Solvent effects on the optical properties of photosynthetic pigments evaluated by evolutionary optimization

With the development of computational technologies, it became possible to do numerical simulations of the optical properties of molecules and organic crystals taking into account the modern sophisticated theoretical approaches. Our work is devoted to the study of carotenoids in different solvents. Carotenoids, being photosynthetic pigments, are located inside pigment-protein complexes and are covalently bound to the proteins. They have a wide variety and despite their relative simple structure, their optical properties remain poorly understood. Thus, their role in photosynthetic machinery is still an open question. In this study, we investigate the properties of carotenoid electronic excitations by fitting their absorption spectra considering pigments in polar and nonpolar solvents. As it is known, carotenoids have four main nuclear vibration modes. Each mode is characterized by three parameters: frequency, Huang-Rhys factor and damping factor. In particular, the Huang-Rhys factor cannot be measured experimentally. To do the modeling, we developed special software to fit experimental data using differential evolution (DE) algorithm, which allows obtaining an unambiguous solution for applied quantum models. With the help of DE, it was possible to estimate the influence of the solvent on the excited states of a pigment. Examining astaxanthin and lycopene spectra in three solvents (chloroform, hexane, ethanol), we have shown that some quantum parameters are very sensitive to the type of solvent and can be considered as markers of polarity.


Introduction
To study the energy levels of biological molecules containing hundreds of atoms by means of modern density functional theory requires the knowledge of their microstructure: the location in space of all atoms is essential for optimizing the molecular structure of the ground and excited states.These numerical simulations are time-consuming and can only be carried out using modern computer facilities.However, the combination of optical experimental techniques and application of semiclassical theories of optical response can greatly simplify the problem [1].
Considering carotenoids as the object of study, we are going to demonstrate how differential evolution (DE), a heuristic algorithm of evolutionary optimization [2], can be an excellent choice for modeling the optical response and simultaneous fitting of experimental data [3,4].All carotenoids have a polyene chain and most of them have end chemical groups.Their optical properties strongly depend on the length of the polyene chain and the type of end groups [5].They are two basic types of carotenoidscarotenes (have no oxygen atoms) and xanthophylls (have oxygen atoms).Despite the relatively simple chemical structure of carotenoids, the dependence of the optical properties on the various modifications of this kind of pigments is very strong and requires specific theoretical studies.Thus, applying DE, we developed an effective method of fitting the absorption spectra of pigments and at the same time gaining the parameters of quantum models [4,6].
One carotene (lycopene, a simple polyene chain without oxygen) and one xanthophyll (astaxanthin, with four oxygen atoms, two in each of the pyrrole rings) were taken for modeling.To study the optical response of a single molecule, it must be isolated from the others using a solvent.Generally, solvents have different effects on absorption spectra of carotenoids, in particular, on the structure of electron clouds of the ground and excited states.Solvents are also divided into two types: polar and nonpolar.The polarity of the solvent depends on the magnitude of the dipole moment, and for the correctness of the model, it is preferable to use solvents of both types.The scheme of the algorithm is shown in Figure 1.The objective function in our simulations is the difference between the calculated and measured spectra that depends on many variables.The algorithm works as follows: at the initialization, a set of vectors is created in the n-dimensional space (n is the number of free parameters) where the objective function is defined.Then, in each of the following iterations, a new generation is created, which is a linear combination of three vectors of the old generation.A crossover operation is made on the mutant vector consisting in the fact that some coordinates are replaced by the coordinates of the original vector with a certain probability, which is also an important parameter of the algorithm (Figure 1).
Finally, we will discuss the obtained results of the lycopene and astaxanthin quantum models.Particularly, the Huang-Rhys factors of the C=C and C-C vibronic modes, and their influence on the allowed (S2) electronic excited state of carotenoids.

