0235-0357$08.00/0 © 1994 American Chemical Society

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BIOMEMBRANE ELECTROCHEMISTRY

T h e sequence o f o p e n a n d closed states o f an individual i o n channel can be revealed by the patch clamp technique developed b y N e h e r , Sakmann, Sigworth, H a m i l l , and others ( I , 2). A small piece of c e l l membrane, w h i c h may contain one i o n channel, is sealed i n the tip of a micropipette. T h e picoampere currents w h e n the channel is o p e n can be resolved. T h u s , as shown i n F i g u r e 1, the durations o f the times spent i n the o p e n a n d closed states can be measured. Because this measurement is p e r f o r m e d o n a single protein molecule, it is a m u c h more sensitive probe o f p r o t e i n kinetics than other b i o c h e m i c a l or biophysical techniques that average their signals over many molecules i n different states. T h e switching between different conformational states results f r o m phys ical processes i n the channel protein. T h u s , the patch clamp data is an important probe o f the mechanisms that cause the channel protein to change its conformational state. W e w i l l review h o w the changing interpretations o f the patch clamp data l e d to a n e w understanding o f the mechanisms that o p e n a n d close the i o n channel.

Hodgkin-Huxley Model I n 1952 H o d g k i n a n d H u x l e y ( 3 ) proposed a mathematical f o r m that c o u l d represent the currents measured across c e l l membranes. T h e y noted (3) that "these equations can be given a physical basis i f we assume that potassium can only cross the m e m b r a n e w h e n four similar particles occupy a certain region o f the m e m b r a n e . . . a n d i f the s o d i u m conductance is assumed to be proportional to the n u m b e r o f sites o n the inside o f the m e m b r a n e w h i c h are

f = 10 Hz c

5 pA

50 msec

Figure 1. The current through an individual ATP-sensitive potassium channel of rat pancreatic Β cells recorded by K. Gillis, L. Falke, and S. Misler. At the bottom, one opening is shown atfinertime resolution to illustrate that it has self-similar fractal properties. (Reproduced with permission from reference 38. Copynght 1990 New York Academy of Sciences.)

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o c c u p i e d simultaneously by three activating molecules b u t are not b l o c k e d b y an inactivating m o l e c u l e . " A l t h o u g h it is n o w k n o w n that channels function as fixed units rather than transitorily assembled forms, the central idea o f this m o d e l that channels have only a few discrete states has remained. It is assumed that the switching between these different states is random; that is, there is a constant probability p e r second o f switching f r o m one state to another. T h i s mathematical formulation is called a M a r k o v m o d e l . T h e physical interpretation of these mathematical assumptions is that the channel protein has a few distinct, independent conformational states, w h i c h are separated by significant activation energy barriers, that are constant i n t i m e .

Fractals T h e psychologist M a s l o w wrote that " i f the only tool y o u have is a h a m m e r , y o u t e n d to treat everything as i f it were a n a i l " (4). M a r k o v processes based o n the H o d g k i n - H u x l e y m o d e l h a d b e e n w i d e l y used to describe ionic currents measured i n many different experiments. H o w e v e r , i n 1986, w e began to use a n e w tool to analyze the patch clamp data. T h e insight gained f r o m this n e w analysis has changed o u r ideas about the processes that o p e n a n d close the i o n channel. T h e n e w tool is based o n fractals. Fractals have fascinating properties that are present i n many natural objects a n d h a d not b e e n incorporated i n previous models o f nature (5, 6). If any small piece o f a fractal is magnified, it appears similar to a larger piece. T h i s property is called self-similarity a n d is illustrated b y the fractal i n F i g u r e 2. Self-similarity can occur only i f structures at a small scale are related to structures at a larger scale. F r a c t a l objects i n c l u d e the repeated bifurcations o f the airways i n the l u n g (7), the distribution o f b l o o d flow i n the ever-smaller vessels i n the heart (8), a n d the ever-finer infoldings o f cellular membranes (9). Quantitatively, self-similarity means that a property L measured at scale χ is proportional to the same property measured at scale ax; namely, L(x)

= kL(ax)

(1)

w h e r e k is a constant. S u c h self-similarity means that a property L measured at resolution scale χ w i l l be a p o w e r law function o f the scale x: L(x)

