Long-distance correlations of rhinovirus capsid dynamics contribute to uncoating and antiviral activity

Abstract

Human rhinovirus (HRV) and other members of the enterovirus genus bind small-molecule antiviral compounds in a cavity buried within the viral capsid protein VP1. These compounds block the release of the viral protein VP4 and RNA from inside the capsid during the uncoating process. In addition, the antiviral compounds prevent “breathing” motions, the transient externalization of the N-terminal regions of VP1 and VP4 from the inside of intact viral capsid. The site for externalization of VP1/VP4 or release of RNA is likely between protomers, distant to the binding cavity for antiviral compounds. Molecular dynamics simulations were conducted to explore how the antiviral compound, WIN 52084, alters properties of the HRV 14 capsid through long-distance effect. We developed an approach to analyze capsid dynamics in terms of correlated radial motion and the shortest paths of correlated motions. In the absence of WIN, correlated radial motion is observed between residues separated by as much as 85 Å, a remarkably long distance. The most frequently populated path segments of the network were localized near the fivefold symmetry axis and included those connecting the N termini of VP1 and VP4 with other regions, in particular near twofold symmetry axes, of the capsid. The results provide evidence that the virus capsid exhibits concerted long-range dynamics, which have not been previously recognized. Moreover, the presence of WIN destroys this radial correlation network, suggesting that the underlying motions contribute to a mechanistic basis for the initial steps of VP1 and VP4 externalization and uncoating.

Human rhinovirus (HRV) infection is the most common cause of respiratory infection in humans (1) and incurs considerable economic expense (2). HRV replication is often restricted to the upper respiratory tract, and the resulting illness is of short duration, but HRV can also invade the lower respiratory tract and cause more serious infections (3, 4). Aided by the structure determination of HRV 14 in 1985 (5), much effort has been made to develop antiviral compounds such as disoxaril (WIN 51711) (6), WIN 52084 (7), WIN 54954 (8), and pleconaril (9). Pleconaril advanced to phase II clinical trials (10); however, no drug against HRV has been approved for distribution. These WIN compounds act primarily by inhibiting the “uncoating” process (11, 12) or release of genomic RNA from inside the host cell. Here we report results that elucidate this antiviral activity.

HRV, a member of the Picornaviridae family and the enterovirus genus, has an icosahedral capsid (Fig. 1A) with 60 copies of the protomer, which comprises four structural proteins—VP1, VP2, VP3, and VP4. A pentameric unit with five protomers is shown in Fig. 1B. The three larger proteins, VP1, VP2, and VP3, shown in blue, green, and red, respectively, in Fig. 1 B and C, share a common structural motif, the eight-stranded, antiparallel β barrel (5). A deep “canyon” on the capsid surface, indicated with a yellow ring in Fig. 1 B and C, circumscribes the fivefold symmetry axis. WIN compounds, including WIN 52084 (Fig. 1 B and C, gold space-filled spheres), bind below this canyon in a hydrophobic cavity, or pocket, in the interior of VP1.

Figure 1.

(A) Schematic of an icosahedron with 20 triangular tiles. Each tile has three asymmetric units corresponding to the protomer of the HRV 14 capsid (blue area). The Z axis is coincident with one of the twofold symmetry axis. In this study the pentamer, or five protomers centered around a fivefold symmetry axis, is the unit cell (blue and yellow areas). Inset shows orientation of one protomer in the unit cell. Neighboring protomers are generated using symmetry operators. (B) Top view of five protomers in the primary unit of simulation. Protomer 1 is shown in space-filling spheres, and protomer 2–5 are shown as ribbon drawings. In protomer 1 and 3 VP1 is shown in blue, VP2 in green, VP3 in red, and VP4 in cyan. WIN 52084 is shown in volume representation in gold. (C) Side view of protomer 1. N termini of VP1 and VP4 are the innermost residues and lie on the inside of the capsid. The N termini from five VP3 molecules together form a β-annulus (not shown) at the fivefold symmetry axis. The fivefold, threefold, and twofold symmetry axes are indicated by the usual symbols. The yellow circle, 35 Å from the fivefold symmetry axis, represents the approximate position of the canyon. Chemical structure of WIN 52084 is shown.

