Comparison of HSE and CN measures
In the previous section, experiments on a small protein show that minimization of the HSE energy in high coordination lattices leads to structures with HSE vectors that are very similar (or equal) to the native structure. In this section, experiments on proteins of varying size are done using the TS heuristic with tabu difference 0.4 and the HC8 lattice. The energy functions are based on the HSE vectors of native structures as described in section HSE energy function. In addition to the HSE energy, the CN energy is considered for comparison. The main purpose of the experiments is to examine the reconstructability of a protein's backbone solely from the information stored in the HSE-/CN vectors.
Each TS run is started from a random structure which is iteratively improved as described in section Heuristics. For these experiments we want to start TS on 100 random structures that are as different from each other as possible. Therefore, to effectively sample the search space, 10000 random conformations are initially generated. Ideally, from this set of 10000 conformations, we would like to choose the set of 100 conformations such that the minimum RMSD between any two conformations is maximized. This problem is generally known as the p-dispersion problem and is NP hard[27]. Solving this problem to optimality is therefore not feasible, so we use a greedy heuristic to find a good set of 100 different random conformations. The greedy heuristic works by first picking a random conformation. The following 99 conformations are then picked one at a time, such the minimimum RMSD to any of the already picked conformations is maximized.
For each protein, the energy function based on its native structure is minimized for each of the 100 random starting conformations and the structures with lowest energy are reported. The search is stopped after 12 hours or if the energy reaches zero. Zero energy means that a structure with exactly the same HSE- or CN vector as the native structure is found (but not necessarily identical structures).
To evaluate the quality of the structures with low energy, the RMSD with the native structure and angle correlation [28,29] is used. Angle correlation is a measure with the following definition. For each Cα, let V→ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWGwbGvgaWcaaaa@2DF3@α be the vector pointing in the side chain direction (see Figure 2). Let Vαmc→ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGwbGvdaqhaaWcbaacciGae8xSdeMaemyBa0Maem4yamgabaGaeyOKH4kaaaaa@3453@ be the vector pointing in the direction of the mass center, and let θα be the angle between V→ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWGwbGvgaWcaaaa@2DF3@α and Vαmc→ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGwbGvdaqhaaWcbaacciGae8xSdeMaemyBa0Maem4yamgabaGaeyOKH4kaaaaa@3453@. The angle correlation measure is the average of the differences in θα between the optimized structure and the native structure. Zero angle correlation is perfect correlation, 90° is random correlation and 180° is perfect 'anti'-correlation. Note that the CN- and HSE vectors of a structure are identical to the vectors of the mirror of the structure. Therefore, in the following results, if the RMSD between a structure and its native mirror image is smaller we report this value instead. All computations were performed on a 236 nodes Dell Optiplex GX260 cluster (2,4 GHz P4, 512 Mb RAM).