PMC:1867812 / 19607-30728 JSONTXT

{"project":"2_test","denotations":[{"id":"17407611-9806935-1692924","span":{"begin":3728,"end":3730},"obj":"9806935"}],"text":"3.1 Data Preprocessing\nSame 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 self-contained, we next briefly go over these preprocessing steps. We also explain the rationale of such steps in the context of protein folding.\n\nContact Map Generation\nWhen generating contact maps, we consider the Euclidean distances between α-carbons (Cα) of each amino acid. Two α-carbons 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 ith and jth α-carbons 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 Cα 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.\n\nIdentifying Maximally Connected Bit-patterns\nEvery bit in a contact map has eight neighbor bits. For an edge position, we assume its out-of-boundary positions contain 0. In a contact map, a connected bit-pattern is a collection of bit-1 positions, where for each 1, at least one of its neighbors is 1. Correspondingly, we define a maximally-connected bit-pattern (also referred to as a bit-pattern 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 maximally-connected patterns in each contact map within the two series of contact maps, corresponding to the two folding trajectories of BBA5. Altogether, we identified 352 maximally-connected bit-patterns in such contact maps. For the GSGS folding data, a total of 50,572 unique bit-patterns are constructed. We then represent each identified bit-pattern as a 6-tuple feature vector consisting of the following attributes:\n• Height: the number of rows contained in the pattern's Minimum Bounding Rectangle (MBR).\n• Width: the number of columns in the pattern's MBR.\n• NumOnes: the number of 1s in the pattern.\n• 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 least-squares method to estimate the slope of a linear regression line. For a pattern containing n 1s, we denote the positions of the 1s as: (x1, y1)...(xn, yn). The least-squares method then estimates the slope β1 as: β1=∑i=1n((xi−x¯)∗(yi−y¯))/∑i=1n((xi−x¯)2) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@5A04@\n• xStdDev: the standard deviation of all the 1s' x-coordinates (this quantifies how the 1s spread along the x dimension).\n• yStdDev: the standard deviation of all the 1s' y-coordinates.\nNote that this feature vector captures the main geometric properties of a bit-pattern.\nAs discussed in the literature [18-21], non-local patterns (where bit-patterns are one type of non-local 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 bit-patterns, one can construct structural signatures for proteins of different classes or folds. In the context of protein folding, we have observed that the above-defined bit-patterns 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 bit-patterns that correspond to \"premature\" α-helices and native-like α-helices respectively. Henceforth, we refer to the 3D structure formed by all the participating residues of a bit-pattern as the 3D motif of the bit-pattern. The relationship between bit-patterns and 3D motifs will be further discussed in the next section.\n\nClustering Bit-patterns into Approximately Equivalent Groups\nIn this step, we partition the extracted bit-patterns into approximately equivalent groups, each of which consists of bit-patterns that exhibit similar geometric properties (e.g., shape and size). To construct such equivalent groups, we run the k-means based clustering algorithm [22] over the bit-patterns' corresponding feature vectors, where k is the number of clusters (or equivalent groups) that will be produced.\nTo 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 bit-patterns. Second, for each clustering scheme, we compute its entropy. Let c1, ..., cl be the l clusters after clustering the set of N bit-patterns. Furthermore, each cluster ci (1 ≤ i ≤ l) has an individual entropy Hi and contains Ni elements, then the total entropy of this clustering is given by the following formula: H=∑i=1kHi∗(Ni/N) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGibascqGH9aqpdaaeWaqaaiabdIeainaaBaaaleaacqWGPbqAaeqaaOGaey4fIOIaeiikaGIaemOta40aaSbaaSqaaiabdMgaPbqabaGccqGGVaWlcqWGobGtcqGGPaqkaSqaaiabdMgaPjabg2da9iabigdaXaqaaiabdUgaRbqdcqGHris5aaaa@3F89@ The entropy of each individual cluster, i.e., Hi , is computed by summing up the entropy of each of the six bit-pattern 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 bit-patterns into 10 clusters (or types). As for the GSGS data, 12 clusters are identified.