=Ax ~ l

d

(2)

where A is a constant and d, w h i c h is between 1 a n d 2, is called the fractal d i m e n s i o n . T h e scaling relation o f e q 2 can be verified b y substituting e q 2 into e q 1. Fractals have some mathematical properties that many scientists may find surprising. F o r example, the m o m e n t o f a set o f measurements, such as the

360


Figure 2, The Sierpinski triangle is an example of a fractal object. It is self similar, that is, small pieces are similar to the whole object. (Reproduced with permission from reference 38. Copynght 1990 New York Academy of Sciences.) mean or variance, exists w h e n it approaches a finite l i m i t i n g value as additional data are analyzed. H o w e v e r , i n a fractal, a m o m e n t may not exist because as more data are analyzed, the value o f the m o m e n t may continually increase o r decrease rather than approach a finite l i m i t i n g value ( 6 ) . T h i s changing m o m e n t value is d u e to the fact that self-similarity means that the small scale deviations are repeated as ever-larger deviations at larger scales. F o r example, examination o f F i g u r e 2 at ever-finer spatial resolution reveals ever m o r e holes, a n d thus the average density approaches zero rather than a finite value. I n a fractal t i m e series, there w i l l b e ever-larger deviations that have the self-similar shape o f the smaller deviations, a n d thus the variance w i l l approach infinity. T h e value o f the signal averaged over short times c a n thus fluctuate widely.

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Single Channel Recordings Have Fractal Properties Researchers w h o do patch c l a m p experiments ( i n c l u d i n g us) often find that the activity o f channel openings a n d closings fluctuates i n time and changes suddenly f r o m periods of great activity to periods o f little activity. These changes are interpreted as reflections o f sudden physical changes i n the channel protein. H o w e v e r , to us these alternations resemble the fluctuations p r o d u c e d b y a fractal process, w i t h infinite variance, w h e r e the u n d e r l y i n g physical process remains absolutely constant. This interpretation is supported b y the self-similar nature o f the data shown i n F i g u r e 1. T h e r e are bursts w i t h i n bursts w i t h i n bursts o f openings and closings. C h a n n e l data collected w i t h i n the u p p e r hierarchies o f bursts w i l l have high channel activity, whereas data collected between the hierar chies w i l l have little channel activity. These impressions were suggestive but only qualitative. T h u s , w e n e e d e d n e w quantitative methods to test i f the channel data were fractal. Previously, people h a d assumed various kinetic schemes and fit these schemes to the data to determine the parameters o f these models. O u r approach was very different. W e developed n e w methods to analyze a n d display the data (10, I I ) . O n these n e w plots, fractal a n d other models obviously have different forms. Thus, w e can plot the data a n d let the f o r m o f the plots reveal the characteristics o f the channel kinetics. W e recorded the current through potassium channels i n the corneal e n d o t h e l i u m i n analog f o r m o n a frequency m o d u l a t i o n ( F M ) tape recorder. W e sampled the data w i t h an analog-to-digital ( A / D ) converter a n d t h e n d e t e r m i n e d the distributions o f closed times o n o u r computer. T o sample the same data at different time scales w e repeated the analysis at different A / D sampling rates. These results are shown i n F i g u r e 3. E a c h histogram has a very different time scale. W h e n the data are sampled very rapidly, the analysis concentrates o n the short closings; w h e n the data are sampled very slowly, the analysis concentrates o n the l o n g closings. Yet, as seen i n F i g u r e 3, the shape o f each distribution is similar. T h i s is self-similarity i n t i m e . T h a t is, the f o r m of the distribution is the same w h e n the data are sampled at different time scales. T h i s observation implies that there exists a fractal scaling relationship that describes h o w the b r i e f closings are related to the l o n g closings. W h e n the data are not fractal, the histograms sampled this way do not have this f o r m (10). T h e self-similarity o f the data suggests another n e w quantitative measure o f the channel kinetics. T h e most frequently used kinetic measure is the kinetic rate constant, w h i c h is the probability p e r second that the channel switches f r o m one state to another. H o w e v e r , the channel must r e m a i n i n a state l o n g enough to be detected i n that state. T h u s , what w e really want to measure is the conditional probability (per second) that i f the channel remains i n a state for at least a certain t i m e t that it w o u l d switch to e f f

