Atom-specific persistent homology and its application to protein flexibility analysis
David Bramer Guo-Wei Wei







Atom-specific persistent homology and its application to protein flexibility analysis 
    David Bramer^1 and
Guo-Wei Wei^1,2,3 [ Address correspondences  to Guo-Wei Wei. E-mail:wei@math.msu.edu]


^1 Department of Mathematics 

Michigan State University, MI 48824, USA

^2  Department of Biochemistry and Molecular Biology

Michigan State University, MI 48824, USA 

^3 Department of Electrical and Computer Engineering 

Michigan State University, MI 48824, USA 


    Received / Accepted
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Recently, persistent homology has had tremendous success in biomolecular data analysis. It works by examining the topological relationship or connectivity of a group of atoms in a molecule at a variety of scales,  then rendering a family of topological representations of the molecule. However, persistent homology is rarely  employed for the analysis of atomic properties, such as biomolecular flexibility analysis or B factor prediction. This work introduces atom-specific persistent homology to provide a local atomic level representation of a molecule via a global topological tool. This is achieved through the construction of a pair of conjugated sets of atoms and corresponding conjugated simplicial complexes, as well as conjugated topological spaces. The difference between the topological invariants of the pair of conjugated sets is measured by Bottleneck and Wasserstein metrics and leads to an atom-specific topological representation of individual atomic properties in a molecule.  Atom-specific topological features are integrated with various machine learning algorithms, including gradient boosting trees and convolutional neural network for protein thermal fluctuation analysis and B factor prediction. Extensive numerical results indicate the proposed method provides a powerful topological tool for analyzing and predicting localized information.  

 
Keywords: Atom-specific topology, Element-specific persistent homology,   Protein flexibility,  Gradient boosting tree, Convolutional neural network.








§ INTRODUCTION


In recent years tools from topology have been successfully applied to protein analysis <cit.>. Topology  offers one of highest level of abstractions of geometric data and allows one to infer high dimensional structure from low dimensional representations. However, conventional topology oversimplifies geometry and thus lacks descriptive power for most real world problems. Persistent homology (PH) overcomes this difficulty by introducing a filtration parameter that describes the geometry in terms of a family of  Betti numbers at various scales known as a barcode  <cit.>.
Indeed, three dimensional (3D) protein spatial information from a protein data bank (PDB) file can be converted into a family of simplicial complexes. One can apply tools from algebraic topology to  convert structural information into global topological invariants that provide a useful representation of biomolecular properties <cit.>. However, for quantitative biomolecular analysis and prediction, persistent homology  alone neglects chemical and biology information. Element-specific persistent homology has been introduced to incorporate chemical and biological information into topological invariants <cit.>. Similarity and differences between barcodes from different molecules can be measured by Wasserstein <cit.> and/or Bottleneck <cit.> distances. However, the previous applications of persistent homology and element-specific persistent homology are for the modeling and prediction of  molecule-level   thermodynamical or structural properties, such as protein-ligand binding affinities  <cit.>,   protein folding free energy changes  upon mutations <cit.>, drug toxicity <cit.>,  solubility, partition coefficient <cit.>, and drug virtual screening (ligand and decoy classification) <cit.>. Essentially, topology is a global tool that examines the connectivity and relationship among many atoms in a neighborhood as a whole. High dimensional topological invariants, such as Betti 1 and Betti 2, describe the collective behavior of many atoms.  Therefore, it is not clear how to represent atomic level property, such as the B factor of an atom, by persistent homology.  
 
 In proteins, beta factor (B factor) or  (Debye-Waller factor is a measure of the attenuation of  X-ray scattering caused by thermal motion. The strength of the thermal motion of an atom is theoretically proportional to its B factor during the structure determination from X-ray  diffraction data. It is well known that biomolecular flexibility provides an important link between its structure and function. In particular, it has been shown that intrinsic structural flexibility correlates to meaningful protein conformational variations, reactivity and enzymatic function <cit.>. As such, the accurate prediction of protein B-factor is essential to our understanding of protein structure, function and dynamics  <cit.>. 




Early methods used to predict protein B factor were derived from Hooke's Law and are known as elastic mass-and-spring networks. In these models, alpha carbons (C_α) of biological macromolecules are treated as a mass and spring network and motions are predicted based on a harmonic potential. Given a protein, each C_α is represented as a node in the network and edges are weighted based on a potential function. Nodes are connected by an edge if they fall within a pre-defined euclidean cutoff distance. This captures the local covalent and non-covalent interactions between an individual atom and nearby atoms. One of the first mass-and-spring methods used for protein B factor prediction is normal mode analysis (NMA). Like most B factor prediction methods, NMA is independent of time and uses a Hamiltonian interaction matrix. Eigenvalues of the matrix system correspond to characteristic frequencies of the protein and these frequencies correlate with protein B factors. Low-frequency modes correlate with cooperative motion and can be useful for hinge detection and domain motion. NMA has also been successfully implemented to understand the deformation of supramolecular complexes. <cit.>


Elastic network model (ENM) was introduced as a more efficient model that significantly reduces computational cost compared to NMA through the use of a simplified spring network <cit.>. A specific example is  anisotropic network model (ANM) <cit.>. Gaussian network model (GNM) further reduces the computational cost by ignoring the anisotropic motion, rendering a more accurate method for protein C_α B factor analysis <cit.>.


All of the aforementioned methods depend on matrix diagonalization, which has the computational complexity of 𝒪(N^3), where is the number of matrix atoms involved in the analysis.  Recently, flexibility and Rigidity Index (FRI) methods have  been proposed as a geometric graph approach to further reduce the computational cost. FRI methods rely on constructing a distance matrix using radial basis functions to scale atom to atom distance non-linearly <cit.>. All versions of FRI produce a flexibility index, that correlates to the B factor, for each C_α. Several versions of FRI have been developed. Among them, fast FRI (fFRI) is of 𝒪(N) in computational complexity <cit.>. FRI methods are also more accurate than all of the earlier  algebraic graph-based methods. Additionally,  anisotropic FRI (aFRI) provides   high quality anisotropic motion analysis  <cit.>. Moreover, using several radial basis functions with different parametrizations,   the multiscale flexibility rigidity index (mFRI) can successfully capture multiscale atomic interactions <cit.>.

More recently, the authors introduced a multiscale weighted colored graph (MWCG) model. The MWCG is another geometric graph theory model that has been shown to be the best B factor prediction model to date. First, element-specific interaction subgraphs are constructed based on selected atomic interactions between certain element types. Atoms are represented as graph nodes and subgraphs are generated using pairs of atoms of certain elements (e. g., carbon, nitrogen, oxygen, sulfur). A centrality metric that uses radial basis functions is applied to pairwise interactions in each subgraph. By varying the parametrization of the radial basis functions the MWCG model can capture multiple protein interaction scales. MWCG is unique in its ability to utilize both element specific and multiscale interactions for improved B factor prediction <cit.>. Most recently, MWCG is incorporated with machine learning algorithms for across-protein blind predictions of protein B factors <cit.>.  


The objective of the present work is to extend the utility of persistent homology for atomic level property modeling and prediction. To this end, we introduce atom-specific persistent homology (ASPH) to create a local atomic representation of an atom using a global topological tool in a novel way. Specifically, ASPH  constructs  a pair of conjugated sets of point clouds or atoms centered around the atom of interest. The first set of a pair of conjugated sets of atoms for a given atom is selected by a local sphere of radius r_c around the atom of interest. The second set of atoms is defined by excluding the atom of interest in the first set.  Conjugated simplicial complexes, conjugated chain groups, conjugated homology groups as well as conjugated persistence barcodes or diagrams are induced by an identical filtration. Conjugated persistence barcodes are compared with Bottleneck and Wasserstein metrics. The resulting distance provides a global topological representation of the localized atomic property, such as protein flexibility analysis and atomic-level protein B-factor information.   
Obviously, the proposed atom-specific topology can be applied to a wide variety of chemical and biological problems where atomic properties are measured, such as the chemical shifts of nuclear magnetic resonance (NMR), the B-factors of X-ray structure determination, and the shift and line broadening of other atomic spectroscopy.  
 
We focus on protein C_α B-factor prediction but the approach provided in this work is a general framework that can be used to predict B factors of any atom in a protein. First, we use the generated atom-specific persistent homology features to fit B factors within a given protein using   linear least squares  minimization. Then the atom-specific persistent homology features are combined with other local and global protein features to construct machine learning models for the blind prediction of protein B factors across different proteins. Additionally,  image-like multiscale atom-specific persistent homology features are generated using an early technique <cit.>.  These image like features, together with other features, are fed into convolutional neural networks (CNN).  Training and validation are carried out using a large and diverse set of proteins from the protein data bank (PDB). The proposed method offers some of the best results for blind B factor predictions of a set of 364 proteins.  
 






§ METHODS AND ALGORITHMS





 §.§ Atom-specific persistent homology



  §.§.§ Overview

Topology describes (continuous) objects in terms of topological invariants, i.e., Betti numbers.  Betti-0, Betti-1, and Betti-2  which  can be interpreted as connected components, rings,  cavities, etc. Table <ref> provides examples of the  Betti numbers of a point, circle, sphere, and torus.








 Given discrete data points, such as a point cloud or the set of atoms in a molecule, we use simplicial complexes to describe the topological relationship, or connectivity of the point cloud, to systematically identify topological invariants.  First, a few simplicial complexes, as shown in Figure <ref>, are made up of vertices, edges, triangles, and tetrahedrons, denoted 0-simplex, 1-simplex, 2-simplex, and 3-simplex, respectively. Homology groups constructed from simplicial complexes give rise topological invariants.   Given discrete dataset or a set of protein atoms, nontrivial topological information is generated by persistent homology. This introduces a filtration parameter to create a family of simplexes, which leads to a family of simplicial complexes, homology groups and associated topological invariants. By continuously varying the filtration parameter over an interval, the topological relationship among a given set of atoms is systematically reset, rendering a family of homology groups and corresponding topological invariants, which can be plotted as a persistence diagram, or a set of  barcodes. Both  persistence diagrams and barcodes  record the birth and death (appearance and cessation) of Betti numbers during the filtration process. Many simplicial complex definitions, which determine the rules of  the corresponding  topological relationship, have been proposed. Commonly used  definitions  include Vietoris-Rips (VR) complex, ech complex, and alpha complex. 

Persistent homology allows the extraction of topological invariants that are embedded in the high dimensional data space of biomolecules.  The resulting topological invariants over the filtration, i.e., persistence diagrams or persistence barcodes of different molecules can be compared using Bottleneck and Wasserstein distances. 

The goal of atom-specific persistent homology is to extract topological information of a given atom in a molecule. To embed local atomic information into a global topological description, we construct a pair of conjugated sets of point clouds, namely the original dataset and a datset excluding the atom of interest. The Bottleneck and Wasserstein distances between these two persistence diagrams reveal the desirable topological information of the given atom.    




  §.§.§ Simplex and  simplicial complex

A (geometric) simplex is a generalization of a triangle or tetrahedron to arbitrary dimensions. A k-simplex is a convex hull of k+1 vertices represented by a set of affinely independent points 
    σ={λ_0u_0+λ_1u_1+…+λ_ku_k|∑λ_i=1,λ_i≥0,i=0,1,…,k},

where {u_0,u_1,…,u_k}⊂ℝ^d with d≥ k is the set of points, σ is the k-simplex, and constraints on λ_i's ensure the formation of a convex hull. An affinely independent combination of points can have at most k+1 points in ℝ^k. For example a 1-simplex is a line segment, a 2-simplex a triangle, and a 3-simplex a tetrahedron. A subset of the k+1 vertices of a k simplex with m+1 vertices forms a convex hull in a lower dimension and is called an m-face of the k-simplex. An m-face is proper is m<k. The boundary of a k-simplex σ, is defined as the formal sum of its (k+1) faces. Given as 
    ∂_kσ=∑_i=0^k
    (-1)^i[u_0,…,û_i,…,u_k],
 where [u_0,…,û_i,…,u_k] denotes the convex hull formed by vertices of σ with the vertex u_i being excluded and ∂_k is called the boundary operator. A collection of finitely many simplicies forms a simplicial complex denoted by 𝒦. All simplicial complexes satisfy the following conditions. 

  * Faces of any simplex in 𝒦 are also simplices in 𝒦.

  * The intersection of any two simplicies σ_1,σ_2∈𝒦 is a face of both σ_1 and σ_2.




  §.§.§ Homology

Given a simplicial complex 𝒦, a k-chain c_k of 𝒦 is a formal sum of the k-simplices in 𝒦 
and is defined as c_k=∑ a_iσ_i where σ_i are the k-simplices and a_i's coefficients. Generally, a_i are element of a field such as ℝ, ℚ, or ℤ_n. Computationally, it is common to choose a_i to be in ℤ_2. The group of k-chains in 𝒦, denoted C_k, forms an Abelian group under addition in modulo two. This allows us to extend the definition of the boundary operator introduced in Eq. (<ref>) to chains.  


The boundary operator applied to a k-chain c_k is defined as 
    ∂_kc_k=∑ a_i∂_kσ_i,
 where σ_i's are k-simplices. The boundary operator is a map from 𝒞_k to 𝒞_k-1, which is also known as a boundary map for chains. Note that in ℤ_2, the boundary operator ∂_k satisfies the property that ∂_k∘∂_k+1σ = 0 for any (k + 1)-simplex σ following the fact that any (k-1)-face of σ is contained in exactly two k-faces of σ. The chain complex is defined as a sequence of chains connected by boundary maps with decreasing dimension and is denoted

    …→𝒞_n(𝒦 )𝒞_n-1(𝒦)…𝒞_0 (𝒦 )0.
 
The k-cycle group and k-boundary group are then defined as kernel and image of ∂_k and ∂_k+1 respectively, and 
    𝒵_k   =   ∂_k={c ∈𝒞_k|∂_kc = 0},
    ℬ_k   =   ∂_k+1={c∈𝒞_k | ∃ d∈𝒞_k+1: c= ∂_k+1d
    
    },

where 𝒵_k is the k-cycle group and ℬ_k is the k-boundary group. Since ∂_k∘∂_k+1= 0, we have ℬ_k⊂𝒵_k⊂𝒞_k. Then the k-homology group is defined to be the quotient group of the k-cycle group modulo the k-boundary group,
 
    ℋ_k=𝒵_k/ℬ_k
where ℋ_k is the k-homology group. The kth Betti number is defined to be rank of the k-homology group as β_k= rank(ℋ_k).



