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Department of Chemistry, University of California at Berkeley, Berkeley, California 94720 USA
Correspondence: Address reprint requests to Richard A. Mathies, Fax: 510-642-3599; E-mail: rich{at}zinc.cchem.berkeley.edu.
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ABSTRACT |
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 TOP ABSTRACT INTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONSummaryACKNOWLEDGEMENTSREFERENCES |
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(
) is reduced by up to 5% at wavelengths to the red of 500 nm but is invariant ( = 0.65 ± 0.01) between 450 and 500 nm (Kim et al., 2001
). To understand this nonstatistical internal conversion process, these results are compared with predictions of a Landau-Zener model for dynamic curve crossing. The initial distribution of excess photon energy in the 28 Franck-Condon active vibrational modes of rhodopsin is defined by a fully thermalized sum-over-states vibronic calculation. This calculation reveals that absorption by high-frequency unreactive modes (e.g., C=C stretches) increases as the excitation wavelength is shifted from 570 to 450 nm whereas relatively less energy is deposited into reactive low-frequency modes. This result qualitatively explains the experimentally observed wavelength dependence of () for rhodopsin and reveals the importance of delocalized, torsional modes in the reactive pathway. |
INTRODUCTION |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONSummaryACKNOWLEDGEMENTSREFERENCES |
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) has inspired a wide variety of experiments to understand the mechanism of this reaction in rhodopsin (Sakmar et al., 2002; Mathies and Lugtenburg, 2000; Mathies, 1999; Kochendoerfer and Mathies, 1995). Dramatic differences in the speed and isomerization quantum yield of the protein-bound 11-cis retinal protonated Schiff base (Schoenlein et al., 1993; Dartnall, 1968) compared to solution (Kandori et al., 1995; Becker and Freedman, 1985) emphasize the unique nature of the excited-state dynamics resulting from protein-chromophore interactions. Specifically, the ultrafast product formation time of 200 fs (Schoenlein et al., 1993) and the observation of coherent vibrational oscillations in the absorption of the impulsively produced photoproduct (Wang et al., 1994) indicate that the photochemistry occurs from nonstationary vibronic states. In addition, the fact that the photoreaction occurs before statistical randomization of energy (Kim and Mathies, 2002) indicates that altering the initial preparation of the excited-state wavepacket should influence the photochemical outcome. This nonstationary vibronic state model for the photochemistry of vision thus motivates our experimental and computational studies of photoproduct formation as a function of excitation wavelength.
The classification of a reaction as statistical or nonergodic was developed primarily through studies of gas-phase unimolecular reactions. A founding theory is that of Rice-Ramsperger-Kassel-Marcus (RRKM) in which the central assumption is that the rate of energy randomization is much faster than chemical reaction (Frost, 1973). Non-RRKM behavior may arise due to a "bottleneck" in phase space (intrinsic non-RRKM) or as a result of an ultrafast reaction rate that prevents the statistical redistribution of internal energy before product formation (apparent non-RRKM) (Bunker and Hase, 1973). Although there are many reports of apparent non-RRKM behavior for gas-phase unimolecular reactions (Lee et al., 2000; Diau et al., 1998; Syage et al., 1984; Kim et al., 1996), very few examples of non-RRKM behavior have been reported for condensed-phase reactions. Given the enhanced rate of intramolecular vibrational energy redistribution (IVR) in condensed phase due to solvent-induced fluctuations of solute levels as well as direct coupling between solute and solvent modes (Bakker, 1993), it is generally thought that S1 IVR rates in liquids are much faster than internal conversion rates. One unique example is trans-stilbene in which the gas-phase reaction exhibits non-RRKM behavior whereas the condensed phase isomerization is consistent with RRKM theory (Syage et al., 1984).
