A Constrained K-shortest Path Algorithm to Rank the Topologies of the Protein Secondary Structure Elements Detected in CryoEM Volume Maps

A Constrained K-shortest Path Algorithm to Rank the Topologies of the Protein Secondary Structure Elements

Detected in CryoEM Volume Maps Kamal Al Nasr

Dept. of Computer Science Tennessee State University

3500 John A Merritt Blvd Nashville, TN 37209

Lin Chen, Desh Ranjan, M. Zubair, Dong Si, and Jing He

Dept. of Computer Science Old Dominion University

Norfolk, VA 23529

ABSTRACT

Although many electron density maps have been produced into

the medium resolutions, it is still challenging to derive the atomic

structure from such volumetric data. Current methods primarily

rely on the availability of an existing atomic structure for fitting or

a homologous template structure for modeling. In the process of

developing a template-free, de novo, method, the topology of the

secondary structure elements need to be resolved first. In this

paper, we extend our previous algorithm of finding the optimal

solution in the constraint graph problem. We illustrate an

approach to obtain the top-K topologies by combining a dynamic

programming algorithm with the K-shortest path algorithm. The

effectiveness of the algorithms is demonstrated from the test using

three datasets of different nature. The algorithm improves the

accuracy, space and time of an existing method.

Categories and Subject Descriptors

E.1 [Data Structures]: Arrays, Graphs and networks, Tables,

Trees. F.1.3 [Complexity Measures and Classes].

General Terms

Algorithms, protein, structure, 3-dimensional image.

Keywords

Electron cryomicroscopy, Graph, K-Shortest paths, Protein,

Topology, Algorithm.

1. INTRODUCTION Electron cryomicroscopy (CryoEM) is a biophysical technique

that has great potential in deriving the three-dimensional structure

of large protein complexes [3-6]. Various aspects in CryoEM have

been improved over the last ten years, and as a result, it is possible

to obtain the electron density maps of a protein in the high

resolution range, such as 3-5Å resolution [7-10]. At this

resolution, the connection between the secondary structures is

mostly distinguishable and the backbone of the structure can be

derived. The number of the atomic structures resolved from the

CryoEM density maps at the high resolution range steadily

increases in the last 3 years [11]. Some of these structures include

GroEL, virus and [5, 7, 8]. Although the structure determination

from high-resolution CryoEM maps is promising, a lot more

proteins are resolved at medium resolutions (5-10 Å resolutions)

than those at the high-resolution range. About 2/3 of the medium-

resolution maps have been resolved using fitting or template-

based homology modeling [12]. At this resolution range, the

location and the orientation of most secondary structures such as

helices and β-sheets are detectable using various computational

tools [2, 13-16]. A helix detected from the density map is

represented as a stick (red in Figure 1A) and a β-sheet appears as

a thin sheet (blue Figure 1A). Due to the medium resolution, the

strands of the β-sheet are often not distinguishable. The

connection between two SSEs is often ambiguous. The major

challenge to derive the protein structure from such CryoEM maps

is that it is not known which segment of the protein sequence

corresponds to which of the SSEs detected from the density map.

A topology of the SSEs refers to the order of the SSEs with

respect to the protein sequence and the direction of each SSE. For

example, the true topology of the protein in Figure 1 presents the

true order of the SSEs as

(Figure 1B). In

principle, each helix stick of the protein

corresponds to a sequence segment that forms a

helix in the structure. The four sequence segments and

correspond to a sheet that can be detected in the density map.

Note that there are two directions to correspond a sequence

segment to (arrows of Figure 1A and dot and cross in Figure

1B).

The medium-resolution density maps contain not only the

secondary structure location information but also the connecting

information among them. The skeleton (blue in Figure 2A) of a

density map represents the medial axis of the map. It can be

detected through a thinning and pruning process using Gorgon

[17]. When the detected secondary structure elements (red sticks

in Figure 2(a)) are overlaid with the skeleton (blue Figure 2a), the

connection relationship among them is reviewed. When the

resolution of the density map is at the medium resolution, the

skeleton can be misleading and incomplete, due to the

experimental factors. For example, the skeleton often contains

gaps (i.e. Figure 2(a)), and misleading points. Therefore, the

skeleton provides the connection information, but it is not

completely reliable.

