Complex homogeneous and heterogeneous fluorescence anisotropy decays: enhancing analysis accuracy

In biological macromolecules, fluorophores often exhibit multiple depolarizing motions that require multiple lifetimes and rotational relaxation times to define fluorescence intensity and anisotropy decays. The related analysis of time-correlated single-photon counting data becomes uncertain due to the multitude of decay parameters and numerical sensitivity to deconvolution of the instrument response function (IRF) via discretization of integrals. By using simulations we show that improved discretizations based on quadratic and cubic local approximations of the IRF yield more accurate estimation of short rotational relaxation times and lifetimes than the commonly used Grinvald-Steinberg discretization, which in turn appears more reliable than two discretizations based on linear local approximations of the IRF. In addition, our simulation suggests that cubic approximation is the most advantageous in discriminating complex heterogeneous and homogeneous anisotropy decay. We show that among three different information criteria, the Akaike information criterion is best suited for detection of heterogeneity in rotational relaxation times. It is capable of detecting heterogeneity even when anisotropy decay appears homogeneous within statistical errors of estimation.     

Analysis of Fluorescence Decay by the Nonlinear Least Squares Method

Fluorescence decay deconvolution analysis to fit a multiexponential function by the nonlinear least squares method requires numerical calculation of a convolution integral. A linear approximation of the successive data of the instrument response function is proposed for the computation of the convolution integral. Deconvolution analysis of simulated fluorescence data were carried out to show that the linear approximation method is generally better when one of the lifetimes is comparable to the time interval between data.     

Rotational dynamics of curved DNA fragments studied by fluorescence polarization anisotropy

The rotational dynamics of short DNA fragments with or without intrinsic curvature were studied using time-resolved phase fluorimetry of intercalated ethidium with detection of the anisotropy. Parameters determined were the spinning diffusion coefficient of the DNA fragments about the long axis and the zero-time ethidium fluorescence anisotropy. We find a significant decrease in the spinning diffusion coefficient for all curved fragments compared to the straight controls. This decrease is likewise evident in rotational diffusion coefficients computed from DNA structures obtained by a curvature prediction program for these sequences. Using a hinged-cylinder model, we can identify the change in rotational diffusion coefficient with a permanent bend of 13-16 degrees per helix turn for the sequences studied. Moreover, for some of the curved fragments an increased flexibility has to be assumed in addition to the permanent bend in order to explain the data.     

Application of method of moments analysis to fluorescence decay lifetime distributions

In fluorescence decay work, distributions of exponential decay lifetimes are anticipated where complex systems are examined. We describe here methods of gaining information on such distributions using the method of moments analysis approach. The information obtained may be as simple as the average and deviation of the lifetime distribution, quantities which we show may be estimated directly from the results of a multiexponential analysis. An approximation to the actual distribution shape may also be obtained using a procedure we call the variable filter analysis (VFA) method without making any assumptions about the shape of the distribution. Tests of VFA using both simulated and experimental data are described. Limitations of this method and of distribution analysis methods in general are discussed. Results of analyses on experimental decays for ethidium intercalated in core particles and in free DNA are reported.     

五种火棘属植物的叶面积回归分析

以采集于贵州、云南、广西、湖南等地的火棘、密花火棘、全缘火棘、细圆齿火棘和窄叶火棘共5种火棘属植物26 401个成熟叶样为材料,利用WinFOLIA软件测量叶的多项形态指标并与叶面积进行11种模拟方程回归分析。结果表明:五种火棘属植物的叶面积(LA)与叶长×叶宽(LW)相关性最高,幂函数方程、三次方程、二次方程和线性方程能较好拟合其关系,且均以幂函数方程的解释程度最高(R²均大于0.970),5个物种的幂函数方程分别为LA=0.743(LW)0.936、LA=0.748(LW)0.936、LA=0.742(LW)0.955、LA=0.732(LW)0.952、LA=0.766(LW)0.954。这说明基于叶长×叶宽的叶面积幂函数方程能很好地来模拟五种火棘属植物的叶面积。     

Analysis of excited-state processes by phase-modulation fluorescence spectroscopy

