Filtering artifacts from lifetime distributions when maximizing entropy using a bootstrapped model

The maximum entropy method (MEM) has been used in many studies to reliably recover effective lifetimes from kinetics, whether measured experimentally or simulated computationally. Here, recent claims made by Mulligan et al. regarding MEM analyses of kinetics (Anal. Biochem. 421 (2012) 181-190) are shown to be unfounded. Their assertion that their software allows "analysis of datasets too noisy to process by existing iterative search algorithms" is refuted with a MEM analysis of their triexponential test case with increased noise. In addition, it is shown that lifetime distributions recovered from noisy kinetics data with the MEM can be improved by using a simple filter when bootstrapping the prior model. When deriving the bootstrapped model from the lifetime distribution obtained using a uniform model, only the slower processes are represented as Gaussians in the bootstrapped model. Using this new approach, results are clearly superior to those of Mulligan et al. despite the presence of increased noise. In a second example, ambiguity in the interpretation of Poisson kinetics in the presence of scattered excitation light is resolved by filtering the prior model.     

Distributions of fluorescence decay times for parinaric acids in phospholipid membranes

Analysis of fluorescence decay data for probes incorporated into model or biological membranes invariably requires fitting to more than one decay time even though the same probe exhibits nearly single-exponential decay in solution. The parinaric acids (cis and trans) are examples of this. Data are presented for both parinaric acid isomers in dimyristoylphosphatidylcholine membranes collected to higher precision than normally encountered, and the fluorescence decays are shown to be best described by a smooth distribution of decay times rather than by a few discrete lifetimes. The temperature dependence of the fluorescence decay reveals a clear shift in the distribution to longer lifetimes associated with the membrane phase transition at 23.5 degrees C. The physical significance is that fluorescence lifetime measurements appear to reflect a physical process with a distribution of lifetimes rather than several distinct physical processes.     

Analyzing complicated protein folding kinetics rapidly by analytical Laplace inversion using a Tikhonov regularization variant

Kinetic experiments provide much information about protein folding mechanisms. Time-resolved signals are often best described by expressions with many exponential terms, but this hinders the extraction of rate constants by nonlinear least squares (NLS) fitting. Numerical inverse Laplace transformation, which converts a time-resolved dataset into a spectrum of amplitudes as a function of rate constant, allows easy estimation of the rate constants, amplitudes, and number of processes underlying the data. Here, we present a Tikhonov regularization-based method that converts a dataset into a rate spectrum, subject to regularization constraints, without requiring an iterative search of parameter space. This allows more rapid generation of rate spectra as well as analysis of datasets too noisy to process by existing iterative search algorithms. This method's simplicity also permits highly objective, largely automatic analysis with minimal human guidance. We show that this regularization method reproduces results previously obtained by NLS fitting and that it is effective for analyzing datasets too complex for traditional fitting methods. This method's reliability and speed, as well as its potential for objective, model-free analysis, make it extremely useful as a first step in analysis of complicated noisy datasets and an excellent guide for subsequent NLS analysis.     

Analyzing the Distribution of Decay Constants in Pulse-Fluorimetry Using the Maximum Entropy Method

The maximum entropy method (MEM) is used to analyze time-resolved pulse-fluorescence spectrometry. The central problem in such analyses is the recovery of the distribution of exponentials describing the decay of the fluorescence (i.e., inverting the Laplace transform) which is, in turn, convolved by the shape of the excitation flash. MEM is shown to give high quality results from both computer-generated “noisy” data and experimental data from chemical and biological molecules.

The use of the Shannon-Jaynes entropy function is justified and both the theoretical and practical advantages of MEM are presented. The MEM results are easy to interpret and can help to overcome some experimental limitations. In particular MEM could be a powerful tool to analyze the heterogeneity of fluorescent emission of biological macromolecules which can be correlated with their conformational dynamics in solution.

     

Similarity of fluorescence lifetime distributions for single tryptophan proteins in the random coil state.

The picosecond time-resolved fluorescence decay data of nine single-tryptophan (trp) proteins and two multi-trp proteins in their native and denatured states were analyzed by the maximum entropy method (MEM). In the denatured state (6 M guanidine hydrochloride) a majority of the single-trp proteins show bimodal (at 25 degrees C) and trimodal (at 85 degrees C) distributions with similar patterns and similar values for average lifetimes. In the native state of the proteins the lifetime distributions were bimodal or trimodal. These results (multimodal distributions) are contradictory to the unimodal Lorentzian distribution of lifetimes reported for some proteins in the native and denatured states. MEM analysis gives a unimodal distribution of lifetimes only when the signal-to-noise ratio is poor in the time-resolved fluorescence decay data. The unimodal distribution model is therefore not realistic for proteins in the native and denatured states. The fluorescence decay components of the bi- or trimodal distribution are associated with the rotamer structures of the indole moiety when the protein is in the random coil state.     

