A new approach for extracting information from protein dynamics

Increased ability to predict protein structures is moving research focus towards understanding protein dynamics. A promising approach is to represent protein dynamics through networks and take advantage of well-developed methods from network science. Most studies build protein dynamics networks from correlation measures, an approach that only works under very specific conditions, instead of the more robust inverse approach. Thus, we apply the inverse approach to the dynamics of protein dihedral angles, a system of internal coordinates, to avoid structural alignment. Using the well-characterized adhesion protein, FimH, we show that our method identifies networks that are physically interpretable, robust, and relevant to the allosteric pathway sites. We further use our approach to detect dynamical differences, despite structural similarity, for Siglec-8 in the immune system, and the SARS-CoV-2 spike protein. Our study demonstrates that using the inverse approach to extract a network from protein dynamics yields important biophysical insights.

Introduction SD1, which do not have large structural changes within protein domains, comparing inferred networks allowed us to identify dynamical changes and contributions to stability.

Protein structures
We retrieved crystal structure for FimH L wild type and mutant, Siglec-8 apo and holo, and SARS-CoV-2 spike protein from the Protein Data Bank, as detailed in Table 1. For FimH L , we used crystal structures of the lectin domain without ligand for both the wild type and the mutant. To compare dynamics with and without the disulfide bond as a local perturbation, we used Visual Molecular Dynamics (VMD) to define the bond or two cystines for FimH L . For Siglec-8, we simulated the ligand 6'S sLe x without the 3-amino-propyl linker, which is not thought to interact with the binding pocket 29 .
For the SARS-CoV-2 spike protein, we started from the refined structure on the CHARMM-GUI archive 30,31 . To focus on one hinge system that is thought to be different between the down and up states, we isolated the receptor binding domain (RBD) and subdomain 1 (SD1) protein subunits without the glycans. For the up state, we used chain A where the RBD is accessible for binding the ACE2 receptor on human cells 30 , and for the down state, we used chain B. We used the structure We prepared all systems using VMD version 1.9.3 32 . We solvated each protein with at least 16Å of TIP3P water molecules on each side to prevent interactions with itself through the periodic boundary conditions. We added sodium and chloride ions to neutralize the system and achieve the desired salt concentration in Table 1.

MD simulations
We performed all-atomistic MD simulations using NAMD 33 , with the CHARMM force field 34 . Our NAMD simulation parameters and system details are listed in Table 2. Table 1. PDB structures studied (RRID:SCR 012820). We studied FimH L in the active (wild type) and inactive (Arg60Pro mutant) states; the human immune-inhibitory protein Siglec-8 in the apo and holo (6'S sLe x -bound) states; and the SARS-CoV-2 spike protein RBD-SD1 domains in the up and down states. After observing differences in correlated protein motions between replicates, we performed three replicates of over 200 ns each for wild type FimH L . Due to the tradeoff between the number replicates and simulation length, we performed six replicates of 20 ns of FimH L to make comparisons of wild type and mutant FimH, as well as wild type FimH with and without the Cys3-Cys44 disulfide bond.
Since twenty lowest-energy structures are reported for Siglec-8, we performed a single replicate of 50 ns for each structure to compare apo and holo Siglec-8. For the spike RBD-SD1 domains, we performed six replicates of 60 ns.

Energy minimization
Conjugate gradient algorithm in NAMD ≥ 10,000 steps with protein fixed ≥ 10,000 steps with protein free

Backbone and sidechain dihedral angle dynamics
We used dihedral angles to capture protein dynamics because dihedral angles identify localized regions responsible for the collective displacement of regions distal from the angular rotation, such as in hinged motion 35 . Dihedral angles are also an internal coordinate system that avoids the structure alignment step when using Cartesian coordinates, which can introduce artifacts 27 . We use both backbone (φ , ψ) and sidechain (χ 1 − χ 5 ) dihedral angles. We extracted protein features with MDTraj 1.9.5.

