Biopolymer Chain Elasticity: a novel concept and a least deformation energy principle predicts backbone and overall folding of DNA TTT hairpins in agreement with NMR distances

Abstract

A new molecular modelling methodology is presented and shown to apply to all published solution structures of DNA hairpins with TTT in the loop. It is based on the theory of elasticity of thin rods and on the assumption that single-stranded B-DNA behaves as a continuous, unshearable, unstretchable and flexible thin rod. It requires four construction steps: (i) computation of the tri-dimensional trajectory of the elastic line, (ii) global deformation of single-stranded helical DNA onto the elastic line, (iii) optimisation of the nucleoside rotations about the elastic line, (iv) energy minimisation to restore backbone bond lengths and bond angles. This theoretical approach called ‘Biopolymer Chain Elasticity’ (BCE) is capable of reproducing the tri-dimensional course of the sugar–phosphate chain and, using NMR-derived distances, of reproducing models close to published solution structures. This is shown by computing three different types of distance criteria. The natural description provided by the elastic line and by the new parameter, Ω, which corresponds to the rotation angles of nucleosides about the elastic line, offers a considerable simplification of molecular modelling of hairpin loops. They can be varied independently from each other, since the global shape of the hairpin loop is preserved in all cases.

INTRODUCTION

Hairpins are one of the fundamental structural units of DNA. They are formed from a single-strand molecule and consist of a base-paired stem structure and a loop sequence with unpaired or mismatched nucleotides as shown in Figure 1. Today, a large number of conformational studies of DNA hairpins, of the order of 100 have been published and are available. These tri-dimensional structures exhibit a considerable wealth of variations, which suggests that single-stranded DNA structures can be as complex as RNA or protein structures. Part of this complexity can be simplified and we propose a new molecular modelling methodology to predict the sugar–phosphate backbone structure of short DNA or RNA hairpins and to rationalise the description of loop bases in the frame of the backbone structure.

Figure 1.

Stereo views of the d(aagc-T_aT_bT_c-gctt) hairpin [JBNMR2K (16)] (A) into the major groove, (B) along the helical axis from the top of the hairpin. The hairpin structure contains the loop fold with the three thymines T_a, T_b and T_c shown in red and a stem with four base pairs shown in yellow. The sugar–phosphate chain is highlighted in bold to show the characteristic ‘S’ shape trajectory and the sharp reversal turn is indicated by an arrow.

Biological importance of DNA hairpins has long been suspected but has been demonstrated only recently. In vivo, DNA hairpins occur as transient single-stranded intermediates in many aspects of DNA metabolism including DNA replication, repair, and recombination and can exist when intrastrand pairing occurs between inverted repeats (1–3). Inverted repeats are widespread in the genomes of both prokaryotes and eukaryotes and are a common feature of control regions, including promoters, terminators and replication origins in viruses, prokaryotes, eukaryotes and mammalian cells (4). They have been shown to be functionally important for the initiation of DNA replication in plasmids, bacteria, eukaryotic viruses and mammalian cells (5). They have the potential to form cruciform structures under conditions of torsional strain on DNA (6). Cruciform formation in vivo has been demonstrated in prokaryotes and mammalian cells (4,5). Inverted repeats are regions of DNA with alternative secondary structure (double helical or cruciform) and pose a barrier to replication fidelity (7): they can facilitate frameshift mutations by bringing the DNA slippage sites, which are direct repeats, into close proximity mediated by stem–loop formation. Depending on whether the hairpin occurred on the parental strand or on the newly replicated strand, it results in addition or deletion mutations. This mechanism causes unstable transmission of genetic material related to several human genetic diseases. Some of the latter are caused by instability of repetitive tracts of simple trinucleotides. This is not due to defective or absent repair protein but to the formation of a secondary structure that allows a DNA loop to escape repair in vivo (8,9).

The recent developments and successes of the aptamer strategy have provided further motivation for understanding DNA hairpin structures. This methodology offers a way to design nucleic acid ligands (DNA or RNA) against a wide range of targets (10). For instance, it has been used to identify DNA ligands of the HIV trans-activation-responsive (TAR) RNA element binding to the viral protein Tat (11). These ligands could hypothetically compete with the TAR binding protein and might prevent the transcription process. The resulting DNA aptamers possess a stem–loop structure and fit to the TAR stem–loop structure, through a complex where the loop bases play a key role in the binding (12). An important area of development is the efforts to use non-pathogenic viruses such as the adenoassociated virus 2 (AAV2) as a gene therapy vector due to its ability to integrate into a specific site on human chromosome 19. Hairpin-like structural elements at the ends of the linear virion DNA are essential both for replication of the virus and its capacity for site-specific integration. It was recently proposed that they adopt a three-way junction with two hairpin arms, each with a 9-bp stem and a T₃ or A₃ loop (13). More precisely, chemical modification experiments indicated that the last two T in the T₃ loop are the most important specific bases involved in the binding of the AAV Rep binding protein to the terminal repeat sequence (14,15). These results motivate our search of the underlying building principles of the T₃ hairpin conformations to account for its efficiency as an element of a gene therapy vector (16).