Although there has been no direct comparison between the reaction rate for isomerization calculated using RRKM theory and the experimental product formation time of 200 fs, the 50-fs excited-state lifetime of rhodopsin (Kochendoerfer and Mathies, 1996) is comparable to or shorter than typical IVR times of large condensed-phase, polyatomic molecules (Elsaesser and Kaiser, 1991). Therefore, the isomerization of the 11-cis retinal chromophore in rhodopsin likely follows a nonergodic pathway. The unthermalized nature of the reaction requires a dynamic model such as that presented by Landau and Zener (Landau, 1932; Zener, 1932; Syage et al., 1984) where the reaction quantum yield depends on the velocity of nuclear motion in the crossing region (Fig. 1) as expressed by the equation 
, where 
is the wavepacket velocity that depends on excess energy, and k reflects the energy difference and shapes of the potential energy surfaces. Therefore, the quantum yield should depend on the excitation energy or . A more sophisticated, multidimensional picture of the isomerization reaction has also been developed that involves a dynamic passage of the wavepacket through a conical intersection (Garavelli et al., 1999a; Garavelli et al., 1998; Vreven et al., 1997; Ben-Nun and Martinez, 1998; Molnar et al., 2000).
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). This work revealed a wavelength-dependent reaction quantum yield between 500 and 570 nm, and a constant quantum yield of 0.65 ± 0.01 between 450 and 500 nm, as shown in Fig. 2. The value of 
= 0.65 ± 0.01 is the most accurate result to date and falls well within the errors of the previous measurements (Dartnall, 1968; Hurley et al., 1977). To more thoroughly understand these results, we present here a complete Franck-Condon analysis of the distribution of excess energy after photoexcitation based on results from resonance Raman intensity analysis (Loppnow and Mathies, 1988; Lin et al., 1998). The resulting knowledge of the wavelength-dependent mode-specific distribution of excess energy in the excited state allows us to quantitatively predict () using a one-dimensional Landau-Zener (LZ) picture. In addition to testing the extent to which the LZ picture is valid, this comparison between observed and predicted () provides us with insight regarding the origin of the wavelength dependence of the isomerization reaction quantum yield. Our results provide a deeper understanding of visual photochemistry by elucidating the contribution of reactive and unreactive degrees of freedom to the absorption, and by revealing how the initial preparation of the excited-state wavepacket affects the photochemical outcome in this nonergodic reaction.
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MATERIALS AND METHODS |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONSummaryACKNOWLEDGEMENTSREFERENCES |
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). Here, we only describe our computational method for analyzing the excited-state Franck-Condon (FC) energy distributions. The absorption spectrum was computed from sums of Franck-Condon factors (Lax, 1952) to determine the contribution of each Franck-Condon active mode to the absorption cross section at a specific excitation wavelength. In this sum-over-states model, the ground and excited state surfaces are assumed to be harmonic and differ from each other only in their equilibrium positions. A set of N normal modes can then be treated as a collection of N independent harmonic oscillators as shown in Fig. 3, and the multidimensional FC factors can be written as products of one-dimensional overlaps. The fully thermalized absorption cross section can then be written as (Myers and Mathies, 1987)
 | (1) |
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is the electronic transition length (Å), 
is the solution index of refraction, 
is a Lorentzian halfwidth at half maximum (HWHM) of the homogeneous linewidth (cm-1), 
is the average 0-0 energy (cm-1), 
is the Gaussian standard deviation for inhomogeneous site broadening (cm-1), 
is the Boltzmann factor for mode j in level i, W is a normalization factor when inhomogenous broadening is utilized (
), 
is the vibrational energy for mode j (cm-1), 
is the excited-state level and 
is the ground-state level for mode j. The Boltzmann term is given as:
 | (2) |
have been derived for the general harmonic case (Manneback, 1951), and we use the recursion relations in the case of equal ground and excited-state frequencies (Myers and Mathies, 1987).
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for a given ground vibrational level 
. Here, k may be any of the 28 normal modes of rhodopsin, 0 < k 
28. Second, the probability for this transition to for a given number of ground- and excited-state quanta in the remaining 27 modes and at a particular excitation wavelength and ground vibrational level was calculated as the complete Boltzmann-weighted, multimode FC product from Eq. 1. A weighted probability for a given transition to is then proportional to the product of the number of quanta of vibrational energy in the excited state, , with the probability for the transition:
 | (3) |
A sum of weighted probabilities for all possible values of for mode k was then determined at all values of , 
, and 
, and normalized by the sum of total weighted probabilities for all modes. The resulting normalized, weighted probability (or energy distribution) for each mode reflects the relative contribution any given mode makes to the overall absorption cross section at a particular excitation wavelength. In this manner, we quantitatively determined the nature of energy partitioning to all FC active modes upon photoexcitation as a function of incident wavelength.
