Sum rules and determination of exciton coupling using absorption and circular dichroism spectra of biological polymers

Abstract

Optical spectra of biological polymers contain important information about their structure and function in living organisms. This information can be accessed by extracting an optical interaction of monomers, i.e., their exciton coupling, from experimental data. This coupling is sensitive to molecular structure, geometry, and conformation and can be used to characterize them. However, the accurate determination of exciton coupling in important biological molecules is difficult because inhomogeneous broadening smears out the monomer interaction. We suggest a way to overcome this problem by applying exact sum rules. These sum rules are derived by establishing a straightforward relationship between integral characteristics of absorption and circular dicroism spectra, and exciton coupling. Exciton coupling between AT pairs in native DNA conformation is estimated by applying these sum rules to DNA hairpin optical spectra as V₀ ∼ 0.035 eV in agreement with the earlier numerical calculations.

Optical properties of biological molecules contain important information about their structure and function (1–5). Proteins and DNA are composed of interacting monomers, i.e., amino acids or base pairs. Interactions of monomer electronic excited states lead to a quantum mechanical delocalization of electronic excitation (exciton). This delocalization affects absorption and circular dichroism (CD) spectra and leads to the fluorescent resonant energy transfer between different monomers or molecules. Based on optical measurements one can attempt to determine exciton coupling strengths of monomers serving as chromophores. This coupling strength depends on the distance between monomers and their relative orientation (6, 7). Thus, by observing changes in DNA and protein optical spectra during chemical and biological processes one can investigate their function in living organisms. In addition to biological applications, exciton coupling is important in optical nanosystems involving DNA and proteins (8, 9).

Optical spectra of biological molecules are crucially sensitive to the interaction with their natural environment of highly polar solvent, i.e., water. Therefore, it is difficult to extract information about monomer interaction by using optical spectra, which are smeared out by the solvent-induced inhomogeneous broadening. Indeed, positions and orientations of water molecules with respect to each solute molecule differ from each other leading to quasi-random shifts of electronic excitation energies. These quasi-random shifts contribute to the inhomogeneous spectral broadening W > 0.1 eV, which usually exceeds an exciton coupling strength V₀ < 0.1 eV (10). Therefore, this broadening can smear out the change of spectra associated with the monomer interaction, which is sensitive to molecular conformation and geometry.

Despite the large inhomogeneous broadening, absorption and CD spectra are still sensitive to interactions of monomers (7). For instance, optical absorption spectra of DNA hairpins made of identical AT base pairs demonstrate the blue shift in the energy corresponding to the absorption spectrum maximum with the number of monomers n (11, 12). Under certain conditions this shift can be used to determine the exciton coupling strength V₀, as was recently demonstrated for DNA hairpins (10). A CD spectrum can be even more sensitive to exciton coupling V₀ than an absorption spectrum, particularly in polymers made of planar monomers located parallel to each other as in DNA. Then the circular dichroism originates entirely from exciton coupling (13) (see also refs. 14 and 15 and references therein), so in contrast to absorption this coupling entirely determines a CD spectrum. However, it is hard to extract exciton coupling from a CD spectrum because of its complicated shape sensitive to the details of inhomogeneous broadening.

In this article we propose an efficient way to overcome the effect of inhomogeneous broadening by using the exact sum rules for CD spectra. These sum rules are based on the Van Vleck theory (16) and its later extension to absorption spectra (17) (see also ref. 18). We show that if the excited electronic state of monomers under consideration is well separated from other states, then one can express the exciton coupling of neighboring monomers V₀ in terms of energy-dependent absorbance A(E) and ellipticity angle Θ(E) characterizing CD strength (expressed in radians)

where n is the number of monomers, ϕ is the angle between dipole moments of adjacent monomers, d is the distance between them, and n_r is the refractive index of the material. This equation applies to a linear system only. The interaction between nonneighboring monomers is ignored because of its rapid decrease with the distance. In this article we derive general expressions for polymer absorption and CD spectra, define the most general model for the molecule affected by the inhomogeneous broadening, derive exact relationships (e.g., Eq. 1) and apply them to determine exciton coupling in DNA hairpins composed of AT base pairs. Using Eq. 1 and experimental data for C₁₂-linked DNA hairpins, we estimated the exciton coupling between adjacent AT pairs in DNA as V₀ ≈ 0.035 eV, which agrees with the semiempirical calculations (10).

