Sensors designed to be activated by light energy will necessarily have some propensity for activation by thermal energy alone. In the visual system, randomly occurring thermal activations of visual pigment molecules constitute an irreducible noise that sets an ultimate limit to the detection of weak light (Autrum, 1943; Barlow, 1956; Aho et al., 1988; Donner, 1992). Thermal activations of the rod visual pigment are very sparse, but can be studied by the electrical signals they generate in the photoreceptor cell. Discrete “bumps” in the receptor current of single rods, recorded, e.g., by the suction-pipette technique, report single-molecule events in a population of >10⁹ rhodopsins. Thus, Baylor et al. (1980, 1984) were first able to show that the rate constant for thermal activation of toad and macaque rhodopsin (~10⁻¹¹ s⁻¹ at 37°C) was of the right magnitude to account for the intrinsic “dark light” invoked to explain the statistics of light detection by dark-adapted humans (Barlow, 1956; cf. Hecht et al., 1942).

Baylor et al. (1980) determined the Arrhenius activation energy of the thermal process from the temperature dependence of the rate constant in toad rhodopsin, obtaining ~22 kcal/mol. This is only about half of the energy needed for activation by light (Lythgoe and Quilliam, 1938; St.George, 1952; Cooper, 1979; Barlow et al., 1993). Because there can be no reasonable doubt that the discrete dark events in the rod current do arise from spontaneous activation of single molecules of visual pigment (cf. Firsov et al., 2002), the current interpretation of this discrepancy is that the photic and thermal modes of activation follow different pathways, the activation energies of which bear no necessary relation.

This view, however, fails to address the striking and biologically important empirical relation between spectral and thermal properties of visual pigments. There is a significant inverse correlation between the dark-event rate and the wavelength of maximum absorbance (λ_max) (Donner et al., 1990; Firsov and Govardovskii, 1990; Fyhrquist, 1999; see Figs. 2 and 3 below). The correlation is qualitatively consistent with the hypothesis that a pigment maximally sensitive to long-wavelength light (i.e., low-energy photons) has a comparatively low-energy barrier for activation and thus a high probability for thermal activation, whereas short-wavelength sensitive pigments have a higher activation energy and are therefore thermally more stable (de Vries, 1949; Barlow, 1957).

FIGURE 2 The relation between the rate of thermal dark events per molecule of visual pigment () and the wavelength of peak absorbance (λmax) in rods (data from Table 1). The Briggsian logarithm of the rate constant k is plotted as a function of 1/λ (more ...)

FIGURE 3 The relation between the rate of thermal dark events per molecule of visual pigment () and the wavelength of peak absorbance λmax in cones (data from Table 2). The Briggsian logarithm of the rate constant k is shown as a function of 1/λ (more ...)

We are faced with a fundamental dilemma: if the processes of activation by light and by heat are uncoupled, the observed correlation between λ_max and thermal noise remains a mystery. To resolve this conflict, we propose a new model, drawing on insights already expressed by St. George (1952) and Lewis (1955). The crucial point is that the thermal behavior of a complex molecule (rhodopsin, or even the 11-cis retinaldehyde chromophore) must be described by Hinshelwood (1933) rather than Boltzmann statistics. This is because thermal activation of a molecule composed of many atoms may be supported by the energy present in a large number of vibrational modes, in addition to the kinetic energy in the relative motion of molecules (translational degrees of freedom). Arrhenius-type estimates for the thermal activation energy will decisively depend on how many modes are assumed to be involved, i.e., on the complexity of the activation process. The estimates of Baylor et al. (1980) for toad rhodopsin and subsequent estimates for other visual pigments (e.g., Matthews, 1984) rely on the arbitrary assumption that the activation process is a very simple one, involving only two (translational) degrees of freedom (n = 2). This undermines the evidence for a large difference between the energies of activation by heat and by light.

Here we test the model under the assumption that the thermal activation energy of each pigment has the same value as its photoactivation energy, this being the simplest possible alternative to the current notion that the two are very different. Some justification for assuming that they may be at least rather close comes from recent work suggesting that the gap between the minimum of the excited-state and the maximum of the ground-state energy surfaces for chromophore isomerization could be as small as 5–6 kcal/mol (Mathies, 1999).

When thermal and photic activation energies are set equal, the model predicts the temperature dependence of dark-event rates observed by Baylor et al. (1980) if the number of vibrational modes involved in the thermal activation process is ~39. Most importantly, with the same number of vibrational modes, the model gives a good fit to the collected data on λ_max versus light-like thermal noise in rods as well as in cones. The model by no means requires that thermal and photic activation energies be identical, but they must show some correlation, and we suggest there are good reasons to expect they do (see Discussion).

