Theoretical and Experimental DNA Computation (Natural ...

5.9 Other Laboratory Implementations 141
Φ =(¬x13 ∨ x16 ∨ x18) ∧ (x5 ∨ x12 ∨¬x9) ∧ (¬x13 ∨¬x2 ∨ x20) ∧ (x12 ∨ x9 ∨
¬x5) ∧ (x19 ∨¬x4 ∨ x6) ∧ (x9 ∨ x12 ∨¬x5) ∧ (¬x1 ∨ x4 ∨¬x11) ∧ (x13 ∨¬x2 ∨
¬x19) ∧ (x5 ∨ x17 ∨ x9) ∧ (x15 ∨ x9 ∨¬x17) ∧ (¬x5 ∨¬x9 ∨¬x12) ∧ (x6 ∨ x11 ∨
x4) ∧ (¬x15 ∨¬x17 ∨ x7) ∧ (¬x6 ∨ x19 ∨ x13) ∧ (¬x12 ∨¬x9 ∨ x5) ∧ (x12 ∨ x1 ∨
x14)∧(x20 ∨x3 ∨x2)∧(x10 ∨¬x7 ∨¬x8)∧(¬x5 ∨x9 ∨¬x12)∧(x18 ∨¬x20 ∨x3)∧
(¬x10 ∨¬x18 ∨¬x16)∧(x1 ∨¬x11 ∨¬x14)∧(x8 ∨¬x7 ∨¬x15)∧(¬x8∨x16 ∨¬x10)
with a unique satisfying assignment of
x1 = F, x2 = T,x3 = F, x4 = F, x5 = F, x6 = F, x7 = T,x8 = T,x9 =
F, x10 = T,x11 = T,x12 = T,x13 = F, x14 = F, x15 = T,x16 = T,x17 =
T,x18 = F, x19 = F, x20 = F .
As there are 20 variables, there are 220 =1, 048, 576 possible truth assignments.
To represent all possible assignments, two distinct 15 base value sequences
were assigned to each variable xk(k =1,...,20), one representing
true (T), X T k , and one representing false (F), XF k
. A mix and split generation
technique similar to that of Faulhammer et al. [58] was used to generate a 300base
library sequence for each of the unique truth assignments. Each library
sequence was made up of 20 value sequences joined together, representing the
20 different variables. These library sequences were then amplified with PCR.
The computation proceeds as follows: for each clause, a glass clause module
is constructed which is filled with gel and contains covalently bound probes
designed to capture only those library strands that do satisfy that clause;
strands that do not satisfy the clause are discarded.
In the first clause module (¬x13 ∨ x16 ∨ x18) strands encoding X F 3 , XF 16 ,
and X T 18 are retained, while strands encoding X T 3 , X T 16, andX F 18 are discarded.
Retained strands are then used as input to the next clause module, for each
of the remaining clauses. The final (24 th ) clause module should contain only
those strands that have been retained in all 24 clause modules and hence
encode truth assignments satisfying Φ.
The experimental results confirmed that a unique satisfying truth assignment
for Φ was indeed found using this method. Impressive though it is, the
authors still regard with scepticism claims made for the potential superiority
of DNA-based computers over their traditional silicon counterparts. However,
“they enlighten us about alternatives to electronic computers and studying
them may ultimately lead us to the true ‘computer of the future’” [33].
5.9.4 Maximal Clique Computation
The problem of finding a Maximal Clique (see Chap. 2) using DNA is addressed
by Ouyang et al. in [115]. We recall that a clique is a fully connected
subgraph of a given graph. The maximal clique problem asks: given a graph,
how many vertices are there in the largest clique? Finding the size of the
largest clique is an NP-complete problem.