Inferring Diffusion Dynamics from FCS in Heterogeneous Nuclear Environments

Fluorescence correlation spectroscopy (FCS) is a noninvasive technique that probes the diffusion dynamics of proteins down to single-molecule sensitivity in living cells. Critical mechanistic insight is often drawn from FCS experiments by fitting the resulting time-intensity correlation function, G(t), to known diffusion models. When simple models fail, the complex diffusion dynamics of proteins within heterogeneous cellular environments can be fit to anomalous diffusion models with adjustable anomalous exponents. Here, we take a different approach. We use the maximum entropy method to show—first using synthetic data—that a model for proteins diffusing while stochastically binding/unbinding to various affinity sites in living cells gives rise to a G(t) that could otherwise be equally well fit using anomalous diffusion models. We explain the mechanistic insight derived from our method. In particular, using real FCS data, we describe how the effects of cell crowding and binding to affinity sites manifest themselves in the behavior of G(t). Our focus is on the diffusive behavior of an engineered protein in 1) the heterochromatin region of the cell’s nucleus as well as 2) in the cell’s cytoplasm and 3) in solution. The protein consists of the basic region-leucine zipper (BZip) domain of the CCAAT/enhancer-binding protein (C/EBP) fused to fluorescent proteins.     

Parametric Models for Estimating the Number of Classes

We consider parametric distributions intended to model heterogeneity in population size estimation, especially parametric stochastic abundance models for species richness estimation. We briefly review (conditional) maximum likelihood estimation of the number of species, and summarize the results of fitting 7 candidate models to frequency‐count data, from a database of >40000 such instances, mostly arising from microbial ecology. We consider error estimation, goodness‐of‐fit assessment, data subsetting, and other practical matters. We find that, although the array of candidate models can be improved, finite mixtures of a small number of components (point masses or simple diffuse distributions) represent a promising direction. Finally we consider the connections between parametric models for abundance and incidence data, again noting the usefulness of finite mixture models. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

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.     

Time series analysis of hybrid neurophysiological data and application of mutual information

Multivariate time series data of which some components are continuous time series and the rest are point processes are called hybrid data. Such data sets routinely arise while working with neuroscience data, EEG and spike trains would perhaps be the most obvious example. In this paper, we discuss the modeling of a hybrid time series, with the continuous component being the physiological tremors in the distal phalanx of the middle finger, and motor unit firings in the middle finger portion of the extensor digitorum communis (EDC) muscle. We employ a model for the two components based on Auto-regressive Moving Average (ARMA) type models. Another major issue to arise in the modeling of such data is to assess the goodness of fit. We suggest a visual procedure based on mutual information towards assessing the dependence pattern of hybrid data. The goodness of fit is also verified by standard model fitting diagnostic techniques for univariate data.     

S-Sample Smooth Goodness of Fit Testing: Rederivation and Monte Carlo Assessment

S-sample smooth goodness of fit tests may be constructed using components from one sample goodness of fit testing. Each sample could be assessed for consistency with a target distribution using these components, although that is not our objective here. Contrasts in the components may be used to assess consistency of the samples with each other. If all the samples are consistent, we could then conveniently perform a one-sample goodness of fit test for the target distribution. If the samples are not consistent, an LSD-type analysis can be performed on the one-sample components to identify where the differences between occur. This approach gives a detailed and informative scrutiny of the data.     

Analysis of phosphorescence in heterogeneous systems using distributions of quencher concentration.

A continuous distribution approach, instead of the traditional mono- and multiexponential analysis, for determining quencher concentration in a heterogeneous system has been developed. A mathematical model of phosphorescence decay inside a volume with homogeneous concentration of phosphor and heterogeneous concentration of quencher was formulated to obtain pulse-response fitting functions for four different distributions of quencher concentration: rectangular, normal (Gaussian), gamma, and multimodal. The analysis was applied to parameter estimates of a heterogeneous distribution of oxygen tension (PO2) within a volume. Simulated phosphorescence decay data were randomly generated for different distributions and heterogeneity of PO2 inside the excitation/emission volume, consisting of 200 domains, and then fit with equations developed for the four models. Analysis using a monoexponential fit yielded a systematic error (underestimate) in mean PO2 that increased with the degree of heterogeneity. The fitting procedures based on the continuous distribution approach returned more accurate values for parameters of the generated PO2 distribution than did the monoexponential fit. The parameters of the fit (M = mean; sigma = standard deviation) were investigated as a function of signal-to-noise ratio (SNR = maximum signal amplitude/peak-to-peak noise). The best-fit parameter values were stable when SNR > or = 20. All four fitting models returned accurate values of M and sigma for different PO2 distributions. The ability of our procedures to resolve two different heterogeneous compartments was also demonstrated using a bimodal fitting model. An approximate scheme was formulated to allow calculation of the first moments of a spatial distribution of quencher without specifying the distribution. In addition, a procedure for the recovery of a histogram, representing the quencher concentration distribution, was developed and successfully tested.     