Photoactivation of a visual pigment molecule requires absorption of a photon with sufficient energy to bridge the gap (E_a) between the ground state and the first excited state. This gap corresponds to the photon energy at some wavelength λ₀ (E_a = hc/λ₀). Stiles (1948) proposed that the reason why spectral sensitivity exhibits no sharp drop at λ₀ is that photon energy may be supplemented by thermal energy of the visual pigment molecule. If a photon at λ > λ₀ encounters a molecule that can contribute enough thermal energy (≥hc(1/λ₀ − 1/λ) to the process, activation may occur. He assumed that the Boltzmann distribution gives the relevant probabilities for encountering visual-pigment molecules at given thermal energy levels. The fraction of molecules with thermal energy greater than some value E is then

(1)

Obviously, the fraction with at least the energy E = hc(1/λ₀ − 1/λ) needed to supplement a photon of wavelength λ > λ₀ decreases proportionally to exp(1/λ). This would explain the fact that the final slope of absorbance spectra at very long wavelengths, plotted on logarithmic ordinates against wavenumber (1/λ), approximates a straight line (Stiles, 1948).

Lewis (1955) noted that spectra are in fact gently concave in the wavenumber domain of interest. He pointed out that for a complex molecule with many vibrational modes (many degrees of freedom), the fraction of molecules with thermal energy >E is given by an expression derived by Hinshelwood (1933):

(2)

The total (maximal) number of quadratic energy terms of a molecule, the number of “internal degrees of freedom,” is given by:

(3)

For purely thermal activation, the rate constant k is proportional to the fraction of molecules that have energy larger than the thermal activation energy, which we denote E_a,H:

(4)

A_H is a proportionality constant, often referred to as the “pre-exponential factor”. The above expression for k derives from the first term of Eq. 2, which is a good approximation for the full series when E RT (≈0.6 kcal/mol at 20°C) (Hinshelwood, 1933). The rate constant based on the Boltzmann distribution is obtained from Eq. 1:

(5)

In this case, we give the pre-exponential factor as well as the activation energy the subscript B to emphasize that these values based on Boltzmann statistics (associated with “conventional” Arrhenius analysis) are different from those based on Hinshelwood statistics, denoted by subscript H. For n = 2, Eq. 4 converges to Eq. 5.

When Eq. 4 replaces Eq. 5, the generally accepted values E_a,B derived from experimental data on the temperature dependence of dark-event rates (Baylor et al., 1980) appear as basically arbitrary. The true relation between the energies of thermal and photic activation remains unresolved. In the following, we study the predictions of the model under the simplest possible assumption (itself not essential to the model) that the two activation energies have the same value, i.e., E_a = E_a,H.

FIGURE 1 The temperature dependence of the rate of thermal dark events per rod (Rh*s−1) in “red” rods of the toad Bufo marinus. The data are from Baylor et al. (1980, their Table 2); each symbol type denotes data from one rod. The (more ...)

The straight line in Fig. 1 also represents the conventional Arrhenius slope for E_a,B = 21.9 kcal/mol, the mean of the individual E_a,B values determined by Baylor et al. (1980) from each of the five rods separately. Our rationale for fitting this model (Eq. 4) has been to require that the predicted temperature dependence with E_a,H = E_a = 44.3 kcal/mol should coincide with this “mean” line. (Obviously, the line deviates somewhat from the outcome of simply applying linear regression analysis to the set of data points.) The fitting entails optimizing the value of the parameter n, which was done as follows. The rate of change of lnk with temperature, d lnk/dT according to the “conventional Arrhenius” model (based on Eq. 5) is:

(6)

The corresponding rate of change according to the “Hinshelwood” model (based on Eq. 4) is:

(7)

Obviously, when n is large, a given empirical temperature dependence is indicative of a larger activation energy according to Eq. 7 than according to Eq. 6, the difference being E_a,H − E_a,B = (n/2 − 1)RT (cf. St. George, 1952). To determine n, we set this expression equal to the difference between the Baylor et al. (1980) estimate 21.9 kcal/mol and our present postulate 44.3 kcal/mol:

The straight line in Fig. 1 represents the identical predictions of, on one hand, the “Arrhenius-Boltzmann” model for E_a,B = 21.9 kcal/mol and on the other hand the “Hinshelwood” model for E_a,H = 44.3 kcal/mol and n = 79. As explained above in connection with Eq. 2, this number of quadratic energy terms (kinetic + potential energy) would arise from half that number of vibrational modes, which (taken as the smallest sufficient integer, i.e., the first integer ≥79/2) would be 39.

We now fix the value n = 79 and study whether the “Hinshelwood” model with this parameter value can account for the observed correlations between 1/λ_max, E_a, and thermal noise in visual pigments.