  §.§.§ Filtration  and persistence

For a simplicial complex 𝒦, we define a filtration of 𝒦 as a nested sequence of subcomplexes of 𝒦, 
    ∅⊆𝒦_0⊆𝒦_1…⊆𝒦_n=𝒦


In persistent homology, the nested sequence of subcomplexes usually depends on a filtration parameter. The persistence of a topological feature is denoted graphically by its life span with respect to filtration parameter. Subcomplexes corresponding to various filtration parameters offer the topological fingerprints over multiple scales. The k^th persistence Betti number  β^i,j_k is given by  the ranks of the k^th homology groups of 𝒦_i that are alive and are defined as 

    β^i,j_k=(ℋ^i,j_k)=(𝒵_k(𝒦_i)/(ℬ_k(𝒦_j)∩𝒵_k(𝒦_i))).


The persistence of Betti numbers over the filtration interval can be recorded in many different ways. The commonly used ones are  persistence barcodes and persistence diagrams. An example of barcodes is provided in Figure <ref>.





  §.§.§ Similarity and distance

In this work, we  use Bottleneck and Wasserstein distances to extract atom-specific topological information and facilitate atom-specific persistent homology.  Let X and Y be multisets of data points, 
the  Bottleneck   and Wasserstein distances of  X and Y are given by  <cit.>

    d_B(X,Y)=inf_γ∈ B(X,Y)sup_x∈ X|| x-γ(x)||_∞,

and <cit.>

    d_W^p(X,Y)=(inf_γ∈ B(X,Y)∑_x∈ X|| x-γ(x)||_∞^p)^1/p,

respectively. Here  B_(X,Y) is the collection of all bijections from X to Y. Note that in our work, topological invariants of different dimensions are compared separately.




  §.§.§ Vietoris-Rips complex

Given a metric space M and a cutoff distance d, a simplex is formed if all points have pairwise distances no greater than d. All such simplices form the Vietoris-Rips (VR) complex. The abstract nature of the VR complex allows the construction of simplicial complexes from a correlation function, which models the pairwise interaction of atoms using a radial basis function versus more standard distance metrics.  The R library TDA  is used to generate persistence barcodes <cit.> . 



  §.§.§ Atom-specific persistent homology and element-specific persistent homology


 Element-specific persistent homology was introduced to embed chemical and biology information into topological invariants <cit.>. Its essential idea is to construct topological representations from subsets of atoms in various element types in a protein. For example, if one selects all carbon atoms in a protein, the resulting persistence barcodes will represent the strength and network of hydrophobicity in the protein.  


In contrast, atom-specific persistent homology is designed to highlight the topological information of a given atom in a biomolecule. It creates two conjugated subsets of atoms centered around the atom of interest, one with and one without the specific atom.  Conjugated simplicial complexes, conjugated homology groups and conjugated topological invariants are generated for the conjugated sets of points clouds.  The difference between the conjugated topological invariants, measured by both Wasserstein and Bottleneck distances, offers a topological representation of the atom of interest.  As shown in Figure <ref>, atom-specific and element-specific conjugated point clouds can be constructed  for a given dataset. 

In this work, we focus on C_α B factor predictions. We use element specific persistent homology to enhance the topological representation of each  C_α neighborhood. Meanwhile, we develop atom-specific persistent homology to pinpoint the topological representation at each C_α atom.   With these selections of subsets, Vietoris-Rips complexes are constructed by contact maps or matrix filtration <cit.>. 

To capture element-specific interactions we consider three subsets of carbon-carbon, carbon-nitrogen, and carbon-oxygen point clouds. This gives us the following element specific pairs,

    𝒫={ CC,CN,CO}.

For a given Protein Data Bank (PDB) file, persistence barcodes are calculated as follows. Given a specific C_α of interest, say r_i^k∈𝒫_k in an element specific set 𝒫_k (𝒫_1= CC, 𝒫_2= CN, and  𝒫_3= CO) ,   a point cloud consisting of all atoms within a pre-defined cutoff radius r_c is selected:  

    ℛ^k_i= {𝐫^k_j| ||𝐫^k_i-𝐫^k_j||<r_c,    r_i^k,  r_j^k∈𝒫_k, ∀ j∈ 1,2,… N},

where N is the number of atoms in the kth element pair 𝒫_k. A conjugated set of point cloud,  ℛ̂^k_i, includes the same set of atoms, except for  r_i^k.    For a given pair of conjugated point clouds  ℛ^k_i and  ℛ̂^k_i, conjugated  simplicial complexes, conjugated homology groups,  and conjugated persistence barcodes  are computed via persistent homology.  We compute  Euclidean  distance based filtration using the Vietoris-Rips complex. Additionally, for a given set of atoms selected according to atom-specific and element specific constructions,    we generate a  family of multiresolution persistence barcodes by a resolution controlled filtration matrix:    <cit.>

    M_nm(ϑ )= 1-Φ(||𝐫_n-𝐫_m||;ϑ),

where ϑ denotes a set of kernel parameters. We have used  both exponential kernels  


    Φ(||𝐫_n-𝐫_m||; η, κ )=e^-(||𝐫_n-𝐫_m||/ η)^κ,    κ>0

and  Lorentz kernels  


    Φ(||𝐫_n-𝐫_m||;η, ν)=11+(||𝐫_n-𝐫_m||/η)^ν,     ν>0

where η κ, and ν are pre-defined constants. This filtration matrix is used in association with the Vietoris-Rips complex to generate persistence barcodes or persistence diagrams. Then these topological invariants are  compared using both Bottleneck and Wasserstein distances. An example of the conjugated persistence barcode pair generated for a C_α atom is illustrated in Figure <ref>.







 §.§ Machine learning models


Topological  features are used for prediction of protein B factor using both least squares fitting and machine learning as described in the following subsections.



  §.§.§ Gradient boosted trees

Gradient boosting is an ensemble method that uses a number of  “weak learners” to construct a prediction model in an iterative manner. The method is optimized via   gradient descent, which minimizes the residuals of  a loss function. At each step of the gradient boosting, gradient boosting trees (GBTs) incorporate decision trees to improve their predictive power. Ensemble methods like GBTs are useful because they can handle  a diverse feature set, have strong predictive power, and are typically robust to outliers and against overfitting.

In this work, we optimize the GBT hyper-parameters using the standard practice of a grid search. The parameters used for testing are provided in Table <ref>. Any hyper-parameters not listed in the table were taken to be the default values provided by the python scikit-learn package.

 




  §.§.§ Deep learning with a convolutional neural network

Neural networks are modeled after the function of neurons in brain. A neural network applies activation functions, called perceptrons, to  inputs. Weights of the network are trained to minimize a loss function over many epochs, or passes of an entire training dataset. When a neural network has several layers of perceptrons we call it a deep neural network (DNN) and the intermediate  layers are known as hidden layers.

Convolutional neural networks (CNNs) have recently had great success in image classification. Using convolutions of a pre-defined filter size and number of filters,  CNNs can automatically extract high-level features from input images. CNNs are advantageous because they can perform as well as other models without training as many parameters as a densely connected deep neural network. By applying several convolutions one can extract high-level features of an image. In this work we generate a image-like heat map by using a range of kernel parameters for atom-specific and element-specific  persistent homology. The CNN output is then flattened and fed as input to a DNN along with global and local protein features. This allows us to use the same feature set as the boosted gradient method as well as the generated PH image data. A diagram of the CNN architecture is provided in Figure <ref>.




For each C_α of the training set, the CNN is passed a three-channel persistent homology image of dimension (8,10,3). The model takes the input image data and applies two convolutional layers with 2x2 filters followed by a dropout of 0.5. The image data is passed through a dense layer, flattened, then joined with the other global and local features to form a dense layer of 218 neurons. This is followed by a dropout layer of 0.5, another dense layer of 100 neurons, a dropout layer of 0.25, a dense layer of 10 neurons, and finishes with a dense layer of  output. Figure <ref> provides an illustration   of the deep CNN used in this work.


The deep convolutional neural network has several hyper-parameters that can be tuned. As with the GBT, the deep convolutional neural network hyper-parameters are optimized using a basic grid search. Table <ref> provides the parameters used for testing. Any hyper-parameters that are not listed below were taken to be the default values provided by the python Keras package.

 




  §.§.§ Consensus method


 In this work, we combine the predictions of two machine learning models to construct a simple consensus model. The consensus prediction used in this work is generated by the average of  C_α B factor values predicted from the GBT and deep CNN models.



 §.§ Machine learning features


A variety of element-specific and atom-specific persistence barcodes were generated using the techniques discussed in Sec. <ref>. In this work, we include 60 topological features. These features are generated in several ways by varying: kernels (Lorentz and exponential), element-specific pairs (CC, CN, CO),   and  distance metrics (Wasserstein-0 and Wasserstein-1, Bottleneck-0 and Bottleneck-1). For this work all  persistent homology features were generated with the cutoff of 11Å.



  §.§.§   Wasserstein and Bottleneck metrics for modified persistence diagrams


The distances evaluated from  Wasserstein and Bottleneck evaluations of persistence diagrams depend on the boundary of the diagrams. Specifically, when two persistence diagrams are compared, the extra events on one diagram that do not match any events on the other diagram might contribute to the final distance by their distances from the boundary. For this reason,  
we create two additional persistence diagrams in which the y-axis is rotated clockwise by  30^∘ or 60^∘, respectively, see Figure <ref>.  This modification changes the Bottleneck and Wasserstein distances and allows the model to recognize elements that have a short persistence (i.e. have a short lifespan). Lastly, we modified the persistence diagram by  reflecting around the diagonal axis. An example of this modification is illustrated in Figure <ref>.  Table <ref> provides  a list of  kernels, kernel parameters, y-axis change,  distance metric, and element-specific pairs used to generate features in machine learning models.




Other features include global features  from PDB files, i.e., R-value, protein resolution, and number of heavy atoms. Additional local features include packing density, amino acid type, occupancy, and secondary structure information generated by STRIDE software <cit.>. 



  §.§.§ Image-like  persistent homology features


Using the process described in Section <ref> we generate 2D image-like  persistent homology   features, F^k_i={f^k_i(η,κ) }, for each C_α of the proteins in the dataset by using various values of η and κ in the kernel function. A cutoff of 11 Å with an exponential kernel and different values of η and κ are used to capture a wide variety of scales. In particular we use 
    η={1,2,3,4,5,10,15,20},
 and 
    κ={1,2,3,4,5,6,7,8,9,10}.
 
The image-like matrix is given by F_i^k in Eq. (<ref>), where each atom F_i^k represents the PH feature of the i^th C_α atom, and k^th atom interaction (C, N, or O). 



    
    . F_i^k = [                             1,1                             1,2                               …                             1,9                            1,10;                             2,1                             2,2                               …                             2,9                            2,10;                               ⋮                                                               ⋮;                            15,1                            15,2                               …                            15,9                           15,10; κ20,1   20,2   …   20,9   20,10 ]}η

 


This results in 2D PH images of dimension (8,10). Images are created for element-specific  C_α interactions with carbon, nitrogen, and oxygen atom giving each image three channels. This results in a final image dimension of (8,10,3) for each C_α atom.
























§ RESULTS





 §.§ Data sets

In this work, we use two data sets, one from Refs. <cit.> and the other from Park, Jernigan, and Wu  <cit.>. The first contains 364 proteins <cit.> and the second contains 3 subsets of small, medium, and large proteins  <cit.>. All sequences have a resolution of 3 Å or higher and an average resolution of 1.3  Å  and the sets include proteins that range from 4 to 3912 residues <cit.>.

For all testing, we exclude protein 1AGN due to known problems with this protein data <cit.>. Proteins 1NKO, 2OCT, and 3FVA are also excluded because these proteins have residues with B factors reported as zero, which is unphysical. For the machine learning results, proteins 1OB4, 1OB7, 2OLX, and 3MD5 are excluded because the STRIDE software is unable to provide secondary features for these proteins. The image like features used in all convolutional neural networks were standardized with mean 0 and variance of 1



 §.§ Evaluation metric
 

  We use the proposed methods to predict B factors of all C_α atoms present in a protein. Linear least square fitting was done using  only topological features. The machine learning models were executed using a leave-one-(protein)-out method to blindly predict the B factors of all C_α atoms in each protein. The machine learning models were trained using the data and features described in Sections <ref>, <ref>, <ref>. For comparison, we include previously existing C_α B factor prediction fitting methods.



To quantitatively assess our method for B factor prediction we use the Pearson correlation coefficient  given by 

    PCC=∑_i=1^N(B_i^e-B̅^e)(B_i^t-B̅^t)[∑_i=1^N(B_i^e-B̅^e)^2∑_i=1^N(B_i^t-B̅^t)^2]^1/2,

where B^t_i,i = 1, 2,…,N 
are  predicted B factors using the proposed method and B^e_i ,i = 1, 2,…,N experimental B factors from the PDB file. The terms B^t_i and B^e_i represent the i^th theoretical and experimental B factors respectively.
Here B̅^e and  B̅^t are averaged B factors.  





 §.§ Cutoff distance




In this work, the optimal cutoff of r_c =11Å is found over a grid search using various cutoff distances. Figure <ref> displays the average Pearson correlation coefficient, obtained via fitting, over an entire dataset of 364 protein using all persistent homology metrics with various point cloud distance cutoffs.

For each protein we use the parameters listed in Table <ref>. 
The values used in this work were determined using the standard practice of a grid search.






