A spatiotemporal mining approach towards summarizing and analyzing protein folding trajectories
 Hui Yang^{1}Email author,
Affiliated with
 Srinivasan Parthasarathy^{2} and
Affiliated with
 Duygu Ucar^{2}
Affiliated with
DOI: 10.1186/1748718823
© Yang et al. 2007
Received: 04 August 2006
Accepted: 04 April 2007
Published: 04 April 2007
Abstract
Understanding the protein folding mechanism remains a grand challenge in structural biology. In the past several years, computational theories in molecular dynamics have been employed to shed light on the folding process. Coupled with high computing power and large scale storage, researchers now can computationally simulate the protein folding process in atomistic details at femtosecond temporal resolution. Such simulation often produces a large number of folding trajectories, each consisting of a series of 3D conformations of the protein under study. As a result, effectively managing and analyzing such trajectories is becoming increasingly important.
In this article, we present a spatiotemporal mining approach to analyze protein folding trajectories. It exploits the simplicity of contact maps, while also integrating 3D structural information in the analysis. It characterizes the dynamic folding process by first identifying spatiotemporal association patterns in contact maps, then studying how such patterns evolve along a folding trajectory. We demonstrate that such patterns can be leveraged to summarize folding trajectories, and to facilitate the detection and ordering of important folding events along a folding path. We also show that such patterns can be used to identify a consensus partial folding pathway across multiple folding trajectories. Furthermore, we argue that such patterns can capture both local and global structural topology in a 3D protein conformation, thereby facilitating effective structural comparison amongst conformations.
We apply this approach to analyze the folding trajectories of two small synthetic proteinsBBA5 and GSGS (or Beta3S). We show that this approach is promising towards addressing the above issues, namely, folding trajectory summarization, folding events detection and ordering, and consensus partial folding pathway identification across trajectories.
1 Background
The three dimensional (3D) native structures of proteins have important implications in proteomics. Understanding such structures enables us to explore the function of a protein, explain substrate and ligand binding, perform realistic drug design and potentially cure diseases caused by protein misfolding. The protein folding problem is therefore one of the most fundamental yet unsolved problems in computational molecular biology. One major challenge in simulating the protein folding process is its complexity. Snow et al. state that performing a Molecular Dynamics (MD) simulation on a miniprotein for just 10 μs would require decades of computation time on a typical CPU [1]. Researchers in the Folding@home project recently proposed a World Wide Webbased computing model to simulate the protein folding process [2].
As the volume of folding trajectories produced from highthroughput simulation tools increases drastically, there is an urgent need to compare, analyze, and manage such data. Previously, researchers have examined several summary statistics (e.g. radius of gyration, root mean square deviation (RMSD)) to identify similar 3D conformations in folding trajectories. Although summary statistics are commonly used for comparison, they can only capture biased and limited global properties of the conformation. Recently, Russel et al. [3] suggested using geometric spanners for mapping a simulation to a more discrete combinatorial representation. They apply geometric spanners to discover the proximity between different segments of a protein across a range of scales, and track the changes of such proximity over time.
To overcome the difficulties in managing and analyzing the large amount of protein folding simulation data, Berrar et al. [4] proposed using a data warehouse system. They embed the warehouse in a grid computing environment to enable data sharing. They also propose implementing a set of data mining algorithms to facilitate commonly needed data analysis tasks.
In this article, we propose a spatiotemporal mining approach to analyze folding trajectories. We extend the spatiotemporal data mining framework that we have developed earlier to analyze and manage such data [5]. This framework is designed to analyze spatiotemporal data produced in several scientific domains. Previously, we have applied this framework to analyze 8732 proteins taken from the Protein Data Bank to identify structural fingerprints for different protein classes (e.g., αproteins) [6]. Each protein is associated with a set of objects that are extracted from its contact map. We then realize the notion of Spatial Object Association Pattern (SOAP) to effectively capture spatial relationships among such objects, Furthermore, by associating SOAPs with proteins in different protein classes, we have identified multiple types of SOAPs that can potentially function as the structural fingerprints for different protein classes. In this article, we extend such strategies to a new application domain: analyzing and characterizing the folding process of a protein.
Clearly, protein folding trajectories consist of both spatial and temporal components. Each protein in a MD simulation is composed of a number of residues spatially located in the 3D space that move over time. Each frame (or snapshot) of the trajectory can be represented as a 2D contact map, which captures the pairwise 3D distances between residues. We extract nonlocal bitpatterns from these contact maps. We then use an entropybased clustering algorithm to cluster such bitpatterns into groups. These bitpatterns are further associated to form spatial object association patterns (SOAPs). By using SOAPs, we are able to effectively summarize and analyze folding trajectories produced by MD simulations. A major advantage of this representation is its appropriateness for crosscomparison across different simulations, as discussed in later sections.
Compared to our previous work on protein structural analysis [6, 7], we have made the following contributions:

