In the loop-lock representation, we consider a protein to be a set of closed loops [5-7]. We name the place where the ends of a loop meet each other a van der Waals lock [10,11]. The lock area is +/– Lr (“the lock radius” [10]) relative to the position of each end of the loop. Lr is equal to 1 amino acid, 2 amino acids or 3 amino acids.

In this paper, we present algorithms to decompose a protein into a set of non-overlapping loops. We implemented two loop decomposition approaches.

The first approach (the geometrical algorithm) is based on minimising lock distances (distance between two ends of the loop) and maximising coverage of the protein with loops. We give priority to loops with smaller distances between the Cα atoms at the ends of the loops and with longer lengths.

The second approach (the physical algorithm) is based on the loop density criteria, i.e., finding a set of loops with the maximum number of internal links (not only links at the ends). Internal links are pairs of Cα atoms separated by a small distance in 3D space and by a long distance along the protein. In both approaches, we bounded the loop size by minimal and maximal values.

Both the discussed algorithms are novel and were not considered previously.

We now describe these two algorithms in detail.

Let us consider a pool of all the possible loops for a given protein chain. The loops can be distinctive or can overlap with each other. We also have the optimal, i.e. maximal weight, set of non-overlapping loops from the loops belonging to the pool. Our objective is to find the second-best set of non-overlapping loops, the third-best set of non-overlapping loops and so on.

The second-best set must be different from the first set by at least one loop. Therefore, we can find the second-best set by excluding from the pool the loops that belong to the optimal set one by one; we then search, as described above, for the maximum-weight, non-overlapping loop set in the pool that is reduced by one loop each time. If there are N loops in the first (best) set, then we have N variants after excluding each loop once. The maximum weight set from the search over all the variants is the second-best loop set. We put the remaining sets that were not chosen into the collection of candidates. Note that if the second-best set is different from the optimal set by more than one loop (the degenerate case), we will obtain the same set a number of times equal to the number of different loops.

The third-best loop set should be different from the first and the second best sets by at least one loop. Hence we exclude from the pool the same loop that was excluded to obtain the second-best set. In addition, we exclude the loops belonging to the second best set, one by one, and renew the search for the maximum-weight, nonoverlapping loop set. The third-best set is the set with the maximum weight from the results of the exclusions at this stage and from the sets within the collection of candidates. We add the sets produced in this stage that were not chosen to the collection of candidates.

If we have the above-defined degenerate case, the thirdbest set will be the same as the second-best set. In this case, we omit this result and begin to look for the nextbest set.

The next-best sets that follow can be extracted by a similar procedure.

We chose the parameters for the algorithms based on the optimal correspondence between an intuitive “human” decomposition and the decomposition with the algorithm. We recommend the following parameters with these criteria:

Visualisation of HoPLLS results requires the Chime program to be downloaded (http://www.umass.edu/microbio/ chime/getchime.htm). This program helps Internet Explorer visualise 3D molecular structures written in pdb format.

Let us explain the structure of our web server. Let us open the main page: http://leah.haifa.ac.il/~skogan/Apache/ mydata1/main.html The “menu” is located on the left side of the page. It includes:

1) Main page: gives links to the page for the loop structure calculation and to the Chime download site;

2) Introduction: explains the algorithm for finding the loop-lock structure;

3) Theory: contains the publications that are most relevant to the topic of the site: 3D protein structure and looplock decomposition;

4) Detect loop (geometrical algorithm): the main part of the site for the loop structure calculation;

5) Example of output: gives an example of the page that results from the “Detect loop” job for the protein 4 tim;

6) References: lists publications related to the topic of the site;

7) Download C++ Program for finding loops: our program for finding loops. We ask the reader to reference this paper if he uses this program for his task;

8) Authors: the creators of the site;

9) Download Chime: download Chime program;

10) Download Acrobat: download Acrobat reader.

Let us choose the most important option: the page of “Detect loop (geometrical algorithm).”

There are three ways to input a protein for the closed loop decomposition (Figure 1):

1) to load the “pdb” file from the user’s computer;

2) to paste the pdb file in the “window” on the site;

3) to input the PDB code (4 characters: 4 tim, for example). The HoPLLS routine loads this protein from the Protein Data Bank (http://www.rcsb.org/pdb/home/home.do).

An example of the output page is shown in Figure 2. The output includes the top five decompositions for every protein chain. The output lists the loops with the numbers of the first and the last amino acids; underneath, the image of the protein is given with the loops shown in different colours. The Chime program allows us to rotate the protein in 3D (in the image) with the computer mouse.

Consider for example the 4 tim protein (Chain A), which forms a TIM-barrel fold, decomposed by HoPLLS with the geometrical algorithm at the recommended parameters. The results for the closed-loop decomposition by the geometrical algorithm are shown in Figure 3.

The results for the same task performed by the DHcL server [9] are shown in Figure 4. This server uses a rather simple procedure for loop selection; it allows overlaps (five amino acids) and provides only one possible set of loops. This server repeatedly starts from the “better” loop (i.e., the tightest loop with the shortest Cα-Cα distance between the ends), and at each iteration, the sequence region corresponding to the mapped loop is excluded from further consideration.