Absorption spectrum calculation
The development of time-dependent (Heller et al., 1982; Lee and Heller, 1979) and time-correlator (Page, 1991; Page and Tonks, 1981) methods have made the calculation of multimode absorption spectra with thermal ground-state excitation very efficient. A full absorption spectrum for the 28-mode rhodopsin system can be completed in less than 1 min. Unfortunately, the partitioning of mode-specific excitation is hidden in these calculations, necessitating a return to the earlier method based on classic perturbation theory. Because this calculation scales as 
, the number of terms in 
for rhodopsin is truly formidable and the calculation time for 28 modes with thermal occupation is reasonable with a single processor only when some approximations are made. For example, it was necessary to utilize cutoff values in our thermalized algorithm to truncate the sum at an appropriate level. Two different cutoff values were implemented: a multimode cutoff value was compared to the calculated Boltzmann-weighted, FC multimode product 
for given values of ij and vj to determine whether the term should be included in , and a Boltzmann cutoff was used to determine the uppermost vibrational ground-state level based on the population. Inclusion of these cutoff values was necessary to make the calculation tractable; out of a possible 
1095 total terms for (50 ground and 50 excited-state levels), our algorithm calculated only the most significant 109 terms for a reasonable thermalized absorption spectrum. We found that over 90% of the true time-dependent calculation was recovered when only those terms in which the Boltzmann weighted, Franck-Condon product was >10-11 and the Boltzmann term 
was >10-6 were included, and this calculation required 95 h on a 1.4-GHz Pentium 4 processor.
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RESULTS |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONSummaryACKNOWLEDGEMENTSREFERENCES |
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0 K with only homogeneous broadening. These unthermalized spectra calculated via either the time-dependent or sum-over-states method are identical, verifying that our algorithm gives accurate results at 0 K. The middle panel (B) presents fully thermalized T = 298 K spectra from sum-over-states and time-dependent calculations in which only homogeneous broadening has been considered. The integrated area of the curve resulting from the sum-over-states calculation is within 90% of the integrated area of the time-dependent calculation. In addition, the spectral features are identical, verifying that our algorithm provides accurate results for 298-K calculations. As more terms were included in the sum-over-states calculation, the resulting spectra continued to converge to the time-dependent result with a dramatic increase in computational time. The spectrum shown in Fig. 4 B required 95 h of continuous calculation. Achieving further significant improvement of the spectrum would require an unreasonable amount of computation time and was not explored. The bottom panel (C) of Fig. 4 presents calculated spectra at T = 298 K with both homogeneous and inhomogeous broadening. Again, the spectrum resulting from a thermalized sum-over-states calculation is within 90% of the spectrum from a time-dependent calculation.
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). The sum-over-states algorithm is intrinsically Lorentzian broadened; therefore a similar comparison could not be performed without major modification of the sum-over-states equation. However, the similarity between the time-dependent and sum-over-states results with Lorentzian broadening supports the use of Eq. 1 to obtain mode-specific information about the distribution of excitation energy.
Mode-specific energy distribution
Normalized, weighted energy distributions were calculated for each mode as a function of incident wavelength for the unthermalized and fully thermalized absorption spectra in Fig. 4 and this result is presented in Fig. 5. The low-frequency modes with large 
's dominate both the unthermalized and thermalized absorption spectra at any given wavelength; the three lowest-frequency modes of 93, 131, and 246 cm-1 contribute 40–72% to the total absorption cross section between 450 and 570 nm. The ethylenic at 1548 cm-1 also contributes significantly to the absorption band, but this contribution is strongly wavelength dependent, ranging from 1 to 8% (thermalized) and 3 to 13% (unthermalized), with the largest contribution made when the excitation energy exceeds the ethylenic 0 
1 and 0 2 transitions. Other high-frequency modes contribute <2% each. The main difference between the normalized, weighted energy distributions calculated at 0 and 298 K is the contribution of low-frequency modes to the overall absorption cross section. The increase in the number of initially populated low-frequency ground-state modes in the thermalized calculation results in an increase in the total number of transitions to the excited state by these low-frequency modes.
