Stretching of proteins in the entropic limit

There is considerable current interest in the mechanical manipulation of single biological molecules. In particular, stretching studies of large proteins with atomic force microscopes (AFM) and optical tweezers [1,2] reveal intricate, specific, and reproducible force (F ) -displacement (d ) curves that call for understanding and theoretical interpretation. The patterns depend on the pulling speed and on the stiffness of the pulling device [3]. They must also depend on the effec-tive temperature given by the ratio of thermal to binding energies. In experiments this ratio can be varied slightly by changing temperature T, and over a large range by changing solvent properties [2], such as pH. While the role of effective temperature in folding is well studied [4,5], its effect on mechanical unfolding is not.

In this paper, we report results of molecular dynamics simulations of simplified models of six proteins that reveal universal trends with increases in the effective temperature. Thermal fluctuations accelerate rupture, lower the peaks in the F -d curves, and reduce the number of peaks. Most inter-estingly, the succession of unfolding events simplifies from a complex pattern determined by the energy landscape into a simple, uniform pattern determined by entropic consider-ations. All of these changes are gradual, and move to completion near the T where the specific heat peaks. In the entropic limit, rupturing events are governed exclusively by the contact order, i.e., by the distance along a sequence between two amino acids which make a contact in the native state. The contact order is also believed to be the major influence in folding to the native state. However, there is no general correlation between folding and the extremely complex unfolding scenarios observed at low T [3].

The models we use are coarse grained and Go-like [6]. Full details are provided in earlier studies of folding [5,7]. Briefly, the amino acids are represented by point particles of mass m located at the positions of the C^a atoms. They are tethered by a strong harmonic potential with a minimum at the peptide bond length. The native structure of a protein is taken from the PDB [8] data bank and the interactions be-tween the amino acids are divided into native and non-native contacts. The distinction is made by taking the fully atomic representation of the amino acids in the native state and then associating native contacts with overlapping amino acids.The criterion for overlaps uses the van der Waals radii of the atoms multiplied by 1.24 to account for the softness of the potential [9].

The interaction between each pair of overlapping acids i and j is described with a 6-12 Lennard-Jones (LJ) potential whose interaction length σ size 12{σ} {}_{i j} (4.4 -12.8 Å) is chosen so that the potential energy minimum coincides with the native C^a -C^a distance. This forces the ground state to coincide with the native state at room temperature T. The non-native contacts are described by a LJ potential with σ= size 12{σ={}} {} 5 Å that is truncated at r . 2 ^{1/6 σ size 12{σ} {}} to produce a purely repulsive force. An energy penalty is added to states with the wrong chirality as described in Ref. [5]. This facilitates folding, but has little effect on mechanical stretching since the protein starts with the correct chirality. All the potentials have a common energy scale e , which is taken as the unit of energy. Time is measured in terms of the usual LJ vibrational time scale τ≡mσ2/ size 12{τ equiv sqrt {mσ rSup { size 8{2} } / in } } {}.

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