Propose a contact mapbased approach to analyze protein folding trajectories: Our previous work focused on identifying structural signatures in native conformation of proteins in different classes or folds. Thus, there is no temporal component involved. In contrast, a folding trajectory has both spatial and temporal components. In addition, bitpatterns in a folding trajectory will interact with each other and evolve over time. Moreover, the proposed approach also effectively integrates 3D structural information in the overall analysis. This is critical in understanding the protein folding mechanism.

Map 2D bitpatterns in contact maps with 3D structural motifs: To better understand and explain the biological meaning of the bitpatterns in contact maps, we have made an effort to establish a mapping between such bitpatterns and wellknown structural motifs (e.g., αhelices and βturns) in 3D conformations. Currently, this task is carried out manually. We are in the process of automating this mapping. Such a 2D3D mapping is essential to folding data analysis due to the following reasons: First, to gain insight into the folding process, it is critical to identify the formation of important local 3D motifs such as βturns. Second, our previous studies show that by associating multiple bitpatterns in contact maps, one can construct effective structural signatures for different protein classes or folds [6]. This leads us to hypothesize that a mapping might exist between 2D bitpatterns in contact maps and 3D local motifs of a protein. In this work, we validate this hypothesis and report the mapping result later. Finally, such a mapping not only enables one to take advantage of the simplicity of working in the 2D space of contact maps, but also allows one to relate to the 3D space of protein conformations. This is important in understanding the protein folding process.

Indirectly capture interactions among structural motifs in 3D space: In our previous work, two bitpatterns are considered spatially proximate if they are located in the same vicinity within a 2D contact map. This is problematic in the context of protein folding, as two bitpatterns can be spatially proximate in a contact map even though their corresponding motifs are distant in the 3D conformation. (See Section 3 for more details.) We address this issue by considering the 3D distance between two bitpatterns.

Propose novel strategies to analyze protein folding trajectories: We propose several novel strategies to analyze protein folding trajectories based on spatial and spatiotemporal association patterns.
In summary, one can benefit from our mining approach in two main aspects:

Effective, informative and scalable representation of folding simulations: We represent each frame by a set of SOAPs, where each SOAP in turn characterizes the spatial relationship (or interactions in the folding case) among multiple bitpatterns. SOAPs are not only easily obtainable but also, as we will show, able to capture folding events along a folding trajectory.

Crossanalysis of trajectories to reveal a consensus partial folding pathway: By representing each frame as a set of SOAPs, one can carry out analysis across different trajectories. Such analysis includes detecting critical events and identifying consensus partial folding pathways across trajectories.
The remainder of the article is organized as follows. In Section 2, we describe the two proteinsBBA5 and GSGSand their trajectories produced from computational simulation. We also identify two main goals to analyze such trajectories. In Section 3, we present a stepbystep description of our analysis approach. We next report the empirical results on analyzing the trajectories of the two proteins in Section 4. We focus on the protein BBA5. Finally we conclude and report several ongoing research directions in Section 5.
2 Analyzing Protein Folding Trajectories
2.1 Protein Folding Trajectories
Advances in highperformance computing technologies and molecular dynamics have led to successful simulations of folding dynamics for (small) proteins at the atomistic level [8]. Such simulations result in a large number of folding trajectories, each of which consists of a series of 3D conformations of the protein under simulation. These conformations are usually sampled regularly (e.g., every 200fs) during a simulation. In this article, we also refer to each conformation as a folding frame or simply a frame. Furthermore, to represent a protein conformation, we adopt one of the commonly adopted representation schemes, where a conformation is represented as a sequence of αcarbons (C _{ α }) located in 3D space.
In this article, we focus on the folding trajectories of two mini proteins: BBA5 (Protein Data Bank ID) [9] and GSGS (orBeta3s) [10, 11]. Such trajectories were produced by the Folding@ home research group at Stanford University [12].
A brief description of the GSGS folding trajectories.
Trajectory ID  Total number of conformations 