Fluorescence phase shift and demodulation methods were used in the analysis of excited-state reactions and to investigate solvent relaxation around fluorophores in viscous solvents. The chosen samples illustrate the results expected for fluorophores bound to biological macromolecules. These moderately simple samples served to test the theoretical predictions described in the preceding paper (J.R. Lakowicz and A.B. Balter, Biophys. Chem. 16 (1982) 99.) and to illustrate the characteristic features of phase-modulation data expected from samples which display time-dependent spectral shifts. The excited-stale protonation of acridine and exciplex formation between anthracene and diethylaniline provided examples of one-step reactions in which the lifetimes of the initially excited and the reacted species were independent of emission wavelength. Using these samples we demonstrated the following: (I) Wavelength-dependent phase shift and demodulation values can be used to prove the occurrence of an excited-state process. Proof is obtained by observation of phase angles (φ) larger than 90° or by measurement of ratios of m/cos φ > 1, where m is the modulation of the emission relative to that of the excitation. (2) For a two-state process the individual emission spectra of each state can be calculated from the wavelength-dependent phase and modulation data. (3) The phase difference or demodulation factor between the initially excited and the reacted states reveals directly the fluorescence lifetime of the product of the reaction. (4) Phase-sensitive detection of fluorescence can be used to prove that the lifetimes of both the initially excited and the reacted states are independent of emission wavelength. (5) If steady-state spectra of the individual species are known, then phase-sensitive emission spectra can be used to measure the lifetimes of the individual components irrespective of the extent of spectral overtap. (6) Spectral regions of constant lifetime can be identified by the ratios of phase-sensitive emission spectra. In addition, we examined 6-propionyl-2-dimethylaminonaphthalene(PRODAN) and N-acetyl-l-tryptophanamide (NATA) in viscous solvents where the solvent relaxation times were comparable to the fluorescence lifetimes. Using PRODAN in n-butanol we used m/cos φ measurements, relative to the blue edge of the emission, to demonstrate that solvent relaxation requires more than a single step. For NATA in propylene glycol we used phase-sensitive detection of fluorescence to directly record the emission spectra of the initially excited and the solvent relaxed states. These measurements can be easily extended to fluorophores which are bound to proteins and membranes and are likely to be useful in studies of the dynamic properties of biopolymers.     

Multiplexed fluorescence detection in microfabricated devices with both time-resolved and spectral-discrimination capabilities using near-infrared fluorescence

We examined the feasibility of using a two-color time-resolved detection scheme with microdevices for DNA sequencing applications. A home-built dual-color optical-fiber-based time-resolved near-infrared (IR) fluorescence microscope successfully coupled lifetime discrimination with color discrimination, increasing fluorescence multiplexing capabilities. The instrument was constructed by using two pulsed-diode lasers (680/780-nm excitation) and two avalanche photodiodes as the basic building blocks. The data were processed using electronics configured in a time-correlated single-photon counting format. The use of near-IR fluorescence detection greatly simplified the hardware and allowed low detection limits (< 0.1nM). We examined the separation of a single-base tract on a microchip and compared the performance with that of conventional capillary gel electrophoresis. The microchip was fabricated in glass and contained an effective separation length of 7.0 cm. It was found that, without incorporating a solid-phase reversible immobilization cleanup procedure, the calculated lifetime of the dye label on the microchip was longer and the standard deviation was larger than those of the same sample analyzed using capillary electrophoresis. Using cleanup steps, the accuracy and precision of the measurements improved. Lifetimes of four near-IR dyes (AlexaFluor680, IRD700, IRD800, and IRD40) used in this study were determined to be 986 ps (RSD=2.1%), 1551 ps (RSD=1.8%), 520 ps (RSD=3.3%), and 788 ps (RSD=4.9%), respectively, in a microchannel filled with poly(dimethylacrylamide) (POP-6) gel. The lifetimes calculated using maximum likelihood estimators provided favorable precision on the microchip, where small numbers of photocounts were collected. An M13mp18 template was sequenced on the microchip using a two-color two-lifetime format with POP-6 as the sieving polymer. Read lengths of 294 bp with calling accuracies of 90.8 and 83.7% were achieved in each color channel. The relatively low calling accuracy and the short read length resulted primarily from the short separation channel, which yielded low electrophoretic resolution.     