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is reduced to two dimensions, we can calculate the wavelength dependence of the quantum yield, 
, using a simple Landau Zener model. Fig. 1 presents a two-dimensional schematic of the internal conversion process for rhodopsin in which absorption of a photon deposits energy (Exs) to the excited-state wavepacket that evolves along reactive and unreactive coordinates before curve crossing to form product. In this model, the quantum yield for isomerization is given as 
where 
depends on the energy difference and slopes of the ground and excited-state potential energy surfaces and 
is the wavepacket velocity at the crossing region. Our key hypothesis is that the wavelength dependence of the quantum yield arises because is proportional to the square root of excess energy deposited into reactive modes. We can rewrite the LZ equation as 
where 
now has an additional factor resulting from the conversion of velocity to energy. Erxt is the excess energy along the reaction coordinate, which is equal to the total excess energy (Exs) times the percentage of energy distributed to the reactive modes. The total excess energy Exs is given as
 | (4) |
We then utilized both the localized and delocalized coordinate partitioning to calculate the wavelength dependence of . A critical aspect of these calculations is the choice of E00. Raman intensity analysis suggests a value of 18,130 cm-1 (550 nm) for E00 (Lin et al., 1998). An earlier temperature-dependent bleaching study suggested an E00 value of 590 nm (St. George, 1952). In our determination of 
we choose the average of the values determined by Raman analysis and temperature-dependence studies (E00 = 17,544 cm-1). Using the FC vibrational energy of 862 cm-1, E00eff has a value of 16,682 cm-1 (600 nm) and the resulting value for Exs is then weighted by our result in Fig. 6 to yield Erxt (500 nm) = 2933 cm-1 and therefore k' = 23 cm-1/2. Values for based on this value of k' in the LZ equation are presented in Fig. 7. Relative quantum yields when the reaction coordinate consists only of localized torsional modes have a stronger wavelength dependence than experimentally observed. As more delocalized torsional modes are included in the reaction coordinate, however, the resulting ratios begin to approach the experimental data. When inhomogeneous (full width at half maximum (FWHM) 1230 cm-1) and homogeneous (FWHM 330 cm-1) broadening mechanisms are considered, the value for E00eff is reduced to 15,122 cm-1 (660 nm) and this new E00eff combined with a delocalized torsional coordinate yields improved values. By reducing E00eff to 725 nm, the calculated results are even closer to the experimental data. No further improvement in the calculated quantum yield was achieved with the inclusion of additional modes in the reactive coordinate, such as single-bond stretches and methyl rocks.
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DISCUSSION |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONSummaryACKNOWLEDGEMENTSREFERENCES |
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, while also indicating the need for more sophisticated models.
Nature of the reaction coordinate
The spontaneous excited-state emission spectrum (Kochendoerfer and Mathies, 1996) and reaction quantum yield (Fig. 2) of rhodopsin are both wavelength dependent, providing strong evidence that the cis-to-trans isomerization occurs before randomization and thermalization of vibrational energy. Recent anti-Stokes Raman data on the primary photoproduct also support this result (Kim and Mathies, 2002). Consistent with the experimental results, our sum-over-states calculation indicates that the distribution of photon energy among the FC accessible vibrational modes varies by as much as 25% as the incident wavelength is changed. Specifically, the high-frequency modes make a significantly larger contribution to the overall absorption cross section with blue excitation; in contrast, low-frequency modes become more dominant in the red region. The strong wavelength dependence of the ethylenic mode is consistent with the picture that vibrational progression in this mode dominates the breadth of the absorption spectrum at energies higher than E00 (Loppnow and Mathies, 1988; Warshel and Karplus, 1974).