T _{1}  25,664 
T _{2}  30,075 
T _{3}  19,649 
T _{4}  25,263 
T _{5}  25,664 
2.2 Comparing Conformations of BBA5 and GSGS Across Trajectories
Although both trajectories of BBA5 start from the same extended conformation as shown in Figure 1(b), when we examine the visualized frames, they seem to identify two very different folding processes. Figures 1(c) and 1(d) illustrate the last frame in the two trajectories T _{23} and T _{24} respectively. This also applies to the five GSGS folding trajectories, where each starts with the same conformation (Figure 2(b)) but ends at a different conformation (Figures 2(c), 2(d) &2(e)).
This seeming difference might be attributed to the stochastic nature of the folding simulation process [8, 9]. However, it is also desirable to characterize the similarities (or dissimilarities) across multiple trajectories.
To compare two trajectories, one must address the following key issue: how can we compare two protein conformations? Several measures have been commonly used towards such a purpose, including RMSD (root mean squared distance) [13], contact order [14], and native contacts [15]. However, all these measures are designed to quantify the global topology of a conformation. Furthermore, based on our empirical analysis of these measures, we notice that they are generally too coarse and thus can often be misleading. Even more importantly, such measures fail to identify similar local structures (or motifs) between conformations. This is especially crucial for small proteins like BBA5. As demonstrated in both experimental and theoretical studies, small proteins often fold hierarchically and begin locally [16]. For instance, it has been shown that BBA5 tends to first form secondary structures such as βturns and αhelices, then conform to its global topology [9]. Finally, as suggested by Pande [8], both sterics (local motifs) and global topology might play an important role in protein folding. Therefore, to compare conformations of (small) proteins, a more reasonable comparison should consider both local and global structures. Moreover, it should also take the native topology of the protein under study into account.
Partitions along the primary sequence of BBA5.
Partition  Amino Acids  Remark 

F _{1}  1–10  βhairpin 
F _{2}  11–23  αhelix 
F _{3}  1–6, 16–23  The 1^{ st } half of F _{1} and the 2^{ nd } half of F _{2} 
F _{4}  6–17  The 2^{ nd } half of F _{1} and the 1^{ st } half of F _{2} 
Partitions along the primary sequence of GSGS.
Partition ID  Amino Acids  Remark 

