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Department of Physiology and Biophysical Sciences, State University of New York at Buffalo, Buffalo, New York
Correspondence: Address reprint requests to Dr. Feng Qin, 124 Sherman Hall, SUNY at Buffalo, Buffalo, NY 14214. Tel.: 716-829-2764; Fax: 716-829-2569; E-mail: qin{at}acsu.buffalo.edu.
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INTRODUCTION |
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Traditionally, single-channel currents are detected by a combination of low-pass filtering and half-amplitude threshold crossing (Sachs et al., 1982
; Sigworth, 1983; Gration et al., 1982). Although conceptually simple, these methods suffer from the problem of band-limiting distortions. Because of the small magnitude of the unitary current, heavy filtering is usually necessary so that different conductance levels can be distinguished unambiguously. The finite time response of the low-pass filter, however, may reduce short transitions below threshold and therefore prevent them from being detected. These missed events result in apparent increases in the duration of the experimentally observed dwell times. In the extreme case of small currents with rapid kinetics, the majority of events may be missed. In addition, the noise in the records can either facilitate or depress the detection of individual events, depending on the direction of the noise, and these effects of noise do not cancel out (Blatz and Magleby, 1986). Large noise peaks may be identified as false events.
The threshold crossing techniques incur their limitations in part because of their simplistic assumption that the data points are independent of each other. In reality, the transitions of the channel are time-dependent, and the closed and open samples tend to occur in long runs. Consequently, an improved detection would necessitate the use of information from adjacent samples. Several methods of this type have been developed. For example, Moghaddamjoo (1991) proposed a segmentation procedure in which sequential samples are processed and an event is detected only if the variation of samples within a class is minimized while the variation between classes is maximized. Fredkin and Rice (1992a) introduced two Bayesian restoration methods based on statistical smoothing through the use of a two-state Markov chain. VanDongen and others considered the use of slope threshold in addition to amplitude threshold to minimize spurious transitions (VanDongen, 1996; Tyerman et al., 1992). Nonlinear filtering techniques such as Hinkley detector, which detects abrupt and time-dependent variations, have also been exploited (Draber and Schultze, 1994; Schultze and Draber, 1993). Besides improvements on temporal resolutions, many of these approaches feature a generalized applicability to ion channels with multiple conductance levels, and sometimes even unknown amplitudes.
Hidden Markov modeling (HMM) provides a general paradigm that takes account of the statistical characteristics of both signal and noise simultaneously. The technique has gained popularity in analysis of single-channel currents (Chung et al., 1990; Fredkin and Rice, 1992b; Qin et al., 2000b; Venkataramanan and Sigworth, 2002). Within this framework, channel activity is modeled as a first-order Markov process to which is added white Gaussian noise. The parameters of the model are estimated by maximizing the a priori probability using either the Baum-Welch reestimations or an optimization-based approach (Qin et al., 2000a). The single-channel current is then uncovered as the most likely state sequence by maximizing the a posteriori probability using the Viterbi algorithm (Forney, 1973). Compared to threshold crossing, the approach has a significantly improved detection performance and is particularly well suited for the case in which the signal/noise ratio is poor. However, the maximization of likelihood and the reestimation of parameters are generally a time-consuming process because of the need to evaluate the probability of each state at each sample point. The standard Baum-Welch reestimation requires a computational load that is quadratically proportional to the complexity of the model and linearly to the length of the dataset.
This work extends the study of hidden Markov modeling for single-channel analysis. A simplified HMM approach for idealization of single-channel currents is presented. The approach is based on the segmental k-means (SKM) method (Rabiner et al., 1986). As an alternative to Baum-Welch reestimations, the algorithm is computationally more efficient, thereby alleviating the problem of heavy computations required by the standard HMM. Yet the algorithm maintains the essence of HMM to allow the statistics of both channel kinetics and noise characteristics to be taken into account explicitly in a natural but concise manner. The method, although it estimates models, is intended for idealization of current traces; after this, more sophisticated dwell-time analysis techniques such as histogram fitting or the full dwell-time maximum likelihood approach (Qin et al., 1996, 1997) can be used for model estimation. In the following, the theory of the algorithm is first described. Some issues on implementation and practical use of the algorithm are then addressed. Finally, a number of examples are chosen to demonstrate its performance as well as its limitations.
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THEORY |
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t, and the diagonal element aii defines the probability to stay in the current state.
