Pipeline overview

The SINATRA Pro pipeline involves five key steps (see Fig 1). First, the algorithm begins by taking aligned structures from two all-atom protein MD simulation trajectories of different phenotypic states (e.g., wild-type versus mutant) as inputs (Fig 1A). In the second step, SINATRA Pro uses the 3D Cartesian coordinates of each atom of the proteins to create mesh representations of their 3D structures (Fig 1B). Here, atoms within a predetermined physical distance cutoff (e.g., ∼6 Ångströms (Å) apart) are connected by “edges” and then triangles enclosed by the connected edges are filled to create “faces.” In the third step, we convert the resulting triangulated meshes to a set of topological summary statistics using an invariant called the “differential Euler Characteristics (DEC)” transform (Fig 1C). In the fourth step, SINATRA Pro implements a nonlinear Gaussian process model to classify the protein structures using the topological summary statistics, with which association measures are computed for each topological feature to provide a statistical notion of “significance” (Fig 1D). In the last step of the pipeline, SINATRA Pro maps the association measures back onto the original protein structures (Fig 1E), which produces “evidence scores” that reveal the spatial locations that best explain the variance between two protein ensembles. Theoretical details of our implementation are fully detailed in the Materials and Methods sections.

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Fig 1. Schematic overview of SINATRA Pro: A novel framework for discovering biophysical signatures that differentiate classes of proteins.

(A) The SINATRA Pro algorithm requires the following inputs: (i) (x, y, z)-coordinates corresponding to the structural position of each atom in every protein; (ii) y, a binary vector denoting protein class or phenotype (e.g., mutant versus wild-type); (iii) r, the cutoff distance for simplicial construction (i.e., constructing the mesh representation for every protein); (iv) c, the number of cones of directions; (v) d, the number of directions within each cone; (vi) θ, the cap radius used to generate directions in a cone; and (vii) l, the number of sublevel sets (i.e., filtration steps) used to compute the differential Euler characteristic (DEC) curve along a given direction. Guidelines for how to choose the free parameters are given in Table 1. (B) Using the atomic positions for each protein, we create mesh representations of their 3D structures. First, we draw an edge between any two atoms if the Euclidean distance between them is smaller than some value r, namely dist|(x₁, y₁, z₁), (x₂, y₂, z₂)| < r. Next, we fill in all of the triangles (or faces) formed by these connected edges. We treat the resulting triangulated mesh as a simplicial complex with which we can perform topological data analysis. (C) We select initial positions uniformly on a unit sphere. Then for each position, we generate a cone of d directions within angle θ using Rodrigues’ rotation formula [86], resulting in a total of m = c × d directions. For each direction, we compute DEC curves with l sublevel sets. We concatenate the DEC curves along all the directions for each protein to form vectors of topological features of length J = l × m. Thus, for a study with N-proteins, an N × J design matrix is statistically analyzed using a Gaussian process classification model. (D) Evidence of association measures for each topological feature vector are determined using relative centrality measures. We reconstruct corresponding protein structures by identifying the atoms on the shape that correspond to “statistically associated” topological features. (E) The reconstruction enables us to visualize the enrichment of biophysical signatures that best explain the variance between the two classes of proteins. The heatmaps display atomic (or residue-level, which we define as a collection of atoms) evidence potential on a scale from [0–100], with a score of 100 meaning most enriched.

https://doi.org/10.1371/journal.pcbi.1010045.g001

Software overview

The software for SINATRA Pro requires the following inputs: (i) 3D Cartesian coordinates corresponding to the atomic positions in each protein structure; (ii) y, a binary vector denoting protein class or phenotype (e.g., y_i = 0 for wild-type or y_i = 1 for mutant); (iii) r, the cutoff distance for simplicial construction (i.e., constructing the mesh representation for every protein); (iv) c, the number of cones of directions; (v) d, the number of directions within each cone; (vi) θ, the cap radius used to generate directions in a cone; and (vii) l, the number of sublevel sets (i.e., filtration steps) used to compute the differential Euler characteristic (DEC) curve along any given direction. In addition to real data analyses, we implement a controlled benchmark simulation study designed to assess SINATRA Pro’s performance at identifyin structurally-perturbed regions in protein dynamics relative to other methods. Results for the controlled benchmark simulations were done using free parameters {r = 1.0Å, c = 20, d = 8, θ = 0.80, l = 120}, and results for the real data analyses were done using parameters {r = 6.0Å, c = 20, d = 8, θ = 0.80, l = 120}. All values were chosen via a grid search. Guidelines for how to choose the free parameters for the software are given in Table 1. Tables detailing the scalability of the current algorithmic implementation of SINATRA Pro can also be found in Supporting Information (see S1–S3 Tables).

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Table 1. General guidelines for choosing values for the free parameters in the SINATRA Pro pipeline software.

The guidelines provided are based off of intuition gained through the simulation studies. In practice, we suggest specifying multiple cones c > 1 and utilizing multiple directions d per cone (see monotonically increasing power in S1 Fig in Supporting Information). While the other two parameters (θ and l) do not have monotonic properties, their effects on SINATRA’s performance still have natural interpretations. Selection of θ ∈ [0.1, 0.8] supports previous theoretical results that cones should be defined by directions in close proximity to each other [13, 15]; but not so close that they explain the same local information with little variation. Note that our sensitivity analyses suggest that the power of SINATRA Pro is relatively robust to the choice of θ. Optimal choice of l depends on the size of the protein molecules that are being analyzed. Intuitively, for rigid proteins, coarse filtrations with too few sublevel sets cause SINATRA Pro to miss or “step over” structural shifts that occur locally during the course of a molecular dynamic (MD) trajectory. In practice, we recommend choosing the angle between directions within cones θ and the number of sublevel sets l via cross validation or some grid-based search.

https://doi.org/10.1371/journal.pcbi.1010045.t001

Performance of SINATRA Pro on benchmark simulations

We implemented a controlled simulation study designed to assess SINATRA Pro’s performance at identifying structurally-perturbed regions in protein dynamics relative to other methods. Here, the premise behind “controlled simulations” is that topological artifacts (i.e., perturbations of atomic positions in a certain region) are manually introduced to a set of protein structures to establish a ground truth and statistically evaluate the concept of power. The original and perturbed structures represent two phenotypic classes and are fed into SINATRA Pro to assess whether it can reliably identify the perturbed regions of interest.

To generate data for these controlled simulations, we use real structural data of wild-type β-lactamase (TEM), an enzyme widely implicated in microbial resistance that has evolved numerous mutations of clinical relevance. In the first phenotypic group (set A), original structures are drawn at 1 nanosecond (ns) intervals over a 100 ns MD trajectory (e.g., t_MD = [0, 1, 2, 3, …, 99] ns + δ, where δ is a time offset parameter). Next, a comparable set of perturbed structures (set B) are drawn at 1 ns intervals but shifted by 0.5 ns with respect to the set A structures (e.g., t_MD = [0.5, 1.5, 2.5, 3.5, …, 99.5] ns + δ) to allow for thermal noise to be introduced. Here, we displace the atomic positions of each atom in the Ω-loop (i.e., the region of interest or ROI) in each perturbed structure within set B by

• a constant Cartesian vector set to (i) 0.5 Å, (ii) 1.0 Å, and (iii) 2.0 Å in each (x, y, z) direction;
• a spherically uniform random vector where each (x, y, z) direction is first drawn from a standard Gaussian distribution and then the vector is normalized to be of length (iv) 0.5 Å, (v) 1.0 Å, and (vi) 2.0 Å.