Modeling of crossbred cattle growth, comparison between cubic and piecewise random regression models

Two analyses, cubic and piecewise random regression, were conducted to model growth of crossbred cattle from birth to about two years of age, investigating the ability of a piecewise procedure to fit growth traits without the complications of the cubic model. During a four-year period (1994-1997) of the Australian "Southern Crossbreeding Project", mature Hereford cows (N = 581) were mated to 97 sires of Angus, Belgian Blue, Hereford, Jersey, Limousin, South Devon, and Wagyu breeds, resulting in 1141 steers and heifers born over four years. Data included 13 (for steers) and eight (for heifers) live body weight measurements, made approximately every 50 days from birth until slaughter. The mixed model included fixed effects of sex, sire breed, age (linear, quadratic and cubic), and their interactions between sex and sire breed with age. Random effects were sire, dam, management (birth location, year, post-weaning groups), and permanent environmental effects and for each of these when possible, their interactions with linear, quadratic and cubic growth. In both models, body weights of all breeds increased over pre-weaning period, held fairly steady (slightly flattening) over the dry season then increased again towards the end of the feedlot period. The number of estimated parameters for the cubic model was 22 while for the piecewise model it was 32. It was concluded that the piecewise model was very similar to the cubic model in the fit to the data; with the piecewise model being marginally better. The piecewise model seems to fit the data better at the end of the growth period.     

Mechanism of the efficient tryptophan fluorescence quenching in human gammaD-crystallin studied by time-resolved fluorescence

Human gammaD-crystallin (HgammaD-Crys) is a two-domain, beta-sheet eye lens protein found in the lens nucleus. Its long-term solubility and stability are important to maintain lens transparency throughout life. HgammaD-Crys has four highly conserved buried tryptophans (Trps), with two in each of the homologous beta-sheet domains. In situ, these Trps will be absorbing ambient UV radiation that reaches the lens. The dispersal of the excited-state energy to avoid covalent damage is likely to be physiologically relevant for the lens crystallins. Trp fluorescence is efficiently quenched in native HgammaD-Crys. Previous steady-state fluorescence measurements provide strong evidence for energy transfer from Trp42 to Trp68 in the N-terminal domain and from Trp130 to Trp156 in the C-terminal domain [Chen, J., et al. (2006) Biochemistry 45, 11552-11563]. Hybrid quantum mechanical-molecular mechanical (QM-MM) simulations indicated that the fluorescence of Trp68 and Trp156 is quenched by fast electron transfer to the amide backbone. Here we report additional information obtained using time-resolved fluorescence spectroscopy. In the single-Trp-containing proteins (Trp42-only, Trp68-only, Trp130-only, and Trp156-only), the highly quenched Trp68 and Trp156 have very short lifetimes, tau approximately 0.1 ns, whereas the moderately fluorescent Trp42 and Trp130 have longer lifetimes, tau approximately 3 ns. In the presence of the energy acceptor (Trp68 or Trp156), the lifetime of the energy donor (Trp42 or Trp130) decreased from approximately 3 to approximately 1 ns. The intradomain energy transfer efficiency is 56% in the N-terminal domain and is 71% in the C-terminal domain. The experimental values of energy transfer efficiency are in good agreement with those calculated theoretically. The absence of a time-dependent red shift in the time-resolved emission spectra of Trp130 proves that its local environment is very rigid. Time-resolved fluorescence anisotropy measurements with the single-Trp-containing proteins, Trp42-only and Trp130-only, indicate that the protein rotates as a rigid body and no segmental motion is detected. A combination of energy transfer with electron transfer results in short excited-state lifetimes of all Trps, which, together with the high rigidity of the protein matrix around Trps, could protect HgammaD-Crys from excited-state reactions causing permanent covalent damage.     

A note on power approximations for the transmission/disequilibrium test

The transmission/disequilibrium test (TDT) is a popular method for detection of the genetic basis of a disease. Investigators planning such studies require computation of sample size and power, allowing for a general genetic model. Here, a rigorous method is presented for obtaining the power approximations of the TDT for samples consisting of families with either a single affected child or affected sib pairs. Power calculations based on simulation show that these approximations are quite precise. By this method, it is also shown that a previously published power approximation of the TDT is erroneous.     