Along with release of RNA, the uncoating process is also associated with the loss of VP4 from the viral capsid (13) as well as externalization of the N terminus of VP1 (14). Antibody recognition (15) and results from limited proteolysis of isolated virus (16) provide evidence that the N termini of VP1 and VP4, which are regions located inside the viral capsid (Fig. 1C), are reversibly exposed in the mature capsid, and suggest this externalization is part of the initial stages of uncoating. Notably, in the presence of WIN compounds these regions are not proteolyzed, and thus the motions leading to the exposure of these N termini are dampened, consistent with the activity of WIN compounds being to inhibit uncoating.

The reversible externalization of interior regions of VP1 and VP4, viral uncoating, and the inhibition of these processes by WIN together infer large-scale, concerted motions of the capsid. Further, RNA, VP4, and the N terminus of VP1 are suggested to exit the capsid at the fivefold symmetry axis (17–21) or an interface near the twofold axis (22–24), so that WIN binding in a pocket distant to these sites indicates a long-range activity. These observations motivated MD studies of the solvated HRV 14 capsid, with and without WIN 52084 bound, to probe functional protein dynamics that might underlie the initial stages of the externalization process and uncoating and to further elucidate the physical basis of the long-distance antiviral activity. The atomic fluctuations observed from MD studies were analyzed for radially correlated motions between capsid residues. Radial cross-correlation was examined rather than the more commonly used displacement vector cross-correlation (DCC) to overcome the dependence of the later on directionality (25) (see SI Appendix). We introduce a framework to analyze long-distance, concerted motions using a network based on radial cross-correlation coefficients. Potential pathways underlying the motions were traced in the radial correlation network by using a modification of Dijkstra’s graph searching algorithm (26). The resulting important paths involve residues at spontaneous drug-resistant mutation sites and were sensitive to the presence of the WIN compound. This framework based on a radial correlation network has allowed the previously undescribed identification of correlated motion over a long distance from an MD simulation, and allowed insights into the molecular mechanism of antiviral activity.

Results and Discussion

To investigate large scale, concerted motions of the HRV 14 capsid, trajectories were computed using rotational boundary symmetry conditions (27–31) with a solvated viral pentamer as the asymmetric unit for simulation. Five copies of the protomer centered at the fivefold axis (Fig. 1 A and B) accurately models microscopic dynamics of not only the WIN-binding pocket but also the region around the fivefold symmetry axis. Trajectories were calculated starting from the crystallographic coordinates of either HRV 14 (32) or HRV 14 with WIN 52084 bound (33) (see SI Appendix). The last 2 ns of 10 3-ns trajectories were used for analysis, for a total time equal to 20 ns for HRV 14 trajectories, referred to as T_nowin, and 20 ns for HRV 14-WIN 52084 trajectories, referred to as T_win. Confidence intervals, CI, of time-averaged quantities were obtained from the bootstrapping method (see SI Appendix).

Long-Range Correlations.

Reasoning that protein externalization and uncoating involve motions with a radial directionality, we examined the correlations in the distance from the capsid center among pairs of C_α atoms. A breathing motion is reflected in such a radial correlation coefficient in contrast to the DCC, which is generally diminished by angular averaging (25) (see SI Appendix). The absolute radial correlation coefficient, or absolute normalized radial covariance, C_R(i,j) (see Materials and Methods), and the associated 95% CI were determined for pairs of C_α atoms from T_nowin and T_win.

To identify concerted motions over large spatial scales, we consider C_R(i,j) values for C_α atoms of noninteracting residue pairs—i.e., those with a minimum distance between residues greater than the nonbond cutoff distance 14 Å. The distributions for the C_R(i,j) values with 95% CI greater than 0.6 from T_nowin and T_win are shown in Fig. 2, blue and red curves, respectively, as a function of the distance between the two C_α atoms. The time-average values for C_R(i,j) are well converged within 1 ns, and only a small fraction of the total C_α-C_α radial correlations are greater than 0.6 (see SI Appendix). We find that a large number of radially correlated C_α-C_α pairs exist for HRV 14 in T_nowin with the majority from pairs separated by 14 to 30 Å. In striking contrast, the corresponding C_R(i,j) distribution for HRV 14-WIN from T_win is overall greatly reduced. Moreover, many of the T_nowin pairs with large radial correlation were separated by distances greater than 45 Å and some with distances as large as 85 to 90 Å. C_R(i,j) values ≥0.8 were observed for C_α-C_α pairs at distances greater than 55 Å. This was not the case for T_win; many fewer C_α-C_α pairs separated by long distances have a 95% CI for C_R(i,j) greater than 0.6. In addition, the radially correlated C_α-C_α pairs separated by distances greater than 45 Å are largely nonoverlapping sets for T_nowin and T_win, as shown by the dotted black line in Fig. 2. This dotted line is the distribution of C_α-C_α pairs present in the T_nowin distribution of Fig. 2 (blue curve) and have a 95% CI for C_R(i,j) that is less than 0.6 in T_win. The dotted black curve nearly overlaps the blue curve, demonstrating that these correlated C_α-C_α pairs in T_nowin are not as correlated in T_win.

