Fourier-transform analysis of radiopharmaceutical data

A modification of a method of Gardner, which employs Fourier-transform techniques, is used to obtain initial estimates for the number of terms and values of the parameters for data which are represented by a sum of exponential terms. New experimental methods have increased both the amount and accuracy of data from radiopharmaceutical experiments. This in turn allows one to devise specific numerical methods that utilize the better data. The inherent difficulties of fitting exponentials to data, which is an ill-posed problem, cannot be overcome by any method. However, we show that the present accuracy of Fourier methods may be extended by our numerical methods applied to the improved data sets. In many cases the method yields accurate estimates for the parameters; these estimates then are to be used as initial estimates for a nonlinear least-squares analysis of the problem.     

Analysis of kinetics using a hybrid maximum-entropy/nonlinear-least-squares method: application to protein folding

A hybrid analysis that combines the maximum entropy method (MEM) with nonlinear least squares (NLS) fitting has been developed to interpret a general time-dependent signal. Data that include processes of opposite sign and a slow baseline drift can be inverted to obtain both a continuous distribution of lifetimes and a sum of discrete exponentials. Fits by discrete exponentials are performed with initial parameters determined from the distribution of lifetimes obtained with the MEM. The regularization of the parameter space achieved by the MEM stabilizes the introduction of each successive exponential in the NLS fits. This hybrid approach is particularly useful when fitting by a large number of exponentials. Revision of the MEM "prior" based on features in the data can improve the lifetime distribution obtained. Standard errors in the mean are estimated automatically for raw data. The results presented for simulated data and for fluorescence measurements of protein folding illustrate the utility and accuracy of the hybrid algorithm. Analysis of the folding of dihydrofolate reductase reveals six kinetic processes, one more than previously reported.     

Linking exponential components to kinetic states in Markov models for single-channel gating

Discrete state Markov models have proven useful for describing the gating of single ion channels. Such models predict that the dwell-time distributions of open and closed interval durations are described by mixtures of exponential components, with the number of exponential components equal to the number of states in the kinetic gating mechanism. Although the exponential components are readily calculated (Colquhoun and Hawkes, 1982, Phil. Trans. R. Soc. Lond. B. 300:1-59), there is little practical understanding of the relationship between components and states, as every rate constant in the gating mechanism contributes to each exponential component. We now resolve this problem for simple models. As a tutorial we first illustrate how the dwell-time distribution of all closed intervals arises from the sum of constituent distributions, each arising from a specific gating sequence. The contribution of constituent distributions to the exponential components is then determined, giving the relationship between components and states. Finally, the relationship between components and states is quantified by defining and calculating the linkage of components to states. The relationship between components and states is found to be both intuitive and paradoxical, depending on the ratios of the state lifetimes. Nevertheless, both the intuitive and paradoxical observations can be described within a consistent framework. The approach used here allows the exponential components to be interpreted in terms of underlying states for all possible values of the rate constants, something not previously possible.     

Kinetic diversity of Na+ channel bursts in frog skeletal muscle

Individual Na+ channels of dissociated frog skeletal muscle cells at 10 degrees C fail to inactivate in 0.02% of depolarizing pulses, thus producing bursts of openings lasting hundreds of milliseconds. We present here a kinetic analysis of 87 such bursts that were recorded in multi-channel patches at four pulse potentials. We used standard dwell-time histograms as well as fluctuation analysis to analyze the gating kinetics of the bursting channels. Since each burst contained only 75-150 openings, detailed characterization of the kinetics from single bursts was not possible. Nevertheless, at this low kinetic resolution, the open and closed times could be well fitted by single exponentials (or Lorentzians for the power spectra). The best estimates of both the open and closed time constants produced by either technique were much more broadly dispersed then expected from experimental or analytical variability, with values varying by as much as an order of magnitude. Furthermore, the values of the open and closed time constants were not significantly correlated with one another from burst to burst. The bursts thus expressed diverse kinetic behaviors, all of which appear to be manifestations of a single type of Na+ channel. Although the opening and closing rates were dispersed, their average values were close to those of alpha m and 2 beta m derived from fits to the early transient Na+ currents over the same voltage range. We propose a model in which the channel has both primary states (e.g., open, closed, and inactivated), as well as "modes" that are associated with independent alterations in the rate constants for transition between each of these primary states.     