In this paper, we will focus on T₃ hairpin structures. Numerous thermodynamical and structural studies of DNA hairpins have been reported (for a review see 17) and T₃ hairpin structures are probably one of the most extensively studied (16,18–21). Furthermore several authors published the relevant distances obtained from NMR data that were used to solve the molecular structures (18,20,21) and all, except Baxter et al., 1993 (19), who found substantial flexibility of the loop region, gave torsion angle information to reconstruct and visualise the molecules. For these reasons, the structures of Boulard et al., 1991 (18), Mooren et al., 1994 (20), Kuklenyik et al., 1996 (21) and Chou et al., 2000 (16), summarised in Table 1, served here as reference structures. They are named after a contraction of the journal name and of the year of publication, respectively: NAR91, Biochem94, EJB96 and JBNMR2K. They are referred collectively as published solution structures, PubNMR.

Table 1. Published hairpin NMR solution structures with TTT loop.

Identification	Authors	DNA sequences	Ta	Tb	Tc
NAR91	Boulard et al., 1991 (18)	d(cgtggatcg-TaTbTc-cgatccgag)	m	m	solv.
	Baxter et al., 1993 (19)	d(cgatcg-TaTbTc-cgatcg)	M	solv.	M
Biochem94 T7	Mooren et al., 1994 (20)	d(tctctc-TaTbTc-gagaga)	m	M	M
Biochem94 T8	Mooren et al., 1994 (20)	d(tctctc-TaTbTc-gagaga)	M	m	M
EJB96	Kuklenyik et al., 1996 (21)	d(gcgc-TaTbTc-gcgc)	m	M	M
JBNMR2K	Chou et al., 2000 (16)	d(gaagc-TaTbTc-gcttc)	m	M	M

Identifications used in this article, original authors, DNA sequences and locations of the loop bases in the major (M) groove, in the minor (m) groove or in the external solvent (solv.).

The loop bases are marked T_a, T_b and T_c in the 5′ to 3′ direction.

Our goal is to produce a new molecular modelling methodology to predict the tri-dimensional structures of backbones of DNA or RNA hairpins and to lay the foundation for qualitative description of locations of loop bases. The word ‘model’ is used with different meanings that must be specified: (i) NMR solution structures are molecular models derived from experiments, (ii) theoretical models aim at structure prediction, and (iii) qualitative models are used to discuss and to organise according to category.

The reference NMR solution structures of Table 1 were obtained by different modelling methods. The NAR91 structure was obtained with a conformational search program that varied all possible torsion angles to systematically generate all conformations. In practice, this search was limited at the mononucleotide level and the structure was built up stepwise while keeping track of combinatorial exploration. The Biochem94, EJB96 and JBNMR2K structures were reached by means of distance geometry algorithms, respectively, DIANA (22), DSPACE (23) and DGII (MSI, Inc.). In all cases, the authors used molecular mechanics force fields [AMBER (24–25) or CHARMm (26)] and restrained molecular mechanics energy minimisations (18,20) combined with restrained molecular dynamics (16,19,21). Modelling methods to derive NMR solution structures have much evolved over the last decades and have now reached a state of maturity.

Different theoretical approaches were used to study or to predict nucleic acid hairpin structures. Monte Carlo simulations have been carried out, directly by random variations of nucleotide conformation angles (27,28), in two steps by building a library of feasible dinucleotide structures (29) or by using a set of reduced coordinates to simplify the description and the modelling of nucleic acid structures (30). Molecular dynamics simulations (31–34) led to structurally stable models that allowed detailed exploration of the dynamics of large nucleic acid molecules. Another modelling method involved an automatic conformational search procedure based on a genetic algorithm (35). A different modelling procedure was based on a combinatorial exploration of the conformational space limited by sets of structural constraints derived from databases of nucleotide conformations and base–base interactions (36,37).

The qualitative discussion of tri-dimensional hairpin structures was initiated very early (38). These authors recognised the existence of qualitative features common to many hairpins. An important idea is the continuation of helical stacking of the stem into the loop on one strand in the 5′→3′ direction for DNA and on the other strand in the 3′→5′ direction for RNA. Another idea is the loss of this continuity in the loop at a turning point called the ‘sharp turn’.