362


A/D s 5000 Hz

100

180

260

340

A/D = 1700 Hz

1122

594

Number 100

χ

Kl

10

0.1

A/D = 500 Hz

to

200

1000

1800

2800

3400

n

100

ft

A/D = 170 Hz

10

0.1

-H-

3300

5940

11220

Closed Time in ms Figure 3. To determine if the current through a potassium channel in the corneal endothelium has fractal properties, we sampled the same analog recording of the current at four different analog-to-digital (A /D) conversion rates. The histogram of closed times determined from each A /D rate are shown. The time axis of each histogram is quite different because the fast A /D sampling rates preferentially sample the brief closings, whereas the slow A /D sampling rates preferentially sample the long closings. Note that the form of each distribution is similar. That is, the form of the closed-time histogram is the same when the data are sampled at different rates, indicating that these data are fractal in time. Thus there exists a fractal scaling relationship that describes how the brief closings are related to the long closings. (Reproduced with permission from reference 38. Copyright 1990 New York Academy of Sciences.)

16.


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363

another state. W e called this measure the effective kinetic rate constant, k (10, I I ) . T h e time f r e q u i r e d for detecting the state o f the channel is the effective time resolution at w h i c h w e measure the data. e{f

e f f

T h e nature o f the channel data can be d e t e r m i n e d b y evaluating h o w the effective kinetic rate constant fc varies as a function o f the effective t i m e scale £ at w h i c h it is measured. W e developed several different methods to determine this function f r o m the experimental data (10, I I ) , a n d subse quently o u r methods have b e e n i m p r o v e d b y others (12, 13). eff

eff

I f the channel openings a n d closings are fractal, t h e n i n analogy to e q 2, w e find that

W

r

) = ^

f

() 3

T h e effective kinetic rate constant k is the probability f o r changing states w h e n w e observe the data at temporal resolution f . N o t e that because 1 < d < 2, fc increases w h e n w e observe the channel at finer t e m p o r a l resolution £ . T h a t is, the faster w e can look, the faster w e see the channel o p e n a n d close. I f l o g k is plotted versus l o g f , t h e n e q 3 is a straight line. W h e n the data are not fractal, this plot has other forms. F o r example, w h e n there are only a f e w discrete states, such as those p r e d i c t e d b y the M a r k o v m o d e l , then there are a few w e l l separated plateaus o n this plot ( 1 0 ) . T h u s , without m a k i n g any a p r i o r i assumptions about the data, w e determine the function fc (i ) a n d thus plot l o g k versus l o g t . T h e f o r m o f this plot c a n thus tell us the characteristics o f the channel kinetics. e(f

e f f

eff

eff

e f f

e{{

eff

eff

eff

eS

A s shown i n F i g u r e 4, f o r the channel i n the corneal e n d o t h e l i u m , w e f o u n d ( 1 0 ) that the logarithm o f the effective kinetic rate constant fc as a function o f the logarithm o f the effective t i m e scale £ is a straight line, w h i c h is consistent w i t h e q 3. T h u s , this channel has fractal kinetics. W e also f o u n d a similar f o r m for the currents r e c o r d e d through channels i n c u l t u r e d hippocampal neurons ( I I ) . eff

eff

I f Pit) is the cumulative probability that a channel remains i n a state for a t i m e t o r longer, then it can b e shown (10, I I ) that

*eff(*eff) =

P(t)]/dt}

~{4ln

t=te({

(4)

T h u s , f o r a c h a n n e l w i t h fractal kinetics, equations 3 a n d 4 i m p l y that the cumulative d w e l l t i m e distribution has the f o r m P(t)=exp{[-A/(2-d)]t - } 2

d

(5)

w h i c h is a f o r m f o u n d i n many fractals a n d is k n o w n b y many different names, i n c l u d i n g stretched exponential, W e i b u l l distribution, a n d W i l l i a m s Watts law il4). W h e n d « 2, t h e n e q 5 is approximately Pit)

= Bt

(6)

