Computational Methods for Protein Structure Prediction & Modeling V1 - Xu Xu and Liang

4.2 Definitions of Structural Domains

There are many definitions of protein domains; while some are based on sequence information only (they are rather cursory), most others (see below) consider structure information in addition to sequence information. Evolutionary information, i.e., recurrence of domains in different proteins, is also employed (Murzin et al., 1995). The most common definition of domains is often referred to as structural domains. The analysis presented here is concerned with structural domains; however, often they will be referred to simply as “domains” throughout this chapter.

Structural domains are units of the structure that (1) are compact, (2) are stable,

(3) contain a hydrophobic core, (4) can fold independently of the rest of the protein, (5) occur in combinations with different domains, and (6) perform a specific function. Right away we see that there are structural/thermodynamics (def. 1–4), evolutionary (def. 5), and functional (def. 6) aspects to structural domains. Methods for partitioning protein structure into domains use these definitions as their guides (for more detailed discussion see Veretnik et al., 2004). There are several manual methods for domain partitioning (SCOP, AUTHORS) (Islam et al., 1995; Murzin et al., 1995), semiautomatic method (CATH) (Orengo et al., 1997), and a score of automatic methods (Table 4.1). Human expertise is nearly always superior to algorithms in the area of domain decompositions. However, computational methods are critical in the current era of structural genomics, when the sheer number of solved structures overwhelms human experts. Moreover, for the theoretical/computational study of proteins it is essential to have a consistent domain partitioning, which only can be achieved with automatic methods. How do computational methods approach this problem and how well do they capture the principles of structural domains? Before addressing this question, let us point out that structure partitioning into domains is not always unequivocally agreed upon even by human experts. The reason for it lies in the underlying complexity of structural domains—they do not always match the above set of definitions: some domains do recur in different combinations, but they are small and are lacking a hydrophobic core, other domains that recur as a single unit are large and can clearly be decomposed into separate structural units. There are many cases in which domains are not globular, but they constitute parts of the compact protein–protein (or nucleic acid–protein) complex; finally there are many cases where the function is carried jointly by two or more domains. In all such cases domains will be defined differently depending on which aspect of the definition— structural, evolutionary, or functional—becomes the most important for a particular research. Among three existing manual or semiautomatic methods, SCOP focuses on evolutionary recurrent units of structures and CATH stresses structural integrity of the units, while the AUTHORS method considers the function as well as the contribution of the structural unit to a protein complex. The corresponding resources, based on these three methods, disagree in over 20% of the cases; the rate of disagreement is particularly high in proteins with complex architectures, indicating that multiple solutions may exist in such cases. This inherent ambiguity of structural

Table 4.1 Summary of domain decomposition methods.

domain definition is one of the difficult issues for computational methods, as we will see below.

4.3 Computational Methods

As a general rule computational methods focus on structural integrity of the resulting domains—this is intuitively clear and practically achievable, while evolutionary and functional information is much trickier to capture, support, and use. Historically the most common principle of structural partitioning is based on the interpretation of the so-called “contact density,” i.e., on the simple fact that there are more residue-to-residue contacts within a structural unit than between structural units. The implementations of this principle can be very different in different methods, but the underlying idea always comes back to finding regions with a high density of interactions. The extensive atomic interactions within structural domain were first pointed out by Wetlaufer (1973) and implemented in the very first domain partitioning algorithm by Rossman and Liljas (1974), who use C –C distance maps to identify structural domains. New computational methods have been appearing ever since; overall approximately 20 different methods had been published in literature. Methodologically they can be separated into first generation (methods published from 1974 to 1993) and second-generation algorithms: from 1994 until now (Wernisch and Wodak, 2003). There were no attempts of formal training and validation of the early algorithms: parameters of the algorithms were tuned using all existing data (meaning overfitting algorithms to a particular data set); no validation on an independent set of data was performed. The lack of solved structures

at that time might be the chief reason. The second-generation methods approach the problem differently. First, algorithms were trained to optimize parameters, and then in a separate step their performance was evaluated. While most of the firstgeneration methods partition structure only into contiguous domains (consisting of a single polypeptide fragment with single starting and ending points in a protein chain), second-generation methods universally allow for noncontiguous domains (consisting of several fragments separated by other residues). Domain partitioning can be divided into two fundamental approaches: top-down (starting from the entire structure and proceeding to partition it iteratively into smaller units) and bottomup (defining very small structural units and assembling them into domains). Some methods use both approaches within their algorithm: first decomposition and then assembly or vice versa. Here we classify each method based on its chief or overall approach to domain identification. Generally, the process of domain decomposition is performed in two steps: (1) tentative domains are constructed (either by splitting the structure into domains or building domains from smaller units) and (2) tentatively defined domains are evaluated in a postprocessing step. Overall an amazing array of approaches has been put forward over the years to solve the domain decomposition problem. In the rest of this section, 16 different domain partitioning methods—from the earliest ones to the latest ones—are discussed in terms of their general approach and methodology used.