Kinetic measurements of T----R structural transitions in insulin.

The T6----T3R3 and T3R3----R6-structural transitions of cobalt insulin hexamers as induced by SCN ions or m-cresol were studied in stopped-flow experiments using the absorption in the visible for monitoring their time course. The T6----T3R3 transition induced by either SCN or limited concentrations of m-cresol is mono-exponential with a rate constant of 0.1 s-1 and 0.4 s-1, respectively. A mono-exponential time course is also encountered for the m-cresol-induced T3R3----R6 transition when starting from the T3R3 state preestablished by either SCN or m-cresol. The corresponding rate constants are 1.3 s-1 and 0.49 s-1, respectively. If m-cresol is used beyond the concentration range where transformation is limited to one trimer, two exponentials are required for fitting the time course. The second exponential corresponds to the T3R3----R6 step with a concentration-independent rate constant of 0.4 s-1. The rate constant for the faster T6----T3R3 transition, however, increases with increasing excess of m-cresol.     

Testing for microscopic reversibility in the gating of maxi K+ channels using two-dimensional dwell-time distributions.

An assumption usually made when developing kinetic models for the gating of ion channels is that the transitions among the various states involved in the gating obey microscopic reversibility. If this assumption is incorrect, then the models and estimated rate constants made with the assumption would be in error. This paper examines whether the gating of a large conductance Ca-activated K+ channel in skeletal muscle is consistent with microscopic reversibility. If microscopic reversibility is obeyed, then the number of forward and backward transitions per unit time for each individual reaction step will, on average, be identical and, consequently, the gating must show time reversibility. To look for time reversibility, two-dimensional dwell-time distributions of the durations of open and closed intervals were obtained from single-channel current records analyzed in the forward and in the backward directions. Two-dimensional dwell-time distributions of pairs of open intervals and of pairs of closed intervals were also analyzed to extend the resolution of the method to special circumstances in which intervals from different closed (or open) states might have similar durations. No significant differences were observed between the forward and backward analysis of the two-dimensional dwell-time distributions, suggesting time reversibility. Thus, we find no evidence to indicate that the gating of the maxi K+ channel violates microscopic reversibility.     

Model-based fitting of single-channel dwell-time distributions

Single-channel recordings provide unprecedented resolutions on kinetics of conformational changes of ion channels. Several approaches exist for analysis of the data, including the dwell-time histogram fittings and the full maximal-likelihood approaches that fit either the idealized dwell-time sequence or more ambitiously the noisy data directly using hidden Markov modeling. Although the full maximum likelihood approaches are statistically advantageous, they can be time-consuming especially for large datasets and/or complex models. We present here an alternative approach for model-based fitting of one-dimensional and two-dimensional dwell-time histograms. To improve performance, we derived analytical expressions for the derivatives of one-dimensional and two-dimensional dwell-time distribution functions and employed the gradient-based variable metric method for fast search of optimal rate constants in a model. The algorithm also has the ability to allow for a first-order correction for the effects of missed events, global fitting across different experimental conditions, and imposition of typical constraints on rate constants including microscopic reversibility. Numerical examples are presented to illustrate the performance of the algorithm, and comparisons with the full maximum likelihood fitting are discussed.     

A Kinetic Model for Human Platelet Survival with Long-Lasting Time-Activity Curves

A model is presented that allows for the interpretation of the time course of the level of radiolabeled platelets in terms of platelet survival times, rate constant for removal from circulation, pooling time in an extra pool, the rate at which platelets re-enter circulation from the extra-pool, and the size of the plasma pool and extra-pools. The tenets of the model are that: (1) platelets leave the circulation at a rate proportional to their number per unit volume; (2) of the leaving platelets, a fraction b goes into a pool from which they return into circulation after a pooling time and another fraction (1 - b) is irreversibly destroyed; (3) the platelets in the extra-pool do not “queue up”, and thus the distribution function describing the probability of return is exponential; and (4) the time activity curve of the radiolabeled platelets can be described by the sum of two exponentials. Under steady state conditions, curve fitting allows determination of the constants determining the time activity curve (the respective amplitudes and rate constants of the two exponentials); mean pooling time; relative pool size; and survival time of platelets. The model is applied to data collected from patients over a period from 10 days following reinjection of autologous radiolabeled platelets.     