In this article we build upon these ideas by using a less constraining hypothesis, which bears solely on the continuity of the sugar–phosphate backbones. We postulate in agreement with reports on the T₃ hairpin (16,18,20,21) (NAR91, Biochem94, EJB96 and JBNMR2K) that the sugar–phosphate backbone of the stem continues in the loop as a right-handed B-DNA helix: it is remarkable that most torsion angles in the loop are close to classical B-DNA values except in the sharp turn region. The JBNMR2K structure shown in Figure 1 provides a view of this continuity. By convention, the three thymines are denoted T_a, T_b and T_c in the 5′→3′ direction. Near the 3′ end of the loop and before reaching the 5′ top of the stem, the backbones take a sharp turn at or near a phosphate group that is called the turning phosphate (17) as shown in Figure 1A and B from two different viewpoints. In these structures, this occurs between the last thymine, T_c, of the loop, and the base at 5′ top of the stem. More generally the positions of turning phosphates are not exactly the same in published models with different hairpin sequences, with hairpins of different lengths, or with different stem sequences.

These observations taken as a starting point suggest that DNA TTT triloop structures might be obtained by bending as smoothly as possible a single-strand B-DNA helix into a hairpin fold. This procedure would preserve most torsion angle values and overall B-DNA geometry, and would also predict a region of relatively sharper curvature related to the sharp turn region. Furthermore the overall ‘S’ shapes (16) or the ‘yin-yang’ shapes (17) of the sugar–phosphate backbones (for a review see 17), suggest that they behave as continuous, rigid, but deformable polymer chains. This remarkable behaviour is due in part to the necessity of bridging, and therefore of straining, the two extremities of the double helical stem with a short chain of 3 nt. The smoothest and simplest bending model of a flexible chain is provided by the theory of elasticity and the bending of thin rigid rods. In this article, we present a new molecular modelling methodology based on the theory of elasticity to fold short single-stranded oligonucleotides into hairpins. Elasticity theory of thin rods has been successfully applied to DNA in a different context: that of long double-stranded helical DNA in the range of hundreds of base pairs. The latter approach should not be confused with our methodology that is applied for the first time at the level of several nucleotides. We proceed to compare, at each construction step, the model structures derived from our theoretical approach to the original NMR data and reference structures summarised in Tables 1 and 2. They indicate that thymines T_a, T_b and T_c are found in three different locations: in the minor groove, in the major groove or in the bulk solution depending on the complete oligonucleotide sequences. Our main focus is to show here whether our theoretical approach is capable of: (i) predicting a priori the tri-dimensional courses of the sugar–phosphate chains from the boundary conditions imposed by the double helical stem and (ii) generating models comparable to the solution structures, from these predictions and from a small set of NMR-derived distances.

Table 2. Total numbers of inter-residues NMR-derived distances between loop residues, given in published articles, and used to build and evaluate BCE theoretical models; numbers in parentheses are, respectively, the detailed numbers of distances between: (deoxyribose, deoxyribose), (deoxyribose, base) and (base, base).

DistNMR (i, j), between protons (i, j)	NAR91	Biochem94 T7	Biochem94 T8	EJB96
Base of 3′ top stem, Ta	3 (1; 2; 0)	5 (0; 4; 1)	6 (0; 5; 1)	7 (3; 4; 0)
Base of 3′ top stem, Tb	None	2 (0; 2; 0)	2 (0; 2; 0)	3 (0; 1; 2)
Ta, Tb	4 (0; 3; 1)	3 (0; 3; 0)	4 (0; 4; 0)	5 (0; 4; 1)
Tb, Tc	None	6 (0; 6; 0)	6 (0; 6; 0)	3 (0; 2; 1)
Tc, base of 5′ top stem	None	3 (2; 1; 0)	3 (2; 1; 0)	1 (0; 1; 0)
Total	7	19	21	19

MATERIALS AND METHODS

Original molecular structures, ‘PubNMR’

Original NMR solution structures were reconstructed by constrained energy minimisation from published torsion angles data (PubNMR) with addition of E = Σk(θ – θ₀)² to the energy function of AMBER. θ is any of the torsion angles α, β, γ, δ, ε, ζ and χ, and the constant k was set to 900 kcal/(mol.rad²). All energy refinements and molecular dynamics were carried out with the program AMBER (24,25,39) as in previous studies (40–42).

Initial stem and loop model building by molecular mechanics

All initial structures were generated from canonical B-DNA (43).

Theoretical molecular structures, ‘BCE’

A computer program S-mol^©, a registered software, was developed under UNIX and Linux environments using Mathematica (44), Geomview (45) and C languages. It contains libraries to build, cut and paste initial DNA or RNA macromolecules in helical conformation and to fold these macromolecules into hairpin loops using prescribed geometric boundary conditions with the Biopolymer Chain Elasticity (BCE) approach. Macromolecules can be visualised and analysed with S-mol^©. S-mol^© is in the initial stage of development and is available through a cooperation agreement. The complete DNA sequences of the theoretical molecular structures were determined so that the sequence in the loop and of the first four base pairs in the stem were identical to those of original molecular structures. The length, L, of the capping rod shown in Figure 2 was obtained by fitting a helical line to the atoms of the main sugar–phosphate backbone (O5′, C5′, C4′, C3′, O3′, P) of three thymidines in B-DNA conformation and by minimising the root-mean-square of the sum of squared distances to the helical line. Radius of helical line was 8.35 Å and its pitch was 33.74 Å/turn.

