Abstract
Ion channels open and close in a stochastic fashion, following the laws of probability. However, distinct from tossing a coin or a die, the probability of finding the channel closed or open is not a fixed number but can be modified (i.e., we can cheat) by some external stimulus, such as the voltage. Singlechannel records can be obtained using the appropriate electrophysiological technique (e.g., patch clamp), and from these records the open probability and the channel conductance can be calculated. Gathering these parameters from a membrane containing many channels is not straightforward, as the macroscopic current I = iNP_{o}, where i is the singlechannel current, N the number of channels, and P_{o} the probability of finding the channel open, cannot be split into its individual components. In this tutorial, using the probabilistic nature of ion channels, we discuss in detail how i, N, and P_{o max} (the maximum open probability) can be obtained using fluctuation (nonstationary noise) analysis (Sigworth FJ. G Gen Physiol 307: 97–129, 1980). We also analyze the sources of possible artifacts in the determination of i and N, such as channel rundown, inadequate filtering, and limited resolution of digital data acquisition by use of a simulation computer program (available at www.cecs.cl).
 noise
 variance analysis
 rundown
 filter
 single channel
Ion channels are molecular machines, perfectly tuned to the transport of ions through the cell membrane, with very high efficiency (10^{6}10^{8} ions/s) [Latorre and Miller (13), Hille (11)]. Ion channels belong to a class of integral membrane proteins, some of which have evolved as highly selective to a given ion. For example, some potassium channels are ∼1,000 times more permeable to K^{+} than to Na^{+}. Amazingly, despite their exquisite ion selectivity, these channels do not lose their high ion throughput. Today, thanks to elucidation of the crystal structure of the K^{+} channel from the bacterium Streptomyces lividans [Doyle et al. (5); MoraisCabral et al. (14)], we understand how this high degree of ion selectivity is achieved. Channel opening and closing is a stochastic process. Transmembrane voltage determines the probability of finding the channels in either state, as in voltagedependent channels, stretch in mechanosensitive channels, neurotransmitters in neurotransmitter receptor channels, or second messengers such as Ca^{2+} in Ca^{2+}activated K^{+} channels (11). Ion selectivity, ion conduction, and channel activation can be characterized using electrophysiological techniques such as patch clamping, which in its different modalities is used to determine singlechannel properties [Neher and Sackmann (15), Sackmann and Neher (18)] and voltage clamping employed when the current induced by a channel population is measured [Hodgkin and Huxley (12)]. Singlechannel recordings contain information about the channel conductance, the probability of finding the channel open, and the distribution of open and closed dwelling times. This information is also present in the macroscopic currents measured in an ensemble of many ion channels. However, the process of retrieving this information from macroscopic currents is not straightforward. In this paper, we describe a procedure to analyze the fluctuations (nonstationary noise) in membrane conductance caused by opening and closing of ion channels. Using this methodology, we can reveal the number of channels (N) present in a membrane preparation, the unitary current (i) carried by a single channel, and the maximum probability of finding the channel open (P_{o max}). This tutorial is aimed at helping graduate students with solid knowledge of electrophysiology to search for these parameters from macroscopic current records when the need arises. We discuss the theory underlying this procedure and some of its limitations, and we present a stepbystep description of the method with examples generated using a computer program simulation. Sigworth (20) developed the nonstationary noise analysis to study Na^{+} channels. The analysis is simple and powerful and much more comprehensible than those that use correlation functions or spectral densities.
We decided to write this tutorial because the apprentice biophysicist usually approaches membrane noise with apprehension, and we thought it convenient to give the interested scientist a friendly approach with emphasis on the concepts rather than the mathematics of the problem. Moreover, as stated in the preface of Louis de Felice’s book (4) “ … in the intervening years membrane noise became a definable subdivision of membrane biophysics.”
This tutorial should be approached as an introductory lecture on noise analysis that should be complemented with the “handson” experience given by the educational computer program that we have developed, which is available at www.cecs.cl (1). Our experience has been that the tutorial is a good primer on the subject and can be handled independently by any Ph.D. student interested in the field and with some mathematics and physics background. Of course, the word “primer” should be taken in sensum strictum, and once the student is enticed by the subject, he/she should continue swimming in the rougher waters of the advanced approaches to noise analysis.
BASIC CONCEPTS
Classical experiments to characterize a voltagedependent ion channel require measurement of current relaxation under voltage clamp. Membrane potential is depolarized for a short time, and the current is recorded. The current record contains information about both channel opening (during the depolarization) and channel closing (upon returning to the holding potential). A family of current records is usually studied using a series of stepped voltage pulses (Fig. 1A).
