Uncertainty analysis of correlated stability constants

Every attempt of using a computer to model reality has two main uncertainties: the conceptual uncertainty and the data uncertainty. The conceptual uncertainty deals with the choice of model selected for the simulation and the data uncertainty is about the precision and accuracy of the input data. They are often determined experimentally and may thus be encumbered by a number of uncertainties. Normally when treating uncertainties in input data these data are treated as independent variables. However, since many of these parameters are determined together they are actually correlated. This paper focuses on chemical stability constants, a most important parameter for chemical calculations based on speciation. Commonly in the literature they are at best given with an uncertainty interval. We propose to also give the covariance matrix thus giving the opportunity to really assess correlations. In addition we discuss the effect of these correlations on speciations.     

Correlations between the activities of a gamma-ray emitter calculated from different peaks in the spectrum

A model for calculating correlations between the activities of the same gamma-ray emitter calculated from different peaks in its spectrum is presented. The correlation coefficients can be expressed by the relative uncertainties of the input quantities. The use of this model in the calculation of the mean activity prevents the calculation of an excessively small uncertainty of the mean, since averaging of correlated uncertainty components is avoided.     

Uncertainty analysis of correlated parameters in automated reaction mechanism generation

Uncertainty analysis is a useful tool for inspecting and improving detailed kinetic mechanisms because it can identify the greatest sources of model output error. Owing to the very nonlinear relationship between kinetic and thermodynamic parameters and computed concentrations, model predictions can be extremely sensitive to uncertainties in some parameters while uncertainties in other parameters can be irrelevant. Error propagation becomes even more convoluted in automatically generated kinetic models, where input uncertainties are correlated through kinetic rate rules and thermodynamic group values. Local and global uncertainty analyses were implemented and used to analyze error propagation in Reaction Mechanism Generator (RMG), an open-source software for generating kinetic models. A framework for automatically assigning parameter uncertainties to estimated thermodynamics and kinetics was created, enabling tracking of correlated uncertainties. Local first-order uncertainty propagation was implemented using sensitivities computed natively within RMG. Global uncertainty analysis was implemented using adaptive Smolyak pseudospectral approximations as implemented in the MIT Uncertainty Quantification Library to efficiently compute and construct polynomial chaos expansions to approximate the dependence of outputs on a subset of uncertain inputs. Cantera was used as a backend for simulating the reactor system in the global analysis. Analyses were performed for a phenyldodecane pyrolysis model. Local and global methods demonstrated similar trends; however, many uncertainties were significantly overestimated by the local analysis. Both local and global analyses show that correlated uncertainties based on kinetic rate rules and thermochemical groups drastically reduce a model's degrees of freedom and have a large impact on the determination of the most influential input parameters. These results highlight the necessity of incorporating uncertainty analysis in the mechanism generation workflow.     

Congestion game scheduling for virtual drug screening optimization

In virtual drug screening, the chemical diversity of hits is an important factor, along with their predicted activity. Moreover, interim results are of interest for directing the further research, and their diversity is also desirable. In this paper, we consider a problem of obtaining a diverse set of virtual screening hits in a short time. To this end, we propose a mathematical model of task scheduling for virtual drug screening in high-performance computational systems as a congestion game between computational nodes to find the equilibrium solutions for best balancing the number of interim hits with their chemical diversity. The model considers the heterogeneous environment with workload uncertainty, processing time uncertainty, and limited knowledge about the input dataset structure. We perform computational experiments and evaluate the performance of the developed approach considering organic molecules database GDB-9. The used set of molecules is rich enough to demonstrate the feasibility and practicability of proposed solutions. We compare the algorithm with two known heuristics used in practice and observe that game-based scheduling outperforms them by the hit discovery rate and chemical diversity at earlier steps. Based on these results, we use a social utility metric for assessing the efficiency of our equilibrium solutions and show that they reach greatest values.     