In contrast, our server, HoPLLS, finds an optimal nonoverlapping set by weighting the loops. In addition, HoPPLS provides several suboptimal sets through the algorithm for finding a Maximum Weight 2—Independent Set on Interval Graphs [12].

Although several of the selected loops are identical, some differences are also present. One of the main differences is the number of selected loops: 7 for HoPLLS and 8 for DHcL. This difference means that, indeed, the decomposition of the 3D protein structure into a closed loop assembly depends on the algorithm for loop se-

lection. In the current case, the main reason for the difference is the following: we do not allow loops to overlap, whereas with DHcL, loop overlaps are possible. This difference explains the larger number of loops obtained with DHcL than obtained with HoPLLS.

Previously, it was proposed that a limited number of ancestral closed loops existed in the form of individual molecules and appear today as the structural elements of modern proteins, although their sequences and 3D structures have undergone significant changes during evolution. Detecting the different fragments in modern proteins that correspond to a closed loop ancestor is a challenging task.

There are generally two main approaches to detect this correspondence.

The first approach is from sequence to structure. Several studies have been published in this field [7,8,13], and several primary ancestral closed loops have been already described.

The second approach is from structure to sequence. This approach requires the extraction of loops from 3D protein structures prior to analysis of the sequences of these structures, as performed in [14].

We have applied our geometrical algorithm for loop extraction to approximately 6000 proteins from the PDB. These proteins have been decomposed into approximately 100,000 loops.

In our investigation, we planned to solve the following tasks:

1) Form a library of all the possible loops from the proteins with known 3D structures;

2) Try to group most of the found loops into a small number of groups (with the same consensus sequence inside of every group), if such groups indeed exist. We name the set of the consensus sequences the “alphabet” of loops;

3) If a small number of such consensus sequences actually exist, we can find some correspondent sequence in a protein and predict the possible position of the loop from the 1D structure.

To select groups of similar loops, the following algorithm was used:

1) The sequence of the first loop in the whole set was compared to the sequences of all the other loops in the set. Two sequences were said to be close if they retained at least 30% similarity. The probability of this occurring randomly is very small and is equal to approximately 1/100,000 [7];

2) All loops that were sequence-wise similar to this loop were placed into the distinct group and were deleted from the set;

3) Return to the first step and repeat until the set is empty.

If such an “alphabet” of loops actually exists, they can be clustered into a small number of groups in such a way that all the loops in the same group are close to each other. Because the probability of random closeness is 1/100,000, in the absence of any “alphabet”, we expect to form about 50,000 groups with a uniform distribution of the number of loops over the groups.

The number of detected groups was about 40,000. Most of the groups included two loops. The maximal number of loops in a group was fifteen. A more complex loop comparison model, which allowed insertions and deletions [15] of amino acids, was also applied. The number of groups found with this model was about

30,000. Most of the groups included three loops. The maximal number of loops in a group was twenty. We also used a “binary” code model [16,17], in which 20 amino acids were divided into two groups that originated from the ALA and GLY amino acids. In the “binary” code case, two sequences were said to be close if they retained at least 90% similarity. Similarly to the 20 amino acids code, the probability of this occurring randomly in the “binary” code case is approximately 1/100,000. The number of groups found by the “binary” code was about 50,000. Most of the groups included two loops. The maximal number of loops in a group was nine.

The results demonstrate that the loop distribution is approximately random and that no “alphabet” of loops actually exists.

We also tried to find an “alphabet” of locks. The lock is defined to be the last three or five amino acids on both ends of the loop. The number of groups found in the lock-clustering experiments was found to be approximately the value expected for a random distribution: indeed, almost all combinations of amino acids were found with an approximately uniform distribution of lock numbers.

These results show that closed loop selection by our algorithm does not make it possible to decompose the complete set of loops into a small number of groups based on sequence similarity. This impossibility may exist because too many evolutionary changes have occurred in the primary closed loop ancestors. It is possible that more complicated procedures than direct sequence comparison should be applied.

It is well known that the same structure and even the same function often can be presented in nature by different sequences [14,18,19]; i.e., the sequence diversity in protein sequences is very high. A promising approach to overcome this problem is to define sequence relatedness in terms of connectivity through sequence similarity networks, as was proposed recently [20]. It was found that the natural sequence fragments form prolonged “walks” (connected through the networks), in which one sequence smoothly transforms to another one while conserving its structure and function. Remarkably, the sequences at the ends of such walks (on the periphery of the networks) are often completely different [20,21]. Several examples of completely unmatched sequences from the same network are documented [21,22], and many more can easily be found in the networks.

In many cases, these networks allow researchers to overcome the problem of sequence diversity and suggest a new criterion for sequence relatedness. In addition, these networks provide an alternative approach to all the methods in which notions of “sequence pattern” or “profile” and PSSM are used, e.g., [23,24]. These notions imply an “average state” of sequences, whereas the network approach simultaneously considers a complete list, often of samples rather than different sequences linked to a common specific structure and function. This list may correspond to different “average states” yet be connected together through chains (walks) of pair-wise close similarities, which suggests evolutionary connections between the “average states”. The same function for a given segment may be encoded by different consensus sequences. Moreover, in some cases a consensus may not exist at all.