Cellular growth

Our simulations include hours of cell culture growth in some cases (simulated experimental time, not computational time). Thus, in addition to the dynamics of the regulatory network, we also need to take into account the effects of cell growth. The volume of a single cell is increasing from the beginning of the cell cycle until the moment it divides into two daughter cells and the molecules included in the initial cell are distributed between the two new ones. As a consequence, the cellular growth introduces a dilution in the concentration of the cellular species, which is represented in our deterministic model by adding a dilution term $- μ \cdot Χ$ in the Ordinary Differential Equations (ODEs) of all species, with the exception of the autoinducer in the environment $C_{e}$.

Parameters

The growth rate of the cells is represented by the parameter Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): μ . The parameter values were derived from published data on S. coelicolor growth rate under different environmental conditions. In order to avoid infinitely slow or fast culture growth and also in order to maintain consistency with the stochastic model, doubling times (τ) rather than growth rates will be employed in the simulations.

Name	Value	Units	Value in previous GBL models	Remarks-Reference
Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): τ	$64.8−297.06$	$min$	$6.7 \cdot 10^{-5} s^{-1} (μ=0.00402 min^{-1}, τ=172.4 min)$ $6 \cdot 10^{-5} s^{-1} (μ=0.0036 min^{-1}, τ=192.5 min)$ Range tested: $0-10^{-4} s^{-1}$ ($μ=0-0.006 min^{-1}, τ=0-115.5 min$) Bistability range: $0-10^{-4} s^{-1}$ ($μ=0-0.006 min^{-1}, τ=0-115.5 min$) and $6 \cdot 10^{-5}-8 \cdot 10^{-5} s^{-1}$ ($μ=0.0036-0.0048 min^{-1}, τ=144.4-192.5 min$)	In a study on comparison between turbimetric and gravimetric techniques for the measurements of specific growth rates of streptomycetes, Flowers et al. reported growth rate values in the range of $0.249-0.642 h^{-1} (0.00415-0.011 min^{-1})$. The reported doubling times were in the range of $1.08-2.79 h (64.8-167.4 min)$. Flowers et al. 1977 Additionally, Melzoch et al. studied the observed growth rates of S. coelicolor in variously limited chemostat cultures and reported a maximum growth rate of $0.14 h^{-1} (0.0023 min^{-1}, τ=297.06 min)$ Melzoch et al. 1997 These values are comparable with the findings of R.A. Cox, who reported genomic properties and macromolecular compositions of Streptomyces coelicolor A3(2). The largest growth rate was $0.3 h^{-1} (0.005 min^{-1},τ=138.6)$. He noted that the maximum specific growth rate, is attained when cells are amply supplied with the most favourable nutrients. Cox et al. 2004

Parameters with uncertainty

When deciding how to describe the uncertainty for this parameter we must take into consideration that some of the parameters were calculated or estimated based on experimental data on S. coelicolor cultures, so there might be some error included in the estimations. Additionally, since the simulations will investigate the switch which regulates antibiotic production in S. coelicolor, it would be preferable to be as close as possible to the growth rate which facilitates this process. These facts influence the quantification of the parameter uncertainty and therefore the shape of the corresponding distribution. By assigning the appropriate weights to the parameter values and using the method described here, the appropriate probability distribution was designed.

In order to keep the parameter values as strictly as possible within the range of the values reported in literature, the prediction interval in this case for the definition of the distribution is set to 95.45% rather than 68.27%. Thus, the weight of the distribution will be put to Failed to parse (Cannot store math image on filesystem.): 99.3 min which is also set as the mode of the log-normal distribution for the Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): μ and the Spread is set to Failed to parse (Cannot store math image on filesystem.): 1.74 , in order to test the system's behaviour under the full range of reported values. This means that the range where 95.45% of the parameter values are found is between Failed to parse (Cannot store math image on filesystem.): 57 and Failed to parse (Cannot store math image on filesystem.): 172.8 min .

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

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The values retrieved from literature and their weights are indicated by the blue dashed lines, and the uncertainty for each value is indicated using a default value of 10% error (orange lines).

The parameter information of the distribution is:

Parameter	Mode	Spread	μ	σ
Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): τ	Failed to parse (Cannot store math image on filesystem.): 99.3	$1.74$	Failed to parse (Cannot store math image on filesystem.): 4.6699	Failed to parse (Cannot store math image on filesystem.): 0.26762

Dynamics of bacterial colony growth

For our simulations we assume that cells are maintained in the exponential phase with doubling time Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): τ . In our virtual colony, each individual cell grows exponentially in time until its division into two daughter cells, as per

$V_{c,i}(t)=V_{0}\cdot 2^{t/τ_{i}}$,

where $V_{0}$ is the volume of a cell at the beginning of the cell cycle (same for all cells), $τ_{i}$ is the duration of the cell cycle of cell i, and t is the time to the precedent division event. When $t = τ_{i}$ the cell i has doubled its volume and a new division takes place. At this timepoint the internal clocks and volumes of the daughter cells are reset to zero and $V_{0}$ respectively. Moreover, when a cell divides, proteins, mRNAs and signalling molecules are binomially distributed between daughter cells and one copy of the DNA is given to each cell. The regulatory complexes bound to the DNA are detached prior to the distribution between daughter cells. The total volume of the culture $V_{tot}$ is constant and is composed by the volume of the total cells and the volume of the medium, as per the equation $V_{tot} = V_{ext} + \displaystyle\sum_{j=1}^{N}V_{c,j}(t)$.

With regards to the effect of the cell volume of individual cells on the diffusion rate of the autoinducer, the coefficient $r$ which contributes to the diffusion of the external autoinducer into the cells, is described by the equation

$\rho_{i}(t)= \frac{V_{c,i}(t)}{V_{tot}- \displaystyle\sum_{j=1}^{N}V_{c,j}(t)}= \frac{V_{c,i}(t)}{V_ext}$.

The duration of the cell cycle, $τ_{i}$, is different for each cell and is set independently after a division according to the stochastic rule

$τ_{i}=λ\cdot τ+ (1-λ)\cdot \tilde{τ}$,

where Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): τ and $\tilde{τ}$ correspond to the deterministic and stochastic parameters of the cell cycle duration respectively, and $λ \in [0, 1]$ is a parameter that weights their relative importance.The stochastic component accounts for the period of time between events described by a Poissonian process according to an exponential distribution,

$\phi(\tilde{τ})=\frac{e^{-\frac{\tilde{τ}}{τ}}}{τ}$.

In this way, variability is enabled from cell to cell with regards to the duration of the cell cycle, yet a minimum cell cycle duration $λ\cdot τ$ is set. Therefore, the average duration and standard deviation of the cell cycle are Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): τ and $(1 − λ) \cdot τ$ respectively.

The values of the cell growth parameters that are used in the model are summarized in the following table:

Name	Description	Value	Remarks-Reference
Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): τ	Cell cycle duration (doubling time)	$64.8−297.06 min$	The doubling time values are the ones reported in the section above.
Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): λ	relative weight between the det.-stoch. parameters of the cell cycle	$0.8$	Single cell level gene regulation studies in E. coli showed a difference of ~0.8 between single cell cycle duration and the mean cell cycle duration, due to periodic oscillations.
$V_{0}$	cell volume at the beginning of cell cycle	$1.2 μm^{3}$	The cell volume is derived by the publication of R.A. Cox (ref. $V_{(av)}$ in the figure in Parameters section) where he reported the average cell volume for S. coelicolor A3(2) grown at 30 oC under different growth rates, between $0.97-1.29 μm^3$
$V_{tot}$	total cell culture volume	$2 \cdot 10^{−4} μl$