If we assume that the singlechannel conductance is independent of the membrane potential, the following relation describes the measured current (1) where I(t) is the timedependent ionic current experimentally observed; N is the number of channels in the preparation; γ is the singlechannel conductance; P_{o}(V,t) is the probability of finding the channel open, which is a function of time and membrane potential; V is the membrane potential, the variable controlled by the voltageclamp system; and V_{x} is the reversal potential of the current. V_{x} is usually found using a doublepulse protocol. The prepulse (first pulse) opens the channels, and the test pulse (second pulse) closes the channels. The current measured during the test pulse is the tail current, which will be positive for V V_{x} or negative for V < V_{x}. In trials of many different test pulse voltages, a test pulse voltage will be found for which there is no tail current. This voltage is exactly V_{x} (Fig. 1B). A plot of I as a function of V crosses the current axis at V = V_{x}. The product NγP_{o}(V,t) is the slope of a cord drawn from V_{x} to a given point on the curve and is called the cord conductance.
Unless N and γ are known, the probability of finding the channel open cannot be determined from a simple analysis of current amplitudes. A classical current relaxation experiment on voltagedependent channels consists of collecting membrane currents elicited by a series of voltage pulses of increasing amplitude. Cord conductance, I(t)/(V − V_{x}), is usually an Sshaped curve reaching an asymptotic value within the limit of very high voltages. Conductance as a function of voltage is customarily normalized as a fraction of this limiting value. According to Eq. 1, the maximum cord conductance is NγP_{o max}, where P_{o max} is the maximum open probability of the channel. P_{o max} does not necessarily equal 1. One obvious mechanism producing a P_{o max} <1 is channel inactivation, as found in classical sodium channels in nerve (20). A P_{o max} <1 can also be found for a voltagedependent channel in which the last step that opens the channel is only weakly voltage dependent or is voltage independent. The product N multiplied by γ cannot be split into the individual terms N and γ in the classical analysis. Nonstationary noise analysis of the macroscopic current records provides tools to separate N from γ, and P_{o max} can be determined as we describe below (20); for reviews see Heinemann (10), Heinemann and Conti (8), Neher and Stevens (16); for more details, De Felice (4) is recommended].
Singlechannel noise.
We consider a hypothetical ion channel with the following properties: 1) the difference in current between closed and open states is directly proportional to membrane potential; 2) the fraction of channels in the open state varies with membrane potential from 0 to 1 as the voltage increases; 3) the statistics of the number of open channels in membranes with few channels follow a binomial distribution; and 4) the voltagedependent opening and closing of the channels explain the voltagedependent properties of a membrane containing many channels. In other words, ion channels act independently of one another [e.g., Ehrenstein et al. (6)].
Figure 2 shows three representative records of a computer simulation of the time course of the current carried by this hypothetical ion channel at three different membrane potentials. Continuous records are split into 10 successive sections as shown. The straight horizontal line in each panel represents the average current recorded when the channel is closed. Channel opening appears as upswings of the current trace. Channels open and close randomly, and it is clear that the probability of finding the channel open is near 0 at 25 mV, about onehalf at 45 mV, and close to 1 at 65 mV. There are two classes of current fluctuations (noise) visible in these records. The first class of noise appears as fluctuations of the trace around the closed channel current. These fluctuations are not related to membrane voltagedependent properties and can be regarded as the equipment background noise.^{1} The amplitude of this noise is the same for both an open and a closed channel. The second class of noise is represented by the current trace that swings upward from the baseline every time the channel opens and that returns to the baseline when the channel closes. This is the noise that is directly related to channel opening and closing. Thus there is a noise impulse (or transition event) every time the channel either opens or closes. The intensity of the noise can be evaluated by counting the number of current transitions observed during a period of time. In Fig. 2, we count 5 openingclosing events in the 25mV record, where the channel is closed most of the time; 17 similar events in the 45mV record, where the channel is open onehalf of the time; and 5 events in the 65mV record, where the channel is open most of the time. From this observation, we conclude that channel noise depends on the probability of finding the channel open [P_{o}(V,t)]. The noise, or the number of transitions, is maximal when the probability of finding the channel open is 0.5, and this must be 0 when the channel is always closed or open. The probability of a transition occurring per unit time is proportional to the total number of channels. Therefore, it is clear that the noise level must depend on N, the number of channels present in the membrane.
