Electronic circular dichroism of proteins computed using a diabatisation scheme

ABSTRACT

Circular dichroism (CD) spectroscopy is a powerful technique employed to study the structure of biomolecules. More accurate calculation of CD from first principles will aid both computational and experimental studies of protein structure and dynamics. We apply a diabatisation scheme to improve the description of nearest neighbour interactions between two electronic transitions (nπ* and π_nbπ*) localised on each individual peptide bond (amide group) in a protein. These interactions are incorporated into DichroCalc, an exciton-based computational method to calculate CD, and yield improvements over the standard DichroCalc parameter set, particularly for calculation of CD for important secondary structural elements such as an α helix.

GRAPHICAL ABSTRACT

KEYWORDS:

• Exciton
• CD spectroscopy
• diamide
• ab initio
• far-UV

Introduction

Calculations from first principles of the electronic circular dichroism (CD) spectroscopy of proteins are challenging. It is a topic that has attracted the interest of many researchers, including the late Nick Besley [1], to whose memory this Special Issue is dedicated. In this paper, we present a new theoretical development, but we begin by surveying some of the contributions of Besley to the field, which underpin the current study. The CD spectra of proteins in the far-ultraviolet (far-UV) have distinct signatures, reflecting the protein secondary structure, which arise from the regular, repeating arrays of backbone amide chromophores. Thus, CD is a widely used biophysical technique for characterising proteins in solution [2]. Often, statistical techniques are used to estimate the secondary structure content from the experimentally determined CD spectrum [3,4]. However, our focus is on the reverse, namely, computing the CD spectrum from the atomic coordinates of the protein.

CD spectra can be computed from protein structures in various ways, e.g. see [5–10], some of which have been reviewed elsewhere [11]. We focus here on the exciton framework, whereby an effective Hamiltonian is constructed through a consideration of localised chromophores within the protein [12,13]. The peptide bond is the most important contributor to the far-UV CD spectra of proteins. A simple effective Hamiltonian would consider the nπ* and π_nbπ* electronic transitions of the amide chromophore, which occur at about 222 and 190 nm, respectively. The π_nb, n and π* molecular orbitals for an example diamide are depicted in Figure 1.

Figure 1. Molecular orbitals for diamide 2a [φ = 180°, ψ = 180°] with an isosurface cut-off of 0.05. Orbitals from a state-averaged CASSCF calculation as described in reference 32. (a) π* for amide 1, (b) n for amide 1, (c) π_nb for amide 1, (d) π* for amide 2, (e) n for amide 2 and (f) π_nb for amide 2.

A desire to improve the quality of protein CD calculations that use the exciton framework motivated several studies of the electronic structure of small amides using ab initio multi-configurational methods, including complete active space self-consistent field (CASSCF) calculations on formamide [14] and N-methylacetamide [15] in the gas phase. Analogous calculations with various models of solvent showed subtle, but important, changes to the electronic transitions, including, for example, differences in the orientation of the π_nbπ* electric transition dipole moment compared to the gas phase [16,17]. This factor was influential in leading to an improvement in the accuracy of calculations of protein CD spectra based on parameterisation of the ab initio descriptions of the amide chromophore [18–20] compared to spectra calculated using parameters from semi-empirical calculations [21]. The computational methodology for protein CD calculations and the chromophore parameters are available via a web-interface [22], which allows the non-expert simply to upload a Protein Data Bank file with the coordinates of a protein structure and to compute the CD spectrum. The theoretical background and related developments have been summarised elsewhere [11,23] and the approach has been used, sometimes in conjunction with molecular dynamics simulations, to study a diverse range of systems, e.g. glycosylated antimicrobial peptides [24], SARS-CoV-2 proteins [25] and chymotrypsin adsorbed on silica [26].

The above discussion has focused on two localised electronic transitions, the nπ* and π_nbπ* excitations, on an individual amide chromophore. Whilst the computational approach based on this approximation has been successful, additional insights have been gained through the study of cyclic [27–29] and linear diamides [30–32]. Electronic excitations with higher transition energies may be important [33] and in this region charge transfer transitions between orbitals on neighbouring peptide groups are also observed. Quantum chemical calculations at the CASSCF level on a linear diamide enabled the generation of parameters describing the charge transfer transitions [31]. Protein CD calculations incorporating these transitions gave an encouraging level of agreement with synchrotron radiation CD spectra [34] measured down to 170 nm. The CASSCF calculations on the diamide provide an additional opportunity, which is the subject of our study, namely, to consider the interaction between two covalently bound amide chromophores at the quantum chemical level using a procedure known as diabatisation.

