LNCS 2950 - Aspects of Molecular Computing (Frontmatter Pages)

The Power of Networks of Watson-Crick D0L Systems 115
2 ≤ i ≤ n, is checked at the node Mn+i. Notice that after checking the occurrence
of the letter ai, the new strings will be forwarded to all nodes of Γ, so the
strings representing all possible paths in γ are checked according to the criteria
of being Hamiltonian. After the (2n)th step, if the node M2n contains a string
v ¯ X2n, wherev = ai1 ...ain, then this string is forwarded to the node M2n+1 and
is rewritten to v. It will represent a Hamiltonian path in γ, where the string of
indices i1,...,in corresponds to the nodes Vi1,...,Vin, visited in this order. ✷
5 Population Growth in DNA Systems
The alphabet in the systems considered above is generally bigger than the fourletter
DNA alphabet. We now investigate the special case of DNA alphabets.
By a DNA system, [18], we mean a Watson-Crick D0L system whose alphabet
equals the four-letter DNA alphabet. In this section we consider networks based
on DNA systems only. It turns out that quite unexpected phenomena occur in
such simple networks. For instance, the population growth can be very weird, a
function that is not even Z- rational, although the simplicity of the four-letter
nodes might suggest otherwise.
We will show in this section that it is possible to construct standard networks
of DNA systems, where the population growth is not Z-rational. (By the population
growth function f(n) we mean the total number of words in all nodes at
the nth time instant, n ≥ 0. We refer to [7] for formal details of the definition.)
The construction given below can be carried out for any pair (p, q) of different
primes. We give it explicitly for the pair (2, 5). The construction resembles the
one given in [18].
Theorem 3 There is a network of standard DNA systems, consisting of two
components, whose population growth is not Z-rational. This holds true independently
of the protocol.
Proof. Consider the network Γ of standard DNA systems, consisting of two
components defined as follows. The first component has the axiom TGand rules
A → A, G → G, T → T 2 , C → C 5 .
The second component has the axiom A and rules
A → A, G → G, T → T, C → C.
Clearly, the second component does not process its words in any way.
The beginning of the sequence of the first component is
TG, A 2 C, T 2 G 5 , T 4 G 5 , A 8 C 5 , T 8 C 25 ,
T 16 G 25 , A 32 C 25 , T 32 G 125 , T 64 G 125 ,A 128 C 125 ,
T 128 G 625 , T 256 G 625 , T 512 G 625 ,A 1024 C 625 ...