Heat maps of the 2D-distributions (A, C, and E) and correlations (B, D, and F) of γ N and γ C for 5-residue ETB segments of proteins chains derived assuming the cutoff for the central θ angle, θ cut = 120° (A and B) and θ cut = 135° (C and D), and for the NETB chains (E and F). The unit of the color scale is 10 −5 . The plots were made with GRI. 48

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... the multitorsional potential of an ETB chain segment depends on the sum of γ angles along the segment (eq 4), it can be expected that, for an ETB chain segment with length m = 5 (i.e., with two consecutive γ angles), the distribution of these angles is narrower along Δγ N = Δγ C and broader along Δγ N = −Δγ C direction, where Δγ is the displacement of the respective angle from distribution center. The reason for this is that, if the changes of the two angles are opposite to each other, the sum of the angles remains constant and, consequently, there is no free-energy cost due to the multitorsional term expressed by eq 4. The respective plots for the ETB chain segments are shown in Figure 5, parts A and C, for θ ext = 120° and 135°, respectively. It can be seen from Figure 5, parts A and C, that the γ N and γ C angles are indeed anticorrelated, the anticorrelation being more pronounced for θ ext = 135°. ...

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... reason for this is that, if the changes of the two angles are opposite to each other, the sum of the angles remains constant and, consequently, there is no free-energy cost due to the multitorsional term expressed by eq 4. The respective plots for the ETB chain segments are shown in Figure 5, parts A and C, for θ ext = 120° and 135°, respectively. It can be seen from Figure 5, parts A and C, that the γ N and γ C angles are indeed anticorrelated, the anticorrelation being more pronounced for θ ext = 135°. The bulk of the distribution is centered at about γ N = 20°, γ C = −110°. ...

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... bulk of the distribution is centered at about γ N = 20°, γ C = −110°. The anticorrelation between the γ N and γ C angles is even more apparent from the respective covariance plots shown in panels B and D of Figure 5. The anticorrelation also results in keeping approximately the same dihedral between the two virtual bonds at the end of the segment and the extended-segment axis (the angle Γ′ shown in Figure 2A). ...

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... also is another distribution center at about γ N = 110°, γ C = −120°, which corresponds to uncorrelated angles. The intensity of the two centers is swapped when θ ext is smaller ( Figure 5A). This is understandable, because the weight of the multitorsional term quickly drops with the interchain-segment θ angle(s) becoming smaller (Figure 4). ...

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... reference, the distribution and covariance of the γ N and γ C angles of the NETB chains are shown in Figure 5, parts E and F, respectively. It can be seen that both the distribution and the covariance are different from those shown in panels A−D of the figure, with two peaks at γ N of about 50° and γ C of about −120° and 40°, respectively, with no correlation between the angles exhibited. ...

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... trend (low PMF for small angles for even n and higher for odd n) seems to persist for n > 4; however, the statistics are poorer in such cases. Therefore, the conclusion from analyzing the γ N , γ C distribution for n = 2 ( Figure 5) that the dihedral angle Γ′ defined by the two flanking backbone virtual bonds and the extended segment axis (Figure 2A) is largely restricted (being small for an even number of dihedrals and extended for an odd number of dihedrals) seems to extend at least until n = 4, although this trend seems to vanish as the length of the extended segment increases. Folded Chain Segments. ...

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... distributions and the PMFs are shown in Figures S2−S4 of the Supporting Information. As can be seen from these figures, there are no qualitative differences between the plots shown in Figure 5 Figure S3 for n > 4 are more rugged compared to those of Figure 6, and most of the heat map of the (γ N , γ C ) distribution for the nonstructured segments with n > 20 ( Figure S4C) shows zero population, because there are very few unstructured chain segments with length greater than 20 and no glycine or proline residues. ...

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... models were in all-atom representation obtained from the UNRES representation by applying the conversion procedure, 23 which is based on the PULCHRA 37 and SCWRL 38 algorithms. The results are shown in Figures S5 and S6 of the Supporting Information, respectively. ...

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... can be seen from Figure S5, parts A and C, that the distribution of the γ N and γ C angles for the ETB chain segments from the UNRES group models are significantly different from those encountered in proteins. There is a maximum at small positive γ angles, which is virtually not present in protein structures ( Figure 5, parts A and C), for which γ C is extended and negative. ...

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... can be seen from Figure S5, parts A and C, that the distribution of the γ N and γ C angles for the ETB chain segments from the UNRES group models are significantly different from those encountered in proteins. There is a maximum at small positive γ angles, which is virtually not present in protein structures ( Figure 5, parts A and C), for which γ C is extended and negative. This feature of the NETB chain segments from the UNRES models is especially visible in Figure S5A, in which the distribution for θ cut = 120° is shown and for which the maximum at small positive γ dihedral angles Figure 8. ...

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... is a maximum at small positive γ angles, which is virtually not present in protein structures ( Figure 5, parts A and C), for which γ C is extended and negative. This feature of the NETB chain segments from the UNRES models is especially visible in Figure S5A, in which the distribution for θ cut = 120° is shown and for which the maximum at small positive γ dihedral angles Figure 8. Heat maps of the 2D-distributions (expressed as probability per degree 2 ) of γ N and θ N and those for γ C and θ C for (A) folded (FD) and (B) nonstructured (NS) segments of protein chains. ...