Comparing approximations to spatio-temporal models for epidemics with local spread

Analytical methods for predicting and exploring the dynamics of stochastic, spatially interacting populations have proven to have useful application in epidemiology and ecology. An important development has been the increasing interest in spatially explicit models, which require more advanced analytical techniques than the usual mean-field or mass-action approaches. The general principle is the derivation of differential equations describing the evolution of the expected population size and other statistics. As a result of spatial interactions no closed set of equations is obtained. Nevertheless, approximate solutions are possible using closure relations for truncation. Here we review and report recent progress on closure approximations applicable to lattice models with nearest-neighbour interactions, including cluster approximations and elaborations on the pair (or pairwise) approximation. This study is made in the context of an SIS model for plant-disease epidemics introduced in Filipe and Gibson (1998, Studying and approximating spatio-temporal models for epidemic spread and control, Phil. Trans. R. Soc. Lond. B 353, 2153–2162) of which the contact process [Harris, T. E. (1974), Contact interactions on a lattice, Ann. Prob. 2, 969] is a special case. The various methods of approximation are derived and explained and their predictions are compared and tested against simulation. The merits and limitations of the various approximations are discussed. A hybrid pairwise approximation is shown to provide the best predictions of transient and long-term, stationary behaviour over the whole parameter range of the model.     

A navigational guide to variable fitness: common methods of analysis,where they break down,and what you can do instead

Our methods for analyzing stochastic fitness are mostly approximations, and the assumptions behind these approximations are not always well understood. Furthermore, many of these approximations break down when fitness variance is high. This review covers geometric mean growth, diffusion approximations, and Markov processes. It discusses where each is appropriate, the conditions under which they break down, and their advantages and disadvantages, with special attention to the case of high fitness variance. A model of sessile and site-attached coastal species is used as a running example, and fully worked calculations and code are provided. Summary: The logarithm of geometric mean growth is usually only appropriate when (a) an invader growth rate is needed and (b) fitness variability is driven by environmental fluctuations. The usual approximation breaks down when fitness variance is high. Diffusion approximations can provide a reasonable guide to the expected change in frequency over a time step if expected fitnesses and fitness variances are appropriately scaled by the average expected fitness. Diffusion approximations can perform less well for fixation probabilities, especially since further approximations may be required. Fixation probabilities can be calculated exactly using a Markov process, regardless of how large fitness variance is, although an analytic expression is frequently not possible. If an analytic expression is desired, it may be worth using a diffusion approximation and checking it with a Markov process calculation.     

Testing environmental and genetic effects in the presence of spatial autocorrelation

Spatial autocorrelation is a well‐recognized concern for observational data in general, and more specifically for spatial data in ecology. Generalized linear mixed models (GLMMs) with spatially autocorrelated random effects are a potential general framework for handling these spatial correlations. However, as the result of statistical and practical issues, such GLMMs have been fitted through the undocumented use of procedures based on penalized quasi‐likelihood approximations (PQL), and under restrictive models of spatial correlation. Alternatively, they are often neglected in favor of simpler but more questionable approaches. In this work we aim to provide practical and validated means of inference under spatial GLMMs, that overcome these limitations. For this purpose, a new software is developed to fit spatial GLMMs. We use it to assess the performance of likelihood ratio tests for fixed effects under spatial autocorrelation, based on Laplace or PQL approximations of the likelihood. Expectedly, the Laplace approximation performs generally slightly better, although a variant of PQL was better in the binary case. We show that a previous implementation of PQL methods in the R language, glmmPQL, is not appropriate for such applications. Finally, we illustrate the efficiency of a bootstrap procedure for correcting the small sample bias of the tests, which applies also to non‐spatial models.     