The effective Hamiltonian matrix in the exciton framework comprises diagonal elements corresponding to the energies of monomer (diabatic) states and off-diagonal elements. The latter are inter-state interactions or couplings. Whilst the isolated monomer, i.e. N-methylacetamide, appears to be a good choice of diabatic state with the couplings computed using a Coulombic model, an alternative approach would be to apply a diabatisation scheme to a full ab initio calculation on a diamide model, whereby the eigenstates of the diamide are transformed to a diabatic basis. This method is capable of treating multiple excited states and also taking into account both short- and long-range effects of the excitonic couplings. Thus, an orthogonal transformation is used to connect the adiabatic excited states with a set of diabatic excited states. There is no single ‘ideal’ set of diabatic states [35], but several strategies can be adopted to define the adiabatic-to-diabatic transformation. The fragment excitation difference scheme uses the so-called excitation density [36]. A variant of this and some extensions are available in the EXAT software [37]. An alternative strategy derives the couplings from the energy splitting between the excited states and the extent of exciton delocalisation between the fragments [38]. Other approaches have maximised the overlap with a reference wavefunction [39] or the similarity with the dipole moment [40]. Diabatisation approaches have been applied to aggregates of anthracene and tetracene [41], to aggregates of propeller-shaped emitters [42] using the fromage software [43], dimers of ethylene and derivatives [44] and to various nucleic acid systems [45,46]. Herein, we apply a diabatisation scheme to compute the CD of some polypeptide structures corresponding to conformations of particular significance, including α-helices, β-strands, polyproline II helices and 3₁₀-helices.

Proteins rich in β-strands can be separated into two distinct classes: β-I and β-II [47]. This classification can be made based on their CD spectra [47,48]. β-I proteins, e.g. concanavalin A, contain regular β strands and give rise to CD spectra with a positive band at 195 nm and a negative band at 215–220 nm. β-II proteins, e.g. elastase, yield CD spectra reminiscent of unordered polypeptides with a negative band around 198 nm [49,50]. It is thought that β-II proteins may adopt more polyproline II conformation [50], in addition to the possibility that they are more conformationally labile. Accurate computation of the CD spectrum of the polyproline II conformation would, thus, help provide useful insight into the spectra of β-II proteins. Another conformation of interest is the 3₁₀ helix, which, despite some structural similarity to the α helix, has a distinct CD spectrum [51].

Methods

Diabatisation

A diabatisation procedure was performed to generate more accurate inter-amide couplings between two electronic transitions (nπ* and π_nbπ*) localised on each of the two amide monomers when in a dimeric system (a diamide). The procedure follows that described by Aragó and Troisi [41], which is based on orthogonal transformation of the adiabatic states (eigenstates of the total Hamiltonian) to diabatic (noninteracting) states for a two-level dimer system and generalisable to systems with an arbitrary number of interacting states (four in this study). For two monomers (amides) labelled 1 and 2 that combine, via a peptide bond, as a dimer (a diamide), consider four singlet excited states on the dimer and two singlet excited states on each of the monomers. The four adiabatic excited states on the dimer are the n₁π₁* and π_nb1π₁* localised on amide 1 in the dimer and the n₂π₂* and π_nb2π₂* localised on amide 2 in the dimer. The two excited states on each of the two isolated monomers are nπ* and π_nbπ*. The diabatic Hamiltonian H^D is found by a transformation of the adiabatic Hamiltonian H^A, thus: (1) ${\mathbf{H}}^D=\mathbf{C}{\mathbf{H}}^A{\mathbf{C}}^{†}$(1) The unitary transformation matrix C relates the diabatic states (on the dimer) to the adiabatic states (on the dimer). It is desirable that the diabatic wave functions be as close as possible to the initial noninteracting wave functions. The unitary matrix C is computed by minimising the difference between the adiabatic transition dipole moments (TDMs) on the dimer and the isolated monomer TDMs in the coordinate frame of the dimer [41]. Computation of matrix C proceeds as follows [41].