These simple, artificial control cases are designed to represent two different forms of structural changes that can happen within protein dynamics. Namely, scenarios (i)-(iii) involve a displacement of atoms by a constant amount in a constant direction, which emulates a static structural change; while, scenarios (iv)-(vi) displace the atoms by a constant amount in a (spherically uniform) random direction, which emulates a dynamic or stochastic structural change. Altogether, we use datasets of N = 1000 protein structures per simulation scenario: 100 ns intervals × 5 different choices of δ = {0.0, 0.1, 0.2, 0.3, 0.4} ns × 2 phenotypic classes (wild-type versus perturbed). We evaluate all competing methods’ abilities to correctly identify perturbed atoms located within the Omega-loop region (Material and methods). Here, we use receiver operating characteristic (ROC) curves that plot true positive rates (TPR) against false positive rates (FPR) (Fig 2). This is further quantified by assessing the area under the curve (AUC). The results presented in the main text reflect using SINATRA Pro with parameters set to {r = 1.0Å, c = 20, d = 8, θ = 0.80, l = 120} chosen via a grid search. Note that additional figures assessing how robust SINATRA Pro is to different free parameter value settings can be found in the Supporting Information (see sensitivity analysis in S1 Fig).

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Fig 2. Receiver operating characteristic (ROC) curves comparing the power and robustness of SINATRA Pro to competing 3D mapping approaches in controlled molecular dynamic (MD) simulations.

To generate data for these simulations, we consider two phenotypic classes using the real structural data of wild-type β-lactamase (TEM). In the first phenotypic class, structural protein data are drawn from equally spaced intervals over a 100 ns MD trajectory (e.g., t_MD = [0, 1, 2, 3, …, 99] ns + δ, where δ is a time offset parameter). In the second phenotypic class, proteins are drawn at 1 ns intervals shifted 0.5 ns with respect to the first set (e.g., t_MD = [0.5, 1.5, 2.5, 3.5, …, 99.5] ns + δ) to introduce thermal noise, and then we displace the atomic positions of each atom in the Ω-loop region by (top row) a constant Cartesian vector of (A) 0.5 Ångströms (Å), (B) 1.0 Å, and (C) 2.0 Å, or (bottom row) by a spherically uniform random vector of (D) 0.5 Å, (E) 1.0 Å, and (F) 2.0 Å. Altogether, we have a dataset of N = 1000 proteins per simulation scenario: 100 ns interval × 5 different choices δ = {0.0, 0.1, 0.2, 0.3, 0.4} ns × 2 phenotypic classes (original wild-type versus perturbed). The ROC curves and corresponding area under the curves (AUC) depict the ability of SINATRA Pro to identify “true class defining” atoms located within the Ω-loop region using parameters {r = 1.0Å, c = 20, d = 8, θ = 0.80, l = 120} chosen via a grid search. We compare SINATRA Pro to four methods: root mean square fluctuation (RMSF) (orange); principal component analysis (PCA) (green); Elastic Net classification (pink); and a Neural Network (brown). For details on these approaches, see Materials and methods. A sensitivity analysis exploring the optimal parameter configurations for PCA and Neural Network can be found in Supporting Information (see S3 and S4 Figs, respectively).

https://doi.org/10.1371/journal.pcbi.1010045.g002

Lastly, we want to point out that we do not realign the proteins after structural perturbation has occurred. Realigning the structures after introducing a perturbation poses a slightly different and notably less controlled simulation study. For example, in the constant displacement case, realigning the structures will shift the whole structure against the perturbing vector and result in an unintentional displacement on the opposite side of the structure. A true positive in this case will not be as well-defined as when we keep the unperturbed structure in place and define the perturbed structure as the ground truth for the positive signal. An example of this can be seen in S2 Fig.

Detecting conformational changes in real protein systems

To examine SINATRA Pro’s ability to identify known structural changes of biological significance in real data, we consider the following five protein systems (Table 2): (1) the wild-type and Arg164Ser mutant of TEM β-lactamase; (2) the wild-type and Ile50Val mutant of HIV-1 protease; (3) the guanosine triphosphate (GTP) and guanosine diphosphate (GDP) bound states of EF-Tu; (4) the wild-type and Met290Ala mutant of the Abl1 tyrosine protein kinase; and (5) unbound and IBB-bound states of importin-β. We choose to analyze these particular systems because they undergo varying degrees of conformational changes that have been well-studied in the literature (again see Table 2). Here, we will treat these previously identified features as ROIs, where the assumed “difficulty” for SINATRA Pro to statistically resolve structural signatures will be based on the stochasticity observed within each protein system. Namely, it will be more difficult to perform feature selection on structural ensembles that are highly dynamic as spurious fluctuations can interfere with detecting signal from the ROI. For each protein system, three MD trajectories were generated. From each of these trajectories, ten different series of structures are sampled, where each series is separated by 0.1 ns and each structure in a series is separated by 1 ns. SINATRA Pro is then implemented on the ten different series of structures extracted from the same trajectory data to mitigate the impact of variability between MD simulations (see Material and methods).

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Table 2. Detailed overview of the different protein systems analyzed in this study.

The columns of this table are arranged as follows: (1) the name of each protein studied; (2) the corresponding Protein Data Bank (PDB) ID for each molecule [92]; (3) the known chemical change or mutation type that is considered; (4) the specific structural signatures that are known to be associated with each chemical change or mutation type; (5) the presumed difficulty level for SINATRA Pro to detect each structural signature based on the homogeneity in shape variation between the wild-type and mutant proteins; and (6) references that have previously suggested some level of association or enrichment between each structural change and the mutation of interest.

https://doi.org/10.1371/journal.pcbi.1010045.t002

Atomic enrichments are illustrated in S5 and S6 Figs, while residue-level structural enrichments are shown in Figs 3–5 and S7–S21 Figs. To quantitatively assess the probability that SINATRA Pro is identifying any given ROI by chance, we implement a null region hypothesis test to estimate a P-value and an approximate Bayes factor (BF) corresponding to our power to reliably and robustly select certain features (Material and Methods). Reported results for the P-values and BF calculations are based on all MD simulated structures and can be found in Table 3. For comparison, we again implement the RMSF (Figs 3–5 and S12, S18, S20 and S21 Figs) and Elastic Net (S8, S11, S14, S16 and S19 Figs) baselines on all atomic positions with correspondences within these same protein systems. Here, we use scatter plots to illustrate the correlation between how each of these methods and SINATRA Pro rank the variable importance of the “pairable” atoms. All results presented in the main text reflect using SINATRA Pro with parameters set to {r = 6.0Å, c = 20, d = 8, θ = 0.80, l = 120} chosen via a grid search.

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Fig 3. Real data analyses aimed at detecting structural changes in the Ω-loop of β-lactamase (TEM) induced by an R164S mutation.

