Excitation Properties of Photopigments and Their Possible Dependence on the Host Star

It is a well-known fact that the excitation of electrons is a vital prerequisite for photosynthesis, as expounded in Lodish et al. (2000, Chapter 16), Falkowski & Raven (2007, Chapter 2), and Blankenship (2014, Chapter 1). A panoply of models have, therefore, sought to relate key characteristics of photopigments with their absorption spectra. A crucial parameter in the model is the number of conjugated π-electrons (N_⋆). Since these electrons can be delocalized and are liable to excitation in response to a flux of photons (Johnson 2016), their number is a determinant of the pigment characteristics.

Attempts to quantify the aforementioned relationship span a broad range of complexity: in this work, we will draw on the basic free-electron model propounded in the 1940s (Bayliss 1948; Kuhn 1948, 1949; Simpson 1949). Our rationale for doing so is twofold. First and foremost, as we shall explicate hereafter, it is endowed with a sufficient degree of accuracy and realism, while retaining simplicity and transparency. Second, as we are adopting an astrobiological perspective where we must deal with a lack of concrete data, it is desirable to keep the model generic and minimize the proliferation of free parameters that could take on arbitrary values. We will accordingly, for the most part, mirror the approach delineated in Phillips et al. (2012, Chapter 18).

We will model the conjugated π-electrons as though they are placed within a "well" of infinite depth, with the photopigment comprising the real-world counterpart of the latter. Although the derivation is fairly elementary and can be found in any standard book on quantum mechanics (e.g., Griffiths & Schroeter 2018, Section 2.2), we will recapitulate the salient steps below for the benefit of the general readership. The wave function ψ(x) for an electron confined within an infinite well obeys the time-independent Schrödinger equation as follows:

$$\begin{eqnarray}&&-\displaystyle \frac{{\hslash }^{2}}{2{m}_{e}}\displaystyle \frac{{d}^{2}\psi }{{{dx}}^{2}}={ \mathcal E }\psi ,\end{eqnarray} \tag{ 1 }$$

where m_e is the mass of the electron and ${ \mathcal E }$ signifies the energy. The two boundary conditions for the system are ψ(0) = 0 and ψ(L) = 0, where L is the width of the infinite well. It is straightforward to verify that the mathematical solution of the above ordinary differential equation (ODE) is given by

$$\begin{eqnarray}&&\psi (x)={C}_{1}\sin \left({kx}\right)+{C}_{2}\cos \left({kx}\right),\end{eqnarray} \tag{ 2 }$$

where $k=\sqrt{2{m}_{e}{ \mathcal E }/{{\hslash }}^{2}}$. The boundary condition at the origin demands C₂ = 0 implying $\psi (x)={C}_{1}\sin \left({kx}\right)$, and the boundary condition at the edge of the box leads to the quantization condition

$$\begin{eqnarray}&&{kL}\,=\,n\pi ,\end{eqnarray} \tag{ 3 }$$

where n is an integer. The resulting expression for the quantized energy of the n-th level (${{ \mathcal E }}_{n}$) is

$$\begin{eqnarray}&&{{ \mathcal E }}_{n}=\displaystyle \frac{{k}^{2}{{\hslash }}^{2}}{2{m}_{e}}=\displaystyle \frac{{n}^{2}{h}^{2}}{8{m}_{e}{L}^{2}}.\end{eqnarray} \tag{ 4 }$$

We will, now, suppose that the conjugated π-electrons are treated as effectively being situated on a chain or loop (Platt 1949; Taubmann 1992), with a mean spacing of ℓ_⋆. By drawing on this assumption, it becomes apparent that $L=\left({N}_{\star }-1\right){{\ell }}_{\star }$ in (4).