Macromolecular crowding and size effects on probe microviscosity

Development of biologically relevant crowding solutions necessitates improved understanding of how the relative size and density of mobile obstacles affect probe diffusion. Both the crowding density and relative size of each co-solute in a mixture will contribute to the measured microviscosity as assessed by altered translational mobility. Using multiphoton fluorescent correlation spectroscopy, this study addresses how excluded volume of dextran polymers from 10 to 500 kDa affect microviscosity quantified by measurements of calmodulin labeled with green fluorescent protein as the diffusing probe. Autocorrelation functions were fit using both a multiple-component model with maximum entropy method (MEMFCS) and an anomalous model. Anomalous diffusion was not detected, but fits of the data with the multiple-component model revealed separable modes of diffusion. When the dominant mode of diffusion from the MEMFCS analysis was used, we observed that increased excluded volume slows probe mobility as a simple exponential with crowder concentration. This behavior can be modeled with a single parameter, β, which depends on the dextran size composition. Two additional modes of diffusion were observed using MEMFCS and were interpreted as unique microviscosities. The fast mode corresponded to unhindered free diffusion as in buffer, whereas the slower agreed well with the bulk viscosity. At 10% crowder concentration, one finds a microviscosity approximately three times that of water, which mimics that reported for intracellular viscosity.     

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.     

Non-linear analysis of GeneChip arrays

The application of microarray hybridization theory to Affymetrix GeneChip data has been a recent focus for data analysts. It has been shown that the hyperbolic Langmuir isotherm captures the shape of the signal response to concentration of Affymetrix GeneChips. We demonstrate that existing linear fit methods for extracting gene expression measures are not well adapted for the effect of saturation resulting from surface adsorption processes. In contrast to the most popular methods, we fit background and concentration parameters within a single global fitting routine instead of estimating the background before obtaining gene expression measures. We describe a non-linear multi-chip model of the perfect match signal that effectively allows for the separation of specific and non-specific components of the microarray signal and avoids saturation bias in the high-intensity range. Multimodel inference, incorporated within the fitting routine, allows a quantitative selection of the model that best describes the observed data. The performance of this method is evaluated on publicly available datasets, and comparisons to popular algorithms are presented.     

Some Formal Approaches to the Analysis of Kinetic Data in Terms of Linear Compartmental Systems

A formal approach to the routine analysis of kinetic data in terms of linear compartmental systems is presented. The methods of analysis are general in that they include much of the theory in common use, such as direct solution of differential equations, integral equations, transfer functions, fitting of data to sums of exponentials, matrix solutions, etc. The key to the formalism presented lies in the fact that a basic operational unit—called “compartment”—has been defined, in terms of which physical and mathematical models as well as input and output functions can be expressed. Additional features for calculating linear combinations of functions and for setting linear dependence relations between parameters add to the versatility of this method. The actual computations for the values of model parameters to yield a least squares fit of the data are performed on a digital computer. A general computer program was developed that permits the routine fitting of data and the evolution of models.     

A comparison of three maximum likelihood models for stage-frequency data

A comparison has been made between the estimates obtained from maximum likelihood estimation of gamma, inverse normal, and normal distribution models for stage-frequency data. Results have been compared for six of sets of test data, and from many sets of simulated data. It is concluded that (1) some estimates may differ substantially between the models, (2) estimates from the correct model have little bias, and estimated standard errors are generally close to theoretical values, (3) there are problems in determining degrees of freedom for chi-squared goodness of fit tests, so that it is best to compare test statistics with simulated distributions, and (4) goodness of fit tests may not discriminate well between the three models.     

种子植物区系属大小与种多度分布格局的相似性

对属大小的分析是植物区系研究的重要组成,属大小的分布格局可以在一定程度上反映植物区系的类群多样性与系统发生多样性。在植物区系研究中,平均属大小指标与植物区系包含的物种数有关,使不同植物区系的研究结果难以进行整合分析。对中国8个地区种子植物区系属大小的分布格局进行研究,对数级数模型取得了良好的拟合结果,拟合优度R²＞0.99,属大小表现出与种—多度相似的分布格局。稀疏分析和相关分析结果表明：抽样大小较小时,平均属大小在稀疏分析前后的相关性不显著;而对数级数模型中α指数与稀疏分析前表现出稳定的相关性。同时,对于相同的抽样大小α指数与平均属大小之间存在显著的相关性。因此,α指数可以作为稳定的指标来描述植物区系的丰富程度。     