Figure 2.

The boundary value problem for the capping of the double helix, shown in yellow, by a rigid rod, in red: (A) view into the major groove; (B) top view along the helical axis; (C) side view along the axis joining the 5′- and 3′-ends of the loop. Arrows in black are the tangents to the extremities of the double helical stem. Extremities of the capping rod in red are clamped rigidly along these tangents to define a continuous and differentiable elastic curve, BCE1curve, at tangent points. Note (A) and (B) are presented from the same point of view as stereo views (A) and (B) of Figure 1.

Optimised molecular structures, ‘BCE3Ωopt’

NMR-derived distances provided in the original articles (PubNMR) and summarised in Table 2 were used at the third stage of theoretical molecular modelling to optimise the rotation angles about the elastic line, Ω, and the glycosidic torsion angles, χ, of each nucleotide in the loop. This was performed by a least square fit on distances to give optimised BCE models, BCE3Ωopt.

Final theoretical molecular structures, ‘BCE4finalrm’

NMR-derived distances provided in the original articles (PubNMR) and summarised in Table 2 were used in the final stages of molecular modelling as restraints in energy refinements with a force constant restraint of 10 kcal/ (mol.Ų) and with a large stopping root-mean-square energy gradient criterion 0.5 kcal/(mol.Å).

RESULTS

Theoretical analysis and comparison to experiments

In this section we describe the four different construction steps of the methodology called BCE and applied to one of the most studied DNA hairpins: the T₃ hairpin. We focus on the molecular structures obtained at each step. They are compared with the available original set of NMR distances and with original solution structures.

First step: the single-stranded DNA or RNA loop treated as a continuous and flexible thin rod → BCE1curve. In this model the single-stranded DNA or RNA backbone of the loop is considered as a thin, inextensible, homogeneous, transversely isotropic and intrinsically straight rod in the absence of stress conditions. Since the rod is thin, it may be regarded as a line, often called an elastic line. To understand the overall shape of macromolecular chains, it has been the habit to schematically represent the backbone of biological macromolecules such as DNA, RNA or proteins by ribbons. Here it is for reasons of visibility that the elastic line is also drawn in most figures as a ribbon. This precaution must be emphasised: unlike usual macromolecular representations, the underlying model of the sugar–phosphate backbone is a continuous and flexible thin rod.

We need to compute the tri-dimensional trajectory of the elastic line that bridges the two extremities of the double helical stem in continuity with the double helical line. The trajectory of this capping rod, shown in Figure 2, follows directly from boundary conditions. More precisely, end conditions are defined by clamping the ends of the capping rod at the extremities of the double helical stem: as shown in Figure 2A–C, the two extremities are considered as fixed and the direction of the two tangents to the rod cannot depart from the tangents at the top of the helical stem. From a physical point of view, the trajectory is defined by its length, L, by its elastic properties, by applied forces and moments at extremities of the rod. From a mathematical point of view, the trajectory can be deduced from its length, L, and from the sole geometry of end conditions. We used the general ideas of Kirchhoff–Clebsch–Love (46), the detailed treatments in Cartesian coordinates of Landau and Lifschitz (47) as well as the recent developments and results of Shi and Hearst (48) and of Tobias et al. (49), Coleman et al. (50) and Swigon et al. (51) to define and to solve this problem of elasticity theory.

Agreement between the tri-dimensional trajectory of the theoretical elastic curve, BCE1curve and published solution structures. Direct and visual comparison is given in Figure 3 of the superposition of the backbones of published solution structures with the elastic curve. An original quantitative means of comparison is provided by computing the distance, d, between the main atoms of these sugar–phosphate chains to the theoretical elastic line as shown in Figure 4. As summarised in Table 3, mean d values are in the range 0.98–1.26 Å for published models. These values are in close agreement with d = 0.76 ± 0.32 Å, the average and standard deviation distance recorded for B-DNA helix, and are in the range of distances found in B-DNA as shown by the solid line and the grey region of Figure 4. Best agreements are obtained for the most recently published solution structures, EJB96 and JBNMR2K, as shown in Table 3A and in Figure 3. Therefore the theoretical elastic line computed in Figure 2 fits the main atoms of the sugar–phosphate chains of published solution structures as well as the helical line fits those of B-DNA.

Figure 3.

