Plaza del Pino I M, Parody-Morreale A, Sanchez-Ruiz J M
Departmento de Quimica-Fisica (Facultad de Ciencias, Universidad de Granada, Spain.
Anal Biochem. 1997 Jan 15;244(2):239-55. doi: 10.1006/abio.1996.9873.
We show that numerical inversion of the Laplace transform by using the maximum entropy method can be successfully applied to the analysis of complex kinetic processes involving chemical and folding-unfolding changes in proteins. First, we present analyses of simulated data which support that: (i) the maximum entropy calculation of rate distributions, combined with Monte Carlo analyses of the associated uncertainties, yields results consistent with the information actually supplied by the data, thus preventing their over-interpretation; (ii) maximum entropy analysis may be used to extract discrete rates (corresponding to individual exponential contributions), as well as broad rate distributions (provided, of course, that the adequate information is supplied by the data). We further illustrate the applicability of the maximum entropy analysis with experimental data corresponding to two nontrivial model processes: (a) the kinetics of chemical modification of sulfhydryl groups in glycogen synthase by reaction with Ellman's reagent; (b) the kinetics of folding of ribonuclease a under strongly folding conditions, as monitored by fluorescence and optical absorption. Finally, we discuss that the maximum entropy approach should be particularly useful in studies on protein folding kinetics, which generally involve the comparison between several complex kinetic profiles obtained by using different physical probes. Thus, protein folding kinetics is usually interpreted in terms of kinetic mechanisms involving a comparatively small number of kinetic steps between well-defined protein states. According to this picture, rate distributions derived from experimental kinetic profiles by maximum entropy analysis are expected to show a small number of comparatively narrow peaks, from which we can determine, without a priori assumptions, the number of exponential contributions required to describe each experimental kinetic profile (the number of peaks), together with their amplitudes (from the peak areas), time constant values (from the peak positions), and associated Monte Carlo uncertainties. On the other hand, recent theoretical studies describe protein folding kinetics in terms of the protein energy landscape (the multidimensional surface of energy versus conformational degrees of freedom), emphasize the difficulty in defining a single reaction coordinate for folding, and point out that individual chains may fold by multiple pathways. This indicates that, in some cases at least, folding kinetics might have to be described in terms of broad rate distributions (rather than in terms of a small number of discrete exponential contributions related to kinetic steps between well-defined protein states). We suggest that the maximum entropy procedures described in this work may provide a method to detect this situation and to derive such broad rate distributions from experimental data.
我们表明，通过使用最大熵方法对拉普拉斯变换进行数值反演能够成功应用于分析涉及蛋白质化学变化以及折叠-去折叠变化的复杂动力学过程。首先，我们对模拟数据进行分析，这些分析支持以下观点：（i）速率分布的最大熵计算，结合对相关不确定性的蒙特卡罗分析，得出的结果与数据实际提供的信息一致，从而避免对数据的过度解读；（ii）最大熵分析可用于提取离散速率（对应于各个指数贡献）以及宽泛的速率分布（当然，前提是数据提供了足够的信息）。我们进一步用对应于两个非平凡模型过程的实验数据来说明最大熵分析的适用性：（a）糖原合酶中巯基与埃尔曼试剂反应的化学修饰动力学；（b）在强折叠条件下核糖核酸酶a的折叠动力学，通过荧光和光吸收进行监测。最后，我们讨论最大熵方法在蛋白质折叠动力学研究中应特别有用，蛋白质折叠动力学通常涉及使用不同物理探针获得的几个复杂动力学曲线之间的比较。因此，蛋白质折叠动力学通常根据涉及在明确界定的蛋白质状态之间相对较少数量动力学步骤的动力学机制来解释。根据这种情况，通过最大熵分析从实验动力学曲线得出的速率分布预计会显示出少量相对较窄的峰，由此我们可以在没有先验假设的情况下确定描述每个实验动力学曲线所需的指数贡献数量（峰的数量），以及它们的幅度（从峰面积）、时间常数（从峰位置）和相关的蒙特卡罗不确定性。另一方面，最近的理论研究根据蛋白质能量景观（能量与构象自由度的多维表面）来描述蛋白质折叠动力学，强调定义单一折叠反应坐标的困难，并指出单个链可能通过多种途径折叠。这表明，至少在某些情况下，折叠动力学可能必须根据宽泛的速率分布来描述（而不是根据与明确界定的蛋白质状态之间动力学步骤相关的少量离散指数贡献）。我们认为，本文所述的最大熵程序可能提供一种方法来检测这种情况，并从实验数据中得出这种宽泛的速率分布。