The detected secondary structure elements (SSEs) provide

relative geometrical relationship among them. However, it is not

known which segment of the protein sequence corresponds to

which secondary structure element detected from the volumetric

density map. The topology of the SSEs refers to the order of the

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ACM-BCB 2013 750

elements with respect to the protein sequence and the direction of

each element. To derive the backbone of the protein, the topology

of the secondary structures has to be determined first and then the

backbone of the protein can be built for further optimization [18-

20].

The topology determination problem combines two sources

of information. One source contains the detected secondary

structures, such as helix sticks (red Figure 1A), from the

volumetric map. The other source contains the predicted

secondary structures from the sequence (red, Figure 1C).

Many methods, such as PSIPRED [21], SSPRO [22] and

Porter [23], have been developed for the secondary structure

prediction from the amino acid sequence of a protein. In

general, the prediction accuracy of these methods is about 70-

80% [24]. Let be the helices of the amino acid

sequence in the protein. Due to the linear nature of the protein

sequence, the sequence segments have a fixed order

. Let { } be the set of sticks detected

from CryoEM volume map. In the context of this paper, we

assume , although vice versa is possible. The topology

determination problem can be described as problem of

matching to { }in an optimal way.

In the assignment, each is assigned to in one of the two

opposite directions. The total number of possible topologies is

( ) . For each sticks picked out of segments, there

are different orders and there are two directions to assign a

sequence fragment to each helix stick. When the assignment

involves -sheets, the total number of possible topologies

becomes (

) (

)

, in which and are the

number of sequence segments for helices and -strands

respectively, and similarly and are the number of the

detected helix and -stand sticks respectively.

Current approaches to find the best topology can be categorized

into three approaches. The direct approach enumerates all the

possible topologies of the SSEs and identifies the best by

comparing all of them [18, 25]. This approach has the limitation

of handling medium to large size proteins due to the huge solution

space. The largest number of secondary structures that can be

handled using a single desktop computer is about 9 helices [18].

Another approach uses Monte Carlo simulation to sample the

solution space [19]. Although this method can work with a large

solution space, the stochastic nature of the approach may miss the

native topology. The third, perhaps the most effective approach is

to translate the topology problem into a graph problem by

exploiting the constraints from a pair of sticks. Gorgon is such a

graph-matching method [26]. It produces two graphs, one

represents to the connectivity relationship of the sticks derived

from the volumetric map, and the other represents the linear

relationship of the segments on the proteins sequence. The

secondary structure assignment problem is then translated to an

inexact graph matching problem. Gorgon uses A* search in

matching the two graphs. The time complexity of A* depends on

the heuristics used. The worst case is the entire solutions space,

but often a significant portion of the entire space is explored.

We previously formulated the SSE topology problem into a

constraint graph problem and gave a dynamic programming

algorithm for the simplest situation in which [27]. More

importantly, our previous algorithm determines the optimal

solution that often fails when the optimal solution is not the true

topology due to the various errors in the data. We noticed that the

true topology is often near the top but not necessarily the top-1.

We illustrate an approach to find the top-K topologies using a

combination of a dynamic programming and the K-shortest paths

algorithms. In reality, determining the topology in a protein with

both α-helices and β-sheets is much harder, although the principle

is the same. The close positioning of the β-strands in a β-sheet

requires additional constraints from knowledge. On the other

hand, the quality of the skeleton plays an important role in the

quality of the results. In this paper, we will use a new tool to

extract the skeletons from CryoEM density maps (being

reviewed). The use of the new tool demonstrates that the quality

of the skeleton is one of the main factors to solve the problem of

the topology determination. Our fast algorithms make it possible

for a generic desktop computer to derive the true topologies

automatically for large proteins such as those containing 20-33

helices. This was not possible previously without user

intervention.

2. METHODS

2.1 Edge weight obtained from tracing the

skeleton

Figure 1. SSEs and the topology. (A) The density map (grey) was simulated to 10 Å resolution using protein 3PBA

from the Protein Data Bank (PDB) and EMAN software [1]. The SSEs (red: helix sticks, blue: sheet) were detected

using SSETracer, an extended version of Helix Tracer [2], and viewed by Chimera. For clear viewing, only SSEs at

the front of the structure are labeled. Arrows: the direction of the protein sequence; (B) The true topology of the sticks

(arrow, cross and dot for directions); (C) H1 to H10 : helix segments; E1 to E4: β-strands; ". . .": loops longer than 2

amino acids.