TABLE 1 Characteristics of the rod visual pigments included in the analysis

The photoactivation energy E_a is known only for some of the pigments. To relate λ_max to E_a for the whole data set, we use the empirical equation obtained by Ala-Laurila et al. (2004) by linear regression of E_a on 1/λ_max in 12 visual pigments:

(8)

Equation 8 is a useful statistical relation, although it should be noted that no tight physical connection exists between E_a and λ_max. Two pigments with the same λ_max may have quite different E_a and vice versa (Koskelainen et al., 2000). However, the regression Eq. 8 explains 73% of the variation in E_a in the sample of visual pigments that have been studied in this respect (Ala-Laurila et al., 2004). In testing this model, we now assume that the same equation also describes the relation between the thermal activation energy E_a,H and λ_max.

The solid straight line in Fig. 2 traces the model prediction on this assumption (for n = 79 and T = 294.15 K). The line has been vertically positioned for best fit to the whole data set, which implies fixing the value of the pre-exponential factor A_H in Eq. 4. Clearly, the model fairly successfully predicts how steeply the thermal activation rate depends on λ_max. For comparison, the dashed line shows the (excessively steep) relation based on the Boltzmann distribution (Eq. 5), as originally suggested by Barlow (1957). The dotted line shows the prediction of this “Hinshelwood” model under the modified assumption that E_a,H is systematically somewhat smaller than E_a (see below).

TABLE 2 Characteristics of the cone visual pigments included in the analysis

The correlation between λ_max and dark-event rates The E_a values for Rana catesbeiana porphyropsin and rhodopsin together with n = 44 in Eq. 4 yields the dark-event rate ratio k_A2/k_A1 = 13.6. Compared with the experimental estimate k_A2/k_A1 ≈ 10, this prediction is about equally good as the value 7.6 obtained under the previous assumption E_a,H = E_a and n = 79.

In Figs. 2 and 3, presenting the full sets of data on rod and cone pigments, the predictions of the “energy offset” modification (E_a − E_a,H = 10 kcal/mol and n = 44) are shown as dotted lines. The fits are slightly less good than those for E_a,H = E_a and n = 79 (solid lines), but still much better than the Boltzmann fits (dashed lines).

Thus, the model works well with a moderate offset between E_a and E_a,H, as long as the two are coupled (see Discussion). On the other hand, the modification does not improve the fits obtained under the simplest assumption E_a,H = E_a. In the model parameters, energy offsets are traded against changes in the apparent number of thermal modes (n/2) involved in the activation process. Due to this trade-off, the model is not very helpful for determining the precise relation between E_a,H and E_a, nor the value of n, from the experimental data. Its value lies in showing that the data are consistent with the idea that photoactivation and thermal activation energies are similar or equal, and in explaining the correlation between E_a,H and λ_max on this basis.

We may finally note that in the “Boltzmann” model (Eq. 5), the introduction of a constant offset ΔE, such that E_a,B = E_a − ΔE, will not at all affect the predicted ratio of dark-event rates in two pigments with different activation energies. In the ratio k_A2/k_A1, the offset ΔE disappears by reduction: if k_A1 exp[−(E_a1 − ΔE)/RT] and k_A2 exp[−(E_a2 − ΔE)/RT], then k_A2/k_A1 = exp[(E_a1 − ΔE − E_a2 + ΔE)/RT] = exp[(E_a1 − E_a2)/RT]. Thus the predicted ratio k_A2/k_A1 is the same as if E_a,B = E_a, implying that the much too steep dependence of dark-event rates on λ_max remains unaffected.

It has been suggested that the discrete photon-like events in the rod dark current might not, in fact, reflect spontaneous activation of single-pigment molecules, but arise at some later point in the transduction machinery (for a discussion, see Barlow et al., 1987). At least in the case of vertebrate rods, however, it is hard to believe that the discrete events could originate elsewhere than in the visual pigment. The shape of the electrical response to photoactivation of a single rhodopsin molecule depends on serial activation of some 100 transducins by the rhodopsin and subsequent suppression of rhodopsin catalytic activity by multiple phosphorylation and arrestin binding, as well as a number of other shut-off reactions (Leskov et al., 2000; Pugh and Lamb, 2000; Fain et al., 2001). It is highly improbable that standardized bumps indistinguishable from this repeatedly arise, e.g., from concerted spontaneous activation of hundreds of transducins followed by quenching that just happens to mimic the shut-off kinetics of the light response.