Energy transfer kinetics in C-phycocyanin from cyanobacterium Westiellopsis prolifica studied by pump-probe techniques

The relaxation processes of C-phycocyanin at different aggregates have been investigated by pump-probe techniques. The lifetimes of ground state recovery measured at various wavelengths are analyzed by computer fitting of the kinetic data to a sum of three and four exponentials for monomers and trimers according to the nonlinear least-square principle, respectively. The shortest lifetime (about 56ps) is due to beta s----beta f transfer in one monomer, that decreases to 31ps in trimer due to the opening of new transfer channels. The second fastest component (about 151ps) in monomer is attributed tentatively to distribution of excitation energy between alpha and beta f chromophores, that decreases to about 117ps in trimer caused by redistribution of excitation energy between them. The two long-lived components (about 690ps and 1385ps for monomer, 620ps and 1320ps for trimer) from some kinds of heterogeneity in some chromophores, such as alpha and beta 1 chromophores which are emitting, show an equal amplitude ratio of 1:2 in both monomer and trimer.     

Exponential Sum-Fitting of Dwell-Time Distributions without Specifying Starting Parameters

Fitting dwell-time distributions with sums of exponentials is widely used to characterize histograms of open- and closed-interval durations recorded from single ion channels, as well as for other physical phenomena. However, it can be difficult to identify the contributing exponential components. Here we extend previous methods of exponential sum-fitting to present a maximum-likelihood approach that consistently detects all significant exponentials without the need for user-specified starting parameters. Instead of searching for exponentials, the fitting starts with a very large number of initial exponentials with logarithmically spaced time constants, so that none are missed. Maximum-likelihood fitting then determines the areas of all the initial exponentials keeping the time constants fixed. In an iterative manner, with refitting after each step, the analysis then removes exponentials with negligible area and combines closely spaced adjacent exponentials, until only those exponentials that make significant contributions to the dwell-time distribution remain. There is no limit on the number of significant exponentials and no starting parameters need be specified. We demonstrate fully automated detection for both experimental and simulated data, as well as for classical exponential-sum-fitting problems.     

Resolvability of fluorescence lifetime distributions using phase fluorometry.

The analysis of the fluorescence decay using discrete exponential components assumes that a small number of species is present. In the absence of a definite kinetic model or when a large number of species is present, the exponential analysis underestimates the uncertainty of the recovered lifetime values. A different approach to determine the lifetime of a population of molecules is the use of probability density functions and lifetime distributions. Fluorescence decay data from continuous distributions of exponentially decaying components were generated. Different magnitudes of error were added to the data to simulate experimental conditions. The resolvability of the distributional model was studied by fitting the simulated data to one and two exponentials. The maximum width of symmetric distributions (uniform, gaussian, and lorentzian), which cannot be distinguished from single and double exponential fits for statistical errors of 1 and 0.1%, were determined. The width limits are determined by the statistical error of the data. It is also shown that, in the frequency domain, the discrete exponential analysis does not uniformly weights all the components of a distribution. This systematic error is less important when probability and distribution functions are used to recover the decay. Finally, it is shown that real lifetime distributions can be proved using multimodal probability density functions. In the companion paper that follows we propose a physical approach, which provides lifetime distribution functions for the tryptophan decay in proteins. In the third companion paper (Alcala, J.R., E. Gratton, and F.J. Prendergast, 1987, Biophys. J., in press) we use the distribution functions obtained to fit data from the fluorescence decay of single tryptophan proteins.     

POTATO: Automated pipeline for batch analysis of optical tweezers data

Optical tweezers are a single-molecule technique that allows probing of intra- and intermolecular interactions that govern complex biological processes involving molecular motors, protein-nucleic acid interactions, and protein/RNA folding. Recent developments in instrumentation eased and accelerated optical tweezers data acquisition, but analysis of the data remains challenging. Here, to enable high-throughput data analysis, we developed an automated python-based analysis pipeline called POTATO (practical optical tweezers analysis tool). POTATO automatically processes the high-frequency raw data generated by force-ramp experiments and identifies (un)folding events using predefined parameters. After segmentation of the force-distance trajectories at the identified (un)folding events, sections of the curve can be fitted independently to a worm-like chain and freely jointed chain models, and the work applied on the molecule can be calculated by numerical integration. Furthermore, the tool allows plotting of constant force data and fitting of the Gaussian distance distribution over time. All these features are wrapped in a user-friendly graphical interface, which allows researchers without programming knowledge to perform sophisticated data analysis.     