F _{1}  1–15  The 1^{ st } βturn 
F _{2}  1–7  The 1^{ st } βstrand 
F _{3}  3–10  Critical region of the 1^{ st } βturn 
F _{4}  6–15  The 2^{ nd } βstrand 
F _{5}  6–20  The 2^{ nd } βturn 
F _{6}  10–18  Critical region of the 2^{ nd } βturn 
F _{7}  14–20  The 3^{ rd } βstrand 
To realize the comparison of conformations, two more issues must still be addressed. First, how can one effectively capture and represent local motifs? Second, how can we represent the global topology of a conformation in terms of local motifs? To address the first issue, we leverage the nonlocal patterns in protein contact maps. For the second, we characterize the spatial arrangement among nonlocal patterns. Please see Section 3 for more details.
2.3 Folding Trajectory Analysis: Objectives
There are two main goals we would like to achieve in analyzing the folding trajectories. First, we would like to address the following issues for individual trajectories: (1) to detect (or predict) significant folding events, including the formation of βturns, αhelices, and nativelike conformations; and (2) to recognize the temporal ordering of important folding events in the trajectory. For instance, between the two secondary structures αhelix and βhairpin in BBA5, which forms earlier? What is ordering of the two events preceding a βhairpin formation: formation of two extended strands or formation of the turn?
In contrast to the first goal, our second goal concerns multiple trajectories. Specifically, we would like to identify a subsequence of similar conformations across trajectories. This subsequence of conformations is referred to as the consensus partial folding pathway. This is analogous to the Longest Common Subsequence (LCS) problem [17], but much more challenging due to the following reasons. First, we are dealing with time series of 3D protein structures. Second, we are looking for similar conformations across trajectories, and our work on mining spatiotemporal data [5].
3 Algorithm
3.1 Data Preprocessing
Same as in our previous studies on protein structural analysis [6, 7], we represent 3D protein conformations by contact maps. In order for this algorithm to be selfcontained, we next briefly go over these preprocessing steps. We also explain the rationale of such steps in the context of protein folding.
Contact Map Generation
When generating contact maps, we consider the Euclidean distances between αcarbons (C _{ α }) of each amino acid. Two αcarbons are considered to be in contact if their distance is within 8.5 Å. Thus, for a protein of N residues, its contact map is an N × N binary matrix, where the cell at (i, j) is 1 if the i ^{ th } and j ^{ th } αcarbons are in contact, 0 otherwise. Since contact maps are symmetric across the diagonal, we only consider the bits below the diagonal. Furthermore, we also ignore the pairs of C _{ α } atoms whose distance in the primary sequence is ≤ 2, as they are sure to be in contact. This step transforms the two BBA5 trajectories into two series of contact maps, with each contact map of size 23 × 23. By the same token, the 5 GSGS trajectories are transformed into 5 sequences of contact maps.
Identifying Maximally Connected Bitpatterns
Every bit in a contact map has eight neighbor bits. For an edge position, we assume its outofboundary positions contain 0. In a contact map, a connected bitpattern is a collection of bit1 positions, where for each 1, at least one of its neighbors is 1. Correspondingly, we define a maximallyconnected bitpattern (also referred to as a bitpattern in this article) to be a connected pattern p where every neighbor bit not in p is 0. We apply a simple region growth algorithm to identify all the maximallyconnected patterns in each contact map within the two series of contact maps, corresponding to the two folding trajectories of BBA5. Altogether, we identified 352 maximallyconnected bitpatterns in such contact maps. For the GSGS folding data, a total of 50,572 unique bitpatterns are constructed. We then represent each identified bitpattern as a 6tuple feature vector consisting of the following attributes:

Height: the number of rows contained in the pattern's Minimum Bounding Rectangle (MBR).

Width: the number of columns in the pattern's MBR.

NumOnes: the number of 1s in the pattern.

Slope: the general linear distribution trend of all the 1s in the pattern within its MBR. To compute the angle of a connected pattern we use the leastsquares method to estimate the slope of a linear regression line. For a pattern containing n 1s, we denote the positions of the 1s as: (x _{1}, y _{1})...(x _{ n }, y _{ n }). The leastsquares method then estimates the slope β _{1} as:

xStdDev: the standard deviation of all the 1s' xcoordinates (this quantifies how the 1s spread along the x dimension).