Figure 2.

Distribution of all nonneighbor C_α pairs with 95% CI of C_R(i,j) > 0.6 from T_nowin (blue) and T_win (red) (see text for explanation of T_nowin and T_win) as a function of intrapair C_α distance.

It has long been suggested that RNA, VP4, and the N terminus of VP1 exit the capsid at the fivefold symmetry axis (17–21), although recent cryoelectron microscopy results on the closely related poliovirus, another member of the enterovirus genus, suggest RNA exits at an interface near the twofold symmetry axes (22–24) and the N terminus of VP1 exits though an opening at the base of the canyon (26). To ask if the long-distance radial correlations shown in Fig. 2 might reflect dynamics associated with breathing, uncoating, and WIN-binding effects, we identify radially correlated pairs that have one C_α atom in the region of the WIN-binding pocket or within 35 Å of the fivefold axis (yellow ring in Fig. 1 B and C). This distribution is shown in Fig. 3A for T_nowin and T_win. While the T_win distribution (red curve) is overall reduced and shows little structure beyond 40 Å, the long-distance radially correlated pairs from T_nowin (blue curve) fall into three groups centered at 46 Å, 64 Å, and 84 Å, designated i, ii, and iii, respectively. These groups are mapped onto the structure in Fig. 3 B and C; for each pair in the group, the C_α near the fivefold axis is indicated with a blue sphere drawn on one of the five protomers, and the second C_α is a sphere colored green for group i, gold for group ii, and red for group iii. Residues surrounding the fivefold axis are highly correlated with residues near the fivefold but in the neighboring protomer (green spheres). Also observed are radial correlation with residues in the vicinity of the closest twofold axis (green spheres) and all other twofold axes across the pentamer (gold and red spheres). The higher densities of long-range radially correlated motions in T_nowin are therefore between residues in the WIN pocket and residues in regions near the fivefold and twofold symmetry axes. Moreover, as noted above, these correlations are lost in the presence of WIN, so that the behavior is suggestive of functional motions related to externalization and uncoating at the fivefold axis as long proposed (17–21) as well as near the twofold axes as more recently reported (22–24).

Figure 3.

(A) Distribution of nonneighbor C_α pairs from Fig. 2 with at least one C_α within 35 Å of the fivefold symmetry axis (the yellow circle in Fig. 1). (D) C_α pairs present in A with at least one C_α from N termini of VP1 (residues 1001–1015) or VP4 (residues 4001–4040) are included. Long-distance radially correlated pairs in A fall into three groups centered at 46 Å, 64 Å, and 84 Å, designated i, ii, and iii, respectively. These groups are mapped onto the structure in B and C. For each pair in the group, the C_α near the fivefold axis is represented by a blue sphere drawn on one of the five protomers, and the second C_α is a sphere colored green for group i, gold for group ii, and red for group iii. D–F are similarly formatted for regions i, ii, and iii in B. Long-distance radial correlation for residues surrounding the fivefold axis with residues near fivefold but in other protomers and with residues in the vicinity of twofold axes as marked with 5f and 2f, respectively.

The radial correlation in groups i, ii, and iii in T_nowin largely involve residues from the reversibly externalized N-terminal regions of VP1 or VP4; however, these radial correlations are not present in T_win. Fig. 3D displays the subset of the distribution in Fig. 3A corresponding to C_α-C_α pairs with one C_α from either VP1 (residues 1001–1015) or VP4 (residues 4001–4040). The curves are colored as in Fig. 3A. Because the dotted black curve in Fig. 3D closely follows the blue curve from T_nowin for distances > 40 Å, most of these long-distance C_α-C_α pairs with large radial correlation coefficients in T_nowin are not radially correlated in T_win. As visualized in Fig. 3 E and F, these VP1 or VP4 residues with long-distance radial correlations in T_nowin but not in T_win are primarily correlated with residues near the neighboring twofold axis (green spheres) or more distant twofold axes (gold and red spheres). These radial correlations are indicative of concerted motions of these innermost regions of VP1 and VP4 spanning distances of 85 Å.