 §.§ Least squares fitting within proteins





The Pearson correlation coefficients using least squares fitting for C_α B factor prediction of small, medium, and large protein subsets are provided in Tables <ref>, <ref>, and <ref> respectively. Results for the all proteins in the dataset are provided in Table <ref>. The average Pearson correlation coefficients for small, medium, large, and superset data sets are provided in Table <ref>. Table <ref> includes fitting results using only Bottleneck, only Wasserstein, and using both Bottleneck and Wasserstein metrics. We also include results using only exponential kernel, only a Lorentz kernel, or both an exponential and Lorentz kernel for fitting. All results reported here PH features generated with a cutoff of 11Å and include three element-specific subsets (carbon-carbon, carbon-nitrogen, carbon-oxygen). Overall fitting methods using the various persistent homology features performed similarly. The best results came from using features generated by both exponential and Lorentz kernels and both Bottleneck and Wasserstein distances. Using both kernels and both distance metrics resulted in an average correlation coefficient of 0.73 for the superset.



 §.§ Blind machine learning prediction across proteins


The aforementioned  least  squares   fitting methods cannot predict the B factors of unknown proteins.  Machine learning methods enable us to blindly predict B factors across proteins. In this section, we utilize  both boosted gradient  and convolutional neural network  algorithms for the blind prediction of B factor across different proteins.  Taken together, the entire dataset contains  more  than  620  000  atoms.  We use a leave-one-protein out cross validation in our prediction. That is, for each protein, the data from a protein whose B factors will be predicted, is excluded from the training data. This gives rise to  a training  set  of  roughly  600  000  data  points  for each protein (i.e., atoms and associated B factors). The Pearson correlation coefficients using boosted gradient (GBT), convolutional neural network (CNN), and consensus method (CON) for C_α B factor prediction of small, medium, and large protein subsets are provided in Tables <ref>, <ref>, and <ref> respectively. Parameters for GBT and CNN methods can be found in Tables <ref> and <ref>. The global and local features used for training and testing are provided in Section <ref>. Results for all proteins are provided in Table <ref>. The average Pearson correlation coefficients for small, medium, large, and superset data sets are  provided in Table <ref>. All results reported here use a cutoff of 11Å and include three element-specific subsets (carbon-carbon, carbon-nitrogen, carbon-oxygen). Kernel parameters for both exponential and Lorentz kernels are provided in Table <ref>. Results from previously existing C_α B factor prediction methods are included for comparison in Table <ref>. Overall both GBT and CNN algorithms perform similarly. As expected, the CNN method outperforms the GBT with average correlation coefficients over the superset of 0.60 and 0.59, respectively. The consensus method improves upon both results with an average Pearson correlation coefficient of 0.61 over the superset. Table <ref> shows that the blind prediction machine learning models perform better than fitting models GNM and NMA and similar to the pFRI fitting model. 








§ CONCLUSION


An essential component of the paradigm of protein dynamics is the correlation between protein flexibility and protein function. The shear complexity and large number of degrees of freedom make quantitative understanding of flexibility and function an inherently difficult problem. Several time-independent methods for predicting protein B factors exist. Examples include NMA<cit.>, ENM <cit.>, GNM <cit.>, and FRI methods <cit.>. None of the methods above are able to blindly predict protein B factors of an unknown protein. We hypothesize that the intrinsic physics of proteins lie in a low-dimensional space embedded in a high-dimensional data space. Based on this hypothesis the authors previously introduced the graph theory based multiscale weighted colored graph (MWCG) <cit.>. The authors showed that MWCG's are able to  successfully blindly predict cross-protein B factors. 

In this work we explore this hypothesis further by creating a B factor predictor using tools from algebraic topology. In order to construct localized topological representations for individual atoms from global topological tools, we propose atom-specific topology    and atom-specific persistent homology. This approach creates two conjugated sets of atoms: the first set is centered around the given atom of interest while the other set is  identical but excludes the atom of interest. Element-specific selections are further implemented to embed biological information into atom-specific persistent homology.  
The distance between the topological invariants generated from these conjugated sets of atoms is used to represent the atom of interest. Both   Bottleneck and Wasserstein metrics are utilized to estimate the topological distances between conjugated barcodes. The Vietoris-Rips complex is employed for topological barcode generation. 
 

To test the proposed method we use over 300 proteins or more than 600,000 B factors. Atom-specific persistent homology features are generated using several element-specific interactions, kernel choices, parametrizations, and barcode distance metrics. First we employ topological features to fit protein B factors using linear least squares. Using topological features our fitting model outperformed previous fitting models with an average Pearson correlation coefficient of 0.73 over the superset of proteins. Next we considered using the topological features to blindly predict protein B factors of C_α atoms. We generated two machine learning models, a gradient boosted tree (GBT) and deep convolutional neural network (CNN). Additionally we averaged the C_α prediction from the two models to generate a more robust consensus model. A variety of local and global features were included in addition to the generated topological features. Our blind prediction consensus model outperformed both GNM and NMA fitting models and produced similar results to the pFRI fitting model.

To the authors' knowledge, this work is the first time persistent homology has been used to predict the B factor of atoms in a protein. This approach is novel because topology is a global property and on its own cannot be used to describe  local atomic information. Our unique approach allows us to create local topological representation  with a variety of customizable parameters using a global mathematical tool. This allows the model to account for multiple spatial interaction scales and element specific interactions. Our results demonstrate that this is a accurate and robust approach. Moreover, the results could easily be improved by including a larger dataset, fine tuning parameters, and exploring different machine learning approaches.

This method can be applied to a variety of interesting applications related to protein dynamics. Examples include  allosteric site detection, computer-aided drug design, hinge detection, hot spot identification,   and protein folding stability changes upon mutation. More generally this method may be amenable to problems outside proteins such as network dynamics and social network centrality measure.



§ ACKNOWLEDGMENT
 

This work was supported in part by  NSF Grants DMS-1721024 and DMS-1761320, and NIH grant  GM126189.  




§ APPENDIX

1010









[H]llccc|llccc
Pearson correlation coefficients for cross protein C_α atom blind B factor prediction obtained by boosted gradient (GBT), convolutional neural network (CNN), and consensus method (CON) for the Superset.
   

1cPDB ID     1cN     1cGBT     1cCNN      1c|CON    
1cPDB ID     1cN     1cGBT     1cCNN     1cCON 

 

10c
  – continued from previous page 
 1cPDB ID     1cN     1cGBT     1cCNN     1c|CON    
1cPDB ID     1cN     1cGBT     1cCNN     1cCON 

 