We are interested in the minimal energy (${\rm{\Delta }}{ \mathcal E }$) theoretically needed for electronic excitation and how that ties in with the longest viable photon wavelength ${\lambda }_{\max }$. With respect to the former, we are essentially attempting to gauge the theoretical limit of the HOMO-LUMO gap (Atkins & de Paula 2009), where these abbreviations stand for highest occupied molecular orbital (HOMO) and lowest unoccupied molecular orbital (LUMO). The wavelength ${\lambda }_{\max }$ in terms of ${\rm{\Delta }}{ \mathcal E }$ is calculated to be

$$\begin{eqnarray}&&{\lambda }_{\max }=\displaystyle \frac{{hc}}{{\rm{\Delta }}{ \mathcal E }}.\end{eqnarray} \tag{ 5 }$$

If we express ${\rm{\Delta }}{ \mathcal E }$ in terms of N_⋆ and ℓ_⋆, our preceding objective is fulfilled. As indicated in the prior paragraph, we must investigate the highest occupied level. As electrons are fermions, each level contains only two of them as per the Pauli exclusion principle, with this duo differing in their spin quantum numbers. Hence, if there exists an even number of electrons in total, the highest occupied level would correspond to n = N_⋆/2, whereas for an odd number of electrons it would be $n=\left({N}_{\star }+1\right)/2$. If N_⋆ is roughly an order of magnitude larger than unity, which turns out to be generally valid as illustrated hereafter, the two cases yield similar values.

Hence, after adopting n = N_⋆/2 and substituting this into (4), we arrive at

$$\begin{eqnarray}&&{{ \mathcal E }}_{{N}_{\star }/2}=\displaystyle \frac{{h}^{2}{N}_{\star }^{2}}{32{m}_{e}{{\ell }}_{\star }^{2}{\left({N}_{\star }-1\right)}^{2}},\end{eqnarray} \tag{ 6 }$$

which is the highest occupied energy level. The energy required to excite an electron from this level to the next level n = N_⋆/2 + 1, which is unoccupied by construction, is computed by first recognizing that

$$\begin{eqnarray}&&{{ \mathcal E }}_{({N}_{\star }/2)+1}=\displaystyle \frac{{h}^{2}{\left({N}_{\star }+2\right)}^{2}}{32{m}_{e}{{\ell }}_{\star }^{2}{\left({N}_{\star }-1\right)}^{2}},\end{eqnarray} \tag{ 7 }$$

which follows from substituting n = N_⋆/2 + 1 in (4). As a consequence, the theoretical minimum excitation energy ${\rm{\Delta }}{ \mathcal E }$ is therefore given by

$$\begin{eqnarray}\begin{array}{rcl}{\rm{\Delta }}{ \mathcal E } & = & {{ \mathcal E }}_{({N}_{\star }/2)+1}-{{ \mathcal E }}_{{N}_{\star }/2}\\ & = & \displaystyle \frac{{h}^{2}\left({N}_{\star }+1\right)}{8{m}_{e}{{\ell }}_{\star }^{2}{\left({N}_{\star }-1\right)}^{2}}.\end{array}\end{eqnarray} \tag{ 8 }$$

It is worth recalling that ${\rm{\Delta }}{ \mathcal E }$ is the theoretical minimum photon energy needed for the requisite excitation, owing to which the ensuing wavelength represents an upper bound; to put it differently, if the wavelength is greater than ${\lambda }_{\max }$, then electronic excitation ought not be feasible. After substituting this expression into (5) and solving for ${\lambda }_{\max }$, we duly end up with

$$\begin{eqnarray}\begin{array}{rcl}{\lambda }_{\max }\, & = & \,\displaystyle \frac{8{m}_{e}c}{h}\,{{\ell }}_{\star }^{2}\,\displaystyle \frac{{\left({N}_{\star }-1\right)}^{2}}{{N}_{\star }+1}\\ & \approx & \,65\,\mathrm{nm}{\left(\displaystyle \frac{{{\ell }}_{\star }}{1.4\,\mathring{\rm A} }\right)}^{2}\displaystyle \frac{{\left({N}_{\star }-1\right)}^{2}}{{N}_{\star }+1},\end{array}\end{eqnarray} \tag{ 9 }$$

where ℓ_⋆ has been normalized by 1.4 Å ≡ 0.14 nm since it corresponds to the characteristic spacing between carbon atoms involved in conjugated bonds, in view of the empirical data adumbrated in Falkowski & Raven (2007, pg. 64) and Atkins & de Paula (2009, pg. 498).