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... becomes the global maximum. It can be noted that the γ angles are anticorrelated for this maximum ( Figure S5, parts A and B). However, in this case, the anticorrelation results from the repulsion of the side chains and peptide groups at the termini of the chain segment. ...

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... the central θ angle is not so extended (which happens for θ cut = 120°, the end groups will overlap unless the end virtual bonds move synchronously. It can be seen from panels C and D of Figure S5 that the anticorrelation disappears for θ cut = 135°, for which the end groups do not overlap significantly even when the end backbone virtual bonds of the segment face each other. The regions of the other distribution maxima do not exhibit any anticorrelation of the γ N and γ C angles. ...

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... regions of the other distribution maxima do not exhibit any anticorrelation of the γ N and γ C angles. It can also be seen from Figure S5, parts E and G that the segments from the UNRES-template group models better reflect the distributions derived from protein structures ( Figure 5, parts A and C) than those from the UNRES group models. However, a significant distribution maximum is still observed for small γ angles ( Figure S5, parts E and G). ...

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... regions of the other distribution maxima do not exhibit any anticorrelation of the γ N and γ C angles. It can also be seen from Figure S5, parts E and G that the segments from the UNRES-template group models better reflect the distributions derived from protein structures ( Figure 5, parts A and C) than those from the UNRES group models. However, a significant distribution maximum is still observed for small γ angles ( Figure S5, parts E and G). ...

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... can also be seen from Figure S5, parts E and G that the segments from the UNRES-template group models better reflect the distributions derived from protein structures ( Figure 5, parts A and C) than those from the UNRES group models. However, a significant distribution maximum is still observed for small γ angles ( Figure S5, parts E and G). Moreover, there is no anticorrelation between the γ N and γ C angles ( Figure S5, parts F and H). ...

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... a significant distribution maximum is still observed for small γ angles ( Figure S5, parts E and G). Moreover, there is no anticorrelation between the γ N and γ C angles ( Figure S5, parts F and H). Therefore, including the information from templates in protein-structure modeling with UNRES seems to help in achieving the correct relative orientation of segment ends but does not seem to capture the concerted change of the γ angles. ...

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... our study, it follows that adding the backbone to the considerations could be beneficial in understanding allosteric interactions. It should be noted at this point that the anticorrelation of the consecutive backbone virtual-bond-dihedral angles γ of extended chain segments shown in Figure 5A−D does not demonstrate allostery as such, because allostery is a causaleffectual phenomenon. However, it strongly suggests that the correlation contribution to the local component of the potential of mean force given by eq 4 provides a smooth road for allosteric interactions to occur. ...

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... limited size of the data sets did not enable us to derive and compare the statistical multitorsional potentials (cf. Figure 6). It can be seen from panels C−H of Figure S7 that the plots corresponding to the randomly selected subsets of proteins do not differ significantly from those of the whole protein set ( Figure 5, parts A and B) and do not differ significantly from each other. In contrast to this, those of the proteins that are involved in allostery do. ...

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... contrast to this, those of the proteins that are involved in allostery do. The region of the main distribution maximum (centered at about γ N = 10°, γ C = −110° for the whole set of proteins) is shifted toward more negative γ C angles and the region of the adjacent distribution maximum, which appears at large positive γ N angles shows anticorrelation between γ N and γ C , which does not occur in the distributions derived from the entire set of proteins ( Figure 5, parts A and B) or from the randomly selected subsets (panels C−H of Figure S7). This result suggests that the anticorrelation between the consecutive backbone-virtual-bond dihedrals is more pronounced for allosteric proteins than for randomly picked proteins. ...

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... of the lowest-order term in multitorsional potentials; Figure S1, histograms of the fractions of specific residue types in the first and second position at the N-and at the C-terminus of (A) ETB and (B) F chains; Figure S2, heat maps of the 2D-distribution of γ N and γ C for ETB chain segments for (A) θ cut = 120°, (B) θ cut = 135° and (C) NETB chain segments obtained after eliminating the entries with the glycine and proline residues at the ends of the segments; Figure S3, potentials of mean force in the sum of virtual-bonddihedral angles γ(Γ) along chain segments with n consecutive backbone-virtual-bond dihedrals for the ETB and NETB segments of protein chains obtained after eliminating the entries with the glycine and proline residues at the ends of the segments; Figure S4, heat maps of the 2D-distributions of γ N and γ C for (A) FD, (B) FH, and (C) NS chains obtained after eliminating the entries with the glycine and proline residues at the ends of the segments; Figure S5, heat maps of the 2D-distributions and correlations of γ N and γ C for 5-residue ETB segments from the CASP14 UNRES group models and the CASP14 UNRES-template models; Figure S6, heat maps of the 2D-distributions of the γ N and γ C corresponding to the CASP14 UNRES group models and CASP14 UNRES-template groups models; Figure S7, heat maps of the 2D-distributions and correlations of γ N and γ C for 5-residue ETB segments from 1101 allosteric proteins and three sets of 1101 proteins each, selected at random from the PDB (PDF) File S1: List of all proteins chains analyzed in this study (TXT) File S2: List of allosteric proteins analyzed in this study (TXT) File S3: Set 1 of the three randomly picked sets of 1,101 proteins each analyzed in this study (TXT) File S4: Set 2 of the three randomly picked sets of 1,101 proteins each analyzed in this study (TXT) File S5: Set 3 of the three randomly picked sets of 1,101 proteins each analyzed in this study (TXT) ...