The matrix M is formed from a product of matrices representing the adiabatic TDMs on the dimer, µ^A, and the isolated TDMs on each of the monomers, µ^ISO, (2) $\mathbf{M}=({\mathit{\mu }}^{\text{A}}{)}^{\ast }{\mathit{\mu }}^{\text{ISO}}$(2) Singular valued decomposition is carried out on matrix M to find the matrices U and V*. These are used to obtain C*^,best, (3) ${\mathbf{C}}^{\ast ,\mathrm{best}}\text{ = }\mathbf{U}{\mathbf{V}}^{\ast }$(3) where C*^,best is the transpose of C^best which is defined as the best unitary transformation satisfying the following equation, (4) ${\mathbf{C}}^{\ast ,\mathrm{best}}=\text{arg}\text{mi}{\text{n}}_{\mathbf{R}}||{\mathit{\mu }}^{\text{A}}\mathbf{R}-{\mathit{\mu }}^{\text{ISO}}||$(4) where $||\cdot ||$ denotes the Frobenius norm.

The adiabatic Hamiltonian is diagonal, where the diagonal elements are the transition energies of the excited states in the dimeric system. The output of the diabatisation is the diabatic Hamiltonian with transition energies on the diagonal and the coupling between excited states in the diabatic representation in the off-diagonal matrix elements. The amide is the monomer and the diamide is the dimer. The monomer and the dimer state properties (TDMs) and adiabatic Hamiltonian are listed as matrices in Table S1. The input properties are from DichroCalc parameter sets for the amide N-methylacetamide (set NMA4FIT2) [17] and the diamide N-acetylglycine N^′-methylamide (13 sets for various φ and ψ dihedral angles of the diamide) [31,32]. The TDMs for the two monomers (from the above parameter set) are transformed onto the coordinate frame of the dimer prior to diabatisation. The 13 pairs of φ and ψ angles and the corresponding parameter sets for the diamide are listed in Table 1. The geometries were chosen to span the Ramanchandran plot. The diabatic Hamiltonian output from the diabatisation contains off-diagonal matrix elements that are the couplings between the states on each monomer in the system of the dimer, and it is these quantities that are used to modify the computation of CD (discussed below). The diabatic Hamiltonian matrices are listed in Table S1.

Table 1. The 13 diamides and their secondary structure and associated DichroCalc parameter set.

φ angle (°)	ψ angle (°)	Secondary structure	Diamide in Oakley et al. [32]	DichroCalc parameter set [22]
180	180	Fully extended	2a	CT01009A
−120	180		2b	CT02009B
−60	180		2c	CT03009A
−135	135	β strand	2d	CTBE009A
−120	120		2e	CT05009A
−120	60		2f	CT08009A
−120	0		2g	CT11009A
−60	0		2h	CT12009A
−74	−4	310 helix	2i	CT31009A*
−48	−57	α helix	2j	CTAL009A
−60	−60	α helix	2k	CT15009A
−62	−41	α helix	2l^#	CTBT009B*
−75	145	Polyproline II	2m^#	CTP2007A

^# Notation following Oakley et al. [32] * Set exclusive to Oakley and Hirst [31].

Model peptides

13 model idealised peptide structures, each comprising 20 L-alanine residues, with different repeating φ and ψ dihedral angles to describe differing proteinaceous secondary structure elements were constructed using the molecular editor Avogadro [52,53]. The 13 model peptides were considered based on the availability of DichroCalc parameter sets that describe intra-amide and inter-amide (charge-transfer) electronic transitions in a diamide (N-acetylglycine N^′-methylamide) of varying φ and ψ angles [31,32], for which the diabatisation procedure was carried out. The diamides are listed in Table 1. The naming of diamides 2l and 2m follows the labelling sequence of Oakley et al. [32], who studied diamides 2a through to 2k (Table 1). The three α-helical diamide structures 2j, 2k and 2l are, respectively, the Corey-Pauling-Branson helix, an idealised helix and the Barlow-Thornton helix [54]. Figure 2 shows a Ramachandran plot of the φ and ψ dihedral angles for the 13 structures. For clarity, we adopt a notation for each diamide and peptide with the φ and ψ dihedral angles in square brackets, e.g. for 2d the notation is 2d [−135°, 135°].

Figure 2. Ramachandran plot of the φ and ψ dihedral angles for the 13 diamide and 13 20 L-Ala peptide structures. The scatter points are labelled by the alphabetical (second) character naming each structure (see Table 1). Inset is the structure of peptide 2k [φ = −60°, ψ = −60°].

