Diffusion of C and Ce

The SCB1 protein (C) diffuses from each cell to the environment and back.

Chemical equation

$C \underset{r \cdot D}{\stackrel{D}{\rightleftharpoons}} C_{e}$

Rate equation

The rate equation is different for the two reactions in order to express the dynamics of the cell density in the environment. Therefore, an additional term $r= V_{cell,tot}/V_{ext}$ is added in the rate of diffusion from the external environment to the cell. This term represents the additional propensity given to the diffusion of extracellular SCB1 by the accumulation of cells in the environment and the consequent increase of cell density. The two rate laws are therefore formed as follows:

$r_{out}= D \cdot ([C_{e}]-[C])$ $r_{in}= r \cdot D \cdot([C]-[C_{e}])$

Parameters

The parameter of this reaction is the degradation rate of C ($d_{C}$).

Name	Value	Units	Origin	Remarks
$d_{C}$	$0-0.001$	$min^{-1}$	Degradation rate of acyl-homoserine lactone (AHL)	The assumption that the degradation of SCB1 is slower than AHL is made, as described in Takano(2006)

Parameters with uncertainty

The most plausible parameter value for the $d_{C}$ is decided to be $0.0002 min^{-1}$ and the confidence interval $2$. This means that the mode of the PDF is 0.0002 and the range where 95% of the values are found is between $0.0001$ and $0.0004 min^{-1}$.

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

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The location and scale parameters of the distribution are:

Parameter	μ	σ
$d_{C}$	$-8.4046$	$0.3356$