Toward a molecular understanding of the anisotropic response of proteins to external forces: insights from elastic network models

Abstract

With recent advances in single-molecule manipulation techniques, it is now possible to measure the mechanical resistance of proteins to external pulling forces applied at specific positions. Remarkably, such recent studies demonstrated that the pulling/stretching forces required to initiate unfolding vary considerably depending on the location of the application of the forces, unraveling residue/position-specific response of proteins to uniaxial tension. Here we show that coarse-grained elastic network models based on the topology of interresidue contacts in the native state can satisfactory explain the relative sizes of such stretching forces exerted on different residue pairs. Despite their simplicity, such models presumably capture a fundamental property that dominates the observed behavior: deformations that can be accommodated by the relatively lower frequency modes of motions intrinsically favored by the structure require weaker forces and vice versa. The mechanical response of proteins to external stress is therefore shown to correlate with the anisotropic fluctuation dynamics intrinsically accessible in the folded state. The dependence on the overall fold implies that evolutionarily related proteins sharing common structural features tend to possess similar mechanical properties. However, the theory cannot explain the differences observed in a number of structurally similar but sequentially distant domains, such as the fibronectin domains.

FIGURE 1

Schematic description of the position vectors at equilibrium, and and their instantaneous deformations induced by mode k, and Stretching of residues i and j (green arrows) gives rise to a deformation vector, d_ij, the direction of which coincidences with that of the equilibrium distance vector, The distance vector induced by mode k is designated

FIGURE 2

Distribution of the deformations in the distance contributed by each mode k (abscissa) in the presence of extensional forces applied to residues i and j. Each panel corresponds to a particular pair of GFP residues examined in previous experiments (17): (A) 3–132, (B) 3–212, (C) 182–212, (D) 132–212, and (E) 117–182, shown on the right panels. Note the distributions sharply skewed toward the lowest frequency modes in B and D, the same trend to a weaker extent in A, and the relatively uniform distribution in C. The first two cases refer to deformation directions that are more “yielding” as they can be accommodated by low-frequency modes. E, on the other hand, points to the involvement of moderate-to-high frequency modes and thereby the need to apply relatively stronger external forces to induce the same level of deformation. The ordinate scale refers to the force constant γ = 0.25 kcal/(mol Ų), and the profiles are independent of γ. The cartoon diagrams here and in all figures were generated using Jmol (68).

FIGURE 3

Correlation between the theoretical effective force constant 〈κ_ij〉 (ordinate) and experimentally reported spring constants (abscissa) (17,69) for the five studied extensions of GFP. Theoretical spring constants are evaluated using Eq. 12. The theory yields spring constants that are about 10 times softer, attributed to local deformations, rather than those, global, experimentally detected. Data points indicated by open circles refer to experiments (40) performed with two permutations of EYFP (cut at 144/145, higher point (outlier); 173/174, lower point). The gray circle refers to the EYFP wild-type protein.

FIGURE 4

Mechanical resistance map for GFP obtained by calculating the effective force constant 〈κ_ij〉 in response to uniaxial extensional forces exerted at each pair of residues. The secondary structure of the protein is shown along the upper abscissa (arrow, β-strand; zigzag line, α-helix). The profile at the lower part of the map displays the mean resistance of each residue, averaged over all values in a given column.

FIGURE 5

Relation between the mechanical resistance (〈κ_ij〉) and the direction of interresidue vector with respect to the cylindrical axis of GFP. (A) Two number distributions are shown for two sets of residue pairs that exhibit opposite behavior: those yielding the lowest 〈κ_ij〉 values in the 1% range (black bars) and those in the upper 10% 〈κ_ij〉 range (gray bars). Clearly, residue pairs which exhibit lower resistance to deformation are oriented along the cylindrical axis (mean at 22°), whereas residue pairs distinguished by their strong resistance are oriented at more perpendicular directions (mean angle of 55° with respect to the cylindrical axis). (B) Illustration of the location of some residue pairs that are predicted to exhibit very low resistance against stretching.

FIGURE 6

Mechanical behavior of Ub. (A) Cartoon representation of Ub. Pulling directions which are relevant to the biological function or examined by Carrion-Vasquez and colleagues (18) are indicated, and the pulled residues are labeled. (B) Mechanical resistance map for Ub, equivalent to the one shown in Fig. 4 for GFP. (C) The effective force constant 〈κ_ij〉 for the extensions in the directions shown in (A).

FIGURE 7

Mechanical resistance of E2lip3. (A) Cartoon representation of E2lip3. Pulling directions examined by Brockwell and colleagues (19) are indicated, and the pulled residues are labeled. (B) The complete resistance map for E2lip3. The secondary structure of the protein is shown along the upper abscissa and right ordinate (arrow, β-strand; zigzag line, α-helix). The mean resistance of each residue is shown in the profile at the lower part of the map. The inset in the top right corner shows the equivalent resistance map obtained with a Gō potential in coarse-grained MD simulations (27). The color code of the matrix is as in Fig. 4, and in the inset, yellow to blue colors indicate high to weak mechanical resistance, respectively (27). The portion of the map corresponding to the diagram in the inset is indicated by the black triangle.

FIGURE 8

Comparison of theoretically predicted force constants (ordinate) and experimentally measured unfolding forces (abscissa) for a series of proteins resolved by x-ray crystallography and subjected to pulling experiments at their N- and C-termini, with a velocity of 0.3–0.6 μm/s. Results are presented for (a) Spectrin (70,71), (b) EYFP (40,72), (c) Fibronectin repeat/domain 10 (¹⁰FNIII) (51), (d) Titin (I1) (73), (e) Protein L (72), (f) ¹³FNIII (12), and (g) ¹²FNIII (12). See Table 1 for the PDB structures used in ANM calculations. The coordinates for ¹²FNIII and ¹³FNIII are taken from the same crystal structure (1fnh), whereas those of ¹⁰FNIII refer to a different PDB structure (1fnf) with considerably smaller temperature factors (52,53).