While it is now widely accepted that the dark events originate in the visual pigment, this has led to another state of nescience: the two processes of thermal and photic activation evidently follow different molecular pathways, but nothing is known about the low-energy thermal pathway. The problem is exacerbated by the fact that 22 kcal/mol is not only radically smaller than the photoactivation energy, but too small to allow the reaction to visit the early postisomerization photoproducts of rhodopsin, notably bathorhodopsin, whose ground-state enthalpy lies ~35 kcal/mol above that of the native pigment (Cooper, 1979; Mathies, 1999; Okada et al., 2001). The very rapid formation of bathorhodopsin (within 200 fs; Schoenlein et al., 1991; Wang et al., 1994) leaves no time for significant conformational changes in the opsin (cf. Filipek et al., 2003). Thus, entropy stays essentially constant up to the bathorhodopsin stage, and the change in free energy is approximately equal to the enthalpy change.

A cogent and testable hypothesis about the identity of the low-energy activation pathway was proposed by Barlow et al. (1993), stating that the thermal events originate in a small subpopulation of rhodopsin molecules where the Schiff base linking the chromophore to the opsin is unprotonated. The calculations of the authors indicate that Schiff-base deprotonation would lower the activation energy for chromophore isomerization from ~45 kcal/mol to about the right level, ~23 kcal/mol. This hypothesis, however, now seems implausible, as recent studies have failed to detect any pH dependence either of dark-event rates in rods (Firsov et al., 2002) or the dark-noise-equivalent rate of quantal activations in cones (Sampath and Baylor, 2002).

It is worth remembering that all experimental evidence for low thermal activation energies comes from measurement of dark-event rates at a few temperatures (cf. Fig. 1). The values extracted from such data depend wholly on the theory applied. We here suggest that the low estimates may be analytical artifacts, due to improper application of conventional Arrhenius analysis to a complex molecule. When the thermal energy present in the vibrational modes of the chromophore is taken into account, the data are consistent with the idea that thermal and photic activation energies are close or even equal, ~40–50 kcal/mol. We propose that a particular “low-energy” thermal activation pathway simply may not exist.

In the formalism of the model, the difference resides in the “pre-exponential factor” A_H (see Eq. 4). We can only speculate on its molecular correlate. Rod opsins evidently have a very “closed” chromophore pocket, which probably imparts high thermal stability to the chromophore. (The price is slow chromophore exchange, thus slow recovery from bleaching.) Regardless of the energy barrier, the possible molecular routes for thermal isomerization of the chromophore may be tightly restricted by the opsin. By contrast, the chromophore pocket of cone pigments is much “looser”, arguably to ensure fast chromophore exchange, thus fast recovery from bleaching. In at least some cone pigments, there is a continuous exchange of chromophore even in darkness (Matsumoto et al., 1975). This seems consistent with the idea that the opsin control is much less restrictive than in rods.

Differences in the pre-exponential factor (A_H) within either group of pigments (rod or cone) will appear as random variation when the dark-event rate is considered as a function of λ_max. The A_H difference is expected to be negligible between the A1 and A2 members of a pigment pair, as both are restricted by the same opsin. For rod pigments with different opsins, the “unexplained” component of experimental variation (cf. Fig. 2) may at present be the only clue to the possible range of variation in A_H. In contrast to the rod opsins, the cone opsins phylogenetically belong to several subfamilies (see, e.g., Yokoyama and Yokoyama, 2000), and one might expect even more variation, e.g., in the topology of the chromophore pocket. In the small experimental material available, however, it is not possible to assess the true degree of “random” scatter of cone dark-event rates.

It is also instructive to apply a teleological argument. A visual pigment ought to combine a high quantum efficiency for photoisomerization with high thermal stability. Thermal stability would be maximized if the ground-state barrier for chromophore isomerization were “infinitely” high, but such a molecule would have zero quantum efficiency, because it would always drop back to the 11-cis ground state from the excited state. High quantum efficieny for photoisomerization requires that the ground-state barrier be equal or lower than the excited-state surface in the direction of the isomerization coordinate. In view of the stability requirement, it is optimal to have a barrier as high as possible consistent with this. To the extent that the properties can be tuned by natural selection, the result would be a close coupling of the photic excitation energy and the thermal energy barrier.

When the internal energy present in the many vibrational modes of a visual pigment molecule is taken into account, the distribution on thermal energy levels follows Hinshelwood (1933) rather than Boltzmann statistics. The large difference between current estimates of thermal activation energies and photoactivation energies then disappears as an analytical artifact. The experimental data are consistent with the two kinds of activation energy being close or even equal. This has at least two important consequences. Firstly, it allows that the molecular pathways for activation by light and by heat may be identical from the initial event of chromophore isomerization in the native (resting) configuration of the opsin. Secondly, it makes understandable the biologically important correlation between high sensitivity to long-wavelength light and high rates of spontaneous activations in visual pigments.

We thank Drs. Robert Barlow and Robert Birge for valuable discussions and Dr. Richard Mathies for constructive comments on the manuscript.

This work was supported by the Academy of Finland (grants 49947 and 206221), the Emil Aaltonen Foundation, and the Ella and Georg Ehrnrooth Foundation.