CD calculations

DichroCalc [22] was used to compute the CD spectra for the 13 model peptides. A modified version of the DichroCalc code was used to read a Hamiltonian matrix where, for each peptide, the nearest neighbour inter-amide interactions from the diabatisation procedure replaced the default off-diagonal matrix elements generated by DichroCalc. The theoretical approach utilised by DichroCalc, as well as alternative approaches to the computation of CD for biomolecules, has been described in detail in a recent review [11]. Briefly, the DichroCalc software implements the exciton framework, with the transition electron densities associated with excitations of biological chromophores represented by a set of point charges (monopoles) that reproduce the electrostatic potential arising from the electron density. The parameters (transition energies, electric and magnetic transition dipole moments, electric permanent dipole moments and monopole charge sets) for the amide backbone are derived from ab initio calculations (parameter set NMA4FIT2) [17] and describe the intra-amide nπ* transition (at 220 nm) and the π_nbπ* transition (at 193 nm). To obtain a CD spectrum, a rotational strength line spectrum, computed by DichroCalc, is convoluted with Gaussian functions of full-width half-maximum (FWHM) of typically, e.g. 9.0 or 12.5 nm. Throughout the rest of the paper, results using the un-modified DichroCalc Hamiltonian are termed ‘ab initio’ and the results obtained from modifying the off-diagonal elements of the DichroCalc Hamiltonian are termed ‘diabatic’.

Results

Figure 3 displays computed CD spectra (using Gaussian functions of FWHM of 9.0 nm) for the six most physically relevant of the 13 model peptide geometries 2d [−135°, 135°], 2i [−74°, −4°], 2j [−48°, −57°], 2k [−60°, −60°], 2l [−62°, −41°] and 2m [−75°, 145°]. Figure S1 displays rotational strength line spectra and CD spectra for all 13 peptides. For seven of the 13 model peptides, there is a marked difference for the rotational strength line spectra and for the CD spectra when the diabatic inter-amide interactions are incorporated in the Hamiltonian. The seven peptides are: 2a [180°, 180°], 2b [−120°, 180°], 2c [−60°, 180°], 2d [−135°, 135°], 2f [−120°, 60°], 2i [−74°, −4°] and 2m [−75°, 145°]. For the other six peptides, more subtle differences are found between the spectra. For the three α-helical peptides (2j [−48°, −57°], 2k [−60°, −60°] and 2l [−62°, −41°]) the diabatic CD spectra feature the double minimum (characteristic of an α-helical spectrum) in the 200–230 nm region; and the band maxima and minima are in similar regions of the spectrum for peptides 2e [−120°, 120°], 2g [−120°, 0°] and 2h [−60°, 0°] but with slightly differing band shape characteristics. Reproduction of the double minimum between 200 and 230 nm represents a qualitative improvement over the ab initio calculated spectra.

Figure 3. CD spectra for the model peptides comprising 20 L-Ala residues and named as the diamides. CD spectra with band shapes from convolution with Gaussian functions of FWHM of 9.0 nm. Spectra from DichroCalc ab initio Hamiltonian (black line) and diabatic Hamiltonian (blue line). (a) 2d [−135°, 135°], (b) 2i [−74°, −4°], (c) 2j [−48°, −57°], (d) 2k [−60°, −60°], (e) 2l [−62°, −41°] and (f) 2m [−75°, 145°].

Table S1 displays, for each of the 13 diamides, the three input matrices for the diabatisation procedure: the monomer TDMs, the dimer TDMs and the adiabatic Hamiltonian of the dimer. Table S1 also shows the outputted diabatic Hamiltonian, where the off-diagonal matrix elements 1–2 and 3–4 are intra-amide coupling interactions (for n₁π₁* with π_nb1π₁* and n₂π₂* with π_nb2π₂* transitions) and where matrix elements 1–3, 1–4, 2–3 and 2–4 are inter-amide coupling interactions (for n₁π₁* with n₂π₂*, n₁π₁* with π_nb2π₂*, π_nb1π₁* with n₂π₂* and π_nb1π₁* with π_nb2π₂* transitions, respectively).

The ab initio and the diabatic inter-amide interactions (off-diagonal elements of the Hamiltonian matrix) are shown in Table 2 for six of the 13 diamide geometries (2d [−135°, 135°], 2i [−74°, −4°], 2j [−48°, −57°], 2k [−60°, −60°], 2l [−62°, −41°] and 2m [−75°, 145°]). Table S2 lists the couplings for all 13 diamides. For the diamide in the β strand arrangement (2d [−135°, 135°]), there are significant differences in the three matrix elements for the 1–4, 2–3 and 2–4 interactions (H₁₄, H₂₃ and H₂₄). The sign is opposite for 2–3 (positive for the ab initio and negative for the diabatic) and the order of magnitude is different for 1–4 (one order larger for the diabatic) and 2–4 (one order larger for the ab initio). The diamide 2i [−74°, −4°] (3₁₀ helix) has differences in all the four off-diagonal matrix elements, with H₂₄ (π_nb1π₁* with π_nb2π₂*) having the largest difference between the ab initio and diabatic Hamiltonians.

