Theoretical and Experimental DNA Computation (Natural ...

4.4 Ogihara and Ray’s Boolean Circuit Model 79
The simulation of gates at level k>0 proceeds as follows. We denote by
i1 and i2 the indices of the gates that supply the inputs for gi.
(1) Input(U)
(2) for k =1tod do
(3) for each gate gi at level k in parallel do
(4) if gi computes ∨ then do
(5) if (σ[i1] ∈ U) or (σ[i2] ∈ U) then U ← U ∪ σ[i]
(6) else if gi computes ∧ then do
(7) if (σ[i1] ∈ U) and (σ[i2] ∈ U) then U ← U ∪ σ[i]
(8) end for
(9) end for
At step (1) the initial tube, U, is created. Then, for each circuit level k>0
(step 2), the gates at level k are simulated in parallel (steps 3 through 8). The
simulation of each gate gi at level k is achieved as follows. If gi is an ∨-gate,
the string σ[i] ismadepresent 1 in U (i.e., gi evaluates to 1) if either of the
strings representing the inputs to gi is present in U (step 5). If gi is an ∧-gate,
the string σ[i] ismadepresentinU (i.e., gi evaluates to 1) if both of the strings
representing the inputs to gi are present in U (step 7). The simulation then
proceeds to the next level. At the termination of the computation, Ogihara
and Ray analyze the contents of U to determine the output of the circuit.
4.4.1 Ogihara and Ray’s Implementation
In this section we describe the laboratory implementation of the abstract
Boolean circuit model just described. The circuit simulated in [113] is depicted
in Fig. 4.2.
Each gate gi is assigned a sequence of DNA, σ[i], of length L, beginning
with a specific restriction site, E. Eachedgei → j is assigned a sequence ei,j
that is the concatenation of the complement of the 3’ L/2-mer of σ[i] and
the complement of the 5 L/2-mer of σ[j]. In this way, ei,j acts as a “splint”
between gi and gj if and only if both σ[i] andσ[j] are present. However, we
later highlight a case where this strand design does not hold.
The simulation of gates at level 0 (i.e., the construction of the initial tube)
proceeds as follows. Begin with a tube of solution containing no DNA. For
each input gate xi that evaluates to 1, pour into the tube a population of
strands representing σ[i]. If xi evaluates to 0, do not add σ[i] tothetube.
We now consider the simulation of gates at level k>0. The simulation of
∨-gates differs from that ∧-gates, so we first consider the case of an ∨-gate,
gj, at level k.
First, pour into the tube strands representing σ[j]. Then, for each edge i → j
pour into the tube strands representing ei,j. Allow ligation to occur. If strands
1 The process by which this is achieved is described in detail in Sect. 4.4.1.