Theory
To calculate the absorption spectra of monomeric pigment molecules, we use the multimode Brownian oscillator model (MBOM).A detailed methodology of the numerical calculations is described in [6].The spectral density function  ′′ () is the foundation stone of our simulations [1]: According to the basic framework of MBOM, the spectral density depends on frequency   , Huang-Rhys factor   and damping factor   of each mode.These three variables {  ,   ,   } characterize the jth mode and, simultaneously, they are free parameters in the optimization procedure.The most difficult to interpret and evaluate is the Huang-Rhys factor, since, unlike the two other variables, this one cannot be found experimentally.Therefore, the exact quantum model and the correct work of the algorithm are necessary conditions for finding it.The overall expression for the absorption spectrum of a carotenoid can be written as where   is the energy of the S2 electronic transition of carotenoids, Δ =  2√2 • 2 ⁄ is a parameter of the inhomogeneous broadening, and () is the lineshape function [1], the correlation function of the S2 transition moment, which depends on the ambient temperature: Sequential evaluation of (1), (3), and finally (2) provides us the theoretical absorption spectrum to compare with the experimental one.

Results and Discussion
To explore the effect of solvents on the optical properties of carotenoids, absorption spectra of astaxanthin and lycopene in chloroform, hexane, and ethanol were fitted by the optimization algorithm.We had a following set of the quantum model parameters (electron transition energy   , the full width at half maximum of the inhomogeneous broadening , four vibronic modes, two overtones, and the sum mode) to be optimized.
The number of vibronic modes depends on the structure of the biological pigment.For example, chlorophylls have 30-40 vibronic modes with different intensity.Carotenoids have four vibronic modes, and therefore, using his example, we can better see the influence of each mode on the spectrum.These modes are: C=C (ν1 = 1523 cm -1 ) and C-C (ν2 = 1157 cm -1 ) stretching modes, methyl rocking vibrations mode (ν3 = 1004 cm -1 ) and hydrogen wagging (ν4 = 962 cm -1 ).Considering that the modes ν1 and ν2 have the greatest influence, in addition to the main modes, we add to the model two overtones (oscillations at double frequencies of ν1 and ν2) and the mode of their sum (ν1 + ν2).DE has 10 different strategies and two parameters to tune the performance of the algorithm: Fthe weighting factor and Crthe crossover rate.Based on our previous calculations [6], the optimal DE strategy is DE/best/1/bin and the optimal values are F = 0.55 and Cr = 0.9.The results of astaxanthin and lycopene spectra modeling are presented in Table 1 1.2e-4 3.5e-4 3.7e-8 9.2e-8 0.25 0.05 0.10 4.1e-5 0.08 8.2e-5 0.05 3.6e-5   1 + 2 1.3e-5 3.6e-5 1.3e-7 3.4e-7 5.7e-6 1.2e-5 0.02 4.3e-5 0.02 2.0e-4 0.04 7.2e-5  2 1 0.12 2.8e-4 0.20 3.4e-5 0.09 0.02 0.07 2.6e-5 0.04 1.2e-4 0.04 4.7e-5 Table 1.The values of microparameters for astaxanthin and lycopene in three solvents obtained.10 runs were made for each set We can see that for astaxanthin the type of solvent turned out to be much more significant than for lycopene.It was shown that the inhomogeneous broadening of astaxanthin is on average 2 times higher than that of lycopene.In lycopene, the influence of double and single bonds is approximately the same; in astaxanthin, the influence of double bonds is bigger than of single bonds.The influence of ν3 and ν4 is negligible for both carotenoids in all solvents (Table 1).Thus, as a result of our simulations, the experimental data were fitted with high accuracy using DE algorithm.This allows concluding that it is possible to find unique quantum models and the values of their microparameters with high accuracy for all types of carotenoids and solvents.

Figure 1 .
Figure 1.A flowchart of optimization of carotenoid absorption spectra modeling.The initialization is in the top panel.The rest of the panel shows the main cycle of DE.The plots present calculated spectral density and absorption spectra .