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 $\mu$. In our model, the number of cells is described by a six-parameter Baranyi–Roberts model. In order to confirm that the estimated doubling time is realistic, we compared it to values derived from published data on S. coelicolor growth rate under different environmental conditions.

Name	Value	Units	Value in previous GBL models	Remarks-Reference
$\tau$	$64.8 - 297.06$	$min$	$6.7 \cdot 10^{-5} s^{-1} (\mu=0.00402 min^{-1}, \tau=172.4 min)$ $6 \cdot 10^{-5} s^{-1} (\mu=0.0036 min^{-1}, \tau=192.5 min)$ Range tested: $0-10^{-4} s^{-1}$ ($\mu=0-0.006 min^{-1}, \tau=0-115.5 min$) Bistability range: $0-10^{-4} s^{-1}$ ($\mu=0-0.006 min^{-1}, \tau=0-115.5 min$) and $6 \cdot 10^{-5}-8 \cdot 10^{-5} s^{-1}$ ($\mu=0.0036-0.0048 min^{-1}, \tau=144.4-192.5 min$)	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 $0.14 h^{-1} (0.0023 min^{-1}, \tau=297.06 min)$ 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 $0.3 h^{-1} (0.005 min^{-1},\tau=138.6)$. 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

In our model, the number of cells is described by a six-parameter Baranyi–Roberts model which takes into account the lag phase by using an adjustment function. In this model, the natural logarithm of the cell concentration is:

$y(t)=y_0+ \mu_{max}A(t)-\frac{1}{m}ln(1+\frac{e^{m \cdot \mu_{max} \cdot A(t)}-1}{e^{m(y_{max}-y_0)}})$

where $A(t)=t- \lambda +\frac{ln(1-e^{-v \cdot t}+e^{-v \cdot (t- \lambda)})}{v}$, $y_0$ is the logarithm of the initial number of cells (inoculum), $\mu_{max}$ is the maximal growth rate, $m$ is the curvature parameter ($m$=1 for logistic growth), $\lambda$ is the lag phase and $v$ is the rate limiting growth (usually assumed to be equal to $\mu_{max}$). The specific growth rate ($\mu$) is the first derivative of this equation and describes the change in the growth rate according to the cell population status. The parameters for the cellular growth were derived from the experimental data for Streptomyces growth reported by Nieselt et al. by performing a nonlinear least squares curve-fitting. By fitting the six-parameter Baranyi–Roberts equation for bacterial growth to the number of cells (approximately calculated from the reported biomass values), the carrying capacity, the initial cell number and the maximum growth rate were estimated. As there were only 11 available experimental data points, a number of different parameter sets that all fitted the Baranyi–Roberts equation were identified, thus defining a confidence interval around the fitted data.

The prediction of the carrying capacity by the logistic equation was $7.14 \cdot 10^{12} - 9\cdot 10^{12}$ cells, which is very close to the $8.24 \cdot 10^{12}$ cells calculated from the final biomass. The parameter values for cellular growth are summarised in the table below.

Parameter	Description	Range of values	Units
$K$	Carrying capacity	$7.14 \cdot 10^{12} - 9\cdot 10^{12}$	cells
$N_0$	Initial number of cells	$5 \cdot 10^{10} - 2\cdot 10^{11}$	cells
$\mu_{max}$	Maximum growth rate	$0.003368 - 0.007499$	$min^{-1}$
$v$	Curvature of the growth curve during the transition from lag to the exponential phase	$0.001482 - 0.5814$	$min^{-1}$
$m$	Curvature of the growth curve during the transition from exponential to the stationary phase	$0.47 - 2.46$	n.a.
$\lambda$	Lag phase	$330.52 - 883$	$min$