Stochastic approach to data analysis in fluorescence correlation spectroscopy

Fluorescence correlation spectroscopy (FCS) has emerged as a powerful technique for measuring low concentrations of fluorescent molecules and their diffusion constants. In FCS, the experimental data is conventionally fit using standard local search techniques, for example, the Marquardt-Levenberg (ML) algorithm. A prerequisite for these categories of algorithms is the sound knowledge of the behavior of fit parameters and in most cases good initial guesses for accurate fitting, otherwise leading to fitting artifacts. For known fit models and with user experience about the behavior of fit parameters, these local search algorithms work extremely well. However, for heterogeneous systems or where automated data analysis is a prerequisite, there is a need to apply a procedure, which treats FCS data fitting as a black box and generates reliable fit parameters with accuracy for the chosen model in hand. We present a computational approach to analyze FCS data by means of a stochastic algorithm for global search called PGSL, an acronym for Probabilistic Global Search Lausanne. This algorithm does not require any initial guesses and does the fitting in terms of searching for solutions by global sampling. It is flexible as well as computationally faster at the same time for multiparameter evaluations. We present the performance study of PGSL for two-component with triplet fits. The statistical study and the goodness of fit criterion for PGSL are also presented. The robustness of PGSL on noisy experimental data for parameter estimation is also verified. We further extend the scope of PGSL by a hybrid analysis wherein the output of PGSL is fed as initial guesses to ML. Reliability studies show that PGSL and the hybrid combination of both perform better than ML for various thresholds of the mean-squared error (MSE).     

Separating structural heterogeneities from stochastic variations in fluorescence resonance energy transfer distributions via photon distribution analysis

We establish a probability distribution analysis (PDA) method for the analysis of fluorescence resonance energy transfer (FRET) signals to determine with high precision the originating value of a shot-noise-limited signal distribution. PDA theoretical distributions are calculated explicitly including crosstalk, stochastic variations, and background and represent the minimum width that a FRET distribution must have. In this way an unambiguous distinction is made between shot-noise distributions and distributions broadened by heterogeneities. This method simultaneously and effectively extracts highly resolved information from FRET distributions. The theoretical histograms match the exact profile of histograms generated from constant transfer efficiency experimental data with a chi2 near unity. The chi2 surface suggests an ultimate level of precision with FRET of < 1% of the F?rster radius. Distributions of FRET signals in donor-acceptor-labeled DNA were unambiguously identified as being broader than shot-noise variations could explain. A model describing a Gaussian distribution of distances was tested with the PDA method and demonstrated 5 A inhomogeneities due to dye motions. The capability of this method to recover quantitative information from FRET distributions has potential applications for studying molecular conformations and dynamics. Potential sources for artifacts such as acceptor photobleaching, spectrally different observation volumes, and fluctuations of the F?rster radius are ruled out.     

Particle size analysis of polymer latexes by light scattering. V. The significance of an assumed distribution in the analysis of angular scattering data

In the analysis of light scattering data from polymer latex systems or other systems of spherical particles, it is necessary to assume a particle size distribution function. Theoretical angular scattering functions based on the assumed distribution and representing a wide range of size distribution parameters are compared to experimental data in order to obtain a best fit. In previous work, it has been shown that as the polydispersity increases beyond certain limits the uncertainty in the assignment of the size distribution parameters (i.e., the best fit) increases. This report is concerned with the analysis of angular scattering from unimodal systems and simulated cases where theoretical scattering functions for wide, negatively skewed distributions are used as “experimental data,” are analyzed by utilizing four different distribution functions. These functions represent different degrees of skewness and include negatively, positively, and normally skewed distributions. The results from the use of the various distribution functions are discussed with respect to the uncertainly in the assignment of distribution parameters resulting from the loss of structure in the angular scattering pattern due to increased polydispersity. Scattering data from the bimodal distribution are analyzed by assuming a unimodel distribution, and the consequences of this assumption are assessed.     

Probability-based protein identification by searching sequence databases using mass spectrometry data

Several algorithms have been described in the literature for protein identification by searching a sequence database using mass spectrometry data. In some approaches, the experimental data are peptide molecular weights from the digestion of a protein by an enzyme. Other approaches use tandem mass spectrometry (MS/MS) data from one or more peptides. Still others combine mass data with amino acid sequence data. We present results from a new computer program, Mascot, which integrates all three types of search. The scoring algorithm is probability based, which has a number of advantages: (i) A simple rule can be used to judge whether a result is significant or not. This is particularly useful in guarding against false positives. (ii) Scores can be compared with those from other types of search, such as sequence homology. (iii) Search parameters can be readily optimised by iteration. The strengths and limitations of probability-based scoring are discussed, particularly in the context of high throughput, fully automated protein identification.     