Identification of Correlation Paths.

What are the underlying origins of the radial correlations occurring over distances as large as 90 Å? That these correlations exist in free HRV 14 but are largely dampened in the presence of WIN 52084 suggests that identifying the origin of the radial correlations could provide information toward the mechanism for transient externalization, or breathing, and perhaps the uncoating process.

To answer the question, we developed a method for detecting the link between two C_α atoms based on a modified version of Dijkstra’s graph searching algorithm (26). The nodes of the network correspond to C_α atoms and edges connecting all C_α atoms belonging to residues within the nonbonded distance of 14 Å (Fig. 4). To capture the radial correlation behavior, an edge between C_αi and C_αj was assigned a value equal to 1 - C_R(i,j). The path between any two nodes of the network, l and m, was determined from the minimum value of the weight (see Materials and Methods). The weight of a path is specified as one minus the product of the radial correlation coefficients for all edges of the path so that the path with the minimum weight is the optimal path in terms of high radial correlation and fewest number of edges. This path is named hereafter the “shortest path.” The shortest path was determined for all C_α-C_α pairs of the pentamer, or approximately 8 × 10⁶ paths. The second step of the method to identify the origin of the radial correlations was to determine which path segments had the highest betweenness (34). Betweenness of an edge is the probability with which the edge occurred in all shortest paths, or the number of occurrences of the edge relative to the total number of paths. From the distribution of betweenness values, shown in SI Appendix, a small number of edges are observed to lie at the far edge of the distribution for T_nowin by having considerably higher betweenness than other edges. In contrast, the betweenness through these edges is substantially reduced in T_win.

Figure 4.

A simplified network. Each node in the network corresponds to a C_α atom of a residue. Two nodes are connected by an edge if atoms in the corresponding residues are within 14 Å during the simulation. Node i is connected with its neighbors through orange color edges and node j through red color edges. All other edges are shown in gray. Weight of an edge , W_ia = 1 - C_R(i,a), is written next to the edge. Weight of path , W_iab ≡ W_ia⊗W_ab ≡ 1 - C_R(i,a)·C_R(a,b). Shortest path between two nodes i and j is , the path with minimum W_i...j.

The component of the radial correlation network with highest betweenness is visualized by mapping the edges with the largest betweenness values onto the structure in Fig. 5A. The edges having a betweenness with a 95% CI > 0.0005 are shown with blue lines of thickness proportional to betweenness. It is noted that all edges within the pentamer asymmetric unit are treated independently so that the near fivefold symmetry apparent in the blue segments of Fig. 5A is evolved from microscopic sampling. High-betweenness edges from T_win using the same cutoff value of 0.0005 are shown with red lines in Fig. 5B. The presence of WIN 52084 very clearly alters the observed patterns of betweenness; fewer red lines are observed and the overall thinner widths indicate smaller betweenness in general for T_win.

Figure 5.

Edges with 95% confidence interval of betweenness > 0.0005 are shown with their thickness proportional to their betweenness for T_nowin in A and for T_win in B. Betweenness is the number of occurrences of an edge relative to the total number of paths. Highest betweenness through any C_α pair are 0.0056 and 0.0027 in T_nowin and in T_win, respectively. (C) A sequence of edges with high betweenness connecting N terminus of VP4 (residue 4008 of protomer 1) near fivefold symmetry axis to a residue across canyon (residue 1254 of protomer 3).

The patterns in Fig. 5 are striking. Each node of the network has close to a hundred or more edges so that the number of possible paths between two C_α atoms is large and increases dramatically as the distance of separation increases. Thus, an edge with highest betweenness, and more importantly, sequential edges of high betweenness are definite outliers. Sequential edges of highest betweenness are a linked set of radially correlated C_α atoms and suggest a mechanism for the long-range connectivity in Fig. 3. Numerous blue path segments are observed from T_nowin (Fig. 5A) and found to be concentrated around the canyon and the fivefold axis. The set of blue lines encircling the fivefold axis corresponds to the N-terminal region of VP3, which forms a well-ordered β-annulus through interprotomer interactions. This annulus is absent from the pattern of highest betweenness paths calculated from T_win when WIN is bound (Fig. 5B).