In this analysis, we compare the molecular dynamic (MD) trajectories of wild-type β-lactamase (TEM) versus R164S mutants [21, 22]. For both phenotypic classes, structural data are drawn from equally spaced intervals over a 100 ns MD trajectory (e.g., t_MD = [0, 1, 2, 3, …, 99] ns + δ, where δ is a time offset parameter). Altogether, we have a final dataset of N = 2000 protein structures in the study: 100 ns long interval × 10 different choices δ = {0.0, 0.1, 0.2, …, 0.9} ns × 2 phenotypic classes (wild-type versus mutant). This figure depicts results after applying SINATRA Pro using parameters {r = 6.0Å, c = 20, d = 8, θ = 0.80, l = 120} chosen via a grid search. The heatmaps in panels (A)-(C) highlight residue evidence potential on a scale from [0–100]. A maximum of 100 represents the threshold at which the first residue of the protein is reconstructed, while 0 denotes the threshold when the last residue is reconstructed. Panel (A) shows the residue-level evidence potential when applying SINATRA Pro to the whole protein, while panels (B) and (C) illustrate results when strictly applying the SINATRA Pro pipeline to atoms in residues 65–230 and 65–213, respectively. Annotated regions of interest are color coded and correspond to the shaded residue windows in panel (D). Panel (D) shows the mean association metrics (and their corresponding standard errors) computed for each residue within each analysis (see Material and methods). Here, the overlap shows the robustness of SINATRA Pro to identify the same signal even when it does not have access to the full structure of the protein. The final row plots the correlation between the SINATRA Pro association metrics and the root mean square fluctuation (RMSF) for all atoms with correspondences in the (E) whole protein, (F) fragment 65–230, and (G) fragment 65–213.

https://doi.org/10.1371/journal.pcbi.1010045.g003

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Fig 4. Real data analyses recover structural changes in the flap region of HIV-1 protease driven by a Ile50Val mutation.

In this analysis, we compare the molecular dynamic (MD) trajectories of wild-type HIV-1 protease versus Ile50Val mutants (i.e., within residues 47–55). For both phenotypic classes, structural data are drawn from from equally spaced intervals over a 100 ns MD trajectory (e.g., t_MD = [0, 1, 2, 3, …, 99] ns + δ, where δ is a time offset parameter). Altogether, we have a final dataset of N = 2000 proteins in the study: 100 ns long interval × 10 different choices δ = {0.0, 0.1, 0.2, …, 0.9} ns × 2 phenotypic classes (wild-type versus mutant). This figure depicts results after applying SINATRA Pro using parameters {r = 6.0Å, c = 20, d = 8, θ = 0.80, l = 120} chosen via a grid search. The heatmaps in panels (A) and (B) highlight residue evidence potential on a scale from [0–100]. A maximum of 100 represents the threshold at which the first residue of the protein is reconstructed, while 0 denotes the threshold when the last residue is reconstructed. Panel (A) shows residue-level evidence potential when applying SINATRA Pro to chain A, while panel (B) depicts results for chain B. Annotated regions of interest are color coded and correspond to the shaded residue windows in panel (C). Panel (C) shows the association metrics (and their corresponding standard errors) computed for each residue in chains A and B, with and without the 50th residue’s side chain being included in the analysis (see Material and methods). Here, the overlap shows the robustness of SINATRA Pro for identifying the same signal even when it does not have access to the full structure of the protein. The final row plots the correlation between the SINATRA Pro association metrics and the root mean square fluctuation (RMSF) for all atoms with correspondences in (D) chain A and (E) chain B, respectively. Highlighted are all atoms with correspondences found in regions of the protein corresponding to the fulcrum (brown), elbow (purple), flap (blue), cantilever (red), and I/V50 (yellow) [29, 31, 32].

https://doi.org/10.1371/journal.pcbi.1010045.g004

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Fig 5. Real data analyses identify enrichment in the N-terminal pocket of the Abl1 Tyrosine protein kinase due to a M290A mutation in the αC helix.

In this analysis, we compare the molecular dynamic (MD) trajectories of wild-type Abl1 versus M290A mutants [2, 43, 89–91]. For both phenotypic classes, structural data are drawn from equally spaced intervals over a 150 ns MD trajectory (e.g., t_MD = [0, 1, 2, 3, …, 99] × 1.5 ns + δ, where δ is a time offset parameter). Altogether, we have a final dataset of N = 3000 proteins in the study: 150 ns long interval × 15 different choices δ = {0.0, 0.1, 0.2, …, 1.4} ns × 2 phenotypic classes (wild-type versus mutant). This figure depicts results after applying SINATRA Pro using parameters {r = 6.0Å, c = 20, d = 8, θ = 0.80, l = 120} chosen via a grid search. The heatmaps in panels (A)-(C) highlight residue evidence potential on a scale from [0–100]. A maximum of 100 represents the threshold at which the first residue of the protein is reconstructed, while 0 denotes the threshold when the last residue is reconstructed. Panel (A) shows residue-level evidence potential when applying SINATRA Pro to the whole protein, while panels (B) and (C) illustrate results when strictly applying the SINATRA Pro pipeline to atoms in residues 242–502 and 242–315, respectively. Annotated regions of interest are color coded and correspond to the shaded residue windows in panel (D). Panel (D) shows the association metrics (and their corresponding standard errors) computed for each residue within each analysis (see Material and methods). Here, the overlap shows the robustness of SINATRA Pro to identify the same signal even when it does not have access to the full structure of the protein. The final row plots the correlation between the SINATRA Pro association metrics and the root mean square fluctuation (RMSF) for all atoms with correspondences in the (E) whole protein, (F) fragment 242–502, and (G) fragment 242–315.

https://doi.org/10.1371/journal.pcbi.1010045.g005

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Table 3. Null hypothesis experiment to evaluate SINATRA Pro’s ability to find regions of interest (ROI) in each of the proteins analyzed in this study.

Here, we assess how likely it is that SINATRA Pro finds the region of interest (ROI) by chance. These ROIs include: (i) the Ω-loop (residues 163–178) in TEM; (ii) the flap region (residues 47–55) in HIV-1 protease; (iii) Domain 2 (residues 208–308) in EF-Tu; and (iv) the DFG motif (residues 381–383) in Abl1. Note that protein structures were only analyzed if they contained an entire ROI. For example, in the context of Importin-β, the superhelix includes the entire structure and so we do conduct a null analysis. In this experiment, to produce the results above, we generate “null” regions on each protein using a K-nearest neighbors (KNN) algorithm on different atoms as random seeds [84], and exclude any generated regions that overlap with the ROI. Next, for each region, we sum the association metrics of all its atoms. We compare how many times the aggregate scores for the ROI are higher than those for the null regions. These “P-values,” and their corresponding calibrated Bayes factors (BF) when the computed P < 1/e, are provided above. Note that P-values less than the nominal size 0.05 and BFs greater than 2.456 are in bold. Results above are based on SINATRA Pro using parameters {c = 20, d = 8, θ = 0.80, l = 120} while varying the radius cutoff parameter r for mesh construction on each protein structure.

https://doi.org/10.1371/journal.pcbi.1010045.t003

Note that we also provide additional figures assessing how robust SINATRA Pro is to different configurations of protein meshes for these data in the Supporting Information. For example, results analyzing SINATRA Pro with protein meshes constructed under different radius cutoffs r values can be found in S22–S26 Figs. To ensure sampling consistency of the 100 ns results, we performed sensitivity analyses based on protein structures drawn from different lengths up to 200 ns MD simulation trajectories (S7, S10, S13, S15 and S17 Figs). Lastly, we test the robustness of SINATRA Pro when the protein structures have been aligned in two different ways (Material and Methods). In the first, we align protein structures by minimizing the root-mean-square distance (RMSD) between their atoms (which we focus on in the main text); while, in the second, we perform sequence-independent structural alignment where we implicitly normalize the 3D protein structures by rotationally aligning their topological summary statistics (i.e., alignment without taking into correspondence between structures). Figures comparing the correlation between SINATRA Pro results on the RMSD pre-aligned protein structures versus using the topological rotational alignment can be found in S27 and S28 Figs.