A data-integrated method for analyzing stochastic biochemical networks

Variability and fluctuations among genetically identical cells under uniform experimental conditions stem from the stochastic nature of biochemical reactions. Understanding network function for endogenous biological systems or designing robust synthetic genetic circuits requires accounting for and analyzing this variability. Stochasticity in biological networks is usually represented using a continuous-time discrete-state Markov formalism, where the chemical master equation (CME) and its kinetic Monte Carlo equivalent, the stochastic simulation algorithm (SSA), are used. These two representations are computationally intractable for many realistic biological problems. Fitting parameters in the context of these stochastic models is particularly challenging and has not been accomplished for any but very simple systems. In this work, we propose that moment equations derived from the CME, when treated appropriately in terms of higher order moment contributions, represent a computationally efficient framework for estimating the kinetic rate constants of stochastic network models and subsequent analysis of their dynamics. To do so, we present a practical data-derived moment closure method for these equations. In contrast to previous work, this method does not rely on any assumptions about the shape of the stochastic distributions or a functional relationship among their moments. We use this method to analyze a stochastic model of a biological oscillator and demonstrate its accuracy through excellent agreement with CME/SSA calculations. By coupling this moment-closure method with a parameter search procedure, we further demonstrate how a model's kinetic parameters can be iteratively determined in order to fit measured distribution data.     

Simple expressions for diffusion coefficient determination of adsorption within spherical and cylindrical absorbents using direct simulation method

Various analytical expressions for solute adsorption kinetics within porous absorbents of defined geometry (planar sheet, cylinder, and sphere) are available in the literature. However, these expressions are limited for practical numerical evaluation because they are based on infinite series. An investigation of these expressions has been carried out and then accurate but simple expressions derived that enable rapid determination of effective diffusion coefficients for adsorption within geometrically categorical absorbents. These involve directly fitting calculated kinetic adsorption curves to experimental ones. A simple one point method is also proposed to estimate the effective diffusion coefficient for an adsorption process within these simple geometrical absorbents as an initial value for a best fit.     

A parallel tabu search for conformational energy optimization of oligopeptides

We have developed and implemented a tabu search heuristic (TS) to determine the best energy minimum for oligopeptides. Our test molecule was Met‐enkephalin, a pentapetide that over the years has been used as a validation model for many global optimizers. The test potential energy function was ECEPP/3. Our tabu search implementation is based on assigning integer values to the variables to be optimized, and in facilitating the diversification and intensification of the search. The final output from the TS is treated with a local optimizer, and our best result competes both in quality and CPU time with those reported in the literature. The results indicate that TS is an efficient algorithm for conformational searches. We present a parallel TS version along with experimental results that show that this algorithm allows significant increases in speed. © 2000 John Wiley & Sons, Inc. J Comput Chem 21: 147–156, 2000     

Global dynamic optimization for parameter estimation in chemical kinetics

We present the first method guaranteed to find the best possible least-squares (chi2) fit of experimental data by a nonlinear kinetic model. Several important advantages of knowing with certainty the best possible fit rather than a locally optimum fit are discussed and demonstrated using data from the recent literature. This is particularly important when the model and the data appear to be inconsistent. With the new method, one can rigorously demonstrate that a nonlinear kinetic model with several adjustable rate parameters is inconsistent with measured experimental data. The numerical method presented is a valuable tool in evaluating the validity of a complex kinetics model.     