Model

In this section we define the model for calculation of absorbance A(E) and circular dichroism Θ(E) spectra by using the Fermi Golden rule (19). Circular dichroism is determined by the difference between absorbances δA(E) of left-hand and right-hand circularly polarized waves, which can be described by their vector potentials A_l,r(r,t) = Re(A₀exp(iqz − iωt)(e^x ∓ ie^y) for left- and right-hand polarized waves, respectively. Here, A₀ is the amplitude of the wave propagating in the z direction, ω and q = n_rω/c are frequency and wavevector, n_r is the medium refractive index, and c is the speed of light. e_x and e_y are unit vectors in x and y directions. The ellipticity angle Θ(E) (in radians) used to characterize CD spectra can be expressed through the absorbance difference δA(E) as Θ(E) = ln(10)δA(E)/4.

Molecules absorb optical energy because of their interaction with light (20)

where ĵ(r) stands for the operator of an electron current density. For a polymer molecule made of n monomers having no remarkable electronic overlap, one can express this interaction Eq. 2 as the sum of interactions with each monomer i = 1,2,…n

Here, r_i and z_i stand for the average coordinate of the monomer i and ρ = r − r_i.

The absorbance of light with the photon energy E = ℏω by a sample of a thickness L with a concentration of solute molecules n_* in the linear regime can be expressed by using the Fermi Golden rule as

where (j_i^p(q))_0a = ∫ dρΨ₀*e^iqρĵ_i^p(ρ)Ψ_a, index a enumerates electronic excited states of the molecule, and wavefunctions Ψ describe molecular electronic eigenstates. Configurational average 〈…〉 is made over different realizations of solvent environment with respect to each molecule.

The system absorbance can be determined ignoring small products qz ∼ qρ ≈ 0 because the wavelength of light λ = 2π/q is much larger than the molecular size. In this approximation there is no difference between right- and left-hand polarizations. After the averaging of Eq. 4 over molecular orientations with respect to the external electromagnetic field we get

where $\widehat{\mu }$_i = i ∫ dρĵ_i/ω stands for the dipole moment operator of the monomer i and n_abs is the concentration of molecules in absorption measurements.

Circular dichroism shows up in the first order in ql < 1, where l ∼ 1 nm is the molecular size. Below we expand Eq. 4 to the first order in ql. One can find 3 contributions associated with 2 different terms in the expansion e^{iq(z_i−z_j)+iq(ρ−ρ′)} − 1 ≈ iq(z_i − z_j) + iq(ρ−ρ′). Both the first and second terms can be different for left- and right-hand polarized light and lead to the circular dichroism. The ellipticity angle determined by the difference of absorbances for right- and left-hand waves can be expressed after averaging over external field directions as

Here, n_CD is the concentration of molecules used in the circular dichroism measurements. The operator $\widehat{r}$_i = $\frac{1}{2c}$ ∫ ρ × ĵ_idρ stands for the monomer magnetic moment and the current operator î_i is the derivative of the dipole moment operator, so that i_i^0a = −i_i^a0 = iωμ_i^0a.

The sum rules can be derived from Eqs. 5 and 6. Before considering sum rules we define the model for the molecule.

The exciton Hamiltonian of a polymer molecule can be described by using the tight binding model coupled to the environment leading to fluctuations of monomer excitation energies φ_i. It can be written as (13)

Here, c_i, c_i⁺ are operators of creation and annihilation of the exciton in a site (monomer) i. Random excitation energies φ_i can be characterized by their configurational averages ϕ_i = 〈φ_i〉 and a distribution function P(φ₁,φ₂,…φ_n). Interaction V_ij characterizes exciton coupling of monomers i and j. This coupling is most important for adjacent monomers. Since the distance between monomers is often comparable to their size (as in DNA) it must include various multipole interactions and cannot be reduced to the dipole–dipole interaction.

All monomers i = 1,2,…,n possess transition dipole moments μ_i and magnetic moments m_i having identical absolute values μ₀, m₀. The operators of transition dipole moments entering definitions Eqs. 5 and 6 can be expressed as

Our model is applicable when the electronic excitation of interest is well separated in energy from other electronic excitations. The overlap of absorption and CD spectra can be ignored when the energy difference Δ between this and the next electronic states exceeds inhomogeneous broadening

The interaction of excitation with quantum vibrational modes ℏω > k_BT can also be included into our consideration in the standard form (21) of the shift of vibrational equilibria for excited states. Since it does not affect our results for sum rules, we will ignore that interaction.

