Biophysical Instrumentation and Methodology

在对来自分析超离心法的数据进行分析的领域中，在应用稳健回归（也称为L-1回归）来拟合超离心吸光度数据方面进行了进一步的改进。 这具有非正态分布的误差分布，正如通常假设的那样，但可以很容易地证明具有非正态偏态柯西型分布。 这些分布被称为“厚尾”分布，之所以这样称呼是因为与总分布的中心部分相比，尾部的偏差分布相对较大。 已经证明，具有这些类型的误差分布的数据通过L-1回归更好地拟合。 该方法利用残差绝对值之和的最小化，以每个点的标准误差的倒数作为其权重。 与最小二乘（L-2）回归相比，L-1回归具有对数据“离群值”异常不敏感的进一步优点。 应用L-1回归分析超吸收光谱的吸光度数据，其中标准误差是径向位置的函数，是非常快速和容易的，并产生了如此突出的上级的结果，它现在是在这个实验室的选择拟合技术。 L-1回归不提供通常（且不适当）与非线性最小二乘回归一起使用的参数误差估计的平均值。 本文提出了一种平衡自举法的算法，它是参数无关的，用于估计由L-1回归得到的拟合参数的标准误差。 这种方法也产生上级误差估计的非线性最小二乘回归。 系统误差对超离心分析得到的平衡常数的自然对数值的影响的研究表明，转子温度误差高达1摄氏度对得到的热力学参数的影响很小，温度变化后不能达到超离心平衡是热力学研究中最主要和最显著的误差来源。 目前正在改进尽量减少非平衡误差的步骤。 进一步完善这些分析的工作仍在继续。 这些方法研究的目的是获得最好的数据的温度依赖性的值的自然平衡常数的分子相互作用的研究。 这些值允许计算作为温度函数的标准吉布斯自由能变化（Δ G）的值。 使用标准热力学函数，作为温度的函数的Δ G的值可以根据焓变（Δ H）、熵变（Δ S）、热容变（Δ C-sub-P）和热容相对于温度的一阶导数的标准值来描述，所有这些都在适当选择的参考温度下。 如果参考温度是固定的（通常是温度范围的中点），那么函数相对于其他热力学参数是线性的，线性最小二乘拟合给出了参数及其标准误差的优化值，但只有当函数正交时。 对于该函数不是这种情况，并且因此，参数互相关系数和所得到的依赖性值非常大，并且该模型在数学上是非常病态的，因为任何参数值的变化都可以通过其他参数值的变化来很好地补偿。 一种方法，使这个函数几乎正交已被开发，给出参数互相关系数和依赖值是足够小的功能可以被认为是良好的行为，和良好的和有意义的值的热力学参数和它们的标准误差可以得到。 目前正在继续努力提高这一程序的效率。 进一步优化热力学分析，给出有意义的参数值，应该允许这些获得的值与分子相互作用机制的性质以及反应物及其复合物的结构和功能相关。 已经开始研究全球分析技术的应用，将来自超离心分析的数据与来自其他技术（如光散射或等温滴定量热法）的数据相结合，以便更严格地定义全球获得的热力学参数，进一步验证分析及其适用性。

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. 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. 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.