Finally, noise must also depend on the amplitude of the unitary current fluctuation, i = γ(V − V_{x}), which is the current carried by a single channel. The following section is the development of a theory that will be used to obtain P(V,t), N, and i from the analysis of current fluctuations in membranes with a homogenous population of channels.
Mean current and variance.
Let us consider a membrane with just one channel. Let us define p as the probability of finding the channel open and q the probability of finding the channel closed. Therefore The mean current <I> passing through the singlechannel membrane is the probability of finding the channel open multiplied by the single channel current i. The variance of the current, σ_{I}^{2} is the sum of squared deviations from the mean, which can be calculated as the sum of each possible deviation multiplied by its probability. In a membrane with N independent channels of the same kind, the mean current and the variance are N times larger than that of a singlechannel membrane (2) (3) Combining Eqs. 2 and 3 gives the following expression (4) Equation 4 is useful for understanding the relationship between noise due to channel gating and the fundamental channel characteristics. Equation 4 is a parabola with roots on <I> = 0 and <I> = iN. There will thus be no noise when all of the channels are closed all of the time (P_{o} = 0) or when they are all open all of the time (P_{o} = 1). The first derivative of the function given in Eq. 4 is (5) This derivative becomes zero for <I> = iN/2. Thus the variance has a maximum when the probability of finding the channel open is 0.5. Equation 5 is the slope of the parabola, and within the limit of very small <I> is the singlechannel current i. When all of the channels are open, the mean current is iN; therefore, the slope is −i. In other words, the singlechannel current can be obtained from the slope in the neighborhood of either root of the parabola.
This is the basis of the method used to calculate singlechannel properties from the current fluctuation in a membrane containing many channels. We simply need to record steadystate current at a fixed potential and to compute the mean and the variance of the current. Once a strategy has been found to change the probability p to produce a well defined parabola, Eqs. 4 and 5 can be used to compute the number of channels in the membrane, the unitary current, and the maximum probability of finding the channel open.
APPLYING THE METHOD TO VOLTAGEDEPENDENT CHANNELS
An efficient strategy to determine the characteristics of voltagedependent channels can be designed, since the probability of finding the channels open can be readily altered. For instance, the P_{o} of K^{+} channels in an excitable cell such as a neuron will be altered from 0 to P_{o max} by changing the membrane potential from −70 mV to +50 mV. The change in P_{o} is not instantaneous, but it takes some time to change the open channel probability from zero to a constant steady value; P_{o} is a function of time and voltage. Figure 3 shows the result of 14 simulations of the current relaxation for an ensemble of 1,000 channels obtained using the simulation procedure that we released on www.cecs.cl. To compute the variance, the current relaxation experiment can be repeated M times, so there will be M measurements of I for each time point, i.e., for each value of the openchannel probability. The set of M points taken at any given time is called an isochrone (iso  ‘equal’, chronos  ‘time’). All points on an isochrone have the same mean value of the open channel probability.
Figure 4 shows nine isochrones taken at different times for 100 current records similar to those of Fig. 3. Dispersion around the mean value is clearly smaller for the lower and upper isochrones where the openchannel probability is near zero or near 1. It is maximal for the isochrone with a mean current of 0.5 nA, which corresponds to a 50% open probability. The mean current <I> and its variance can be calculated for the points on each isochrone, as the expected probability of the open channel is the same for all the points on a given isochrone. The smooth line of Fig. 5 is a plot of the mean current of each of the 1,000 isochrones collected during the noise simulation as a function of the time after the start of the depolarization pulse at which the isochrone was measured. The jagged line is a plot of the variance of each isochrone as a function of time. The variance has a maximum value at a time at which the mean current is at onehalf of its steadystate value. Figure 6 displays the variance as a function of the mean current for all of the 1,000 isochrones collected during the channel simulation. Parameters i and N can be obtained by nonlinear curve fitting using Eq. 4. The maximum open probability can be calculated from the maximum mean current recorded, <I>_{max}, divided bythe mean current expected for a membrane with N open channels (6)
How the variance determination is affected by channel rundown.