Superposed backbones of original published hairpin loops, NAR91 (green), Biochem94 T7 (orange), Biochem94 T8 (light cyan), EJB96 (pink), JBNMR2K (purple) and of the elastic rigid rod shown in red. (A) View into the major groove; (B) top view along the helical axis. Note the two most recently determined backbones of EJB96 (pink) and of JBNMR2K (purple) are best fits to the elastic rod.

Figure 4.

Plot of the distance, d, between backbone atom positions and their projections on the theoretical elastic line as a function of atom position along the loop sequence in the 5′→3′ direction. Best fits of Figure 3 are represented: EJB96 (pink) and JBNMR2K (purple). Average and standard deviation values are in Table 3A. Distances of backbone of B-DNA helix (or equivalently of BCE2basicxyz of Fig. 5) are shown in black for reference. The grey region corresponds to the range of minimal and maximal distances found in B-DNA. Horizontal line d = 0.76 Å is the corresponding average measured for a B-DNA helix.

Table 3. Mean distance between main chain backbone atoms and their projections onto (or closest positions to) the theoretical elastic curve, BCE1curve.

	Mean distance	Set of atoms in the loop	NAR91	Biochem94 T7	Biochem94 T8	EJB96	JBNMR2K
A	‘PubNMR’ and theoretical elastic curve	Backbone	1.15 ± 0.56	1.17 ± 0.46	1.26 ± 0.58	0.98 ± 0.49	1.03 ± 0.49
B	‘BCE4finalrm’ and theoretical elastic curve	Backbone	0.83 ± 0.37	0.83 ± 0.38	0.80 ± 0.41	0.86 ± 0.33	*

‘Backbone’ is the set of main atoms of the chain sugar–phosphate backbone O5′, C5′, C4′, C3′ O3′ and P of the 3 nt in the loop and of the first base pair in the stem. ‘PubNMR’, published solution structures from NMR experiments (16,18,20,21). ‘BCE4finalrm’, final theoretical molecular model computed with the BCE approach. Asterisk, NMR-derived distances were not available in the published data.

Second step: global deformation of DNA or RNA polymer chains in the framework of the thin rod model → BCE2basicxyz. At this stage we know the tri-dimensional trajectory of the elastic curve. As postulated in the introduction, we need to relocate the atoms of the helical chain onto the elastic line by a geometrical transformation to construct a molecular structure called hereafter, BCE2basicxyz. A natural local coordinate system at point M of arclength s on the elastic curve is provided by the Frenet trihedron of the three orthogonal unit vectors, {t→(s), n→(s), b→(s)} as shown in Figure 5. In this notation, t→(s) is the tangent vector to the elastic line, the normal vector, n→(s), defines along with t→(s) the osculating plane, and b→(s) is the binormal vector. When the three thymidines are in the starting B-DNA helical conformation, each atom of any given thymidine can be conveniently attached to a point on the helical line as follows. The main atoms of the sugar–phosphate backbone (O5′, C5′, C4′, C3′, O3′, P) are first attached to their closest points on the helical line. These points are taken as the origins of local reference frames, in which are computed the local coordinates of the remaining atoms. This is accomplished by forming blocks of atoms, e.g. H5′ and H5″ are considered in the same block as suggested by chemical bondings and are both computed in the local reference frame of C5′. When the helical line is deformed into the capping elastic line as shown in Figure 5, the main sugar–phosphate backbone atoms are moved to their new positions by a translation-rotation coordinate transformation:

Figure 5.

Schematic view of the smooth deformation of the three thymines helical segment, shown in yellow, into the capping rod elastic curve, in red, with ends clamped at the tangents of the double helix stem. The folded molecular structure, BCE2basicxyz, shown in red, is computed after global deformation by keeping track of the translations and the rotations required to maintain atoms in their local reference frames along the elastic rod. For clarity nucleoside bases are not represented. Helical axis, z, of the double helix, and its perpendicular plane at the top of the stem are represented by the global reference frame at the bottom left-hand side.

where R is the transposed matrix of (n→(s), b→(s), t→(s)).

This transformation preserves the distances of main sugar–phosphate backbone atoms to the elastic line, d = 0.76 ± 0.32 Å, in close agreement with values found for published solution structures of Table 3A.