Results

Sum rules can be expressed as the relationships for various momenta of the absorbance and ellipticity angle functions Eqs. 5 and 6. Consider the integral ∫₀^+∞dEA(E)/E using the definition Eq. 5. The integral can be evaluated as

Using the definition of transition dipole moment operator Eq. 8 we got Σ_a〈$\widehat{\mu }$_i^a0$\widehat{\mu }$_j^a0〉 = μ₀²〈0|c_ic_j^†|0〉 = μ₀²δ_ij, where δ_ij is the Kronecker symbol (i.e., δ_ij = 1 if i = j and 0 otherwise). Then Eq. 10 takes the form

Next, consider the integral ∫₀^+∞dEA(E). Similarly to the previous derivation one can evaluate it into the form

Since quantum states a are eigenstates of the system Hamiltonian one can rewrite the sum over states a in Eq. 12 as the double sum

Because of its invariance with respect to the basis a one can change it to the basis of single-site excitations |i〉. Then one can rewrite Eq. 12 as

where ϕ_ij is the angle between transition dipole moments of i th and j th monomers and 〈φ〉 is the average monomer absorption energy. It is convenient to consider the ratio of Eqs. 14 and 11. This ratio can be expressed as

where ϕ is twisting angle for adjacent monomers. This result coincides with the sum rule derived in ref. 17.

Since the interaction V_ij decreases rapidly with the distance r_ij one can assume that only nearest-neighbor exciton coupling V₀ = V_i,i+1 is important. Then one can express it as

This result is independent of inhomogeneous broadening and can be used to study exciton coupling in various polymer molecules made of identical monomers. The dependence on the number of monomers and their coupling is similar to the one for the maximum of a rescaled absorption β(E) = A(E)/E (10). However, the shift of maximum in β(E) depends on the correlations between different site energies and the shape of the distribution of their fluctuations. In the absence of correlations and for the Gaussian distribution of fluctuations the shift of maximum in β(E) is twice as large as that in Eq. 16. One should notice that in contrast to Eq. 16, this shift can be sensitive to vibrational interaction.

Despite a stronger universality of the sum rule Eq. 16 compared with the maximum change described in ref. 10 it is more difficult to apply this method to real spectra because the effect of interest represents a small correction. The overlap with other absorption bands can be comparable to it because it is not associated with weak exciton coupling. We were not able to apply this sum rule to the absorption spectra of stilbene-linked hairpins (11) because the contribution of stilbene to the tails of integrals in Eq. 16 could not be neglected in contrast to its effect on the maximum position (10). The situation is better for circular dichroism spectra, where exciton coupling can be responsible for the major contribution.

Consider sum rules for CD spectra. First, we evaluate the integral ∫₀^∞dEΘ(E)/E² by using Eq. 6. Similarly to Eq. 10 only the diagonal term with i = j contributes to this integral. There, diagonal term contribution in Θ₁ vanishes because the vector product of 2 identical vectors is equal to zero (μ_i × μ_i = 0). The evaluation of the second term results in the expression

If dot products of the transition current (the transition dipole moment multiplied by the frequency) and the transition magnetic moment (mj*) for each monomer differ from 0, then the circular dichroism is determined mostly by individual monomers. We do not consider this situation, which can be handled by first-principle methods (22).

Further consideration is restricted to the case mj* = mμ = 0 for each monomer. The latter condition can be fulfilled for the systems having a special symmetry. For instance, if the monomer has a planar geometry as in DNA, then its transition dipole moment can also be parallel to that plane, while the magnetic moment is perpendicular to that plane. This takes place for electronic excitations of interest in DNA base pairs as well as in some amino acids (2, 6). In these systems the CD spectrum is determined by monomer interaction and Eq. 17 takes the form

Second, we evaluate the integral ∫₀^∞dEΘ(E)/E. This integral can be calculated similarly to Eq. 14. After evaluation of matrix elements we get

Exciton coupling can be conveniently expressed through the ratio of Eqs. 19 and 14. This ratio reads

Further simplification is possible if all monomers have planar structure and these planes are parallel to each other, as in DNA. Then all dot products of matrix elements for transition currents and magnetic moments are zeros and only the first term survives in Eq. 20. The second term can also be neglected compared with the first term when the distance between adjacent monomers exceeds their characteristic sizes. Assuming that the first condition is satisfied and leaving only nearest-neighbor interaction V₀ one can express the identity Eq. 20 in the form equivalent to Eq. 1

remember that the length d is the distance between adjacent monomers.