A Method to Explore Protein Side Chain Conformational Variability Using Experimental Data

Experimentally measured values of molecular properties or observables of biomolecules such as proteins are generally averages over time and space, which do not contain su?cient information to determine the underlying conformational distribution of the molecules in solution. The relationship between experimentally measured NMR ³J‐coupling values and the corresponding dihedral angle values is a particularly complicated case due to its nonlinear, multiple‐valued nature. Molecular dynamics (MD) simulations at constant temperature can generate Boltzmann ensembles of molecular structures that are free from a priori assumptions about the nature of the underlying conformational distribution. They suffer, however, from limited sampling with respect to time and conformational space. Moreover, the quality of the obtained structures is dependent on the choice of force ?eld and solvation model. A recently proposed method that uses time‐averaging with local‐elevation (LE) biasing of the conformational search provides an elegant means of overcoming these three problems. Using a set of side chain ³J‐coupling values for the FK506 binding protein (FKBP), we ?rst investigate the uncertainty in the angle values predicted theoretically. We then propose a simple MD‐based technique to detect inconsistencies within an experimental data set and identify degrees of freedom for which conformational averaging takes place or for which force ?eld parameters may be de?cient. Finally, we show that LE MD is the best method for producing ensembles of structures that, on average, ?t the experimental data.     

Finite-state and reduced-parameter representations of protein backbone conformations

This article studies the representation of protein backbone conformations using a finite number of values for the backbone dihedral angles. We develop a combinatorial search algorithm that guarantees finding the global minima of functions over the configuration space of discrete protein conformations, and use this procedure to fit finite-state models to the backbones of globular proteins. It is demonstrated that a finite-state representation with a reasonably small number of states yields either a small root-mean-square error or a small dihedral angle deviation from the native structure, but not both at the same time. The problem can be resolved by introducing limited local optimization in each step of the combinatorial search. In addition, it is shown that acceptable approximation is achieved using a single dihedral angle as an independent variable in local optimization. Results for 11 proteins demonstrate the advantages and shortcomings of both the finite-state and reduced-parameter approximations of protein backbone conformations. © 1994 by John Wiley & Sons, Inc.     

Optimization of the separation of chlorophenols with stepwise gradient elution in reversed phase liquid chromatography

An optimization technique based on gradient elution was used to separate eleven chlorophenols by reversed phase liquid chromatography. The separation was based on gradient elution with a stepwise variation pattern of the volume fraction of organic modifier, phi, in the mobile phase. Initially, two-, three-, and four-parameter equations which describe the dependence of ln k' upon phi, were examined for their ability to fit the experimental data. It was found that, among these equations, the four-parameter equation gave the best fit of the experimental data. In addition to separation optimization, a non-linear least squares program with a grid search for initial estimates was used to determine the best variation pattern. The best variation pattern was obtained with phi(1)= 0.27, phi(2)= 0.39, phi(3)= 0.62, t(1) = 33 min, and t(2) = 11 min. This pattern allowed the chromatographic separation of the chlorophenols with a good resolution and a total analysis time of 51 min. Good agreement was observed between predicted and experimental values of the retention times under optimal condition.     

Interatomic contributions to high-energy electron-molecule scattering

Within the independent atom scattering model, we derive an approximate formula for the rotationally and vibrationally averaged three-atom terms in the series expansion of the electron scattering cross section. This formula uses the atomic scattering factors as well as the atomic scattered wavefunctions as input, and rotational averaging is performed numerically. We compare our results to previous theoretical multiple scattering approaches for the molecules F(3), F(4), and SF(6) and to experimental data for TeF(6). Our results are consistent with those of previous calculations and inclusion of the three-atom term produces a dramatically better least squares fit of the TeF(6) data. The algorithm presented here is sufficiently fast and simple to be incorporated easily into existing electron diffraction codes.     

Population-based models of thermoplastic degradation: Using optimization to determine model parameters

A comparison of different model forms for non-oxidative thermoplastic degradation rate is presented. A physically meaningful model is derived from a three-step radical depolymerization mechanism. A moment-based approximation to the population balance equations is derived and implemented to predict the thermal degradation of polyethylene (PE) and poly(methyl methacrylate) (PMMA). In the absence of accurate molecular dynamical calculations to predict the controlling model parameters, it is necessary to use experimental data to find the best fit parameter values. A sequential quadratic programming algorithm (SQP) was used to find the set of parameters that minimized the error between the model and sets of non-isothermal thermogravimetric (TG) data at different heating rates for PE and PMMA.     