Third step: local rotations of nucleosides about the elastic curve and of bases → BCE3Ωopt. The two previous modelling steps provide the overall geometry of the sugar–phosphate backbone and a set of local reference frames on the elastic line. These reference frames can be conveniently used to rotate the local blocks of atoms about their closest tangents on the elastic line. By definition, Ω is the angle of rotation of a nucleoside block (i.e. base and sugar atoms) about the elastic curve as shown in the top part of Figure 6. Ω is defined positive as indicated in Figure 6. By definition, Ω is zero in canonical B-DNA (43) and in the bent conformation of the loop, BCE2basicxyz; this point will be discussed in a separate article. Bases can rotate with glycosidic angle, χ (about C1′–N1 in pyrimidines or C1′–N9 in purines) as shown in the bottom left of Figure 6. Note that the computations of the elastic line, BCE1curve, and of the molecular structure, BCE2basicxyz, require no experimental data since they are solely derived from the theory of elasticity of thin rods. To remain in this framework of construction, we want to determine whether simple nucleoside rotations, Ωs, are sufficient to bring loop nucleosides into positions and conformations close to those observed in PubNMR structures. Note this methodology restricts the number of possible conformations to sterically allowed (Ω, χ) conformations, but does not determine nucleoside locations and conformations. To do so, NMR-derived distances are required as explained below.

Figure 6.

Schematic view of nucleoside block rotations with angle, Ω, about the elastic rod curve (top part), and of base rotations with glycosidic angle, χ, about the glycosidic bond, C1′-N1 in pyrimidines or C1′-N9 in purines (bottom left part). Ω is defined positive if rotation is clockwise upon progression in the 5′→3′ direction as in a right-handed reference frame. Colour codes and global reference frame are as in legend to Figure 5. The 5′ and 3′ labels refer to extremities of the loop. Optimized rotations yield BCE3Ωopt models.

Agreement of theoretical models with NMR-derived distances at different stages of optimisation. Nucleosides T_a, T_b and T_c were rotated in different stages about their associated tangent lines by respectively varying Ω_a, Ω_b and Ω_c to fit the published NMR-derived distances given in Table 2, distNMR_(i,j), between the protons i and j in the loop. At first, optimal combinations of rotations (Ω_a, Ω_b) are found when Sdist, the square root of the sum of all squared differences between computed distances, distModel_(i,j) and published distances, distNMR_(i,j), is the lowest. This is accomplished with fixed glycosidic angles, χ, as shown by the contour maps of Sdist in Figure 7. Three sterically allowed local minima are found for NAR91, whereas four local minima are observed for Biochem94 T7, Biochem94 T8 and EJB96. All four minima of Biochem94 T7 are sterically allowed, but the lowest minimum at 0.36 is ruled out because it is incompatible with data used to position the third T in the loop. Note that the local minima of Biochem94 T8 at 0.36 and at 0.42 and of EJB96 at 0.85 exhibit very severe clashes with the stem and that all other minima are sterically allowed. Good global optima are reached with only two parameters and a single contour map as shown with a cross in Figure 7 and summarised in the first line of Table 4. Further improvements are gained with the global optimisation of Ω_a, Ω_b, χ_a and χ_b (second line of Table 4). This operation is repeated for Ω_a, Ω_b and Ω_c and then for Ω_a, Ω_b, Ω_c, χ_a, χ_b and χ_c. Ω_c values are found in the same range, 95° ± 20°, for all molecules (maps not shown). The best combinations of angles (Ω_a, Ω_b, Ω_c, χ_a, χ_b and χ_c), are used to construct the BCE3Ωopt structures.

Figure 7.

Contour maps of RMSd, Sdist, computed from experimental sets of atomic distances for the BCE2basicxyz theoretical models obtained from the elastic curve approach, as a function of nucleotide angles Ω_a (T_a) and Ω_b (T_b). The experimental sets of NMR distances correspond to the lists of Table 2 without the third loop nucleotide: NAR91, Biochem94 T7, Biochem94 T8 and EJB96. Glycosidic angles of T_a and T_b are set to standard B-DNA values: χ_a = χ_b = –102° in NAR91, Biochem94 T7 and EJB96, and are set to χ_a = 178° and χ_b = –102° in Biochem94 T8 to match an unusual value proposed for Biochem94 T8. Values of local minima and maxima are indicated on the map. Sterically allowed global optima for nucleotide angles Ω_a and Ω_b are marked with a white cross and are reported in Table 6.

Table 4. RMSd in Å (root-mean-square deviation) computed from sets of NMR-derived reference distances.

	RMSd computed from	Set of atoms in the loop	NAR91	Biochem94 T7	Biochem94 T8	EJB96
A	Optimisation of Ωa, Ωb	Compare with NMR experiments	0.26	0.44	0.21	0.58
B	Optimisation of Ωa, Ωb, χa and χb	Compare with NMR experiments	0.26	0.32	0.15	0.37

Ω refers to simple rotations of each loop nucleoside about the elastic curve solution to match NMR-derived distance constraints. Optimisation is reached in several steps: (A) Ω_a and Ω_b for T_a and T_b, then (B) Ω_a, Ω_b, χ_a and χ_b.