Bayesian Total Internal Reflection Fluorescence Correlation Spectroscopy Reveals hIAPP-Induced Plasma Membrane Domain Organization in Live Cells

Amyloid fibril deposition of human islet amyloid polypeptide (hIAPP) in pancreatic islet cells is implicated in the pathogenesis of type II diabetes. A growing number of studies suggest that small peptide aggregates are cytotoxic via their interaction with the plasma membrane, which leads to membrane permeabilization or disruption. A recent study using imaging total internal reflection-fluorescence correlation spectroscopy (ITIR-FCS) showed that monomeric hIAPP induced the formation of cellular plasma membrane microdomains containing dense lipids, in addition to the modulation of membrane fluidity. However, the spatial organization of microdomains and their temporal evolution were only partially characterized due to limitations in the conventional analysis and interpretation of imaging FCS datasets. Here, we apply a previously developed Bayesian analysis procedure to ITIR-FCS data to resolve hIAPP-induced microdomain spatial organization and temporal dynamics. Our analysis enables the visualization of the temporal evolution of multiple diffusing species in the spatially heterogeneous cell membrane, lending support to the carpet model for the association mode of hIAPP aggregates with the plasma membrane. The presented Bayesian analysis procedure provides an automated and general approach to unbiased model-based interpretation of imaging FCS data, with broad applicability to resolving the heterogeneous spatial-temporal organization of biological membrane systems.     

Bayesian Total Internal Reflection Fluorescence Correlation Spectroscopy Reveals hIAPP-Induced Plasma Membrane Domain Organization in Live Cells

Amyloid fibril deposition of human islet amyloid polypeptide (hIAPP) in pancreatic islet cells is implicated in the pathogenesis of type II diabetes. A growing number of studies suggest that small peptide aggregates are cytotoxic via their interaction with the plasma membrane, which leads to membrane permeabilization or disruption. A recent study using imaging total internal reflection-fluorescence correlation spectroscopy (ITIR-FCS) showed that monomeric hIAPP induced the formation of cellular plasma membrane microdomains containing dense lipids, in addition to the modulation of membrane fluidity. However, the spatial organization of microdomains and their temporal evolution were only partially characterized due to limitations in the conventional analysis and interpretation of imaging FCS datasets. Here, we apply a previously developed Bayesian analysis procedure to ITIR-FCS data to resolve hIAPP-induced microdomain spatial organization and temporal dynamics. Our analysis enables the visualization of the temporal evolution of multiple diffusing species in the spatially heterogeneous cell membrane, lending support to the carpet model for the association mode of hIAPP aggregates with the plasma membrane. The presented Bayesian analysis procedure provides an automated and general approach to unbiased model-based interpretation of imaging FCS data, with broad applicability to resolving the heterogeneous spatial-temporal organization of biological membrane systems.     

Goodness of Fit of Probability Distributions for Sightings as Species Approach Extinction

Estimating the probability that a species is extinct and the timing of extinctions is useful in biological fields ranging from paleoecology to conservation biology. Various statistical methods have been introduced to infer the time of extinction and extinction probability from a series of individual sightings. There is little evidence, however, as to which of these models provide adequate fit to actual sighting records. We use L-moment diagrams and probability plot correlation coefficient (PPCC) hypothesis tests to evaluate the goodness of fit of various probabilistic models to sighting data collected for a set of North American and Hawaiian bird populations that have either gone extinct, or are suspected of having gone extinct, during the past 150 years. For our data, the uniform, truncated exponential, and generalized Pareto models performed moderately well, but the Weibull model performed poorly. Of the acceptable models, the uniform distribution performed best based on PPCC goodness of fit comparisons and sequential Bonferroni-type tests. Further analyses using field significance tests suggest that although the uniform distribution is the best of those considered, additional work remains to evaluate the truncated exponential model more fully. The methods we present here provide a framework for evaluating subsequent models.     

Analysis of simulated and experimental fluorescence recovery after photobleaching. Data for two diffusing components.

