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Table 1 The network topology formulated as a system of chemical equations. Note that R 2=∅ and is not represented in the actual model but merely introduced here for ease of notation. The factor L is defined in Eq. 1. The symbols p i represent the gene from which the porins are transcribed. The factor \( \mathcal {H}:= {E_{0}^{2}}/(0.001^{2} + {E_{0}^{2}})\) denotes a Hill function; it prevents gene expression when there is no internal energy in the cell. In all experiments reported here \(d_{\protect \{P_{i},E_{i}\protect \}}\) were set proportional to the growth rate, i.e. the reaction rate of R.VIII, while d R is an evolvable parameter acting on a much faster time-scale

From: In silico evolution of diauxic growth

Substrate

Product

Reaction rate

Reaction number

N i

E i +R i

\( k_{N_{i}} \frac {N_{i}}{N_{i} + K_{N_{i}}} P_{i}\, bm \)

R.I

E i

E 0

\( k_{E_{i}} E_{i}\)

R.II

p 1+E 0

p 1+P 1

\(\left (\textrm {leak1} + k_{P_{1}} {\frac {E_{1}^{H}}{{E_{1}^{H}} + K_{P_{1}}^{H}}} E_{0}\right) L \, bm \, \mathcal {H} \)

R.III

p 2+E 0

p 2+P 2

\(\left (\textrm {leak2} + k_{P_{2}} {\frac {E_{2}^{H}}{{E_{2}^{H}} + K_{P_{2}}^{H}}} {\frac {K_{R}^{H}}{R^{H} + {K_{R}^{H}}}} E_{0}\right)L\, bm \, \mathcal {H}\)

R.IV

R 1+P 2

B 2

k b R 1 P 2

R.V

B 2

R 1+P 2

k ub B 2

R.VI

{P i ,E i ,R 1}

∅

\(d_{\{P_{i},E_{i}, R_{1}\} }\)

R.VII

E 0

b m

\( g\, bm \frac {E_{0}}{E_{0} + {K_{g}}}\)

R.VIII