0.9 Protein-ligand docking, including flexible receptor-flexible  (Page 9/9)

One of the main advantages of using multiple structures instead of using a selection of degrees of freedom to represent protein flexibility is that the flexible region is not limited to a specific small region of the protein. Multiple structures allow the consideration of the full flexibility of the protein without the exponential blow up in terms of computational cost that would derive from including all the degrees of freedom of the protein. On the other hand, flexibility is modeled implicitly and as such only a small fraction of the conformational space of the receptor is represented. In addition, the method by which the multiple receptor structures are combined has a drastic influence on the possible results of the docking computation.

Molecular simulations

To simulate the binding process with as much detail as possible and avoid some of the limitations of previous flexibility models one can use force field based atomistic simulation methods such as Monte Carlo or molecular dynamics (see Figure 9). Whereas molecular dynamics applies the laws of classical mechanics to compute the motion of the particles in a molecular system, Monte Carlo methods are so called because they are based on a random sampling of the conformational space. The main advantage of Monte Carlo or molecular dynamics flexibility representations in docking studies is that they are very accurate and can model explicitly all degrees of freedom of the system including the solvent if necessary. Unfortunately, the high level of accuracy in the modeling process comes with a prohibitive computational cost. For example, in the case of molecular dynamics, state of the art protein simulations can only simulate periods ranging from 10 to 100 ns, even when using large parallel computers or clusters. Given that diffusion and binding of ligands takes place over a longer time span, it is clear that these simulations techniques cannot be used as a general method to screen large databases of compounds in the near future. It is however possible to carry out approximations that reduce the computational expense and lead to insights that would be impossible to gain using less flexible receptor representations. The cost of carrying out the computational approximations is usually a loss in accuracy.

Collective degrees of freedom

An alternative representation for protein flexibility is the use of collective degrees of freedom. This approach enables the representation of full protein flexibility, including loops and domains, without a dramatic increase in computational cost. Collective degrees of freedom are not native degrees of freedom of molecules. Instead they consist of global protein motions that result from a simultaneous change of all or part of the native degrees of freedom of the receptor.

Collective degrees of freedom can be determined using different methods. One method is the calculation of normal modes for the receptor [17] . Normal modes are simple harmonic oscillations about a local energy minimum, which depends on the structure of the receptor and the energy function. For a purely harmonic energy function, any motion can be exactly expressed as a superposition of normal modes. In proteins, the lowest frequency modes correspond to delocalized motions, in which a large number of atoms oscillate with considerable amplitude. The highest frequency motions are more localized such as the stretching of bonds. By assuming that the protein is at an energy minimum, we can represent its flexibility by using the low frequency normal modes as degrees of freedom for the system. Zacharias and Sklenar [18] applied a method similar to normal mode analysis to derive a series of harmonic modes that were used to account for receptor flexibility in the binding of a small ligand to DNA. This in practice reduced the number of degrees of freedom of the DNA molecule from 822 (3 × 276 atoms – 6) to approximately 5 to 40.

An alternative method of calculating collective degrees of freedom for macromolecules is the use of dimensional reduction methods. The most commonly used dimensional reduction method for the study of protein motions is principal component analysis (PCA). This method was first applied by Garcia [19] in order to identify high-amplitude modes of fluctuations in macromolecular dynamics simulations. It has also been used to identify and study protein conformational substates, as a possible method to extend the timescale of molecular dynamics simulations and as a method to perform conformational sampling. In the next module, we present a protocol [20] based on PCA to derive a reduced basis representation of protein flexibility that can be used to decrease the complexity of modeling protein/ligand interactions. The most significant principal components have a direct physical interpretation. They correspond to a concerted motion of the protein where all the atoms move in specific spatial directions and with fixed ratios in overall displacement. An example is provided in Figure 10 where the directions and ratios are indicated by the direction and size of the arrows, respectively. By considering only the most significant principal components as the valuable degrees of freedom of the system, it is possible to cut down an initial search space of thousands of degrees of freedom to less than fifty. This is achievable because the fifty most significant principal components usually account for 80-90% of the overall conformational variance of the system. The PCA approach avoids some of the limitations of normal modes such as deficient solvent modeling and existence of multiple energy minima during a large motion. The last limitation contradicts the initial assumption of a single well energy potential.

Recommended reading and resources:

• A review of flexible receptor methods, as of 2003: Teodoro, M.L. and L.E. Kavraki. [HTML] . (2003). Conformational Flexibility Models for the Receptor in Structure Based Drug Design. Current Pharmaceutical Design, 9, 1635-1648.
• A comprehensive overview of existing docking software, as of 2006: Sousa, S.F., Pedro Fernandes A., and Ramos, M. J. [HTML] . (2006). Protein-ligand docking: Current status and future challenges. Proteins: Structure, Function, and Bioinformatics, 65(1) 15-26.