Conformational changes in the active site and regulatory Ω-loop of Arg164Ser TEM β-lactamase

Previous studies suggest that the Arg164Ser mutation in β-lactamase (TEM) induces structural changes in a highly plastic region known as the Ω-loop (residues 163–178), which plays a major role in the regulation of enzymatic activity [21, 22]. In wild-type β-lactamase, Arg164 makes a salt bridge with Asp179 that “pins down” the Ω-loop. Mutating Arg164 to serine breaks this salt bridge and disrupts a vast network of electrostatic and hydrogen interactions, dramatically affecting the dynamical behavior of the area surrounding the loop, parts of the active site, and potentially other protein domains [23]. These dynamical rearrangements confer multi-drug resistance to bacteria expressing TEM Arg164Ser, allowing them to hydrolyze a large number of cephalosporin antibiotics such as ceftazidime, cefixime, and cefazolin, in lieu of hydrolyzing ampicillin [24]. Given the enormous burden of multiresistant bacteria on public health, it is important that we understand the molecular mechanisms behind the structural rearrangements responsible for the transition to the cephalosporinase phenotype in order to orient future antibiotic design. Although previous studies have probed these rearrangements with varying approaches [23], the full mechanism remains elusive, highlighting the need for novel sampling and analytical methods that can detect the very slight changes in TEM’s active site topology that lead to drug resistance in the Arg164Ser and similar mutants. To help bridge this gap in understanding, we ran all-atom MD simulations of unbound TEM-1 and its Arg164Ser mutant, generated by homology modeling, and analyzed the results using SINATRA Pro, RMSF, and the Elastic Net baselines. Here, we expect SINATRA Pro to reveal new insights about the molecular mechanisms underlying the specificity shift precipitated by the Arg164Ser mutation, due its ability to detect both minute static and stochastic changes in topology that elude traditional methods.

We compare the MD trajectories of wild-type and mutant TEM using aligned structures of the whole protein (Fig 3A and S5(A) Fig), residues 65–230 (Fig 3B and S5(B) Fig), and residues 65–213 (Fig 3C and S5(C) Fig. In all three cases, statistical association measures from SINATRA Pro suggest that there are indeed significant structural changes in the Ω-loop (residues 163–178) relative to the rest of the regions in the protein (Fig 3D), especially on residues 164 and 176–179, which are involved in the electrostatic interaction networks disrupted by the arginine to serine substitution. This ROI is not as prominently identified by the RMSF (see scatter plots in Fig 3E–3G) or the Elastic Net baselines (S8(B)–S8(D) Fig). Alternatively, all three approaches were able to identify the region harboring residues 213–230, which undergoes a noticeable dynamic shift over the course the MD trajectory. These results are consistent with our controlled simulations, which showed that the only time that RMSF and the Elastic Net both exhibit relatively decent power for stochastic changes is when large structural deviations are introduced (e.g., see power comparisons in Fig 2F).

To more thoroughly assess if the Arg164Ser mutation contributes to the detected changes, we removed the Arg/Ser164 sidechain, as well as the whole residue (backbone and side chain), from our analyses. With the Arg/Ser164 atoms removed, association metrics of Arg/Ser164 and residues 176–179 diminished, which implies that signals pertaining to the dynamical contributions from the electrostatic interaction networks mediated by the side-chains of Arg164 and Glu179 are lost due to the missing topology. However, enrichment in the Ω-loop persisted, affirming that the identified topological differences are not just due to changes in these atoms. The null region test showed that the Ω-loop is indeed a robust significant structural feature in TEM, with P = 5.63 × 10⁻² and BF = 2.27 when the whole TEM protein is analyzed (Table 3 with r = 6.0 Å), P = 6.85 × 10⁻² and BF = 2.00 when residues 65–230 are analyzed, and P = 7.22 × 10⁻⁴ and BF = 70.4 when residues 65–213 are analyzed. We hypothesize that the ROI P-value is larger than the nominal 0.05 level for analyses with the whole structure and residues 65–230 because movement in the Ω-loop occurs jointly with moderate fluctuations in the region harboring residues 210–230. Overall, when we limit our scope to just residues 65–213, the region test robustly rejects the null hypothesis of the Ω-loop being identified by chance. It is worth noting that these results also highlight that there can be variations in the strength of signal detected by SINATRA Pro depending on if one carries out analyses on whole protein structures versus on different fragments. These variations are most likely due to bias caused by different structural segments having different centers of mass and reference points during alignment. This will cause differences in the topological summary statistics that are computed for each structure and will lead to slightly varying signals in the biophysical signatures detected in downstream SINATRA Pro analyses.

Our results are particularly interesting for the TEM β-lactamase example because they highlight the importance of codon positions 164 and 179 in controlling Ω-loop dynamics, which contributes to modulating activity. Moreover, SINATRA Pro correctly captures the topological effects of the disruption of the electrostatic network formed by Arg164, Arg178, and Asp179 due to the Arg164Ser mutation. In addition to reaffirming previously observed phenomena, SINATRA Pro also identified meaningful shifts in the 210–230 segment in response to the resistance-granting Arg164Ser mutation. This suggests that the topology of the 210–230 segment, which forms the upper boundaries of the active site, is tightly correlated with shifts in the Ω-loop. Our results suggest an additional potential mechanism for activity modulation by Ω-loop fluctuations, where topological changes propagate from regulatory loops to parts of the active site, suggesting potential allosteric couplings between the Ω-loop and the 210–230 segment. These results function as a testament to SINATRA Pro’s capacity for distinguishing meaningful topological differences from the random fluctuations introduced by disorder-inducing mutations such as Arg164Ser, which obfuscates traditional analyses pipelines.

Changes in the flap region of HIV-1 protease driven by the Ile50Val mutation

Our next analysis focuses on the HIV-1 protease, an enzyme that is essential for viral reproduction and is a well-established target for controlling HIV infections [25]. In vivo, the protease cleaves the HIV polyproteins Gag and Gag-Pol at multiple sites, creating the mature protein components of an HIV virion [26]. Over the past 25 years, ten HIV protease inhibitors have been approved for human use by the Food and Drug Administration (FDA), with many more undergoing clinical trials [27]. Similar to TEM, point mutations in the protease gene lead to products that are considerably less susceptible to inhibition by current drugs, generating drug-resistant HIV variants that pose a considerable risk [28]. Many hypotheses have been proposed for the molecular mechanisms underlying the most common resistance-granting mutations, and recent studies have used sophisticated geometric analyses to classify conformational ensembles of mutant structures based on their influences on the dynamics [29]. Structurally, the HIV protease forms a homodimer with highly ordered domains [30]. Most resistance-granting mutations, such as the Ile50Val substitution, are thought to mainly affect the cross-correlated fluctuations of the flaps (residues 47–55), imparting minute changes to the fulcrum and lateral topology [29, 31, 32]. These findings suggest that mutations such as Ile50Val effectively rewire residue communication networks, significantly reducing its affinity for binding the inhibitor. These structural rearrangements lead to surprisingly nuanced changes to the topology, which as discussed previously, require refined quantitative methods to be detected. To test SINATRA Pro’s performance in detecting these small changes, we ran all-atom molecular dynamics simulations of “protein and ligand complex in water” systems containing HIV Protease or its Ile50Val mutant complexed with the antiviral drug Amprenavir [33]. We then followed that with analysis using either SINATRA Pro, RMSF, or the Elastic Net, with the objective of measuring each routine’s capacity for detecting and reporting the topological changes induced by the mutation.