Flash spectroscopy of purple membrane.

Flash spectroscopy data were obtained for purple membrane fragments at pH 5, 7, and 9 for seven temperatures from 5 degrees to 35 degrees C, at the magic angle for actinic versus measuring beam polarizations, at fifteen wavelengths from 380 to 700 nm, and for about five decades of time from 1 microsecond to completion of the photocycle. Signal-to-noise ratios are as high as 500. Systematic errors involving beam geometries, light scattering, absorption flattening, photoselection, temperature fluctuations, partial dark adaptation of the sample, unwanted actinic effects, and cooperativity were eliminated, compensated for, or are shown to be irrelevant for the conclusions. Using nonlinear least squares techniques, all data at one temperature and one pH were fitted to sums of exponential decays, which is the form required if the system obeys conventional first-order kinetics. The rate constants obtained have well behaved Arrhenius plots. Analysis of the residual errors of the fitting shows that seven exponentials are required to fit the data to the accuracy of the noise level.     

A Fourier method for the analysis of exponential decay curves.

A method based on the Fourier convolution theorem is developed for the analysis of data composed of random noise, plus an unknown constant "base line," plus a sum of (or an integral over a continuous spectrum of) exponential decay functions. The Fourier method's usual serious practical limitation of needing high accuracy data over a very wide range is eliminated by the introduction of convergence parameters and a Gaussian taper window. A computer program is described for the analysis of discrete spectra, where the data involves only a sum of exponentials. The program is completely automatic in that the only necessary inputs are the raw data (not necessarily in equal intervals of time); no potentially biased initial guesses concerning either the number or the values of the components are needed. The outputs include the number of components, the amplitudes and time constants together with their estimated errors, and a spectral plot of the solution. The limiting resolving power of the method is studied by analyzing a wide range of simulated two-, three-, and four-component data. The results seem to indicate that the method is applicable over a considerably wider range of conditions than nonlinear least squares or the method of moments.     

Estimating kinetic parameters for single channels with simulation. A general method that resolves the missed event problem and accounts for noise.

Analysis of currents recorded from single channels is complicated by the limited time resolution (filtering) of the data which can prevent the detection of brief intervals. Although a number of approaches have been used to correct for the undetected intervals (missed events) when identifying kinetic models and estimating parameters, none of them provide a general method which takes into account the true effects of noise and limited time resolution. This paper presents such a method. The approach is to use simulated single-channel currents to incorporate the true effects of filtering and noise on missed events and interval durations. The simulated currents are then analyzed in a manner identical to that used to analyze the experimental currents. An iterative search process using likelihood comparison of two-dimensional dwell-time distributions obtained from the simulated and experimental single-channel currents then allows the most likely rate constants to be determined. The large errors and false solutions that can result from the more typically applied assumptions of no noise and an absolute dead time (idealized filtering) are excluded by the iterative simulation method, and the correlation information contained in the two-dimensional distributions should increase the ability to distinguish among different gating mechanisms. The iterative simulation method is generally applicable to channels which typically open to a single conductance level. For these channels the method places no restrictions on the proposed gating mechanism or the form of the predicted dwell-time distributions.     

Global Kinetic Explorer: A new computer program for dynamic simulation and fitting of kinetic data

We describe a new dynamic kinetic simulation program that allows multiple data sets to be fit simultaneously to a single model based on numerical integration of the rate equations describing the reaction mechanism. Unlike other programs that allow fitting based on numerical integration of rate equations, in the dynamic simulation rate constants, output factors, and starting concentrations of reactants can be scrolled while observing the change in the shape of the simulated reaction curves. Fast dynamic simulation facilitates the exploration of initial parameters that serve as the starting point for nonlinear regression in fitting data and facilitates exploration of the relationships between individual constants and observable reactions. The exploration of parameter space by dynamic simulation provides a powerful tool for learning kinetics and for evaluating the extent to which parameters are constrained by the data. This feature is critical to avoid overly complex models that are not supported by the data.     

