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.     

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.     

Estimation of blood flow distribution in skeletal muscle from inert gas washout

A new method is evaluated for the estimation of blood flow-to-volume distribution in skeletal muscle from inert gas washout kinetics. Acetylene washout from the isolated, blood-perfused canine gracilis muscle was measured continuously with a blood gas catheter in combination with a mass spectrometer. The washout curves were transformed to flow-to-volume ratio distributions by means of a 50-compartment model. The algorithm fits the expression for the washout curve derived from the model by a least-squares method with enforced smoothing. The algorithm was evaluated using computer simulations in which artificial washout curves were generated by a multicompartment model with a known flow distribution. A wide range of given flow distributions could be recovered from the simulated data. The data were also analyzed using a linear programming technique. Analysis of the experimental data with the least-squares method showed that there is considerable heterogeneity in the distribution of perfusion in resting gracilis muscle. The distribution is characterized by at least two modes and a single compartment with a very low perfusion-to-volume ratio. Experimental noise made it impossible to obtain feasible flow distributions by means of linear programming.     

Determination of rate distributions from kinetic experiments.

Rate processes in proteins are often not adequately described by simple exponential kinetics. Instead of modeling the kinetics in the time domain, it can be advantageous to perform a numerical inversion leading to a rate distribution function f(lambda). The features observed in f(lambda) (number, positions, and shapes of peaks) can then be interpreted. We discuss different numerical techniques for obtaining rate distribution functions, with special emphasis on the maximum entropy method. Examples are given for the application of these techniques to flash photolysis data of heme proteins.     

基于神经网络原理的荧光寿命成像研究

荧光寿命成像技术(fhlorescence lifetime imaging,FLIM)是一种新颖且功能强大的、能用于复杂生物组织和细胞结构与功能分析的生物组织成像技术。传统的时域荧光寿命成像数据分析方法,由于没有考虑荧光分子团之间以及他们与周围环境的相互作用,可能导致复杂的连续分布荧光寿命这一实际情况,因此对生物组织中自发荧光发光强度衰减过程的实验数据拟合效果欠佳。文章提出利用人工神经网络(artificial neural network,ANN)原理拟合算法来计算生物荧光分子团衰减动力过程,该方法能有效地建立生物荧光分子团衰减动力过程的非线性模型,并且具有处理非线性模型能力强、鲁棒性好、拟合精度高和所需计算时间少等优点。通过计算证明,相对于单参量指数与多参量指数衰减函数,这种数据拟合方法对于某些荧光分子团的多槽基面效价测定样品(multi-well plate assays)的数据有更好的一致性和更小的计算量。同时在文章中讨论了将该拟合算法应用于荧光寿命成像的前景。     

Time-resolved room temperature protein phosphorescence: nonexponential decay from single emitting tryptophans.

The single room temperature phosphorescent (RTP) residue of horse liver alcohol dehydrogenase (LADH). Trp-314, and of alkaline phosphatase (AP), Trp-109, show nonexponential phosphorescence decays when the data are collected to a high degree of precision. Using the maximum entropy method (MEM) for the analysis of these decays, it is shown that AP phosphorescence decay is dominated by a single Gaussian distribution, whereas for LADH the data reveal two amplitude packets. The lifetime-normalized width of the MEM distribution for both proteins is larger than that obtained for model monoexponential chromophores (e.g., terbium in water and pyrene in cyclohexane). Experiments show that the nonexponential decay is fundamental; i.e., an intrinsic property of the pure protein. Because phosphorescence reports on the state of the emitting chromophore, such nonexponential behavior could be caused by the presence of excited state reactions. However, it is also well known that the phosphorescence lifetime of a tryptophan residue is strongly dependent on the local flexibility around the indole moiety. Hence, the nonexponential phosphorescence decay may also be caused by the presence of at least two states of different local rigidity (in the vicinity of the phosphorescing tryptophan) corresponding to different ground state conformers. The observation that in the chemically homogeneous LADH sample the phosphorescence decay kinetics depends on the excitation wavelength further supports this latter interpretation. This dependence is caused by the wavelength-selective excitation of Trp-314 in a subensemble of LADH molecules with differing hydrophobic and rigid environments. With this interpretation, the data show that interconversion of these states occurs on a time scale long compared with the phosphorescence decay (0.1-1.0 s). Further experiments reveal that with increasing temperature the distributed phosphorescence decay rates for both AP and LADH broaden, thus indicating that either 1) the number of conformational states populated at higher temperature increases or 2) the temperature differentially affects individual conformer states. The nature of the observed heterogeneous triplet state kinetics and their relationship to aspects of protein dynamics are discussed.     

