Algorithm begins with initial population

A self-organizing system operates based on information received locally without the guidance of the external environment. It is an adaptive methodology adopted for GA to discover the interaction of operators to collect the required information to set the parameter values in an automated way. The SOGA was developed with full knowledge of the test problem and parameter values which are incremented uniformly in each generation. Self-organization contributes desirable factors to GA by adding knowledge to guide and assist the choice of right parameter values.

Generally, knowledge regarding the interaction of parameters leads to better global solutions in less time, thereby making GA more robust. A lack of robustness in the choice of design parameters always leads to search imbalance, causing premature convergence and lower performance, which is completely eliminated by the developed algorithm given below:

The chromosome representation varies for different problems, and as a set, it constitutes an individual structure called a gene, a set of solution representatives. In general, Cartesian coordinates, templates, torsion angles, side chains, and simplified residues are considered for chromosome representation in PSP [8]. In the present work, torsion [22] and side chain angles [23] were assumed to be best for encoding the chromosomes since both contribute much to the consistency and confirmation of protein structure.

Self-organizing mutation and crossover rates are uniformly increased until they reach the optimal values; in turn the termination condition decides the number of generations.

Population is created using frequently occurring torsion angles [15] which have the degree of freedom for adequate adaptability of GA; it helps in determining the 3D conformation, which is chosen for population size.

The population created in each generation is evaluated by fitness function for selection of the elite [24]. It projects the nature of each chromosome, and the energy value that determines the elite is calculated using the CHARMM force field chosen as the objective function:

The selection operator elitism, selects the best chromosome with minimal energy as elite. In each generation the elite may be replaced by the low-energy chromosome for genetic operations.

To refine the search space, the crossover operator which is normally fixed in standard genetic algorithm (SGA) is self-organized. The least optimal rate is initially chosen for execution without termination at the convergence point, and it is increased uniformly until it reaches the optimal upper limit [25], or the rates are modified from high to low [26]. The reported range of the rate (0.70 to 0.85) is considered for self-organization.

The working principle of SOCO

The crossover operation is pictorially represented in Figure 2.

The purpose of mutation is to transform the chromosome to change the conformation and to prevent the population from stagnating at any local optimum. With a fixed mutation rate, the search space is compressed, which leads to premature convergence, and it may not be possible to predict the optimal solution. Self-organization is used to avoid this, and the mutation rate is self-configured. Only the constrained values are considered for the test problem since slight changes may occur in the sequential order of amino acids, which in turn may change the entire structure [17].

The working principle of the SOMO operator is:

The mutation operation is pictorially represented in Figure 3.

Met-enkephalin (PDB ID: 1PLW) was chosen because it plays a crucial role in generating tissue damage signals as initiated by both peripheral and central nervous system (CNS). Also, it is a simple pentamer used as a test case to measure the efficiency of various ab initio methods. Unlike natural crossover and mutation observed in real cellular chromosomes, randomness is restricted in SGA, where it is designed to propagate in a defined architecture due to fixed values. This limitation is relaxed by defining the rate range so that better results can be produced by switching between the intervals. The rate ranges of 0.75-0.85 and 0.025-0.075 were adopted for crossover and mutation to execute the developed SOGA, whereas the rates are 0.85 and 0.075 in SGA. Accordingly, three generations were produced using predefined rates as shown in Tables 1 and 2. With the inherent limitations of SGA in choosing a crossover rate, peptide conformations are generated, and a huge deviation of energy is observed as seen in the plot. However, in the SOGA approach, predicted conformations are maintained in a standard low-energy distribution as shown in Figures 4 and 5. The best values scored by the protein structure indicate that it provides much better confirmation against a native-structure using NMR. Also similar backbone confirmation is indicated by a root mean square deviation (RMSD) value of 0.0009. Eventually, differences in side chain orientations are also clearly seen; this improves the backbone dihedral angles, which is very clear from Ramachandran plot shown in Table 3 and Figure 6 [28]. On closer view, differences in the orientation of side chains (Y1, F3, and M5) of the predicted and native structures clearly indicate better side chain selection. The overall results confirm the performance efficiency of SOGA-PSP to predict refined peptide structures free from steric hindrances.

NNFGAIL is a fragment of human islet amyloid polypeptide (IAPP or amylin) of 37-residue hormone found in pancreatic extracts of type 2 diabetics. It plays a role in the function of the pancreas and contributes to glycemic control [29]. The structure of the peptide was predicted using SOGA; as a result, the energy of the structure was comparatively minimal which indicates better conformation as shown in Table 4 and Figure 7. On superimposing with the native structure, an RMSD value of 0.939 was obtained and 5 hydrogen bonds were formed, a most important factor for structure stability. Furthermore, as an advantage out of two hydrogen bonds of X-ray structure, one is conserved. This conservation is also seen in the Ramachandran plot (Table 5 and Figure 8).On further examination, the differences in the orientation of the side chains of the predicted and native structures indicate that better side chain selection is achieved with the proposed approach.