We have quantified the energy distribution in the 28 Franck-Condon modes as a function of excess energy using an E00eff value of 16,682 cm-1 (600 nm). As the photon energy is increased beyond E00eff, absorption to the S1 state occurs and excess energy is primarily deposited in low-frequency modes. The relative distribution of energy in the low-frequency modes remains essentially unchanged as the photon energy increases. As the photon wavelength decreases from 570 to 530 nm, there is <1% change in probability for absorption by low-frequency modes (<500 cm-1). When the excitation wavelength is 520 nm and lower, however, a different trend emerges in which the relative probability for absorption by the low-frequency modes decreases with increasing photon energy; the relative distribution of excess energy in the low-frequency modes is decreased by 15% with 450-nm excitation relative to a 520-nm photon. This occurs because absorption by the ethylenic mode becomes more significant once the 0 1 transition becomes energetically accessible at 510 nm.
Our model using two partitions of the torsional modes helps elucidate the degrees of freedom that contribute to the photoisomerization, and explains the transition at 500 nm from a wavelength-dependent to flat quantum yield. The reactive pathway is likely dominated by torsional modes. Localized torsions that contribute to the reaction coordinate have been identified as the 970 cm-1 C11=C12 hydrogen out-of-plane (HOOP) mode (Eyring et al., 1982; Garavelli et al., 1999b), the 568 cm-1 C11=C12 torsion (Lin et al., 1998; Loppnow and Mathies, 1988; Birge et al., 1982), and the 260 cm-1 C10C13 skeletal torsion (Lin et al., 1998). However, a reaction coordinate consisting solely of these three localized torsions results in a much stronger wavelength dependence in the 450–570 nm region than observed. The strong wavelength dependence results because the excess energy in these modes depends only on the absorbed photon energy. When other delocalized low-frequency torsions are included in the calculation, these additional modes modulate the resulting distribution of excess energy along the reactive coordinate and yield a result that is more consistent with our measured wavelength dependence of as shown in Fig. 7. The fact that the distribution of energy along the reactive coordinate is flat over the range 520–570 nm leads to a decline in the calculated quantum yield over the same wavelength region due to the decline in photon energy. This is consistent with the experimentally observed variation of quantum yield and our basic hypothesis on the role of excess energy in the LZ isomerization process. In the 510–450 nm region, however, the increase in photon energy is offset by the fact that less of the available energy is deposited into the reactive coordinates; consequently, there is a less pronounced effect of incident wavelength on the calculated quantum yield in this spectral region. This finding is qualitatively consistent with our experimental result in which the quantum yield becomes wavelength independent in the blue region.
The simple comparison of the wavelength dependence of the energy distribution for the two different reaction coordinates illustrates the mode-specific nature of the multidimensional reactive pathway. A reaction coordinate defined solely by three localized torsional motions about the C11=C12 bond is inconsistent with experimental observation. However, when the reaction coordinate is expanded and consists of both localized and delocalized torsions, there is qualitative agreement between the calculated wavelength dependence of the energy distribution and the observed . Therefore, it appears that delocalized torsional modes must play a significant role in the reaction coordinate.
We have modeled the wavelength dependence of the quantum yield using the LZ formula and find qualitative agreement with our experimental results, especially when inhomogenous and homogeneous broadening of the absorption process are considered. Fig. 7 shows that the calculated quantum yield (E00eff = 660 nm) increases by 18% with a change in excitation wavelength from 570 to 500 nm, but only increases by 5% as the photon energy is further increased to exc = 450 nm; the predicted change in quantum yield between 570 and 500 nm is 4 times the change predicted between 500 and 450 nm. The relative change in quantum yield is similar to what we observe experimentally; the change in quantum yield observed between 570 and 500 nm (5%) is at least 5 times greater than the change observed between 500 and 450 nm (<1%). When E00eff is further reduced to 725 nm, the calculated ratio begins to quantitatively approach the experimental result. Overall, these calculations allow us to understand why the quantum yield for rhodopsin is wavelength dependent in red but constant in blue spectral regions.
Because the excited-state isomerization dynamics in rhodopsin are influenced by the initially prepared state, a direct probe of the mode-specific nature of the reaction coordinate should be possible. Our calculation indicates that most of the excess energy is localized in low-frequency torsional motion, especially when the incident wavelength is red (>520 nm). This suggests that it should be possible to selectively excite reactive torsional modes using, for example, an infrared pump to study the effect of mode-specific preparation on the isomerization quantum yield and dynamics, as observed for the unrelaxed photoisomerization of gas-phase trans-stilbene (Syage et al., 1984).