In view of the simplifications invoked in deriving (10), it is natural to inquire whether this model is accurate. A bevy of studies have established that the free-electron paradigm constitutes a tenable approximation of more sophisticated treatments (Platt 1949; Simpson 1949; Taubmann 1992). More specifically, we note that this model has been compared against empirical data for molecules such as α-oligothiophenes and exhibited good agreement (de Melo et al. 1999). Furthermore, the quantitative scaling of ${\rm{\Delta }}{ \mathcal E }\propto 1/{N}_{\star }$ expected for large N_⋆—refer to (8)—is confirmed by experiments and density functional theory calculations for pyrroles, thiophenes, and furans (e.g., Hutchison et al. 2003, their Figure 2).

At this stage, it is worth reiterating the significance of ${\lambda }_{\max };$ it lies in the fact that photons with $\lambda \gt {\lambda }_{\max }$ would not possess the requisite energy to excite electrons, and therefore cannot effectuate photosynthesis. We have plotted ${\lambda }_{\max }$ as a function of N_⋆ in Figure 1. ⁶ It is apparent from inspecting this plot that ${\lambda }_{\max }\propto {N}_{\star }$ for N_⋆ ≫ 1 and that ${\lambda }_{\max }\approx 1.1\,\mu {\rm{m}}$ for N_⋆ = 20; we shall return to this latter feature in Section 3. Conversely, if ${\lambda }_{\max }$ is deduced through some other empirical or theoretical method, one may potentially utilize Figure 1 to infer N_⋆ of the photopigments.

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Until now, we have emphasized ${\rm{\Delta }}{ \mathcal E }$, which theoretically embodies the minimum energy for electronic excitation. In reality, however, this extreme limit is unlikely to suffice for excitation of electrons due to a number of intricate biochemical and biophysical processes. Laboratory assays suggest that, for the likes of chlorophylls, the excitation energy must be approximately twice that of ${\rm{\Delta }}{ \mathcal E }$ (Serlin et al. 1975; Grimm 2006; Hedayatifar et al. 2016; Buscemi et al. 2021), which is verifiable after substituting ℓ_⋆ = 1.4 Å (Falkowski & Raven 2007, pg. 64) and N_⋆ = 20 (Rabinowitch & Govindjee 1969, pg. 110) in (8). Hence, it seems credible to assume that the actual wavelength needed for robust excitation is roughly estimated by means of $\lambda \approx {hc}/\left(2{\rm{\Delta }}{ \mathcal E }\right)$, which is expressible in terms of N_⋆ as

$$\begin{eqnarray}\begin{array}{rcl}\lambda \, & = & \,\displaystyle \frac{4{m}_{e}c}{h}\,{{\ell }}_{\star }^{2}\,\displaystyle \frac{{\left({N}_{\star }-1\right)}^{2}}{{N}_{\star }+1}\\ & \approx & \,33\,\mathrm{nm}{\left(\displaystyle \frac{{{\ell }}_{\star }}{1.4\,\mathring{\rm A} }\right)}^{2}\displaystyle \frac{{\left({N}_{\star }-1\right)}^{2}}{{N}_{\star }+1}.\end{array}\end{eqnarray} \tag{ 10 }$$

