Dundee 2014

Modelling PQS

Sigmoidal Signal Response Revealed


We wanted to find a method of increasing the expression of mCherry in our engineered PQS system.

Sigmoidal Expression of mCherry in PQS System

When our PQS system was induced with synthetic PQS, no mCherry was expressed. We attempted to use mathematical modelling to find out why, and how the situation could be resolved.

After constructing a series of ordinary differential equations1 we established the following relationship between PQS and mCherry:

The above equation was then analysed in MAPLE2 for varying PQS concentrations using the parameters in table 1.

Fig 1. shows how the concentration of mCherry increases over time. The general trend is that the rate of expression of mCherry is proportional to PQS signal strength. Most striking is the rapid switch from low expression to high expression as the signal is increased.

When the gradients of mCherry were plotted against PQS concentration (Fig 2), the result was a sigmoidal curve with sharp transition located between signal values of 0.1µM and 1µM. Below this, expression was effectively zero, and above it, expression was uniformly high.

These results suggest that the levels of synthetic PQS investigated in our engineered pathway were too low and that the ten fold increases used in the experiments would yield no measurable change in mCherry expression, provided the signal level was below the threshold predicted by the model. To investigate this further we went on to carry out stochastic simulations of the system.

Stochastic Simulations Elucidates the “Switch” Behaviour

The lab experiments resulted in no measurable mCherry expression in the PQS system. However, our ODE model predicts that even for very low levels of PQS some mCherry is expressed. These levels could represent below-measurable output, thus are essentially zero. We investigated this further at the “single cell” level.

We constructed a stochastic simulation algorithm (SSA)7,8 in order to visualise signal-response at the single cell level. Using the SSA we were able to simulate mCherry expression of 1000 cells to varying PQS signals over a period of one cell cycle.

At first sight, the stochastic mean in A appeared too low in comparison to the example realisations. However, further analysis revealed that this low mean is caused by many realisations in which the expression of mCherry was zero (i.e. the promoter never fired). As shown in Fig 4, in 77% of the realisations for 0.02μM PQS, no mCherry was produced. Increasing the PQS concentration through the previously predicted threshold increased the expression rate to 90%. For PQS concentrations greater than 2μM, all realisations produced mCherry in the given time frame.

What is interesting to note is that once expression started, mCherry was produced at essentially the same rate, as shown by the parallel trajectories in Fig 3. We can conclude that a major rate limiting step in the system is the probability of the initiation of mCherry expression.

These stochastic simulations enhanced the deterministic results. The switch behaviour between low and high levels mCherry expression resulted in predicted corresponding molecule numbers shown in Table 2.

Increasing Promoter Numbers Increases mCherry Expression

We concluded from the SSA that firing of the mCherry promoter was a major rate limiting step, but wanted to provide the wet team with greater understanding of how to increase mCherry production. Subsequently, after testing several model hypotheses, it was found that increasing the number of promoters had the greatest impact on the expression of mCherry (varying other experimentally controllable parameters were predicted to have essentially no effect on expression level).

When the number of promoters in the model was increased by 10-fold (30 to 300) there was a corresponding 10-fold increase in the “high” mCherry expression, as shown in Fig 5. We suggested that increasing the number of promoters would be the only practical method of significantly increasing mCherry expression. However it was subsequently discovered that the number of promoters in our chassis could not be increased significantly as high numbers proved toxic.


Our models revealed a distinct switch behaviour in the signal-response curve for mCherry expression. Using this analysis we put forward the hypothesis that the synthetic PQS concentrations used in our experiments were not sufficiently high and, although these experiments were conducted using a range of synthetic PQS levels, this range corresponded to fold changes in the low-response region shown in Fig 2.

The predicted switch behaviour has important and positive consequences for the L.A.S.S.O. - we predict that the mCherry output level would be distinctly bimodal, avoiding the necessity for sensitive calibration of the detecting device - low signals would result in approximately no response, whereas supercritical signals would induce maximal response from the sample.


3Zender, H. et al. (2013) Journal of Medicinal Chemistry, 56(17):6761–6774.
4Ilangovan, A. et al. (2013) PLoS Pathog, 9(7):e1003508.
5Leake, M. et al. (2008) Proceedings of the National Academy of Sciences USA 105, 15376-15381.
6Andrea J. Twigg & David Sherratt (1980) Nature 283, 216 - 218.