ACM-BCB 2013 751

We previously translated the SSE topology problem using a

weighted directed graph [27]. Briefly, there are

regular nodes in the graph, where is the number of SSE

segments on the protein sequence and is the number of SSE

sticks detected from the density map. Each node represents one

possible assignment between one SSE sequence segment and one

SSE stick in one of the two directions. Most of the edge weights

in the graph were assigned by tracing the skeleton. For any

possible edge, the weight of the edge is the absolute difference

between the virtual length of the loop (the number of Since the

skeleton contains the connection information between the SSEs,

the length from tracing the skeleton can be used as strong

constraints in matching the SSEs. However, the skeleton often

contains gaps and misleading points. In order to estimate the

length along the skeleton correctly, detailed analysis is needed.

The skeleton is represented as a set of voxel points. The main

idea of the tracing algorithm is to translate each voxel point at the

skeleton to a node in undirected graph. The voxel points (nodes)

represent the end of secondary structures are also marked. The

edges between any two nodes depend on the distance between the

two original voxel points. If the distance is less than 3.0 Å, the

two nodes are considered neighbors and an edge is created to

connect them. The weight of the edge equals to the distance

between the two corresponding voxel points. Bron-Kerbosch

algorithm [28] then applied to the graph to find the cliques of at

least size 3. The purpose of finding the cliques is to find the

crowded regions on the graph. The set of nodes involved in the

clique are replaced with one central node, the geometrical central

of all voxels of the clique. The depth first search (DFS) was used

to find the paths between a pair of ending points of two SSEs.

Some of the paths are complete paths. An incomplete path will be

found when a gap exists in the skeleton. For example, in Figure

2B, there are 3 complete paths from node P and one incomplete

path <P, R, S>. All paths found for each SSE end is saved in a

list 𝑒𝑛𝑑𝐿 𝑠𝑡 , where 𝑡 ∈ { }𝑎𝑛𝑑 ≤ ≤ . The variable 𝑡

represents which end on the stick the paths starts from. The length

of each path is simply the summation of weights of edges along

the path.

The process of edge update starts once all lists are built for all

SSE ends. For each edge 𝑒( 𝑡 𝑡 ) on the

graph, we find the complete path in 𝑒𝑛𝑑𝐿 𝑠𝑡 𝑡 or the two

incomplete paths in 𝑒𝑛𝑑𝐿 𝑠𝑡 𝑡 𝑎𝑛𝑑 𝑒𝑛𝑑𝐿 𝑠𝑡

𝑡 that best fit the

number of amino acids on the loop. 𝑡 is the complement

of 𝑡 denotes the other end of the stick . We simply search for a

complete or incomplete path with a length that best fit the

estimated length of the loop on the sequence. The estimated length

of the loop on the sequence is calculated by multiplying the

number of amino acids by 3.8. In both cases, complete or

incomplete, the length of the path should not exceed the estimated

length of the loop plus e=5 Å. Verifying complete paths against

loop is very simple. We trivially compare the two lengths. For

incomplete paths, we try all combination of incomplete paths

between the two lists that are at most 15Å apart. The length of the

new path produced from the two incomplete paths is the

summation of incomplete paths, one from each list, and the gap

between them. For example, in Figure 2B, the two incomplete

paths <P, R, S> and <T, Q> form one complete path <P, R, S, T,

Q>. The new weight of the edge is the absolute difference

between the length of the loop on the sequence and the best path

from the list. The weight of any edge does not have a proper path

(complete or incomplete) on the CryoEM volume map is changed

to ∞.

2.2 K-shortest paths satisfying the constraints The shortest valid path represents, in theory, the best match

between the SSEs on the protein sequence and those in the 3-

dimensional image. The K-shortest paths here refer to the K valid

paths, the score of which are in non-decreasing order. The main

constraint in the topology graph requires that a valid path cannot

visit the same row twice, neither can it visit the same column

twice. Due to the topology constraints, we cannot apply directly

the available K-shortest path algorithms. Instead, we combined the

concept of the “generalization of the Yen’s algorithm” of paper

[29] with our dynamic programming method to find the

constrained K-shortest paths.

Let the th shortest valid path from <START> to <END> be

represented by

. Let

be the subpath of that includes the consecutive vertices of

between vertex and ≤ ≤ . In order to find

the -th shortest valid path, a “reverse pseudo tree” of the shortest valid paths, , is maintained. As mentioned in the

original K-shortest algorithm, is a pseudo tree because there

might be repeated nodes in the tree. We call it the reverse pseudo

tree since the root is in our case. The method to build

the pseudo tree was detailed in [29]. As an example (Figure 3), the

reverse pseudo tree maintains the “coninciding nodes” where

joins one of the paths and never deviates.