Tethered particle motion as a diagnostic of DNA tether length

The tethered particle motion (TPM) technique involves an analysis of the Brownian motion of a bead tethered to a slide by a single DNA molecule. We describe an improved experimental protocol with which to form the tethers, an algorithm for analyzing bead motion visualized using differential interference contrast microscopy, and a physical model with which we have successfully simulated such DNA tethers. Both experiment and theory show that the statistics of the bead motion are quite different from those of a free semiflexible polymer. Our experimental data for chain extension versus tether length fit our model over a range of tether lengths from 109 to 3477 base pairs, using a value for the DNA persistence length that is consistent with those obtained under similar solution conditions by other methods. Moreover, we present the first experimental determination of the full probability distribution function of bead displacements and find excellent agreement with our theoretical prediction. Our results show that TPM is a useful tool for monitoring large conformational changes such as DNA looping.     

Flexible protein‐protein docking based on Best‐First search algorithm

We developed a new high resolution protein‐protein docking method based on Best‐First search algorithm that loosely imitates protein‐protein associations. The method operates in two stages: first, we perform a rigid search on the unbound proteins. Second, we search alternately on rigid and flexible degrees of freedom starting from multiple configurations from the rigid search. Both stages use heuristics added to the energy function, which causes the proteins to rapidly approach each other and remain adjacent, while optimizing on the energy. The method deals with backbone flexibility explicitly by searching over ensembles of conformations generated before docking. We ran the rigid docking stage on 66 complexes and grouped the results into four classes according to evaluation criteria used in Critical Assessment of Predicted Interactions (CAPRI; “high,” “medium,” “acceptable,” and “incorrect”). Our method found medium binding conformations for 26% of the complexes and acceptable for additional 44% among the top 10 configurations. Considering all the configurations, we found medium binding conformations for 55% of the complexes and acceptable for additional 39% of the complexes. Introducing side‐chains flexibility in the second stage improves the best found binding conformation but harms the ranking. However, introducing side‐chains and backbone flexibility improve both the best found binding conformation and the best found conformation in the top 10. Our approach is a basis for incorporating multiple flexible motions into protein‐protein docking and is of interest even with the current use of a simple energy function. © 2010 Wiley Periodicals, Inc. J Comput Chem, 2010     

ANALYSIS OF FLASH PHOTOLYSIS DATA BY A GLOBAL FIT WITH MULTI-EXPONENTIALS–I. DETERMINATION OF THE MINIMAL NUMBER OF INTERMEDIATES IN THE PHOTOCYCLE OF BACTERIORHODOPSIN BY THE 'STABILITY CRITERION'

A major problem in establishing a kinetic scheme for photo-induced reaction sequences is the determination of the unknown number of important intermediates. It is shown that in addition to the χ²-value and the distribution of the residuals obtained from a least-squares fit of the experimental curves by a sum of exponentials, the stability of the apparent rates towards an increase of the number of exponentials with arbitrary test values can be used as an additional criterion. Experimental data describing the photocycle of bacteriorhodopsin after the appearance of intermediate K ( t > 1 μs) are analysed with the described algorithm. It is found that 5 exponentials are necessary and sufficient for the fit.     

Estimating the size of laterally phase separated cholesterol domains in model membranes with Förster resonance energy transfer: a simulation study

In this work, we use two vertically-coupled square two-dimensional lattices to simulate membrane bilayers containing a uniform size distribution of cholesterol immiscible domains of a predetermined size distribution. We substitute cholesterols and phospholipids with their fluorescent analogs and calculate the efficiency of energy transfer as a function of acceptor concentration for four membrane configurations. The simulated efficiency of energy transfer as a function of acceptor concentration data is then fit with an analytical FRET model to estimate the domain size, in the same manner in which experimental FRET data is analyzed. The fitted model parameters (domain size and donor partition coefficient) are compared to the simulation inputs to test the applicability of the FRET model to estimating the size of laterally phase separated cholesterol domains. We show that the FRET model yields good size estimates for domains that range between 1 and 25 nm. We also find that the assumed fluorophore configuration in the FRET model leads to a constant under-prediction of these values. Finally, we demonstrate that when two parameters are open to the fit, the FRET model adequately predicts the donor partition coefficient in addition to the domain size.     

Shot-noise limit for optical polarimeters

The methods of Jones's calculus are used to derive expressions for the signal-to-noise ratio in polarimetric measurements. The formulae show that in shot-noise limited systems there is an optimum angle between the polarization planes of the polarizer and analyser and that, once this angle is employed, the signal-to-noise ratio of the system cannot be appreciably improved by improving the quality (i.e, lowering the extinction ratio) of the polarizers used. The main way to improve the signal-to-noise ratio is to use an intense light source, large optical aperture and a long integration time.