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effective time scale t(eff) in msec Figure 4. The effective kinetic rate constant is the probability that the channel changes state when the current through the channel is measured at an effective time resolution t ^. The data shown were measured from a potassium channel in the corneal endothelium and have the power law form of eq 3, which is indicative of self-similar fractal behavior. (Reproduced with permission from reference 38. Copynght 1990 New York Academy of Sciences.) e

where α a n d Β are constant. T h e data f r o m many different channels have this p o w e r law f o r m (15, 16). T h e data f r o m some channels are m o r e complex than the forms o f equations 5 o r 6. F r e n c h a n d Stockbridge {12, 13) f o u n d a surprising a n d interesting f o r m o n their plots o f l o g fc

eff

versus l o g i

e f f

f r o m potassium

channels i n fibroblasts. These plots were p o w e r laws at short times w i t h a plateau at l o n g times. T h u s , the kinetics were fractal at short time scales a n d had a single w e l l - d e f i n e d discrete M a r k o v state at l o n g time scales. T h e kinetics o f the channel changed w h e n the voltage across the membrane o r the c a l c i u m concentration

i n the solution was changed. W h e n F r e n c h a n d

Stockbridge fitted their data w i t h a M a r k o v m o d e l w i t h three closed and two o p e n states, the many kinetic rate constants o f that m o d e l showed n o consistent trends w i t h voltage o r c a l c i u m concentration. H o w e v e r , plots o f l o g

k

eff

versus l o g f

e f f

revealed that the kinetics changes c o u l d b e characterized

by a change only i n the value o f the plateau o f the M a r k o v state. T h i s discovery o f unanticipated forms shows the value o f using these plots. M o r e o v e r , because this description c a n characterize

the changes i n the

kinetics i n such a simple way, it may lead us to a better understanding o f the underlying mechanisms.

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Inspection o f a large amount o f p u b l i s h e d data suggests that some channels are best described b y fractal scalings, some channels are best described b y discrete single M a r k o v states, a n d other channels show fractal behavior at some t i m e scales a n d discrete-state M a r k o v behavior at other t i m e scales.

Protein Properties Suggested by the Fractal Interpretation T h e M a r k o v description o f i o n channel kinetics, originally d e r i v e d f r o m the H o d g k i n - H u x l e y m o d e l , implies that the i o n channel p r o t e i n has certain physical properties. 1. T h e M a r k o v m o d e l assumes that the channel protein can have only a f e w conformational states. T h e structure o f a protein c a n be represented b y a potential energy function. Stable conforma tional states correspond to local m i n i m a i n that potential func tion. T h u s , the w e l l - d e f i n e d discrete states o f this m o d e l sug gest that there are a few deep local minima i n the potential that are w e l l separated f r o m other local m i n i m a . 2. T h e fact that the kinetic rates that connect different states are d e t e r m i n e d as independent parameters f r o m the data suggests that the physical processes that cause these transitions are independent. 3. Because the probability to switch f r o m one state to another is assumed to b e constant i n time, this m o d e l also suggests that the energy structure remains constant. T h e discovery o f the fractal properties o f the single-channel recordings n o w suggests a different picture o f the physical properties o f the i o n channel protein than the foregoing three properties that were suggested b y the Markov model. 1. T h e continuous nature o f the effective kinetic rate constant function k ( f ) suggests that there is actually a b r o a d contin u u m o f many channel states. That is, the energy structure o f the channel must have a very large number of shallow local minima, rather than the f e w deep m i n i m a suggested b y the Markov model. e { {

e f f

2. T h e fractal nature o f the single-channel data means that there is a relationship between short a n d l o n g d w e l l times i n a state. That is, r a p i d processes are related to slow processes, rather than b e i n g independent. R a p i d processes are transitions over

366


small activation energy barriers, whereas slow processes are transitions over large activation energy barriers. T h u s , the small activation energy barriers are related to the large activation energy barriers w i t h i n the channel protein. E a c h transition over an activation energy barrier can be represented b y a kinetic rate constant. H e n c e , the kinetic rate constants that connect