3|r|Continued on next page 
 





1ABA   	87   	0.73   	0.71   	0.74   	2X5Y   	185   	0.76   	0.68   	0.76

1AHO   	66   	0.66   	0.66   	0.7   	2X9Z   	266   	0.49   	0.52   	0.52

1AIE   	31   	0.75   	0.7   	0.78   	2XHF   	310   	0.58   	0.57   	0.58

1AKG   	16   	0.27   	0.32   	0.29   	2Y0T   	111   	0.71   	0.71   	0.74

1ATG   	231   	0.55   	0.51   	0.55   	2Y72   	183   	0.65   	0.71   	0.69

1BGF   	124   	0.61   	0.58   	0.62   	2Y7L   	323   	0.66   	0.66   	0.68

1BX7   	51   	0.74   	0.74   	0.76   	2Y9F   	149   	0.74   	0.75   	0.76

1BYI   	238   	0.61   	0.5   	0.6   	2YLB   	418   	0.67   	0.66   	0.7

1CCR   	109   	0.55   	0.6   	0.59   	2YNY   	326   	0.65   	0.71   	0.69

1CYO   	88   	0.64   	0.7   	0.68   	2ZCM   	348   	0.33   	0.38   	0.36

1DF4   	57   	0.85   	0.85   	0.88   	2ZU1   	360   	0.66   	0.66   	0.68

1E5K   	188   	0.74   	0.72   	0.74   	3A0M   	146   	0.53   	0.6   	0.59

1ES5   	260   	0.65   	0.62   	0.66   	3A7L   	128   	0.44   	0.61   	0.53

1ETL   	12   	0.37   	0.82   	0.55   	3AMC   	614   	0.68   	0.64   	0.69

1ETM   	12   	0.37   	0.63   	0.43   	3AUB   	124   	0.5   	0.5   	0.55

1ETN   	12   	0.07   	0.48   	0.13   	3B5O   	249   	0.49   	0.55   	0.52

1EW4   	106   	0.59   	0.6   	0.61   	3BA1   	312   	0.62   	0.59   	0.63

1F8R   	1932   	0.52   	0.54   	0.54   	3BED   	262   	0.45   	0.53   	0.5

1FF4   	65   	0.61   	0.66   	0.64   	3BQX   	136   	0.56   	0.55   	0.58

1FK5   	93   	0.59   	0.6   	0.61   	3BZQ   	99   	0.45   	0.53   	0.49

1GCO   	1044   	0.47   	0.47   	0.5   	3BZZ   	103   	0.38   	0.51   	0.44

1GK7   	39   	0.77   	0.9   	0.82   	3DRF   	567   	0.51   	0.45   	0.52

1GVD   	56   	0.71   	0.55   	0.69   	3DWV   	359   	0.63   	0.55   	0.63

1GXU   	89   	0.67   	0.68   	0.69   	3E5T   	268   	0.44   	0.48   	0.46

1H6V   	2927   	0.26   	0.34   	0.34   	3E7R   	40   	0.72   	0.66   	0.77

1HJE   	13   	0.84   	0.75   	0.9   	3EUR   	150   	0.36   	0.42   	0.38

1I71   	83   	0.53   	0.58   	0.56   	3F2Z   	148   	0.73   	0.76   	0.75

1IDP   	441   	0.62   	0.6   	0.63   	3F7E   	261   	0.65   	0.69   	0.68

1IFR   	113   	0.7   	0.64   	0.7   	3FCN   	185   	0.63   	0.65   	0.66

1K8U   	87   	0.57   	0.6   	0.59   	3FE7   	89   	0.52   	0.55   	0.54

1KMM   	1499   	0.64   	0.51   	0.63   	3FKE   	250   	0.51   	0.51   	0.54

1KNG   	144   	0.5   	0.52   	0.51   	3FMY   	75   	0.65   	0.67   	0.68

1KR4   	107   	0.56   	0.71   	0.63   	3FOD   	48   	0.45   	0.57   	0.54

1KYC   	15   	0.62   	0.69   	0.66   	3FSO   	238   	0.72   	0.75   	0.74

1LR7   	73   	0.62   	0.61   	0.64   	3FTD   	257   	0.64   	0.68   	0.67

1MF7   	194   	0.65   	0.66   	0.67   	3G1S   	418   	0.6   	0.57   	0.61

1N7E   	95   	0.63   	0.58   	0.65   	3GBW   	170   	0.74   	0.74   	0.75

1NKD   	59   	0.7   	0.7   	0.72   	3GHJ   	129   	0.58   	0.56   	0.59

1NLS   	238   	0.55   	0.57   	0.57   	3HFO   	216   	0.51   	0.57   	0.54

1NNX   	93   	0.78   	0.79   	0.8   	3HHP   	1314   	0.61   	0.65   	0.65

1NOA   	113   	0.55   	0.53   	0.56   	3HNY   	170   	0.61   	0.6   	0.62

1NOT   	13   	0.69   	0.96   	0.8   	3HP4   	201   	0.56   	0.58   	0.58

1O06   	22   	0.94   	0.93   	0.95   	3HWU   	155   	0.58   	0.65   	0.62

1O08   	221   	0.49   	0.47   	0.49   	3HYD   	8   	0.99   	0.74   	0.99

1OPD   	85   	0.42   	0.34   	0.41   	3HZ8   	200   	0.45   	0.54   	0.48

1P9I   	29   	0.73   	0.73   	0.74   	3I2V   	127   	0.44   	0.52   	0.48

1PEF   	18   	0.79   	0.82   	0.82   	3I2Z   	140   	0.6   	0.6   	0.6

1PEN   	16   	0.36   	0.74   	0.44   	3I4O   	154   	0.62   	0.72   	0.66

1PMY   	123   	0.59   	0.7   	0.65   	3I7M   	145   	0.44   	0.57   	0.49

1PZ4   	113   	0.72   	0.8   	0.77   	3IHS   	173   	0.61   	0.62   	0.64

1Q9B   	44   	0.59   	0.85   	0.67   	3IVV   	168   	0.83   	0.82   	0.84

1QAU   	112   	0.51   	0.59   	0.57   	3K6Y   	227   	0.56   	0.57   	0.58

1QKI   	3912   	0.34   	0.45   	0.38   	3KBE   	166   	0.56   	0.64   	0.6

1QTO   	122   	0.53   	0.48   	0.54   	3KGK   	190   	0.76   	0.8   	0.78

1R29   	122   	0.56   	0.59   	0.59   	3KZD   	94   	0.55   	0.67   	0.6

1R7J   	90   	0.71   	0.77   	0.75   	3L41   	219   	0.61   	0.64   	0.64

1RJU   	36   	0.6   	0.46   	0.58   	3LAA   	176   	0.35   	0.49   	0.42

1RRO   	108   	0.4   	0.45   	0.43   	3LAX   	118   	0.74   	0.69   	0.74

1SAU   	123   	0.54   	0.66   	0.59   	3LG3   	846   	0.45   	0.51   	0.5

1TGR   	111   	0.66   	0.69   	0.69   	3LJI   	270   	0.57   	0.55   	0.58

1TZV   	157   	0.74   	0.77   	0.76   	3M3P   	244   	0.53   	0.59   	0.57

1U06   	55   	0.44   	0.4   	0.45   	3M8J   	178   	0.72   	0.71   	0.74

1U7I   	259   	0.71   	0.74   	0.74   	3M9J   	250   	0.56   	0.52   	0.56

1U9C   	220   	0.57   	0.59   	0.59   	3M9Q   	190   	0.4   	0.48   	0.45

1UHA   	82   	0.71   	0.74   	0.73   	3MAB   	180   	0.63   	0.63   	0.65

1UKU   	102   	0.75   	0.76   	0.77   	3MD4   	13   	0.88   	0.96   	0.96

1ULR   	87   	0.54   	0.53   	0.56   	3MEA   	170   	0.62   	0.63   	0.63

1UOY   	64   	0.72   	0.7   	0.76   	3MGN   	277   	0.08   	0.09   	0.09

1USE   	47   	0.05   	0.32   	0.12   	3MRE   	446   	0.54   	0.54   	0.57

1USM   	77   	0.73   	0.72   	0.75   	3N11   	325   	0.51   	0.47   	0.52

1UTG   	70   	0.62   	0.64   	0.66   	3NE0   	208   	0.67   	0.73   	0.71

1V05   	96   	0.6   	0.64   	0.63   	3NGG   	97   	0.72   	0.75   	0.75

1V70   	105   	0.63   	0.62   	0.64   	3NPV   	500   	0.51   	0.5   	0.54

1VRZ   	13   	0.54   	0.34   	0.54   	3NVG   	6   	0.51   	0.63   	0.71

1W2L   	97   	0.43   	0.5   	0.47   	3NZL   	70   	0.56   	0.58   	0.57

1WBE   	206   	0.6   	0.56   	0.6   	3O0P   	197   	0.68   	0.72   	0.71

1WHI   	122   	0.59   	0.56   	0.6   	3O5P   	147   	0.6   	0.59   	0.61

1WLY   	322   	0.64   	0.62   	0.66   	3OBQ   	150   	0.59   	0.57   	0.59

1WPA   	107   	0.65   	0.65   	0.67   	3OQY   	236   	0.66   	0.59   	0.66

1X3O   	80   	0.41   	0.43   	0.44   	3P6J   	145   	0.66   	0.72   	0.69

1XY1   	16   	0.82   	0.75   	0.83   	3PD7   	216   	0.68   	0.7   	0.71

1XY2   	8   	0.79   	0.82   	0.81   	3PES   	166   	0.56   	0.54   	0.57

1Y6X   	86   	0.5   	0.46   	0.51   	3PID   	387   	0.48   	0.3   	0.45

1YJO   	6   	0.7   	-0.06   	0.57   	3PIW   	161   	0.72   	0.77   	0.75

1YZM   	46   	0.69   	0.64   	0.7   	3PKV   	229   	0.52   	0.51   	0.53

1Z21   	96   	0.68   	0.65   	0.69   	3PSM   	94   	0.8   	0.77   	0.82

1ZCE   	139   	0.7   	0.74   	0.73   	3PTL   	289   	0.53   	0.55   	0.55

1ZVA   	75   	0.7   	0.7   	0.71   	3PVE   	363   	0.61   	0.61   	0.63

2A50   	469   	0.6   	0.54   	0.6   	3PZ9   	357   	0.61   	0.58   	0.63

2AGK   	233   	0.67   	0.63   	0.67   	3PZZ   	12   	0.94   	0.85   	0.93

2AH1   	939   	0.48   	0.55   	0.54   	3Q2X   	6   	0.95   	0.72   	0.93

2B0A   	191   	0.62   	0.59   	0.63   	3Q6L   	131   	0.47   	0.53   	0.52

2BCM   	415   	0.5   	0.51   	0.52   	3QDS   	284   	0.62   	0.62   	0.63

2BF9   	35   	0.48   	0.79   	0.58   	3QPA   	212   	0.55   	0.67   	0.59

2BRF   	103   	0.72   	0.77   	0.75   	3R6D   	222   	0.65   	0.74   	0.69

2C71   	225   	0.57   	0.6   	0.6   	3R87   	148   	0.47   	0.45   	0.48

2CE0   	109   	0.6   	0.66   	0.64   	3RQ9   	165   	0.46   	0.4   	0.46

2CG7   	110   	0.3   	0.32   	0.32   	3RY0   	128   	0.41   	0.49   	0.46

2COV   	534   	0.74   	0.72   	0.75   	3RZY   	151   	0.65   	0.62   	0.66

2CWS   	235   	0.61   	0.47   	0.6   	3S0A   	132   	0.53   	0.49   	0.54

2D5W   	1214   	0.54   	0.64   	0.59   	3SD2   	100   	0.56   	0.56   	0.57

2DKO   	253   	0.78   	0.78   	0.8   	3SEB   	238   	0.63   	0.6   	0.63

2DPL   	565   	0.41   	0.36   	0.42   	3SED   	126   	0.53   	0.52   	0.55

2DSX   	52   	0.34   	0.34   	0.36   	3SO6   	157   	0.65   	0.65   	0.66

2OCT   	439   	0.64   	0.67   	0.67   	3SR3   	657   	0.5   	0.46   	0.5

2E3H   	81   	0.65   	0.68   	0.67   	3SUK   	254   	0.58   	0.59   	0.6

2EAQ   	89   	0.57   	0.63   	0.61   	3SZH   	753   	0.69   	0.67   	0.71

2EHP   	246   	0.66   	0.62   	0.67   	3T0H   	209   	0.71   	0.7   	0.73

2EHS   	75   	0.62   	0.67   	0.65   	3T3K   	122   	0.76   	0.76   	0.78

2ERW   	53   	0.12   	0.24   	0.16   	3T47   	145   	0.51   	0.62   	0.57

2ETX   	390   	0.49   	0.48   	0.51   	3TDN   	359   	0.47   	0.49   	0.49

2FB6   	129   	0.73   	0.75   	0.75   	3TOW   	155   	0.61   	0.63   	0.63

2FG1   	176   	0.57   	0.61   	0.59   	3TUA   	226   	0.62   	0.56   	0.63

2FN9   	560   	0.57   	0.54   	0.58   	3TYS   	78   	0.66   	0.74   	0.72

2FQ3   	85   	0.77   	0.82   	0.81   	3U6G   	276   	0.53   	0.46   	0.52

2G69   	99   	0.62   	0.5   	0.6   	3U97   	85   	0.67   	0.72   	0.71

2G7O   	68   	0.72   	0.86   	0.8   	3UCI   	72   	0.42   	0.42   	0.43

2G7S   	206   	0.55   	0.58   	0.58   	3UR8   	637   	0.64   	0.6   	0.64

2GKG   	150   	0.56   	0.64   	0.59   	3US6   	159   	0.61   	0.63   	0.64

2GOM   	121   	0.69   	0.59   	0.69   	3V1A   	59   	0.57   	0.27   	0.55

2GXG   	140   	0.65   	0.67   	0.68   	3V75   	294   	0.49   	0.56   	0.53

2GZQ   	203   	0.34   	0.4   	0.37   	3VN0   	193   	0.85   	0.85   	0.86

2HQK   	232   	0.77   	0.77   	0.78   	3VOR   	219   	0.47   	0.48   	0.48

2HYK   	237   	0.65   	0.63   	0.65   	3VUB   	101   	0.59   	0.55   	0.59

2I24   	113   	0.44   	0.46   	0.46   	3VVV   	112   	0.56   	0.57   	0.57

2I49   	399   	0.65   	0.61   	0.66   	3VZ9   	163   	0.72   	0.64   	0.72

2IBL   	108   	0.65   	0.66   	0.67   	3W4Q   	826   	0.65   	0.6   	0.66

2IGD   	61   	0.57   	0.56   	0.58   	3ZBD   	213   	0.55   	0.49   	0.55

2IMF   	203   	0.53   	0.58   	0.56   	3ZIT   	157   	0.52   	0.42   	0.5

2IP6   	87   	0.6   	0.66   	0.63   	3ZRX   	241   	0.54   	0.6   	0.58

2IVY   	89   	0.51   	0.45   	0.51   	3ZSL   	165   	0.49   	0.57   	0.53

2J32   	244   	0.75   	0.79   	0.79   	3ZZP   	74   	0.38   	0.48   	0.42

2J9W   	203   	0.64   	0.58   	0.64   	3ZZY   	226   	0.65   	0.65   	0.68

2JKU   	38   	0.57   	0.71   	0.66   	4A02   	169   	0.59   	0.65   	0.62

2JLI   	112   	0.62   	0.68   	0.65   	4ACJ   	182   	0.62   	0.66   	0.64

2JLJ   	121   	0.71   	0.71   	0.74   	4AE7   	189   	0.65   	0.7   	0.68

2MCM   	112   	0.71   	0.77   	0.75   	4AM1   	359   	0.54   	0.52   	0.55

2NLS   	36   	0.23   	0.47   	0.29   	4ANN   	210   	0.44   	0.43   	0.45

2NR7   	193   	0.78   	0.76   	0.79   	4AVR   	189   	0.56   	0.53   	0.56

2NUH   	104   	0.72   	0.56   	0.7   	4AXY   	56   	0.59   	0.65   	0.62

2O6X   	309   	0.76   	0.76   	0.78   	4B6G   	559   	0.69   	0.68   	0.71

2OA2   	140   	0.54   	0.55   	0.56   	4B9G   	292   	0.74   	0.74   	0.76

2OHW   	257   	0.56   	0.46   	0.54   	4DD5   	412   	0.61   	0.62   	0.63

2OKT   	377   	0.42   	0.42   	0.43   	4DKN   	423   	0.66   	0.64   	0.68

2OL9   	6   	0.94   	0.85   	0.94   	4DND   	93   	0.62   	0.67   	0.65

2PKT   	93   	0.01   	-0.04   	-0.01   	4DPZ   	113   	0.7   	0.74   	0.72

2PLT   	98   	0.52   	0.53   	0.54   	4DQ7   	338   	0.55   	0.6   	0.57

2PMR   	83   	0.6   	0.63   	0.63   	4DT4   	170   	0.67   	0.69   	0.69

2POF   	428   	0.62   	0.6   	0.66   	4EK3   	313   	0.6   	0.58   	0.61

2PPN   	122   	0.64   	0.54   	0.63   	4ERY   	318   	0.57   	0.59   	0.59

2PSF   	608   	0.42   	0.42   	0.43   	4ES1   	96   	0.69   	0.69   	0.71

2PTH   	193   	0.69   	0.7   	0.71   	4EUG   	225   	0.56   	0.55   	0.58

2Q4N   	1208   	0.44   	0.43   	0.45   	4F01   	459   	0.35   	0.26   	0.33

2Q52   	3296   	0.55   	0.28   	0.52   	4F3J   	143   	0.58   	0.63   	0.62

2QJL   	107   	0.54   	0.57   	0.56   	4FR9   	145   	0.6   	0.56   	0.61

2R16   	185   	0.44   	0.49   	0.46   	4G14   	5   	-0.28   	0.45   	0.04

2R6Q   	149   	0.63   	0.62   	0.65   	4G2E   	155   	0.75   	0.72   	0.76

2RB8   	93   	0.67   	0.7   	0.7   	4G5X   	584   	0.71   	0.73   	0.74

2RE2   	249   	0.65   	0.66   	0.68   	4G6C   	676   	0.56   	0.54   	0.58

2RFR   	166   	0.61   	0.69   	0.66   	4G7X   	216   	0.45   	0.4   	0.45

2V9V   	149   	0.53   	0.52   	0.54   	4GA2   	183   	0.61   	0.53   	0.61

2VE8   	515   	0.55   	0.55   	0.58   	4GMQ   	94   	0.76   	0.67   	0.76

2VH7   	94   	0.75   	0.56   	0.73   	4GS3   	90   	0.61   	0.56   	0.61

2VIM   	114   	0.44   	0.47   	0.47   	4H4J   	278   	0.75   	0.74   	0.77

2VPA   	217   	0.66   	0.75   	0.71   	4H89   	175   	0.53   	0.58   	0.56

2VQ4   	106   	0.7   	0.75   	0.72   	4HDE   	167   	0.66   	0.72   	0.7

2VY8   	162   	0.77   	0.68   	0.76   	4HJP   	308   	0.68   	0.6   	0.67

2VYO   	207   	0.6   	0.63   	0.63   	4HWM   	129   	0.54   	0.6   	0.57

2W1V   	551   	0.64   	0.69   	0.66   	4IL7   	99   	0.55   	0.55   	0.56

2W2A   	350   	0.59   	0.6   	0.61   	4J11   	377   	0.58   	0.49   	0.58

2W6A   	139   	0.71   	0.69   	0.72   	4J5O   	268   	0.67   	0.68   	0.69

2WJ5   	110   	0.45   	0.53   	0.48   	4J5Q   	162   	0.72   	0.74   	0.74

2WUJ   	103   	0.35   	0.54   	0.45   	4J78   	305   	0.63   	0.6   	0.64

2WW7   	161   	0.36   	0.35   	0.37   	4JG2   	202   	0.72   	0.72   	0.73

2WWE   	120   	0.49   	0.55   	0.53   	4JVU   	207   	0.7   	0.7   	0.72

2X1Q   	240   	0.44   	0.5   	0.47   	4JYP   	550   	0.59   	0.67   	0.65

2X25   	167   	0.5   	0.57   	0.55   	4KEF   	145   	0.48   	0.53   	0.51

2X3M   	175   	0.64   	0.65   	0.65   	5CYT   	103   	0.39   	0.34   	0.39

       	      		   		   		   	6RXN   	45   		0.59   	0.6   	0.61




																							
																









*11c

Pearson correlation coefficients of least squares fitting C_α B factor prediction of all proteins using 11Å cutoff. Two Bottleneck (B) and Wasserstein (W) metrics using various kernel choices are included.  