Addressing the challenges of modeling the scattering from bottlebrush polymers in solution

Small-angle scattering measurements of complex macromolecules in solution are used to establish relationships between chemical structure and conformational properties. Interpretation of the scattering data requires an inverse approach where a model is chosen and the simulated scattering intensity from that model is iterated to match the experimental scattering intensity. This raises challenges in the case where the model is an imperfect approximation of the underlying structure, or where there are significant correlations between model parameters. We examine three bottlebrush polymers (consisting of polynorbornene backbone and polystyrene side chains) in a good solvent using a model commonly applied to this class of polymers: the flexible cylinder model. Applying a series of constrained Monte-Carlo Markov Chain analyses demonstrates the severity of the correlations between key parameters and the presence of multiple close minima in the goodness of fit space. We demonstrate that a shape-agnostic model can fit the scattering with significantly reduced parameter correlations and less potential for complex, multimodal parameter spaces. We provide recommendations to improve the analysis of complex macromolecules in solution, highlighting the value of Bayesian methods. This approach provides richer information for understanding parameter sensitivity compared to methods which produce a single, best fit.     

Application of the CPA equation of state to reservoir fluids in presence of water and polar chemicals

The complex phase equilibrium between reservoir fluids and associating compounds like water, methanol and glycols has become more and more important as the increasing global energy demand pushes the oil industry to target reservoirs with extreme or complicated conditions, such as deep or offshore reservoirs. Conventional equation of state (EoS) with classical mixing rules cannot satisfactorily predict or even correlate the phase equilibrium of those systems. A promising model for such systems is the Cubic-Plus-Association (CPA) EoS, which has been successfully applied to well-defined systems containing associating compounds. In this work, a set of correlations was proposed to calculate the CPA model parameters for the narrow cuts in ill-defined C₇₊ fractions. The correlations were then combined with either the characterization method of Pedersen et al. or that of Whitson et al. to extend CPA to reservoir fluids in presence of water and polar chemical such as methanol and monoethylene glycol. With a minimum number of adjustable parameters from binary pairs, satisfactory results have been obtained for different types of phase equilibria in reservoir fluid systems and several relevant model multicomponent systems. In addition, modeling of mutual solubility between light hydrocarbons and water is also addressed.     

Influence of environment induced correlated fluctuations in electronic coupling on coherent excitation energy transfer dynamics in model photosynthetic systems

Two-dimensional photon-echo experiments indicate that excitation energy transfer between chromophores near the reaction center of the photosynthetic purple bacterium Rhodobacter sphaeroides occurs coherently with decoherence times of hundreds of femtoseconds, comparable to the energy transfer time scale in these systems. The original explanation of this observation suggested that correlated fluctuations in chromophore excitation energies, driven by large scale protein motions could result in long lived coherent energy transfer dynamics. However, no significant site energy correlation has been found in recent molecular dynamics simulations of several model light harvesting systems. Instead, there is evidence of correlated fluctuations in site energy-electronic coupling and electronic coupling-electronic coupling. The roles of these different types of correlations in excitation energy transfer dynamics are not yet thoroughly understood, though the effects of site energy correlations have been well studied. In this paper, we introduce several general models that can realistically describe the effects of various types of correlated fluctuations in chromophore properties and systematically study the behavior of these models using general methods for treating dissipative quantum dynamics in complex multi-chromophore systems. The effects of correlation between site energy and inter-site electronic couplings are explored in a two state model of excitation energy transfer between the accessory bacteriochlorophyll and bacteriopheophytin in a reaction center system and we find that these types of correlated fluctuations can enhance or suppress coherence and transfer rate simultaneously. In contrast, models for correlated fluctuations in chromophore excitation energies show enhanced coherent dynamics but necessarily show decrease in excitation energy transfer rate accompanying such coherence enhancement. Finally, for a three state model of the Fenna-Matthews-Olsen light harvesting complex, we explore the influence of including correlations in inter-chromophore couplings between different chromophore dimers that share a common chromophore. We find that the relative sign of the different correlations can have profound influence on decoherence time and energy transfer rate and can provide sensitive control of relaxation in these complex quantum dynamical open systems.     

