LNCS 2950 - Aspects of Molecular Computing (Frontmatter Pages)

Digital Information Encoding on DNA 157
by grouping (see below) DNA bases into two groups, the 0-group (say a, t) and
the 1-group (say c, g), corresponding to the bits of T , and then rewriting the
codewords in C so that their bits are interpreted as one (a or c) or the other
(g or t). Codewords are generated in such a way that the distances would be
essentially preserved (in proportion of 1 : 1), as shown in Table 1(b).
The quality of the code set produced naturally depends on the quality of selfdistance
of T and the error-correcting capability of C. If a minimum self-distance
is assured, the method can be used to produce codewords with appropriate c/g
content to have nearly uniform melting temperatures. In [2], an exhaustive search
was conducted of templates of length l ≤ 32 and templates with minimum selfdistance
about l/3. However, the thermodynamical analysis and lab performance
are still open problems. To produce a point of comparison, the template method
was used, although with a different seed code (32-bit BCH codes [26]). The
pairwise Gibbs energy profiles of the series of template codeword sets obtained
using some templates listed in [2] are shown below in Fig. 2(a).
3.2 Tensor Products
A new iterative technique is now introduced to systematically construct large
code sets with high values for the minimum h-distance τ between codewords. The
technique requires a base code of short biners (good set of short oligos) to seed
the process. It is based on a new operation between strands, called the tensor
product between words from two given codes of s- andt-biners, to produce a
code of s + t-biners. The codewords are produced as follows. Given two biners
x = ab, andy fromtwocodingsetsC, D of s- andt-biners, respectively, where
a, b are the corresponding halves of x, new codewords of length s + t are given
by x ⊘ y = a ′ y ′ b ′ , where the prime indicates a cyclic permutation of the word
obtained by bringing one or few of the last symbols of each word to the front.
The tensor product of two sets C ⊘D is the set of all such products between pairs
of words x from C and y from D, where the cyclic permutation is performed once
again on the previously used word so that no word from C or D is used more
than once in the concatenation a ′ y ′ b ′ . The product code C ⊘D therefore contains
at least |C||D| codewords. In one application below, the construction is used
without applying cyclic permutations to the factor codewords with comparable,
if lesser, results.
Size/τ Best codes
5/1 10000,10100,11000
6/2 100010.100100,110000
7/2 1000010,1001000,1100000
8/2 11000100,11010000,11100000
BNA DNA base
000,010/111,101 a/t
011,100/001,110 c/g
Table 1. (a) Best seed codes, and (b) mapping to convert BNA to a DNA strand.