Lifetime of actin-dependent protein nanoclusters

Protein nanoclusters (PNCs) are dynamic collections of a few proteins that spatially organize in nanometer-length clusters. PNCs are one of the principal forms of spatial organization of membrane proteins, and they have been shown or hypothesized to be important in various cellular processes, including cell signaling. PNCs show remarkable diversity in size, shape, and lifetime. In particular, the lifetime of PNCs can vary over a wide range of timescales. The diversity in size and shape can be explained by the interaction of the clustering proteins with the actin cytoskeleton or the lipid membrane, but very little is known about the processes that determine the lifetime of the nanoclusters. In this paper, using mathematical modeling of the cluster dynamics, we model the biophysical processes that determine the lifetime of actin-dependent PNCs. In particular, we investigated the role of actin aster fragmentation, which had been suggested to be a key determinant of the PNC lifetime, and we found that it is important only for a small class of PNCs. A simple extension of our model allowed us to investigate the kinetics of protein-ligand interaction near PNCs. We found an anomalous increase in the lifetime of ligands near PNCs, which agrees remarkably well with experimental data on RAS-RAF kinetics. In particular, analysis of the RAS-RAF data through our model provides falsifiable predictions and novel hypotheses that will not only shed light on the role of RAS-RAF kinetics in various cancers, but also will be useful in studying membrane protein clustering in general.     

Estimation of ion channel kinetics from fluctuations of macroscopic currents

For single channel recordings, the maximum likelihood estimation (MLE) of kinetic rates and conductance is well established. A direct extrapolation of this method to macroscopic currents is computationally prohibitive: it scales as a power of the number of channels. An approximated MLE that ignored the local time correlation of the data has been shown to provide estimates of the kinetic parameters. In this article, an improved approximated MLE that takes into account the local time correlation is proposed. This method estimates the channel kinetics using both the time course and the random fluctuations of the macroscopic current generated by a homogeneous population of ion channels under white noise. It allows arbitrary kinetic models and stimulation protocols. The application of the proposed algorithm to simulated data from a simple three-state model on nonstationary conditions showed reliable estimates of all the kinetic constants, the conductance and the number of channels, and reliable values for the standard error of those estimates. Compared to the previous approximated MLE, it reduces by a factor of 10 the amount of data needed to secure a given accuracy and it can even determine the kinetic rates in macroscopic stationary conditions.     

Conformational fluctuations and protein reactivity. Determination of the rate-constant spectrum and consequences in elementary biochemical processes

When a protein's active site happens to be strongly coupled with the protein structure, the rate constant of the reaction may eventually be modulated by the conformational fluctuations. Evidence for this effect has long been provided by extensive flash photolysis investigations of liganded hemoproteins and more recently of the non-heme respiratory protein hemerythrin in hydro-organic solvents. Within a given protein conformational substate, an elementary reaction step is characterized by one single free energy barrier and by a first-order rate constant, k, which changes with temperature according to an Arrhenius law. At physiological temperature and low viscosity, ultrafast conformational relaxation causes efficient averaging of the reaction rates and the protein displays exponential kinetics with an average rate constant (k). Under sufficiently general conditions, it can be shown that (k) also follows a simple Arrhenius law with 'effective' values of the pre-exponential factor Aeff and activation enthalpy Heff. It is found that Aeff strongly depends on the overall shape of the rate constant distribution and that Heff actually corresponds to the lower limit of the enthalpy of activation, i.e. the value associated with the highest possible reaction rate. The underlying distribution of rate constants can be reconstructed from a set of experiments in which the kinetics depart from an exponential, i.e. at low temperature and high viscosity. The most probable distribution of exponentials consistent with the observed kinetics of the geminate recombinations of oxygen with photodissociated hemerythrin has been determined by using a new approach, known as the maximum entropy method. The results are consistent with a single pre-exponential value and a distributed enthalpy spectrum. As expected, Heff does not coincide either with the most probable nor with the average value of the enthalpy. The most salient findings are that the probability for any protein molecule to have an enthalpy of activation equal to the effective value Heff vanishes and that Aeff differs by nearly three orders of magnitude from the true value A0. Biochemical reaction rates are actually average values, since protein reactions are measured under physiological conditions, where conformational relaxation is always fast. Our understanding of the significance of Aeff and Heff is therefore entirely dependent on the knowledge of the distribution function of the rate constants. In particular, enthalpy and entropy terms of similar reactions performed by different proteins cannot be compared as long as the distribution of the rate constants remains unknown.     