Inverse of the covariance matrix
In the literature, the covariance matrix is one approach used to identify protein regions with motions that are related to the motions of many other regions; in particular it is used to identify correlated motions between distant regions in allostery 22,36 . However, constructing networks from the covariance matrix, even with a threshold to remove weak correlations, is susceptible to induced correlations when two nodes (e.g. A and C) are not directly connected but share a connection with a third node (B) 14 . Borrowing from the field of network reconstruction, we use the inverse of the covariance matrix to identify the connections and weights, or edges, between nodes 14 . This approach is consistent with finding the inverse of a covariance matrix based on Cα positions, which fits the Hessian matrix describing an elastic spring network with anisotropy 25,26 . We have found that the anisotropic elastic network model (ANM) has large errors when used to describe the motion of FimH, which is consistent with errors for hinge-motion described in literature 37 . As a result, we use dihedral angles. This approach is similar to the torsional network model (TNM) which uses equal spring constants to describe dihedral angles across the protein 38 .
In contrast, the inverse of the covariance matrix uses the variances and the covariances of angles to calculate spring constants for a network of torsional springs. The nodes in our network are dihedral angles, the edges are like linearly coupled torsional springs, and the inverse of the covariance matrix is the Hessian matrix for a TNM.
To construct our network, we calculate the Moore-Penrose pseudo-inverse of the covariance matrix using both the backbone and sidechain dihedral angles. We use this approach to understand the relative contributions of backbone and sidechain dynamics to collective motion. Since the sign describes whether the angles turn in the same direction, we take the absolute value to get the interaction strength. We do not apply distance filters. While we use the 97 th percentile as a value threshold for selecting strong interactions or visualizing the network on the protein, we do not use any thresholds for network comparisons. More generally, we recommend caution for applying thresholds to these networks for analysis.

Comparing networks of inferred interactions
To identify interactions that are stronger in one protein state than another, we compare each edge. We select for large differences between groups, relative to the variability within each group. To do this, we filter for differences larger than twice the standard deviation for each group. To compare an edge e between states a and b, each with an ensemble of m and n networks, this is | e a m − e b n | > 2σ e a and > σ e b . We apply this rule without determining statistical significance with corrections for multiple comparisons, in order to see the full effects of comparing all interactions on the matrix. In Supplementary  Figure 11, we also show an example of comparisons without filtering for large differences, in order to illustrate the persistence of the contact-map pattern. We perform the network comparisons in two ways: 1) for every edge on the network (see Supplementary Figure 11), 2) accounting for the multi-layer structure of the network by collapsing the backbone-backbone interactions into residue-residue interactions ( Figure  4). We analyzed data in python, using SciPy (RRID:SCR 008058) and custom packages.

Results
We present results below for these three proteins ( Figure 1A). We first focus on the well-characterized allosteric protein FimH L . Separation of the FimH L domain from its connecting domain (bottom in all figures) is thought to induce an allosteric conformational change on the opposite end of the protein (top in figures) 39,40 . This changes the binding pocket from a state with low affinity for the ligand to one with high affinity ( Figure 1B) 39 . While wild type FimH L is trapped in the high-affinity state, a single-amino acid mutation (Arg60Pro) stabilizes FimH L in the low-affinity state 41,42 . The mutant FimH L is of interest because it undergoes an allostery-like conformational change upon binding mannoside ligands and has been proposed as a minimal model of allostery 42 .
Like FimH L , Siglec-8 binds a carbohydrate ligand and has an immunoglobulin-like fold with two β -sheets ( Figure 1C). Functionally however, Siglec-8 binds to specific sugars found uniquely in human airway tissues to prevent autoimmunity 29,43 . For specific binding, Siglec-8 has a surprisingly rigid binding pocket loop, leading to similar structures for the apo and holo states 29 . The SARS-CoV-2 spike protein RBD binds the human ACE2 receptor with a "hook" region 44 . The hook becomes accessible in the up state when the hinge between the RBD and its connector opens ( Figure 1A). While the hook and interdomain hinge regions are flexible, the bulk of the RBD is thought to be structurally similar in the up and down states. Both Siglec-8 and the spike RBD-SD1 present challenges for detecting differences in inferred interactions because the protein state changes without major structural changes within domains.