yStdDev: the standard deviation of all the 1s' ycoordinates.
Note that this feature vector captures the main geometric properties of a bitpattern.
As discussed in the literature [18–21], nonlocal patterns (where bitpatterns are one type of nonlocal patterns,) in contact maps can effectively capture the secondary structure of proteins. Our previous work [6, 7] demonstrated that by characterizing the spatial relationship among the above described bitpatterns, one can construct structural signatures for proteins of different classes or folds. In the context of protein folding, we have observed that the abovedefined bitpatterns are also capable of capturing a wide range of local 3D structural motifs. They can even approximately measure the strength of secondary structure propensity in a conformation. For instance, we have identified bitpatterns that correspond to "premature" αhelices and nativelike αhelices respectively. Henceforth, we refer to the 3D structure formed by all the participating residues of a bitpattern as the 3D motif of the bitpattern. The relationship between bitpatterns and 3D motifs will be further discussed in the next section.
Clustering Bitpatterns into Approximately Equivalent Groups
In this step, we partition the extracted bitpatterns into approximately equivalent groups, each of which consists of bitpatterns that exhibit similar geometric properties (e.g., shape and size). To construct such equivalent groups, we run the kmeans based clustering algorithm [22] over the bitpatterns' corresponding feature vectors, where k is the number of clusters (or equivalent groups) that will be produced.
To determine an optimal value of k, we take the following three steps. First, we run the clustering algorithm on different k values. This produces different clustering schemes for the same set of bitpatterns. Second, for each clustering scheme, we compute its entropy. Let c _{1}, ..., c _{ l } be the l clusters after clustering the set of N bitpatterns. Furthermore, each cluster c _{ i } (1 ≤ i ≤ l) has an individual entropy H _{ i } and contains N _{ i } elements, then the total entropy of this clustering is given by the following formula: The entropy of each individual cluster, i.e., H _{ i } , is computed by summing up the entropy of each of the six bitpattern attributes such as its height and width. For an attribute, we compute its entropy in a cluster according to the procedure explained by Shannon [23]. In the third and final step, we plot the entropy against the number of clusters, i.e., k, and choose a value k where the entropy plot begins to show a linear trend. For the BBA5 folding data, this clustering step groups the 352 bitpatterns into 10 clusters (or types). As for the GSGS data, 12 clusters are identified.
We also observed a similar scenario for the 12 types of bitpatterns identified in the GSGS trajectories. For instance, the typical 3D motifs of type 0 bitpatterns resemble the native conformation of GSGS (See Figure 2(a)); whereas those of type 6 identify with αhelices.
This demonstrates, to a certain extent, the advantage of using 2D contact maps to analyze 3D protein conformations. Undoubtedly, using contact maps greatly reduces the computational complexity, though at the cost of loss in structural information. However, some of this information loss is recompensated by mapping bitpatterns to structural motifs in 3D conformations. More importantly, by exploiting different features in contact maps (bitpatterns in this work), we are able to connect 2D features with features in 3D space. In the BBA5 case, by identifying 10 types of bitpatterns in contact maps, we indirectly recognize 10 different 3D structural motifs in the folding conformations.
Relabeling Bitpatterns with The Corresponding Cluster Label
In this step, we relabel all the previously identified bitpatterns with their corresponding cluster label. Let p be a labeled bitpattern. It can be represented as follows: p = (trajID, frameID, listC _{ α }, label). Here, trajID identifies a folding trajectory, and frameID indicates the frame where p occurs, listC _{ α } consists of all participating αcarbons of p, identified by their position in the primary sequence. Finally, label is the cluster label of p. For BBA5, label ∈ {g _{0}, g _{1}, ⋯, g _{9}}, corresponding to the 10 approximately equivalent groups (or types).
3.2 Mining Spatiotemporal Object Association Patterns
The preprocessing steps transform a 3D protein conformation into a set of labeled 2D bitpatterns, that indirectly capture the local 3D structural characteristics of the conformation. For the two BBA5 trajectories, each conformation contains an average of 6 bitpatterns. As for the five GSGS trajectories, the average number of bitpatterns in each conformation is 4.