Efficient approximations for learning phylogenetic HMM models from data

MOTIVATION: We consider models useful for learning an evolutionary or phylogenetic tree from data consisting of DNA sequences corresponding to the leaves of the tree. In particular, we consider a general probabilistic model described in Siepel and Haussler that we call the phylogenetic-HMM model which generalizes the classical probabilistic models of Neyman and Felsenstein. Unfortunately, computing the likelihood of phylogenetic-HMM models is intractable. We consider several approximations for computing the likelihood of such models including an approximation introduced in Siepel and Haussler, loopy belief propagation and several variational methods. RESULTS: We demonstrate that, unlike the other approximations, variational methods are accurate and are guaranteed to lower bound the likelihood. In addition, we identify a particular variational approximation to be best-one in which the posterior distribution is variationally approximated using the classic Neyman-Felsenstein model. The application of our best approximation to data from the cystic fibrosis transmembrane conductance regulator gene region across nine eutherian mammals reveals a CpG effect.     

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.     

An Alternative to Moment Closure

Moment closure methods are widely used to analyze mathematical models. They are specifically geared toward derivation of approximations of moments of stochastic models, and of similar quantities in other models. The methods possess several weaknesses: Conditions for validity of the approximations are not known, magnitudes of approximation errors are not easily evaluated, spurious solutions can be generated that require large efforts to eliminate, and expressions for the approximations are in many cases too complex to be useful. We describe an alternative method that provides improvements in these regards. The new method leads to asymptotic approximations of the first few cumulants that are explicit in the model’s parameters. We analyze the univariate stochastic logistic Verhulst model and a bivariate stochastic epidemic SIR model with the new method. Errors that were made in early applications of moment closure to the Verhulst model are explained and corrected.     

On imitative behavior

It is shown that the linear equations of the formal theory of mass behavior proposed by the author in 1939 are first-order approximations of the biophysical theory proposed later. The second approximation contains cubic terms. It permits the expression of the time course of the change of behavior in terms of a quadrature.     

Time-resolved fluorescence of apoferritin and its subunits

The decay of the intrinsic fluorescence of the apoferritin polymer and its subunits has been studied by pulse and phase shift techniques. Both techniques show that the fluorescence decay of all the samples tested cannot be described by a single exponential function. The fluorescence decay data of the apoferritin subunits obtained with either technique can be fitted satisfactorily with a function resulting from the sum of two exponential components. However, the polymer data obtained with the high resolution phase shift technique operated either by synchrotron radiation or by a mode-locked argon ion laser can be fitted better using a bimodal gaussian continuous distribution of lifetime components. The molecular basis for this distribution of lifetime values may lie in the heterogeneity of the tryptophan environment generated by the assembly of the subunits into the polymer. The binding of the first 100 irons to apoferritin quenches the intrinsic fluorescence without affecting the lifetimes in a proportional way. This finding may be taken as an indication that the quenching of the tryptophan fluorescence induced by the binding of iron has both static and dynamic components.     

Numerical method for equilibrium fluorescence polarization

A computer program for making first approximations of equation constants, and for fitting the equilibrium fluorescence polarization equation to data by successively improving the constant values is described. The techniques used in making the first approximations represent a substantial improvement over methods previously available.     

Maximum likelihood method for the analysis of time-resolved fluorescence decay curves

The usefulness of fluorescence techniques for the study of macromolecular structure and dynamics depends on the accuracy and sensitivity of the methods used for data analysis. Many methods for data analysis have been proposed and used, but little attention has been paid to the maximum likelihood method, generally known as the most powerful statistical method for parameter estimation. In this paper we study the properties and behavior of maximum likelihood estimates by using simulated fluorescence intensity decay data. We show that the maximum likelihood method provides generally more accurate estimates of lifetimes and fractions than does the standard least-squares approach especially when the lifetime ratios between individual components are small. Three novelties to the field of fluorescence decay analysis are also introduced and studied in this paper: a) discretization of the convolution integral based on the generalized integral mean value theorem: b) the likelihood ratio test as a tool to determine the number of exponential decay components in a given decay profile; and c) separability and detectability indices which provide measures on how accurately, a particular decay component can be detected. Based on the experience gained from this and from our previous study of the Padé-Laplace method, we make some recommendations on how the complex problem of deconvolution and parameter estimation of multiexponential functions might be approached in an experimental setting. Offprint requests to: F. G. Prendergast