Even though MD simulations are performed on the protein’s native dimeric form, chains A and B were separately selected and aligned before being input into each statistical method to avoid alignment bias due to inter-chain orientation. This focuses SINATRA, RMSF, and the Elastic Net on identifying the structural differences within each chain (e.g., Fig 4A and 4B and S5(D) and S5(E) Fig). However, note that results for the analyses on the dimeric structures can be found in the Supporting Information (S9 Fig). Overall, our analyses reveal that chains A and B seem to respond asymmetrically to the backbone effects of the mutation within the timeframe of the simulations (Table 3). This is not unexpected, as during the course of the simulations, the inhibitor Amprenavir affects dynamics asymmetrically by interacting more significantly with residues of Chain A. The change in RMSF for most of the residues in the flap are shown to be greater than 0.2 Å for chain A and smaller than 0.2 Å for chain B, indicating that the flap became more dynamic in the MD simulations when the Ile50Val mutation was introduced into chain A (Fig 4D–4E and S20(B) Fig). Meanwhile, the Elastic Net shows larger nonzero coefficients in the fulcrum for chain A than in chain B (S11(A) Fig).

While association metrics from SINATRA Pro identify structural changes in the flap for both chains (Fig 4C), they also capture the geometric shifts within the fulcrum for chain A. We hypothesize that the coexistence of the two changes (flap and fulcrum) in chain A contributes to a smaller peak (i.e., a weaker signal) in the association metrics produced by SINATRA Pro in the flap for chain A than in chain B. This asymmetry is confirmed by the null test, as topological changes in the flap appear to be less statistically significant in chain A (P = 2.95 × 10⁻¹ and BF = 1.022) than in chain B (P = 8.14 × 10⁻⁴ and BF = 63.554) for this MD simulation data. Similar to β-lactamase, we assess if the Ile50Val mutation contributes to these detected topological changes. Upon removing the Ile/Val50 side-chain, the signal observed by SINATRA Pro in the flap drops with the missing topology, but still displays a significant peak relative to the rest of the protein, which implies that the change in association scores is not solely due to the structural differences upon introducing the Ile50Val mutation (Fig 4C). Although SINATRA Pro clearly identified the effects of the isoleucine to valine substitution in flap topology and fulcrum dynamics, SINATRA Pro did not detect other previously elucidated structural signatures of the mutation, such as lateral extension [29]. As the baseline approaches also failed to identify these features, it is likely that their absence stems from sampling limitations inherent to the brute-force and relatively short production dynamics used to generate the conformational datasets.

The HIV protease system presents a good test case for SINATRA Pro due to its relative structural simplicity and the symmetry of the dimer. Encouragingly, SINATRA Pro’s results closely match those observed in previous studies that sought to characterize deltas in the backbone dynamics of the HIV protease in response to resistance-granting mutations [29].

Domain 2 in EF-Tu undergoes structural changes upon GTP hydrolysis

In our third analysis, we focus on EF-Tu (elongation factor thermo unstable), which is a G-protein that is responsible for catalyzing the binding of aminoacyl-tRNAs to the ribosome in prokaryotes. After binding GTP and a given aa-tRNA, EF-Tu strongly interacts with the ribosomal A site [34]. Following productive aa-tRNA binding, EF-Tu is released upon GTP hydrolysis [35]. The resultant GDP molecule is exchanged for GTP with EF-Ts (elongation factor thermo stable), allowing elongation to continue. Structurally, EF-Tu is composed of a Ras-like catalytic domain (RasD), common to G-proteins, and two beta-barrel domains (D2 and D3) [36]. Previous studies probing dynamic fluctuations of GTPases have identified that, after hydrolysis (in the GDP-bound state), EF-Tu shows considerably increased flexibility of backbone atoms belonging to Domains 2 and 3, which are downstream of the nucleotide binding site in RasD [37]. These fluctuations are thought to be correlated with conformational rearrangements required for the exchange of GDP for GTP [37]. The conformational rearrangements are thought to occur on multiple millisecond timescales [37], presenting an obstacle for their study using all-atom molecular dynamics simulations. While the full relaxations associated with the change in ligand chemistry are challenging to sample effectively, we expect SINATRA Pro to have the ability to detect subtle topological differences arising from rearrangements of residue interacting networks in response to the chemical change in the ligand.

To compare our method’s performance to that of alternative techniques (S12 and S14 Figs), we run SINATRA Pro, RMSF, and the Elastic Net on the whole structure (S5(F) and S12A(A) Figs) and fragment windows limited to residues 208–308 (S5(G) and S12(B) Figs) and 311–405 (S5(H) and S12(C) Figs). Note that all figures displaying the enrichment of structural features are projected onto the GTP-bound structures. The evidence scores from SINATRA Pro reveal significant structural changes at Domain 2, with minimal structural changes in the majority of the Ras-like Domain, which agrees with findings in previous studies [37]. The null region test shows that Domain 2 is indeed an important structural feature in EF-Tu identified by SINATRA Pro with P = 9.30 × 10⁻⁴ and BF = 56.657, which robustly rejects the null hypothesis of the ROI being identified by chance (Table 3).

The chemical changes associated with the substitution of GTP with GDP in the EF-Tu system are thought to have significant impacts on backbone topology, making this a particularly interesting use case for SINATRA Pro. Despite the challenges associated with the considerable noise inherent to the complex EF-Tu system, SINATRA Pro succeeded in identifying the meaningful topological deltas that are thought to be important for function and that were elucidated in previous studies.