Fourth step: restoring bond lengths and bond angles by a short restrained minimisation → BCE4finalrm. Note that the blocks of atoms such as phosphate residues, C5′H5′H5″ groups or nucleoside bases are transported as single units and are not deformed. In contrast, geometrical parameters between main sugar–phosphate backbone atoms such as bond lengths, bond angles and torsion angles are not constrained. However, since the deformation is global and the transportation is on smooth curves, most local deformations are small as postulated in elasticity theory of thin rods. Most of these geometrical parameters are fully restored after the very short restrained minimisation by molecular mechanics (less than 100 steps or <2 s on SGI R10000 195 MHz) that is performed at the end of the next modelling step. This step is examined in the following article in this issue (52). Displacements of backbone atoms (Table 5A) and overall loop atoms (Table 5B) that occur during this step (BCE3Ωopt → BCE4finalrm), are small, respectively, smaller than 0.67 Å and 1.19 Å. This result indicates that the molecular structures remain very close in space and that most of the adjustment was done in the previous optimisation steps.

Table 5. RMSd in Å (root-mean-square displacement) to compare atom displacements between: (A–B) BCE building steps ‘BCE3Ωopt’ and ‘BCE4finalrm’; (C–D) final theoretical molecular models, BCE4finalrm, versus published reference molecular structures, PubNMR.

	RMSd computed from	Set of atoms in the loop	NAR91	Biochem94 T7	Biochem94 T8	EJB96
A	‘BCE3Ωopt’ versus ‘BCE4finalrm’	Backbone	0.59	0.67	0.56	0.61
B		All	0.74	1.19	0.95	1.00
C	‘BCE4finalrm’ versus ‘PubNMR’	Backbone	1.37	1.26	1.41	0.81
D		All	1.52	1.72	1.71	1.18

‘All’, include all loop atoms. ‘BCE4finalrm’, ‘BCE’ molecular model where bond lengths and bond angles are restored by a short energy minimisation with NMR-derived distance restraints.

Final comparison of theoretical and published molecular models

Three different methods were used to compare theoretical models, BCE4finalrm, with published models, PubNMR. They show a very good agreement between the two types of models. The first means of comparison is the mean distance of backbone atoms of BCE4finalrm to the elastic solution curve, given in Table 3B. Values obtained are in the range 0.80– 0.86 Å, very close to the characteristic values of B-DNA, and agree with values found for published solution structures. The second means of comparison is the computed RMSd: they are 0.8–1.4 Å for backbone atoms, and 1.2–1.7 Å for all loop atoms (Table 5C and D). Direct and visual comparisons are given in Figure 8 with the complete set of superpositions of the experimental and theoretical structures. Visual qualities of superposition are very good, as shown in Figure 8, and as easily checked by computer graphics from all viewpoints. All rotation angles, Ω, for the final theoretical models are given in Table 6. Negative values of Ω indicate a minor groove location, whereas positive values correspond to major grooves. Ω values are a convenient way to quantify minor or major groove location in agreement with values of Table 1.

Figure 8.

Superposed views of hairpin loop molecular structures derived from NMR data by original authors, in yellow, and of the final molecular structures built from the elastic curve approach, in red: NAR91, Biochem94 T7, Biochem94 T8, EJB96, JBNMR2K. Experimental and theoretical sugar–phosphate chains are highlighted in light bold and bold, respectively, and elastic rod curve is shown in dashed line. Left: view into the major groove. Right: top view along the helical axis. Thymine bases with black or white tags are in the minor or major groove, respectively.

Table 6. Final rotation angles, Ω_x, in degrees of the loop bases, T_x, about the elastic line found in the theoretical molecular models, BCE3Ωopt, for the corresponding molecular structures: (A) NAR91, (B) Biochem94 T7, (C) Biochem94 T8, (D) EJB96.

BCE3Ωopt molecular models	(A) NAR91	(B) Biochem94 T7	(C) Biochem94 T8	(D) EJB96
Ωa	–85° (m)	–53° (m)	81° (M)	–66° (m)
Ωb	–64° (m)	68° (M)	–41° (m)	71° (M)
Ωc	120° (M)	80° (M)	80° (M)	100° (M)

Ω_a is the rotation angle of Ta, Ω_b of T_b and Ω_c of T_c.

DISCUSSION

Diversity of experimental molecular models

At the present time, a minimal number of two classes is required to account for the solution conformations found from NMR data and presented in Table 1. When the loop is closed by a C·G base pair as in C-TTT-G containing sequences (Biochem94, EJB96 and JBNMR2K), the consensus conformation, where the first thymine, T_a, is in the minor groove, and where the last two, T_b and T_c, are in the major groove, holds true, although a conformational equilibrium (T7, T8) was suggested for Biochem94 and is discussed below. When the loop is closed by a G·C base pair (18,19) no consensus has yet been reached and more experiments are needed.

