400 500 600 700
wavelength [nm]
0 20 40 60 80
relative absorption
BR HR SRII retinal
Figure 1.11. Absorption spectra of archaeal rhodopsins. Experimental absorption spectra of BR, HR and SRII [87, 101, 102]. The black line shows the absorption spectrum of the protonated retinal Schiff base in methanol solution [103].
S0-S1-transition energy, or preferably with the S1-state, thereby stabilizing the S1-state and lowering the S0-S1-transition energy. BR, HR and SRII lower the S0-S1-transition en-ergy. The absorption maximum of their retinal chromophore is, therefore, red-shifted to light of lower energy compared to the absorption maximum of 440 nm of the protonated retinal Schiff base in polar solution. The opsin-shift of BR and HR is, however, much higher than that of SRII.
1.5 O UTLINE OF THE T HESIS
This thesis aims at advancing the understanding of rhodopsin proteins by theoretical calculations. Since high-resolution X-ray structures suitable for theoretical studies are available, the archaeal rhodopsins, BR, HR and SRII, have been chosen as model systems.
These proteins are representatives of both ion pumps and photoreceptors and aspects of both functional mechanisms,i.e., ion transport and light sensing, are analyzed.
All calculations performed in this work are based on continuum electrostatics given by the Poisson-Boltzmann equation. Chapter 2 introduces the theoretical framework and the numerical methods used in this thesis.
Chapter 3 concentrates on the high-resolution X-ray structures of the archaeal rhodop-sins. The preparation of these structures for the performed electrostatic calculations are described and the system parameters used,e.g., atomic partial charges, are given.
An analysis of the absorption shift of archaeal rhodopsins is presented in Chapter 4.
First, the electrostatic potential that BR, HR and SRII cause at the retinal is compared.
Thereafter, the potential is decomposed to assign the observed differences to individual
protein residues. A generalized model of a quantum mechanical particle in a box is used to qualitatively describe the absorption maxima in dependence of the electrostatic potential.
In Chapter 5, the probability of functionally relevant protonation states of multiple BR structures is presented. Furthermore, the protonation behavior and the correlation of the key residues of proton transfer is analyzed.
Chapter 6 introduces a novel algorithm, termed extended dead-end elimination (X-DEE), which is effective in generating gap-free lists of lowest energy states. X-DEE is applicable to various systems. In this work, X-DEE is implemented for protonation state calcula-tions. A first application is presented in Chapter 7, where gap-free lists of protonation states of BR are analyzed.
A conclusion with respect to the presented results is given at the end of the Chapters 4 to 7. In Chapter 8, a general conclusion of this work is drawn. An outlook is given with respect to further investigations and new applications of the methods used in this thesis.
C HAPTER 2
C ONTINUUM E LECTROSTATICS
Truth is much too complicated to allow anything but approximations.
John von Neumann
Electrostatic interactions play a central role in the stabilization and function of biomole-cules. They are involved in virtually all biological processes such as molecular recogni-tion, ion transport and enzyme catalysis [53, 113–116]. Electrostatics also have a major impact in absorption processes of light. Polar and charged groups constitute a main part of the building blocks of biological macromolecules. In addition, they are often complexed with ions,e.g., Mg2+ ions stabilize the DNA and RNA macro-ions and Ca2+ ions are in-volved in signaling cascades. An understanding of electrostatic interactions is, thus, of major interest in structural biology.
In this thesis a theoretical approach is used to analyze the electrostatics of archaeal rhodopsins. Mathematical models and computer calculations can complement exper-imental research when direct measurements are either impractical or impossible. To successfully describe the system under investigation a suitable model has to be formu-lated based on physical laws and transformu-lated into computer language. In the next section the steps involved in building a computer model will briefly be discussed.
The main part of the chapter presents the conceptual, mathematical and computa-tional models employed in this work. Continuum electrostatics based on the Poisson-Boltzmann equation is used to model the protein-membrane-solvent system of the ar-chaeal rhodopsins. Here, the underlying theory and the numerical solution of the Poisson-Boltzmann equation are outlined. Subsequently, the calculation of protonation equilibria from electrostatic potentials is discussed. Last, the Metropolis Monte Carlo algorithm is explained.
2.1 B UILDING C OMPUTER M ODELS
Exact mathematical descriptions of processes in Nature seldom exist. Instead, the em-phasis in formulating models lies on developing useful approximations that allow to gain
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REAL SYSTEM
CONCEPTUAL MODEL
MATHEMATICAL MODEL
MODEL SYSTEM
COMPUTER MODEL
formal theory
numerical solution (discretization)
calculation, simulation experiment
ana lyti cal sol utio n
Figure 2.1. Computer models. Theory as well as experimental research utilizes mod-els to explain observed phenomena with scientific laws. This diagram shows the main steps towards a computer model of a given system. Computational calculations and simulations give insight into the system under investigation.
insight into the described process. As Samuel Karlin stated,‘the purpose of models is not to fit the data but to sharpen the questions’.
The diagram in Figure 2.1 shows the principal steps of building computational models.
The first and crucial challenge lies in the formulation of a conceptual model that trans-lates the real system into the idealized model system. For this purpose, the problem should be clearly posed, i.e., the first step is to ask ‘What is the Question?’ The model should describe the phenomena under investigation, reproduce already obtained knowl-edge,e.g., experimental data, and allow further predictions for the system under obser-vation. The conceptual model should be as complicated as necessary, but also as simple as possible.
The conceptual model provides a qualitative understanding of the real system. To obtain a quantitative picture, formal physical theory is applied to translate the conceptual into a mathematical model. The same principles as for the formulation of the conceptual model,e.g., available data should be reproduced and it should be possible to predict the outcome of new experiments, apply also to the mathematical model. The level of theory should be chosen with respect to the size of the system, the time-span to be observed and the phenomenon to be analyzed.
An analytical solution of the mathematical model is generally feasible only for few and simple cases. More complex applications require a discretization of the mathematical model leading to the computer model. The resulting calculated data is verified with available experimental data whenever possible and analyzed with respect to phenomena that are as yet and may never be accessible to experimental research.