Helix-sheet packing in proteins

Abstract

The three-dimensional structure of a protein is organized around the packing of its secondary structure elements. Although much is known about the packing geometry observed between alpha-helices and between beta-sheets, there has been little progress on characterizing helix-sheet interactions. We present an analysis of the conformation of alphabeta(2) motifs in proteins, corresponding to all occurrences of helices in contact with two strands that are hydrogen bonded. The geometry of the alphabeta(2) motif is characterized by the azimuthal angle theta between the helix axis and an average vector representing the two strands, the elevation angle psi between the helix axis and the plane containing the two strands, and the distance D between the helix and the strands. We observe that the helix tends to align to the two strands, with a preference for an antiparallel orientation if the two strands are parallel; this preference is diminished for other topologies of the beta-sheet. Side-chain packing at the interface between the helix and the strands is mostly hydrophobic, with a preference for aliphatic amino acids in the strand and aromatic amino acids in the helix. From the knowledge of the geometry and amino acid propensities of alphabeta(2) motifs in proteins, we have derived different statistical potentials that are shown to be efficient in picking native-like conformations among a set of non-native conformations in well-known decoy datasets. The information on the geometry of alphabeta(2) motifs as well as the related statistical potentials have applications in the field of protein structure prediction.

Figure 1. Illustration of the geometric measures characterizing a αβ₂ unit

The position of the helix is defined with respect to a co-ordinate system anchored on the β-sheet. s₁ is the mean vector representing the two strands, while s₂ is a vector perpendicular to s₁ in the plane containing the sheet. The orientation of the helix axis h is defined according to a polar co-ordinate system: the azimuthal angle θ is the angle between s₁ and the projection of h in the plane defined by s₁ and s₂, while the elevation angle ψ is the angle between this plane and h. D measures the distance between the helix and the sheet (see text for details).

Figure 2. Geometric definitions of the axes of a helix and of a strand

A) The axis h of an helix is computed in two steps. First, a local axis is derived for each four consecutive residues that belong to the helix. For the four residues whose Cαs are Cα1, Cα2, Cα3 and Cα4, the local helix axis h₁ is given by h₁ = v₁ × v₂, where × is the cross-product, v₁ is the vector from Cα2 to the mid-point between Cα1 and Cα3, and v₂ is the vector from Cα3 to the middle of Cα2 and Cα4. Second, the axis of the helix is set to the average of all its local axes. B) The axis b of a strand is set to be the average of the two local axes b_odd and b_even corresponding to its odd and even residues, respectively. The local axis b_odd corresponds to the direction of the line of best fit over all odd C_αs in the strand (i.e. in positions 1,3,… where the index of the first residue in the strand is 1). A similar definition is used for the even axis b_odd.

Figure 3. Representatives of the four different patterns of αβ₂ units in proteins

The P0T0, P0T1, P1T0, and P1T1 example units are extracted from proteins 1VQ1(A226-249), 2O1N(116-145), 1COZ(1-28), and 2P7H(A21-46) respectively. P indicates if the two strands are parallel or antiparallel to each other, while T relates to the relative position of the strands and the helix along the protein sequence. Figure drawn with Pymol.

Figure 4

The distributions of angle θ defining the azimuthal angle between the axis and the plane of the β sheet in a αβ₂ motif See text and Figure 3 for a definition of the four sub-classes of the motifs.

Figure 5

The distributions of angle ψ defining the elevation angle between the axis of the helix and the plane containg the β sheet in a αβ₂ motif. See text and Figure 3 for a definition of the four sub-classes of the motifs. All four distributions are centered on 0°, corresponding to the helix being parallel to the plane containing the two strands.

Figure 6

The distributions of distance D defining the mean separation between the axis of the helix and the plane containing the β sheet in a αβ₂ motif See text and Figure 3 for a definition of the four sub-classes of the motifs.

Figure 7

Log of the propensities of finding a contact of type (i, j) at the interface between a helix and a strand in an αβ₂ motif. For clarity of presentation, amino acids on the rows and columns of this matrix are ordered based on contact similarity as follows. Each amino acid type is characterized by a twenty dimensional vector (for example a column in the contact matrix). These vectors are used to compute a similarity matrix that is given as input to a metric multi dimensional scaling (MDS) algorithm . The latter assigns a position to each amino acid type in a 1D space and these positions define the ordering of the columns of the contact matrix. The same procedure is used to order the rows.

Figure 8. Cartoon representation of the native structure of chain A of the urease 4ubp

The central antiparallel β sheet (in blue) defines two αβ₂ motifs, the first one involving the helix h2 (residues 31-49), the second one involving the helix h3 (residues 52-61). Figure drawn with Pymol.

Figure 9. Score-RMSD scattered plots for the 4ubpA decoy set analysed with four statistical potentials specific to their αβ₂ motifs

These four potentials capture the azimuthal angle between the helix and strands (θ), the elevation angle between the plane containing the sheet and the helix (ψ), the distance D between the helix and the sheet, and the propensity of the aminoacid pairs in the interface between the helix and the strands (PR). In all four panels, the native protein is shown as a square.

Figure 10. Different packing styles observed in αβ₂ motifs in proteins

5NLL (left panel), abacterial flavodoxin is a small protein of 138 residues formed of a parallel β-sheet with 5 strands surrounded by 5 helices. 5NLL contains 9 αβ₂ motifs (8 within the P1 T0 category, and 1 in the P1T1 category), all with parallel strands. In all these motifs, the angle θ is close to 170°. 1THX (right panel), a theoredoxin is another small αβ protein whose structure includes an anti-parallel β-sheet covered with 3 helices, forming 6 αβ₂ motifs, two with parallel strands and 4 with antiparallel strands. Note that 1THX contains a fourth helix (shown in green) that does not participate in any of the αβ₂ motifs. The three helices present different orientation with respect to the sheet: helix 2 for example (shown in magenta) is perpendicular to the two strands it is packed on. Figure drawn with Pymol.