Contour maps of the fit criterion to NMR-derived distances as a function of (Ω_a, Ω_b) [see Fig. 7 (Biochem94 T7 and T8)] might explain in part the difficulties encountered to reach consensus conformations. They suggest the existence of four local minima that are closely related since experimental distances to locate the third nucleoside are either absent or insufficient. In Figure 7 (Biochem94 T7 and T8), two models were retained, one for Biochem94 T7 and one for Biochem94 T8, respectively consistent χ_a = χ_b = –102°, or χ_a = 178° and χ_b = –102°. These models are derived from the same set of distances, and the limited number of experimental inter-residues distances may therefore be the source of degeneracy. An alternative interpretation is that both conformations exist in solution (20). As exemplified by these adjustment maps, our modelling approach can help to explore more exhaustively relevant conformational possibilities.

NMR-derived distances

More recent determinations, EJB96 and JBNMR2K, appear to resolve this intrinsic degeneracy by very detailed analyses of NMR data (16,21) as shown for EJB96 in Figure 7, where the fourth local minima at 1.24 tends to disappear. These solution structures, EJB96 and JBNMR2K, are close to Biochem94 T7 as shown in Table 6, and in Figure 8.

The number and quality of experimental data to calculate the distances are important. The approximate symmetry of the two major local minima about a line parallel to a diagonal in the adjustment map of Figure 7 (NAR91) is due to the limited number of distances between loop bases and stem. The shapes of the local minima regions are stretched along this line because most inter-residues distances are between T_a and T_b, and agreement with NMR-derived distances tends to be preserved when (Ω_a–Ω_b) is constant. With larger sets of distances, adjustment maps display better-determined local minima with Cartesian symmetry [Fig. 7 (Biochem94 T7 and T8, EJB96)]. Finally, accurate locations of the third base in the loop were found important to help resolve molecular modelling degeneracy. As shown in Table 2, the third base is the least characterised and the most deformed in the region of the sharp turn.

Torsion angles

Experimental determination of torsion angles from NMR is semi quantitative for β and γ torsion angles, impractical for ζ and α, and χ angles were set in the range –100 ± 30° (16). Sugar puckers add another degree of flexibility. Torsion angle values in BCE4finalrm models are generally close to those of B-DNA except in the region of sharp turn where deviations are more important. There are no abnormal angles, e.g. cis, but there are one or two sterically hindered combinations g+/g– (53) as also encountered in published NMR structures. In summary, overall torsion angles agreement between BCE4finalrm and most recent PubNMR structures is very good although less well defined in the region of sharp turn. Their detailed study is not treated here because it would require extensive molecular dynamics in explicit solvent.

Nucleotide parameter angles Ω_a, Ω_b and Ω_c on the elastic line

As shown by the simple optimisation steps that readily match NMR-derived distances, parameter angles Ω_a, Ω_b and Ω_c appear to behave as natural descriptors of nucleoside locations in hairpin loops. They can serve to rapidly identify optimal combinations of nucleoside rotations as shown in Figure 7 and in Tables 4 and 6, since the number of relevant angles is reduced from six to two: (α, β, γ, δ, ε, ζ, χ) → (Ω, χ) when sugar puckers are known. Unlike sugar–phosphate torsion angles, they can be varied independently from each other, since the global shape of the hairpin loop is preserved in all cases. As a result, exhaustive global conformational search can be performed with greater ease than before, which should be useful to study hairpin structures and mechanics.

CONCLUSION

Single-stranded trinucleotide TTT B-DNA can be deformed into hairpin loops that match four different sets of NMR data or five different molecular conformations of different authors. Therefore single-stranded B-DNA behaves in this case study to first approximation as a continuous, unstretchable and flexible thin rod where most torsion angle values are preserved in the folding process. This is not very surprising since the average value of the B-DNA backbone length was estimated to represent 90% of the length of the fully extended chain (54,55) [see Flory (54), p. 207] and since further stretching would force all backbone torsion angles to flip into trans conformation. Note as a correlate the helical shape of single-stranded B-DNA is implicitly recognised as a trivial solution in elasticity theory of thin rods. This is the first case where the B-DNA chain is shown to fold into a hairpin as an elastic biopolymer chain. ‘Elastic’ is taken in the sense of the elasticity theory of thin rods and in the sense of the least deformation energy principle. Our theoretical approach is a molecular modelling methodology capable of predicting the tri-dimensional course of the sugar–phosphate chain from boundary conditions and, using NMR-derived distances, of generating models very close to published solution structures. The natural description of loop folding with the new parameter angles, Ω, offers a considerable simplification of the molecular modelling of hairpin loops and a reduction in number of conformation parameters. This methodology opens the way to many perspectives of development. Among them, two of the most important questions to address are: (i) how to adapt the method so that BCE models can be compared directly and more automatically to PDB solution structures and (ii) does the BCE methodology apply to hairpins of different lengths and to DNA and RNA hairpins?