Model

In our design, integrases compute the output of the logic gates. Integrases allow to flip one fragment of DNA. The model we developped is described here.

Chemical Species

Modeling DNA-binding sites

Each dimer of integrases can specifically bind to a DNA binding site. As the flipping is irreversible, these DNA binding sites can be three possible states:

• SI_{IntegraseName}: inactive DNA binding site. No dimer is bound to this site, which has never been flipped.

• SA_{IntegraseName}: active DNA binding site. A dimer is to this site.

• SF_{IntegraseName}: flipped DNA binding site. This DNA binding site has been used by a flipping.

The three different states of DBxb1-DNA binding sites.

Reactions

For Bxb1

$$ \begin{align} Bxb1 + Bxb1 &\leftrightarrow DBxb1 \\ DBxb1 + SI_{Bxb1} & \leftrightarrow SA_{Bxb1}\\ Bxb1 &\rightarrow \\ DBxb1 &\rightarrow \end{align}$$

For ΦC31

\begin{align} \phi C31 + \phi C31 &\leftrightarrow D\phi C 31 \\ D\phi C 31 + SI_{\phi C31} & \leftrightarrow SA_{\phi C31}\\ \phi C31 &\rightarrow \\ D\phi C31 &\rightarrow \end{align}

Differential Equations

Applying mass action kinetic laws, we obtain the following set of differential equations for Bxb1.

$$\frac{d[Bxb1]}{dt}=-2 k_{DBxb1}[Bxb1]^2+ 2 k_{-DBxb1}[DBxb1]-d_{Bxb1}[Bxb1]$$

$$\frac{d[DBxb1]}{dt}=-k_{SABxb1}[DBxb1][SI_{Bxb1}]+k_{-SABxb1}[SA_{Bxb1}]+k_{DBxb1}[Bxb1]^2-k_{-DBxb1}[DBxb1]-d_{DBxb1}[DBxb1]$$

$$\frac{d[SA_{Bxb1}]}{dt}=k_{SABxb1}[DBxb1][SI_{Bxb1}]-k_{-SABxb1}[SA_{Bxb1}]$$

Replacing every occurence of Bxb1 by ΦC31 gives the set of differential equations for ΦC31.

State of the Art

Replacing every occurence of Bxb1 by ΦC31 gives the set of parameters for ΦC31. The same status can be applied to those parameters.

Even if degradation rates were not determined specifically for the serine integrases and their dimerized form, degradation rates of proteins in E. coli are available. To characterize integrases behavior, we focus on finding the parameters for dimerization and DNA-binding.

Characterization: K_SABxb1

Data

The parameter fitting is based on Bonnet's paper^[9]. Their experimental setup is different from ours. However, they experimentally retrieve a transfer function between aTc and Bxb1 switching rate. Here is the figure of interest in our case.

Transfer function from aTc to Bxb1. The experimental data corresponds to the points. They fitted this data with their own model. Supplementary figure S4 of Bonnet's paper Amplifying Genetic Logic Gates^[9]

To use their experimental data, we have to model their induction mechanism using aTc. We use a simplify version of it proposed by the 2013 iGEM team UCSF. They modeled the induction with aTc as a leaky Hill function. In the following table are the set of parameters introduced to model induction.

We obtain the following set of differential equations:

$$\begin{align*} \frac{d[Bxb1]}{dt} &= k_{mRNA_{Bxb1}}*(A_L + B_L * \frac{[aTc]^n}{[aTc]^n+K_L^n})- 2 k_{DBxb1}*[Bxb1]^2 + 2k_{-DBxb1}*[DBxb1] - d_{Bxb1}*[Bxb1] \\ \frac{d[DBxb1]}{dt} &= k_{DBxb1}*[Bxb1]^2 - k_{-DBxb1}*[DBxb1] - k_{SABxb1}*[DBxb1]*[SI_{Bxb1}] + k_{-SABxb1}*[SA_{Bxb1}] - d_{DBxb1}*[DBxb1] \\ \frac{d[SA_{Bxb1}]}{dt} &= k_{SABxb1}*[DBxb1]*[SI_{Bxb1}] - k_{-SABxb1}*[SA_{Bxb1}] \end{align*}$$

Assumptions

Assumption A

The back-reaction from DBxb1 binding to site inactive is considered to be negligible, compared to the flipping rate. That is to say that once a site is active, it can only be flipped. Thus, an active site would only be a transitional state in our whole cell model. As the flipping is not modeled in our integrase subsystem, active sites are not transformed into flipped sites at the end of the information pipeline. Thus, we consider that we can express the switching rate given active site concentration.

Assumption B

As switching needs two active sites to be effective (for more information on flipping, check the XOR gate page), the switching rate is approximated to: $${(\frac{SA_{Bxb1}}{S_{TOT}})}^2$$ This approximation is understated by statistical considerations.