the

states are related, rather than b e i n g independent o f each other. T h u s , even though there are very many kinetic rate constants, w e do not n e e d a m o d e l w i t h very many adjustable parameters, because the kinetics can be described b y a fractal scaling w i t h a small n u m b e r o f parameters. T h i s fractal scaling determines the relationship between the kinetic rate constants. Because dif ferent processes are interrelated, the o p e n i n g or closing o f the channel depends o n the coordinated interaction o f these many different physical processes; that is, they function cooperatively. T h i s mechanism is a very different physical picture f r o m that suggested by the M a r k o v m o d e l i n w h i c h the different physical processes, represented b y the different kinetic rate constants, are assumed to be completely independent. B o t h the fractal and M a r k o v interpretations can have many states. T h e differ ence i n the physical pictures suggested by these models is that the fractal m o d e l implies that the transition rates between these states are causally related because o f the physical structure a n d dynamics i n the channel protein. T h e relationships between the kinetic rate constants that are n e e d e d to result i n kinetics w i t h fractal properties were d e r i v e d b y L i e b o v i t c h ( 1 7 ) , M i l l h a u s e r et al. (18), L e v i t t (19), a n d K i e n k e r (20). 3. E a c h change i n channel conformation can be described as the transition across an activation energy barrier. T h e sequence o f such barriers that are crossed thus describes the kinetics. A sequence of different barriers can be equivalently thought o f as a single barrier whose value is different at different times. Thus, the cooperativeness o f the different physical processes i n the channel protein, suggested b y the fractal properties, can also be equivalently described as transitions across an activation energy barrier that varies i n time. T h i s interpretation suggests that the energy structure

of the channel protein

is

changing,

rather than constant i n time. S u c h transitions can be described by a time-dependent rate constant. T h e time dependence

of

such kinetic rates that are n e e d e d to result i n kinetics w i t h fractal properties was d e r i v e d b y L i e b o v i t c h et al. (10, 11, and Lâuger (21).

17)

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T h e different physical properties o f the i o n channel p r o t e i n suggested b y the M a r k o v m o d e l and fractal properties o f the data can also b e described another way. T h e channel kinetics can b e described as transitions between open a n d closed substates separated b y activation energy barriers (22, 23). W h a t distribution o f activation energy barriers is n e e d e d to explain t h e statistical properties o f the single-channel data? T h e M a r k o v m o d e l assumes that this distribution is a small set o f D i r a c delta functions, as shown i n F i g u r e 5 . That is, there are a few such activation energy barriers a n d each occurs only at one discrete energy rather than b e i n g spread over a distribu tion o f energies. T h i s set o f energy barriers is assumed to r e m a i n constant i n t i m e . T h e M a r k o v m o d e l , a p r i o r i , imposes a distribution o f this f o r m . T h i s distribution is specified b y its parameters, w h i c h are the n u m b e r o f activation energy barriers, their activation energies, a n d their relative strengths. T o fit the data, only the parameters o f this distribution are adjusted. This f o r m o f the distribution o f activation energy barriers implies (23) that the cumulative probability distribution P(t) that the channel remains i n a state for a time t or longer has the f o r m P(t)

= a e' * Y

h

+ a e' 2

+ ··· +a e~ »

b 2 t

n

h

(7)

t

where η is the n u m b e r o f discrete activation energy barriers, b are the time constants o f the exponential terms, and a are the relative weights o f the exponential terms. n

n

O n the other hand, rather than impose a functional f o r m , w e can determine the distribution o f activation energy barriers f r o m the observed channel kinetics. L i e b o v i t c h a n d T o t h showed (23) that w h e n the channel data have fractal properties, this distribution has a b r o a d peak. T h e broader this peak, the more fractal the data; the narrower the peak, the more the data resemble that of a single M a r k o v state. R u b i n s o n (24) and C r o x t o n (25) have also derived distributions of activation energy barriers that result i n kinetics w i t h fractal properties. A s noted i n the previous section, many channels seem to have fractal properties at some time scales and yet single-state M a r k o v properties at other t i m e scales. F r a c t a l kinetics result f r o m physical processes that span large ranges o f time scales a n d thus have very b r o a d distributions o f activation energy barriers. Single-state M a r k o v processes result f r o m physical processes w i t h sharply defined time scales a n d thus very narrow distributions o f activation energy barriers. T h e fractal scaling arises w h e n a continuous distribution o f activation energy barriers spans a large range i n energy. A similar scaling w i l l occur i f there are a large n u m b e r o f discrete activation barriers that nearly u n i f o r m l y span a large range i n energy. T h u s , fractal scalings can also b e constructed f r o m discrete M a r k o v - t y p e models w i t h appropriately chosen energy barriers; namely, w h e n there are many states that have enough activation energy barriers to nearly u n i f o r m l y cover a large range i n energy (18, 19). T h i s n e w type o f M a r k o v m o d e l , however, is qualitatively different a n d has a very