A case study of linear versus non-linear modelling

Theories for the facilitation of neurotransmitter release are discussed in a case study of the properties of linear and non-linear models for a phenomenon whose time course can be represented by a sum of decaying exponentials. Particular attention is paid to the effects of a "key factor" on the slopes and amplitudes of the exponentials that can be derived from semilog plots of the data. It is shown that the presence of such effects can give strong evidence for the inappropriateness of linear models. A non-linear model is demonstrated to be capable of describing the changes with extracellular Ca concentration of straight line segments that fit data in semilog plots of facilitation as a function of time. The conclusion is reached that even if data seems to be representable by several independently alterable exponentials one must be cautious in drawing inferences concerning the number, linearity, or independence of the underlying processes.     

How Many M Forms are there in the Bacteriorhodopsin Photocycle?

On capturing a quantum of light, the bacteriorhodopsin of Halobacterium halobium undergoes a photocycle involving different intermediates. The exact scheme of the photocycle and especially the number of M intermediates are subjects of debate. For a quantitative analysis of many effects connected with the photocycle, e.g. the effect of the membrane potential on the kinetics of M decay (Groma et al., 1984. Biophys. J. 45:985-992), a knowledge of the exact photocycle is needed. In the present work sophisticated measurements were made on the decay kinetics of the M forms in cell envelope vesicles, purple membrane suspension and purple membrane fragments incorporated in polyacrylamide gel. The experimental data were analyzed by fitting one, two, and three discrete exponentials. Three different real components were found in the M decay of cell envelope vesicles in 4 M NaCl. All of them exhibited a temperature-dependence obeying the Arrhenius law. Two real components were found for the purple membrane in suspension and in gel in NaCl-free medium. The third phase appeared when the gel was soaked in 4 M NaCl. As an independent means of analysis, a continuous distribution of exponentials was also fitted to the M decay kinetics in cell envelope vesicles. This calculation also resulted in three processes with distinct rates or alternatively two processes with distributed rates.     

Inferring the Rate-Length Law of Protein Folding

We investigate the rate-length scaling law of protein folding, a key undetermined scaling law in the analytical theory of protein folding. Available data yield statistically significant evidence for the existence of a rate-length law capable of predicting folding times to within about two orders of magnitude (over 9 decades of variation). Unambiguous determination of the functional form of such a law could provide key mechanistic insight into folding. Four proposed laws from literature (power law, exponential, and two stretched exponentials) are tested against one another, and it is found that the power law best explains the data by a modest margin. We conclude that more data is necessary to unequivocally infer the rate-length law. Such data could be obtained through a small number of protein folding experiments on large protein domains.     

Fluorescence decay of sarcoplasmic reticulum ATPase. Ligand binding and hydration effects

Multifrequency phase-modulation lifetime data were acquired for sarcoplasmic reticulum Ca2+-ATPase. The intrinsic tryptophan fluorescence decay was complex and was fitted either with three exponentials or with bimodal Lorentzian distributions of lifetimes. Ca2+ binding to the high affinity sites in the ATPase produced an increase of 11% in the center of the main component of the bimodal distribution, shifting the lifetime from 4.04 to 4.50 ns. The effects of solvent on the ATPase were studied with the enzyme dissolved in reverse micelles of detergent bis-(2-ethylhexyl)sulfosuccinate in hexane. Increasing amounts of water up to a water/bis-(2-ethylhexyl)sulfosuccinate molar ratio of 4 produced marked changes in the fluorescence emission of the protein. Comparison of data obtained for micellar solutions of tryptophan or ATPase indicated that the tryptophan residues in the protein are protected from exposure to water. Correlation of water effects on emission intensity and lifetimes suggested that interaction with solvent may result in structural changes that cause a mixture of dynamic and static quenching of ATPase intrinsic fluorescence. Evidence for an effect of hydration on the structure of the active site was obtained by measurements of the fluorescence properties of fluorescein isothiocianate-labeled ATPase in reverse micelles.     

