Artificial networks

The artificial networks used in this work consist of reactions which perform transformations between artificial compounds. These compounds are repre-sented by sets of building blocks. In a specific network there exists a finite setB of building blocks. Each compoundc_j can contain at most N_i units of the building blockiand is uniquely defined by the set of building blocks and the corresponding numbers of unitsb_ij, i∈B. Hence the number of possible compounds cin the network is

c=

B

Y

i

(N_i+ 1)

!

−1. (A.11)

Reactions are required to conserve the building blocks, i.e. the number of units of any building block i in the substrates must be the same as the number of units of the building block in the products. For a reaction with the set of substrates Qand the set of products P this means

X

k∈Q

−s_kbik =^X

l∈P

slbil,∀i∈B, (A.12) with b_ij is the number of building blocks of type i in compound j and s_j being its stoichiometry in this reaction. (Note: for compounds occurring on

both sides of a reaction, the stoichiometries can be modified in such a way that this compounds remains on only one side)

The number of all possible reactions between the above defined com-pounds obeying the conservation relation is infinite. Clearly, the set of sub-strates can be chosen from the 2^c possible sets of compounds. Still, for each compound in this set the stoichiometry s_j can be freely chosen.

However, reactions of type q↔p, with q being the number of substrates and p the number of products and q, p > 1, can be replaced by a set of reactions of type 2 ↔ 1: A reaction of type q → p transfers from each of their q substrates a set of building blocks to each of their p products.

Some of these sets may be empty. This splitting-up of the substrates can be performed by at most q×(p−1) reactions of type 1 → 2. Each of the p products receives a set of building blocks from each of the q substrates.

Again, some of these sets may be empty. This assembly of the products can be performed by at mostp×(q−1) reactions of type 2→1. By utilizing the split-up-reactions before the assembly-reactions it can be assured that the intermediates do not violate any size limitations on the compounds as long as the substrates and products obey them.

Compounds with stoichiometries s_j larger than one can be treated as s_j separate compounds. Hence, with the above substitution, reactions of type q↔p with arbitrary stoichiometries can be represented as a set of reactions of type 2↔1 with all stoichiometries being 1.

Reactions of type 1↔1 with stoichiometries equal to 1 do not play a role for these networks as the construction rule for the compounds does not allow for any distinguishable isomers. 1 ↔1 reactions with larger stoichiometries can be accordingly transformed. Consequently, the set of 2 ↔ 1 reactions contains as a special case reactions of type 2c₁ *) c₂.

The number of 2 ↔1 reactionsrwith unity stoichiometries in a network with size limited compounds, as defined above, is finite. The number of reactions r can be expressed as follows:

r=

c

X

j=1

R(c_j), (A.13)

where R(c_j) is the number of possible reactions that can split up c_j into c_u and c_v:

c_j *) c_u+c_v (A.14)

R(c_j) is dependent on the number of units of a specific building block i in the compound j, b_ij. With equation A.12 it follows:

b_ij =b_iu+b_iv,∀i∈B. (A.15)

Apparently, any set of the b_iu with 0 ≤ b_iu ≤ b_ij yields a possible split-up-reaction of c_j as in equation A.14. The compound c_v is then defined by b_iv = b_ij −b_iu. The two cases where b_iu = 0 for all i ∈ B and b_iv = 0 for alli ∈ B will be excluded, as the corresponding reactions do not perform a transformation. The number of possible sets of the b_iu is

"

This is, however, not the number of possible split-up-reactions. Clearly, swapping c_u and c_v yields the same reaction. Hence, except for the case where biu = biv for all i ∈ B each reaction is generated twice. The case b_iu =b_iv for all i∈B can only occur if all b_ij are even. Hence, R(c_j) can be

0 , otherwise (A.17) This expression can be inserted into equation A.13. As the c_j are defined by the b_ij the sum over all compounds can be replaced by a sum over all combinations of the b_ij. The indexj numerating the compounds disappears, as the b_i are now enumerated directly:

r = Here, |B| is the number of building block types. n_even is the number of combinations of theb_i for which all b_i are even, including the case b_i = 0 for alli∈B. Equation A.19 can be rewritten as:

r= where N_i div 2 denotes the largest integer smaller than or equal to N_i/2.

The 3rd term in equation A.20 is by about a factor of 2^|B|+1 smaller than the

2nd term. Hence, for a sufficiently large number of building block types |B|, g(b_i) may be neglected:

r≈



 1

2

(|B|+1) |B|

Y

i=1

(N_i+ 1)(N_i+ 2)



−

|B|

Y

i=1

(N_i+ 1) (A.21) The 2nd term in equation A.20 is approximately the number of possible compounds c in the network (cf. equation A.11). For sufficiently largeNi, the number of possible reactions r is much larger than the number of compounds c and also this term can be neglected:

r≈

1 2

(|B|+1)|B|

Y

i=1

(N_i+ 1)(N_i+ 2) (A.22) Even though it is not a problem to calculate equation A.20, it should be noted that the approximations A.21 and A.22 give already good results for the networks used in this work. For the network defined by N(A,B,C,D,E) = (6,4,4,3,2), (A.20) yields 186972, (A.21) 186900 and (A.22) 189000 possible reactions. For the case that the N_i ≈ N from equation A.11 and A.22 can be seen that the number of possible compounds in a network is of the order of N^|B| and the number of possible reaction of the order ofN^2|B|/2^|B|+1.