N-pocket enlargement and αC helix displacement in Met290Ala Abl1

Protein tyrosine kinases (TKs) such as Abl1 and Src play significant roles in eukaryotic life, as phosphorylation of tyrosine residues in key proteins act as on/off switches that regulate a plethora of cellular processes and allow for efficient message passing [38]. Deregulation of the activity of these enzymes due to mutations is usually associated with severe forms of cancer and other chronic diseases, posing a grave public health problem [39]. Due to their physiological importance, the enzymatic activity of tyrosine kinases is tightly regulated by a series of structural elements that fluctuate among metastable conformations between the active and inactive states [40]. This highly dynamic behavior has been exploited for the development of TK inhibitors, such as the widely-known anticancer drug Imatinib, which exclusively targets the “DFG out” state of Abl1 [40–42]. In this conformation, the phenylalanine residue of the region known as the DFG motif (comprised of Asp381, Phe382, and Gly383) occupies Abl1’s ATP binding site, preventing substrate binding and inactivating the enzyme [43]. Other TK inhibitors such as Dasatinib are capable of binding to Abl1’s “DFG-in” conformation, in which the positions of the aspartic acid and phenylalanine side-chains are inverted with respect to their positions in DFG-out conformations, activating the enzyme (i.e., making it capable of productive phosphorylation) [42]. The transition from the DFG-in to the DFG-out state is thought to happen on the multi-millisecond timescale, which presents a challenge for capturing it with unbiased atomistic MD simulations [43]. As a workaround, previous studies have used an engineered Abl1 mutant, Met290Ala, in which the energy barrier for the DFG flip is considerably reduced, as the steric effect presented by the bulky methionine is removed [2]. Although the sampling of the entire DFG flip is a rare event that happens in the milisecond timescale and thus outside of the scope of this work, we hypothesize that the Met290Ala mutation should induce minute topological changes around the DFG motif even in shorter simulations, due to the removal of the steric hindrances associated with the bulky methionine and that SINATRA Pro is capable of detecting these changes in contrast to the wild-type dynamics. To test this hypothesis and further measure our method’s capacity for detecting localized topological changes, we ran molecular dynamics simulations on the TK domain of the unbound state of Abl1 and its Met290Ala mutant. Specifically, we run SINATRA Pro, RMSF, and the Elastic Net on the whole structure (Fig 5A and S6(A) Fig), fragments limited to residues 242–502 (Fig 5B and S6(B) Fig), and the N-lobe spanning residues 242–315 (Fig 5C and S6(C) Fig).

Since we only simulated the kinase domain of Abl1, both the N-terminal and C-terminal domains are shown to be highly dynamic as they are no longer stabilized by the mass of the entire protein. As a result, whole structural changes are overshadowed by large and noisy fluctuations, and competing methods (Elastic Net and RMSF) have a difficult time identifying enrichment in the DFG motif (S16(A) and S21A(A) Figs). Nonetheless, SINATRA Pro is able to identify the enrichment in the DFG motif regardless of the inclusion or exclusion of the N- and C-terminal (Fig 5). The signal in the DFG motif ROI becomes better statistically resolved when we remove some of the structural noise and concentrate on regions spanning residue fragments 242–502 and 242–315 (i.e., the N-lobe). The null region test results for SINATRA Pro show that the DFG motif is indeed an important structural feature in Abl1: P = 8.86 × 10⁻³ and BF = 8.783 for the whole structure analysis (i.e., including the termini) and P = 2.50 × 10⁻⁴ and BF = 178 for the analysis on residues 242–502 (i.e., excluding the termini), both of which reject the null hypothesis of the ROI being identified by chance (Table 3). In these analyses, the structural differences around the DFG motif between unbound Abl1 and its Met290Ala mutant were large enough for both RMSF and the Elastic Net to have power. As a comparison, SINATRA Pro not only robustly identifies residues associated with the greater N-pocket cleft as being statistically significant (i.e., the DFG motif), but also the αC helix spanning residues 281–293 as a moderately enriched region (Fig 5).

From the SINATRA Pro output, we can postulate hypotheses regarding the involvement of specific codon positions outside of the DFG motif in the concerted motions that culminate in the flip, such as the two peaks of signal surrounding it (residues 350–360 and 390–400). These interesting results show that even short simulations can prove useful for gaining mechanistic insights regarding long-timescale macromolecular relaxations, as long as the heuristics employed to analyze the resulting trajectories are capable of detecting the often minute signals associated with these topological shifts.

Opening of superhelix differentiates unbound and IBB-bound Importin-β

Our last analysis focused on the karyopherin Importin-β, an essential member of the nuclear import complex in eukaryotes, as it mediates the transportation of cargo from the cytosol to the nucleus [44]. Molecular recognition by Importin-β often requires the cooperative binding of molecular adaptors that recognize and bind to nuclear localization sequences (NLS)—structural motifs present in cargo destined for the nucleus [44]. Structurally, Importin-β is organized as a superhelix composed of up to 20 tandem HEAT repeats, each of which contain two antiparallel alpha helices linked by a turn [45]. This highly ordered structure is further stabilized by interactions with Importin-β-binding (IBB) domains of transport adaptors such Importin-α or Snurportin 1, which attach very strongly to Importin-β [46]. The release of IBB peptides after successful transport across the nuclear pore leads to large structural rearrangements and fluctuations that are propagated across most of Importin-β’s backbone [47]. Although not difficult to detect with traditional analysis pipelines, such as calculating per-residue root-mean-square fluctuations or the backbone’s radius of gyration, the pseudo-global nature of these rearrangements is diametrically opposite to most of the previously explored examples, presenting an important test for SINATRA Pro. Considering this, we ran MD simulations of unbound and IBB-bound Importin-β and, as with the previous examples, analyzed the resulting trajectories with SINATRA Pro to compare against standard methods. The structural features identified are projected onto the IBB-bound form (S6(D) and S18A(A) Figs). Association metrics from SINATRA Pro, RMSF, and the Elastic Net all indicate large-scale conformational changes occur upon IBB release that involve the majority of the importin-β structure (S18(B), S18(C) and S19 Figs).

Since Importin-β functions as a molecular spring due to its supercoiled structure and extensive interactions with targets for transport, the sudden removal of the bound IBB domain to generate the unbound structure leads to extensive and drastic fluctuations across most of the backbone during production dynamics, originating from multiple highly-correlated nodes in each HEAT repeat. These drastic rearrangements translate to significant deltas in the topology and per-residue fluctuations that are readily detected by all tested methods. Importantly, the SINATRA Pro output replicates the expected results for the IBB bound/unbound Importin-β system, demonstrating the method’s capacity for picking up relevant structural determinants not only for localized changes, but also for backbone-wide large-scale fluctuations.

Molecular dynamics simulations

The protein structure data used in the current study are a result of molecular dynamic (MD) simulations. For large systems (i.e., IBB-bound Importin-β, unbound Importin-β) and those containing small-molecule ligands (i.e., wild-type HIV Protease, Ile50Val HIV protease, GTP-bound EF-Tu, and GDP-bound EF-Tu), we used Schrödinger’s Desmond (release 2020–1) [55] to run three independent 100 nanosecond (ns) simulations for each system. This decision is rooted in Desmond’s high performance when dealing with hundreds of thousands of atoms, and the extensive validation of the small-molecule parameters contained in the OPLS3e force-field [56]. The systems were built within a dodecahedron box extending 1 nanometer (nm) beyond the solute in all three dimensions and solvated with water molecules using the SPC model [57]. Charges were neutralized by replacing a varying number of solvent molecules with sodium and chloride ions. Before production dynamics, all systems were relaxed and equilibrated with Desmond’s standard relaxation protocol, which first performs energy minimization with 50 kcal/mol/Å2 restraints on the protein’s heavy atoms, followed by an extensive equilibration protocol. This protocol is detailed below:

1. NVT equilibration at 10 K for 12 ps
2. NPT equilibration at 10 K for 12 ps
3. NPT equilibration at 300K with harmonic restraints on the protein’s heavy atoms for 120 ps
4. NPT equilibration at 300 K, unrestrained, for 240 ps,

where NVT denotes constant temperature and volume and NPT denotes constant temperature and pressure. After equilibration, unrestrained NPT production simulations were conducted at 300 K and 1 atm for 100 ns for each system, to ensure consistency among the trajectories. Time steps for all simulations were set to their default values: 2:2:6 fs (bonded:near:far).