The paths connect radially correlated C_α-C_α pairs separated by long distances (Fig. 3) and are a likely origin for these radial correlations. An example of one such path is given in Fig. 5C, which shows sequential edges connecting VP4 residue 4008 near the fivefold symmetry axis with the VP1 residue 1254 across the canyon. As such, this path provides a linkage between residues of C_α-C_α pairs in group i Fig. 3C (green with blue spheres Fig. 3 E and F). Moreover, this path does not exist in the presence of WIN 52084.

The 10 edges with the highest betweenness from T_nowin are listed in Table 1 by the residue numbers of the nodes delimiting the edge. These edges have a 95% CI of betweenness greater than 0.001. In contrast, their values from T_win are approximately two orders of magnitude smaller. All the the residues in Table 1, other than 2241, 2242, 3005, and 3006, are conserved or vary by one amino-acid type substitution among HRV-B serotypes, of which HRV 14 is a member. The underlined residues in Table 1 are similarly conserved in HRV-A. Of the 15 residues in Table 1, two (1199 and 1219) are in the WIN-binding pocket of the capsid, and an additional eight (1071, 1173, 1249, 1250, 3005, 3006, and 3007) are within 14 Å of WIN 52084 bound in the cavity.

Table 1.

Top 10 unique edges with lower and upper 95% CI of betweenness in T_nowin and corresponding betweenness in T_win

Res #*	Res #*	Betweenness† (×100)
		Tnowin‡	Twin‡
4026	4027	0.330.560.82	0.010.090.19
3006	3007	0.430.550.71	0.000.010.05
1012	4026	0.330.490.66	< 0.001
3005	3006	0.250.370.49	0.000.020.04
1173	1219	0.220.220.23	< 0.001
1249	1250	0.140.210.32	0.080.100.13
3005	3019	0.180.200.22	< 0.001
1199	4015	0.190.190.22	< 0.001
2241	2242	0.100.180.33	0.000.020.04
1071	3100	0.160.170.18	0.000.000.00

^*Italicized residues are within nonbond cutoff distance of drug in T_win. All the residues, except 2241, 2242, 3005, and 3006, are conserved or vary by one residue in HRV-B. Underlined residues are conserved or vary by one residue in HRV-A. Bold faced residues are drug resistant mutants.

^†Betweenness is the number of occurrences of an edge relative to the total number of paths. Variability in betweenness was calculated using bootstrap method.

^‡The 95% CI is shown in superscript.

Conclusion

The uncoating process, a critical part of enveloped picornaviral infection, requires large changes in the capsid conformation. WIN antiviral compounds disrupt uncoating and protect virus particles against thermal loss of VP4 (7) and thus are presumed to prevent these conformational changes. Several experimental observations of the native-state capsid detect large-scale reversible fluctuations that are likely relevant to this uncoating process (16, 19). Antibody epitope sites (15) and limited proteolysis (16) indicate that the N-terminal regions of VP1 and VP4 are reversibly exposed from inside the viral capsid and provide evidence that the mature intact viral capsid undergoes large spatial-scale dynamics. The uncoating process involves the externalization of these same protein regions; the small VP4 protein is lost in its entirety during infection (13, 20), and the N terminus of VP1 becomes exposed (14). The exposed peptides are thought to facilitate membrane permeability (14), and the N terminus of VP1 was shown to insert into membranes and considered important for tethering the virus particle during endocytosis (35). Further the WIN antiviral compound diminish the large-scale motions of the capsid detected by proteolysis, which also links the breathing motion to uncoating. Together, these results argue that the reversible large-scale, breathing motions of the intact native virus particle are relevant to the uncoating process and that WIN compounds can be used to interrogate these dynamics.

Here, an analysis of MD simulations of HRV 14, without and with WIN 52084 bound, revealed large-scale concerted motions of the capsid that reasonably contribute to the initial events that lead to breathing and potentially uncoating. Radial motions of residues of HRV 14 without WIN separated by distances as large as 90 Å were found to be highly correlated (Fig. 2). Many of these radial correlations occur between residues within 35 Å of the fivefold axis, including the canyon region, and either a residue in the vicinity of the fivefold axis or a residue near a twofold axis that is nearby or across the pentameric unit. Sequential edges of the correlation paths with highest betweenness were found to connect these distant regions in HRV 14 (blue lines in Fig. 5A). These pathways are concentrated around the fivefold axis and the canyon region and extend to regions near the twofold symmetry axes regions where the RNA and VP4 are considered likely to exit the capsid (17–24).