1c    3cB & W    3cB    3cW
 PDB ID       N   	Exp	   	Lor	   	Both	   	Exp	   	Lor	   	Both	   	Exp	   	Lor	   	Both
 11c
  – continued from previous page 
 1c    3cB & W    3cB    3cW
 PDB ID       N   	Exp	   	Lor	   	Both	   	Exp	   	Lor	   	Both	   	Exp	   	Lor	   	Both
 

1ABA	   	87	   	0.67	   	0.67	   	0.76	   	0.54	   	0.62	   	0.68	   	0.56	   	0.63	   	0.70	

1AHO	   	66	   	0.75	   	0.78	   	0.88	   	0.72	   	0.73	   	0.79	   	0.53	   	0.65	   	0.75	

1AIE	   	31	   	0.97	   	0.88	   	0.99	   	0.78	   	0.64	   	0.90	   	0.90	   	0.77	   	0.96	

1AKG	   	16	   	0.82	   	0.66	   	1.00	   	0.60	   	0.53	   	0.72	   	0.53	   	0.56	   	0.87	

1ATG	   	231	   	0.50	   	0.50	   	0.61	   	0.45	   	0.47	   	0.53	   	0.38	   	0.48	   	0.51	

1BGF	   	124	   	0.75	   	0.70	   	0.82	   	0.64	   	0.54	   	0.75	   	0.68	   	0.61	   	0.75	

1BX7	   	51	   	0.86	   	0.74	   	0.89	   	0.79	   	0.68	   	0.82	   	0.81	   	0.69	   	0.82	

1BYI	   	238	   	0.50	   	0.51	   	0.58	   	0.41	   	0.46	   	0.49	   	0.44	   	0.48	   	0.54	

1CCR	   	109	   	0.65	   	0.66	   	0.71	   	0.53	   	0.56	   	0.65	   	0.43	   	0.58	   	0.63	

1CYO	   	88	   	0.71	   	0.69	   	0.78	   	0.66	   	0.58	   	0.68	   	0.65	   	0.59	   	0.67	

1DF4	   	57	   	0.93	   	0.92	   	0.97	   	0.92	   	0.89	   	0.95	   	0.88	   	0.91	   	0.94	

1E5K	   	188	   	0.67	   	0.68	   	0.74	   	0.66	   	0.67	   	0.68	   	0.63	   	0.67	   	0.69	

1ES5	   	260	   	0.58	   	0.57	   	0.65	   	0.51	   	0.55	   	0.58	   	0.44	   	0.56	   	0.60	

1ETL	   	12	   	1.00	   	1.00	   	1.00	   	0.68	   	0.87	   	1.00	   	0.95	   	0.98	   	1.00	

1ETM	   	12	   	1.00	   	1.00	   	1.00	   	0.45	   	0.74	   	0.86	   	0.70	   	0.83	   	1.00	

1ETN	   	12	   	1.00	   	1.00	   	1.00	   	0.96	   	0.92	   	0.99	   	0.70	   	0.92	   	1.00	

1EW4	   	106	   	0.58	   	0.60	   	0.73	   	0.52	   	0.51	   	0.55	   	0.55	   	0.55	   	0.62	

1F8R	   	1932	   	0.61	   	0.63	   	0.70	   	0.59	   	0.62	   	0.63	   	0.50	   	0.62	   	0.65	

1FF4	   	65	   	0.77	   	0.72	   	0.80	   	0.70	   	0.65	   	0.75	   	0.68	   	0.68	   	0.76	

1FK5	   	93	   	0.53	   	0.59	   	0.71	   	0.49	   	0.50	   	0.58	   	0.49	   	0.50	   	0.55	

1GCO	   	1044	   	0.63	   	0.64	   	0.66	   	0.59	   	0.63	   	0.63	   	0.53	   	0.63	   	0.65	

1GK7	   	39	   	0.95	   	0.94	   	0.98	   	0.91	   	0.93	   	0.95	   	0.88	   	0.92	   	0.94	

1GVD	   	56	   	0.75	   	0.68	   	0.84	   	0.67	   	0.63	   	0.69	   	0.61	   	0.62	   	0.66	

1GXU	   	89	   	0.75	   	0.78	   	0.82	   	0.72	   	0.61	   	0.75	   	0.69	   	0.72	   	0.77	

1H6V	   	2927	   	0.29	   	0.31	   	0.33	   	0.28	   	0.29	   	0.30	   	0.23	   	0.29	   	0.30	

1HJE	   	13	   	1.00	   	1.00	   	1.00	   	0.72	   	0.79	   	1.00	   	0.67	   	0.57	   	1.00	

1I71	   	83	   	0.44	   	0.66	   	0.76	   	0.41	   	0.46	   	0.56	   	0.38	   	0.58	   	0.59	

1IDP	   	441	   	0.48	   	0.47	   	0.55	   	0.43	   	0.45	   	0.47	   	0.39	   	0.46	   	0.48	

1IFR	   	113	   	0.65	   	0.59	   	0.73	   	0.56	   	0.54	   	0.65	   	0.47	   	0.53	   	0.62	

1K8U	   	87	   	0.72	   	0.74	   	0.85	   	0.67	   	0.64	   	0.71	   	0.65	   	0.67	   	0.75	

1KMM	   	1499	   	0.57	   	0.54	   	0.59	   	0.49	   	0.53	   	0.54	   	0.36	   	0.53	   	0.57	

1KNG	   	144	   	0.52	   	0.51	   	0.61	   	0.43	   	0.47	   	0.51	   	0.43	   	0.50	   	0.53	

1KR4	   	107	   	0.57	   	0.48	   	0.60	   	0.39	   	0.47	   	0.53	   	0.45	   	0.45	   	0.54	

1KYC	   	15	   	0.96	   	0.99	   	1.00	   	0.92	   	0.93	   	0.99	   	0.88	   	0.88	   	1.00	

1LR7	   	73	   	0.61	   	0.62	   	0.71	   	0.57	   	0.55	   	0.63	   	0.46	   	0.56	   	0.58	

1MF7	   	194	   	0.56	   	0.59	   	0.67	   	0.55	   	0.57	   	0.59	   	0.50	   	0.58	   	0.59	

1N7E	   	95	   	0.67	   	0.71	   	0.80	   	0.54	   	0.68	   	0.72	   	0.54	   	0.63	   	0.73	

1NKD	   	59	   	0.73	   	0.69	   	0.89	   	0.56	   	0.58	   	0.63	   	0.55	   	0.65	   	0.75	

1NLS	   	238	   	0.81	   	0.78	   	0.86	   	0.75	   	0.65	   	0.83	   	0.80	   	0.72	   	0.82	

1NNX	   	93	   	0.84	   	0.84	   	0.88	   	0.81	   	0.79	   	0.83	   	0.81	   	0.81	   	0.86	

1NOA	   	113	   	0.63	   	0.65	   	0.72	   	0.60	   	0.57	   	0.63	   	0.53	   	0.57	   	0.59	

1NOT	   	13	   	1.00	   	1.00	   	1.00	   	0.82	   	0.86	   	1.00	   	0.86	   	0.81	   	1.00	

1O06	   	22	   	0.98	   	0.97	   	1.00	   	0.96	   	0.92	   	0.97	   	0.97	   	0.94	   	0.98	

1O08	   	221	   	0.46	   	0.48	   	0.56	   	0.44	   	0.42	   	0.50	   	0.37	   	0.45	   	0.48	

1OB4	   	5	   	1.00	   	1.00	   	1.00	   	1.00	   	1.00	   	1.00	   	1.00	   	1.00	   	1.00	

1OB7	   	5	   	1.00	   	1.00	   	1.00	   	1.00	   	1.00	   	1.00	   	1.00	   	1.00	   	1.00	

1OPD	   	85	   	0.35	   	0.29	   	0.57	   	0.25	   	0.21	   	0.36	   	0.29	   	0.19	   	0.36	

1P9I	   	29	   	0.89	   	0.88	   	0.98	   	0.87	   	0.82	   	0.92	   	0.87	   	0.84	   	0.89	

1PEF	   	18	   	0.96	   	0.97	   	1.00	   	0.88	   	0.94	   	0.96	   	0.92	   	0.94	   	0.96	

1PEN	   	16	   	0.96	   	0.90	   	1.00	   	0.60	   	0.67	   	0.83	   	0.47	   	0.73	   	0.94	

1PMY	   	123	   	0.71	   	0.70	   	0.76	   	0.62	   	0.59	   	0.67	   	0.68	   	0.69	   	0.71	

1PZ4	   	113	   	0.88	   	0.82	   	0.93	   	0.86	   	0.74	   	0.89	   	0.85	   	0.76	   	0.88	

1Q9B	   	44	   	0.79	   	0.76	   	0.94	   	0.58	   	0.59	   	0.69	   	0.69	   	0.57	   	0.71	

1QAU	   	112	   	0.59	   	0.61	   	0.66	   	0.57	   	0.55	   	0.58	   	0.55	   	0.57	   	0.58	

1QKI	   	3912	   	0.38	   	0.42	   	0.45	   	0.34	   	0.38	   	0.41	   	0.32	   	0.38	   	0.40	

1QTO	   	122	   	0.59	   	0.59	   	0.65	   	0.48	   	0.46	   	0.53	   	0.55	   	0.52	   	0.56	

1R29	   	122	   	0.71	   	0.56	   	0.76	   	0.55	   	0.35	   	0.69	   	0.69	   	0.43	   	0.72	

1R7J	   	90	   	0.88	   	0.86	   	0.91	   	0.83	   	0.76	   	0.87	   	0.81	   	0.79	   	0.86	

1RJU	   	36	   	0.81	   	0.74	   	0.91	   	0.75	   	0.69	   	0.81	   	0.62	   	0.65	   	0.72	

1RRO	   	108	   	0.39	   	0.35	   	0.56	   	0.31	   	0.23	   	0.45	   	0.33	   	0.19	   	0.45	

1SAU	   	123	   	0.76	   	0.75	   	0.81	   	0.70	   	0.73	   	0.75	   	0.68	   	0.74	   	0.76	

1TGR	   	111	   	0.77	   	0.76	   	0.83	   	0.72	   	0.70	   	0.74	   	0.74	   	0.73	   	0.75	

1TZV	   	157	   	0.76	   	0.78	   	0.83	   	0.73	   	0.71	   	0.77	   	0.69	   	0.70	   	0.74	

1U06	   	55	   	0.50	   	0.52	   	0.72	   	0.37	   	0.36	   	0.52	   	0.46	   	0.39	   	0.55	

1U7I	   	259	   	0.71	   	0.71	   	0.73	   	0.62	   	0.68	   	0.70	   	0.53	   	0.67	   	0.71	

1U9C	   	220	   	0.66	   	0.65	   	0.74	   	0.61	   	0.57	   	0.64	   	0.61	   	0.60	   	0.67	

1UHA	   	82	   	0.70	   	0.75	   	0.82	   	0.69	   	0.68	   	0.74	   	0.67	   	0.69	   	0.73	

1UKU	   	102	   	0.80	   	0.81	   	0.84	   	0.78	   	0.80	   	0.80	   	0.74	   	0.80	   	0.80	

1ULR	   	87	   	0.56	   	0.53	   	0.68	   	0.49	   	0.50	   	0.59	   	0.44	   	0.50	   	0.61	

1UOY	   	64	   	0.73	   	0.72	   	0.83	   	0.65	   	0.66	   	0.69	   	0.65	   	0.69	   	0.73	

1USE	   	47	   	0.66	   	0.75	   	0.91	   	0.50	   	0.52	   	0.72	   	0.46	   	0.53	   	0.64	

1USM	   	77	   	0.62	   	0.61	   	0.81	   	0.57	   	0.53	   	0.66	   	0.61	   	0.58	   	0.65	

1UTG	   	70	   	0.57	   	0.53	   	0.68	   	0.51	   	0.49	   	0.60	   	0.49	   	0.49	   	0.56	

1V05	   	96	   	0.67	   	0.66	   	0.72	   	0.60	   	0.61	   	0.65	   	0.52	   	0.61	   	0.65	

1V70	   	105	   	0.64	   	0.65	   	0.75	   	0.56	   	0.60	   	0.66	   	0.51	   	0.58	   	0.62	

1VRZ	   	13	   	1.00	   	1.00	   	1.00	   	0.92	   	0.92	   	1.00	   	0.77	   	0.85	   	1.00	

1W2L	   	97	   	0.72	   	0.72	   	0.79	   	0.60	   	0.63	   	0.69	   	0.56	   	0.61	   	0.69	

1WBE	   	206	   	0.53	   	0.47	   	0.63	   	0.43	   	0.38	   	0.55	   	0.36	   	0.42	   	0.48	

1WHI	   	122	   	0.57	   	0.55	   	0.63	   	0.42	   	0.44	   	0.57	   	0.34	   	0.43	   	0.55	

1WLY	   	322	   	0.62	   	0.64	   	0.67	   	0.59	   	0.62	   	0.63	   	0.54	   	0.62	   	0.64	

1WPA	   	107	   	0.70	   	0.69	   	0.79	   	0.61	   	0.52	   	0.71	   	0.66	   	0.56	   	0.70	

1X3O	   	80	   	0.66	   	0.66	   	0.72	   	0.62	   	0.60	   	0.65	   	0.62	   	0.64	   	0.67	

1XY1	   	16	   	0.97	   	0.96	   	1.00	   	0.73	   	0.66	   	0.87	   	0.81	   	0.89	   	0.99	

1XY2	   	8	   	1.00	   	1.00	   	1.00	   	0.99	   	0.95	   	1.00	   	0.91	   	0.91	   	1.00	

1Y6X	   	86	   	0.56	   	0.53	   	0.62	   	0.50	   	0.49	   	0.59	   	0.50	   	0.52	   	0.56	

1YJO	   	6	   	1.00	   	1.00	   	1.00	   	1.00	   	1.00	   	1.00	   	1.00	   	1.00	   	1.00	

1YZM	   	46	   	0.87	   	0.90	   	0.95	   	0.82	   	0.72	   	0.88	   	0.86	   	0.84	   	0.90	

1Z21	   	96	   	0.70	   	0.73	   	0.82	   	0.61	   	0.63	   	0.64	   	0.64	   	0.69	   	0.72	

1ZCE	   	139	   	0.84	   	0.83	   	0.88	   	0.83	   	0.77	   	0.85	   	0.81	   	0.78	   	0.82	

1ZVA	   	75	   	0.85	   	0.85	   	0.94	   	0.84	   	0.78	   	0.92	   	0.83	   	0.81	   	0.86	

2A50	   	469	   	0.64	   	0.63	   	0.70	   	0.54	   	0.60	   	0.67	   	0.41	   	0.58	   	0.67	

2AGK	   	233	   	0.65	   	0.65	   	0.69	   	0.61	   	0.64	   	0.65	   	0.55	   	0.63	   	0.67	

