Genetic circuit

Equations

In the absence of IPTG

Equation 1: Equation describing the rate of transcription of LacI MRNA (M_LacI):

k_mlacI is the transcription rate of M_lacI (a measure of promoter strength) and d_mlacI is the degradation rate. At steady state:

Equation 2: Equation describing the rate of translation of LacI protein (P_lacI)

At steady state:

where k_lacI is the translation rate of lacI protein and d_placI is the degradation rate of P_lacI.Equations 3 and 4 describe the transcription and translation of the protein of interest P_out.

Equation 3: Transcription of P_out

Unlike in the previous case, the output promoter is inducible. In the absence of further information, we model the effect of LacI on transcription/ POPS activity with a Hill function, which represses when amounts are above the threshold K, and activates when P_lacI amounts fall below threshold. Such assumption can be revised in the light of contradicting experimental data.

At steady state:

Equation 4: Equation describing the rate of translation of protein of interest P_out:

At steady state:

When IPTG is introduced

When IPTG is added into the system, LacI can bind to it, forming an intermediate complex [IPTG-LacI]:

k1 and k2 are the dissociation constants of the forward and reverse reactions. Therefore, we can modify Equation 2 to include the effects of IPTG on the LacI system as follows:

Equation 2 modified:

Note that now, the effects of IPTG have been included. The –k2[Plac][IPTG] term contributes negatively, as it removes PLac from the system. The k1[IPTG-Plac] term contributes positively, as it re-stores the levels of PLac in the system. The equations that describe the evolution of IPTG and IPTG-LacI are-

Equation 5: Rate of change of IPTG in the system:

Equation 6: Rate of change of intermediate [IPTG-Plac] complex in the system.

Parameters