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 $r= V_{cell}/V_{ext}$ 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 SCBs by the accumulation of cells in the environment and the consequent increase of cell density. The two rate laws are therefore formed as follows:

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

Further information on the coefficient $r$ and the dynamics of cell growth and division is provided here.

Parameters

The parameter of this reaction is the diffusion rate of C ($D$). In bacteria communication, the autoinducer can either passively diffuse in and out of the cell or, in cases of larger signalling molecules, can be actively transported through the cell membrane. Generally, the diffusion rates when driven by active processes are four orders of magnitude smaller than the passive diffusion rates of small molecules . In this model, active transport is not explored, as SCBs are small molecules that can easily diffuse though the cell membrane. Moreover, transport driven by SCB concentration differences due to increase of the cell density in the environment, is simulated. This means that as cell density grows, more autoinducer molecules enter the cell from the outside. The parameter values were derived from measurements and estimations of different autoinducers' passive diffusion and active transport rates in bacteria.

Name	Value	Units	Value in previous GBL models	Remarks-Reference
$D$	$0.1-24$	$min^{-1}$	$8.3 \cdot 10^{-2} s^{-1}$ (Range tested: $0-0.4 s^{-1}$) (Bistability range: $0.02-0.097 s^{-1}$ and $8.3 \cdot 10^{-6}-4.2 s^{-1}$)	Kaplan et al. conducted experiments on how AHL exits and enters the cell in Vibrio fischeri and reported that the equimolar internal and external concentrations were established within $20 s$ (estimated rate ~$3 min^{-1}$), suggesting a very fast diffusion rate. If the signal concentration in the medium is sufficiently large, the time for intracellular concentration to reach equilibrium with the extracellular, is $~\frac{1}{D}$. Therefore, the rate constant of diffusion can be estimated $~ \frac{1}{20}s^{-1}=3 min^{-1}$. Kaplan et al. 1985 Goryachev et al. have published a stochastic model on quorum sensing in Agrobacterium tumefaciens, where they investigated the role of active transport and passive diffusion of the autoinducer AAI in and out of the cell. The values used for their simulations were $0.1-0.4 s^{-1} (6-24 min^{-1})$ for the passive diffusion and $0.02-0.6 s^{-1} (1.2-36 min^{-1})$. An estimation within the same range was made by Groisman et al. in a study using a microfluidic chemostat to analyze the cell response to an exogenously added autoinducer, where the reported time span for diffusive exchange of small molecules between different compartments was $~40 s$. Therefore, the resulting diffusion rate is $~ \frac{1}{40}s^{-1}=1.5 min^{-1}$. Supplementary material from Goryachev et al. 2005-(Supporting Information) Groisman et al. 2005 Finally, Weber et al. reported some cases of active transport where the diffusion rates are much faster, between $~0.01$ and Failed to parse (Cannot store math image on filesystem.): ~0.1 min^{-1}) . Weber et al. 2011

Parameters with uncertainty

The most plausible parameter value for the $D$ is decided to be Failed to parse (Cannot store math image on filesystem.): 8 min^{-1} and the confidence interval Failed to parse (Cannot store math image on filesystem.): 1.5 . This means that the mode of the PDF is 8 and the range where 95% of the values are found is between Failed to parse (Cannot store math image on filesystem.): 4 and Failed to parse (Cannot store math image on filesystem.): 12 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
$D$	Failed to parse (Cannot store math image on filesystem.): 2.1267	Failed to parse (Cannot store math image on filesystem.): 0.2174