Table 2. Inter-amide interactions. Off-diagonal elements of the Hamiltonian (cm⁻¹). DichroCalc ab initio matrix elements in the first row and diabatic matrix elements in the second row for each diamide geometry. The ab initio parameters have nπ* and π_nbπ*transition energies at 220 nm (45455 cm⁻¹) and 193 nm (51813 cm⁻¹), respectively.

Matrix element index	H13	H14	H23	H24
Diamide (φ, ψ)	n1π1*:n2π2*	n1π1*:πnb2π2*	πnb1π1*:n2π2*	πnb1π1*:πnb2π2*
2d (−135°, 135°)	72	−20	292	−907
	109	−279	−244	−89
2i (−74°, −4°)	15	−36	161	−662
	289	−148	230	284
2j (−48°, −57°)	−18	−17	−131	−272
	97	−808	0	−191
2k (−60°, −60°)	−42	−29	−180	−83
	−21	−1707	267	173
2l (−62°, −41°)	−17	−37	−52	−292
	−359	−660	−131	−363
2m (−75°, 145°)	31	−7	16	−971
	81	356	145	29

For the three α-helical diamides (2j [−48°, −57°], 2k [−60°, −60°] and 2l [−62°, −41°]), the largest difference between the ab initio and the diabatic matrix elements are for the n₁π₁* with π_nb2π₂* interaction (H₁₄), where the diabatic interactions are one order of magnitude greater than the ab initio for diamides 2j [−48°, −57°] and 2l [−62°, −41°] and two orders of magnitude for diamide 2k [−60°, −60°], with all these matrix elements having the same sign (negative). This is clearly an important factor leading to the distinctive double minimum in the α-helical spectra. The diamide representing a polyproline II helical conformation (diamide 2m [−75°, 145°]), has large differences between ab initio and diabatic Hamiltonians for matrix elements H₁₄, H₂₃ and H₂₄, with the largest differences between H₁₄ and H₂₄.

The Coulombic coupling of two electronic transitions is related to the dot product of the transition dipole moments located on monomers A and B [55], (5) $V_{\text{dd}}=\frac{{\mathit{\mu }}_{\text{A}}{\mathit{\mu }}_{\text{B}}}{R_{\text{AB}}^3}-\frac{3({\mathit{\mu }}_{\text{A}}{\overrightarrow{R}}_{\text{AB}})({\mathit{\mu }}_B{\overrightarrow{R}}_{\text{AB}})}{R_{\text{AB}}^5}$(5) V_dd is the coupling between the two dipoles µ_A and µ_B, and R_AB is the distance between the two monomers A and B. If the diabatic coupling is different from the ab initio coupling and the dot product of the transition dipole moments is similar in magnitude, then the coupling may have a short-range non-Coulombic interaction that is not described by the interaction of two dipole moments (or two sets of monopole charges) from a monomeric calculation. The diabatic coupling is from diabatisation of the adiabatic states on a complete dimer, including all the electrons in the system, using a basis set (for the diabatic states) of the monomer transition dipole moments. For cases where the ab initio and diabatic coupling differs and the magnitude of the dot product of the transition dipole moments differ, the diabatic states may contain a more accurate description of Coulombic interactions between the two transitions, due to the more accurate wave function of the adiabatic states, i.e. for the complete dimer.

Table 3 displays the dot products of the un-normalised transition dipole moments for the monomer (in the arrangement of the dimer) and the dimer for the subset of six diamides (2d [−135°, 135°], 2i [−74°, −4°], 2j [−48°, −57°], 2k [−60°, −60°], 2l [−62°, −41°] and 2m [−75°, 145°]). Data for all 13 diamides are in Table S5.

Table 3. Dot products of the un-normalised transition dipole moments (D²) for six of the 13 diamide structures (Table S1). Upper triangle of Hamiltonian H_ij. Intra-amide interactions are 1–2 and 3–4. Inter-amide interactions are 1–3, 1–4, 2–3 and 2–4. Transition dipole moments are not normalised.