Mechanism of codon recognition by transfer RNA and codon-induced tRNA association

The steps of UUC recognition by tRNAPhe were analysed by temperature-jump measurements. At ion concentrations close to physiological conditions we found three relaxation processes, which we assigned to (1) formation of codon-anticodon complexes, (2) a conformational change of the anticodon loop coupled with Mg2+ binding, and (3) codon-induced association of tRNA. The relaxation data were evaluated both by the usual procedure (fitting the exponentials evaluated from the individual experiments of a set to a reaction model) and by "global fitting", i.e. fitting a set of relaxation curves obtained at various concentrations directly to a reaction model, thus leaving out the intermediate exponential fitting step. The data can be represented quantitatively by a three-step model: the codon binds to the anticodon at a rate of 4 X 10(6) to 6 X 10(6) M-1S-1 as is usual for the formation of oligomer helices; the conformation change of the anticodon loop is associated with inner sphere complexation of Mg2+ at a rate of 10(3) S-1; the codon-tRNA complexes form dimers at a rate of 5 X 10(6) to 15 X 10(6) M-1S-1. A similar mechanism is found for the binding of the wobble codon UUU to tRNAPhe at increased concentrations of Mg2+. Measurements at different Mg2+ concentrations demonstrate the distinct role of this ion in the codon recognition and the codon-induced tRNA dimerization. We propose a simple mechanism, based upon the special properties of magnesium ions, for long-distance transfer of reaction signals along nucleic acid chains.     

Application of the stretched exponential function to fluorescence lifetime imaging

Conventional analyses of fluorescence lifetime measurements resolve the fluorescence decay profile in terms of discrete exponential components with distinct lifetimes. In complex, heterogeneous biological samples such as tissue, multi-exponential decay functions can appear to provide a better fit to fluorescence decay data than the assumption of a mono-exponential decay, but the assumption of multiple discrete components is essentially arbitrary and is often erroneous. Moreover, interactions, both between fluorophores and with their environment, can result in complex fluorescence decay profiles that represent a continuous distribution of lifetimes. Such continuous distributions have been reported for tryptophan, which is one of the main fluorophores in tissue. This situation is better represented by the stretched-exponential function (StrEF). In this work, we have applied, for the first time to our knowledge, the StrEF to time-domain whole-field fluorescence lifetime imaging (FLIM), yielding both excellent tissue contrast and goodness of fit using data from rat tissue. We note that for many biological samples for which there is no a priori knowledge of multiple discrete exponential fluorescence decay profiles, the StrEF is likely to provide a truer representation of the underlying fluorescence dynamics. Furthermore, fitting to a StrEF significantly decreases the required processing time, compared with a multi-exponential component fit and typically provides improved contrast and signal/noise in the resulting FLIM images. In addition, the stretched-exponential decay model can provide a direct measure of the heterogeneity of the sample, and the resulting heterogeneity map can reveal subtle tissue differences that other models fail to show.     

A Markov model for interspike interval distributions of auditory cortical neurons that do not show periodic firings

Spontaneous firing properties of individual auditory cortical neurons are interpreted in terms of local and global order present in functioning brain networks, such as alternating “up” and “down” states. A four-state modulated Markov process is used to model neuronal firings. The system alternates between a bound and an unbound state, both with Poisson-distributed lifetimes. During the unbound state, active and closed states alternate with Poisson-distributed lifetimes. Inside the active state, spikes are generated as a realization of a Poisson process. This combination of processes constitutes a four-state modulated Markov process, determined by five independent parameters. Analytical expressions for the probability density functions (pdfs) that describe the interspike interval (ISI) distribution and autocorrelation function are derived. The pdf for the ISI distribution is shown to be a linear combination of three exponential functions and is expressed through the five system parameters. Through fitting experimental ISI histograms by the theoretical ones, numerical values of the system parameters are obtained for the individual neurons. Both Monte Carlo simulations and goodness-of-fit tests are used to validate the fitting procedure. The values of the estimated system parameters related to the active-closed and bound–unbound processes and their independence on the neurons’ mean firing rate suggest that the underlying quasi-periodic processes reflect properties of the network in which the neurons are embedded. The characteristic times of autocorrelations, determined by the bound–unbound and active-closed processes, are also independent of the neuron’s firing rate. The agreement between experimental and theoretical ISI histograms and autocorrelation functions allows interpretation of the system parameters of the individual neurons in terms of slow and delta waves, and high-frequency oscillations observed in cortical networks. This procedure can identify and track the influence of changing brain states on the single-unit firing patterns in experimental animals.     