Bayesian uncertainty quantification of recent shock tube determinations of the rate coefficient of reaction H + O2→ OH + O

We analyze the ignition delay in hydrogen–oxygen combustion and the important chain ‐branching reaction H + O₂→ OH + O that occurs behind the shock waves in shock tube experiments. We apply a stochastic Bayesian approach to quantify uncertainties in the theoretical model and experimental data. The approach involves a statistical inverse problem, which has four “components” as input information: (a) model, (b) prior joint probability density function (PDF) of the uncertain parameters, (c) experimental data, and (d) uncertainties in the scenario parameters. The solution of this statistical inverse problem is a posterior joint PDF of the uncertain parameters from which we can easily extract statistical information. We first perform a parametric study to investigate how the level of the total uncertainty (which we define as the sum of model uncertainty and experimental uncertainty) affects the uncertainty in the rate coefficient k₁ of the reaction H + O₂→ OH + O, which is “most likely” expressed by k₁=1.73×10²³T^?2.5exp(?11550/T) cm³ mol^?1 s^?1 over the experimental temperature range 1100–1472 K. We also introduce the idea of “irreducible” uncertainty when considering other parameters in the system. After statistically calibrating the parameters modeling the rate coefficient k₁, we predict its 95% confidence interval (CI) for different temperature regimes and compare the CI against the values of k₁ obtained deterministically. Our results show that a small uncertainty in gas temperature (±5 K) introduces appreciable uncertainty in k₁. © 2012 Wiley Periodicals, Inc. Int J Chem Kinet 44: 586–597, 2012     

Measurement uncertainty distributions and uncertainty propagation by the simulation approach

A complete and accurate evaluation of measurement uncertainty requires the knowledge of the uncertainty distributions. The latter are rarely determined or verified experimentally, and hence up to now only crude estimates or assumptions based on intuition have been used. The simulation of experimental results is readily accessible and provides a more relia-ble solution to this problem. When using an appropriate model of measurement and after determina-tion of input value parameters by present state-of-the-art techniques, simulation data supply reliable information about the distribution of the output results of a complex measurement. The method permits simple variation of preposition and therefore ready analysis of various features influencing the measurement of uncertainty intervals. In the paper we described examples of such evaluations related to the preparation of certified reference materials, where there is excellent agreement between the traditional and simulation approaches. And evaluation of more complex measurements of diffusion coefficients by the open capillary method, where uncertainty of the simulated result is more realistic than the re-sult from the traditional error method due to non-linearity and probably Cauchy distribution in some steps. Received: 17 March 2000 Accepted: 25 July 2000     

On the statistics of dense,disordered assemblies: Equilibrium and dynamics

We consider implications of a lattice model which operates with a vacancy fraction h as a measure of structural disorder. Consequences for the configurational thermodynamics of one- and multicomponent systems and their phase equilibria are briefly indicated. The principal topic is the glassy state under steady state conditions, as well as the kinetics of relaxational processes toward equilibrium. The central role of the h-function in its dependence on variables of state is made evident among others by the connection between equation of state and thermo-elastic properties. Moreover, a dynamics of volume relaxation can be treated by means of a corresponding theory for the h-function. Applications of this theory to isothermal annealing below the glass temperature T_g, the response to a constant cooling rate of the melt through the transition zone, and the computation of a complex compression modulus are reviewed. The implications of vacancy cluster distributions for the analysis of positron spectroscopy are pointed out. Finally we indicate the basis for the development of correlations between T_g and structural parameters.     

The Importance of Relative Reaction Rates in the Optimization of Detailed Kinetic Models

Numerous mathematical tools intended to adjust rate constants employed in complex detailed kinetic models to make them consistent with multiple sets of experimental data have been reported in the literature. Application of such model optimization methods typically begins with the assignment of uncertainties in the absolute rate constants in a starting model, followed by variation of the rate constants within these uncertainty bounds to tune rate parameters to match model outputs to experimental observations. The present work examines the impact of including information on relative reaction rates in the optimization strategy, which is not typically done in current implementations. It is shown that where such rate constant data are available, the available parameter space changes dramatically due to the correlations inherent in such measurements. Relative rate constants are typically measured with greater relative accuracy than corresponding absolute rate constant measurements. This greater accuracy further reduces the available parameter space, which significantly affects the uncertainty in the model outcomes as a result of kinetic parameter uncertainties. We demonstrate this effect by considering a simple example case emulating an ignition event and show that use of relative rate measurements leads to a significantly smaller uncertainty in the output ignition delay time in comparison with results based on absolute measurements. This is true even though the same range of absolute rate constants is sampled in each case. Implications of the results with respect to the maintenance of physically realistic kinetics in optimized models are discussed, and suggestions are made for the path forward in the refinement of detailed kinetic models.     