Transitions	1–2	1–3	1–4	2–3	2–4	3–4
Diamide (φ, ψ)	Monomer
2d (−135°, 135°)	0.0	−0.032	−0.350	−0.342	−2.037	0.0
2i (−74°, −4°)	0.0	−0.013	−0.454	0.615	−2.985	0.0
2j (−48°, −57°)	0.0	−0.002	−0.595	0.593	1.312	0.0
2k (−60°, −60°)	0.0	0.009	−0.563	0.578	3.020	0.0
2l (−62°, −41°)	0.0	−0.003	−0.547	0.629	0.984	0.0
2m (−75°, 145°)	0.0	−0.002	0.005	−0.619	−2.865	0.0
Diamide (φ, ψ)	Dimer
2d (−135°, 135°)	0.284	−0.036	0.407	−0.399	2.842	−0.052
2i (−74°, −4°)	0.065	0.011	−0.280	−0.730	3.175	−0.512
2j (−48°, −57°)	−0.445	−0.040	−0.020	0.759	1.140	−0.127
2k (−60°, −60°)	−0.155	−0.040	−0.296	0.390	−5.282	0.529
2l (−62°, −41°)	−0.389	−0.035	−0.116	0.618	−3.997	0.434
2m (−75°, 145°)	−0.953	0.048	0.543	−0.610	−4.660	0.637

Discussion

α helices

The experimental spectrum for an α helix possesses an intense positive band peak at 190 nm and a double minimum band with the negative band peaks at 208 and 220 nm. The CD spectra computed for the α-helical peptides (2j [−48°, −57°], 2k [−60°, −60°] and 2l [−62°, −41°]) using the ab initio Hamiltonians and the diabatic Hamiltonians all feature a positive band peak around 190 nm (Figure 3), with very little difference between the ab initio and diabatic spectral bands in this region. Differences in the computed spectra are apparent at wavelengths greater than 200 nm where the double minimum is expected. For peptide 2j [−48°, −57°] the negative peak at 205 nm in the ab initio spectrum is slightly less intense and red shifted to 206 nm in the diabatic spectrum, and the negative band shoulder around 220 nm in the ab initio spectrum is both more pronounced and intense in the diabatic spectrum. At wavelengths greater than 200 nm for the peptide 2k [−60°, −60°], the ab initio spectrum has a negative band peak at 208 nm and a shoulder around 220 nm. The diabatic spectrum for this peptide displays the characteristic double minimum with negative peaks at 207 and 222 nm, with the latter peak being slightly more intense than the former. For peptide 2l [−62°, −41°] and in the ab initio spectrum, the negative band is at 205 nm and the band shoulder is around 220 nm, whereas the diabatic spectrum features a double minimum with negative peaks at 205 and 224 nm, with the former peak being more intense than the latter peak (vice versa to peptide 2k [−60°, −60°]).

For the discussion we focus on the third α-helical dimer 2l [−62°, −41°], the Barlow-Thornton helix whose geometry is representative of structures in the Protein Data Bank. For dimer 2l [−62°, −41°] the H₁₃, H₁₄, H₂₃ and H₂₄ couplings are –17, −37, −52 and –292 cm⁻¹ for the ab initio and −359, −660, −131 and –363 cm⁻¹ for the diabatic calculation. The dot product of the un-normalised TDMs for these couplings are, respectively, −0.003, −0.547, 0.629 and 0.984 D² for the monomer and −0.035, −0.116, 0.618 and −3.997 D² for the dimer. Using the rationale outlined in the Methods section, H₁₃, H₁₄, H₂₃ and H₂₄ may have non-Coulombic interactions in the diabatic representation, with H₁₄ having the largest non-Coulombic interaction, and a significant Coulombic interaction, in the diabatic representation. Coupling 2–4 may also have a significant Coulombic interaction in the diabatic representation. These coupling and dot product values are close to those for dimer 2k [−60°, −60°], reflecting the closeness of the secondary structure of dimers 2k [−60°, −60°] and 2l [−62°, −41°] (both have a φ angle of –60°), except for couplings 1–3 and 1–4. For dimer 2l [−62°, −41°] the extent of non-Coulombic interaction in the H₁₄ coupling may be less than for dimer 2k [−60°, −60°], with the H₁₃ coupling in dimer 2l [−62°, −41°] having more non-Coulombic interaction than in dimer 2k [−60°, −60°].