Define and validate network inference from inverse covariance analysis
Our approach for constructing a network representation of the dynamics of a given protein is comprised of three steps. In the first step, we obtain temporal dynamics for the nodes, which are the backbone (φ , ψ) and sidechain (χ 1−5 ) dihedral angles for each residue 24,35 . In the second step, we calculate the circular covariance for dihedral angles 35 , which can be thought of as the linearization of the interactions captured by mutual information. In the third step, we invert the covariance matrix using the Moore-Penrose pseudo-inverse to calculate the best fit for a linear coupling system that can give rise to the observed covariance matrix 14 .
Similar to mutual information calculations conducted on other proteins 17,24 , we find that backbonebackbone interactions computed from the covariance are weak compared to sidechain-sidechain interactions (compare red and blue boxes in Figure 1B and the mutual information in Figure S4). In these networks, some dihedral angles have a banding pattern, suggesting long-range interactions with many other dihedral angles ( Figure 1B and Figure S1). In contrast, the inverse covariance matrix has localized and specific interactions. In addition, the stronger backbone-backbone interactions have a repeating pattern that resembles the contact map of the protein, and this pattern appears to repeat more weakly in backbone-sidechain interactions.
The banding pattern in the covariance matrix is also widespread in mutual information matrices, where they have been interpreted as long-range interactions important in protein allostery 17,24 . The large number of long-distance edges produce "hairball" networks, which led to the use of pruning algorithms 45,46 Figure S1, S2, and S3 for the other proteins). While many studies rely on the analysis of the covariance matrix, our data clearly shows that the structure of the covariance matrix is dominated by artifacts (vertical and horizontal lines) which are stronger for side chain interactions (red square for χ 1 − χ 1 ). In contrast, the inverse covariance matrix clearly reveals a structure reminiscent of a contact map and is dominated by backbone interactions (blue square for ψ − ψ).
distance filters 22,36,47 in prior studies, in order to make network analysis tractable. Thus, we wondered whether the long-range interactions are capturing a physical feature of the dynamics. To answer this question, we investigate the reproducibility of the covariance, correlation, and inverse covariance matrices extracted from different replicates of MD simulations.
In Figure 2A, we contrast the matrix for one replicate in the upper-diagonal with the second replicate in the lower-diagonal and quantify the similarity in Figure 2B. For replicate MD simulations, we used the same initial protein structure with randomized solvation and initial velocities. Both covariance matrices have banding patterns suggesting hotspots that interact with many residues across the protein. However, each replicate has its own banding pattern, with interaction strengths that are over ten times greater than those found in the other replicate, indicating high variability in the networks that one would construct from replicate simulations.

Cov Corr InvCov Cov Corr InvCov
Cov Corr InvCov Figure 2. Inverse covariance matrix is robust across replicates whereas covariance and correlation matrices are not. (A) Triangular regions above and below the matrix diagonal show results from two replicates of wild type FimH L starting from the same protein structure. We show ψ − ψ interactions. We show interaction strength in blue with a normalization for each triangular region made based on the 97 th percentile of observed strengths. In red, we show the ratio for interaction strength between the two replicates. Purple indicates strong interactions that are not reproduced in the other replicates. (See Figure S5 for weaker interactions visible when normalized to the 95 th percentile for ψ − ψ and χ 1 − χ 1 .) For covariance and correlation matrices, we find that backbone-backbone interactions are mostly quite weak, but the strong interactions (darker blue) vary drastically between replicates. In contrast, for the inverse covariance matrix, the strongest backbone-backbone are symmetric across the diagonal. (B) To evaluate the robustness of network inference, we calculate the Jaccard similarity coefficient for the covariance, correlation, and inverse covariance analyses methods across three simulation replicates. We define edges above the threshold of ≥97 th percentile. (See Figure S7 for other thresholds.) In grey scale, we show similarity separately for ψ − ψ and χ 1 − χ 1 interactions. Darker grey indicates results are similar across replicates for the inverse covariance approach and much less similar for the other two methods. (C) The inverse covariance matrix resembles the contact map. We show the Cα inter-residue distance from the crystal structure. Darker grey indicates shorter distance.
Since the banding pattern is associated with dihedral angles with high variance, we also consider the correlation matrix, which normalizes the covariance matrix by the variance of each dihedral angle. It is visually apparent that the banding in the covariance matrix is not simply due to high variance because there are still bands in the correlation matrix. While normalizing to the correlation matrix uncovers some interactions in a contact map pattern, they are weak compared to the banding pattern (see Figure S5 for a lower maximum value on the color map scale).
In contrast to the irreproducible results obtained with the covariance matrix, for the inverse of the covariance matrix, we find a pattern that is visually similar to the 12Å contact map ( Figure 2C). The diagonally symmetric contact map pattern in blue indicates similarly strong interactions for two replicates ( Figure 2A). After quantifying the robustness across three replicates using the Jaccard similarity index, we find that the inverse covariance has higher similarity (59-72% shared edges) than the covariance (8-10%) or the correlation (13-16%) for ψ − ψ backbone interactions ( Figure 2B). The χ 1 − χ 1 similarity values for inverse covariance are lower, but still higher than for the other two methods. These data clearly demonstrate that networks inferred from inverse covariance analysis are more robust than networks constructed from correlational measures.