The transition matrix A completely determines the kinetics of the channel, assuming a memoryless system, i.e., the transition at any time is only a function of the current state independent of the previous history. It is related to the rate constant matrix Q by
 | (1) |
In accordance with observations, each state of the channel is designated with a conductance. Assume a total number of M different conductance levels, and let Ii, i = 1,2...M, be the corresponding current amplitudes. Some of the states may possess an identical conductance. Therefore, there may be more states than conductance levels, i.e., M 
N. In the simplest case, there are only two conductance levels, corresponding to closed and open, respectively. However, the model itself is general, without any restriction on the number of subconductance levels. The time series of the observed samples will be denoted by yt, t = 1...T, and the underlying state sequence by st, t = 1...T.
The transitions of the channel are not only masked by aggregation of multiple kinetic states into the same conductance, but also by noise. It is assumed that the noise is additive, white, and follows a Gaussian distribution. The noise can be state-dependent, but for practical consideration, it is only assumed to be conductance-specific. States within the same conductance class have the same noise. By doing so, excessive open noise is allowed. Let 
i2 denote the variance of the noise at the ith conductance level. Then, given the channel being in conductance i at time t, the resultant observation has a probability distribution
 | (2) |
The idealization of the currents can be considered as a restoration problem, i.e., to uncover the underlying state sequence st values from the observations yt values. Apparently, there are many possible solutions, depending on which criterion is used. The idealization considered here is sought to maximize the a posteriori probability of the state sequence, i.e.,
 | (3) |
where 
= {aij's, Ii's, i's} designates all model parameters. The probability is also called the likelihood of the idealization.
Making use of the probabilistic model for the channel and the noise, the likelihood can be formulated explicitly. According to the Bayes law, it can be cast into the probability of the state sequence itself, multiplied by the probability observing the samples given that state sequence, leading to
 | (4) |
, multiplied by the subsequent transition probabilities through the entire sequence. The problem of idealization is then to choose among all possible choices a state sequence s and a set of model parameters so that the probability is maximal.
The problem involves optimization on two categories of unknowns: the state sequence and the model parameters. The first is discrete, although the number of choices may be astronomically large, whereas the second is continuous in values. One approach to the problem is to treat the two types of variables separately and optimize them alternately over each domain,
 | (5) |
and then the state sequence is fixed to optimize the model . The first corresponds to idealization of the data with a known model, and the second reestimation of model parameters based on a known idealization along with the given observations. The idealization of the state sequence given a model is essentially a discrete optimization problem. Given N possible states at each time, there are a total of NT permutations of state sequences. Since the probability of each sequence can be calculated readily using Eq. 4, it is conceivable to attempt an exhausted search, i.e., to enumerate all state sequences, compare their probabilities, and then determine the one giving the maximal probability. Unfortunately, the strategy is practically unrealistic. Even in the simplest case with two states, there are 2T sequences for T samples. A small number of 100 samples will result in 1030 state sequences, for which a simple enumeration would take >1023 years on a computer operating at 1 GHz.
A realistic approach to the problem is the Viterbi algorithm (Forney, 1973), which exploits the unique structure of the problem in combination with the power of dynamic programming (Cormen et al., 1998). The algorithm is recursive and proceeds as follows. Let 
1(i) = ibi(y1) for 1 i N. Then the following recursion for 2 t T and 1 j N is
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 | (7) |
t(i). Upon termination, the likelihood is given by
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as follows. Let sT = i*, which maximizes T(i). Then for T 
t 2, st-1 = t(st). The basic idea of the Viterbi algorithm is schematically illustrated in Fig. 1. It performs the idealization through time successively. At each time t, it keeps track of the optimal state sequences (pathways) leading to all possible states at that point. Then, an optimal sequence up to the next time t + 1 is constructed by examining all existing N sequences up to time t in combination with an appropriate transition from time t to t + 1. Because the probabilities of the state sequences up to time t are remembered, the construction of the new extended sequences requires only N2 computations, as implied by Eq. 6. The idealization of the entire dataset therefore takes on the order of N2T operations, which is quadratic on the number of states and linear on the number of samples, as opposed to the exponential dependence required by an exhaustive search.
