New Approaches to in silico Design of Epitope-Based Vaccines

66 CHAPTER 6. EPITOPE ASSEMBLY
dummy epitope by (6.1d). (6.1e) requires all epitopes except for the dummy epitope to
have a higher ordinal number than their N-terminal neighbor. If the edge between two
nodes a, b ∈ E is included in the solution, i.e., xab = 1, (6.1e) becomes ub ≥ ua + 1: the
ordinal number of b is required to be greater than the ordinal number of a. Otherwise,
(6.1e) becomes ua − ub ≤ |E ′ | − 2, which is always true due to the (6.1d). This way the
only possibility to form a cycle and thus to fulfill the first two constraints is by inclusion of
the dummy epitope. Given these constraints and the graph G ′ in Figure 6.2B the solution
P 3 − P 1 − P 3 , P 2 − ε − P 2 is not feasible: In the cycle P 3 − P 1 − P 3 , P 3 is the N-terminal
neighbor of P 1 and P 1 is the N-terminal neighbor of P 3 . According to the fifth set of
constraints, the ordinal number of P 1 has to be greater than the ordinal number of P 3 and
vice versa, which is a contradiction. However, the solution ε − P 3 − P 1 − P 2 − ε, which
corresponds to a Hamiltonian cycle, is feasible: assigning the ordinal numbers 2, 3, and
4 to P 3 , P 1 , and P 2 , respectively, every node except for the dummy node has a higher
ordinal number than its N-terminal neighbor. Since (6.1e) does not apply to the dummy
node, the solution is a feasible solution for the ILP.
minimize
ILP 6.1: ILP formulation of the epitope ordering problem.
a,b∈E ′ wab xab
subject to ∀a ∈ E ′ :
∀a ∈ E ′ :
b∈E ′ xab = 1 (6.1a)
b∈E ′ xba = 1 (6.1b)
uε = 1 (6.1c)
∀a ∈ E: 2 ≤ ua ≤ |E ′ | (6.1d)
∀a, b ∈ E, a = b: ua − ub + 1 ≤ (|E ′ | − 1)(1 − xab) (6.1e)
Definitions
E Set of epitopes
ε Dummy epitope
E ′ Set of epitopes including dummy epitope (E ′ = E ∪ {ε})
Parameters
wab
Variables
Negative of the logarithm of the probability that epitopes a and b will be
processed properly when a is the N-terminal neighbor of b in the polypeptide
xab = 1 if epitope a is the N-terminal neighbor of epitope b; otherwise xab = 0
ua
6.2.2 Heuristic
Position of epitope a in the optimal polypeptide (1 ≤ ua ≤ |E ′ |)
We use a TSP heuristic proposed by Lin & Kernighan [116]. The Lin-Kernighan heuristic
(LKH) is based on the λ-opt algorithm, which again is based on the λ-opt concept: A tour
is λ-optimal if no shorter tour can be obtained by replacing λ links. The larger the value
of λ the more likely it is for a λ-optimal tour to be optimal. Starting from a randomly