For comparatively small systems without ligands (i.e., wild-type TEM, Arg164Ser TEM, wild-type Abl1, and Met290Val Abl1), we used GROMACS (release 2018–2) [58] to run three independent 100 ns simulations for each system (150 ns for Abl1). Simulations were conducted with a 2 fs time step using the Amberff14SB force-field [59] and the TIP3P water model [57]. As with the Desmond simulations, the systems were built within a dodecahedron box and charges neutralized by replacing a number of solvent atoms with sodium and chloride ions. For each system, energy was minimized using a steepest-descent algorithm until the maximum force on any given atom was less than 1000 kJ/mol/min. Solvent atoms were equilibrated in sequential 0.5 ns NVT and NPT simulations with solute heavy atoms restrained by a spring constant of 1,000 kJ/mol/nm² using the LINCS algorithm [60]. After equilibration, production dynamics were conducted sans the position restraints. All simulations were conducted at 300 K and 1 atm. Lastly, using Visual Molecular Dynamics (VMD) (version 1.9.3) [61], we converted all trajectories employed in this study to a DCD file format and stripped solvent atoms to facilitate downstream computational analyses.

Protein structure alignment

In the current study, protein structures are aligned in one of two ways. In the first procedure, we align protein structures by minimizing the root-mean-square distance (RMSD) between the atoms on their backbone alpha-carbons (which we denote in shorthand by C_α). The first frame of the MD simulation is chosen as the reference structure. Next, all other frames (i.e., the mobile and fluctuating structures) in the dataset are aligned to this reference frame by (i) first superposing the center of mass of the C_α atoms to the same origin and then (ii) minimizing the RMSD rotation matrix. This calculation is performed using the MDAnalysis software package in Python (see Data and Software Availability) [6, 62–64]. For inter-class alignment when comparing protein class A to class B (e.g., mutants versus wild-type), all frames in the trajectory of class A and B are aligned to the first frame in the trajectory of class A. Note that, in our current study, there is only one point mutation in each protein. In this case, determining correspondences between sequences in the trajectories of class A and B is trivial because mutated residues in the mutant structures just correspond to the un-mutated residue in the wild-type (e.g., Arg164 in WT versus Ser164 in Arg164Ser mutant in β-lactamase). In the controlled simulation experiments, perturbed structures were obtained by directly modifying the atomic coordinates of the pre-aligned proteins; therefore, the perturbed structures do not need further alignment since their unperturbed regions remain aligned after the controlled modifications.

In the second alignment procedure, we perform sequence-independent structural alignment where we implicitly normalize the 3D protein structures by rotationally aligning their topological summary statistics. To carry out this alignment procedure, we first take each pair of protein structures and superpose the center of mass of the C_α atoms to the same origin. Next, we compute topological summary statistics over the mesh representation of each structure in m = 500 spherically uniformly distributed directions (see the remaining sections of Material and Methods). We take the squared Euclidean distance between any two directions to be the cost needed to align structures via their topological summaries; and we determine the “optimal” direction alignment by finding the rotation that minimizes the cumulative cost of aligning all directional pairs between proteins. Here, we use the random sample consensus (RANSAC) method to determine the rotational matrix that aligns the angle between any two directions to be within an error threshold of 0.9 [65]. More specifically, we require that the dot product between two directions has to be larger than 0.9 to be considered aligned in RANSAC. Figures comparing the correlation between SINATRA Pro results on the five real protein systems using the RMSD pre-alignment versus using the topological rotational alignment can be found in the Supporting Information (S27 and S28 Figs). There are large agreements between both alignment procedures in almost all protein systems. The one exception being HIV-1 protease where we believe that the difference in results is driven by the fact that the small dynamic topology of protein makes it difficult to identify optimal stable reference signatures for alignment in summary statistic space.

Converting protein structure data to 3D Mesh representations

To convert protein structures into a mesh representation, in the first step of the SINATRA Pro pipeline, we make use of a technique which we refer to as a “simplicial construction” (Fig 1B). In this procedure, we treat the atomic positions for the protein as vertices on a 3D shape or surface. First, we draw an edge between any two atoms if their Euclidean distance is smaller than some radius cutoff r, namely dist|(x₁, y₁, z₁), (x₂, y₂, z₂)| < r. Next, we fill in all of the triangles (or faces) formed by these connected edges. The resulting triangulated meshes are then normalized to the unit sphere, which means that the coordinates for all atoms are scaled with respect to the mesh with the largest radius. We treat the normalized meshes as simplicial complexes which we then use to compute topological summary statistics.

Topological summary statistics for protein Mesh representations

Adopted from its predecessor [13], the second step of the SINATRA Pro pipeline uses a tool from integral geometry and differential topology called the Euler characteristic (EC) transform [14–17]. As a brief overview of this approach, given the mesh representation of a protein structure, the Euler characteristic is an accessible topological invariant defined as (1) where the collection denotes the number of vertices (atoms), edges (connections between atoms), and faces (triangles enclosed by edges) of the mesh, respectively. An EC curve tracks the change in the Euler characteristic with respect to a given filtration of length l in some direction ν. Theoretically, this is done by first specifying a height function h_ν(x) = x^⊺ν for some atomic position in direction ν. This height function is then used to define sublevel sets (or subparts) of the mesh in direction ν, where h_ν(x) ≤ a. In practice, the EC curve is computed over a range of l filtration steps in direction ν. The corresponding EC transform is defined as the collection of EC curves across a set of ν = 1, …, m directions, and maps onto a 3D protein structure as a concatenated J = (l × m)-dimensional feature vector to be used for statistical analyses.

In previous studies, it has been observed that the Euler characteristic can be a less-than-optimal shape summary statistic when inter-class variation between 3D objects is high and driven by local fluctuations in morphology [13, 16, 17, 52, 66]. Given that this situation can be quite common in molecular dynamics, we introduce a new topological invariant which we refer to as the differential Euler characteristic (DEC) (see Fig 1C). As an alternative to Eq (1), the DEC is computed as the following (2) where, for some lag parameter t, we define , and . In this study, we set t = 1 such that, intuitively, the DEC tracks the changes (i.e., the local appearance or disappearance of topological features) in the number of vertices, edges, and faces from one sublevel set to the next. Much like with the original Euler characteristic, the DEC curve is computed over a range of l filtration steps in a given direction ν and the DEC transform is similarly defined as the collection of DEC curves across a set of ν = 1, …, m directions. Overall, for each dataset with N total proteins, an N × J design matrix X is statistically analyzed, where the columns denote the differential Euler characteristic computed at a given filtration step and direction. Each sublevel set value, direction, and set of atomic positions used to compute a DEC curve are stored by the algorithm for the association mapping and projection phases of the pipeline.

Probabilistic model for protein structure classification

In the third step of the SINATRA Pro pipeline, we use (weight-space) Gaussian process probit regression model to classify protein structures based on their topological summaries generated by the DEC transformation via Eq (2). Here, we specify the following probabilistic hierarchical model (Fig 1D) [69–73] (3) where y is an N-dimensional vector of Bernoulli distributed phenotypic class labels (e.g., mutants versus wild-type), π is an N-dimensional vector representing the underlying probability that a shape is classified as a “class” (e.g., y = 1 if “mutant”), g(⋅) is a probit link function with Φ(⋅) the cumulative distribution function (CDF) of the standard normal distribution, and f is an N-dimensional vector estimated from the data. We take a classic kernel regression approach [72, 74–76] where we posit that f lives within a reproducing kernel Hilbert space (RKHS) defined by some (nonlinear) covariance function, which implicitly accounts for higher-order interactions between features, leading to more complete classifications of structural data [77–79]. To this end, we assume f is normally distributed with mean vector 0 and covariance matrix K with elements defined by the radial basis function K_ii′ = exp{−ϑ‖x_i − x_i′‖²} with bandwidth ϑ set using the “median criterion” approach to maintain numerical stability and avoid additional computational costs [80]. Here, x_i and x_i′ denote the collection of topological features for the i-th and i′-th observation in X, respectively. The full model specified in Eq (3) is commonly referred to as “Gaussian process classification” (GPC).

