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Large-scale motions of biomolecules involve linear elastic deformations along low-frequency normal modes, …


Biology Articles » Biophysics » Nonlinear elasticity, proteinquakes, and the energy landscapes of functional transitions in proteins » Figures

Figures
- Nonlinear elasticity, proteinquakes, and the energy landscapes of functional transitions in proteins

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Fig. 1. Comparison of energy surfaces with different models. When the energy surface is perfectly harmonic as a function of Cartesian coordinates and the conformational change path is linear, the energy surface along the conformational change path can be derived from the normal mode frequency of mode 1 at the initial state (solid line). However, the energy surface along the normal mode 1 (x) computed explicitly with the full anharmonic Tirion potential yields a much higher energy than the harmonic approximation owing to nonlinearities. The energy surface along the nonlinear conformational change path generated by using an iterative method with one mode ({circ}) agrees with the harmonic approximation quite well up to 4 Å of rmsd but exceeds the harmonic result beyond this point. The energy using three modes to connect initial and final states ({square}) is also shown.

 

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Fig. 2. Residue strain energy of structures along nonlinear conformational change path. (a) The change of the strain energy localized in individual residues as the structure is deformed is shown. The rmsd of each structure from the open structure is indicated on the right. Residues in blue have no strain energy while red residues have high strain energy. The maximum strain (indicated in red) is 0.5 kcal/mol. (b) The residue strain energy of the structure after 15 steps of iteration is shown in a 2D plot. The secondary structure of this protein is indicated on the top of the plot. (c) The structure after 15 steps of iteration is shown along with the residue strain energy. The residues again are colored according to the strain energy; blue corresponding to no strain and red residues corresponding to high strain energy as before. c was prepared with VMD (43) and RASTER3D (44).

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Fig. 3. The energy profiles for open and closed states with and without cracking. Calculations of free energy profiles without cracking and with cracking for two values of and 0.1 kcal/mol are shown. The threshold of the closed state, , is set so that the open and the closed states have the same energy at the totally unfolded state. The strain energy computed from the open form is shown in black. The one computed from the closed form is shown in red for free energy change of {Delta}Geq = 0 kcal/mol and is shown in blue for {Delta}Geq = –3 kcal/mol. Results computed without allowing cracking are shown as solid lines, and the broken lines correspond to a threshold for cracking of . The dotted lines use a cracking threshold of .


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Fig. 4. The transition-state barrier dependence on the reaction driving force. Calculations of how the transition-state barrier, ΔG*, depends on the reaction driving force, (–ΔGeq), both without cracking (a) and with cracking (c) are shown. The corresponding energy surfaces at the different driving forces (–ΔGeq) without cracking (b) and with cracking (d) are also shown. A quadratic curvature is observed for the fully elastic model without cracking. Including the cracking effect makes the barrier dependence on driving force linear for a very large range of driving forces.

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Fig. 5. The transition-state barrier dependence on the cracking threshold. Shown is how the transition-state barrier, ΔG*, depends on the cracking threshold, . A driving force of ΔGeq =–3 kcal/mol is used. The data ({circ}) are fitted by a hyperbolic relation, ΔG* = -0.23/( -0.03) +21. From the slope of the line, the Tafel coefficient a = ¶DG*/¶ is estimated as 47 at  = 0.1 kcal/mol and 16 at  = 0.15 kcal/mol. The experimentally determined urea dependence of the stability, m = ¶ΔGD–N/[urea] = 2.9 kcal/mol per M (31), here M is the molar concentration of urea, corresponds to a change of cracking threshold with the value m/Nres{approx}0.014 kcal/mol per M, where the total number of residues Nres = 214. Combining these, the urea dependence of the transition barrier {partial}ΔG*/{partial}[urea] is {approx}0.65 kcal/mol per M if  = 0.1 kcal/mol and {approx}0.22 kcal/mol per M if  = 0.15 kcal/mol is used.

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