Inverse covariance analysis yields structural networks
Prompted by the strong visual resemblance between the inverse covariance network and the 12Å contact map and the complete absence of this pattern in the covariance network, we wondered if the inverse covariance matrix could be used to identify specific physical interactions. To answer this question, we overlaid the strongest edges (≥ 97 th percentile) on the 12Å contact map, highlighting the backbonebackbone edges as blue dots and the sidechain-sidechain edges as red crosses ( Figure 3A and Figure  S10).
To correct for the high variability across replicates we previously found for the covariance networks, we averaged networks across the three replicates. Despite the averaging, the covariance network shows strong interactions across distant protein regions and are dominated by Cys3-Cys44  Figure S6 for other cutoffs. We averaged edge weight across three replicates. The blue dots represent the average of backbone-backbone interactions by residue. The red crosses represent sidechain-sidechain interactions (χ 1 − χ 1 ). The inverse covariance network are predominantly backbone interactions that fall within the 12Å contact map. There is a ≥ 99 th percentile χ 1 − χ 1 interaction the Cys3-Cys44 disulfide bond (red arrow). However, this interaction is only 80 th percentile in strength (green diamond and arrow) for the covariance matrix, which is dominated by other χ 1 − χ 1 interactions. (B) Since the covariance matrix has many long-range interactions, we only show the backbone interactions and the sidechain interactions for Lys4. In contrast, the inverse covariance network has mostly short-range interactions, including the disulfide bond. We show backbone interactions on the Cα atoms and sidechain interactions on the the 4 th χ 1 atoms. We show backbone interactions in blue and sidechain interactions in red. Darker colors indicate stronger interactions.
To understand how these contrasting patterns affect interpretation, we next visualize strong interactions as edges drawn on the protein structure ( Figure 3B). Since drawing all edges would make the covariance network indecipherable, we only show the edges originating from Lys4. In contrast, the inverse covariance network uncovers edges that mostly connect physically close residues. Specifically, we find a χ 1 − χ 1 edge between Cys3-Cys44 for the sole disulfide bond in FimH L , and that this edge is missing from the covariance network (red vs green arrow in Figure 3A).
Examining the contact map pattern of the inverse covariance network in more detail, we compare edge weight with the distance between Cα atoms ( Figure S9). We find that for backbone-backbone interactions, the strongest interactions are between residues connected by a peptide bond, followed by hydrogen bonds within β sheets, and then non-bonding interactions.
After examining backbone-backbone interactions, we next looked at the progressively weaker interactions involving sidechains distal from the backbone ( Figure 1B and Figure S10). We find the contact map pattern is still apparent for φ − χ 1 or ψ − χ 1 interactions, but becomes very weak for φ − χ 2 or ψ − χ 2 interactions, and becomes indistinguishable from noise for interactions between proximal and distal sidechain dihedrals. The inverse covariance analysis thus suggests that backbone dihedral motion is most strongly coupled to nearby backbone dihedrals and has more dissipated effects on sidechain dihedrals. This relationship is consistent with how backbone motions can sterically trap or free sidechains, whereas sidechain motions are more limited in their impact on backbone motion 48 .
The different strengths of interactions for backbone-backbone and backbone-sidechain edges suggests that qualitatively different types of interactions have different properties. For two residues i and j, the backbone-backbone edge φ − ψ[i, j] are larger than the backbone-sidechain edge , which is consistent with the physical differences between these two edge types. Moreover, most backbonebackbone edges within the same residue, φ − ψ[i, i], are stronger than backbone edges connecting to other Figure S9).