β strand

The experimental CD spectrum for a β strand features a positive band at around 195 nm and a negative band at around 215–220 nm. The 2d [−135°, 135°] peptide has a computed CD spectrum using the diabatic inter-amide interactions that does not possess this feature in the 215–220 nm region, whereas the CD spectrum using the ab initio inter-amide interactions displays a degree of the correct character in this region (Figure 3) as well as a positive band at 186 nm.

Due to the arbitrariness of the phase of the wave functions, the signs of the excitonic coupling from the diabatisation are not well defined. A bimodal distribution (plus or minus) for the excitonic coupling is often observed (see, e.g. Figure 14 from Jiang et al. [56]). Thus, we usually select the signs of the diabatic coupling such that the computed diabatic CD spectra best agree with experimental and/or CD spectra computed using DichroCalc. To examine the differences between the two predicted spectra, the sign and/or magnitude of the three coupling interactions discussed above (1–4, 2–3 and 2–4; respectively, n₁π₁* with π_nb2π₂*, π_nb1π₁* with n₂π₂* and π_nb1π₁* with π_nb2π₂*) in the diabatic Hamiltonian were varied so that the negative band at 215–220 nm could be observed in the computed CD spectra (Figure 4). Table 4 lists the ab initio, diabatic and the matrix elements of the diabatic Hamiltonian that were modified to compute the spectra for peptide 2d [−135°, 135°] shown in Figure 4. Modifying H₁₄ only gives a less intense CD spectrum (compared to the diabatic), centred around 206 nm and where the negative band at 193 nm and the positive band at 221 nm are present (Figure 4, red line). Modifying H₂₃ yields a spectrum similar to that computed from the modification of H₁₄ but less intense with the negative band slightly red shifted to 194 nm (Figure 4, green line). Modifying H₂₄ only gives a spectrum that is like that of the diabatic spectrum but where the negative band is blue shifted (from 192 to 187 nm), and the positive band (at 221 nm) is less intense (Figure 4, magenta line). A positive band at 191 nm and a negative band at 221 nm are computed when both H₁₄ and H₂₃ are modified (Figure 4, yellow line). When, in addition, H₂₄ is modified the computed spectrum resembles that of the spectrum computed using the ab initio Hamiltonian (Figure 4, cyan and black lines).

Figure 4. Rotational strength line spectra (upper panel) and CD spectra with band shapes from convolution with Gaussian functions of FWHM of 9.0 nm (lower panel) for peptide 2d [φ = −135°, ψ = 135°]. Spectra computed using the ab initio (black line), diabatic (blue), diabatic with H₁₄ modified (red), diabatic with H₂₃ modified (green), diabatic with H₂₄ modified (magenta), diabatic with H₁₄ and H₂₃ modified (yellow) and diabatic with H₁₄, H₂₃ and H₂₄ modified (cyan) Hamiltonians.

Table 4. Inter-amide interactions (cm⁻¹ rounded to integer values) for diamide 2d [φ = −135°, ψ = 135°] and used to compute the spectra in Figure 2.

Matrix element index	H13	H14	H23	H24
Hamiltonian	n1π1*:n2π2*	n1π1*:πnb2π2*	πnb1π1*:n2π2*	πnb1π1*:πnb2π2*
Ab initio	72	−20	292	−907
Diabatic	109	−279	−244	−89
Modified diabatic	109	−3	244	−894

H₁₃, H₁₄, H₂₃ and H₂₄ have ab initio couplings of 72, –20, 292 and –907 cm⁻¹, and diabatic couplings of 109, –279, −244 and –89 cm⁻¹ (Table 2). The dot product of the un-normalised transition dipole moments for these couplings are, respectively, −0.032, −0.350, −0.342 and –2.037 D² for the monomer and –0.036, 0.407, −0.399 and 2.842 D² for the dimer (Table 3). Using the rationale outlined in the Methods section, H₁₄, H₂₃ and H₂₄ may have non-Coulombic interactions in the diabatic representation, with H₂₄ having significant non-Coulombic interactions and additional Coulombic interactions in the diabatic representation. As there are little differences in the coupling values and dot products for the H₁₃ coupling, this is likely a Coulombic interaction.

310 helix

The two computed spectra for diamide 2i [−74°, −4°] both feature an intense couplet centred around 197 nm for the ab initio spectrum and centred around 190 nm for the diabatic spectrum (Figure 3), which has the relatively more intense couplet of the two spectra. For the ab initio spectrum, the couplet has a positive peak at 189 nm and a negative peak at 203 nm, which has a shoulder region around 220 nm. The couplet in the diabatic spectrum consists of a negative peak at 184 nm and a positive peak at 197 nm. This spectrum also features a relatively shallow and broad negative band around 218 nm (where the shoulder in the ab initio spectrum is situated).

