Cartoon displaying the electrostatic perturbation induced by a half redox reaction of transferring electron to the heme in the protein used as an example. The elastic deformation of the protein not only shifts the ionized surface residues, shown by charges at the surface, but also results in adiabatic movements of water dipoles solvating them (shown by arrows).
Cartoon showing propagation of piezoelectric perturbation caused by binding an ion to residue i and producing a displacement of residue j. The electric force exerted by the ion is propagated throughout the protein as an elastic deformation indicated by chains of arrows. In contrast, the displacements of ionized surface residues are propagated as water-mediated, dipole-correlated surface motions. Ionized surface residues combine into a global, correlated net for transmitting signals, which does not necessarily require a specific binding site.
Free energy surfaces representing the free energy penalty (reversible work) of changing the charge at a binding site of an allosteric protein. The calculations are done for attaching carbamoylphosphate (q 02 = −2) to the bacterial enhancer-binding protein NtrC (Fig. 8 ). The curves refer to DENM in the unphosphorylated state (q 01 = 0) and to DENM and sDENM in the phosphorylated state (q 02 = −2). The elastic network is defined with k B T/C = 0.75 Å2 and ε = 125; the cutoff radius is 15 Å. The free energy of binding ΔF is unknown and was set at −0.5 eV for the purpose of illustration.
Mean-square displacements of cytochrome B562 (cytB) Cα's from MD simulations, DENM and sDENM calculations. The network parameters in the DENM/sDENM calculations are: k B T/C = 0.75 Å2, ε = 125, and the cutoff radius is 15 Å. The solvent accessible surface for the loop residues labeled in Fig. 5 is scaled down to 45 Å2.
Cartoon of cytochrome B562 (cytB) showing the positions of Cα (spheres) colored by residue charge: charged (green) and uncharged (red). The charged residues marked green (also green points in Fig. 4 ) are also required to have α i greater than 0.16, used as a threshold number. The remaining Cα's are marked red. The side chain atoms are colored by charge: negative (red), positive (blue), and neutral (white). The heme iron is colored brown while the remaining atoms of the heme are blue. Numbers label the unstable residues in the loop for which the water-exposed surface was scaled down to 45 Å2 in order to maintain the network stability.
Loss spectra and for cytB. Compared are MD, DENM, and sDENM calculations. To show the sensitivity of sDENM calculations to solvation of the loop residues in cytB, the results of choosing the solvent-accessible area of a i = 45 Å2 and of a i = 50 Å2 are shown. The two-Debye relaxation parameters are ζ l = 30 ns, ζ h = 0.006ζ l , and a = 0.35 [Eq. (18) ]. The elastic network is defined with k B T/C = 0.75 Å2, ε = 125, and the cutoff radius of 15 Å.
for cytB. The parameters of the network are the same as in Fig. 6 .
Superimposed structures of the bacterial enhancer-binding protein NtrC in dephosphorylated (light blue, PDB entry 1DC7) and phosphorylated (dark blue, PDB entry 1DC8) states. 37,77 The nuclear magnetic resonance structure was determined 77 with carbamoylphosphate binding to Asp54. The displacement of Glu124 in response to a probe charge at Asp54 is shown in Fig. 9 .
Frequency-dependent allosteric displacement [Eq. (27) ] for residue i = 124 (Glu) of NtrC in the unphosphorylated (NtrC) and phosphorylated (P-NtrC) states. The results of calculations within DENM and sDENM are compared as shown in the plot.
Displacements Δr i between two equilibrium structures (NtrC and P-NtrC) of the NtrC protein. δr i (0) shows the displacement of residue i in response to placing a unitary probe charge q ω = 1 at the position of Cα of the binding site (Asp54, Fig. 8 ).
Integrals entering Eq. (A1) calculated with the longitudinal structure factor of TIP3P water. 28 The labeling in the plot point to the integration with j 0(rk) (first equation in Eq. (A1) , labeled as “0”) and with j 2(rk) (second equation in Eq. (A1) , labeled as “2”). The integrals are calculated as the function of r/s with fixed s = 4.4 Å.
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