Conformational differences between wild type and mutant
As a way to further validate our approach, we next test if we are able to identify the well-characterized differences between wild type FimH L and the Arg60Pro mutant. To compare inferred networks for the wild type and mutant proteins, we identified edges where the average difference was larger than two times the standard deviation across each group of replicates. We performed this analysis once with the entire set of edges (Figure S11), and again with only the backbone interactions collapsed into a residue interaction network. For our comparisons and the matrix visualization of the differences, we do not apply a distance filter or a threshold for the edge-strength. However, for the visualization on the protein, we only show differences with magnitude larger than the 97 th percentile.
The visualization of the differences enables us to identify several interactions stronger in either the mutant or the wild type proteins (red or blue patches, respectively, in Figure 4A). This is consistent with the difference in initial structure (RMSD=3.15Å), dihedral dynamics 28 , and the residues that are rearranged in the allosteric conformational change 41,49 .  For concreteness, we focus on two regions at the edge of the protein structure that are easier to visualize: the binding pocket zipper at the top of FimH L , and the insertion loop at the bottom. In the pocket zipper, we found much stronger interactions for the mutant protein (median:2.8-fold, IQR:2.0-5.2-fold), which correspond to smaller dihedral fluctuations 28 . On the other hand, in the insertion loop, we identified changes in interaction that were stronger in the wild type than the mutant protein (3.2-fold, 2.2-3.7-fold). Structurally, this is consistent with how the insertion loop is stabilized in the wild type structure and exposed to solvent in the mutant protein 42,50 . Dynamically, stronger interactions within the insertion loop is consistent with smaller dihedral fluctuations in the wild type protein 28 .
We further identify differences at the β -bulge, α-switch, and swing loop regions of the allosteric pathway, consistent with structural differences between wild type and mutant proteins. The mutant protein has stronger interactions within the loop formed by the β -bulge (2.0-fold, 1.2-2.3-fold) and also with a nearby loop. In the wild type protein, the loop is smoothed out into a β -strand. The wild type protein has stronger interactions (1.8-fold, 1.5-2.6-fold) in the α-helix, compared to the 3 10 -helix in the mutant protein, which is probably due to different hydrogen bonding patterns. In the swing loop, we again find stronger interactions in the wild type protein (2.0-fold, 1.6-2.7-fold).
For these allosteric pathway landmarks, it is visually apparent that we detect large differences in inferred interactions when structures are closer in one state and stretched apart in the other state. Beyond these regions, there are several other regions with similarly large changes in interaction between the wild type and mutant proteins, shown in blue and red patches ( Figure 4A).

Quantifying the impact of disulfide bond reduction
We next used wild type FimH L to explore the impact of reducing the single disulfide bond between Cys3-Cys44 in silico on fast, nanosecond-timescale dynamics. Using the inverse covariance analysis, we correctly identified the 6-fold stronger χ 1 -χ 1 interactions in the presence of the disulfide bond, which was the largest difference detected ( Figure 4D). This matches our expectations because the covalent bond between the most distal atoms forming the χ 1 rotamer angle directly couples χ 1 dynamics. Together, these analyses show that the inverse covariance analysis method is sensitive to both local differences and conformational differences.