The identities Eqs. 21 and 16 are the main results of this work. Below we use them to determine the exciton coupling strength V₀ between AT base pairs in DNA and compare this result with the numerical calculations and experimental definition by using a maximum in rescaled absorption spectrum (10).

Discussion

We discuss absorption and circular dichroism data in C₁₂-linked hairpins H1–H4 (Fig. 1). These hairpins are more convenient for optical measurements than stilbene-linked hairpins because, in contrast to stilbene, a C₁₂ linker has no absorption band overlapping with the low energy absorption band of AT base pairs. The syntheses and characterization of hairpins H2 and H4 have recently been reported (23). A minimized structure for H2 obtained from molecular dynamics simulations is shown in Fig. 2. Salient features of this structure are the extended all-anti conformation of the alkane linker and the B-DNA structure of the base pair domain, which is similar to that recently reported for a hexa(ethylene glycol)-linked hairpin (24).

Fig. 1.

Structure of C₁₂-linked DNA hairpins.

Fig. 2.

A minimized structure for H2 obtained from molecular dynamics simulations (24).

The absorption spectra of the DNA hairpins are shown in Fig. 3. The absorbance per base pair decreases slightly with increasing length for hairpins H1 > H2 > H3 and is greater for the alternating sequence H4 vs. the homopolymeric sequence H2. These trends are consistent with increased hypochromism with increased A-tract length (25). The low-energy absorption peak at the wavelength 260 nm corresponds to the photon energy E₁ ≈ 4.5 eV. It is separated from the next peak at E₂ ≈ 5.8 eV by the energy on the order of 1 eV. Inhomogeneous broadening of the first peak estimated using the Gaussian fit is on the order of 0.4 eV so the peaks are separated. Therefore, one can try to use sum rules to determine the exciton coupling for electronic excitations, corresponding to the first peak.

Fig. 3.

Absorption spectra of C₁₂-linked hairpin sequences H1–H4 in 10 mM phosphate buffer (pH 7.2) containing 100 mM NaCl normalized for the number of base pairs.

Fig. 4 shows the CD spectra of hairpin sequences H1–H4. A careful analysis of the long-wavelength region (230–300 nm) of the CD spectrum provides a clear understanding on the contribution of the DNA bases on the first excited state. The positive Cotton effect with a zero crossing at 260 nm near absorption maximum could be attributed to the fact that the CD spectrum is determined by the difference of left- and right-hand polarized waves. Since the DNA double helix is right-handed and shorter wavelengths correspond to a nearly parallel orientation of dipole moments of interacting base pairs in a collective exciton state (10) the right-hand polarized wave will be absorbed more strongly here. Therefore, the difference between absorptions of left- and right-hand polarized waves is negative. However, the longer wavelength region of the spectrum contains dipole moments of different base pairs that are oriented nearly opposite to each other, which leads to the stronger absorption of left-hand polarized waves and consequently positive Cotton effect.

Fig. 4.

Circular dichroism band of C₁₂-linked hairpin sequences H1–H4 in 10 mM phosphate buffer (pH 7.2) containing 100 mM NaCl normalized for the number of base pairs.

The experimental data for ellipticity angle Ψ in Fig. 4 are given in millidegrees. One can reexpress the identity Eq. 1 in terms of experimental units using the definition Ψ = Θ·180 · 1000/π, DNA parameters ϕ = 36° for poly(A)–poly(T) sequences (H1, H2, H3) and d = 3.4A, water refractive index n_r = 1.33 and experimental ratio n_abs/n_CD ≈ 10 as

Exciton coupling V₀ is expressed in electron volts.

To estimate the exciton coupling strength Eq. 22 one has to perform the numerical integration of absorption and CD spectra (Figs. 3 and 4) over the part of the spectrum associated with the first electronic excited state, i.e., the longest-wavelength absorption band. This band is essentially symmetric with respect to the maximum until large energies on the order of 5.1 eV (short wavelengths), where other electronic excitations become important. The part of the absorption spectrum on the long-wavelength side of the maximum is less affected by other excitations. Therefore one can estimate the integral in the denominator of Eq. 22 as the doubled integral over the low-energy (long-wavelength) side. Performing the integration we evaluated I_X^abs = ∫₀^+∞ dEA(E)/E as I_H1 ≈ 0.156, I_H2 ≈ 0.17, I_H3 ≈ 0.168, I_H4 ≈ 0.14 (see Fig. 1 for definitions of DNA hairpins H1, H2, H3, and H4).