Given the complete specification of the GPC, we use Bayesian inference to draw samples from the posterior distribution of the latent variables, which is proportional to p(f|y) ∝ p(y|f) × p(f). Here, p(y|f) denotes the likelihood of the observed binary labels given the functions (i.e., the Bernoulli distribution), and p(f) is the prior distribution for the latent variables (i.e., the multivariate normal distribution). The probit likelihood in Eq (3) makes it intractable to estimate the posterior distribution p(f|y) via a closed-form solution. We instead use a Markov chain Monte Carlo (MCMC) method called “elliptical slice sampling” to conduct posterior inference [81].

Feature selection of topological summary statistics

After implementing the elliptical slice sampling algorithm to estimate the posterior distribution of the latent variables f in Eq (3), we define a nonparametric effect size for each topological summary statistic via the following standard projection [78, 82] (4) where M⁺ is used to denote the generalized inverse of a matrix M, and each element in β details the nonlinear relationship between the DEC topological summary statistics and the variance between protein structures. In order to determine a statistical rank ordering for these effect sizes, we assign an information theoretic-based measure of relative centrality to each j-th topological feature using Kullback-Leibler divergence (KLD) [79] (5) for j = 1, …, J topological features. Finally, we normalize to obtain an association metric (Fig 1D), (6)

There are two key takeaways from this scaled formulation. First, the KLD is non-negative, and it equals zero if and only if the posterior distribution of β_−j is independent of the effect β_j. Intuitively, this is equivalent to saying that removing an unimportant topological feature has no impact on explaining the variance between different protein structure. Second, γ = (γ₁, …, γ_J) is bounded on the unit interval [0, 1] with the natural interpretation of providing relative evidence of association for each DEC statistic (where values close to 1 suggest greater importance). From a classical hypothesis testing point-of-view, the null hypothesis for Eq (6) assumes that every DEC feature equally contribute to the total variance between proteins, while the alternative hypothesis proposes that some DEC features are better associated with biophysical changes in protein structures than others [13, 79].

Reconstruction and visualization of biophysical signatures

After obtaining association measures γ for each topological feature computed via Eq (6), in the fourth step of the SINATRA Pro pipeline, we map this information back onto the original structures to visualize topological differences between the protein classes. The main idea is that we want to select or prioritize atoms that correspond to the topological features with the greatest association measures. To do this, we perform a criterion-based reconstruction algorithm [13]. In each direction, each atom (i.e., vertex) lies along a filtration step that corresponds to a γ_j value. Therefore, each atom corresponds to m = (c × d) values in γ. To perform the reconstruction, we sort the values in γ from smallest to largest and continuously increase a threshold. If all of the γ values corresponding to an atom are larger than the threshold, the atom is considered “alive”. As the threshold is increased, when the criterion is no longer satisfied, the atom is considered “dead” and that minimum value below the threshold (which we will denote by ) is assigned the atom as its evidence score. This calculation is repeated for each frame in the dataset. For atomic-level evidence scores (e.g., S5 and S6 Figs), the values are ranked among all atoms and scaled from 0 (lowest) to 100 (highest) to facilitate the visualization and interpretation of structural and biophysical enrichment. To compute residue-level evidence scores (e.g., Figs 3–5, and S7, S9, S10, S12, S13, S15, S17 and S18 Figs), we take the average of the values for all atoms within a residue which are then also ranked and scaled from 0 to 100.

Performance assessment for controlled simulation study

We demonstrate the power of the SINATRA Pro pipeline for identifying biophysical signatures in protein dynamics via multiple controlled simulations studies using the sequential procedure:

1. Fit the Gaussian process classification (GPC) model using elliptical slice sampling and compute relative centrality association measures γ_j for each j-th topological feature (i.e., differential Euler characteristic or DEC per sublevel set filtration). Recall, the total number of features J = c × d × l is a product of (i) c, the number of cones of directions; (ii) d, the number of directions within each cone; and (iii) l, the number of sublevel sets (i.e., steps in the filtration) used to compute the DEC along a given direction.
2. Sort the topological features from largest to smallest according to their association measures γ₁ ≥ γ₂ ≥ ⋯ ≥ γ_p.
3. By iteratively moving through the sorted measures T_k = γ_k (starting with k = 1), we reconstruct the atoms corresponding to the topological features with {j: γ_j ≥ T_k}.

An atom is “detected” when the sublevel set in which it resides is selected across all of the directions within a particular cone. We form a union of the set of detected atoms across all cones to construct the set of reconstructed vertices at a given level T_k. Using this set of vertices, we compute the true positive rate (TPR) and false positive rate (FPR) by assessing overlap with the set of truly associated (i.e., perturbed) atoms used to generate the protein classes: (11) where TP is the number of correctly detected true atoms, P is the total number of causal atoms, TN stands for the true negatives detected by the SINATRA Pro pipeline, and N stands for the total number of non-causal atoms. In this manner, we obtain a receiver operating characteristic (ROC) curve for the simulation studies (see Fig 2 and S1 Fig).

ROI null experiment and statistical assessment

To statistically assess whether SINATRA Pro is identifying the known regions of interest (ROI) in proteins by chance (see Table 2), we use a previously developed null-based scoring method [13]. The goal of this analysis is to estimate the probability of obtaining a result from SINATRA Pro under the assumption that the null hypothesis H₀ of there being no structural differences between mutant and wild-type proteins is true. Here, we treat the K atoms located within each ROI of every mutant protein as a landmark. We construct a test statistic τ* for each ROI by summing the association metric scores of every atom it contains. To construct a “null” distribution and assess the strength of any score τ*, we randomly select T “seed” atoms across the mesh outside the ROI for each mutant protein and uniformly generate T-“null” regions that are also K-atoms wide. We then compute similar (null) scores τ₁, …, τ_T for each randomly generated region. A “P-value”-like quantity (for the i-th mutant protein) is then generated by: (12) where is an indicator function, and a smaller P_i means more confidence in either method’s ability to find the desired paraconid landmark. To ensure the robustness of this analysis, we generate the N-random null regions using a K-nearest neighbors (KNN) algorithm on each of the T-random seed vertices [84]. We also use a calibration formula to transform each P-value to an approximate Bayes factor (BF) [85], which is defined as the ratio of the marginal likelihood under the alternative hypothesis H₁ (i.e., that there is indeed a structural difference between phenotypic classes) versus the null hypothesis H₀: (13) for P_i < 1/e and BF(P_i)₁₀ is an estimate of , where is again used to denote the protein meshes. We take the median of the P_i and BF(P_i)₁₀ values in Eqs (12) and (13) across all mutant proteins, respectively, and report them in Table 3.