Protein versus solution photochemistry
The binding pocket environment significantly enhances the photoisomerization reaction in opsin relative to a solvent such as methanol. For example, the reaction quantum yield for the 11-cis retinal protonated Schiff base is 0.2 and independent of wavelength in at least four different solvents (Becker and Freedman, 1985). In addition, the product formation time of 10 ps in methanol (Kandori et al., 1995) is much slower than that for the chromophore in protein (200 fs). These differences are consistent with the idea that the protein-binding pocket accelerates the initial torsional dynamics of the chromophore. A comparison of the Raman spectra of the chromophore in methanol and in rhodopsin reveals that a number of the low-wavenumber reactive modes, namely the 11=12 A2 HOOP at 970 cm-1 and C10–C13 A2 skeletal torsion at 260 cm-1, have significantly greater Raman intensity in the protein (Lin et al., 1998). These larger intensities indicate steeper excited-state slopes along these modes for the protein-bound chromophore, revealing the mechanistic reason for the more efficient isomerization reaction in the protein.
Limitations of the LZ model
Not surprisingly, the LZ model used here does not yield quantitative agreement with experiment. A central assumption in our calculation is of constant nuclear velocity, or that energy remains localized in each of the FC modes on the excited state. Because our calculation yields relative quantum yields that have a greater wavelength dependence than what is experimentally observed, it is likely that some, but not complete, IVR occurs before curve crossing, and the difference between calculated and measured quantum yields may reflect the extent of IVR on S1. Low-frequency reactive modes are likely to be highly anharmonic and therefore play a significant role in such IVR. Other simplifications are inherent in the LZ model, such as the use of a classical one-dimensional reaction coordinate and an unchanging difference in slopes of the diabatic surfaces in the crossing region. Several studies have addressed these limitations and expanded the LZ model to incorporate a multidimensional surface (Zhu et al., 1997) as well as quantum mechanical wavepacket dynamics (Henriksen, 1992) and stochastic energy fluctuations (Kayanuma, 1984). Other models, such as the nonequilibrium golden rule formula (Coalson et al., 1994; Cho and Silbey, 1995) or Redfield theory (Jean, 1996), may also be used to quantitate the rapid, unrelaxed internal conversion process in rhodopsin. More recent ab initio calculations (Garavelli et al., 1999a; Garavelli et al., 1998; Vreven et al., 1997; Ben-Nun and Martinez, 1998) have explored the idea that the cis-to-trans curve crossing process involves true, as opposed to weakly avoided, surface crossings (Bonacic-Koutecky et al., 1984). These and other quantum calculations (Hahn and Stock, 2000; Molteni et al., 1999; La Penna et al., 1998; Ben-Nun and Martinez, 1999) provide a multidimensional picture of the ultrafast isomerization reaction that builds upon the intuitive picture provided by the LZ model. It will be interesting to see how well these more sophisticated computational approaches are able to reproduce the wavelength-dependent quantum yield observed for rhodopsin.
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Summary |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONSummaryACKNOWLEDGEMENTSREFERENCES |
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ACKNOWLEDGEMENTS |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONSummaryACKNOWLEDGEMENTSREFERENCES |
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This work was supported by a grant from the National Institutes of Health (EY-02051)
Submitted on October 7, 2002; accepted for publication November 26, 2002.
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. Excess energy deposited in reactive degrees of freedom results in an increased wavepacket velocity along the reaction coordinate and hence, a greater probability for curve crossing.
. The experimental absorption spectrum for rhodopsin is also displayed (solid curve, right axis).
j and are denoted 
and 
, respectively. The harmonic surfaces are displaced by a dimensionless parameter 
L = 165 cm-1, 
inh = 0; (B) fully thermalized (298 K) sum-over-states (solid) and time-dependent (dashed) calculations with 
) and the unreactive pathway consists of the remaining 25 modes (x). (b) Results for delocalized reactive coordinate (93, 131, 246, 260, 320, 410, 446, 568, 970 cm-1) in which the reactive pathway is made up of nine delocalized torsional modes (
). Wavelength dependence of the quantum yield for a delocalized reaction coordinate with E00eff = 600, 660, or 725 nm is given (dashed line, x), (dashed line, 
); and (dashed line,