As BBA5 and GSGS fold, the dynamics among their residues is constantly changing until it reaches an equilibrium. This means that two residues previously in contact may become out of contact later. As a result, bitpatterns present in one conformation may be absent in the next. The evolving nature of contacting residues and in turn bitpatterns, is essentially the consequence of a variety of weak interactions among amino acids at different levels. Such weak interactions include hydrogen bonds, electrostatic interactions, van der Waal's packing and hydrophobic interactions [24]. To capture these (potential) interactions, a simple yet effective method is to consider how close two amino acids are located from each other in 3D. We also adopt this method here. Specifically, we consider interactions between local 3D motifs captured by labeled bitpatterns. We denote such interactions as "interactions among bitpatterns". Let p _{ i } and p _{ j } be two bitpatterns in a protein conformation, and p _{ i } .listC _{ α } and p _{ j } .listC _{ α } be the list of αcarbons involved in p _{ i } and p _{ j }, respectively. We define p _{ i }and p _{ j } as interacting bitpatterns if at least one pair of αcarbons, each from p _{ i } .listC _{ α } and p _{ j } .listC _{ α } are located within a short distance δ. Note that the value of δ should be greater than the distance that is being used to identify contacting αcarbons when generating contact maps. In our analysis, we set δ = 10 Å.
So far, we have discussed our approach of using bitpatterns in contact maps to characterize local 3D motifs and further represent a protein conformation during folding. We also define the notion of interacting bitpatterns in the folding context. We are now ready to present our method of summarizing folding trajectories to fulfill the two objectives described in Section 2.3. The main idea is that we can summarize a folding trajectory by characterizing the evolutionary behavior of interactions among different types of bitpatterns and in turn, the interactions among local 3D motifs.
Definition of (minLink = 1) SOAP
As proposed in our previous work [5, 25], such interactions can be modeled and captured by discovering different types of spatial object association patterns (SOAPs). Essentially, SOAPs characterize the specific way that objects, bitpatterns in this case, are interacting with each other at a given time. Among the proposed SOAP types, after a careful evaluation, we empirically select (minLink = 1) SOAPs to model the interacting bitpatterns in the folding process. Let p = (g _{1}, g _{2}, ⋯, g _{ k }) be a (minLink = 1) SOAP of size k, where g _{ i } is one of the 10 types of bitpatterns described above. In the context of folding trajectories, p prescribes that there exists k bitpatterns b _{1}, b _{2}, ..., b _{ k } in a conformation, where b _{ i }.label = g _{ i } (1 ≤ i ≤ k). Furthermore, for each b _{ i }, it interacts with at least one of the remaining (k  1) bitpatterns. Note that the k labels in p are not mutually exclusive. For instance, one can have SOAPs such as (7 9 9), which involves one type 7 bitpattern and two type 9 bitpatterns.
We further restrict ourselves to SOAPs that occur frequently during the folding process (frequent SOAPs). However, we are not ruling out rarelyoccurring SOAPS in our future studies. A SOAP is said to be frequent if it appears in no fewer than minSupp frames in a trajectory. In our studies, we set minSupp = 5 for BBA5 and 10 from GSGS.
SOAP Episodes
The next step is to capture the evolutionary nature of the folding process. We do this by identifying the evolutionary nature of SOAPs. As mentioned earlier, small proteins like BBA5 and GSGS often fold hierarchically and begin with local folded structures. As they fold, new SOAPs can be created and existing one can dissipate. To capture such evolutionary behavior, we proposed the concept of SOAP episodes, which provide an effective approach to model the evolution of interactions among spatial objects over time [5]. To reiterate, a SOAP episode E is defined as follows: E = (p, F _{ beg }, F _{ end }), where p is a SOAP composed of one or more bitpatterns, p was created in frame F _{ beg } and persisted till frame F _{ end }. Note that for a given p, it can be created more than once during protein folding, and thus can have more than one episode. To discover frequent (minLink = 1) SOAPs and their episodes in the trajectories of BBA5 and GSGS, we apply our SOAP mining algorithm as explained in our previous work [5].
In summary, this mining phase produces the following results: (i) A list of (minLink = 1) SOAPs of bitpatterns that appeared in at least 5 conformations in each folding trajectories for the protein BBA5 and 10 for GSGS; and (ii) A list of episodes, ordered by beginning frame F _{ beg }, associated with each of these SOAPs.
3.3 Folding Trajectory Analysis
In this section, we describe our strategy on utilizing SOAPs to summarize a folding trajectory and address the two folding analysis issues described in Section 2.3.
