Biophysical Instrumentation and Methodology

Biomedical Imaging & Bioengineering

Investigators

Abstract

In the areas of the analysis of data from analytical ultracentrifugation, further refinements have been made in the application of robust regression (also known as L-1 regression) for the fitting of ultracentrifugal light absorbance data. Such has an error distribution that is not normally distributed, as is commonly assumed, but can be readily demonstrated to possess a logarithmically-skewed Cauchy-type distribution. These are characterized as "fat-tailed" distributions, so called because of the relatively large distribution of deviations in the tails when compared to those in the central portion of the total distribution. It has been demonstrated that data with these types of error distribution are better fit by the L-1 regression. This method utilizes minimization of the sum of the absolute values of the residuals with the reciprocal of the standard error of each point as its weight. L-1 regression has the further virtue of being singularly insensitive to data "outliers," when compared to least-squares (L-2) regression. The application of L-1 regression to absorbance data from the analytical ultracentrifuge, where the standard error is a function of radial position, is singularly rapid and easy and has yielded such outstandingly superior results that it is now the fitting technique of choice in this laboratory. L-1 regression does not provide the means of parameter error estimation normally (and inappropriately) used with non-linear least-squares regression. An algorithm for performing a balanced bootstrap procedure which is parameter independent for the estimation of the standard errors of the fitting parameters obtained by L-1 regression has been developed. This method also yields superior error estimates for non-linear least-squares regression. Investigation of the effects of systematic errors on the values of the natural logarithm of the equilibrium constant obtained by ultracentrifugal analysis has demonstrated that rotor temperature errors as large as one degree Celsius have very little effect on the obtained thermodynamic parameters and that failure to attain ultracentrifugal equilibrium following a temperature change was the dominant and the most marked source of error in thermodynamic studies. Steps to minimize non-equilibrium error are in the process of being refined. Work on further refinements in these analyses is continuing.[unreadable] [unreadable] The objective of these methodological studies has been to obtain the best possible data for the temperature dependence of the values of the natural logarithms of the equilibrium constants for the molecular interactions being studied. These values permit calculation of the values of the standard Gibbs free energy changes (delta G) as a function of temperature. Using standard thermodynamic functions, the value of delta G as a function of temperature can be described in terms of the standard values of the changes of enthalpy (delta H), entropy (delta S), heat capacity (delta C-sub-P), and the first derivative of the heat capacity with respect to temperature, all at an appropriately selected reference temperature. If the reference temperature is taken as fixed (usually the mid-point of the temperature range), then the function is linear with respect to the other thermodynamic parameters and linear least-squares fitting gives optimized values for the parameters and their standard errors, but only if the function is orthogonal. This is not the case for this function and, as a result, the parameter cross-correlation coefficients and the resulting dependency values are very large, and the model is mathematically very ill-conditioned, since changes in any of the parameter values can be very well compensated by changes in the other parameter values. A method for making this function almost orthogonal has been developed, giving parameter cross-correlation coefficients and dependency values that are small enough that the function may be considered to be well-behaved, and good and meaningful values for the thermodynamic parameters and their standard errors can be obtained. Work is continuing in an effort to increase the efficiency of this procedure. Further optimization of thermodynamic analysis giving meaningful parameter values should permit these obtained values to be correlated to the nature of the mechanism(s) of the molecular interaction(s) and to the structure and function of the reactants and their complexes.[unreadable] [unreadable] Studies have been initiated on the application of global techniques of analysis, combining data from ultracentrifugal analysis to be coanalyzed with data from other techniques, such as light scattering or isothermal titration calorimetry, so that the globally obtained thermodynamic parameters are more rigorously defined, further validating the analyses and their applicability.