We assumed that the first bright electronic excitation of AT pair can be approximately separated from other excitations at least for the lower-energy part of the spectrum. This expectation is justified by our calculations of AT pair excitation spectrum preliminary reported in ref. 10. According to these calculations there is the one lowest energy bright state at the excitation energy 4.2 eV. This is a reasonable estimate for the observed absorption peak at 4.5 eV. Next bright excited state of AT pair has been found at energy 0.3 eV higher. Therefore, we assume that the Cotton effect is due to the lowest-energy bright transition and the contribution of the second transition can be almost ignored for the low-energy wing of the CD band. Although separate A and T bases have almost identical excitation energies of their first bright excited states (see refs. 26 and 27, and references therein) the situation can be different for AT pairs, for instance, due to electrostatic interactions of A and T bases. Below, the reasonable agreement of estimates for exciton coupling of adjacent AT pairs obtained using sum rules and numerical calculations is demonstrated. This agreement also supports our consideration of a single transition. The additional features in the CD band, including the inflections around 265 nm, can be due to the next-neighboring AT pair interactions; their analysis is beyond the accuracy of our approach.

It is more difficult to estimate the integral over the ellipticity angle in the numerator in Eq. 22. Its short-wavelength (high-energy) part is clearly affected by high-energy electronic excitations. To evaluate it approximately, using only the low-energy part of the spectrum similarly to the absorption spectrum, one can redefine variables as E = E₀ + ξ, where E₀ is the energy of spectrum maximum E₀ corresponding to the zero circular dichroism. Then, the integral reads

The first term in Eq. 23 is equal to zero according to the identity Eq. 18. Two remaining integrals must be approximately equal to each other for the single electronic excitation according to the perturbation theory (13, 14) for symmetric absorption spectra. Since the absorption spectrum density A(E)/E is approximately symmetric with respect to its maximum at E = E₀ and the perturbation theory (13) in V/W should be applicable for W ∼ 0.4 eV and V₀ ∼ 0.05 eV (10) one can approximate the integral in the numerator of the identity in Eq. 22 as

Numerical evaluation of integrals for different hairpins (Fig. 1) yields I_H1^CD ≈ −0.0368, I_H2^CD ≈−0.045, I_H3^CD ≈−0.0433, I_H4^CD ≈−0.024.

Using these results one can estimate exciton coupling for different hairpins. The first 3 hairpins H1, H2, and H3 are poly (A)–poly(T) sequences of n AT pairs with n = 4, 6 and 8, respectively. Using Eq. 22 we obtained three estimates V₀^H1 = 0.035 eV, V₀^H2 = 0.036 eV, and V₀^H3 = 0.033 eV. The average coupling V₀ = 0.035 eV is close to the result of semiempirical INDO calculations using Gaussian software (22) V₀ ≈ 0.04 eV reported in ref. 10, which is a reasonably good agreement.

The fourth hairpin H4 is made of n = 6 alternate AT-TA base pairs. Exciton coupling of nearest neighbors AT-TA should differ from AT-AT coupling V₀ estimated above. Indeed we obtained V₀^TA,AT ≈ 0.023 · sin(36°)/sin(ϕ′) eV, where ϕ′ is the angle between transition dipole moments of adjacent AT and TA pairs. It is not surprising that V₀^TA,AT < V₀ because the distance between electronic excitations is larger in the alternate sequence. The exciton coupling V₀^TA,AT and the angle ϕ′ were estimated using INDO method similarly to ref. 10 as V₀^TA,AT = 0.021 eV, sin(ϕ′) ∼ 0.64. This estimate agrees very well with the one based on the sum rules. Thus, our theoretical analysis is consistent with the calculations of exciton coupling.

Conclusion

In this article we derived the sum rules for absorption and circular dichroism spectra of polymer molecules. Using these sum rules one can express exciton coupling strength through the integral properties of absorption and CD spectra. This result can be used in the experimental investigation of structural changes in biological molecules during chemical or biological processes.

The results are insensitive to the inhomogeneous broadening. However, they are applicable only if other electronic excited states are well separated from the state under consideration.

Using the sum rules applied to DNA hairpin absorption and CD spectra, we estimated exciton coupling between AT-AT and AT-TA base pairs to be 0.035 eV and 0.023 eV, respectively. This result agrees with the numerical calculations. This analysis can be easily generalized to other DNA conformations. Thus our method establishes the direct link between absorption and CD spectra, and DNA structural and conformational changes through exciton coupling strength of base pairs. This link can be used to investigate chemical and biological processes involving DNA using optical methods.