Network inference detects key mechanisms involved in Siglec-8 binding
Like FimH L , the human immune cell adhesion protein, Siglec-8, is also a lectin with an immunoglobulinlike fold with a single disulfide bond. For Siglec-8, we compare the apo (no ligand) and holo (bound to the native 6'S-sLe x ligand) states. Due to the rigid binding pocket loop that only differs by a few sidechain rearrangements, apo and holo Siglec-8 have extremely similar structures 29 . Intriguingly, the rigidity of the CC' binding pocket loop in apo Siglec-8 occurs in the absence of stabilizing secondary structure motifs 29 and plays a major role in recognizing specific ligands in the airway to avoid autoimmunity 43 .
One hypothesized mechanism for stabilizing the CC' loop in apo Siglec-8 is that the Arg70 sidechain forms hydrogen bonds with the loop backbone at Pro57 and Asp60 ( Figure 5A) 29 . We identified Arg70-Pro57 or Arg70-Asp60 hydrogen bonds with occupancy between 10-33% in seven out of 20 simulations from the ensemble of NMR structures, and higher occupancy (80 and 83%) in only two simulations. This suggests the hydrogen bond is rarely present. Nonetheless, the unstructured CC' loop has surprisingly small fluctuations across the ensemble of MD simulations ( Figure 5B-E).  Figure 1A), while the structures with the RBD hidden are extremely similar (down1, down2, and off conformation). Comparing inferred network interactions, we detect differences near the hinge, including the Cys336-Cys361 disulfide bond (black spheres) and nearby α-helices. For these regions, we show the interactions that are stronger in the down2 (blue) and off (red) conformations on the corresponding structure. See Figure S15 for other comparisons.
Using our inferred interaction networks, we identified strong interactions within the CC' loop, as well as interactions from outside the loop to its hinges at Ala53 and Pro62 ( Figure S12). Consistent with the hydrogen bonding analysis, we only identified strong interactions with Arg70 in a few MD simulations, which disappeared after averaging the networks across the ensemble. Counter-intuitively, in the two structures where Arg70 does form stable hydrogen bonds with Pro57 and Asp60, the CC' loop has paradoxically larger fluctuations, especially at Tyr58 and Gln59 at the loop tip ( Figure S12). Taken together, we suspect that steric hindrance plays a role in stabilizing the CC' loop internally and externally at the hinge edges.
Of note, this example demonstrates an instance where the inferred network approach provides more clarity than contact or interaction energy networks, because the inverse of the covariance matrix identifies the degrees of freedom that most strongly affect the fluctuations at the CC' loop.
After examining why the CC' loop is similar in apo and holo Siglec-8, we next compared inferred networks for the apo and holo states. The holo state has stronger χ 1 -χ 1 interactions corresponding to the Cys31-Cys91 disulfide bond ( Figure 5C and Figure S13). The disulfide bond is conserved in the siglec family and is located on the sheet of the β -sandwich opposite the binding pocket 29 . Nearby, we also observe other changes in interaction strength involving Asp90. Although distant from the binding site, the Asp90-Cys91-Ser92 motif in Siglec-8 is a variant of the Asn-Cys-Ser or -Thr motif that is conserved in the rest of the siglec family 51 . Differences in interaction strength between apo and holo states identify changes in the dynamics of this evolutionarily conserved region during ligand-binding.
In contrast, reducing the Cys31-Cys91 disulfide bond in the holo state in silico has a different pattern ( Figure S13). Both ligand-binding and the presence of the disulfide bond correspond to stronger a χ 1 -χ 1 interaction at Cys31-Cys91. Both conditions also stabilize Cys31, indicated by decreased backbone dihedral fluctuations and increased duration within an extended secondary structure as assigned by the DSSP algorithm, and shorter Cys31-Cys91 Cα distance. Taken together, we find that the network rearrangement that occurs with ligand-binding increases Siglec-8 conformational stability near an evolutionarily conserved disulfide bond, even though the region is not near the binding site.
Comparison of the spike protein RBD-SD1 fragments in the up, down, and off states shows network rearrangements without large structural differences Next, we investigated whether there are differences in the networks inferred from the 'up', 'down', and 'off' states of the RBD-SD1 domains of the SARS-CoV-2 spike protein ( Figure 1A). The RBD connects to SD1 (Fig 5E) via two hinge-like loops that are more flexible in the up state than in the down or off states 52 . While the up state has a different orientation around the hinge than the down and off states, they all have similar SD1 structures (Cα-RMSD ≤ 0.64Å) and somewhat similar RBD structures (Cα-RMSD ≤ 1.55Å).
Opening the hinge angle in the down-to-up transition is thought to make the binding site on the RBD available to attach to human ACE2 30,44 . To focus on the RBD-SD1 hinge, we isolated these domains from the rest of the spike protein and ignored glycosylated sugars. While these simplifications limit the strength of our conclusions into the function of the spike protein, the RBD-SD1 structures nonetheless provide a useful system for comparing dynamics in a system that initially resembles rigid-body motion around a hinge.
Since the trimeric spike protein with one exposed RBD has two hidden RBDs, we isolated one RBD-SD1 fragment in the 'up' state, and two fragments in the 'down' state. We obtained another RBD-SD1 domain in the 'off' state from a structure with all RBDs hidden and 3-fold rotational symmetry. We first compared the RBD-SD1 fragments in the down and off conformations. Despite the structural similarity, we nevertheless detect differences in the inferred networks (Fig 5E) based on dynamics. We find the networks for the two down conformations are more similar to each other than to the networks for the off conformation ( Figure S15).
We next compared RBD-SD1 fragments in the down and off conformations to the fragment in the up conformation ( Figure S15). Surprisingly, we find that the differences in inferred networks among the down and up conformations are more localized, while the networks for the off conformation are distinct from the others. The localized differences are in the hook of the RBD that is exposed in the up conformation, and at an α-helix near the RBD-SD1 hinge. Intriguingly, we also identified differences in the structural orientation and the inferred network interaction at the Cys336-Cys361 disulfide bond. In this region, one down conformation is more similar to the up conformation, while the other down conformation resembles the off conformation ( Fig 5E).
As a proof-of-concept, we investigated whether it is possible to infer an interaction network for the trimeric SARS-CoV-2 spike protein, even though it is much larger than the RBD-SD1 domains. However, the amount of data required for network inference using the inverse covariance method scales faster than the number of residues. As a result, our approach required more data than was available from the two sets of publicly-accessible simulations for which the up and down states are labeled 53,54 . Instead, we chose the longer simulations that explore the transition among the down, up, and open states published by the Bowman lab 55 . Using 100,000 snapshots, we were able to infer an interaction network for the 3,363 residues of the spike protein ( Figure S14). The number of snapshots is an order of magnitude larger than those currently available for labeled states. Thus, it may be feasible to compare states for the entire S1/S2 complex of the spike protein if there is sufficient data, or by choosing a less data-intensive method for solving the inverse problem.