Under certain experimental conditions, the current recorded from a membrane may decrease unexpectedly during data collection; this problem is called channel rundown. A simulation of a membrane with channel rundown is shown in Fig. 7. In this simulated experiment, 200 sweeps were collected. The membrane contained 1,000 channels at the beginning of the experiment and only 800 at the end. Figure 7A shows selected isochrones. We can clearly see the scattering of the points along the isochrones and that it is clearly smaller for the lowermost and uppermost traces than for those obtained at relative currents between 0.3 and 0.8. However, this “bird’s eye” appreciation of the noise structure in the isochrones is not consistent with the variance calculated using the standard formula as shown in Fig. 7B. Figure 7D is a closer display of an isochrone, and in Fig. 7E the deviations around the mean current are plotted. All deviations are positive for sweeps 1–100 and negative for sweeps 100–200. Because the dispersion is measured with respect to the mean of all data points, the variance computed from these deviations will reflect mostly the rundown of the channels rather than the random variance of the number of open channels. A better method to calculate the variance relies on measurements of the differences in the current measured on successive sweeps. This method [Heinemann and Conti (8), Sigg et al. (19)] uses the differences (y_{i}) of successive points along the isochrone to calculate the variance, i.e. (7) where x_{i} is the i^{th} point along the isochrone. By use of this transformation, the variance is now given by the expression (8) Figure 7F shows the deviations of the differences with respect to the mean of the differences y_{i}. Positive and negative deviations are evenly distributed and represent the random variations of the number of open channels. Figure 7C shows that the variance vs. mean current plot is now a parabola and that the values of i and N obtained from the fit to the data using Eq. 8 are the expected ones.
How many channels in the membrane and how many repetitions are required for reliable measurements of i and N?
In measuring macroscopic currents, we know that the relative current noise increases as the magnitude of the measured current decreases. Having large currents is thus convenient when the objective is a “clean” current record, but this can be a disadvantage when determining i and N by use of noise analysis. This is because we are measuring the difference between the mean current and the current measured at a given time in the different current sweeps, and this difference, as we show below, will vanish as the <I> becomes very large. Recalling Eqs. 2 and 3, and assuming that P_{o} = 0.5 for the sake of simplicity, we have This implies that, for a very large N, the standard deviation of I at any given time will be too small compared with <I> to be measured accurately. This is especially important when using analogtodigital conversion, since the minimum current difference that can be measured is one bit. In a 12bit system, this is 1 part in 4,096. For example, let us consider a membrane with 10 channels, and we adjust our data acquisition system so the full amplitude of the macroscopic current is represented in 1,000 digital counts. Therefore, the unitary current is represented by 100 digital counts. To calculate the variance of the isochrones, we calculate the differences of the current recorded on successive sweeps. Figure 8A illustrates the differences recorded in the 10channel membrane. The first six differences are 1, −1, 1, 0, 2, and −4 channels, which are accurately represented by 100, −100, 100, 0, 200 and −400 digital counts. Let us now consider a 10,000channel membrane, in which we adjust the system so that the macroscopic current is represented again by 1,000 digital counts. In this case, each digital count will represent 10 channels. Figure 8B shows the current differences computed for this membrane, and we can see that all of the differences are multiples of 10 channels. This is an error introduced by the analogtodigital conversion, since the actual differences are any integer number of channels. In this situation, our noise analysis will be inaccurate.
Our experience tells us that the accuracy of a determination depends on the size of the sample. This is also true for the meanvariance noise analysis. The accuracy of the determinations depends strongly on the number of sweeps collected during the experiments, i.e., the number of data points on each isochrone. As an example, we simulated the channel noise of a membrane with 1,000 channels and collected 100 sweeps. The standard deviation of a sample of 10 determinations was 20% of the central value for both the unitary current and the number of channels. Repeating the same procedure, but collecting 1,000 sweeps on each trial, reduced the standard deviation to 4%.
How reliable is the analysis when the experimental data cover only part of the parabola?
Figure 9 shows a series of simulations in which the openchannel probability was explored to different extents. Figure 9A represents the ideal case, where the openchannel probability changed from 0 to 1, and we thus have experimental points over the complete parabola. In Fig. 9B, the openchannel probability changed only from 0 to 0.1. In this case, the data look like a straight line rather than a parabola. In other words, only the slope of the variancemean relation can be accurately calculated in this case. As mentioned earlier, the absolute value of the slope of the parabola near the roots is the unitary current i. Therefore, in this case, we can determine the unitary current but not the number of channels in the membrane. While producing the simulations for this tutorial, we were surprised to find that the unitary current can be estimated fairly accurately even for very limited explorations of the open probability space as shown in Fig. 9, C, D, and E. However, the number of active channels in the membrane, N, cannot be calculated accurately for situations where the complete parabola cannot be experimentally attained.
In Fig. 9F, we show a case in which the openchannel probability changed from 0.8 to 1. Despite the limited data points used to search for the best parabola, the fit gave accurate values for i and N (see Fig. 9A).