SOAPbased Trajectory Summarization
The previous mining phase discovers a collection of frequent (minLink = 1) SOAPs and the associated episodes in each trajectory. Therefore, it identifies all the conformations in the trajectories that contain at least one frequent (minLink = 1) SOAPs. For instance, the last conformation in trajectory T23 (Figure 1(c)) has two SOAPs of size 2:(5 8) (i.e., association of a type 5 and a type 8 bitpattern) and (7 8), and three SOAPs of size 1: (5), (7), and (8), while the last conformation in trajectory T24 has three SOAPs: (7 8), (7) and (8). This leads to our SOAPbased approach for folding trajectory summarization.
To summarize a folding trajectory, we perform the following three steps. First, for each conformation, we identify all the frequent SOAPs that appear in it and use these SOAPs to represent this conformation. Note that not every conformation contains frequent SOAPs, especially when minSupp is set high. Second, for each SOAPrepresentable conformation, we carry out two tasks on its associated SOAPs. We next use the folding trajectories of BBA5 to explain how these two tasks are carried out.
Detecting Folding Events and Recognizing Ordering Among Events
Once each folding trajectory is summarized into generalized SOAPs, it is fairly straightforward to detect folding events such as the formation of αhelix or βturn like local structures. This can be done by simply locating the frames that contain the local motif(s) of interest. We can also easily identify nativelike conformations, by finding those that contain the generalized SOAP (β.1 α.2). Finally, based on the summarization, one can quickly identify the ordering of folding events in a trajectory. For instance, to check which secondary structure forms more rapidly, αhelix or βhairpin, one can simply compare the first occurrence of these structures in the summarized trajectory (Figure 9(b)).
Identifying the Consensus Partial Folding Pathway Across Trajectories
To do this, we simply compute the longest common subsequence (LCS) [17] between two summarized trajectories. One can utilize the summarization either before the 3D motif generalization (Figure 9(a)) or after (Figure 9(b)). We use the latter in our analysis. Based on the LCS of generalized SOAPs, we construct the consensus folding pathway by identifying pairs of conformations, one from each trajectory, along the LCS of two summarized trajectories. In other words, the resulting consensus pathway consists of a sequence of conformationpairs of similar 3D structures. Notice here that the comparison between 3D protein conformations (as described in Section 2.2) is done by using bitpatterns to model local structural motifs, and associations of bitpatterns (SOAPs) to characterize the global structure. This forms a hierarchical comparison and is in accordance with the hierarchical folding process of small proteins.
4 Results
A summary of the BBA5 folding trajectories.
Protein  PDB Identifier: BBA5; Primary sequence: 23 residues; Designed protein; Native fold: Nterminal 1–10 β hairpin, Cterminal 11–23 αhelix 
Trajectory  Two trajectories: T23 and T24; T23: 192 conformations; T24: 150 conformations 
Contact map  Based on contacts between αcarbons. Two αcarbons are in contact if their Euclidian distance is ≤ 8.5 Å 
Bitpatterns  A total of 352 unique maximally connected bitpatterns were identified from all conformations; Average number of bitpatterns per conformation is 6; Bitpatterns are further classified into 10 approximately equivalent types 
Interacting bitpatterns  If at least one pair of αcarbons, one from each bitpattern, is of Euclidian distance ≤ 10 Å 
Frequent SOAPs  A SOAP is frequent if it appears in ≥ 5 conformations; A total of 444 frequent SOAPs identified in trajectory T23, and 258 in T24 
Consensus partial folding pathway  We identified a consensus partial folding pathway across the two trajectories. It is composed of 71 pairs of similar conformations, one from each trajectory 
A summary of the GSGS folding trajectories.
Protein  Name: GSGS or Beta3s; Primary sequence: 20 residues; Designed protein; Native fold: three stranded antiparallel βsheets with turns at 6–7 and 14–15 
Trajectory  Five trajectories: T1, T2, T3, T4 and T5; T1 : 25, 664 conformations; T2 : 30, 075 conformations; T3 : 19, 649 conformations; T4 : 25, 263 conformations; T5 : 25, 664 conformations; 
Contact map  Based on contacts between αcarbons. Two αcarbons are in contact if their Euclidian distance is ≤ 8.5 Å 
Bitpatterns  A total of 50, 572 unique maximally connected bitpatterns were identified from all conformations; Average number of bitpatterns per conformation is 4; Bitpatterns are further classified into 12 approximately equivalent types 
Interacting bitpatterns  If at least one pair of αcarbons, one from each bitpattern, is of Euclidian distance ≤ 10 Å 