368

BIOMEMBRANE

ELECTROCHEMISTRY

Markov

distribution of activation energy

activation energy

Fractal

distribution of activation

Figure 5. The kinetics of the channel switching between different states can be described by the distribution of activation energy barriers that connect the different states. Top: The Markov model assumes that this distribution is a small set of Dirac delta functions at discrete activation energies. That is, each activation energy barrier is sharply defined at only one unique value. Bottom: The published data from many channels suggest that the distribution of activation energy barriers consists of sets of activation energy barriers, each of which is a distribution with a finite width. At time scales corresponding to activation energies that have wide distributions, the kinetics have very strong fractal properties, which correspond to physical processes extending over many time scales. At time scales that correspond to activation energies that have narrow distributions, the kinetics are more like that of a single Markov state, which corresponds to physical processes that occur only over a small range of time scales.

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different interpretation than the traditional M a r k o v m o d e l , w h i c h has only a few states whose rate constants are independent o f each other. B o t h the continuous fractal distribution a n d the n e w type of M a r k o v m o d e l w i t h many states w i t h dependent rate constants i m p l y that there are many states a n d that the f o r m o f the distribution o f energy barriers arises f r o m some physical mechanism w i t h i n the channel p r o t e i n . T h e p u b l i s h e d data suggest that most channels have different processes that span different ranges of time scales. Some processes extend only over a l i m i t e d range o f t i m e scales, have relatively narrow distributions o f activation energy barriers, a n d thus are m o r e M a r k o v - l i k e . O t h e r processes extend over a very w i d e range o f time scales, have very b r o a d distributions o f activation energy barriers, a n d thus are more fractal-like. C h a n n e l proteins display a collection of different types of physical processes w i t h these different charac teristics. H e n c e , the distribution o f activation energy barriers that w o u l d fit the channel data is the distribution schematically shown i n F i g u r e 5.

Fractal Interpretation Consistent with Known Properties of Globular Proteins T h e biophysical properties o f m e m b r a n e - b o u n d proteins, such as i o n chan nels, are m u c h m o r e difficult to determine than the properties o f globular proteins because the structure a n d function o f m e m b r a n e proteins d e p e n d o n their unique location: partially e m b e d d e d i n the h y d r o p h o b i c l i p i d m e m brane, yet reaching into the h y d r o p h i l i c aqueous solution. T h e structure o f m e m b r a n e - b o u n d proteins significantly changes w h e n they are r e m o v e d f r o m the l i p i d m e m b r a n e . T h u s , few m e m b r a n e proteins have b e e n crystallized for analysis b y X - r a y diffraction to determine their three-dimensional structure. L a c k of this structural i n f o r m a t i o n also has l i m i t e d theoretical simulation techniques, such as molecular dynamics, w h i c h require such experimental information as a starting point. H e n c e , the detailed biophysical properties o f i o n channels are not yet k n o w n . H o w e v e r , w e expect that i o n channel proteins share m a n y properties w i t h their cousins the globular proteins. T h u s , w e w i l l briefly review the properties o f globular proteins a n d compare t h e m to the properties suggested by the M a r k o v a n d fractal interpretations of the single-channel data. Calculations o f the potential energy f u n c t i o n o f a large n u m b e r o f different globular proteins demonstrate that these proteins all have a very large n u m b e r o f shallow local energy m i n i m a (26). T h i s analysis is consistent w i t h the physical properties of i o n channel proteins suggested by the fractal properties o f the channel data a n d inconsistent w i t h the few deep m i n i m a p r e d i c t e d by the M a r k o v m o d e l . T h e t i m e course o f the b i n d i n g o f C O to m y o g l o b i n is d e t e r m i n e d b y the distribution of activation energy barriers that the C O must cross to reach its b i n d i n g site. T h u s , experiments that measure the n u m b e r o f m y o g l o b i n molecules that have b o u n d C O after a given time can be used to calculate the