Replica exchange with nonequilibrium switches: enhancing equilibrium sampling by increasing replica overlap

We describe a replica exchange strategy where trial swap configurations are generated by nonequilibrium switching simulations. By devoting simulation time to the switching simulations, one can systematically increase an effective overlap between replicas, which leads to an increased exchange acceptance rate and less correlated equilibrium samples. In this paper, we derive our method for a general class of stochastic dynamics, and discuss various strategies for enhancing replica overlap through novel dynamical schemes and prudent choices of switching protocols. We then demonstrate our method on a model system of alanine dipeptide in implicit solvent, characterizing decreases in data correlations and gains in sampling efficiency.     

Experimental studies of uncertainties associated with chromatographic techniques.

The paper describes experiments for the evaluation of uncertainties associated with a number of chromatographic parameters. Studies of the analysis of vitamins by HPLC illustrate the estimation of the uncertainties associated with experimental "input" parameters such as the detector wavelength, column temperature and mobile phase flow-rate. Experimental design techniques, which allow the efficient study a number of parameters simultaneously, are described. Multiple linear regression was used to fit response surfaces to the data. The resulting equations were used in the estimation of the uncertainties. Three approaches to uncertainty calculation were compared--Kragten's spreadsheet, symmetric spreadsheet and algebraic differentiation. In cases where non-linearity in the model was significant, agreement between the uncertainty estimates was poor as the spreadsheet approaches do not include second-order uncertainty terms.     

Stochastic ensembles of thermodynamic potentials

The provision of uncertainty estimates along with measurement results or values computed thereof is metrologically mandatory. This is in particular true for observational data related to climate change, and thermodynamic properties of geophysical substances derived thereof, such as of air, seawater or ice. The recent International Thermodynamic Equation of Seawater 2010 (TEOS-10) provides such properties in a comprehensive and highly accurate way, derived from empirical thermodynamic potentials released by the International Association for the Properties of Water and Steam (IAPWS). Currently, there are no generally recognised algorithms available for a systematic and comprehensive estimation of uncertainties for arbitrary properties derived from those potentials at arbitrary input values, based on the experimental uncertainties of the laboratory data that were used originally for the correlations during the construction process. In particular, standard formulas for the uncertainty propagation which do not account for mutual uncertainty correlations between different coefficients tend to systematically and significantly overestimate the uncertainties of derived quantities, which may lead to practically useless results. In this paper, stochastic ensembles of thermodynamic potentials, derived from randomly modified input data, are considered statistically to provide analytical formulas for the computation of the covariance matrix of the related regression coefficients, from which in turn uncertainty estimates for any derived property can be computed a posteriori. For illustration purposes, simple analytical application examples of the general formalism are briefly discussed in greater detail.     

Vibrational state-dependent predissociation dynamics of ClO (A (2)Pi(3/2)): Insight from correlated fine structure branching ratios

We have studied the v'-dependent predissociation dynamics of the ClO A (2)Pi(3/2) state using velocity-map ion-imaging. Experimental final correlated state branching ratios, i.e. Cl((2)P(J=3/2,1/2)) + O((3)P(J=2,1,0)) channels, have been measured for v' = 6-11. We find that the branching ratios are highly variable and depend strongly on v', providing a window into the v'-dependent predissociation mechanism. A comparison of the experimental results with the recent model of Lane et al. (I. C. Lane, W. H. Howie and A. J. Orr-Ewing, Phys. Chem. Chem. Phys., 1999, 1, 3087) in both the diabatic and adiabatic limits suggests that the dynamics are closer to the diabatic limit. The overall Cl((2)P(J)) branching ratios are in good agreement with the diabatic model results. There are significant differences, however, between theory and experiment at the correlated state level, demonstrating the sensitivity of correlated measurements to the role of the exit channel coupling in the predissociation dynamics. The results highlight the need for more sophisticated quantum dynamical calculations to describe the correlated fine structure branching ratios in this system.     