2AH1	   	939	   	0.45	   	0.47	   	0.49	   	0.42	   	0.45	   	0.46	   	0.33	   	0.46	   	0.48	

2B0A	   	191	   	0.59	   	0.60	   	0.69	   	0.50	   	0.58	   	0.62	   	0.48	   	0.59	   	0.63	

2BCM	   	415	   	0.46	   	0.41	   	0.50	   	0.39	   	0.39	   	0.40	   	0.35	   	0.39	   	0.45	

2BF9	   	35	   	0.94	   	0.73	   	0.97	   	0.70	   	0.65	   	0.78	   	0.89	   	0.71	   	0.92	

2BRF	   	103	   	0.74	   	0.73	   	0.76	   	0.74	   	0.71	   	0.74	   	0.72	   	0.72	   	0.75	

2C71	   	225	   	0.45	   	0.38	   	0.56	   	0.29	   	0.33	   	0.42	   	0.23	   	0.30	   	0.48	

2CE0	   	109	   	0.77	   	0.79	   	0.86	   	0.75	   	0.73	   	0.80	   	0.71	   	0.77	   	0.79	

2CG7	   	110	   	0.32	   	0.44	   	0.63	   	0.29	   	0.31	   	0.36	   	0.30	   	0.33	   	0.41	

2COV	   	534	   	0.66	   	0.64	   	0.70	   	0.63	   	0.64	   	0.67	   	0.57	   	0.64	   	0.67	

2CWS	   	235	   	0.59	   	0.55	   	0.66	   	0.53	   	0.52	   	0.54	   	0.40	   	0.52	   	0.55	

2D5W	   	1214	   	0.52	   	0.52	   	0.54	   	0.49	   	0.52	   	0.52	   	0.41	   	0.52	   	0.53	

2DKO	   	253	   	0.75	   	0.72	   	0.79	   	0.72	   	0.69	   	0.75	   	0.68	   	0.69	   	0.72	

2DPL	   	565	   	0.35	   	0.36	   	0.41	   	0.30	   	0.32	   	0.35	   	0.24	   	0.33	   	0.37	

2DSX	   	52	   	0.54	   	0.50	   	0.78	   	0.37	   	0.30	   	0.56	   	0.41	   	0.36	   	0.55	

2E10	   	439	   	0.60	   	0.59	   	0.65	   	0.51	   	0.58	   	0.61	   	0.43	   	0.57	   	0.62	

2E3H	   	81	   	0.66	   	0.71	   	0.82	   	0.62	   	0.69	   	0.76	   	0.56	   	0.69	   	0.78	

2EAQ	   	89	   	0.81	   	0.77	   	0.86	   	0.78	   	0.72	   	0.81	   	0.77	   	0.76	   	0.82	

2EHP	   	246	   	0.63	   	0.65	   	0.71	   	0.58	   	0.62	   	0.65	   	0.52	   	0.62	   	0.64	

2EHS	   	75	   	0.75	   	0.73	   	0.81	   	0.72	   	0.71	   	0.74	   	0.69	   	0.71	   	0.73	

2ERW	   	53	   	0.62	   	0.41	   	0.84	   	0.33	   	0.26	   	0.60	   	0.31	   	0.28	   	0.49	

2ETX	   	390	   	0.54	   	0.54	   	0.57	   	0.52	   	0.53	   	0.56	   	0.47	   	0.51	   	0.54	

2FB6	   	129	   	0.71	   	0.66	   	0.76	   	0.67	   	0.63	   	0.69	   	0.65	   	0.63	   	0.74	

2FG1	   	176	   	0.55	   	0.56	   	0.62	   	0.54	   	0.52	   	0.58	   	0.52	   	0.54	   	0.57	

2FN9	   	560	   	0.51	   	0.49	   	0.62	   	0.44	   	0.47	   	0.55	   	0.41	   	0.46	   	0.55	

2FQ3	   	85	   	0.78	   	0.76	   	0.82	   	0.75	   	0.75	   	0.79	   	0.68	   	0.75	   	0.78	

2G69	   	99	   	0.59	   	0.65	   	0.76	   	0.42	   	0.50	   	0.66	   	0.47	   	0.45	   	0.60	

2G7O	   	68	   	0.89	   	0.91	   	0.95	   	0.85	   	0.79	   	0.88	   	0.76	   	0.82	   	0.87	

2G7S	   	206	   	0.63	   	0.60	   	0.66	   	0.59	   	0.58	   	0.63	   	0.54	   	0.59	   	0.63	

2GKG	   	150	   	0.77	   	0.71	   	0.83	   	0.74	   	0.65	   	0.78	   	0.76	   	0.67	   	0.78	

2GOM	   	121	   	0.47	   	0.52	   	0.64	   	0.42	   	0.42	   	0.45	   	0.44	   	0.47	   	0.53	

2GXG	   	140	   	0.74	   	0.72	   	0.79	   	0.71	   	0.68	   	0.72	   	0.69	   	0.68	   	0.73	

2GZQ	   	203	   	0.45	   	0.40	   	0.60	   	0.38	   	0.34	   	0.48	   	0.24	   	0.29	   	0.31	

2HQK	   	232	   	0.80	   	0.79	   	0.83	   	0.70	   	0.74	   	0.80	   	0.68	   	0.76	   	0.81	

2HYK	   	237	   	0.59	   	0.58	   	0.63	   	0.51	   	0.55	   	0.59	   	0.43	   	0.54	   	0.60	

2I24	   	113	   	0.47	   	0.44	   	0.69	   	0.40	   	0.40	   	0.48	   	0.45	   	0.40	   	0.49	

2I49	   	399	   	0.54	   	0.53	   	0.62	   	0.43	   	0.51	   	0.56	   	0.41	   	0.49	   	0.58	

2IBL	   	108	   	0.69	   	0.71	   	0.75	   	0.66	   	0.67	   	0.70	   	0.65	   	0.68	   	0.71	

2IGD	   	61	   	0.67	   	0.72	   	0.84	   	0.61	   	0.64	   	0.74	   	0.61	   	0.66	   	0.74	

2IMF	   	203	   	0.61	   	0.65	   	0.71	   	0.59	   	0.56	   	0.60	   	0.59	   	0.59	   	0.64	

2IP6	   	87	   	0.72	   	0.66	   	0.82	   	0.66	   	0.58	   	0.73	   	0.64	   	0.64	   	0.78	

2IVY	   	89	   	0.43	   	0.53	   	0.69	   	0.35	   	0.45	   	0.48	   	0.34	   	0.42	   	0.57	

2J32	   	244	   	0.77	   	0.72	   	0.85	   	0.73	   	0.68	   	0.77	   	0.73	   	0.68	   	0.77	

2J9W	   	203	   	0.59	   	0.60	   	0.70	   	0.55	   	0.59	   	0.64	   	0.51	   	0.59	   	0.62	

2JKU	   	38	   	0.89	   	0.75	   	0.95	   	0.85	   	0.65	   	0.88	   	0.83	   	0.60	   	0.88	

2JLI	   	112	   	0.87	   	0.81	   	0.90	   	0.82	   	0.70	   	0.85	   	0.85	   	0.78	   	0.86	

2JLJ	   	121	   	0.78	   	0.75	   	0.80	   	0.71	   	0.65	   	0.74	   	0.74	   	0.71	   	0.76	

2MCM	   	112	   	0.80	   	0.80	   	0.85	   	0.78	   	0.77	   	0.81	   	0.75	   	0.77	   	0.82	

2NLS	   	36	   	0.75	   	0.66	   	0.88	   	0.61	   	0.32	   	0.76	   	0.49	   	0.47	   	0.69	

2NR7	   	193	   	0.75	   	0.75	   	0.79	   	0.74	   	0.72	   	0.76	   	0.71	   	0.73	   	0.77	

2NUH	   	104	   	0.77	   	0.74	   	0.85	   	0.73	   	0.63	   	0.81	   	0.75	   	0.66	   	0.80	

2O6X	   	309	   	0.74	   	0.75	   	0.78	   	0.70	   	0.73	   	0.75	   	0.65	   	0.73	   	0.75	

2OA2	   	140	   	0.63	   	0.64	   	0.70	   	0.55	   	0.49	   	0.60	   	0.60	   	0.63	   	0.67	

2OHW	   	257	   	0.35	   	0.39	   	0.48	   	0.29	   	0.32	   	0.35	   	0.27	   	0.34	   	0.38	

2OKT	   	377	   	0.43	   	0.37	   	0.49	   	0.31	   	0.36	   	0.40	   	0.22	   	0.33	   	0.46	

2OL9	   	6	   	1.00	   	1.00	   	1.00	   	1.00	   	1.00	   	1.00	   	1.00	   	1.00	   	1.00	

2OLX	   	4	   	1.00	   	1.00	   	1.00	   	1.00	   	1.00	   	1.00	   	1.00	   	1.00	   	1.00	

2PKT	   	93	   	0.44	   	0.39	   	0.69	   	0.40	   	0.35	   	0.55	   	0.36	   	0.36	   	0.43	

2PLT	   	98	   	0.66	   	0.63	   	0.72	   	0.57	   	0.59	   	0.67	   	0.52	   	0.59	   	0.66	

2PMR	   	83	   	0.69	   	0.68	   	0.80	   	0.59	   	0.62	   	0.68	   	0.65	   	0.65	   	0.69	

2POF	   	428	   	0.62	   	0.56	   	0.66	   	0.48	   	0.55	   	0.60	   	0.44	   	0.54	   	0.63	

2PPN	   	122	   	0.57	   	0.61	   	0.74	   	0.51	   	0.59	   	0.63	   	0.44	   	0.57	   	0.63	

2PSF	   	608	   	0.43	   	0.45	   	0.53	   	0.41	   	0.44	   	0.45	   	0.37	   	0.42	   	0.44	

2PTH	   	193	   	0.71	   	0.71	   	0.77	   	0.65	   	0.70	   	0.73	   	0.61	   	0.69	   	0.72	

2Q4N	   	1208   	0.65	   	0.62	   	0.68	   	0.58	   	0.55	   	0.59	   	0.55	   	0.57	   	0.61	

2Q52	   	3296   	0.65	   	0.66	   	0.70	   	0.62	   	0.56	   	0.64	   	0.63	   	0.57	   	0.65	

2QJL	   	107	   	0.45	   	0.52	   	0.63	   	0.42	   	0.46	   	0.50	   	0.41	   	0.49	   	0.51	

2R16	   	185	   	0.50	   	0.51   	0.66	   	0.46	   	0.45	   	0.51	   	0.45	   	0.46	   	0.52	

2R6Q	   	149	   	0.71	   	0.72   	0.76	   	0.66	   	0.68	   	0.70	   	0.62	   	0.65	   	0.67	

2RB8	   	93	   	0.81	   	0.78   0.84   0.78   0.75   0.80	   	0.74	   	0.76	   	0.81	

2RE2	   	249	   	0.64	   	0.65	   	0.70	   	0.57	   	0.59	   	0.61	   	0.59	   	0.60	   	0.63	

2RFR	   	166	   	0.73	   	0.66	   	0.80	   	0.68	   	0.57	   	0.74	   	0.72	   	0.59	   	0.74	

2V9V	   	149	   	0.60	   	0.51	   	0.66	   	0.53	   	0.48	   	0.56	   	0.55	   	0.50	   	0.62	

2VE8	   	515	   	0.46	   	0.48	   	0.55	   	0.42	   	0.41	   	0.44	   	0.40	   	0.43	   	0.47	

2VH7	   	94	   	0.59	   	0.54	   	0.68	   	0.52	   	0.49	   	0.63	   	0.42	   	0.49	   	0.54	

2VIM	   	114	   	0.38	   	0.33	   	0.52	   	0.29	   	0.28	   	0.41	   	0.24	   	0.31	   	0.40	

2VPA	   	217	   	0.73	   	0.75	   	0.78	   	0.72	   	0.71	   	0.73	   	0.68	   	0.73	   	0.74	

2VQ4	   	106	   	0.56	   	0.54	   	0.64	   	0.43	   	0.49	   	0.56	   	0.35	   	0.46	   	0.58	

2VY8	   	162	   	0.47	   	0.46	   	0.58	   	0.38	   	0.42	   	0.46	   	0.38	   	0.42	   	0.49	

2VYO	   	207	   	0.68	   	0.70	   	0.77	   	0.64	   	0.66	   	0.72	   	0.59	   	0.68	   	0.70	

2W1V	   	551	   	0.69	   	0.67	   	0.77	   	0.63	   	0.63	   	0.70	   	0.56	   	0.64	   	0.68	

2W2A	   	350	   	0.60	   	0.59	   	0.65	   	0.57	   	0.56	   	0.59	   	0.54	   	0.57	   	0.60	

2W6A	   	139	   	0.59	   	0.59	   	0.64	   	0.51	   	0.52	   	0.54	   	0.52	   	0.56	   	0.60	

2WJ5	   	110	   	0.63	   	0.55	   	0.79	   	0.59	   	0.52	   	0.68	   	0.59	   	0.53	   	0.64	

2WUJ	   	103	   	0.69	   	0.68	   	0.79	   	0.62	   	0.52	   	0.65	   	0.67	   	0.59	   	0.71	

2WW7	   	161	   	0.44	   	0.48	   	0.60	   	0.40	   	0.42	   	0.50	   	0.33	   	0.43	   	0.49	

2WWE	   	120	   	0.71	   	0.71	   	0.83	   	0.62	   	0.62	   	0.75	   	0.61	   	0.58	   	0.73	

2X1Q	   	240	   	0.48	   	0.44	   	0.54	   	0.38	   	0.39	   	0.46	   	0.34	   	0.37	   	0.47	

2X25	   	167	   	0.62	   	0.61	   	0.73	   	0.56	   	0.57	   	0.64	   	0.57	   	0.57	   	0.64	

2X3M	   	175	   	0.61	   	0.61	   	0.69	   	0.60	   	0.55	   	0.64	   	0.57	   	0.57	   	0.60	

2X5Y	   	185	   	0.67	   	0.63	   	0.71	   	0.60	   	0.59	   	0.64	   	0.53	   	0.58	   	0.69	

2X9Z	   	266	   	0.50	   	0.42	   	0.54	   	0.37	   	0.38	   	0.42	   	0.38	   	0.39	   	0.51	

2XHF	   	310	   	0.62	   	0.62	   	0.67	   	0.58	   	0.56	   	0.60	   	0.55	   	0.62	   	0.63	

2Y0T	   	111	   	0.69	   	0.68	   	0.83	   	0.60	   	0.61	   	0.68	   	0.56	   	0.64	   	0.70	

2Y72	   	183	   	0.71	   	0.71	   	0.78	   	0.69	   	0.69	   	0.72	   	0.66	   	0.70	   	0.71	