Cell responses to single pheromone molecules may reflect the activation kinetics of olfactory receptor molecules

Olfactory receptor cells of the silkmoth Bombyx mori respond to single pheromone molecules with "elementary" electrical events that appear as discrete "bumps" a few milliseconds in duration, or bursts of bumps. As revealed by simulation, one bump may result from a series of random openings of one or several ion channels, producing an average inward membrane current of 1.5 pA. The distributions of durations of bumps and of gaps between bumps in a burst can be fitted by single exponentials with time constants of 10.2 ms and 40.5 ms, respectively. The distribution of burst durations is a sum of two exponentials; the number of bumps per burst obeyed a geometric distribution (mean 3.2 bumps per burst). Accordingly the elementary events could reflect transitions among three states of the pheromone receptor molecule: the vacant receptor (state 1), the pheromone-receptor complex (state 2), and the activated complex (state 3). The calculated rate constants of the transitions between states are k(21)=7.7 s(-1), k(23)=16.8 s(-1), and k(32)=98 s(-1).     

Using a second‐order differential model to fit data without baselines in protein isothermal chemical denaturation

In vitro protein stability studies are commonly conducted via thermal or chemical denaturation/renaturation of protein. Conventional data analyses on the protein unfolding/(re)folding require well‐defined pre‐ and post‐transition baselines to evaluate Gibbs free‐energy change associated with the protein unfolding/(re)folding. This evaluation becomes problematic when there is insufficient data for determining the pre‐ or post‐transition baselines. In this study, fitting on such partial data obtained in protein chemical denaturation is established by introducing second‐order differential (SOD) analysis to overcome the limitations that the conventional fitting method has. By reducing numbers of the baseline‐related fitting parameters, the SOD analysis can successfully fit incomplete chemical denaturation data sets with high agreement to the conventional evaluation on the equivalent completed data, where the conventional fitting fails in analyzing them. This SOD fitting for the abbreviated isothermal chemical denaturation further fulfills data analysis methods on the insufficient data sets conducted in the two prevalent protein stability studies.     

Picosecond tryptophan fluorescence of thioredoxin: evidence for discrete species in slow exchange

The steady-state tryptophan fluorescence and time-resolved tryptophan fluorescence of Escherichia coli thioredoxin, calf thymus thioredoxin, and yeast thioredoxin have been studied. In all proteins, the tryptophan residues undergo strong static and dynamic quenching, probably due to charge-transfer interactions with the nearby sulfur atoms of the active cysteines. The use of a high-resolution photon counting instrument, with a time response of 60 ps full width at half-maximum, allowed the detection of fluorescence lifetimes ranging from a few tens of picoseconds to 10 ns. The data were analyzed both by classical nonlinear least squares and by a new method of entropy maximization (MEM) for the recovery of lifetime distributions. Simulations representative of the experimental data were used to test the MEM analysis. Strong support was obtained in this way for a small number of averaged discrete species in the fluorescence decays. Wavelength studies show that each of these components spreads over closely spaced excited states, while the temperature studies indicate that they do not exchange significantly on the nanosecond time scale. The oxidized form of thioredoxin is characterized by a high content of a very short lifetime below 70 ps, the amplitude of which is sharply decreased upon reduction. On the other hand, the fluorescence anisotropy decays indicate that reduction causes an increase of the very fast tryptophan rotations in an otherwise relatively rigid structure. While the calf thymus and E. coli proteins have mostly similar dynamical fluorescence properties, the yeast thioredoxin differs in many respects.     

Excited‐state lifetime studies of the three tryptophan residues in the N‐lobe of human serum transferrin

The energy transfer from the three Trp residues at positions 8, 128, and 264 within the human serum transferrin (hTF) N-lobe to the ligand to metal charge transfer band has been investigated by monitoring changes in Trp fluorescence emission and lifetimes. The fluorescence emission from hTF N-lobe is dominated by Trp264, as revealed by an 82% decrease in the quantum yield when this Trp residue is absent. Fluorescence lifetimes were determined by multifrequency phase fluorometry of mutants containing one or two Trp residues. Decays of these samples are best described by two or three discrete lifetimes or by a unimodal Lorentzian distribution. The discrete lifetimes and the center of the lifetime distribution for samples containing Trp128 and Trp264 are affected by iron. The distribution width narrows on iron removal and is consistent with a decrease in dynamic mobility of the dominant fluorophore, Trp264. Both the quantum yield and the lifetimes are lower when iron is present, however, not proportionally. The greater effect of iron on quantum yields is indicative of nonexcited state quenching, i.e., static quenching. The results of these experiments provide quantitative data strongly suggesting that Förster resonance energy transfer is not the sole source of Trp quenching in the N-lobe of hTF.