Now, let us consider a few specific photopigments on Earth and assess the reliability of our modeling. To err on the side of caution, we restrict our attention to pigments explicitly involved in photosynthesis. For this reason, we do not explicitly consider microbial rhodopsins (e.g., proteorhodopsins)—which harvest as much (or more) solar energy as chlorophylls in Earth's marine environments (Gómez-Consarnau et al. 2019; Hassanzadeh et al. 2021)—because they do not take part in photoautotrophy. The rhodopsin family, which exhibits absorption maxima of ≲ 550 nm (Man et al. 2003; Inoue et al. 2021) that are distinctly lower than those corresponding to certain (bacterio)chlorophylls, is posited to have been widespread on early Earth and similar worlds (DasSarma & Schwieterman 2021), and might even be capable of photoautotrophy in principle (Larkum et al. 2018). Since there is no such concrete empirical evidence hitherto for the latter, we do not analyze this pigment hereafter as stated above; however, if and when appropriate, it is straightforward to use the fact that rhodopsin has six conjugated bonds (Sen et al. 2021).

The pigment β-carotene, which is widespread in photosynthetic organisms, is endowed with 11 conjugated bonds (Rabinowitch & Govindjee 1969, Figure 9.6). Therefore, upon specifying N_⋆ = 22 in (10), we obtain λ ≈ 633 nm, which is ∼1.5 times higher than the absorption maxima of β-carotene at ∼425, 450, and 480 nm (Rabinowitch & Govindjee 1969, Table 9.2). Next, on turning our attention toward chlorophyll a, a predominant pigment in oxygenic photosynthesis (Nishio 2000), this molecule has 10 conjugated bonds (Rabinowitch & Govindjee 1969, pg. 110). After specifying N_⋆ = 20 in (10), we end up with λ ≈ 567 nm; this value is broadly compatible with the absorption peaks at ∼435 nm and ∼670–700 nm documented for chlorophyll a (Schwieterman et al. 2018, pg. 684).

Thus, the λ predicted by our approach, and the wavelengths of the absorption peaks of Earth-based photopigments are manifestly not far removed from one another. For the sake of argument, suppose that certain abiotic factors govern the wavelengths at which photopigments absorb strongly. If this wavelength regulated by such nonbiological processes (denoted by λ_opt) were known, one could accordingly constrain N_⋆ by invoking (10). The premise implicit herein is that these molecules were gradually adapted over the course of evolution such that their eventual absorption was rendered efficacious at λ_opt. ⁷ Therefore, if we presume that this picture is tenable, it may become feasible to establish a direct connection between putative abiotic phenomena and the structure of extrasolar photopigments, which we examine further in the upcoming Sections 2.2 and 3.

Needless to say, this potential link should not be viewed as exact because evolution is patently not a "perfect" optimization process; in other words, it is by no means assured that N_⋆ for the potential photopigments would be precisely identical to what one will obtain from (10) after equating it with λ_opt.

The question of what wavelengths may be typically associated with the absorption peaks of photopigments on exoplanets has not been extensively investigated, barring a few notable publications (Kiang et al. 2007a, 2007b; Lehmer et al. 2018, 2021; Lingam & Loeb 2021a). One key problem is that most of these studies involve several parameters that are currently indeterminate or poorly resolved on other worlds: examples include atmospheric composition, the so-called relative cost parameter of Marosvölgyi & van Gorkom (2010), and the availability of cofactors such as NADP⁺ (Milo 2009). Hence, we opt to construct here a simpler model with no unknown parameters; while it trades some precision, it is still fairly accurate (as expounded hereafter).