2Y7L	   	323	   	0.68	   	0.70	   	0.72	   	0.66	   	0.68	   	0.69	   	0.58	   	0.69	   	0.69	

2Y9F	   	149	   	0.75	   	0.72	   	0.78	   	0.65	   	0.69	   	0.71	   	0.58	   	0.70	   	0.74	

2YLB	   	418	   	0.55	   	0.52	   	0.63	   	0.46	   	0.49	   	0.52	   	0.34	   	0.49	   	0.59	

2YNY	   	326	   	0.63	   	0.67	   	0.75	   	0.60	   	0.62	   	0.63	   	0.56	   	0.63	   	0.66	

2ZCM	   	348	   	0.42	   	0.39	   	0.49	   	0.34	   	0.35	   	0.40	   	0.24	   	0.32	   	0.43	

2ZU1	   	360	   	0.61	   	0.61	   	0.68	   	0.53	   	0.58	   	0.63	   	0.45	   	0.58	   	0.63	

3A0M	   	146	   	0.74	   	0.76	   	0.84	   	0.68	   	0.70	   	0.72	   	0.61	   	0.73	   	0.78	

3A7L	   	128	   	0.69	   	0.61	   	0.78	   	0.52	   	0.45	   	0.59	   	0.62	   	0.54	   	0.67	

3AMC	   	614	   	0.54	   	0.53	   	0.64	   	0.47	   	0.50	   	0.54	   	0.37	   	0.51	   	0.57	

3AUB	   	124	   	0.36	   	0.41	   	0.53	   	0.31	   	0.26	   	0.41	   	0.32	   	0.32	   	0.37	

3B5O	   	249	   	0.55	   	0.58	   	0.66	   	0.52	   	0.56	   	0.63	   	0.46	   	0.55	   	0.57	

3BA1	   	312	   	0.67	   	0.66	   	0.72	   	0.64	   	0.65	   	0.68	   	0.60	   	0.65	   	0.70	

3BED	   	262	   	0.61	   	0.55	   	0.67	   	0.53	   	0.53	   	0.56	   	0.44	   	0.53	   	0.61	

3BQX	   	136	   	0.52	   	0.50	   	0.54	   	0.47	   	0.48	   	0.51	   	0.41	   	0.46	   	0.51	

3BZQ	   	99	   	0.57	   	0.62	   	0.69	   	0.50	   	0.55	   	0.61	   	0.47	   	0.55	   	0.59	

3BZZ	   	103	   	0.60	   	0.63	   	0.68	   	0.51	   	0.58	   	0.61	   	0.45	   	0.50	   	0.59	

3DRF	   	567	   	0.32	   	0.32	   	0.38	   	0.27	   	0.29	   	0.33	   	0.22	   	0.30	   	0.34	

3DWV	   	359	   	0.67	   	0.63	   	0.69	   	0.62	   	0.62	   	0.66	   	0.54	   	0.62	   	0.65	

3E5T	   	268	   	0.55	   	0.52	   	0.60	   	0.51	   	0.51	   	0.56	   	0.38	   	0.50	   	0.55	

3E7R	   	40	   	0.81	   	0.86	   	0.96	   	0.78	   	0.77	   	0.81	   	0.73	   	0.82	   	0.88	

3EUR	   	150	   	0.49	   	0.46	   	0.53	   	0.39	   	0.43	   	0.47	   	0.31	   	0.42	   	0.47	

3F2Z	   	148	   	0.76	   	0.78	   	0.84	   	0.75	   	0.76	   	0.78	   	0.69	   	0.77	   	0.78	

3F7E	   	261	   	0.66	   	0.65	   	0.71	   	0.61	   	0.64	   	0.65	   	0.47	   	0.63	   	0.69	

3FCN	   	185	   	0.60	   	0.65	   	0.75	   	0.56	   	0.59	   	0.64	   	0.54	   	0.59	   	0.67	

3FE7	   	89	   	0.69	   	0.65	   	0.76	   	0.58	   	0.60	   	0.67	   	0.54	   	0.63	   	0.70	

3FKE	   	250	   	0.47	   	0.42	   	0.52	   	0.40	   	0.36	   	0.49	   	0.34	   	0.36	   	0.45	

3FMY	   	75	   	0.71	   	0.69	   	0.79	   	0.66	   	0.64	   	0.70	   	0.66	   	0.66	   	0.71	

3FOD	   	48	   	0.48	   	0.47	   	0.82	   	0.42	   	0.33	   	0.55	   	0.38	   	0.35	   	0.48	

3FSO	   	238	   	0.82	   	0.82	   	0.85	   	0.77	   	0.74	   	0.77	   	0.77	   	0.81	   	0.82	

3FTD	   	257	   	0.60	   	0.57	   	0.67	   	0.49	   	0.52	   	0.59	   	0.41	   	0.52	   	0.60	

3G1S	   	418	   	0.44	   	0.51	   	0.68	   	0.41	   	0.45	   	0.51	   	0.38	   	0.45	   	0.49	

3GBW	   	170	   	0.77	   	0.78	   	0.84	   	0.64	   	0.74	   	0.79	   	0.51	   	0.71	   	0.81	

3GHJ	   	129	   	0.71	   	0.71	   	0.81	   	0.65	   	0.67	   	0.72	   	0.65	   	0.68	   	0.72	

3HFO	   	216	   	0.75	   	0.72	   	0.82	   	0.70	   	0.63	   	0.75	   	0.65	   	0.69	   	0.74	

3HHP	   	1314	   	0.61	   	0.62	   	0.68	   	0.57	   	0.59	   	0.62	   	0.52	   	0.59	   	0.63	

3HNY	   	170	   	0.59	   	0.56	   	0.64	   	0.47	   	0.52	   	0.57	   	0.42	   	0.49	   	0.56	

3HP4	   	201	   	0.60	   	0.61	   	0.72	   	0.57	   	0.54	   	0.64	   	0.43	   	0.56	   	0.62	

3HWU	   	155	   	0.60	   	0.69	   	0.81	   	0.57	   	0.61	   	0.63	   	0.50	   	0.61	   	0.68	

3HYD	   	8	   	1.00	   	1.00	   	1.00	   	1.00	   	1.00	   	1.00	   	1.00	   	1.00	   	1.00	

3HZ8	   	200	   	0.58	   	0.59	   	0.66	   	0.55	   	0.53	   	0.56	   	0.52	   	0.54	   	0.58	

3I2V	   	127	   	0.57	   	0.58	   	0.66	   	0.51	   	0.53	   	0.61	   	0.40	   	0.48	   	0.53	

3I2Z	   	140	   	0.58	   	0.59	   	0.65	   	0.52	   	0.54	   	0.56	   	0.56	   	0.57	   	0.61	

3I4O	   	154	   	0.63	   	0.64	   	0.73	   	0.58	   	0.59	   	0.60	   	0.56	   	0.63	   	0.66	

3I7M	   	145	   	0.58	   	0.62	   	0.71	   	0.53	   	0.55	   	0.58	   	0.49	   	0.58	   	0.64	

3IHS	   	173	   	0.62	   	0.67	   	0.74	   	0.58	   	0.54	   	0.60	   	0.58	   	0.60	   	0.62	

3IVV	   	168	   	0.80	   	0.80	   	0.89	   	0.75	   	0.76	   	0.83	   	0.68	   	0.74	   	0.79	

3K6Y	   	227	   	0.53	   	0.53	   	0.60	   	0.48	   	0.49	   	0.52	   	0.42	   	0.50	   	0.55	

3KBE	   	166	   	0.62	   	0.61	   	0.65	   	0.57	   	0.60	   	0.62	   	0.52	   	0.60	   	0.61	

3KGK	   	190	   	0.79	   	0.80	   	0.84	   	0.77	   	0.79	   	0.81	   	0.68	   	0.79	   	0.80	

3KZD	   	94	   	0.79	   	0.72	   	0.83	   	0.55	   	0.68	   	0.77	   	0.47	   	0.66	   	0.78	

3L41	   	219	   	0.61	   	0.62	   	0.71	   	0.59	   	0.60	   	0.66	   	0.57	   	0.59	   	0.67	

3LAA	   	176	   	0.70	   	0.66	   	0.80	   	0.68	   	0.56	   	0.76	   	0.69	   	0.60	   	0.77	

3LAX	   	118	   	0.81	   	0.81	   	0.86	   	0.80	   	0.76	   	0.83	   	0.77	   	0.78	   	0.82	

3LG3	   	846	   	0.40	   	0.38	   	0.41	   	0.36	   	0.37	   	0.40	   	0.32	   	0.37	   	0.41	

3LJI	   	270	   	0.53	   	0.53	   	0.62	   	0.47	   	0.52	   	0.58	   	0.45	   	0.52	   	0.56	

3M3P	   	244	   	0.47	   	0.44	   	0.69	   	0.40	   	0.40	   	0.58	   	0.25	   	0.35	   	0.48	

3M8J	   	178	   	0.74	   	0.72	   	0.75	   	0.69	   	0.69	   	0.73	   	0.67	   	0.70	   	0.73	

3M9J	   	250	   	0.57	   	0.56	   	0.59	   	0.53	   	0.54	   	0.56	   	0.39	   	0.53	   	0.56	

3M9Q	   	190	   	0.53	   	0.52	   	0.59	   	0.50	   	0.51	   	0.53	   	0.46	   	0.50	   	0.51	

3MAB	   	180	   	0.57	   	0.56	   	0.62	   	0.52	   	0.47	   	0.55	   	0.56	   	0.51	   	0.56	

3MD4	   	13	   	1.00	   	1.00	   	1.00	   	0.91	   	0.94	   	1.00	   	0.93	   	0.99	   	1.00	

3MD5	   	14	   	1.00	   	1.00	   	1.00	   	0.98	   	0.93	   	1.00	   	0.94	   	0.92	   	1.00	

3MEA	   	170	   	0.58	   	0.58	   	0.68	   	0.57	   	0.57	   	0.64	   	0.48	   	0.57	   	0.59	

3MGN	   	277	   	0.33	   	0.32	   	0.47	   	0.26	   	0.28	   	0.30	   	0.16	   	0.29	   	0.39	

3MRE	   	446	   	0.40	   	0.38	   	0.45	   	0.32	   	0.36	   	0.40	   	0.24	   	0.35	   	0.41	

3N11	   	325	   	0.43	   	0.45	   	0.51	   	0.42	   	0.44	   	0.45	   	0.38	   	0.44	   	0.45	

3NE0	   	208	   	0.77	   	0.79	   	0.84	   	0.75	   	0.70	   	0.77	   	0.70	   	0.76	   	0.82	

3NGG	   	97	   	0.80	   	0.81	   	0.85	   	0.72	   	0.74	   	0.78	   	0.74	   	0.76	   	0.80	

3NPV	   	500	   	0.44	   	0.44	   	0.50	   	0.40	   	0.42	   	0.44	   	0.36	   	0.43	   	0.47	

3NVG	   	6	   	1.00	   	1.00	   	1.00	   	1.00	   	1.00	   	1.00	   	1.00	   	1.00	   	1.00	

3NZL	   	70	   	0.68	   	0.61	   	0.84	   	0.53	   	0.49	   	0.66	   	0.59	   	0.55	   	0.67	

3O0P	   	197	   	0.62	   	0.64	   	0.71	   	0.59	   	0.62	   	0.64	   	0.53	   	0.62	   	0.64	

3O5P	   	147	   	0.64	   	0.60	   	0.71	   	0.55	   	0.57	   	0.60	   	0.53	   	0.56	   	0.64	

3OBQ	   	150	   	0.59	   	0.59	   	0.66	   	0.46	   	0.49	   	0.58	   	0.53	   	0.56	   	0.58	

3OQY	   	236	   	0.71	   	0.66	   	0.73	   	0.63	   	0.64	   	0.70	   	0.60	   	0.64	   	0.72	

3P6J	   	145	   	0.75	   	0.73	   	0.81	   	0.69	   	0.71	   	0.73	   	0.61	   	0.71	   	0.75	

3PD7	   	216	   	0.65	   	0.66	   	0.72	   	0.62	   	0.60	   	0.65	   	0.60	   	0.61	   	0.65	

3PES	   	166	   	0.70	   	0.72	   	0.79	   	0.58	   	0.63	   	0.70	   	0.52	   	0.60	   	0.66	

3PID	   	387	   	0.50	   	0.49	   	0.56	   	0.44	   	0.48	   	0.53	   	0.37	   	0.46	   	0.51	

3PIW	   	161	   	0.66	   	0.67	   	0.78	   	0.60	   	0.63	   	0.70	   	0.56	   	0.63	   	0.72	

3PKV	   	229	   	0.50	   	0.52	   	0.63	   	0.43	   	0.48	   	0.53	   	0.35	   	0.50	   	0.57	

3PSM	   	94	   	0.83	   	0.78	   	0.88	   	0.79	   	0.77	   	0.83	   	0.68	   	0.76	   	0.79	

3PTL	   	289	   	0.50	   	0.50	   	0.53	   	0.49	   	0.49	   	0.50	   	0.43	   	0.49	   	0.50	

3PVE	   	363	   	0.45	   	0.45	   	0.59	   	0.37	   	0.39	   	0.44	   	0.41	   	0.42	   	0.45	

3PZ9	   	357	   	0.51	   	0.45	   	0.57	   	0.36	   	0.38	   	0.42	   	0.34	   	0.39	   	0.50	

3PZZ	   	12	   	1.00	   	1.00	   	1.00	   	0.95	   	0.90	   	1.00	   	0.94	   	0.80	   	1.00	

3Q2X	   	6	   	1.00	   	1.00	   	1.00	   	1.00	   	1.00	   	1.00	   	1.00	   	1.00	   	1.00	

3Q6L	   	131	   	0.39	   	0.44	   	0.56	   	0.33	   	0.31	   	0.37	   	0.34	   	0.37	   	0.42	

3QDS	   	284	   	0.63	   	0.62	   	0.69	   	0.59	   	0.59	   	0.65	   	0.51	   	0.59	   	0.64	

3QPA	   	212	   	0.68	   	0.66	   	0.78	   	0.45	   	0.45	   	0.47	   	0.59	   	0.59	   	0.65	

3R6D	   	222	   	0.65	   	0.66	   	0.73	   	0.62	   	0.63	   	0.65	   	0.53	   	0.64	   	0.69	

3R87	   	148	   	0.48	   	0.47	   	0.55	   	0.41	   	0.44	   	0.48	   	0.40	   	0.45	   	0.47	

3RQ9	   	165	   	0.51	   	0.47	   	0.61	   	0.41	   	0.44	   	0.52	   	0.39	   	0.45	   	0.56	

3RY0	   	128	   	0.44	   	0.45	   	0.54	   	0.40	   	0.40	   	0.47	   	0.41	   	0.42	   	0.47	

3RZY	   	151	   	0.65	   	0.65	   	0.84	   	0.59	   	0.54	   	0.65	   	0.57	   	0.51	   	0.59	