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 | (10) |
i denotes the states of the ith conductance class, and the denominator represents the number of samples that are idealized into i. Similarly, the transition probability can be estimated by counting the number of transitions occurring from each state, i.e.,
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Ideally, the new estimates of the model parameters should agree with those that initiate the idealization. When the model is unknown, however, they may not be equal, in which case the estimates can be used to upgrade the model. This leads to an iterative loop as shown in Fig. 2, where an initial model, 0, is chosen, and the Viterbi algorithm is used to find an optimal idealization from which the model parameters are reestimated. The iteration continues until it converges, for example, when the difference of the parameter values in two consecutive iterations becomes less than a preset small tolerance. This is the essence of the segmental k-means method (Rabiner et al., 1986). Convergence of the algorithm is assured (Juang and Rabiner, 1990), and the reestimated model parameters always give rise to an improved likelihood value at each iteration. As illustrated by examples in the following, the convergence is generally fast, taking only a few iterations.
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IMPLEMENTATION |
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t(i), one calculates lnt(i). The recursion Eq. 6 becomes
 | (12) |
t(i) can be calculated from its precedent. More importantly, the recursion now involves no multiplication. The algorithm can be implemented with only additions, which is efficient to compute. The use of log probabilities also avoids the problem of numeric overflow. The term t(i) can be considered as a product of probabilities over samples, and if the number of samples is large, the product will eventually exceed the range of computer precision. With the log representation, the problem is avoided. The other maneuver that can improve the efficiency of the algorithm is to calculate the Gaussian distributions before recursions. This can be done using a lookup table. During recursions, the deviation of a sample from a conductance level can be calculated and discretized to find the appropriate distribution values from the table. This alleviates the expensive computation of exponential functions involved in the distributions.
Application to single-channel currents
The algorithm described above has been tested extensively in the context of idealization of single-channel currents. In the following, a few representative examples are presented. These examples are intended to illustrate the basic performance of the idealization. The algorithm has many features—for example, the allowance for multiple conductance levels or multiple channels, excessive opening noise, constraints on parameters, and so on. The usage of these features is straightforward and will not be discussed here.
The algorithm was implemented in C/C++ language with a Windows graphical interface to support user-interactive manipulations. The interface provides convenient tools for initialization of current amplitudes and noise variances, which are typically "grabbed" from a highlighted region on a data trace. A state model with appropriately specified rate constants is used for initializing the transition probabilities. The program is available through the IcE/QuB software suite (www.qub.buffalo.edu).
Sensitivity to noise and channel kinetics
One advantage of the algorithm is its tolerance for noise. For certain types of channels, good idealization can be achieved at a signal/noise ratio as low as i/ = 2. As an example, consider the data shown in Fig. 3 A (noisy trace), which were simulated from a two-state model (Scheme I) with current amplitude i = 1 pA and noise standard deviation = 0.5 pA. The rate constants of the model were k12 = k21 = 100 s-1. The data were sampled at 100 µs, and a total of 1,000,000 samples were generated. Fig. 3 A (bottom trace) shows the resultant idealization by the algorithm, which agrees well with the true currents, as shown on the top. In total, the simulation resulted in 9967 dwell-times with a mean duration of 10.03 ms. The idealization recovered 9067 events giving a mean duration of 11.03 ms. The error rate of the idealization was therefore within 10% for both the number of events and the mean dwell-time duration. The algorithm was insensitive to the starting values of the model. Repeat of the idealization with different starting values led to comparable results. Fig. 3 B shows the convergence of the log likelihood through iterations. The algorithm generally converged in a few iterations.
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), but the segmental k-means method is more efficient, especially when the model has a high complexity.
Although the algorithm worked well with a noise level as high as = 0.5 pA in the above test, this cannot be taken as a general criterion. Its performance also relies on channel kinetics. As the kinetics get fast, the tolerance for noise declines. Fig. 4 A shows the error rates of the idealization as a function of noise level at several kinetic settings. The results were obtained using the same simulation conditions as stated above, except for the rates and noise that were subject to examination. Although the errors were <10% at k x t = 0.01 with noise up to = 0.55 pA, the same accuracy could only be achieved with = 0.35 pA when the kinetics became 10 times faster.