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distribution o f activation energy barriers ( 2 2 ) . T h e time course o f the flipping o f a tryptophan ring, as reported b y fluorescence, c a n also b e used to determine the distribution o f activation energy barriers through w h i c h the ring moves ( 2 7 ) . These examples, a n d many other experiments, demonstrate that globular proteins have a b r o a d continuous distribution o f activation energy barriers. S u c h distributions are consistent w i t h the physical properties o f i o n channel proteins suggested b y the fractal properties o f the c h a n n e l data a n d inconsistent w i t h the set o f a f e w D i r a c delta functions p r e d i c t e d b y the Markov model. M o l e c u l a r dynamic simulations o f the fluctuations i n proteins show that the energy structure varies i n time a n d that such variations i n the activation energy barriers are crucial i n the functioning o f enzymes. F o r example, the energy barrier d e t e r m i n e d f r o m the static crystallographic structure o f m y o globin is so large that oxygen w o u l d never b e expected to reach the b i n d i n g site i n the p r o t e i n . H o w e v e r , as time goes by, small fluctuations i n the structure o p e n u p a passageway for oxygen to reach the b i n d i n g site ( 2 8 ) . T h e time dependence o f the energy structure is consistent w i t h the physical properties o f i o n channel proteins suggested b y the fractal properties o f the channel data a n d is inconsistent w i t h the static structure p r e d i c t e d b y the Markov model. T h u s , the biophysical studies demonstrate that globular proteins have (1) a very large n u m b e r o f conformational states corresponding to many shallow local m i n i m a i n the potential energy function, (2) very b r o a d continuous distributions o f activation energies, a n d (3) time-dependent activation energy barriers. A l l these properties are consistent w i t h the physical properties o f i o n channels d e r i v e d f r o m the fractal properties observed i n the channel data a n d are inconsistent w i t h the physical properties d e r i v e d f r o m the M a r k o v m o d e l .

Controversy T h e physical properties o f the i o n channel protein suggested b y the M a r k o v m o d e l differ f r o m the properties suggested b y the fractal properties o f the single-channel data. T h e M a r k o v m o d e l suggests that (1) there are a f e w discrete conformational states corresponding to w e l l - d e f i n e d energy m i n i m a , (2) the processes that cause transitions between these states are independent, (3) the energy structure o f the c h a n n e l remains constant i n time, a n d (4) the distribution o f activation energy barriers that separate these states are a f e w D i r a c delta functions at discrete activation energies. T h e fractal properties o f the single-channel data suggest that (1) there is a b r o a d c o n t i n u u m o f conformational states corresponding to a large n u m b e r o f shallow energy m i n i m a , (2) the processes that cause transitions between states are related, (3) the energy structures, such as the activation energy barriers, vary i n time, and (4) the distribution o f activation energy barriers consists o f some distribu tions that are very b r o a d a n d some that are narrow.

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T h e r e is still some question as to whether the i o n channel p r o t e i n has the physical properties suggested by the M a r k o v m o d e l or the fractal properties o f the single-channel data. F o r example, note the lively exchange o f views i n the letters i n the Biophysical Journal b y L i e b o v i t c h (29), H o r n and K o r n (30), a n d M c M a n u s et al. (31). T h e M a r k o v models can fit the single-channel data w i t h great accuracy. H o w e v e r , this does not prove the validity o f these models because the M a r k o v models have a very large n u m b e r o f adjustable parameters; sometimes as many as 16. W i t h such a large n u m b e r o f ad justable parameters, even an invalid m o d e l is likely to fit any data. A r g u m e n t s have b e e n made that some M a r k o v models are a statistically better fit to the single-channel data than some fractal models (32-34). H o w e v e r , the mathe matical justification that is the basis for this statistical comparison has b e e n shown to be invalid (35). T h e extensive experimental a n d theoretical studies o f globular proteins support the physical properties suggested b y the fractal interpretation. H o w e v e r , it is not yet k n o w n i f there are important differences between the properties o f globular proteins a n d m e m b r a n e proteins such as ion channels. W e think that i o n channels have the physical properties suggested b y the fractal interpretation because the fractal properties are so clearly present i n the experimental data, and these properties are consistent w i t h the extensive experimental a n d theoretical studies of globular proteins. H o w e v e r , this issue can only be resolved by future w o r k that includes X - r a y diffraction a n d Ν M R , w h i c h are n e e d e d to determine the three-dimensional structure o f the channel p r o t e i n at h i g h resolution; molecular dynamic simulations, w h i c h are n e e d e d to determine the dynamics o f the motions w i t h i n the channel; a n d molecular biology, w h i c h can test o u r ideas o f channel structure a n d d y n a m ics by p u r p o s e f u l alterations of the channel.