Discussion
We identify some of the shortcomings of correlation-based approaches for network inference from protein dynamics using the covariance, correlation, and mutual information matrices. These networks have low reproducibility among replicates and exhibit long-range connections that are difficult to tie to physical explanations. To address these shortcomings, we use the inverse of the covariance matrix. This is a well-established technique from network inference 14 for solving the inverse problem for a system that can produce the observed correlated dynamics. Our approach builds networks where each node is a dihedral angle, including both the backbone (φ , ψ) and the sidechains (χ 1−5 ), and edges are inferred from the coupling interactions between angles. We chose the internal coordinate system of dihedral angles 35 to easily include sidechain dynamics, localize hinges that drive distal dynamics 28 , and to avoid the alignment step in Cartesian coordinates that introduces 'artifacts' in hinged motion 37 .
Using the inverse covariance approach, we detected differences in conformation, subtle differences between protein states without large conformational changes, and localized perturbations in the structure of biomedically important proteins. The inverse covariance networks capture a hierarchy of interactions that resemble that qualitatively different types of interactions, suggesting a multi-layer network structure. The strongest edges connect dihedral angles with covalently-bonded atoms, with weaker interactions for greater distances. Moreover, the contact map-like pattern found in backbone-backbone interactions are repeated more weakly in backbone-sidechain and sidechain-sidechain interactions. This hierarchy of inferred interactions is consistent with the view that smaller backbone rearrangements related to larger sidechain motions 48 .
Our results suggest that solving the inverse problem uncovers the underlying interactions that ultimately drive protein dynamics, but are not well-captured by cataloguing the observed correlated motions or comparing static structures. Notably, inverting the covariance matrix is the simplest of a variety of tools available for network inference from dynamics used in network science 14,56 . While the simplicity of inverting the covariance matrix increases accessibility, there are some obvious limitations 14 . We calculate the circular covariance matrix on dihedral angle distributions that are multi-modal. The inverse of the covariance matrix is analogous to linearly coupled torsional springs, which do not represent the complexity of atomic interactions within a protein. Moreover, network inference by inverting the covariance matrix requires a large amount of data 14 . Our work establishes a baseline approach, which can be easily built upon by incorporating more sophisticated 14,18 -and yet more involved -approaches that better describe dihedral distributions 57 or account for nonlinear interactions 58 .
Despite these limitations, our approach yielded significant insights for three adhesion proteins. Comparing the networks inferred for two protein states at a time, we were able to tie differences in inferred network structure to structural and dynamical differences. For FimH L , a comparison of inferred networks for the wild type and mutant proteins identifies protein regions with conformational changes consistent with the allosteric pathway sites 41 . For Siglec-8 and the SARS-CoV-2 RBD-SD1 construct, we were able to detect network rearrangements despite the similar structures of Siglec-8 in the apo and holo states, and of the individual RBD and SD1 domains in the up and down states. In Siglec-8, we were also able to use strong interactions identified by the network to propose a new mechanism for stabilizing an unstructured loop in the binding pocket. This serves as an example where the network inference approach has an advantage over contact and interaction energy networks. Taken together, our results show that the network inference approach can identify protein regions of interest based on dynamical differences that are rooted in physically interpretable interactions.

Selected examples of the covariance matrix and its inverse
To demonstrate that the different patterns found between the two matrices apply to multiple MD trajectories, we show the covariance matrix diagonally across from its inverse for FimH (Supplementary Figure 1), Siglec-8 (Supplementary Figure 2), and the SARS-CoV-2 spike protein RBD-SD1 domains (Supplementary Figure 3). For each, we set the color scale maximum to the 97 th percentile. We show these in the same format as in Figure 1.