The idea of finding the next shortest path is that the th

shortest path is not too different from the previous shortest

paths. It is at least one edge different from each of the previous

shortest paths. At each cycle, new candidates for the th

shortest path are generated in an edge deletion process and are

deposited in , a set of the candidate paths. The th shortest

path is to be selected as the shortest path from X at iteration k+1.

Figure 2. The skeleton and tracing the skeleton to derive

the edge weight for the topology graph. A: The density map

(gray) of protein (PDB ID:3IXV_A) was extracted from the

entire density map (EMDB ID:5100) and is superimposed

with the skeleton (blue) and the true atomic structure (green

ribbon). The region of the skeleton containing gap is

highlighted with a box. B: Automatic tracing of skeleton to

overcome a gap between point S and point T.

A B

ACM-BCB 2013 752

2.3 β-Sheet Constraints

We designed the following constraints to bias towards the

popular topologies such as antiparallel strands (Figure 4(a)) with

short loops.

Short loop and strand spacing - This constraint reflects the fact

that two consecutive -strands on the protein sequence are more

likely to be neighboring strands in the density map. Let (E1, E2, ..,

) be the list of -strands from the N-terminal to the C-terminal

on the protein sequence and (B1, B2, .., ) be the list of -sticks

in the density map. When the loop connecting two β-strands has

less than five amino acids, this constraint applies. We require that

𝑎 𝑎 , in which 𝑎 ,

≤ ≤ ≤ and 𝑎 ⌊ ⌋ ≤

≤ . is a tolerance parameter. is the measured

shortest Euclidian distance between two -sticks Bk, and Bl. We

set a penalty of 𝑎 𝑎 to the edge weight if

two connected nodes have 𝑎 𝑎 .

Antiparallel strands - When two consecutive SSEs in sequence

are assigned to two sticks that are immediate neighbors, we bias

towards antiparallel strands when the loop is not long enough to

make a parallel relationship. When the loop is shorter than the

length of the second β-stick, we require (Figure 4(c)).

A penalty of 150 was given for the violation.

Three strands - For most of the popular topologies, three

consecutive strands form an antiparallel relationship. A penalty

was given if and 𝑑 or if

and 𝑑 𝑎 .

Neighboring strands - This constraint awards the assignment of

two consecutive β-strands on the sequence towards two neighbors.

When the loop between the β-strands is less than 5 amino acids,

we set a reward of .

Long Helix Matching - If the length between a long helix in the

sequence and that of the α-stick is less than 15% of the stick, a

reward of -5 was given.

3. RESULTS We tested the top-K SSE matching algorithm using three data

sets. The first data set contain proteins with only -helices and the

skeleton was generated using Gorgon [17]. The second dataset

contains -proteins, but the skeleton was generated using another

method [30]. The third dataset contains proteins with both -

helices and -sheets. All the tests were run on a generic PC – Dell

Optiplex 980 machine at 2.8 GHz and 8 GB of memory. All the

tests were run on a generic PC – Dell Optiplex 980 machine at 2.8

GHz and 8 GB of memory.

3.1 Test using α-proteins The first dataset contains eight proteins, among which six are

simulated density maps and two are experimentally derived

CryoEM maps. The simulated density maps were produced to

10Å resolution using the protein structures from the PDB and

EMAN software [1]. Two experimentally derived density maps

EMDB_5100 (6.8Å ) and EMDB_5030 (6.4Å) were downloaded

from the EMDB [11] . The N-terminal portion of the structure

(222 amino acids) contains only α-helices and its corresponding

portion of the density map (EMDB_5100) was used. For each

density map, we used SSETracer [13] to detect the helix sticks.

We built the topology graph and assigned the edge weight by

tracing the skeleton. The top 35 ranked topologies were produced

for each protein using our constrained K-shortest path algorithm.