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t 
0.1, beyond which a further increase of kinetics led to a reduction on the errors. One possible explanation for this biphasic dependence is that with extremely fast kinetics, the correlation between adjacent samples becomes weakened and the channel activity behaves statistically more like a white noise. As a result, the detection of the currents can be actually reinforced by the presence of noise. From the figure, it is evident that the detection degraded rapidly with the increase of either kinetics or noise, consistent with the previous observations in Fig. 4 A. The algorithm compares favorably with the threshold detection. Fig. 5 shows a direct comparison between the two methods in terms of the number of events and the mean dwell-time duration. As expected, the SKM method exhibits a much higher tolerance for noise. In addition, the erroneous events resulting from SKM appear to be fundamentally different from those obtained with threshold detection. Although the absolute errors of the idealization increase with noise in both cases, they proceed in opposite directions. The SKM method always underestimates the total number of events, whereas the threshold detection overestimates it. Consistently, the mean dwell-time duration is underestimated with SKM but overestimated with threshold. This suggests that the errors involved in SKM idealization are primarily due to missed events whereas threshold detection results in false events. To this extent, the SKM detection is advantageous since the missed events, but not the false events, can be corrected during the stage of dwell-time analysis. To avoid the problem of false events, the threshold analysis has to rely on heavy filtering to minimize errors.
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t. Furthermore, the distribution appeared to decay exponentially. Beyond 2t, the errors reached a plateau and became essentially invariant to kinetics and noise. Therefore, most of the errors arising in the idealization by the algorithm can be attributed to the events with durations <2–3 t. This relatively fixed range of missed events offers the possibility for specification of a constant dead time irrespective of underlying channel kinetics or noise level, which is in contrast to threshold detection, where the dead time increases with the level of low-pass filtering.
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). Table 1 lists the values of the rate constants used in simulation. The model has aggregated states, three closed and two open, and the lifetimes of the states span a broad range from tens of milliseconds to submilliseconds. For testing, a total of 1,000,000 samples were simulated, which gave rise to 14,788 dwell-times (16,414 before sampling), with mean closed and open durations of 3.52 and 3.25 ms, respectively. Data was sampled at 50 µs. The unitary current of the channel was 1 pA, and the noise standard deviation = 0.25 pA.
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C = 3.61 and O = 3.33 ms, which differed from their simulated values by 
5%. The three-state model, despite its reduced complexity, performed remarkably well; the differences from the full model for both the number of dwell-times and the mean durations were 1%. The two-state model, however, showed an error >10%, as compared to the simulated data.
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The insensitivity of the algorithm to model topology can also be seen from the reestimates of kinetic parameters. For the full model given above, the transition probability matrix was estimated as
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Note that a12 = 1 and a21 = 0, as if the first closed state was absent. This was the case because the lifetimes of C1 and C2 were long as opposed to the sampling duration, and collectively they could be represented by a single state. In general, the algorithm performs poorly on the estimation of rates especially between aggregated states, indicating that a model with reduced complexity suffices for a valid idealization.
Since the idealization is the first stage of data analysis, it becomes an issue how to choose a model with adequate complexity. A practical solution to the problem is to develop the model and perform the idealization retrospectively. First, a simplistic model can be used to obtain a relatively coarse idealization. The resulting dwell-time distributions can then be explored for extra components, based on which a model with sufficient complexity can be established. As an example, the data simulated above was first idealized with a two-state model. Fitting of the resultant distributions resolved three closed and two open states. Since the true model topology was unknown, a five-state uncoupled model, as shown in Scheme IV, was used to refine the idealization. A model of this type possessed the maximal complexity for a given number of states that could be possibly resolved for a binary channel based on single-channel measurements (Hui et al., 2003). With the model, the data was reidealized. The results, as shown in Table 2, were improved and became comparable to those obtained with the full model.
Low-pass filtering
In this example, the performance of the algorithm in the presence of low-pass filtering is examined. The two-state model in Scheme I was again used. The simulation conditions were chosen similar to typical experimental settings. The channel had a lifetime of 1 ms for both openings and closures, corresponding to a rate of 1000 s-1. The noise had a standard deviation of = 0.5 pA, relative to a 1-pA unitary current. The sampling rate was 50 kHz. A total of 1,000,000 samples were generated. Data were filtered to different extents before idealization. Standard Gaussian digital filters with specified cutoff frequencies were used.
Fig. 8 summarizes the idealization performance as a function of filtering frequency. Also shown are the results from noise-free data to isolate the effect of filtering from that of noise. In the absence of noise, the idealization exhibited a plateau of performance over a wide range of frequency extending from 25 to 5 kHz. Within this range, the error rates on the number of events and the mean dwell-time durations were both <10%. The algorithm started to show significant errors <5 kHz. Further filtering resulted in rapid degradation on idealization. At 1 kHz, the error rate reached nearly 50% for the number of events and >90% for the mean dwell-time duration.