Thresholding for visualization
While we do not use thresholds for comparing inferred interactions, we do use thresholds to visualize networks as the adjacency matrix and on the protein. In the matrix visualizations, we set the color scale maximum to the 97 th percentile. However, to better illustrate the pattern at lower values in the correlation matrix in Figure 2A, we also show a lower color scale maximum set to the 95 th in Figure S5. In contrast, for the inverse covariance matrix, showing lower values does not have a profound effect, since the contact map pattern is formed by inferred interactions with high values.
We also show how thresholds affect the Jaccard similarity. In Figure 2B, we used a threshold of 97 th percentile to create a mask where every edge above the threshold is set to 1, and edges below the threshold are set to 0. Here we show masks for thresholds at the 50, 95, 97, and 99 th percentiles, for the covariance, mutual information, and inverse covariance matrices ( Figure S8). In Figure S7, we show that for strong interactions, the inverse covariance matrix has high similarity across replicates, while the other matrices show decreasing similarities. While other matrices have higher similarity around the 50 th percentile, the masks in Figure S8 show the extremely large number of edges included at these low thresholds.
The banded pattern indicates edges connecting one dihedral angle to many others, and an excessive number of these edges produces hairball networks. Thus, for the covariance, correlation, and mutual information matrices, there is a tradeoff where low thresholds with higher similarity produce hairball networks. In contrast, for the inverse covariance network, the high thresholds that produce higher similarity select for stronger interactions, producing physically interpretable networks that resemble the contact map.  Figure 6. The inverse covariance matrix has a "contact map"-like network at multiple thresholds. We show the covariance (top left) and inverse covariance (bottom right) at the : a, 90 th , b, 95 th , and c, 98 th percentiles to complement the 97 th percentile threshold in Figure 3. We use the same representation scheme as in Figure 3.   Figure 8. Representative thresholds for defining unweighted networks used to calculate the Jaccard Index (JI) for the inverse covariance, covariance, and mutual information matrices. We directly compare two replicates in the top left and bottom right triangles. Edges above the threshold in both replicates are shown in grey, while those only present in one replicate are shown in red. For each replicate pair, we list JI for ψ − ψ (blue box) and χ 1 − χ 1 (yellow box) submatrices.   Figure 10. Sidechain interactions become weaker further away from the backbone. Grey scale indicates absolute value of the interaction strength for the inverse covariance for one replicate of wild type FimH L . a, Backbone dihedral (ψ) interactions with a distal sidechain dihedral (χ 2 ) is weaker than b, interactions with a more proximal one (χ 1 ). ψ − χ 1 interactions still retains the contact map pattern. c, While interactions between distal sidechains χ 2 − χ 2 do not have a clear pattern, d, yet, there is a still a slight pattern between proximal and distal sidechains (χ 1 − χ 2 ). a b Supplementary Figure 11. Comparison of inferred networks for wild type and mutant FimH L using all inferred edges. For wild type -mutant, we show edges stronger in the wild type in blue, and those for the mutant in red (colorbar). We show two versions: a, without thresholding and b, requiring differences to be larger than twice the standard deviation of technical replicates within each group. We do not apply any filters based on distance, secondary structure, or other structural information.  Figure 13. Impact of ligand-binding and reducing the Cys31-Cys91 disulfide bond on Siglec-8. Difference in inferred network interactions for a, apo vs holo states, and b, the holo state with and without the disulfide bond. Due to the difference in inferred interaction strength at the disulfide bond between apo and holo Siglec-8, we also compared c, the apo state with the disulfide bond intact vs the holo state without the disulfide bond. We use the same representation scheme as in Figure 4. Differences in the backbone-backbone interactions are shown in the top left in dots; sidechain-sidechain interactions in the bottom right as crosses. On the adjacency matrix, we show all differences between groups larger than 2σ , but we only draw differences larger than the 97 th threshold for interaction strength on the protein structure. We show the secondary structure on the top and right borders. We show landmarks on the bottom and left borders: CC' loop (orange), GG' loop (purple), and the evolutionarily conserved Asp90-Cys91-Ser92 motif (green). Backbone-backbone interactions for the trimer, with ψ-ψ interactions highlighted by the blue box. The first protomer is shown with a purple box, to highlight the pattern that repeats three times within the blue box. The other backbone-backbone interactions also show a pattern that repeats three times. b, Zoomed-in view of the first protomer's ψ-ψ interactions shows a contact-map pattern.