Our algorithm was able to find the native topology within the top-

35 ranked topologies for all of the eight cases. The native

<START>

<END>

(1,1,1)

(2,2,1)

(5,3,0)

p1

<START>

<END>

(1,1,1)

(2,2,1)

(5,3,0)

p1q1p1 q2p1q3p1

Coin(q2p1)

C=6C=1 C=1

Coin(q3p1)

<START><START>

(4,2,1)

(2,1,0)(1,1,0)

C=2

Coin(q1p1)

<START>

(5,1,1)

(4,2,0)

(2,3,0)

C=2

<START>

<END>

(1,1,1)

(2,2,1)

(5,3,0)

p1q1p1 q2p1q3p1

Coin(q2p1)

C=6C=1

Coin(q3p1)

<START><START>

(4,2,1)

(2,1,0)(1,1,0)

C=2

Coin(q1p1)

<START>

(5,1,1)

(4,2,0)

(2,3,0)

C=2

p2

q2p2 q3p2

Coin(q2p2)

Coin(q3p2)

<START><START>

(3,3,1)

(1,2,1)

C=3

N/A

C=∞

(1,2,0)

<START>

Coin(q1p2)

(5,1,0)

(2,3,0)

q1p2C=2

p3p2

Figure 3: An example of the reverse pseudo tree for the first three shortest paths.

Figure 4. Popular topologies and -sheet constraints. (a)

probability popular topology; (b) a rare topology; The

beginning and ending of a strand is labeled in (c). The

diagonal DEE is generally longer than the side of a rectangle

DES in this case.

ACM-BCB 2013 753

topology was ranked top-1 for four of the eight protein density

maps (Table 1). This dataset contains five large proteins with 20-

33 helices. We selected this dataset to see how well our algorithm

performs on the large and complicated density maps. The largest

protein (row 6 of Table 1) has 585 amino acids in length, 33

helices, among which 20 helices were detected by SSETracer, yet

the native topology was ranked the 4th. This suggests that although

the SSE detection affects the accuracy of our algorithm, it is the

overall SSEs that matter. The time and memory (Table 1) includes

that for building the graph and for finding the top-35 assignments

between the sequence segments and the sticks. The major time in

searching for the top-ranked topologies is to build the dynamic

programming tables in the order of . Once the tables

are built, it takes to find the top-K topologies where

is (analysis details in a separate paper currently under

review). In comparison to Gorgon, a popular interactive tool, our

method uses less time and memory yet rank the native topology

higher. Note that the experiment was done using the same

skeleton.

Table 2 shows the evaluation results when the skeleton was

generated using a recently developed method (paper under

review). The new skeleton appear to have less gaps that the

skeleton Gorgon produced that was used in the results of Table 1.

Gorgon performs better for this dataset when the new skeleton is

used. Gorgon was able to find the true topology for 20 out of 22

proteins in the data set, although our method works slightly better

in ranking the native topologies.

3.2 Test with β-sheets The topology graph and the dynamic programming algorithm

apply, in principle, to both -proteins and / proteins. In

practice, it is more challenging to derive topologies for proteins

with -sheets due to the close spacing of about 4.5 between two

-strands. We applied additional constraints to bias towards

known popular topologies of -sheets. We used seven simulated

density maps (8 resolution) and two experimentally derived

maps in the test. The -strand locations were visually detected

since there is no automatic tool to detect β-strands from a β-sheet.

Our program produced a list of candidate topologies that are

ranked by the score.

It appears that the framework of the top-K topology algorithm

generally applies to the proteins with both -helices and -sheets.

It was able to rank the native topology among the top 25 for 7 out

of 9 proteins when no constraints were added for β-sheets (column

6, Table 3). The β-sheet constraints are effective in identifying the

native topology. For example, EMDB ID-1733 has 17 sticks with

5 α-sticks and 12 β-sticks. The native topology was not found

within the top 100 topologies without β-constraints, but was

ranked the 13th out of 7.5e+15 total possible topologies after using

the constraints. Although there are

different topologies, the ones that satisfy the density requirement

and the -sheet constraints can be quite limited. The results in this

paper further demonstrated our previous finding about the

amazing properties of SSE topologies that is the native topology is

Table 1: Improved accuracy, space and time for topology

identification.

No

. IDa #AA

#hlces

b

#stick

sc

Our algorithm Gorgon

Space/timed Ranke space/timed Ranke

1 1FLP 142 7 7 0.004/<=2 1 0.64/<=2 1

2 1Z1L 345 23 14 18.59/2.4 1 >934.6/42.6 N/A

3 3ODS 415 21 16 42.34/2.9 2 377.4/15.2 23

4 1HZ4 373 21 19 273.00/14.7 3 458.8/40.3 N/A

5 3HJL 329 20 20 236.9/<=2 1 N/A N/A

6 2XVV 585 33 20 1225.1/126 4 >1312.9/276 N/A

7 3IXV

222 14 10 0.004/4.2 1 >922.0/30.3 N/A 5100

8 3FIN

117 4 4 0.004/<=2 4 0.48/<=2 N/A 5030

a: The PDB ID/EMDB ID of the protein/CryoEM volume map. b: The number of actual helices in the protein.

c: The number of detected helices from volume map.

d: The space (in MB) and time (in Sec.) needed to rank top 35

topologies. The sign > means that the task could not be

completed. e: The rank of the true topology within top 35 topologies. N/A

means the true topology could not be ranked within top 35

topologies.