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10 kHz before entering the rapidly degrading phase. This arose presumably as a result of tradeoff between signal/noise ratio and band-limiting distortions. With little filtering, the noise was high, thereby limiting detections of short events. As filtering increased, the signal/noise ratio improved, and so did the idealization. With further filtering, however, the distortions introduced by low-pass filtering became significant, which caused reduction on the accuracy of the detection again. This biphasic trend suggests that there exists an optimal filtering frequency in practice, although its precise value is less certain pending on the level of noise and the kinetics of the channel.
Application to experimental data
As the final example, the applicability of the algorithm to real experimental data is demonstrated. The data was recorded from a mutant, recombinant mouse n-methyl-d-aspartate (NMDA)-activated receptor expressed in Xenopus oocytes. The recordings were made from outside-out patches (Fig. 9 A, top trace). In these patches, only a single channel was active. The data were digitized at a sampling rate of 20 kHz and low-pass filtered to 10 kHz. A total of 100,000 samples were analyzed. Channel opening is indicated by upward deflection of the signal.
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DISCUSSION |
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The segmental k-means method is closely related to another HMM technique, namely, the Baum-Welch algorithm (Rabiner and Juang, 1986). The latter seeks a model to maximize the probability of the observed samples given the model. Mathematically, this is equivalent to summing up the probability over all possible state sequences. The Baum-Welch algorithm uses the forward-backward procedure to evaluate likelihood and Baum's reestimations to optimize model parameters. Upon determination of the model, a most likely state sequence can be restored using the Viterbi algorithm. The Baum-Welch algorithm is superior to the segmental k-means method as it is a full likelihood approach and produces asymptotically unbiased and efficient estimates of model parameters. The segmental k-means method does not have these properties. But it has the advantage of being computationally efficient. This is particularly true when the Viterbi algorithm is implemented with only addition operations. Furthermore, despite its theoretical inferiority, the method produces adequate idealizations, which are comparable to those obtained with the Baum-Welch algorithm. Therefore, the method can be considered as a good tradeoff between theoretical optimality and practical applicability yet without compromising accuracy of idealization.
The weakness of the segmental k-means method is primarily on its estimation of kinetic parameters of the channel. The method is more sensitive to amplitude variables than to transition probabilities. Examples suggested that the estimates of amplitudes and noise variances are in good agreement with their true values to a good precision. The estimates on kinetic rates, however, tend to be biased when the model has aggregated states. In these cases, the program sometimes simply sets the transition probabilities between aggregated states to zero. This kinetic insensitivity is believed to be a result of the simplistic use of the probability of a single state sequence as the likelihood. The sequence, although most likely, contains only a limited amount of information on the transitions of the channel. In contrast, the Baum-Welch algorithm makes use of both the most likely and less likely sequences, which, collectively, provide a large context of kinetic information. Therefore, a reliable estimation of kinetics requires use of the ultimate full likelihood approach. The estimation of the transitional probabilities resulting from the segmental k-means idealization may be biased.
The relative insensitivity of the segmental k-means method to channel kinetics, on the other hand, provides ease for selection of models. In many cases, a nonaggregated model in which each state corresponds to a conductance level, proved adequate. For binary channels, a simple two-state model may suffice. There are cases where an aggregated model is necessary. This is particularly true when the channel contains dwell-times that are orders-of-magnitude different in durations. Under such conditions, introducing a new state can greatly improve the likelihood as well as idealization, particularly the detection of the fast transitions. In practice, an adequate model can be obtained retrospectively. The data can be first idealized with a relatively simple model. Then the resultant dwell-time distributions can be explored for additional components. Once the number of components is determined, a fully connected and uncoupled model can be used for full idealization. Such a model assures adequate complexity as it has as many parameters as the two-dimensional dwell-time distributions, which are known to contain all the information in the data (Fredkin et al., 1985).
Subsequent analysis of idealized dwell-times requires knowledge of dead time, the minimal duration of the events that can be reliably detected, to correct for effects of missed events. With half-amplitude threshold detection, the dead time is primarily determined by the rise time of the filter. When a Gaussian filter is used, the rise time can be explicitly given as a function of its 3dB cutoff frequency, tr = 0.3321/fc (Colquhoun and Sigworth, 1995). The segmental k-means method, on the other hand, does not have such a simple rule. An event is detected not only based on its amplitude but also on its kinetics. A short-lived event may be detected even though its amplitude is under the half-amplitude threshold. This is especially true when the noise is high. Nevertheless, examples suggest that the events that go undetected in the segmental k-mean idealization are mostly the brief ones with duration 2t. The limit appears to hold for a large range of data with different kinetics and noise levels. Furthermore, the number of missed events appears to decrease exponentially with duration. Therefore, for the dwell-times resulting from the segmental k-means method, the dead time is relatively fixed, intrinsically limited by the method.
