Comment on Activation of Visual Pigments by Light and Heat (Science 332, 1307-312, 2011)

It is known that the Arrhenius equation, based on the Boltzmann distribution, can model only a part (e.g. half of the activation energy) for retinal discrete dark noise observed for vertebrate rod and cone pigments. Luo et al (Science, 332, 1307-312, 2011) presented a new approach to explain this discrepancy by showing that applying the Hinshelwood distribution instead the Boltzmann distribution in the Arrhenius equation solves the problem successfully. However, a careful reanalysis of the methodology and results shows that the approach of Luo et al is questionable and the results found do not solve the problem completely. One Sentence Summary: Retinal discrete dark noise cannot be completely explained by thermal activation based on the approach of Luo et al. The application of the Arrhenius equation, which is based on the simple Boltzmann distribution, to model the temperature dependence of the dark events results in the fact that the predicted thermal activation energy is only being about half of the photo-isomerization activation energy measured experimentally (1,2,3). This leads to the conclusion that the molecular pathway due to spontaneous thermal activation is different from that due to photoactivation. Recently, the use of the Boltzmann distribution has been debated (4,5) and the idea has been put forward by Luo et al (4) that due to thermal activation of the low energy vibrational modes, the Hinshelwood distribution should be used instead of the Boltzmann distribution. Luo et al determined the number of vibrational modes (m) to be 45 in order to fill the gap between the thermal activation energy obtained from the Arrhenius analysis and the activation energy caused by light. After carefully reviewing the approach of Luo et al we come to the conclusion that there are three shortcomings of this approach, which question to validity of explaining the dark noise of rods and cones by only assuming a thermal activation energy process. Our arguments were as follows: (1) It has to be noted that the application of the Hinshelwood distribution to model one molecule is only valid in the classical limit where the thermal energy scale is much larger than the energy level spacing (ε) of the quadratic modes of the molecule (i.e. kT ˃˃ ε , with k is the Boltzmann constant, T is the absolute temperature). Hence, assuming that the room temperature at which the thermal energy is about 25 meV, there must exist many modes with much less energies than this value. However, the opposite is true since the resonance Raman excitation of rhodopsin reveals that the Raman lines corresponds to several tens of modes with energies varying from 98 cm to 1655 cm (corresponding to ~10 to ~200 meV, respectively) which are in order or larger than the scale of the thermal energy (6, 7, 8). Moreover, Luo et al obtained 45 modes were found to have equal energy values, kT (“[...] each vibrational mode of the molecule contributing a nominal energy of kT”(4)) in which the 45 modes all are activated and each energy mode has exactly the same energy as the thermal energy. As a conclusion, the equipartition theorem (9) cannot be applied for these modes; hence the application of the Hinshelwood distribution to model the dark noise of photoreceptors is questionable. (2) Even if we agree that the Hinshelwood distribution is applicable for photoreceptors then the methodology and the obtained results by Luo et al can be questioned. The authors determined that the number of modes (i.e. m = 45) is generally valid for the all values of λ!"# (see Fig 4C’ and Fig S8’) while this value is obtained only via a simple equation (4, 10) for Bufo red rhodopsin with λ!"# = 500 nm based on the apparent thermal activation energy of 21.9 kcal/mol. If we consider mouse rhodopsin with the same λ!"# = 500 nm and use the apparent thermal activation energy of 14.54 kcal/mol (obtained from Fig. S4C’) we find m = 58 (10) according to the methodology used by Luo et al. This indicates that the statement of Luo et al regarding the general validity of the parameter value m = 45 is not supported by experimental findings. To show discrepancy more clearly, the rate constant diagrams based on m = 45 and our obtained m values (i.e. m=49 for rod cells and m=42 for cone cells) based on the fitting method (10) are compared with the experimental data for different rod and cone cells (10, 3) in Fig. 1. The results indicate that m=45 is not an exclusive value and has a significant deviation relative to the experimental data. Moreover, our obtained preexponential factors (A) deviate from the A value used by Luo et al (see Table S4’) which was obtained by the authors by simple averaging and not by fitting, which is imprecise as well. If we apply the average of m=42 and m=49 as m=45 with a single A value for a combined datasets I and II for both rod and cone cells then the amount of deviation from experimental 1 The primed numbers for tables and figures refer to the paper of Luo et al (4). data will be very large. To check this high deviation see below the comment 3 about predictions of the Table 1'. As a result, rod and cone cells should be investigated separately with different m values. Fig. 1. Fitted functions (rate constant vs. λmax) according to Equation 1’ of (4) with optimal m-values (red) and m = 45 (blue) as predicted by Luo et al (4). Shown are the results with a linear (a, b) and logarithmic scaling (c, d). (a, b): data set I (rhodopsins), (c, d): data set II (cone pigments) (10). (3) Another problem in the paper of Luo et al appears in their predictions given in Table 1’. There, the authors claimed that the ratios of rate constants, k, are the ratios of their distribution functions, f ≥Ea T (“[...] We began with A being the same for the all pigments [...] thus the predicted thermal rate ratio between two pigments is simply their f ≥Ea T ratio”(4)). However, the pre-exponential factor A varies with m (e.g. see the caption of FigS8’) and it varies for different pigments, so it is unfortunately erroneous to compare the distribution ratios (as predicted rate constant ratios) with the measured rate-constant ratios in Table1’ while the A values are not equal even for the same number of modes (i.e. m = 45) for cone and rod cells (see Table S4’ and also the average A and SD values for rod cells). Moreover, these preexponential factors of Table S4’ are obtained directly from the measured rate constants themselves which causes an unfair comparison. The authors have mentioned that there is about 26-fold difference between A values of rods and cones. To check this claim, we compared these ratios for different samples. The results are shown in Table1 in which large discrepancies between theory and experiment can be recognized, indicating that the distribution ratios are not equal to the rate constant ratios. Table 1. Comparison between theoretical predictions offered by Luo et al (for m = 45) and the measurements of rate constants of visual pigments (10). Even for similar λmax values of rods (Bufo and mouse) and cones (human and turtle), 16 and 83 fold difference appeared respectively. For other comparisons between rods and cones the differences are very large numbers which indicates that predicted and measured rate constants are not comparable. Pigment λmax (nm) Ea(kcal mol ) f≥EaT Predicted rate constant ratio (Luo et al.’s approach) Measured rate constant (s) Measured rate constant ratio Bufo rhodopsin 500 48.03 3.65×10 1 4.18×10 1 16 mouse rhodopsin 500 48.03 3.65×10 6.64×10 human red cone 617 38.93 2.44×10 1 6.70×10 1 83 Turtle (Trachemysscriptaelegans) L-cone 617 38.93 2.44×10 -3 5.28×10 Larval tiger salamander (Ambystomatigrinum) rod 521 46.10 1.67×10 -5 1