A neural network approach to infrared spectrum interpretation

The simple linear neural network model was investigated as a method for automated interpretation of infrared spectra. The model was trained using a database of infrared spectra of organic compounds of known structure. The model was able to learn, without any prior input of spectrum-structure correlations, to recognize and identify 76 functional groupings with accuracies ranging from fair to excellent. The effect of network input parameters and of training set composition were studied, and several sources of spurious correlations were identified and corrected.Dedicated to Professor W. Simon on the occasion of his 60th birthday     

Non-equilibrium capillary electrophoresis of equilibrium mixtures--appreciation of kinetics in capillary electrophoresis

We describe a new electrophoretic method (patent pending), Non-Equilibrium Capillary Electrophoresis of Equilibrium Mixtures (NECEEM), and demonstrate its application to the study of protein-DNA interactions. A single NECEEM experiment allows for the determination of equilibrium and kinetic parameters of protein-DNA complex formation. The equilibrium mixture is prepared by mixing protein and DNA; it contains three components: free protein, free DNA, and the protein-DNA complex. A small plug of such a mixture is injected onto a capillary and the three components are separated under non-equilibrium conditions using a run buffer that does not contain the components of the equilibrium mixture. The protein-DNA complex decays during the NECEEM separation; the resulting electropherograms contain characteristic peaks and exponential curves. A simple analysis of a single electropherogram reveals two parameters: the equilibrium dissociation constant of the protein-DNA complex and the monomolecular rate constant of complex decay. The bimolecular rate constant of complex formation can then be calculated as the ratio of the two experimentally-determined constants. NECEEM was applied to find the equilibrium and kinetic parameters of interaction between an E. coli single-stranded DNA binding protein and a fluorescently-labeled oligonucleotide. The constants determined by NECEEM are in good agreement with those obtained by other methods. The new method is simple, fast, and accurate. It can be equally applied to other non-covalent molecular complexes.     

Estimation of uncertainty in p<Emphasis Type="Italic">K</Emphasis><Subscript>a</Subscript> values determined by potentiometric titration

A procedure is presented for estimation of uncertainty in measurement of the pK(a) of a weak acid by potentiometric titration. The procedure is based on the ISO GUM. The core of the procedure is a mathematical model that involves 40 input parameters. A novel approach is used for taking into account the purity of the acid, the impurities are not treated as inert compounds only, their possible acidic dissociation is also taken into account. Application to an example of practical pK(a) determination is presented. Altogether 67 different sources of uncertainty are identified and quantified within the example. The relative importance of different uncertainty sources is discussed. The most important source of uncertainty (with the experimental set-up of the example) is the uncertainty of pH measurement followed by the accuracy of the burette and the uncertainty of weighing. The procedure gives uncertainty separately for each point of the titration curve. The uncertainty depends on the amount of titrant added, being lowest in the central part of the titration curve. The possibilities of reducing the uncertainty and interpreting the drift of the pK(a) values obtained from the same curve are discussed.     

Investigation of the effect of correlated uncertain rate parameters via the calculation of global and local sensitivity indices

Applications of global uncertainty methods for models with correlated parameters are essential to investigate chemical kinetics models. A global sensitivity analysis method is presented that is able to handle correlated parameter sets. It is based on the coupling of the Rosenblatt transformation with an optimized Random Sampling High Dimensional Model Representation method. The accuracy of the computational method was tested on a series of examples where the analytical solution was available. The capabilities of the method were also investigated by exploring the effect of the uncertainty of rate parameters of a syngas–air combustion mechanism on the calculated ignition delay times. Most of the parameters have large correlated sensitivity indices and the correlation between the parameters has a high influence on the results. It was demonstrated that the values of the calculated total correlated and final marginal sensitivity indices are independent of the order of the decorrelation steps. The final marginal sensitivity indices are meaningful for the investigation of the chemical significance of the reaction steps. The parameters belonging to five elementary reactions only, have significant final marginal sensitivity indices. Local sensitivity indices for correlated parameters were defined which are the linear equivalents of the global ones. The results of the global sensitivity analysis were compared with the corresponding results of local sensitivity analysis for the case of the syngas–air combustion system. The same set of reactions was indicated to be important by both approaches.