\nIntuitively, the 3D motifs of the bit-patterns in a cluster will also have similar 3D geometric properties. This is verified based on our manual analysis on the BBA5 trajectories. Figure 5 illustrates the representative 3D motifs.corresponding to the 9 of 10 types of bit-patterns identified in BBA5 trajectories. We omit type 0, as bit-patterns of this type, unlike the others, correspond to a wide variety of 3D motifs.\nFigure 5 Mapping between 2D bit-patterns and 3D sub-structures. The figure visualizes the representative 3D sub-structures corresponding to the 10 classes of bit-patterns identified in the contact maps along BBA5's two folding trajectories. The bit-patterns shown here are randomly selected from their respective group for illustration purpose. We also observed a similar scenario for the 12 types of bit-patterns identified in the GSGS trajectories. For instance, the typical 3D motifs of type 0 bit-patterns resemble the native conformation of GSGS (See Figure 2(a)); whereas those of type 6 identify with α-helices.\nUpon a closer look at this 2D-3D mapping illustrated in Figure 5, one can observe the following interesting aspects. First, multiple types of bit-patterns can be associated with a single type of 3D motif. For instance, there are 3 types of bit-patterns are mapped to an α-helical motif. Second, contrary to a commonly accepted belief that β-turns or β-sheets cannot be captured by maximally connected bit-patterns as defined earlier, our analysis shows that this belief does not stand. To illustrate this point, we take two examples. The first example, illustrated in Figure 6, corresponds to the β-turn structure. As shown in Figure 6(b), the β-turn formed by the first 10 Cα atoms of BBA5 can be captured by the maximally connected bit-pattern shown in Figure 6(a). The second example, shown in Figure 7, illustrates that a two turn β-sheet (Figure 7(b)) can also be captured by a bit-pattern (Figure 7(a)). Finally, not every type of bit-patterns can be mapped to a typical 3D motif. This might be attributed to our entropy-based criteria for selecting an \"optimal\" value of the parameter k in the clustering task.\nFigure 6 β-turns vs. maximally connected bit-patterns: an example. (a) A type 8 bit-pattern is identified in the 166th frame of the BBA5 T23 trajectory. This bit-pattern corresponds to the the connected 1s in the table, where a '1' indicates two corresponding Cα atoms are in contact,'-' otherwise. This pattern consists of the first 10 Cα atoms. (b) The 3D conformation of this frame, where the first 10 Cα atoms resembles a β-turn.\nFigure 7 β-sheets vs. maximally connected bit-patterns: an example. (a) A type 0 bit-pattern is identified in the 24201th frame of the GSGS T1 trajectory. This bit-pattern corresponds to the the connected 'x'-es in the table, where an 'x' indicates two corresponding Cα atoms are in contact,'-' otherwise. It consists of Cα atoms from 5 through 20. (b) The 3D conformation of this frame, where the 5–20 Cα atoms resembles a β-sheet of two turns. 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 re-compensated by mapping bit-patterns to structural motifs in 3D conformations. More importantly, by exploiting different features in contact maps (bit-patterns in this work), we are able to connect 2D features with features in 3D space. In the BBA5 case, by identifying 10 types of bit-patterns in contact maps, we indirectly recognize 10 different 3D structural motifs in the folding conformations.\n\nRe-labeling Bit-patterns with The Corresponding Cluster Label\nIn this step, we re-label all the previously identified bit-patterns with their corresponding cluster label. Let p be a labeled bit-pattern. It can be represented as follows: p = (trajID, frameID, listCα, label). Here, trajID identifies a folding trajectory, and frameID indicates the frame where p occurs, listCα consists of all participating α-carbons of p, identified by their position in the primary sequence. Finally, label is the cluster label of p. For BBA5, label ∈ {g0, g1, ⋯, g9}, corresponding to the 10 approximately equivalent groups (or types)."}