4.3.1 Rossman and Liljas (Rossman and Liljas, 1974)

Method: C –C distance plots of the structure against itself are generated and contours of interactions are examined. Domains are recognized as a series of shorter interactions closer to the diagonal. Similar pattern of contours indicates similar domain structure.

Results: Recurrence of different combinations of domains in various proteins is pointed out.

4.3.2 Crippen (Crippen, 1978)

Method: Method is based on the assumption that the stable conformation of a protein is largely due to the energetically favorable interactions of the residues which are frequently distant in sequence, i.e., long-range interactions. In the first step, protein chain is divided into segments—short stretches of polypeptide chain whose residues have no long-range interactions with any other residues. Long-range interactions between two residues are defined as follows: sequential distance is at least seven

˚

residues, distance between C –C atoms is <9 A. The segments (which correspond frequently to isolated secondary structures or coiled regions) are then hierarchically clustered, using contact density criteria (contact density is normalized by the size of the individual segments). The process is repeated iteratively first joining segments in the most intimate contact, while last clusters have few contacts relative to the number of residues involved.

Results: The method allows one to view the crystal structure of a protein as a packing tree, with individual secondary structures at the bottom, structural domains close to the top, and the subdomain structure in order of increasing complexity in the intermediate levels of the tree. The author hypothesizes that the steps of the folding mechanism can possibly be discerned from such packing tree. Folding intermediates correspond to internal nodes of the packing tree; the order of their appearance can be predicted by following the packing tree from the bottom level upwards.

4.3.3Rose (Rose, 1979)

Method: The protein is treated as a rigid body and a set of Cartesian axes are drawn through the center of its mass: in classical mechanics these are referred to as “principal axes” and they correspond to eigenvectors of inertia tensors. The threedimensional structure of a protein is then reduced to two-dimensional space by projecting C atoms of each residue onto the so-called domain “disclosing plane.” The plane is determined by two of the three principal axes: those with the larger moments of inertia. The C atoms are connected in sequential fashion. An arbitrary line is drawn through the plane to divide protein into at least two pieces. The best line is searched according to two criteria: first, the two parts of a protein must be contiguous fragments and separated as much as possible by the line. In the ideal situation the line will completely separate the two parts; deviation from this ideal case is measured by the length by which one part of the protein extends into the other (the length is defined as the sum of distances between C atoms on their projection onto the disclosing plane). Second, the length of two resulting domains is compared; in the ideal case the domains will be of identical length; deviation from this is measured with a nonlinear function. The entire process is then repeatedly applied to each of the two domains until termination criteria are met: the projection of the chain on the disclosing plane does not close upon itself, which indicates either very short domains or a lack of compactness within a domain.

Results: The author points out that the hierarchy of domains resulting from such a process, when placed in ascending order, is in accordance with the local process of the automata theory, in which current state can be derived from the previous state. The hierarchical domain structure might then reflect the folding process of the protein, which proceeds by hierarchical condensation. During hierarchical condensation the nearby hydrophobic elements coalesce to form folding primitives which in turn coalesce to form larger module until the entire hydrophobic core is complete.

4.3.4Wodak and Janin (Wodak and Janin, 1981)

Method: The method evaluates size and properties of domain interfaces: a domain interface is defined as a surface area buried in contacts between two groups of residues. Minima in the interface area are found for the entire protein structure and then recursively for each of the partitioned units. The process stops when the significance of interface minima drops below a predefined threshold.