The present implementation of the algorithm assumes unfiltered data, to best match the first-order condition of the standard HMM. Despite the assumption, the method is shown to work well with filtered data at a cutoff frequency up to 5 kHz. Over this range, the performance of the idealization remains relatively invariant. Since the method allows for a high level of noise, it generally requires less filtering than this limit for typical patch-clamp data, and therefore is devoid of filtering problems as encountered by other methods such as threshold crossing. For moderately filtered data, it is possible to restore its Nyquist bandwidth with an appropriately designed inverse filter. But for heavily filtered data, the correction is limited. Therefore, it is important to acquire data at a high bandwidth and perform low-pass-filtering offline.
Theoretically, the method can be extended to take account of filtering explicitly. A filtered Markov process exhibits a skewed Gaussian distribution, known as the ß-distribution. Therefore, some extent of filtering effect can be taken into account by substituting Gaussian distributions with ß-functions. In the paradigm of Markov modeling, it is possible to extend the model to cope with the correlations introduced by filtering. A common practice is to define a meta-state that includes both the current state of the channel and its precious histories (Qin et al., 2000b; Venkataramanan and Sigworth, 2002). If the filter has a response extending over a length of p samples, the meta-states consist of p-tuples as
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 | (13) |
specifies a meta-state and {hi,1 i p} is the impulse response of the filter. The idealization can be done by applying the segmental k-means method to the meta-state Markov process. The reestimations of the current amplitudes and noise variances become
 | (14) |
 | (15) |
Although the method has been described in the context of ion channel modeling, it is applicable to other types of single-molecule data as well. Common to all these data is noise contaminating a signal that involves discrete jumps between states. These are essentially the same characteristics of single-channel currents. Therefore, it is anticipated that the method can be used to restore the discrete jumps in these applications. The benefits that have been observed in single-channel analysis, such as the allowance for sublevels and high bandwidths, are expected in those applications as well.
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This work was supported by grants R01-RR11114 and R01-GM65994 from the National Institutes of Health.
Submitted on September 29, 2003; accepted for publication November 26, 2003.
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 W. Zhang, Y. Cho, E. Lolis, and J. R. Howe Structural and Single-Channel Results Indicate That the Rates of Ligand Binding Domain Closing and Opening Directly Impact AMPA Receptor Gating J. Neurosci., January 23, 2008; 28(4): 932 - 943. [Abstract] [Full Text] [PDF]
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 A. Jha, D. J. Cadugan, P. Purohit, and A. Auerbach Acetylcholine Receptor Gating at Extracellular Transmembrane Domain Interface: the Cys-Loop and M2 M3 Linker J. Gen. Physiol., November 26, 2007; 130(6): 547 - 558. [Abstract] [Full Text] [PDF]
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 D. J. Cadugan and A. Auerbach Conformational Dynamics of the {alpha}M3 Transmembrane Helix during Acetylcholine Receptor Channel Gating Biophys. J., August 1, 2007; 93(3): 859 - 865. [Abstract] [Full Text] [PDF]
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S. Elenes, Y. Ni, G. D. Cymes, and C. Grosman Desensitization Contributes to the Synaptic Response of Gain-of-Function Mutants of the Muscle Nicotinic Receptor J. Gen. Physiol., November 1, 2006; 128(5): 615 - 627. [Abstract] [Full Text] [PDF]
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L. S. Milescu, A. Yildiz, P. R. Selvin, and F. Sachs Extracting Dwell Time Sequences from Processive Molecular Motor Data Biophys. J., November 1, 2006; 91(9): 3135 - 3150. [Abstract] [Full Text] [PDF]
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Y. Purohit and C. Grosman Estimating Binding Affinities of the Nicotinic Receptor for Low-efficacy Ligands Using Mixtures of Agonists and Two-dimensional Concentration-Response Relationships J. Gen. Physiol., May 30, 2006; 127(6): 719 - 735. [Abstract] [Full Text] [PDF]
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