Figure 1.
Schematic showing single cell bistability and population bimodal distribution.
For cCF10 pheromone (signaling molecule) induced pCF10 conjugation system, concentration of PrgB protein indicates the level of conjugation. Our model indicates that (A) due to different plasmid copy number in culture of cells, bimodal population distribution can arise from cells without bistability and (B) due to interaction with each other, cells with bistability can abandon bimodal population distribution.
Figure 2.
Schematic drawing of gene regulation and population balance model.
(A) The gene reaction network of pCF10 based conjugation system. The prgQ-prgX gene pair regulates conjugation. While pheromone cCF10 is released by recipient cells in the extracellular environment, the inhibitor iCF10 is encoded from both QS and QL RNA, products of the prgQ gene. Both iCF10 and cCF10 compete for binding to PrgX protein which is assumed to exist at a constant concentration in this study. In the off state, iCF10-bound PrgX tetramers repress prgQ gene expression via formation of a DNA loop. Under these conditions, the nascent prgQ transcript Qpre interacts with the non-coding antisense RNA, Anti-Q, to give rise to shorter QS RNA. In the on state PrgX-cCF10 dimers relieve prgQ repression to give rise to increased production of Qpre, which tends to titrate Anti-Q, allowing production of the longer QL RNA and consequently PrgB protein. The concentration of PrgB protein indicates the level of conjugation. (B) the DNA configuration of on state. (C) the DNA configuration of off state. (D) Schematic depicting the Population Balance Model (PBM) with stochastic gene regulation. Grey color indicates a cell in off state and green indicates on state. Properties of the PBM include I) Uneven distribution of plasmids to daughter cells. II) Cells with different plasmid copy number or different states act differently and influence each others. III) A cell acts random according to stochastic intracellular gene regulation.
Table 1.
Nomenclature of pCF10 system.
Figure 3.
(A) System demonstrates bistability using the parameter values as shown in Table 2. (B) The possibility of bistability can be analytically excluded by setting the parameter value of K4,2 = K4,1 independent of other parameters. For both (A) and (B), plasmid copy number k is equal to 5.
Figure 4.
Bimodal distribution results from no bistability.
The parameters used in this simulation are shown in Table 2 except for K4,2 = K4,1 = 0.001 (1/s). (A) The stationary distribution responding to different concentrations of extracellular pheromone, cCF10. As the concentration of pheromone increases, cell population migrates from state of low PrgB, viewed as off state, to state of high PrgB, viewed as on state. When pheromone concentration reaches 30 nM, all cells stay at on state. (B) The change of PrgB protein distribution for cCF10 concentration from 6 nM to 9 nM. (C) The dynamic behavior responding to constant concentration of pheromone, cCF10. This simulation is done with extracellular pheromone concentration maintained at 10 nM.
Figure 5.
Population distribution for a system with bistability.
Both (A) and (B) use the same parameters shown in Table 2. Extracellular pheromone concentration is maintained at 13.5 nM for both simulations. Only noise term of protein and peptides are taken into consideration [48], [49]. Thirty thousand cells are used. (A) Outcome from single cell stochastic model described in Eq.(30). Due to stochasticity, bimodal distribution comes from initial unimodal distribution. Single cell stochastic model ignores the interaction between cells. Extracellular inhibitor is used only by the donor cell secreting it. (B) Outcome from PBM described in Eq.(29). The extracellular inhibitor is utilized by the whole population. The simulation outcome shows that the population effect leads to unimodal distribution.
Figure 6.
Population distribution, started with bimodal distribution, merging into a unimodal distribution due to population effect.
Bimodal distribution generated from single cell stochastic model with extracellular pheromone concentration equal to 13.5 nM served as initial distribution of PBM. Simulation was done with plasmid copy number equal to 5 and parameter values listed on Table 2. Population effect of extracellular inhibitor causes two modes to merge into one.
Table 2.
Values of Parameters of pCF10 system.
Figure 7.
Reaction network for gene regulation in toy example.
After uptake by membrane protein, the precursor of signal molecule matures into signal molecule. When signal molecule binds to DNA, it favors the expression of xp gene encoding product and represses the transcription of xi gene which encodes the inhibitor. The binding reaction of inhibitor to signal molecule is fast and irreversible so either signal molecule or inhibitor dominates.
Figure 8.
Comparison of the outcome from population balance model and single cell stochastic model in toy example.
Thirty thousand cells are used with constant concentration, 10 arbitrary unit/volume, of add-in signal molecular precursor. (A) Bimodal distribution is observed from single cell stochastic model. (B) Unimodal distribution is observed from population balance model.
Figure 9.
Bimodal distribution from no bistability.
Thirty five thousand cells are used for simulation. (A) Although there is no true steady state, the effect from the increment of on standard deviation of the product protein distribution is small enough to consider the system as pseudo steady state. (B) The plot of product protein distribution with different add-in signal molecular precursor.
Table 3.
Parameters of toy example.