The couplings listed in Table 2 show a difference for all four between the ab initio and the diabatic, where H₁₃, H₁₄, H₂₃ and H₂₄ are 15, −36, 161 and −662 cm⁻¹ for the former and 289, −148, 230 and 284 cm⁻¹ for the latter. The dot product of the un-normalised transition dipole moments for these couplings are, respectively, −0.013, −0.454, 0.615 and –2.985 D² for the monomer and 0.011, −0.289, −0.730 and 3.175 D² for the dimer. Thus, H₁₄, H₂₄, H₂₃ and H₂₄ may have non-Coulombic interactions in the diabatic representation.

The experimental CD spectrum of an octameric sequence of L amino acids in a right-handed 3₁₀-helical structure (with mean φ and ψ angles of −54° and −32°, respectively) displays a negative band at 207 nm with a shoulder centred near 222 nm [51]. In addition to a slight positive band at 195 nm and negative band at 184 nm. The ab initio spectrum presented here is closest to the experimental spectrum. We note that the octamer has a structure close to diamide 2j [−48°, −57°] (discussed above) and not diamide 2i [−74°, −4°], and that the diabatic CD spectrum for 2j [−48°, −57°] is closer to the experimental CD spectrum of the 3₁₀-helical octamer.

Polyproline II

The ab initio spectrum for diamide 2m [−75°, 145°] features an intense couplet centred around 188 nm with a negative band at 181 nm and a positive band at 194 nm (Figure 3). The spectrum also has a shallow negative band at 208 nm and a weak and broad positive band at 220 nm. In contrast, the diabatic spectrum possesses a positive band at 186 nm and two negative bands at 199 and 220 nm with the latter negative band being broader and shallower than the former.

The coupling values for H₁₃, H₁₄, H₂₃ and H₂₄ are, respectively, 31, −7, 16 and −971 cm⁻¹ for the monomer and 81, 356, 145 and 29 cm⁻¹ for the dimer (Table 2). The dot product of the un-normalised TDMs for these couplings are, respectively, −0.002, 0.005, −0.619 and –2.865 D² for the monomer and 0.048, 0.543, −0.610 and −4.660 D² for the dimer (Table 3). As noted for the β strand dimer (2d [−135°, 135°]), H₁₄, H₂₃ and H₂₄ may have non-Coulombic interactions in the diabatic representation, with H₂₄ having significant non-Coulombic interactions and additional Coulombic interactions in the diabatic representation. As there are little differences in the coupling values and dot products for the H₁₃ coupling, this is likely a Coulombic interaction.

The experimental CD spectrum of a polyproline II structure (poly-L-proline) features a negative peak in the region around 200 nm and a positive peak at 220 nm [57]. The computed diabatic CD spectrum features a negative band at 199 nm, in agreement with experiment, and a negative band at 220 nm, in disagreement with experiment.

Future work and outlook

Future work will involve derivation of an empirical correction term (for inter-amide couplings) to incorporate into the DichroCalc NMA4FIT2 parameter set, based on the diabatisation work presented here. The empirical correction will follow the work of Scholes and Ghiggino [58], who defined a distance-based exponential term for short-range, non-Coulombic interactions between two chromophores, and Voityuk [55], who derived expressions to compute the electronic coupling based on vertical transition energies, transition dipole moments (of the system and of individual chromophores) and the relative orientation of the chromophores or individual transition dipole moments. The correction term will account for the short-range, non-Coulombic interactions between excited states on neighbouring amides, an interaction currently neglected in the NMA4FIT2 parameter set and which has been shown, in this study, to be a significant contributor toward computing more accurate CD spectra, e.g. the double minimum for an α–helical polypeptide.

Acknowledgements

We are grateful for access to the University of Nottingham’s high-performance computer. J. H. is supported by the Royal Academy of Engineering under the Chairs in Emerging Technologies scheme. H.D. is supported by the NSFC Research Fund for International Young Scientists (grant number 21850410456).

Data availability statement

The DichroCalc code is available at https://github.com/dmrogers75/DichroCalc. Data, example files and other code used in this study are available in a folder named ‘diabatisation’.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported by National Natural Science Foundation of China: [Grant Number 21850410456]; Royal Academy of Engineering: [Grant Number CiET2021_17].