Robust reconstruction of the rate constant distribution using the phase function method

Many biological processes exhibit complex kinetic behavior that involves a nontrivial distribution of rate constants. Characterization of the rate constant distribution is often critical for mechanistic understandings of these processes. However, it is difficult to extract a rate constant distribution from data measured in the time domain. This is due to the numerical instability of the inverse Laplace transform, a long-standing mathematical challenge that has hampered data analysis in many disciplines. Here, we present a method that allows us to reconstruct the probability distribution of rate constants from decay data in the time domain, without fitting to specific trial functions or requiring any prior knowledge of the rate distribution. The robustness (numerical stability) of this reconstruction method is numerically illustrated by analyzing data with realistic noise and theoretically proved by the continuity of the transformations connecting the relevant function spaces. This development enhances our ability to characterize kinetics and dynamics of biological processes. We expect this method to be useful in a broad range of disciplines considering the prevalence of complex exponential decays in many experimental systems.     

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.     

Analog modelling of cochlear adaptation

Adaptation phenomena in the peripheral hearing organ originate from the properties of the haircell-neuron synaps as revealed by their temperature dependency. In order to verify this hypothesis a model experiment is set up. The model consists of programming on an analog computer the equations describing the reaction kinetics from the frogs myoneural junction. A model neuron is attached to the synapsmodel in which two independent noise sources are incorporated. This noise addition serves to simulate the stochastic behavior of the transmitter release in the synaps resulting in a fluctuating generator potential and independently thereof to reflect the varying threshold of the nerve fiber. The reaction rate constants in the synaptic model were modified with respect to the original ones in order to get a coincidence of in vivo- and model results. The compound model primarily is used to simulate the quite different synchronization between the auditory nerve fibers occurring during intensity changes and during changes of the stimulus repetition rate. These results were known to be different in the animal experiments and were also generated by the model. It is shown clearly that neither synaptic noise alone nor membrane noise alone can account for the observations made in the animal experiments. Therefore a combination of both types of noise is used in this model. The model experiment also visualizes that amplitude changes for compound AP's during adaptation can be explained in toto by a decreasing synchronization, i.e. the broadening of the latency distribution function for the nerve fibers.     

Universal features of fluctuations in globular proteins

Using data from 2000 non‐homologous protein crystal structures, we show that the distribution of residue B factors of proteins collapses onto a single master curve. We show by maximum entropy arguments that this curve is a Gamma function whose order and dispersion are obtained from experimental data. The distribution for any given specific protein can be generated from the master curve by a linear transformation. Any perturbation of the B factor distribution of a protein, imposed at constant energy, causes a decrease in the entropy of the protein relative to that of the reference state. Proteins 2016; 84:721–725. © 2016 Wiley Periodicals, Inc.     