Evidence for multiple open states of sodium channels in neuroblastoma cells

Summary Open times of voltage-gated sodium channels in neuroblastoma cells were measured during repolarization (following a short depolarizing conditioning pulse) and during moderate depolarization. Conditional and unconditional channel open-time histograms were best fitted by the sum of two exponentials. (The conditional open time was measured from the end of the conditioning pulse until an open channel shuts provided it was open att=0). Time constants of both histograms depended on the postpulse and were shifted to more positive potentials with increasing conditioning pulse potential. This shift could be explained by assuming more than two time constants in the histograms, which could not be separated. Channel open-time histograms from single-pulse experiments showed a maximum att>0. These histograms could be best fitted by an exponential function with three time constants. One term of this function included the difference of two exponentials resulting in a maximum att>0. Open-time histograms showed a definite time dependence. At 2 to 6.5 msec after the beginning of the depolarization the best fit could be obtained by the difference of two exponentials. To these components another term had to be added at 0 to 2 msec. Between 6.5 and 14.0 msec the sum of two exponentials, and after 14.0 msec a single exponential resulted in a good fit. The results support the hypothesis that sodium channels in neuroblastoma cells may have multiple open states. Two of these states are irreversibly coupled.     

Some pitfalls in curve-fitting and how to avoid them: A case in point

When curve-fitting is used to support a complex nonlinear model containing several exponential terms, some of which have closely-spaced time constants, a particular burden of proof must be assumed. Most important, the uniqueness of the solution must be explored and discussed. Statistical tests for the degree of error and independence of the parameters should be provided, as well as information relating to the steps actually used in the fitting procedures. As an example of the need for the procedures we recommend in this communication, we have chosen an important case in point that has been published recently, and which deals with the kinetics of electron transfer from fully-reduced cytochrome oxidase to O₂, analyzed by the method of SVD-based least squares. The problems we deal with in this case are applicable to a wide variety of other cases that involve curve-fitting to mathematical models.     

Temperature dependence of photovoltages generated by bacteriorhodopsin.

The temperature dependence of the photovoltage developed by a model membrane containing bacteriorhodopsin (BR) is studied. The model membrane is formed by first coating a thin Teflon sheet with lipid and then fusing BR vesicles to it. The time course of the photoresponse is resolved down to 1 microsecond. The photoresponse is taken to be a sum of exponentials. Exponential time constants and amplitudes are determined by an analysis of the photoresponse with a photovoltage vs. log time plot, correlation filter, and nonlinear least-squares routine. The photovoltage is taken to be the sum of three exponentials but only two of the three time constants are resolved. Both are temperature dependent and indicate a thermally activated transport process. The corresponding activation energies are 55 kJ/mol and 62 kJ/mol. Since the photovoltage is proportional to charge times displacement the corresponding charge displacements are 11 and 34 A assuming a total displacement of 45 A. The remaining exponential term corresponds to a small negative transient in the photovoltage that has a rise time less than 1 microsecond even at -20 degrees C. The calculated charge displacement is estimated to be less than 2 A.     

A nonstationary Poisson point process describes the sequence of action potentials over long time scales in lateral-superior-olive auditory neurons

The behavior of lateral-superior-olive (LSO) auditory neurons over large time scales was investigated. Of particular interest was the determination as to whether LSO neurons exhibit the same type of fractal behavior as that observed in primary VIII-nerve auditory neurons. It has been suggested that this fractal behavior, apparent on long time scales, may play a role in optimally coding natural sounds. We found that a nonfractal model, the nonstationary dead-time-modified Poisson point process (DTMP), describes the LSO firing patterns well for time scales greater than a few tens of milliseconds, a region where the specific details of refractoriness are unimportant. The rate is given by the sum of two decaying exponential functions. The process is completely specified by the initial values and time constants of the two exponentials and by the dead-time relation. Specific measures of the firing patterns investigated were the interspike-interval (ISI) histogram, the Fano-factor time curve (FFC), and the serial count correlation coefficient (SCC) with the number of action potentials in successive counting times serving as the random variable. For all the data sets we examined, the latter portion of the recording was well approximated by a single exponential rate function since the initial exponential portion rapidly decreases to a negligible value. Analytical expressions available for the statistics of a DTMP with a single exponential rate function can therefore be used for this portion of the data. Good agreement was obtained among the analytical results, the computer simulation, and the experimental data on time scales where the details of refractoriness are insignificant. For counting times that are sufficiently large, yet much smaller than the largest time constant in the rate function, the Fano factor is directly proportional to the counting time. The nonstationarity may thus mask fractal fluctuations, for which the Fano factor increases as a fractional power (less than unity) of the counting time.     

