Synthesis of C

SCB (C) is synthesized by glycerol derivative and b-keto acid derivative precursors in a reaction catalyzed by ScbA (A).

Chemical equation

$A \xrightarrow{S} A + C$

Rate equation

In our model we assume that there is an infinite amount of substrate S for the SCB synthesis and that the rate of the reaction is dependent only on the concentration of the catalyst A. Therefore, the rate of the reaction is:

$r= K_{C}\cdot [A]$

Parameters

The parameter of this reaction is the synthesis rate of SCB ($K_{C}$). The parameter values were chosen with the assumption that GBL production involves a one rate limiting step involving the Afsa-like protein and then a spontaneous cyclization.

Name	Value	Units	Value in previous GBL models	Remarks-Reference
$K_{C}$	$0.00315-85.5$	$min^{-1}$	$0.74 s^{-1} (44.4 min^{-1})$ Range tested: $0-1.7 s^{-1}$ ($0-102 min^{-1}$) Bistability range: $0.67-1.7 s^{-1}$ ($40.2-102 min^{-1}$) and $0.074-37 s^{-1}$ ($4.44-2220 min^{-1}$)	Chen et al. calculated a synthesis rate of $0.189 h^{-1}$ ($0.00315 min^{-1}$) for the autoinducer PAI2 in Pseudomonas aeruginosa at stationary phase, by subtracting the $k_{deg}$ from the net decay constant ($k_{deg}-k_{syn}$). Chen et al. 2005 Additionally, Garcia-Ojalvo et al. modelled repressilators coupled by quorum sensing and reported a value for the synthesis rate of Acyl homoserine lactone (AHL) in E. coli of $0.01 min^{-1}$. Garcia-Ojalvo et al. 2004 Finally, Parsek et al. reported a range of values for the $V_{max}$ of AHL in E. coli between 2 and 16 (mol of products $\cdot min^{-1} \cdot mol^{-1}$ RhlI). By using the enzyme concentration and $M_r$, the $k_{cat}$ of the AHL synthesis can be calculated ($0.1-85.5 min^{-1}$). Parsek et al. 1999

Parameters with uncertainty

When deciding how to describe the uncertainty for each parameter we must take into consideration that the reported values are either calculated or derived with approximation from experiments. Additionally, they correspond to synthesis rates of autoinducers in other bacteria (E. coli, P. aeruginosa) and not to butyrolactones. This means that there might be a notable difference between actual parameter values and the ones reported in literature. These facts influence the quantification of the parameter uncertainty and therefore the shape of the corresponding distributions. By assigning the appropriate weights to the parameter values and using the method described here, the appropriate probability distribution was designed.

Therefore, the weight of the distribution is kept at $0.094 min^{-1}$ which is set as the mode of the log-normal distribution for $K_{C}$ and the Spread is set to $60.49$. In this way, the range where 68.27% of the values are found is between $0.00155$ and $5.67 min^{-1}$.

The probability distribution for the parameter, adjusted accordingly in order to reflect the above values, is the following:

The parameter information of the distribution is:

Parameter	Mode	Spread	μ	σ
$K_{C}$	$0.094$	$60.49$	$0.87914$	$1.8017$