Fluorescence recovery after photobleaching has been a popular technique to quantify the lateral mobility of membrane components. A variety of analysis methods have been used to determine the lateral diffusional mobility, D. However, many of these methods suffer from the drawbacks that they are not able to discern two-component diffusion (i.e., three-point fit), cannot solve for two components (linearization procedures), and do not perform well at low signal-to-noise. To overcome these limitations, we have adopted the approach of fitting fluorescence recovery after photobleaching curves by the full series solution using a Marquardt algorithm. Using simulated data of one or two diffusing components, determinations of the accuracy and reliability of the method with regard to extraction of diffusion parameters and the differentiation of one- versus two-component recovery curves were made under a variety of conditions comparable with those found in actual experimental situations. The performance of the method was also examined in experiments on artificial liposomes and fibroblast membranes labeled with fluorescent lipid and/or protein components. Our results indicate that: 1) the method was capable of extracting one- and two-component D values over a large range of conditions; 2) the D of a one-component recovery can be measured to within 10% with a small signal (100 prebleach photon counts per channel); 3) a two-component recovery requires more than 100-fold greater signal level than a one-component recovery for the same error; and 4) for two-component fits, multiple recovery curves may be needed to provide adequate signal to achieve the desired level of confidence in the fitted parameters and in the differentiation of one- and two-component diffusion.     

An evaluation of fitting methods for the sum of two hyperbolas: application to uptake studies

Analysis of data in terms of the sum of two rectangular hyperbolas is frequently required in solute uptake studies. Four methods for such analysis have been compared. Three are based on least-squares fitting whereas the fourth (partition method I) is an extension of a single hyperbola fitting procedure based on non-parametric statistics. The four methods were tested using data sets which had been generated with two primary types of random, normal error in the dependent variable: one of constant error variance and the other of constant coefficient of variation. The methods were tested on further data sets which were obtained by incorporating single 10% bias errors at different positions in the original two sets. Partition method I consistently gave good estimates for the four parameters defining the double hyperbola and was highly insensitive to the bias errors. The least-squares procedures performed well under conditions satisfying the least-squares assumptions regarding error distribution, but frequently gave poor estimates when these assumptions did not hold. Our conclusion is that in view of the errors inherent in many solute uptake experiments it would usually be preferable to analyse data by a method such as partition method I rather than to rely on a least-squares procedure.     

Analysis of a combined F1/F2 diallel cross in wheat

Analysis of a conventional diallel cross offers only perfect-fit estimates of the genetic components of variation, but no test for the goodness of fit of the model based on these estimates. When F₂ progenies are available, however, combining F₁ and F₂ diallels in a single experiment overcomes these problems. Least-squares estimates of these components can be calculated, errors attached to them and the goodness of fit of the resultant model tested. This analysis was applied to data on the severity of yellow rust infection in an F₁/F₂ half-diallel cross among eight bread wheat lines adapted to the East African highlands. After removing two interacting arrays, genetic analysis indicated that an additive/dominance model of gene action satisfactorily explained the variation observed among the remaining six parents and their progenies, in both the individual F₁ and F₂ diallels and the combined F₁/F₂ diallel. Resistance to yellow rust was dominant to susceptibility and genes for increased resistance were more frequent. Received: 25 May 2000 / Accepted: 31 July 2000     

Theory of ligand binding to heterogeneous receptor populations: Characterization of the free-energy distribution function

We present a method for characterizing the free-energy and affinity distributions of a heterogeneous population of molecules interacting with a homogeneous population of ligands, by driving expressions for the moments as functions of experimental binding curve characteristics, and then constructing the distribution as an expansion over a Gaussian basis set. Although the method provides the complete distribution in principle, in practice it is restricted by experimental noise, inaccuracies in data fitting, and the severity with which the distribution deviates from a Gaussian. Limitations imposed by experimental inaccuracies and the requirement of an appropriate analytic function for data fitting were evaluated by Monte Carlo simulations of binding experiments with various degrees of error in the data. Thus a distribution was assumed, binding curves with random errors were generated, and the technique was applied in order to determine the extent to which the characteristics of the assumed distribution could be recovered. Typical inaccuracies in the first two moments fell within experimental error, whereas inaccuracies in the third and fourth were generally larger than standard deviations in the data. The accuracy of these higher-order moments was invarient for experimental errors ranging from 2 to 10% and may thus be limited, within this range, primarily by the curve fitting procedure. The other aspect of the problem, accurate inference of the distribution, is limited in part by inaccuracies in the moments but more importantly by the extent to which the distribution deviates from a Gaussian. The extensive statistical literature on the problem of inference enables the delineation of specific criteria for estimating the efficiency of construction, as well as for deciding whether certain features of the inferred distribution, such as bimodality, are artifacts of the procedure. In spite of the limitations of the method, the results indicate that the mean and standard deviation are obtainable with greater accuracy than by a Sipsian analysis. This difference is particularly important when the distribution is narrow and width detection is beyond the sensitivity of the Sips plot. The method should be more accurate than the latter as an assay for homogeneity as well as for characterizing the moments, though equally easy to apply.