Human antibody Fc deamidation in vivo

Protein and peptide deamidation occurs spontaneously in vitro under relatively mild conditions. For antibodies and other therapeutic proteins, great effort is placed in manufacturing and storage to minimize this form of degradation. Concern has been especially great in cases where deamidation has been shown to impact protein activity. Here we monitored asparagine deamidation from a recombinant human antibody in humans and found that among the conserved sites, only Asn 384 deamidated at an appreciable rate. Under physiological temperature and pH conditions, in vitro antibody deamidation followed similar kinetics, indicating that simple incubation reactions may be used to model in vivo behavior. Endogenous IgG isolated from human serum possessed 23% deamidation at this site, further demonstrating that this modification is naturally occurring. Thus, deamidation generated in manufacturing and storage does not fully determine the patient exposure to the attribute. Instead, pharmacokinetic data along with the deamidation kinetics are combined to predict patient exposure. The deamidation rate can also be used to estimate the serum lifetime of antibodies. This approach could potentially be used to estimate turnover for other cellular or extracellular proteins.     

A Lorentz model for weak magnetic field bioeffects: Part II—Secondary transduction mechanisms and measures of reactivity

In Part I it was shown that the thermal component of the motion of a charged particle in an oscillator potential, that is, within a molecular binding site, rotates at the Larmor frequency in an applied magnetic field. It was also shown that the Larmor angular frequency is independent of the thermal noise strength and thus offers a mechanism for the biological detection of weak (µT‐range) magnetic fields. Part II addresses the question of how the Larmor trajectory could affect biological reactivity. The projection of the motion onto a Cartesian axis measures the nonuniformity of the Larmor trajectory in AC and combined AC/DC magnetic fields, suggesting a means of assessing resonances. A physically meaningful measure of reactivity based upon the classical oscillator trajectory is suggested, and the problem of initial conditions is addressed through averaging over AC phases. AC resonance frequencies occur at the Larmor frequency and at other frequencies, and are dependent upon the ratio of AC/DC amplitudes and target kinetics via binding lifetime. The model is compared with experimental data reported for a test of the ion parametric resonance (IPR) model on data from Ca²⁺ flux in membrane vesicles, neurite outgrowth from PC‐12 cells and a cell‐free calmodulin‐dependent myosin phosphorylation system, and suggests Mg²⁺ is the target for these systems. The results do not require multiple‐ion targets, selection of isotopes, or additional curve fitting. The sole fitting parameter is the binding lifetime of the target system and the results shown are consistent with the literature on binding kinetics. Bioelectromagnetics 30:476–488, 2009. © 2009 Wiley‐Liss, Inc.     

Combined affinity and rate constant distributions of ligand populations from experimental surface binding kinetics and equilibria

The present article considers the influence of heterogeneity in a mobile analyte or in an immobilized ligand population on the surface binding kinetics and equilibrium isotherms. We describe strategies for solving the inverse problem of calculating two-dimensional distributions of rate and affinity constants from experimental data on surface binding kinetics, such as obtained from optical biosensors. Although the characterization of a heterogeneous population of analytes binding to uniform surface sites may be possible under suitable experimental conditions, computational difficulties currently limit this approach. In contrast, the case of uniform analytes binding to heterogeneous populations of surface sites is computationally feasible, and can be combined with Tikhonov-Phillips and maximum entropy regularization techniques that provide the simplest distribution that is consistent with the data. The properties of this ligand distribution analysis are explored with several experimental and simulated data sets. The resulting two-dimensional rate and affinity constant distributions can describe well experimental kinetic traces measured with optical biosensors. The use of kinetic surface binding data can give significantly higher resolution than affinity distributions from the binding isotherms alone. The shape and the level of detail of the calculated distributions depend on the experimental conditions, such as contact times and the concentration range of the analyte. Despite the flexibility introduced by considering surface site distributions, the impostor application of this model to surface binding data from transport limited binding processes or from analyte distributions can be identified by large residuals, if a sufficient range of analyte concentrations and contact times are used. The distribution analysis can provide a rational interpretation of complex experimental surface binding kinetics, and provides an analytical tool for probing the homogeneity of the populations of immobilized protein.     

Measuring kinetics of complex single ion channel data using mean-variance histograms.