3S0A	   	132	   	0.39	   	0.43	   	0.52	   	0.33	   	0.34	   	0.38	   	0.32	   	0.31	   	0.37	

3SD2	   	100	   	0.65	   	0.67	   	0.77	   	0.64	   	0.63	   	0.69	   	0.56	   	0.63	   	0.67	

3SEB	   	238	   	0.63	   	0.66	   	0.77	   	0.62	   	0.61	   	0.68	   	0.61	   	0.62	   	0.67	

3SED	   	126	   	0.39	   	0.45	   	0.55	   	0.28	   	0.29	   	0.38	   	0.33	   	0.33	   	0.40	

3SO6	   	157	   	0.67	   	0.71	   	0.78	   	0.63	   	0.69	   	0.73	   	0.55	   	0.64	   	0.70	

3SR3	   	657	   	0.45	   	0.44	   	0.48	   	0.43	   	0.41	   	0.45	   	0.39	   	0.43	   	0.44	

3SUK	   	254	   	0.53	   	0.54	   	0.64	   	0.46	   	0.48	   	0.54	   	0.47	   	0.49	   	0.57	

3SZH	   	753	   	0.53	   	0.53	   	0.57	   	0.51	   	0.51	   	0.52	   	0.45	   	0.52	   	0.53	

3T0H	   	209	   	0.76	   	0.73	   	0.78	   	0.72	   	0.69	   	0.74	   	0.68	   	0.71	   	0.76	

3T3K	   	122	   	0.66	   	0.66	   	0.72	   	0.55	   	0.62	   	0.68	   	0.48	   	0.60	   	0.68	

3T47	   	145	   	0.54	   	0.54	   	0.78	   	0.45	   	0.45	   	0.62	   	0.43	   	0.47	   	0.54	

3TDN	   	359	   	0.47	   	0.43	   	0.53	   	0.43	   	0.42	   	0.44	   	0.38	   	0.43	   	0.49	

3TOW	   	155	   	0.66	   	0.65	   	0.74	   	0.58	   	0.61	   	0.66	   	0.53	   	0.60	   	0.65	

3TUA	   	226	   	0.57	   	0.55	   	0.63	   	0.52	   	0.50	   	0.55	   	0.45	   	0.52	   	0.54	

3TYS	   	78	   	0.78	   	0.58	   	0.86	   	0.67	   	0.48	   	0.73	   	0.70	   	0.46	   	0.75	

3U6G	   	276	   	0.44	   	0.39	   	0.54	   	0.39	   	0.37	   	0.45	   	0.27	   	0.35	   	0.48	

3U97	   	85	   	0.78	   	0.78	   	0.84	   	0.77	   	0.73	   	0.80	   	0.77	   	0.76	   	0.80	

3UCI	   	72	   	0.67	   	0.64	   	0.72	   	0.48	   	0.53	   	0.57	   	0.55	   	0.56	   	0.63	

3UR8	   	637	   	0.52	   	0.53	   	0.60	   	0.49	   	0.51	   	0.55	   	0.45	   	0.52	   	0.53	

3US6	   	159	   	0.60	   	0.56	   	0.67	   	0.55	   	0.49	   	0.62	   	0.53	   	0.46	   	0.59	

3V1A	   	59	   	0.74	   	0.57	   	0.95	   	0.51	   	0.53	   	0.77	   	0.39	   	0.46	   	0.68	

3V75	   	294	   	0.50	   	0.49	   	0.57	   	0.48	   	0.46	   	0.53	   	0.47	   	0.47	   	0.53	

3VN0	   	193	   	0.87	   	0.88	   	0.90	   	0.86	   	0.87	   	0.88	   	0.79	   	0.88	   	0.89	

3VOR	   	219	   	0.64	   	0.58	   	0.70	   	0.56	   	0.52	   	0.63	   	0.53	   	0.55	   	0.63	

3VUB	   	101	   	0.65	   	0.60	   	0.71	   	0.60	   	0.56	   	0.61	   	0.61	   	0.57	   	0.64	

3VVV	   	112	   	0.64	   	0.64	   	0.79	   	0.55	   	0.48	   	0.65	   	0.57	   	0.49	   	0.58	

3VZ9	   	163	   	0.65	   	0.64	   	0.70	   	0.60	   	0.55	   	0.63	   	0.60	   	0.60	   	0.67	

3W4Q	   	826	   	0.61	   	0.60	   	0.68	   	0.56	   	0.59	   	0.61	   	0.47	   	0.60	   	0.64	

3ZBD	   	213	   	0.36	   	0.47	   	0.74	   	0.24	   	0.28	   	0.34	   	0.25	   	0.31	   	0.36	

3ZIT	   	157	   	0.51	   	0.47	   	0.59	   	0.36	   	0.39	   	0.47	   	0.47	   	0.41	   	0.52	

3ZRX	   	241	   	0.56	   	0.56	   	0.63	   	0.49	   	0.52	   	0.53	   	0.46	   	0.52	   	0.56	

3ZSL	   	165	   	0.39	   	0.39	   	0.54	   	0.28	   	0.22	   	0.40	   	0.31	   	0.24	   	0.37	

3ZZP	   	74	   	0.40	   	0.30	   	0.47	   	0.19	   	0.27	   	0.31	   	0.12	   	0.22	   	0.40	

3ZZY	   	226	   	0.65	   	0.67	   	0.69	   	0.63	   	0.63	   	0.64	   	0.59	   	0.63	   	0.64	

4A02	   	169	   	0.61	   	0.56	   	0.66	   	0.49	   	0.52	   	0.57	   	0.31	   	0.51	   	0.60	

4ACJ	   	182	   	0.55	   	0.59	   	0.75	   	0.55	   	0.58	   	0.61	   	0.51	   	0.59	   	0.60	

4AE7	   	189	   	0.69	   	0.67	   	0.74	   	0.63	   	0.61	   	0.65	   	0.63	   	0.65	   	0.69	

4AM1	   	359	   	0.57	   	0.54	   	0.59	   	0.53	   	0.52	   	0.53	   	0.46	   	0.53	   	0.55	

4ANN	   	210	   	0.50	   	0.48	   	0.57	   	0.42	   	0.43	   	0.48	   	0.36	   	0.42	   	0.47	

4AVR	   	189	   	0.57	   	0.57	   	0.70	   	0.53	   	0.51	   	0.59	   	0.49	   	0.53	   	0.57	

4AXY	   	56	   	0.55	   	0.60	   	0.76	   	0.47	   	0.48	   	0.63	   	0.47	   	0.50	   	0.62	

4B6G	   	559	   	0.70	   	0.71	   	0.75	   	0.67	   	0.69	   	0.72	   	0.60	   	0.69	   	0.73	

4B9G	   	292	   	0.81	   	0.82	   	0.85	   	0.78	   	0.80	   	0.81	   	0.71	   	0.82	   	0.83	

4DD5	   	412	   	0.60	   	0.63	   	0.71	   	0.57	   	0.59	   	0.63	   	0.51	   	0.61	   	0.66	

4DKN	   	423	   	0.59	   	0.58	   	0.63	   	0.52	   	0.54	   	0.56	   	0.42	   	0.55	   	0.61	

4DND	   	93	   	0.75	   	0.66	   	0.82	   	0.67	   	0.64	   	0.75	   	0.61	   	0.64	   	0.74	

4DPZ	   	113	   	0.68	   	0.70	   	0.79	   	0.65	   	0.64	   	0.67	   	0.62	   	0.64	   	0.69	

4DQ7	   	338	   	0.45	   	0.46	   	0.51	   	0.37	   	0.44	   	0.49	   	0.29	   	0.40	   	0.46	

4DT4	   	170	   	0.76	   	0.74	   	0.78	   	0.70	   	0.68	   	0.72	   	0.70	   	0.70	   	0.73	

4EK3	   	313	   	0.58	   	0.63	   	0.65	   	0.55	   	0.56	   	0.58	   	0.53	   	0.59	   	0.60	

4ERY	   	318	   	0.61	   	0.60	   	0.67	   	0.59	   	0.59	   	0.64	   	0.52	   	0.59	   	0.65	

4ES1	   	96	   	0.76	   	0.77	   	0.86	   	0.69	   	0.73	   	0.78	   	0.57	   	0.74	   	0.83	

4EUG	   	225	   	0.61	   	0.61	   	0.67	   	0.54	   	0.60	   	0.62	   	0.51	   	0.58	   	0.62	

4F01	   	459	   	0.38	   	0.37	   	0.47	   	0.32	   	0.34	   	0.37	   	0.22	   	0.34	   	0.39	

4F3J	   	143	   	0.57	   	0.63	   	0.66	   	0.52	   	0.59	   	0.61	   	0.47	   	0.58	   	0.60	

4FR9	   	145	   	0.65	   	0.62	   	0.78	   	0.63	   	0.58	   	0.70	   	0.58	   	0.57	   	0.64	

4G14	   	5	   	1.00	   	1.00	   	1.00	   	1.00	   	1.00	   	1.00	   	1.00	   	1.00	   	1.00	

4G2E	   	155	   	0.75	   	0.64	   	0.85	   	0.59	   	0.61	   	0.74	   	0.68	   	0.61	   	0.80	

4G5X	   	584	   	0.71	   	0.69	   	0.80	   	0.69	   	0.64	   	0.74	   	0.64	   	0.67	   	0.72	

4G6C	   	676	   	0.43	   	0.44	   	0.50	   	0.40	   	0.44	   	0.46	   	0.24	   	0.43	   	0.45	

4G7X	   	216	   	0.53	   	0.47	   	0.61	   	0.41	   	0.31	   	0.47	   	0.51	   	0.37	   	0.53	

4GA2	   	183	   	0.55	   	0.56	   	0.70	   	0.52	   	0.53	   	0.57	   	0.49	   	0.53	   	0.60	

4GMQ	   	94	   	0.73	   	0.77	   	0.84	   	0.68	   	0.66	   	0.72	   	0.67	   	0.63	   	0.72	

4GS3	   	90	   	0.65	   	0.68	   	0.74	   	0.60	   	0.64	   	0.68	   	0.51	   	0.66	   	0.70	

4H4J	   	278	   	0.67	   	0.67	   	0.82	   	0.63	   	0.64	   	0.75	   	0.57	   	0.66	   	0.69	

4H89	   	175	   	0.39	   	0.50	   	0.67	   	0.33	   	0.37	   	0.39	   	0.35	   	0.40	   	0.42	

4HDE	   	167	   	0.63	   	0.55	   	0.75	   	0.59	   	0.52	   	0.69	   	0.59	   	0.51	   	0.67	

4HJP	   	308	   	0.62	   	0.61	   	0.65	   	0.57	   	0.55	   	0.59	   	0.58	   	0.58	   	0.62	

4HWM	   	129	   	0.69	   	0.66	   	0.71	   	0.66	   	0.60	   	0.68	   	0.68	   	0.63	   	0.70	

4IL7	   	99	   	0.63	   	0.63	   	0.65	   	0.60	   	0.59	   	0.62	   	0.57	   	0.61	   	0.62	

4J11	   	377	   	0.66	   	0.63	   	0.68	   	0.62	   	0.61	   	0.63	   	0.63	   	0.61	   	0.66	

4J5O	   	268	   	0.77	   	0.76	   	0.82	   	0.71	   	0.62	   	0.77	   	0.75	   	0.66	   	0.77	

4J5Q	   	162	   	0.65	   	0.63	   	0.75	   	0.57	   	0.56	   	0.66	   	0.59	   	0.57	   	0.64	

4J78	   	305	   	0.48	   	0.48	   	0.56	   	0.43	   	0.44	   	0.50	   	0.38	   	0.47	   	0.53	

4JG2	   	202	   	0.63	   	0.63	   	0.74	   	0.61	   	0.61	   	0.64	   	0.58	   	0.60	   	0.63	

4JVU	   	207	   	0.67	   	0.64	   	0.75	   	0.57	   	0.58	   	0.66	   	0.59	   	0.60	   	0.67	

4JYP	   	550	   	0.59	   	0.60	   	0.69	   	0.52	   	0.57	   	0.61	   	0.38	   	0.58	   	0.61	

4KEF	   	145	   	0.52	   	0.49	   	0.65	   	0.40	   	0.42	   	0.49	   	0.27	   	0.45	   	0.56	

5CYT	   	103	   	0.53	   	0.52	   	0.65	   	0.49	   	0.46	   	0.54	   	0.43	   	0.48	   	0.50	

6RXN	   	45	   	0.74	   	0.63	   	0.86	   	0.59	   	0.48	   	0.76	   	0.49	   	0.49	   	0.76	








































10

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Z. X. Cang and G. W. Wei, “Analysis and prediction of protein folding energy
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Z. X. Cang and G. W. Wei, “Integration of element specific persistent
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K. Wu and G. W. Wei, “Quantitative Toxicity Prediction Using Topology Based
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K. Wu, Z. Zhao, R. Wang, and G. W. Wei, “TopP-S: Persistent Homology-Based
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KLXia:2013f
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Opron:2014
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Opron:2015a
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DBramer:2018a
D. Bramer and G. W. Wei, “Weighted multiscale colored graphs for protein
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D. Bramer and G. W. Wei, “Blind prediction of protein b-factor and
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KLXia:2015c
K. L. Xia and G. W. Wei, “Multidimensional persistence in biomolecular data,”
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Opron:2016a
K. Opron, K. L. Xia, Z. Burton, and G. W. Wei, “Flexibility-rigidity index for
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