4.3.5 Holm and Sander (Holm and Sander, 1994)

Method: The method identifies folding units by finding groups of residues that have longest intergroup oscillation time. Oscillation time is proportional to the center of masses and inversely proportional to the interface strength, which is determined by nonbonded atomic interactions at the interface between two units (two atoms

≤ ˚

are interacting if their distance is 4.0 A). Folding units are found by first creating a contact matrix and then the rendering of the contact matrix in such a way that strongly interacting residues are grouped together (using a reciprocal averaging) and rows/columns 1 through k belong to one unit while rows and columns of k + 1 through L(length of the protein) belong to the other unit. Bisection of the ordered contact matrix can continue recursively for each of the resulting folding units until some limit on unit size is reached (10–19 residues). At this point each autonomous folding unit becomes a domain. An additional hierarchical five-level filtering is applied during the process (once the higher level filter is met, the partition is accepted):

(1) Domains should have 40 or more residues. Thus, units smaller than 80 residues are never cut. (2) Highly flexible units are always cut. (3) -Sheets are never cut.

(4) A cut is acceptable if both resulting units have a high globularity/compactness value. (5) A cut that produces a nonglobular domain with less than 40 residues is accepted on condition that the larger domain in the cut will be split into two domains upon recursive application of the filters.

4.3.6 Swindells (Swindells, 1995b)

Method: Hydrophobic cores of the protein structure are defined using a set of rules based on solvent exposure, minimal size of a core, fraction of the spatially adjacent residues, etc. The hydrophobic cores become the centers of the domains, which grow by iteratively including residues spatially proximal to the core until most of the residues are assigned. Isolated residues that generate contradictory results are removed from the assignment. In the final step the unassigned residues are assigned by extending domains to both ends of the structure and/or to the ends of the appropriate secondary structure.

4.3.7 Islam, Luo, and Sternberg (Islam et al., 1995)

Method: The protein chain is cut iteratively into domains by finding the minima in the interdomain contact density. The cutting is stopped when density of contacts reaches an empirically determined threshold F. The series of contiguous segments is evaluated in a postprocessing step: every two noncontiguous segments are tentatively combined, and if their contact density is over the threshold F, the segments are clustered together. This process is repeated iteratively until no more clustering of the segments occurs. If any of the remaining segments are less than 32 residues long, they are merged with most appropriate larger segment. At this point each segment becomes a domain.

minimum contacts between them, repeating the process until either defined termination criteria are met or no cut can be made any further based on compactness criteria. The search for substructures with minimum contact area begins by systematically subtracting residues (one at a time) from one subset (V ) and moving them to its complement U (V + U contain all residues in the structure) in such a way that contact between U and V increases minimally. After each move described above, a process is initiated in which all possible switches among residues in the two sets are tested. The arrangement which produces the lowest number of contacts between two sets is selected. The process is repeated by moving subsequent residues from V to U until N /2 residues are moved. From all partitions of residues between U and V the one with lowest contact number is chosen as the final prediction and the resulting domains go through a postprocessing step in which the validity of domains (based on various heuristics for compactness) is assessed. The entire process repeats for each domain.

The contact area between two sets of residues in U and V is based on the sum of contact areas of all residues in U and V , where the contact area between any two residues is based on the pairwise sum of all of its interacting atoms. Interaction between atoms is determined using Voronoi cell, which in turn is based on the van der Waals radii of the atoms.

4.3.12Guo, Xu, and Xu (Guo et al., 2003; Xu et al., 2000)

Method: The DomainParser method uses a graph-theoretical approach to find the best partitioning of a given structure into two parts. A protein structure is modeled as a graph in which nodes represent residues and edges connecting nodes represent interactions between residues. The weight of connection is proportional to the strength of interaction between two residues. A minimum cut splitting the network into two is found by determining the maximum flow through the network, using the Ford–Fulkerson algorithm (a more detailed description is given below in Section 4.4).

The process is repeated iteratively for each of the resulting domains until stopping criteria are met. Multiple minima partitions are performed at each step; the best partition is determined during a post-processing step by checking several basic properties of the resulting domains. The properties include hydrophobic moment, the number of noncontiguous fragments, domain size, compactness, and relative motion of domains. The “acceptable” range for each property as well as a set of stopping criteria is determined in advance using a neural network trained on the set of known structures.

4.3.13Xuan, Ling, and Chen (Xuan et al., 2000)

Method: The method (no name given) uses fuzzy clustering analysis for domain recognition. In the first step the chain is partitioned into elementary fragments which consist of residues adjacent in sequence and in similar “contact environment.” The contact environment of a residue is defined in terms of all other residues which

interact with that residue. In the next step the initial fragments are clustered with other fragments using fuzzy logic and lower threshold on the criterion of similarity of contact environment. At this point the rudimentary domains are formed. During the last step the final domains are assembled from the rudimentary domains using principles of minimum domain size, minimum fragment size, and integrity of secondary structure joins.