Assumption C

Given the normalization of the paper, basal rate of production of Bxb1 is not taken into account. Thus, we consider that $$A_L = 0$$

Assumption D

The activation by aTc is assumed to be dominant over degradation of Bxb1 and dimerization of Bxb1. It is supposed to be valid on the range of aTc concentration considered.

$$\frac{k_{mRNA_{Bxb1}}*(A_L + B_L * \frac{[aTc]^n}{[aTc]^n+K_L^n})}{d_{Bxb1}K_{DBxb1}} >> 1 $$ with $$K_{DBxb1} = \frac{k_{-DBxb1} + d_{DBxb1}}{k_{DBxb1}} $$

Parameter fitting

We consider the system at steady-state. After derivation, the following explicit equation can be retrieved:

$$[SR]_{qss} = {(\frac{B_{L} * [aTc]^{n}}{ \lambda_1 K_L^{n} + (B_{L} +\lambda_1) [aTc]^{n}})}^2 $$

where $$ \lambda_1 = \frac{2*d_{Bxb1}*K_{SABxb1}}{k_{mRNA_{Bxb1}}} ; K_{SABxb1} = \frac{k_{-SABxb1}}{k_{SABxb1}} $$

Using the Least Absolute Residual method, we determined the lumped parameter λ₁. Here is the value with its 95% confidence bounds:

$$\lambda_1 = 1.82e-07 (1.649e-07, 1.992e-07)$$

Parameter fitting of the dissociate rate constant of K_SABxb1

We assume that:

• d_Bxb1 corresponds to the order of magnitude of 10^-2 min^-1, as most of the protein in E. coli ^[17].
• k_mRNABxb1 is of the order of magnitude 10^-1 min^-1 mRNA^-1. We estimated to be a low value because the starting codon of Bxb1 is GTG (and not ATG) and this parameter also takes into account folding time.

Thus, K_SABxb1's order of magnitude is 10^-6 nM. The interpretation of this dissociation constant is that the DNA binding reaction is really specific, as it can be expected for integrases.

Characterization: K_DBxb1

Data

The parameter fitting is based on Bonnet's paper^[9]. Their experimental setup is different from ours. However, they experimentally retrieve a transfer function between aTc and Bxb1 switching rate. Here is the figure of interest in our case.

Transfer function from aTc to Bxb1. The experimental data corresponds to the points. They fitted this data with their own model. Supplementary figure S4 of Bonnet's paper Amplifying Genetic Logic Gates^[9]

We do the same modeling steps, as for the previous characterization.Thus, we obtain the following set of differential equations:$$\begin{align*} \frac{d[Bxb1]}{dt} &= a_{}*(A_L + B_L * \frac{[aTc]^n}{[aTc]^n+K_L^n})- 2 k_{DBxb1}*[Bxb1]^2 + 2k_{-DBxb1}*[DBxb1] - d_{Bxb1}*[Bxb1] \\ \frac{d[DBxb1]}{dt} &= k_{DBxb1}*[Bxb1]^2 - k_{-DBxb1}*[DBxb1] - k_{SABxb1}*[DBxb1]*[SI_{Bxb1}] + k_{-SABxb1}*[SA_{Bxb1}] - d_{DBxb1}*[DBxb1] \\ \frac{d[SA_{Bxb1}]}{dt} &= k_{SABxb1}*[DBxb1]*[SI_{Bxb1}] - k_{-SABxb1}*[SA_{Bxb1}] \end{align*}$$

Assumptions

Assumption A

The back-reaction from DBxb1 binding to site inactive is considered to be negligible, compared to the flipping rate. That is to say that once a site is active, it can only be flipped. Thus, an active site would only be a transitional state in our whole cell model. As the flipping is not modeled in our integrase subsystem, active sites are not transformed into flipped sites at the end of the information pipeline. Thus, we consider that we can express the switching rate given active site concentration.

Assumption B

As switching needs two active sites to be effective (for more information on flipping, check the XOR gate page), the switching rate is approximated to: $${(\frac{SA_{Bxb1}}{S_{TOT}})}^2$$ This approximation is understated by statistical considerations.

Assumption C

Given the normalization of the paper, basal rate of production of Bxb1 is not taken into account. Thus, we consider that $$A_L = 0$$

Assumption E

We neglect the degradation rate of DBxb1, d_DBxb1, with respect to the dimerization reaction.

Parameter fitting

We consider the system at steady-state. After derivation, the following explicit equation can be retrieved:

$$[SR]_ = {(\frac{(B_{L} * [aTc]^{n})^2}{((B_{L} * [aTc]^{n})^2 + \lambda_2 * \lambda_1 (K_{L}^{n} + [aTc]^{n})^2))}})^2$$

where $$ \lambda_1 = \frac{2*d_{Bxb1}*K_{SABxb1}}{k_{mRNA_{Bxb1}}} ; \lambda_2 = \frac{d_{Bxb1}*K_{SABxb1}}{2 *k_{mRNA_{Bxb1}}}; K_{DBxb1} = \frac{k_{-DBxb1}}{k_{DBxb1}}; K_{SABxb1} = \frac{k_{-SABxb1}}{k_{SABxb1}} $$

As the value of λ₁ was derived in the previous characterization step, we use the Least Absolute Residual method to determine the lumped parameter λ₂. Here is the value with its 95% confidence bounds:

$$\lambda_2 = 8.211e-07 (7.421e-07, 9.001e-07))$$

Parameter fitting of the dissociation rate constant of K_DBxb1

We assume that:

• d_Bxb1 corresponds to the order of magnitude of 10^-2 min^-1, as most of the protein in E. coli ^[17].
• k_mRNABxb1 is of the order of magnitude 10^-1 min^-1 mRNA^-1. We estimated to be a low value because the starting codon of Bxb1 is GTG (and not ATG) and this parameter also takes into account folding time.

Thus, K_DBxb1's order of magnitude is 10^-6 nM. The interpretation of this dissociation rate constant is that the dimerization reaction is really specific, as it can be expected for integrases.

Results

The figures show the predicted transfer function for the ratios of active sites of Bxb1 (SABxb1) to total number of sites (SBxb1TOT) and active sites of ΦC31 (SAΦC31) to total number of sites (SΦC31TOT) as a function of their respective integrase concentrations based on the fitted parameters. With the fitted parameters, we found the Km value to be 10 nM.

Predicted transfer function for Bxb1 module.

Predicted transfer function for ΦC31 module.