Method for comparative protein structure modeling by MODELLER

MODELLER implements an automated approach to comparative protein structure modeling by satisfaction of spatial restraints (Figure 1.1) [Š ali & Blundell, 1993]. The method and its applications to biological problems are described in detail in references listed in Section 1.2. Briefly, the core modeling procedure begins with an alignment of the sequence to be modeled (target) with related known 3D structures (templates). This alignment is usually the input to the program. The output is a 3D model for the target sequence containing all mainchain and sidechain non-hydrogen atoms. Given an alignment, the model is obtained without any user intervention. First, many distance and dihedral angle restraints on the target sequence are calculated from its alignment with template 3D structures (Figure 1.2). The form of these restraints was obtained from a statistical analysis of the relationships between many pairs of homologous structures. This analysis relied on a database of 105 family alignments that included 416 proteins with known 3D structure [Š ali & Overington, 1994]. By scanning the database, tables quantifying various correlations were obtained, such as the correlations between two equivalent ${C}_\alpha$ - ${C}_\alpha$ distances, or between equivalent mainchain dihedral angles from two related proteins. These relationships were expressed as conditional probability density functions (pdf's) and can be used directly as spatial restraints. For example, probabilities for different values of the mainchain dihedral angles are calculated from the type of a residue considered, from mainchain conformation of an equivalent residue, and from sequence similarity between the two proteins. Another example is the pdf for a certain ${C}_\alpha$ - ${C}_\alpha$ distance given equivalent distances in two related protein structures (Figure 1.2). An important feature of the method is that the spatial restraints are obtained empirically, from a database of protein structure alignments. Next, the spatial restraints and CHARMM energy terms enforcing proper stereochemistry [MacKerell et al., 1998] are combined into an objective function. Finally, the model is obtained by optimizing the objective function in Cartesian space. The optimization is carried out by the use of the variable target function method [Braun & Gõ, 1985] employing methods of conjugate gradients and molecular dynamics with simulated annealing (Figure 1.3). Several slightly different models can be calculated by varying the initial structure. The variability among these models can be used to estimate the errors in the corresponding regions of the fold.

There are additional specialized modeling protocols, such as that for the modeling of loops (Section 2.3).

Figure 1.1: First, the known, template 3D structures are aligned with the target sequence to be modeled. Second, spatial features, such as ${C}_\alpha$ - ${C}_\alpha$ distances, hydrogen bonds, and mainchain and sidechain dihedral angles, are transferred from the templates to the target. Thus, a number of spatial restraints on its structure are obtained. Third, the 3D model is obtained by satisfying all the restraints as well as possible.

Figure 1.2: A restraint on a given ${C}_\alpha$ - ${C}_\alpha$ distance, $d$ , is expressed as a conditional probability density function that depends on two other equivalent distances ($d' = 17.0$ and $d'' = 23.5$ ): $p(d/d',d'')$ . The restraint (continuous line) is obtained by least-squares fitting a sum of two Gaussian functions to the histogram, which in turn is derived from the database of alignments of protein structures. In practice, more complicated restraints are used that depend on additional information, such as similarity between the proteins, solvent accessibility, and distance from a gap in the alignment [Š ali & Blundell, 1993].

Figure 1.3: Optimization of the objective function (curve) starts with a distorted average of template structures. In this run, the first $\sim 2,000$ iterations correspond to the variable target function method relying on the conjugate gradients technique. This approach first satisfies sequentially local restraints and slowly introduces longer range restraints until the complete objective function is optimized. In the last 4,750 iterations for this model, molecular dynamics with simulated annealing is used to refine the model. Typically, a model is calculated in the order of minutes on a PC workstation.