Variance reduction by simultaneous multi-exponential analysis of data sets from different experiments

The analysis of experimental data from the photocycle of bacteriorhodopsin (bR) as sums of exponentials has accumulated a large amount of information on its kinetics which is still controversial. One reason for ambiguous results can be found in the inherent instabilities connected with the fitting of noisy data by sums of exponentials. Nevertheless, there are strategies to optimize the experiments and the data analysis by a proper combination of well known techniques. This paper describes an applicable approach based on the correct weighting of the data, a separation of the linear and the non-linear parameters in the process of the least squares approximation, and a statistical analysis applying the correlation matrix, the determinant of Fisher's information matrix, and the variance of the parameters as a measure of the reliability of the results. In addition, the confidence regions for the linear approximation of the non-linear model are compared with confidence regions for the true non-linear model. Evaluation techniques and rules for an optimum experimental design are mainly exemplified by the analysis of numerically generated model data with increasing complexity. The estimation of the number of exponentials significant for the interpretation of a given set of data is demonstrated by using records from eight absorption and photocurrent experiments on the photocycle of bacteriorhodopsin. Offprint requests to: K.-H. Müller     

Initial conditions and the kinetics of the sodium conductance in Myxicola giant axons. II. Relaxation experiments

The time-course of the decay of INa on resetting the membrane potential to various levels after test steps in potential was studied. The effects of different initial conditions on these Na tail currents were also studied. For postpulse potentials at or negative to -35 mV, these currents may be attributed nearly entirely to the shutdown of the activation process, inactivation being little involved. Several relaxations may be detected in the tail currents. The slower two are well defined exponentials with time constants of approximately 1 ms and 100 mus in the hyperpolarizing potential range. The fastest relaxation is only poorly resolved. Different initial conditions could alter the relative weighting factors on the various exponential terms, but did not affect any of the individual time constants. The activation of the sodium conductance cannot be attributed to any number of independent and identical two-state subunits with first order transitions. The results of this and the previous paper are discussed in terms of the minimum kinetic scheme consistent with the data. Evidence is also presented suggesting that there may exist a small subpopulation of channels with different kinetics and a faster rate of recovery from TTX block than the rest of the population.     

Data transformations for improved display and fitting of single-channel dwell time histograms.

A.L. Blatz and K.L. Magleby (1986a. J. Physiol. [Lond.]. 378:141-174) have demonstrated the usefulness of plotting histograms with a logarithmic time axis to display the distributions of dwell times from recordings of single ionic channels. We derive here the probability density function (pdf) corresponding to logarithmically binned histograms. Plotted on a logarithmic time scale the pdf is a peaked function with an invariant width; this and other properties of the pdf, coupled with a variance-stabilizing (square root) transformation for the ordinate, greatly simplify the interpretation and manual fitting of distributions containing multiple exponential components. We have also examined the statistical errors in estimation, by the maximum-likelihood method, of kinetic parameters from logarithmically binned data. Using binned data greatly accelerates the fitting procedure and introduces significant errors only for bins spaced more widely than 8-16 per decade.     

Kinetic time constants independent of previous single-channel activity suggest Markov gating for a large conductance Ca-activated K channel

Models for the gating of ion channels usually assume that the rate constants for leaving any given kinetic state are independent of previous channel activity. Although such discrete Markov models have been successful in describing channel gating, there is little direct evidence for the Markov assumption of time-invariant rate constants for constant conditions. This paper tests the Markov assumption by determining whether the single-channel kinetics of the large conductance Ca-activated K channel in cultured rat skeletal muscle are independent of previous single-channel activity. The experimental approach is to examine dwell-time distributions conditional on adjacent interval durations. The time constants of the exponential components describing the distributions are found to be independent of adjacent interval duration, and hence, previous channel activity. In contrast, the areas of the different components can change. Since the observed time constants are a function of the underlying rate constants for transitions among the kinetic states, the observation of time constants independent of previous channel activity suggests that the rate constants are also independent of previous channel activity. Thus, the channel kinetics are consistent with Markov gating. An observed dependent (inverse) relationship between durations of adjacent open and shut intervals together with Markov gating indicates that there are two or more independent transition pathways connecting open and shut states. Finally, no evidence is found to suggest that gating is not at thermodynamic equilibrium: the inverse relationship was independent of the time direction of analysis.