Chaos T h e entire previous discussion assumed that the switching o f the i o n channel f r o m one conformational state to another is an inherently stochastic or r a n d o m process. T h e kinetic rate constant, or the effective kinetic rate constant, tells us the probability p e r second that the channel w i l l switch f r o m one state to another, but it does not t e l l us exactly w h e n this switch w i l l occur. A r e these transitions f r o m one state to another really stochastic? W e n o w k n o w that there are processes, w h i c h are not stochastic, whose output mimics stochastic behavior. This p h e n o m e n o n is n o w called chaos. Chaos is a jargon w o r d that means that a system has certain mathematical properties. It s h o u l d not be confused w i t h its nontechnical h o m o n y m that means confusion or disorder. A chaotic system can be described b y a set o f nonlinear difference or differential equations that have a small n u m b e r o f independent variables. Because these equations can be integrated i n time, the future values o f the variables are completely d e t e r m i n e d b y their past values.

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That is, the system is completely deterministic; there is no randomness at a l l . H o w e v e r , the evolution o f the values o f the variables is exquisitely sensitive to their exact initial values. T h a t is, very slightly different initial values o f the variables w i l l soon evolve i n very different ways. Because these initial conditions can only be specified w i t h finite accuracy, the exact long-term behavior o f the system is unpredictable, although its statistical properties can often be d e t e r m i n e d . T h u s , chaotic systems are deterministic, yet unpre dictable i n the l o n g r u n . W e have shown that chaotic models can describe the qualitative features a n d statistical properties o f the single-channel data (36, 3 7 ) . I n these models the switching between different conformational states is a completely deter ministic rather than a stochastic process. T h e i o n channel is a mechanical structure o f masses a n d springs, called atoms a n d bonds. These chaotic models suggest that such a mechanical structure can f u n c t i o n as a nonlinear oscillator to amplify its o w n motions, to force itself f r o m one conformational state to another. Alternatively, one c o u l d say that the channel is organizing the nonperiodic thermal fluctuations o f its structure into coherent patterns o f m o t i o n . T h u s , the motions w i t h i n channels that change the conformational state may be more organized than the stochastic switching models (previously described) suggest. T h e s e ideas are still i n their infancy, but they may ultimately change the way that w e v i e w a n d understand channel structure a n d dynamics.

Summary T h e patch c l a m p technique can measure the sequence o f o p e n a n d closed times i n a single i o n channel. T h i s allows us the unprecedented opportunity to study h o w an individual p r o t e i n switches between different conformational states. T h i s single-channel data has fractal properties a n d is self-similar; that is, the statistical properties o f the switching between o p e n a n d closed states, v i e w e d at one-time resolution, are similar to the properties v i e w e d at finer time resolution. T h u s , the short d w e l l times i n a state are related to the l o n g d w e l l times. These fractal properties suggest that the i o n channel protein has certain physical properties: 1. T h e r e is a b r o a d c o n t i n u u m o f conformational states corre sponding to a large n u m b e r o f shallow energy m i n i m a . 2. T h e processes that cause transitions between states are related. 3. T h e energy structure, such as the activation energy barriers, varies i n t i m e . 4. T h e distribution o f activation energy barriers consists o f distri butions that are very b r o a d a n d distributions that are narrow.

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It h a d always b e e n assumed that the changes i n conformational state are inherently stochastic processes. H o w e v e r , n e w chaotic models suggest that channel proteins m a y organize fluctuations i n structure into coherent patterns of motion, so that the switching between

conformational states may be

deterministic rather than stochastic.

Acknowledgments T h i s w o r k was done d u r i n g the tenure o f a n Established Investigatorship f r o m the A m e r i c a n H e a r t Association a n d was also supported i n part b y grants from the W h i t a k e r F o u n d a t i o n a n d f r o m the N a t i o n a l Institutes o f Health (EY6234).

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RECEIVED for review January 2 9 , 1 9 9 1 . ACCEPTED revised manuscript June 18, 1992.

Biomembrane Electrochemistry - American Chemical Society

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