Towards understanding the fast isomerization of rhodopsin
In biochemistry one encounters small light-harvesting molecules or ``chromophores'' that trigger a (not yet fully determined) sequence of steps after photo-excitation. A specific example of such a chromophore is the relatively small conjugated molecule 11-cis-retinal that has a carbon backbone of five (C-C=C)-units, and which is bound inside the protein opsin to form the light-sensitive rhodopsin. Rhodopsin is present in membranes of the rod cells of vertebrate retina, thereby enabling perhaps the most important sense: vision. Photo-excitation of this chromophore leads to an intermediate state (which is called bathorhodopsin) on an extremely short time scale, of the order of two hundred femtoseconds. On this time scale the chromophore undergoes a cis-to-trans type of isomerization. In an idealized picture, the isomerization essentially amounts to a twist of the molecule around the 11-12 bond. All other processes, which eventually lead to the triggering of a nerve signal, happen on much longer time scales. Taken together, these observations indicate that the fast isomerization is a consequence of quantum-mechanically coherent dynamics of the molecule. The challenge is to understand this dynamics.
With Daniel Aalberts, who was a postdoc at the Instituut-Lorentz in 1995 and 1996, and
Ferry Vos (then a graduate student)
we have started to develop a tight binding type
of model aimed at understanding this fast isomerization of
rhodopsin. Our model is essentially an extension of the
the Su-Schrieffer-Heeger (SSH) Hamiltonian
which has proven to be a
successful theoretical framework for understanding conjugated polymer
chains like polyacetylene. In this tight-binding model one
focuses on the coupling between the 
-electrons that constitute
the valence band, and the ionic motions along the one-dimensional
polymeric chain. As is well known, this model exhibits a rich variety
of nonlinear phenomena and topological excitations (solitons) coupling
the two possible and equivalent configurations of bond-length
alternation in the Peierls distorted ground state, and has been able
to explain a large number of fundamental issues concerning the physics
of quasi one-dimensional conducting polymers.
As a first step towards developing a model in the same spirit for studying the fast isomerization dynamics of molecules like rhodopsin, we have started to build up analytical and numerical experience with the SSH model. As a by-product of our exploratory work in this direction, we then found a physically appealing and exact way to calculate the renormalization of the speed of sound due to the electron-phonon interaction.
While the original SSH model is a one-dimensional model, in our extension the three-dimensional conjugated motion can be analyzed: it naturally involves twist and bending effects. We have calculated vibrational modes of the model, and comparison of these with typical Ramann frequencies of conjugated systems like polyacetylene, enables us to fix the parameters in our Hamiltonian. Moreover, preliminary tests on simple molecules like cyclotetraoctaene indicated that we have to include electron correlation effects, modelled, e.g., by a Hubbard-U term, in order to properly address the fast cis-trans isomerization of rhodopsin.
Very recently, this line of research has started to pay off: by including the Hubburd U correlation term, we arrived at a very appealing physical picture of how the fast isomerization occurs: in the cis state, the torsional stiffness of a molecule like rhodopsin results from the balance between the stabilizing electronic degrees of freedom and the destabilizing steric effects. When the molecule is electronically excited, the contribution to the torsional stiffness from the electronic degrees of freedom is so much reduced that the balance is tipped in favor of the destabilizing steric forces. In other words, the molecule is torsionally unstable! This simple scenario can even be demonstrated analytically for etylene, and our preliminary results indicate that it also applies to rhodopsin.
References
D. P. Aalberts, F. L. J. Vos, and W. van Saarloos, Towards Understanding Ultra-Fast Dynamics of Rhodopsin,
Pure and Appl. Chem. 69, 2099-2104 (1997).
F. L. J. Vos, D. P. Aalberts and W. van Saarloos, A simple method for calculating the speed of sound in tight-binding models: application to the SSH model, Phys. Rev. B 53 R5986-R5989 (1996).
F. L. J. Vos, D. P. Aalberts and W. van Saarloos, The Su-Schrieffer-Heeger model applied to chains of finite length, Phys. Rev. B 53, 14922
(1996).
D. P. Aalberts, M. S. L. du Croo de Jongh, B. F. Gerke and W. van
Saarloos, Towards Understanding Quantum Coherent Dynamics of
Molecules: A Simple Scenario for Ultrafast Photoisomerization,
submitted to
Phys. Rev. Lett., June 1999.
General reference on the SSH model
A. J. Heeger, S. Kivelson, J. R. Schrieffer and W.-P. Su, Solitons in Conducting Polymers, Rev. Mod. Phys. 60 781 (1988).
July 13, 1999
[Correlated systems] [Wim van Saarloos] [Instituut-Lorentz]