The measurement of single ion channel kinetics is difficult when those channels exhibit subconductance events. When the kinetics are fast, and when the current magnitudes are small, as is the case for Na+, Ca2+, and some K+ channels, these difficulties can lead to serious errors in the estimation of channel kinetics. I present here a method, based on the construction and analysis of mean-variance histograms, that can overcome these problems. A mean-variance histogram is constructed by calculating the mean current and the current variance within a brief "window" (a set of N consecutive data samples) superimposed on the digitized raw channel data. Systematic movement of this window over the data produces large numbers of mean-variance pairs which can be assembled into a two-dimensional histogram. Defined current levels (open, closed, or sublevel) appear in such plots as low variance regions. The total number of events in such low variance regions is estimated by curve fitting and plotted as a function of window width. This function decreases with the same time constants as the original dwell time probability distribution for each of the regions. The method can therefore be used: 1) to present a qualitative summary of the single channel data from which the signal-to-noise ratio, open channel noise, steadiness of the baseline, and number of conductance levels can be quickly determined; 2) to quantify the dwell time distribution in each of the levels exhibited. In this paper I present the analysis of a Na+ channel recording that had a number of complexities. The signal-to-noise ratio was only about 8 for the main open state, open channel noise, and fast flickers to other states were present, as were a substantial number of subconductance states. "Standard" half-amplitude threshold analysis of these data produce open and closed time histograms that were well fitted by the sum of two exponentials, but with apparently erroneous time constants, whereas the mean-variance histogram technique provided a more credible analysis of the open, closed, and subconductance times for the patch. I also show that the method produces accurate results on simulated data in a wide variety of conditions, whereas the half-amplitude method, when applied to complex simulated data shows the same errors as were apparent in the real data. The utility and the limitations of this new method are discussed.     

Conformational dynamics and intersubunit energy transfer in wild-type and mutant lipoamide dehydrogenase from Azotobacter vinelandii. A multidimensional time-resolved polarized fluorescence study.

Time-resolved fluorescence and fluorescence anisotropy data surfaces of flavin adenine dinucleotide bound to lipoamide dehydrogenase from Azotobacter vinelandii in 80% glycerol have been obtained by variation of excitation energy and temperature between 203 and 303 K. The fluorescence kinetics of a deletion mutant lacking 14 COOH-terminal amino acids were compared with the wild-type enzyme to study a possible interaction of the COOH-terminal tail with the active site of the enzyme. The flavin adenine dinucleotide fluorescence in both proteins exhibits a bimodal lifetime distribution as recovered by the maximum entropy method of data analysis. The difference in standard enthalpy and entropy of associated conformational substates was retrieved from the fractional contributions of the two lifetime classes. Activation energies of thermal quenching were obtained that confirm that the isoalloxazines in the deletion mutant are solvent accessible in contrast to the wild-type enzyme. Red-edge spectroscopy in conjunction with variation of temperature provides the necessary experimental axes to interpret the fluorescence depolarization in terms of intersubunit energy transfer rather than reorientational dynamics of the flavins. The results can be explained by a compartmental model that describes the anisotropy decay of a binary, inhomogeneously broadened, homoenergy transfer system. By using this model in a global analysis of the fluorescence anisotropy decay surface, the distance between and relative orientation of the two isoalloxazine rings are elucidated. For the wild-type enzyme, this geometrical information is in agreement with crystallographic data of the A. vinelandii enzyme, whereas the mutual orientation of the subunits in the deletion mutant is slightly altered. In addition, the ambiguity in the direction of the emission transition moment in the isoalloxazine ring is solved. The anisotropy decay parameters also provide information on electronic and dipolar relaxational properties of the flavin active site. The local environment of the prosthetic groups in the deletion mutant of the A. vinelandii enzyme is highly inhomogeneous, and a transition from slow to rapid dipolar relaxation is observed over the measured temperature range. In the highly homogeneous active site of the wild-type enzyme, dipolar relaxation is slowed down beyond the time scale of fluorescence emission at any temperature studied. Our results are in favor of a COOH-terminal polypeptide interacting with the active site, thereby shielding the isoalloxazines from the solvent. This biological system forms a very appropriate tool to test the validity of photophysical models describing homoenergy transfer.