Results and discussion
The results of the HSE and CN comparisons are shown in Table 3. The table shows how many of the 100 HSE/CN minimized conformations are below a certain RMSD threshold. The associated RMSDs and energy values of the 100 conformations are also shown. In Figures 8 to 12, histograms show the RMSD and energy distribution of the CN- or HSE-optimized structures. The histograms reveal that most of the lowest energy structures are similar to the native structure. This trend is much more prevalent for the HSE-optimized structures. Based on the histograms, we conclude that the CN-/HSE-energy functions have a large smooth minimum around the structure of the native state and few smaller local minima scattered around the conformational space.
Table 3  Comparison of the HSE- and CN measures for various proteins.
Residues  Measure  < 7 Å RMSD  < 6 Å RMSD  < 5 Å RMSD  < 4 Å RMSD  < 3 Å RMSD  < 2 Å RMSD  lowest RMSD  lowest energy
Human Endothelin (1EDN)
21  CN  100  100  98  60  18  0   2.09   0.00
HSE  100  100  100  93  65  37   0.88   0.00
Tryptophan Zipper 1(1LE0)
13  CN  100  100  100  100  100  22   1.38   0.00
HSE  100  100  100  100  100  67   0.95   0.00
Third Zinc Finger (1SRK)
35  CN  60  42  17  (1SRK) 1  0  0   3.52   0.00
HSE  56  33  13  5  0  0   3.02   0.33
Mu-Conotoxin GIIA (1TCH)
23  CN  100  100  97  63  23  5   1.58   0.00
HSE  100  100  100  97  61  38   0.91   0.00
Pandinus Toxin (2PTA)
35  CN  59  32  14  3  0  0   3.17   0.00
HSE  58  44  17  11  2  0   2.66   0.33
Figure 8  Human Endothelin (1EDN), 21 residues. In the energy versus RMSD plot, the CN values have an offset of 0.01 for better illustration.
Figure 9  Third Zinc Finger (1SRK). 35 residues.
Figure 10  Mu-Conotoxin GIIA (1TCH). 23 residues. In the energy versus RMSD plot, the CN values have an offset of 0.01 for better illustration.
Figure 11  Pandinus Toxin (2PTA). 35 residues.
Figure 12  Tryptophan Zipper 1 (1LEO). 13 residues. All optimized structures have zero energy. Scatter plots show the angle correlation vs. RMSD. The Figures also show the best HSE- and CN-optimized structures superimposed on the native structure. The yellow backbone is the native structure, the red backbone is the best HSE optimized structure and the green backbone is the best CN optimized structure.
The CN and HSE comparisons show that low HSE-energy structures are generally closer to the native structure than low CN-energy structures, this both in terms of RMSD and angle correlation. A backbone structure with a good angle correlation implies that the general orientation of the residues is accurate. The plots show that this property is much more prevalent in HSE-optimized structures. Existing protein structure prediction methods that use the CN measure could therefore benefit from using the HSE measure instead of the CN measure.
Here we have developed a lattice model for protein structure prediction using the CN-/HSE energy functions. The search heuristic is based on TS with a novel tabu definition and the results indicate that TS performs better than MCS for this problem. TS with this new tabu definition might also be applied with success for other protein structure optimization problems.
Lattice experiments suggest that near zero energy structures only exists in high coordination lattices. Therefore, when using the HSE measure the model should have a high degree of freedom. All results are found using small proteins (the largest protein has 35 amino acids). When using larger proteins, it becomes very time consuming to find low energy structures and they are often not native like.
We have shown that it is possible to reconstruct the backbone of small proteins using the HSE vector of the native structure. Obviously, a predicted HSE vector would have some errors or noise as compared to the exact HSE vector. A future research project could therefore be to analyze the reconstructability of a protein backbone using HSE vectors with various degree of noise. Other directions could be to consider a more detailed energy function using other predictable information such as secondary structure. Another option could be to enforce protein-like geometry, using for example angular constraints.
In this article, we only considered lattice models. However, off-lattice models and other conformational search heuristics such as replica exchange MCMC[30] could be considered as well.