Table 2: Improved accuracy with the new skeletons.

No. IDa #A

A

#hlces

b

#stick

sc

Our algorithm

Rankd

Gorgon

Rankd

1 3THG 107 4 4 1 1 2 3IEE 270 9 8 1 16

3 1HG5 289 11 9 1 1

4 2OVJ 201 12 9 2 2

5 2XB5 207 13 9 2 1

6 1P5X 245 13 9 6 22

7 3HJL 329 20 20 1 1

8 1BZ4 144 5 5 1 1

9 1HZ4 373 21 19 17 1

10 1I8O 114 6 5 30 N/A

11 1JMW 146 6 4 1 1

12 1LWB 122 6 6 2 1

13 1NG6 148 9 7 1 3

14 1XQO 256 14 14 14 N/A

15 2IU1 208 13 10 4 2

16 2PSR 100 5 4 5 10

17 2PVB 108 8 5 15 28

18 2VZC 131 7 6 1 24

19 2X3M 239 12 8 2 1

20 3ACW 293 17 14 3 2

21 3HBE 204 11 9 2 7

22 3LTJ 201 16 12 1 1

a: The PDB ID of the protein.

b: The number of actual helices in the protein.

c: The number of detected helices from volume map.

e: The rank of the true topology within top 35 topologies.

Figure 5. An example of a / proteins. (a) Protein 1ICX,

including PDB structure (yellow), density map (transparent

yellow), and new skeleton (red). (b) SSE sticks and the loop

traces between SSEs for the true topology (Rank 1). The loop

traces are high-lighted with red color. Purple sticks for beta-

strand sticks and cyan sticks for alpha-helix sticks. (c) SSE

sticks and the loop traces between SSEs for the wrong

topology (Rank 25). The correct loop traces are high-lighted

with red color. The wrong traces are marked with yellow,

green and blue respectively. Purple sticks for beta-strand

sticks and cyan sticks for alpha-helix sticks.

ACM-BCB 2013 754

near the top of the entire topological space [31]. Figure 5 shows

an example of the correct and incorrect topology.

4. CONCLUSION The topology of the secondary structure elements detected from

the density map is a critical piece of information for deriving the

atomic structures from such density maps. This paper illustrated

the K-shortest paths approach to the SSE topology problem. The

algorithms were tested using data sets involving proteins with

large number of SSEs in both α-proteins and α/β proteins. The

effectiveness of the algorithms was demonstrated in the ability of

ranking the native topologies to the near top for α-proteins up to

33 helices in which 20 of them were detected in the density map

and α/β-proteins up to 12 β-strands and 5 α-helices. The results

represent a major improvement in the ability to derive the

secondary structure topology automatically for large and

complicated density maps.

5. ACKNOWLEDGMENT Correspondence to: Jing He, [email protected].

The funding of this work is in part by NSF-CREST HRD-

0420407, the start-up fund and MSF fund of Old Dominion

University.

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Table 3. The evaluation of / proteins.

ID/EMDB

#H

elicesa

#S

trand

b

Sh

eet_ID

c

#T

otal

d

NO

PT

e

Ran

kf

5030 4/3 4/3 A 3.7e+04 1 1

1733 5/5 12/12 O,P,Q 7.5e+15 -/100 13

1OZ9 5/5 5/4 A 7.7e+05 25 7

2KUM 2/2 3/3 A 3.8e+02 5 1

2KZX 3/3 3/3 A 2.3e+03 10 10

2L6M 2/2 3/3 A 3.8e+02 6 6

1BJ7 5/1 9/9 A 1.9e+09 -/100 4

1ICX 6/3 7/7 A 6.2e+08 2 1

1JL1 4/4 5/5 A 1.5e+06 22 16

a: The number of -helices in the protein sequence and the

number of -sticks detected from the density map;

b: The number of -strands in the protein sequence and the

number of -strands visually detected;

c: -Sheet ID;

d: The number of total possible topologies;

e: The rank of the native topology without β-constraints; -

/100: the native topology not found in top 100 topologies

f: The rank of the native topology with β-constraints

ACM-BCB 2013 755

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