LNCS 2950 - Aspects of Molecular Computing (Frontmatter Pages)

272 Satoshi Kobayashi, Takashi Yokomori, and Yasubumi Sakakibara
4 Minimum Free Energy Under Boundary
Constraints
Let S be a finite set of strings of length n over Σ and consider the configurations,
⎛
−→
α (i)
v1 = ⎝ ·
β
←− (j)
⎞
⎠ and
� −→γ
(k)
v2 = ·
←−
δ (l)
�
,
where α, β, γ, δ ∈ S. We define minI(v1,v2) as the minimum free energy of 2cycles
among all 2-cycles with the boundary configuration (v1,v2) in a structured
string x(T )withx ranging over S + .ByminH(v1) wedenotetheminimumfree
energy of 1-cycles among all 1-cycles with the boundary configuration v1 in
a structured string x(T )withx ranging over S + . The notation minD(v1) is
defined to be the minimum free energy of free end structures among all free end
structures with the boundary configuration v1 in a structured string x(T )with
x ranging over S + .
For a string α ∈ S and an integer i with 1 ≤ i ≤ n, we introduce the notation
α:i which indicates the ith position of the string α.
Then, for a set S, we define:
V (S) ={α:i | α ∈ S, 1 ≤ i ≤ n }.
Consider two elements α:i, β:j in V (S).
(1) In case that either α �= β or i �< jholds, we define:
W (α:i, β:j) ={α[i, n]γβ[1,j] | γ ∈ S ∗ }.
(2) In case that α = β and i