Frequent SOAPs  A SOAP is frequent if it appears in ≥ 10 conformations; 
4.1 Detecting and Ordering Folding Events
We summarize both folding trajectories of BBA5 into a sequence of SOAPs as illustrated in Figure 9. Coincidently, both summarized trajectories consist of 64 conformations.
Based on these summarized trajectories, we can quickly identify all the conformations where the first αhelixlike or βturnlike local motifs were formed. For trajectory T _{23}, the first αhelixlike motif was identified in frame 26, and the first βturnlike local motif was formed in frame 63. For the other trajectory T _{24}, the frames were 29 and 38. This is in accordance with experimental results that αhelices generally fold more rapidly than βturns. However, since we only consider frequent SOAPs, it is very possible that we might miss the actual first formation of such local motifs. To address this issue, we might need to consider rarely occurring SOAPs. We plan to investigate this in the future. For the two events related to βturn formation, formation of two extended strands and formation of the turn, we found that for both trajectories, the formation of extended strands preceded the formation of the turn.
4.2 Consensus Partial Folding Pathway Across Trajectories
Based on the generalized trajectory summarization of BBA5, we identify a consensus partial folding pathway of length 71. In other words, 71 pairs of conformations, one from each trajectory, are considered similar to each other. Figure 3 displays four such pairs along this consensus folding pathway. For instance, the two conformations shown in Figure 3(c), corresponding to the 182^{ th } frame in the T _{23} trajectory and the 116^{ th } frame in the T _{24} trajectory of BBA5 respectively, are considered structurally similar, since both conformations exhibit an αhelix in the left half of the backbone, and a βturn in the right half.
Currently, we rely on visual tools to justify these consensus pathways. We did attempt to use several measurements that have been used previously to quantify the similarity between 3D protein conformations, but to no avail. These measurements include RMSD, contact order, and native contacts. If we identify the pathway based on the best match given by any of the above measurements, we often ended up with a very short consensus pathway (as short as 10 frames). Two conformations are said to be a best match if they have the lowest RMSD or have the smallest difference in contact order or native contacts. Moreover, different bestmatched measurements rendered very different consensus pathways. Finally, we notice that the bestmatched conformations based on any of such measurements can often exhibit very different structural characteristics. We are investigating alternative methods for quantitative validation of our results.
5 Conclusions and Ongoing Work
In this article, we present a novel approach to analyze protein folding trajectories and a case study on the small proteins BBA5 and GSGS. We capture a variety of structural motifs in the 3D protein conformations by nonlocal bitpatterns identified in their 2D contact maps. By modeling the interactions or spatial relationships among bitpatterns as SOAPs and SOAP episodes, we effectively characterize the evolutionary nature of the folding process. We also describe two methods to summarize folding trajectories by superimposing protein specific information and 3D motifs onto SOAPs. Utilizing the summarized trajectories, we demonstrate that one can detect folding events and the temporal order among events. We also show that through comparing such summarized trajectories, one can identify a partial folding pathway common to multiple trajectories.
We realize that it is a very hard and challenging task to understand the folding mechanism of proteins. Based on our analysis results over a small protein, we are not in the position to make any general comments on the protein folding problem. However, the approach presented here is general and applicable to any folding trajectories.
Presently, we are in the process of addressing several other related issues. First, we are automating the mapping between 2D bitpatterns and 3D motifs. Second, we are further analyzing the identified consensus folding pathways and validating them through other means. Third, it is wellknown that the side chains of a protein play a crucial role in the folding process. We are currently investigating different approaches to involve side chains in our analysis. Finally, we are investigating whether bitpatterns can be used to index and manage protein folding simulation data.
Declarations
Acknowledgements
We thank Dr. Yusu Wang at The Ohio State University for providing the folding simulation data and sharing many constructive and insightful thoughts with us.
Authors’ Affiliations
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