The essence of our working hypothesis, paralleling Kiang et al. (2007a) and Lingam & Loeb (2021a, Chapter 4.3.5), lies in positing that the peak absorbance of photopigments occurs in the vicinity of the wavelength where the spectral photon flux (n_λ ) of the host star attains its maximal value; the star is treated as a blackbody with temperature T_⋆. We remark that this criterion is consistent with the penultimate paragraph of Section 2.1 since the abiotic factor would be the stellar electromagnetic spectrum. The spectral photon flux for the star consequently is evaluated using

$$\begin{eqnarray}&&{n}_{\lambda }=\displaystyle \frac{2c}{{\lambda }^{4}}{\left[\exp \left(\displaystyle \frac{{hc}}{\lambda {k}_{B}{T}_{\star }}\right)-1\right]}^{-1}.\end{eqnarray} \tag{ 11 }$$

Since λ_opt is the maximum of n_λ , it is determined by solving ${{dn}}_{\lambda }/d\lambda =0$, that simplifies to

$$\begin{eqnarray}&&{\lambda }_{\mathrm{opt}}\approx 3.67\times {10}^{6}\,\mathrm{nm}{\left(\displaystyle \frac{{T}_{\star }}{1\,{\rm{K}}}\right)}^{-1}.\end{eqnarray} \tag{ 12 }$$

It is necessary to assess how realistic our model is when it comes to actually predicting the peak absorbance of photopigments, which we tackle below.

Naturally the Sun is our fiducial star—on substituting T = T_⊙ ≡ 5780 K (viz., the temperature of the Sun), we end up with λ_opt ≈ 635 nm. This value constitutes a good match with the empirical absorption peaks of ∼630–700 nm confirmed for chlorophylls a, b, and c (Schwieterman et al. 2018, Table 1). We have emphasized this trio of molecules because they are widely prevalent in oxygenic photoautotrophs. The latter, in turn, comprises the predominant source of biomass on Earth (Bar-On et al. 2018), and the evolution of oxygenic photosynthesis proved to be a major evolutionary innovation that transformed Earth's biosphere (Knoll 2015). However, it is worth recognizing that the absorption maxima of chlorophyll d and f occur at slightly longer wavelengths of 710–740 nm (Mielke et al. 2013; Gan et al. 2014; Nürnberg et al. 2018).

Now, let us turn our attention to other stars, with the corresponding spectral type provided in parentheses. By applying (12), we obtain λ_opt ≈ 515 nm for T_⋆ = 7120 K (F2V), λ_opt ≈ 635 nm for T_⋆ = T_⊙ (G2V), λ_opt ≈ 794 nm for T_⋆ = 4620 K (K2V), and λ_opt ≈ 1375 nm for T_⋆ = 2670 K (M7V). To put it another way, the absorption peak may switch to bluish wavelengths for hotter stars and near-infrared (near-IR) wavelengths for cooler stars. In order to assess how good a predictor (12) is, we compare the formula with absorption maxima extracted from the detailed numerical model of Lehmer et al. (2021, Table 1), which was itself partly predicated on Marosvölgyi & van Gorkom (2010).

The numerical model of Lehmer et al. (2021) cited in the preceding paragraph yielded λ_opt ≈ 468 and 476 nm for T_⋆ = 7120 K, λ_opt ≈ 644 and 672 nm for T_⋆ = 5780 K, λ_opt ≈ 675, 711, and 746 nm for T_⋆ = 4620 K, and λ_opt ≈ 987 and 1050 nm for T_⋆ = 2670 K, which clearly agrees with our analytical model. We caution that (12) might be somewhat inaccurate for early M-dwarfs (Lehmer et al. 2021, Table 1), although the potential discrepancy is diminished if we compare our calculations instead with Lehmer et al. (2018, Figure 2).

As remarked a couple of paragraphs before, when one considers lower stellar temperatures, λ_opt shifts to near-IR wavelengths as per (12), while the opposite tendency is forecast for hotter temperatures. Aside from the quantitative comparison furnished above, we note that this qualitative trend is consistent with previous studies, which have generally suggested that photosynthesis on M-dwarfs might operate at near-IR wavelengths and that the spectral edge of photosynthetic organisms may be manifested in this regime (Heath et al. 1999; Wolstencroft & Raven 2002; Kiang et al. 2007b; Lingam & Loeb 2019c); however, refer to Takizawa et al. (2017) and Gale & Wandel (2017) for contrasting takes.