Importantly, the presence of WIN 52084 dampens the radial correlations and destroys many of the highest-betweenness paths surrounding the fivefold axis (red lines, Fig. 5B), as would be predicted from the antiviral activity of inhibiting the uncoating process and blocking the breathing motion. The results therefore suggest the long-range basis for the WIN antiviral activity is to interrupt local interactions that are necessary for the highest-betweenness correlation paths. This effect of WIN on the large-scale capsid dynamics could act in concert with an entropic effect on the virus capsid predicted previously (36) from MD simulation studies of the HRV capsid. That is, binding the hydrophobic WIN compound in the buried pocket of VP1 was predicted to entropically stabilize the capsid, a prediction later supported by experiment (37). With the reasoning that long-range concerted motions are lower entropy and that disruption of the correlation paths by WIN could lead to an increase in entropy, the possibility of an entropic effect upon WIN binding was examined using a quasi-harmonic analysis of the configurational entropy (38) of T_nowin and T_win. Although the error from such an estimate is large, we find that the HRV14 capsid with WIN bound is higher entropy (SI Appendix, Fig. G1). Thus, WIN binding could both stabilize the intact capsid by increasing the configurational entropy and also disrupt capsid interactions essential for the concerted dynamics leading to uncoating.

The paths of sequential edges with highest betweenness illustrated by the blue lines in Fig. 5A encompass a number of conserved residues (Table 1). Two of these, 1199 and 1219, are associated with antiviral activity and uncoating (39) as well as known to alter breathing (16). Together, the pattern of radial correlations and highest-betweenness paths composed of conserved residues, and the loss of this pattern when WIN is bound strongly suggest these paths are the origins of the radial correlations and part of the functional motion leading to uncoating.

Long-distance, correlated motions are a candidate mechanism for allostery in proteins. Nonetheless, such motions are often elusive in simulation studies (40); DCC are a function of the angle between centered vector variables (25) and insensitive to orthogonal displacements (41) (see SI Appendix A). Cross-correlation coefficients derived from dihedral angles are even less sensitive for detecting correlated motion (42). To circumvent this problem, we used radial cross-correlation coefficients, which defined numerous very long-distance correlations in capsid protein dynamics.

The insensitivity of displacement vector cross-correlation to identify long-distance effects motivated the development of network based approaches. Networks built from protein structures alone provide topological analysis that can give information on structural stability and binding sites (43). Another approach that has been used to investigate allostery and long-distance communication is a network using low-frequency fluctuations from elastic network normal mode analysis (ENMA) (44, 45). However ENMA is most useful for such analysis of binding effects in the presence of large structural changes (46). This is not the case with WIN binding HRV 14; the root mean square deviation between C_α atoms of average structures from T_nowin and T_win is only 1.32 Å. Indeed, our efforts to use elastic network normal mode calculations were not fruitful and found no significant difference in low-frequency modes of HRV and HRV-WIN 52084. Finally, a network derived from interaction energy correlations was reported to identify coupling over long-distance in signaling proteins (47, 48). Correlations in energy reflect interactions that reasonably lead to concerted motion but do not provide any direct information on that motion. The radial correlation network approach reported here offers an alternative method to query long-distance correlation paths from concerted motions.

Materials and Methods

Initial coordinates were from crystallography (PDB ID code 4RHV for HRV 14 and 1RUD for HRV 14-WIN). Modeling of missing coordinates and protocol of MD simulation are detailed in SI Appendix.

Each C_α atom was represented by a node in the network. WIN 52084 was represented by the first carbon atom on the right of the benzene ring in Fig. 1C. Two nodes are connected by an edge if any atom of the corresponding residues are within 14 Å in at least one frame of the 10 trajectories. An edge is assigned a weight calculated from the radial correlation of the C_α atoms, C_R(i,j),

[1]

where, R_i is the distance of the C_α atom of the ith residue from the origin at an instance and denotes the ensemble average. The weight of an edge connecting two neighboring nodes i and j, W_ij, is 1 - C_R(i,j).

A path in the network between two C_α atoms, or nodes, consists of one or more edges (Fig. 4). If nodes i and j are not neighbors and there exists a path , where a₁ ..a_n are nodes between i and j and n≥1, then the weight of the path is

[2]

The shortest path or path with least weight, between all nonneighbor pairs of nodes is found using an algorithm developed by Dijkstra (26) with edge weights modified to include the radial correlation defined by Eq. 2. The betweenness of an edge is the fraction of all possible shortest-paths that include the edge.