Spectral Diffusion in Proteins:  A Simple Phenomenological Model

Abstract

Previous experimental studies on a modified cytochrome c have shown that optical hole widths have a powerlaw dependence on waiting time. We show that a phenomenological model, which assumes Gaussian random frequency fluctuations whose two-point time-correlation function is a stretched exponential, is consistent with the experimental data.Optical spectroscopy has proven to be a useful technique for probing the dynamics of proteins at low temperatures. 1 In this paper, we are concerned with recent optical hole burning experiments on protoporphyrin IX-substituted cytochrome c in dimethylformamide/glycerol glass, performed by Fritsch et al. 2 at 4 K. In these experiments, the optical transition of the chromophore (protoporphyrin IX) is inhomogeneously broadened and a narrow-band laser is used to selectively excite only a small fraction of the chromophores (those on resonance with the laser). Some of the excited molecules undergo a photochemical reaction, and the photoproduct absorbs light in a spectral region different from the chromophore. Therefore, when the chromophore's absorption line shape is subsequently scanned, a dip or "hole" appears at the frequency of the burning laser.In a waiting time experiment, the width of the hole is measured as a function of the waiting time, t w , between burning and scanning. Typically, the hole width increases with t w as a result of "spectral diffusion". That is, the transition frequency of an individual chromophore is not static in time, but rather fluctuates due to changes in the chromophore's local environment. As the ensemble of chromophores evolves in time, this leads to a broadening of the hole. In particular, this experiment measures the convolution of the initial hole shape with the waiting-time-dependent "spectral diffusion kernel". The latter is the conditional probability density that a chromophore has a transition frequency ν at time t w given that it had transition frequency ν 0 at time 0.In the simplest scenario, both the initial hole and the (spectral diffusion) kernel are Lorentzian functions of frequency, in which case one can simply subtract the initial hole width from the hole width at t w to obtain the width of the kernel. And there is at least one physical model that leads to a Lorentzian kernel: when the chromophore is interacting in a dipolar manner with a collection of point defects 3 whose internal states change with time. A particular realization of this picture, believed to be appropriate for chromophores in glasses, takes the point defects to be two-level systems (TLS) whose energy asymmetries and tunneling matrix elements are widely distributed. [4][5][6][7] In the usual TLS model, it is known that the width of the kernel increases logarithmically with t w , and indeed, this t w dependence has been seen for many different chromophore/glass systems. 6 The protein experiments, when analyzed by assuming a Lorentzian kernel, show a power-law dependence of the width of the kernel on t w , with an exponent of about 1/2, 2 rather than the logarithmic depen...

Protein dynamics at low temperatures

“…24 The present experiments are well described by ␣/2 ϭ0.27Ϯ0.02. An important question concerns the final width 0 which the spectral diffusion width would eventually reach after an infinitely long waiting time.…”

“…Another, quite different approach, is the diffusion model. 9,23,24 In the diffusion model it is assumed that conformational dynamics is governed by diffusion processes. In this case, the appropriate equation for the time evolution of the conditional probability in conformation space is a Fokker-Planck-type equation.…”

Solvent effects on conformational dynamics of proteins: Cytochrome c in a dried trehalose film

“…The respective coupling parameter scales with the inhomogeneous width, which is much larger in trehalose than in a glycerol/water glass. 5,[10][11][12] Trehalose, a disaccharid, is a very interesting solvent for biological materials. [13][14][15][16] For instance, some organisms can survive in the dry for quite a long time if embedded in a trehalose-rich environment.…”

Direct observation of tiers in the energy landscape of a chromoprotein: A single-molecule study

“…As a consequence, the pigments react to conformational changes of the protein with changes of their electronic energies, making them suited to monitor the dynamics of a protein with optical spectroscopy. Valuable information about the dynamics of proteins and the related time scales has been obtained by persistent spectral hole-burning spectroscopy (7,(12)(13)(14).Here, we report the observation of CSs in single-molecule experiments on light-harvesting complex 2 (LH2) complexes from Rhodospirillum molischianum. The salient feature of this technique is that it allows us to elucidate information that is commonly washed out by ensemble averaging in bulk measurements, by allowing direct observation of the dynamics of molecular processes without the need for synchronization of events.…”

“…As a consequence, the pigments react to conformational changes of the protein with changes of their electronic energies, making them suited to monitor the dynamics of a protein with optical spectroscopy. Valuable information about the dynamics of proteins and the related time scales has been obtained by persistent spectral hole-burning spectroscopy (7,(12)(13)(14).…”