4.3.14Alexandrov and Shindyalov (Alexandrov and Shindyalov, 2003)

Method: The PDP method partitions a polypeptide chain into two parts by introducing either a single cut through the chain which results in two contiguous domains or a double cut, which will produce a contiguous and a noncontiguous domain (the two cuts must be at least 35 residues apart). The optimal position of the cut(s) is chosen by minimizing the number of contacts between two resulting domains. Calculation of the number of contacts takes into account the sizes of resulting domains. The threshold for domain contacts is set to be one-half of the expected number of contacts. The expected number of contacts between domains is considered to be proportional to their surface area. The calculation of the surface of the domain assumes the shapes of the resulting domains to be close to spherical; thus, it is proportional to n2/3, where n is the number of residues in a domain. The process of division is repeated iteratively for each of the resulting domains. A post-processing step involves rechecking each of the contacts among the resulting domains: filtering out very small domains (less than 30 residues) and combining domains together if they exhibit a high number of contacts.

4.3.15Berezovsky (Berezovsky, 2003)

Method: Initially a protein chain is partitioned into segments based on the threshold (referred to as “potential barriers”) between local maximum and minimum of van der Waals energies calculated using 6-12 Lennard-Jones potential. Next, the pairwise interactions between isolated segments are evaluated; segments are either joined together or granted the status of a domain. A segment becomes a domain if its van der Waals energy internal to the segment is at least 3 times higher than the sum of its interaction energies with all other segments (bottom up step). The chain segmentation step can be performed at different thresholds of “potential barrier”: lowering the potential barrier increases the number of segments and decreases their size.

Results: Multiple solutions to the protein partitioning can be produced by applying a series of thresholds—a unique approach so far among existing methods. Multiplicity of solutions reflects observations that for some proteins there are several possible domain decompositions—all of which appear to be biologically meaningful. The authors also point out that the resulting domains are made of closed loops— contiguous subtrajectories of the folded chain with a short C –C distance between

their ends. Such closed loops are proposed to be the elementary units of structural domains.

4.3.16Kundu, Sorensen, and Phillips (Kundu et al., 2004)

Method: This approach to decomposition of the structure into domains assumes a semi-independent motion and local compactness of the domains. A concept of Gaussian Network Model (GNM) is employed to find structurally compact clusters which are connected internally, but motionally decoupled from the other parts of the structural units. The method employs connected graphs in which each residue is represented by a node and connection between any two residues is determined by C distances of the residues. The structure is recursively partitioned into domains by constructing Laplacian matrix, performing a single value decomposition to find the lowest eigenvalue which corresponds to the slowest motion within the structure. After each decomposition step, potential domains are evaluated using a layer of filters which include minimum size of the domain, minimum length of discontinuous fragments comprising the domain, and integrity of -sheets.

4.4In-depth Look into Algorithmic Domain Decomposition

To gain a better insight into how domain decomposition methods work, two methods are discussed here in detail—one is from the first generation of algorithms and the other is a recent method.

Crippen’s method (Crippen, 1978) is one of the earliest methods published. It uses a bottom-up approach of clustering secondary structures and is actually unique among first-generation methods as it allows construction of noncontiguous domains. Twenty-five structures were analyzed. The same 25 proteins are employed for tuning several parameters used in the algorithm (described below). During the first step, protein structure is decomposed into so-called “basic units”: each unit consists of consecutive residues that do not have long-range interactions with other residues in the unit. Long-range interaction between two residues is defined with the following

˚

condition: distance <9A between C atoms of two residues that are at least seven residues apart in sequence. The definition of long-range interactions comes from fitting secondary structure assignments of myoglobin to the segments generated; it matches the definition of secondary structures by Levitt and Greer (1977) rather well. The assigned basic units are permitted to overlap during the assignment process; the final boundaries between the segments are drawn in the middle of units overlap. Once an entire structure is broken down into basic units, the process of clustering begins: units with the highest contact density are paired first to form a new larger unit. The contact density is determined by calculating all pairwise contacts between all possible pairs of residues in two units; this value is then normalized by the