Bacteriorhodopsin photocycle kinetics analyzed by the maximum entropy method

Switch from conventional to distributed kinetics in the bacteriorhodopsin photocycle

Below 195 K, the bacteriorhodopsin photocycle could not be adequately described with exponential kinetics [Dioumaev, A. K., and Lanyi, J. K. (2007) Proc. Natl. Acad. Sci. U.S.A. 104, 9621-9626] but required distributed kinetics, previously found in hemoglobin and myoglobin at temperatures below the vitrification point of the surrounding solvent. The aim of this study is to determine which factors cause the switch from this low-temperature regime to the conventional kinetics observed at ambient temperature. The photocycle was monitored by time-resolved FTIR between 180 and 280 K, using the D96N mutant. Depending on the temperature, decay and temporal redistribution of two or three intermediates (L, M, and N) were observed. Above approximately 245 K, an abrupt change in the kinetic behavior of the photocycle takes place. It does not affect the intermediates present but greatly accelerates their decay. Below approximately 240 K, a kinetic pattern with partial decay that cannot be explained by conventional kinetics, but suggesting distributed kinetics, was dominant, while above approximately 250 K, there were no significant deviations from exponential behavior. The approximately 245 K critical point is >/=10 K below the freezing point of interbilayer water, and we were unable to correlate it with any FTIR-detectable transition of the lipids. Therefore, we attribute the change from distributed to conventional kinetics to a thermodynamic phase transition in the protein. Most probably, it is related to the freezing and thawing of internal fluctuations of the protein, known as the dynamic phase transition, although in bacteriorhodopsin the latter is usually believed to take place at least 15 K below the observed critical temperature of approximately 245 K.

Bayesian maximum entropy (two-dimensional) lifetime distribution reconstruction from time-resolved spectroscopic data

Time-resolved spectroscopy is often used to monitor the relaxation processes (or reactions) of physical, chemical, and biochemical systems after some fast physical or chemical perturbation. Time-resolved spectra contain information about the relaxation kinetics, in the form of macroscopic time constants of decay and their decay associated spectra. In the present paper we show how the Bayesian maximum entropy inversion of the Laplace transform (MaxEnt-iLT) can provide a lifetime distribution without sign-restrictions (or two-dimensional (2D)-lifetime distribution), representing the most probable inference given the data. From the reconstructed (2D) lifetime distribution it is possible to obtain the number of exponentials decays, macroscopic rate constants, and exponential amplitudes (or their decay associated spectra) present in the data. More importantly, the obtained (2D) lifetime distribution is obtained free from pre-conditioned ideas about the number of exponential decays present in the data. In contrast to the standard regularized maximum entropy method, the Bayesian MaxEnt approach automatically estimates the regularization parameter, providing an unsupervised and more objective analysis. We also show that the regularization parameter can be automatically determined by the L-curve and generalized cross-validation methods, providing (2D) lifetime reconstructions relatively close to the Bayesian best inference. Finally, we propose the use of MaxEnt-iLT for a more objective discrimination between data-supported and data-unsupported quantitative kinetic models, which takes both the data and the analysis limitations into account. All these aspects are illustrated with realistic time-resolved Fourier transform infrared (FT-IR) synthetic spectra of the bacteriorhodopsin photocycle.

Practical aspects of the maximum entropy inversion of the laplace transform for the quantitative analysis of multi-exponential data

The number, position, area, and width of the bands in a lifetime distribution give the number of exponentials present in time-resolved data and their time constants, amplitudes, and heterogeneities. The maximum entropy inversion of the Laplace transform (MaxEnt-iLT) provides a lifetime distribution from time-resolved data, which is very helpful in the analysis of the relaxation of complex systems. In some applications both positive and negative values for the lifetime distribution amplitudes are physical, but most studies to date have focused on positive-constrained solutions. In this work, we first discuss optimal conditions to obtain a sign-unrestricted maximum entropy lifetime distribution, i.e., the selection of the entropy function and the regularization value. For the selection of the regularization value we compared four methods: the chi2 criterion and Bayesian inference (already used in sign-restricted MaxEnt-iLT), and the L-curve and the generalized cross-validation methods (not yet used in MaxEnt-iLT to our knowledge). Except for the frequently used chi2 criterion, these methods recommended similar regularization values, providing close to optimum solutions. However, even when an optimal entropy function and regularization value are used, a MaxEnt lifetime distribution will contain noise-induced errors, as well as systematic distortions induced by the entropy maximization (regularization-induced errors). We introduce the concept of the apparent resolution function in MaxEnt, which allows both the noise and regularization-induced errors to be estimated. We show the capability of this newly introduced concept in both synthetic and experimental time-resolved Fourier transform infrared (FT-IR) data from the bacteriorhodopsin photocycle.

Transformation of time-resolved spectra to lifetime-resolved spectra by maximum entropy inversion of the laplace transform

We present a method for the analysis of time-resolved spectroscopic data following first-order kinetics. The time traces at all the available spectroscopic channels (e.g., wavelength or wavenumber) are inverse Laplace transformed. The transformation is stabilized by the maximum entropy method generalized for solutions without sign-restriction. In this way, time-resolved spectra can be converted to lifetime-resolved spectra, where bands appear at coordinates corresponding to their spectroscopic maxima and time constant of appearance (negative amplitude) or disappearance (positive amplitude). From the lifetime-resolved spectra, the number of exponentially decaying components, their time constants, and their decay-associated spectra are readily available. Moreover, since bands are spread in two dimensions extra band-resolution is possible. We named this method of transforming time-resolved spectra into lifetime-resolved spectra multi-spectroscopic channel maximum entropy inversion of the Laplace transform (M-MaxEnt-iLT). The basis of M-MaxEnt-iLT is presented in detail and its properties and limitations are thoroughly discussed. We also show how the combination of M-MaxEnt-iLT with spectral smoothing or deconvolution can improve the appearance and/or band resolution of the obtained lifetime-resolved spectra.