Dynamics of bacterial colony growth

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For our simulations we assume that cells are maintained in the exponential phase with doubling time \tau. 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/ \tau_{i}},

where V_{0} is the volume of a cell at the beginning of the cell cycle (same for all cells), \tau_{i} is the duration of the cell cycle of cell i, and t is the time to the precedent division event. When t = \tau_{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 [1] 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).

Colony.png

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, \tau_{i}, is different for each cell and is set independently after a division according to the stochastic rule [2]

\tau_{i}=\lambda \cdot \tau+ (1-\lambda)\cdot \tilde{\tau},

where \tau and \tilde{\tau} correspond to the deterministic and stochastic parameters of the cell cycle duration respectively, and \lambda \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{\tau})=\frac{e^{-\frac{\tilde{\tau}}{\tau}}}{\tau}.

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 \lambda\cdot \tau is set. Therefore, the average duration and standard deviation of the cell cycle are \tau and (1 - \lambda) \cdot \tau respectively. [3]

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

Name Description Value Remarks-Reference
\tau Cell cycle duration (doubling time)  64.8-297.06 min The doubling time values are the ones reported in the section above.
\lambda relative weight between the det.-stoch.

parameters of the cell cycle

0.8 [1][4] 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 \mu m^{3} [5] 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 \mu m^3
V_{tot} total cell culture volume 2 \cdot 10^{−4} \mu l

References