Figure 10 shows another situation where P_{o} never reaches 1.0 because of channel inactivation. In this case, the kinetic scheme has three states: closed, open, and inactivated, in which only the open state carries current. Upon depolarization, channels are briefly in the open state and then proceed to the inactivated state. The time course of the current is represented in Fig. 10, in which the currents measured during several repeat experiments are superimposed. Current rises quickly and then decreases slowly to a steadystate value. The variance is plotted as a function of the mean current for each isochrone in Fig. 10B. It is clear from this figure that the data fit using Eq. 4 effectively reproduces i but not N (actual number of channels is 1,000). It is interesting to note that time does not appear in the equation relating variance to the mean current. This means that the parabola can be drawn from P_{o} = 0 to P_{o} = 1 as well as from P_{o} = 1 to P_{o} = 0. This is why the sparse points belonging to the fast upstroke of the current as well as those closely spaced points belonging to the slower inactivation all fall on the same parabola.
Beware of the filter!
It is our daily experience that passing the current record through a lowpass filter will reduce the noise. Thus we can anticipate that filtering will interfere with noise analysis. This is illustrated in Fig. 11, in which we filtered currents before performing noise analysis. The model we used is a twostate channel with a relaxation time of 1 ms. Filters with time constants of 0.3, 0.1, and 0.03 times the channel time constants were used. The distortion of the parabola and the errors in i and N are apparent when compared with the nofilter analysis. The lesson that we have drawn from of this example is that the filter must be set to a time constant well below those that describe channel gating. Because the rate of opening and closing of the channels may be unknown, demonstration of the stability of the results using different filters is mandatory.
The estimations of both i and the P_{o max} can be seriously affected by choosing an incorrect filtering procedure, for example, in the case of a channel with a fast (flickering) between the open state and the closed state. The kinetic scheme of the simulation shown in Fig. 12 has two closed and one open state, C_{1}, C_{2}, and O. Exchange between the C_{2} and O states is much faster than that between the C_{1} and C_{2} states. Upon depolarization, channels shift from state C_{1} to C_{2} and then reaches the state O. Figure 12A shows noise analysis performed in the absence of filtering. The analysis retrieves the correct values for i, N, and P_{o max}. Introducing a 10kHz filter has the effect of halving i and increasing the P_{o max} with almost no effect on the number of channels (Fig. 12B). The filter in this case eliminates the flicker collapsing the channel current, and since the filter eliminates the fast closing events the mean open time increases with consequent increase in P_{o max}.
NOISE ANALYSIS OF MACROSCOPIC CURRENTS OBTAINED USING THE PATCHCLAMP TECHNIQUE
We close this tutorial by showing real data obtained using the patch clamp technique. The macroscopic current record shown in Fig. 13A is from a membrane macropatch in Xenopus laevis oocyte. In this case, the oocyte expressed calciumactivated K^{+} channels (K_{Ca}, human Slowpoke). Using the same technique, we recorded outward currents flowing through ShakerH4Δ(6–46) K^{+} channels (Fig. 13D). Plots of variance vs. current corresponding to the raw data shown in Figs. 13, B and E and plots of variance vs. mean current are given in Figs. 13, C and F. The solid line is a fit to the data by using Eq. 4, yielding an estimate of the singlechannel current, the number of channels contained in the patches, and the maximum probability of opening for K_{Ca} and Shaker K^{+} channels. These values can be converted to an estimate of the singlechannel conductance dividing them by the driving force, which is 120 mV, because the internal and external K^{+} concentration were the same. The values of singlechannel conductances obtained for the K_{Ca} and the Shaker channel were 175 pS and 12 pS, respectively. These values compare well with those measured directly from unitary currents.
Acknowledgments
This work was supported by Chilean grants Fondo Nacional de Investigacion Científica y Tecnológica 1000890 (R. Lattore) and Cátedra Presidencial, a Human Frontiers in Science Program, a group of Chilean companies (Compañía del Cobre, Dimacofi, Empresas CMPC, MASISA, and Telefónica del Sur). The Centro de Estudios Científicos is a Millennium Science Institute.
Address for reprint requests and other correspondence: R. Latorre, Centro de Estudios Científicos, Avenida Arturo Prat 514, Valdivia, Chile (Email: ramon{at}cecs.cl).
Footnotes

↵1 However, the current noise in an open channel can be much greater than the baseline noise. This can be caused by open and closed transitions too fast to be followed by the currentmeasuring system or by the shot noise induced by the statistical motion of ions as they flow through the open channel. These cases will not be discussed here, but the interested reader may consult Heinemann and Sigworth (9, 10).
 © 2002 American Physiological Society