Computational Methods in Systems Biology

Strong Turing Completeness 119

Fig. 2.Differential and stochastic simulation traces of the compiled reactions for gen-
erating the cosine function as a function of time.

Algorithm 1.Transformation of a PIVP that generates a functionf(t)ina
PIVP that computes the functionf(x) for anyxasf(γ(t)).

1. replacetbyγ(t) in the ODE that generates the functionf(t);


2. multiply all the terms of the ODE byx−γ(t);


3. add the equationγ′=x−γ;


4. initializeγtox 0 and the result variable tof(x 0 ).

For instance, the compilation of the cosine functioncos(x) for any input
concentrationxgenerates the following elementary synthesis reaction system,
where the first four reactions computeγ(t)ingpandgm(withλ= 1), and
the other reactions result from the multiplication byx−γof the ODE terms for
cos(t) which basically translates to the addition of catalystsxpandgmto the
reactions forcos(t):

biocham: compile_fromexpression(cos, x, r).
= [g_m] => gp.
= [x_p] => gp.
= [g_p] => gm.
= [x_m] => gm.
= [g_m+z4_p] => rp.
= [g_p+z4_m] => rp.
= [x_m+z4_m] => rp.
= [x_p+z4_p] => rp.
= [g_m+z4_m] => rm.
= [g_p+z4_p] => rm.
= [x_p+z4_m] => rm.
= [x_m+z4_p] => rm.
= [g_m+r_m] => z4p.
= [g_p+r_p] => z4p.
= [x_p+r_m] => z4p.
= [x_m+r_p] => z4_p.