The Bell model can be applied to protein unfolding lifetimes under constant force. In this model, the average protein lifetime ηB is given by(170):
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T k Fx G B u B o B exp (4.5)where ηB is the unfolding time for a protein under a given force (F), ηo is the inverse of the energy well escape attempt frequency of the protein (~10-13s), ΔGB is the unfolding energy barrier in the absence of force, and xu is distance between the unfolded energy state and the top of the unfolding energy barrier. (See Figure 4.6) Applying a constant force to a protein can be seen as lowering the energy barrier for unfolding. At the point where F*xu ≈ΔGB, the probability for the protein to be found in the folded and unfolded state are equal. The force at which F*xu ≈ΔGB has been termed the critical force FC.
Figure 4.6 Unfolding Energy Barrier: Protein folding and unfolding can be thought of as an energy landscape. In its folded state, the protein resides at the lowest free energy. The energy required to kick it out of its folded state is the unfolding energy barrier (ΔGB) and the extension in length to reach to height of the energy barrier is (xu). By applying a constant force to a protein, one is essentially reducing the energy barrier by an amount F*xu. At the critical force (FC), the protein is equally probable to be found in a folded or unfolded state. (Reprinted with modifications from Kesner, B.A., Ding, F., Temptle, B.R., Dokholyan, N.V. “N-terminal Strands of Filamin IG domains act as a Conformational Switch Under Biological Forces” Proteins 2010, 78:12-24. Used with Permission from John Wiley and Sons. © 2009)
To estimate the critical force for coiled-coil unfolding, the simulations were allowed to run for 1,000,000 DMD time units (corresponding to 50ns), and then the protein length vs. force was plotted (See Figure 4.7). The plot indicates that the protein
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remains folded at 100pN of force, but at 150pN the coiled coil region has unfolded a length of 12nm. This distance corresponds to the distance needed to unfold the triple helix portion of the coiled coil (Figure 4.8). This provides an estimate for the critical unfolding force (FC) of the coiled coil of 125±25pN which is fairly consistent with the 94pN reported by Brown et. al in AFM experiments.(75) An estimate of xu can be made by making a histogram of the time series of end-to-end distances of the coiled coil during extension at 100pN and 150pN of force (on either side of the critical force). Because the protein is unlikely to remain at the top of an unstable maxima in the energy landscape, xu can be approximated as the extension at which the coiled-coil is least likely to reside (Figure 4.8). Using an xu of 1.5nm gives a ΔGB of 187.5pN·nm (27kcal/mol or 45kBT), indicating a fairly high stability. To further refine this critical force value, a series of simulations should be run at a constant force of 125pN with different initial velocities. A series of unfolding trajectories will be generated and the critical residues in the unfolding process can be determined, allowing for further refinement of both FC and xu. The part of the coiled coil with a 4th coil was only observed unfolding in simulations at forces above 225pN, indicating a critical force between 150pN and 225pN. Extending the fourth coil can result in an additional 9nm of extension, giving a total extension for the coiled-coil region of 21nm.
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Figure 4.7: Coiled coil unfolded distance vs. Force: (Top) End to end distance (calculated as the distance between γCys19 and αCys161) after ~50ns was calculated for runs at each force. The critical force estimated to lie between 100pN and 150pN of force given the sudden jump in protein length at this juncture. At 150pN the triple helix of the coiled coil was unfolded, but the 4th coil remained folded. At higher forces, the 4th coil unraveled as well. (See Figure 4.8) (Bottom) Histogram of the end-to end distances of the coiled-coil during the 100pN run (remained folded) and the 150pN run (unfolded). The critical unfolding length (xu) can be estimated as the place of lowest frequency occurrence of end to end lengths because the protein resides at the top of an unstable energy landscape (Figure 4.6). Given the distributions, it appears that the critical distance occurs at 195Å extension. Assuming an unstretched coiled-coil length of 180Å gives xu~15Å of extension.
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These simulations also predict an intra-chain α-helix to inter-chain β-sheet hydrogen bond re-orientation upon unfolding of the coiled-coil region (See Figure 4.8). While this transformation has been speculated due to the similarity of the coiled coil structure to that of keratin, these simulations provide the first computationally based prediction of the effect.(65, 67) As discussed in Ch. 3, these results constrain the coiled coil region in its functionality as a reversible spring. Simulations are currently ongoing, but after 100ns of simulation time in the absence of force, the energy of the β-sheet conformation appears stable indicating that the region may not re-fold into the coiled coil structure after the removal of force.
Figure 4.8 α-helix to β-sheet transition: As the α-helical coiled-coil region is stretched, the first part to open is the “kink” in the middle. Next, the N-terminal portion of the coiled-coil opens, and finally, the C-terminal portion with a 4th coil unravels. Interestingly, as the triple-helix unravels, inter chain β sheets begin to form. As can be seen in the plot on the right, The coiled-coil structure transitions from intra-chain contacts to inter-chain contacts indicating the α-helix to β-sheet transition. We define a contact when the distance between two Cβ carbons is less than 7.5Å.