Model For Vaccine Design By Prediction Of B-Epitopes Of IEDB Given ...

2.2. Electronegativity Perturbation Model for Prediction of B-Epitopes

Very recently Gonzalez-Diaz et al. [58] formulated a general-purpose perturbation theory or model for multiple-boundary QSPR/QSAR problems. We adapted here this new theory or modeling method to approach to the peptide prediction problem from the point of view of perturbation theory. Let be a set of ith peptide molecules denoted as m i with a value of efficiency ε ij as epitopes experimentally determined under a set of boundary conditions c j ≡ (c 0, c 1, c 2, c 3,…, c n). We put the main emphasis here on peptides reported in the database IEDB. In this sense, the boundary conditions c j used here are the same reported in this database, c 0 = is the specific peptide, c 1 = soj, c 2 = hoj, c 3 = ipj, and c 4 = tqj. In general, so is the organism that expresses the peptide (but it can include also artificial peptides, cellular lines, etc.), ho is the host organism exposed to the peptide by means of the bp detected with tq. As our analysis, based on the data reported by IEDB we are unable to work with continuous values of epitope activity ε ij. Consequently, we have to predict the discrete function of B-epitope efficiency λ(ε ij) = 1 for epitopes reported in the conditions c j and λ(ε ij) = 0, otherwise. Our main aim is to predict the shift or change in a function of the output efficiency Δλ(ε ij) = λ(ε ij)ref − λ(ε ij)new that takes place after a change, variation, or perturbation (ΔV) in the structure and/or boundary conditions of a peptide of reference. But we know the efficiency of the process of reference λ(ε ij)ref in addition to the molecular structure and the set of conditions c j for initial (reference) and final processes (new). Consequently, to predict Δλ(ε ij) we have to predict only λ(ε ij)new the efficiency function of the new state obtained by a change in the structure of the peptide and/or the boundary conditions. Let ΔV be a perturbation in a function λ; we can define V ij as the state information function for the reference and new states. According to our recent model [58], we can write V ij as a function of the conditions and structure of the peptide m i as follows. In fact, the variational state functions V ij have to be written in pairs in order to describe the initial (reference) and final (new) states of a perturbation, as follow:

Vij=λ(εij)new−∑j=14(λ(mi)−λ(cij)avg),Vqr=λ(εqr)ref−∑r=14(λ(mq)−λ(cqr)avg).(2)

The state function n V ij is for the ith peptide measured under a set of c ij boundary conditions in output, final, or new state. The conjugated state function r V qr is for the qth peptide measured under a set of c qr boundary conditions for the input, initial, or reference state. The difference ΔV between the new (output) state and the reference (input) state is the additive perturbation [58]. Consider

ΔV=Vij−Vqr=[λ(εij)new−∑j=14(λ(mi)−λ(cij)avg)]−[λ(εqr)ref−∑r=14(λ(mq)−λ(cqr)avg)].(2)

Equation (3) described before opens the door to test different hypothesis. A simple hypotheses is H0: existence of one small and constant value of the perturbation function ΔV = e 0 for all the pairs of peptides and a linear relationship between perturbations of input/output boundary conditions with coefficients a ij, b ij, c qr, and d ij. Consider

e0=ΔV=[aij·λ(εij)new−∑j=14bij·(λ(mi)−λ(cj)avg)]−[cqr·λ(εqr)ref−∑r=14dqr·(λ(mq)−λ(cr)avg)].(2)

We can use elemental algebraic operations to obtain from these equations an expression for efficiency as epitope of the peptide λ(ε ij)new. In this case, considering b ij ≈ d qr, we can obtain the different expressions; the last may be very useful to solve the QSRR problem for the large datasets formed by IEDB B-epitopes. Consider

λ(εij)new=(cqraij)·λ(εqr)ref+[∑j=14(bqraij)·(λ(mi)−λ(cj)avg)new]−[∑r=14(dqraij)·(λ(mq)−λ(cr)avg)ref]+(e0aij),λ(εij)new=c0′· λ(εqr)ref+∑j=14dij′· Δ(λ(mi)−λ(cj)avg)+e0′,λ(εij)new=c0′· λ(εqr)ref+∑j=14dij′· ΔΔλijqr+e0′,λ(εij)new=c0′· λ(εqr)ref+∑j=14dij′· ΔΔχijqr+e0′.(2)