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Team:Dundee/Modeling
From 2014.igem.org
Modeling
Maths.. maths is fun!
Modeling and Analysis of Signaling Pathways
Introduction
In order to help analyze, construct and optimise the biochemical pathways in the Lung Ranger, we used a variety of mathematical tools to create algorithms and simulations.This allowed us to accelerate the development and testing of various hypotheses.
Methodology
As shown in figure 1 each of the approaches; ordinary differential equations (ODEs), stochastic simulation algorithm (SSA) and NetLogo, provided a different understanding of each system.
PQS
Objectives
We wanted to find a method of increasing the expression of mCherry in our engineered PQS system.
Results
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 equations (full derivation can be found in the appendix) we established the following relationship between PQS (Se) and mCherry:
Equation (1) was then analysed in MAPLE 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 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) 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 with 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 Figure 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 a more understanding of how to increase mCherry production. Subsequently, after testing several model l 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.
Conclusions
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.(link to lasso page) - 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.
BDSF
Objectives
Results
BCAM0228 phosphorylated by unknown compound
When the Burkholderia system was tested by the wet team, they found that even in the absence of BDSF, GFP was still expressed and needed our help for answers.
Before we could hypothesise why the system wasn’t working we had to gain an understanding of what should be happening.
Our deterministic model (figure 8) shows that in the absence of signal, no GFP is produced - the result the wet team were hoping to see. Conversely, when BDSF is present GFP is produced, shown in figure 9.
From our system of equations we know that the expression of GFP is dependant on the levels of BCAM0228[P]. We hypothesised that one of the ways our cell could be expressing GFP in the absence of signal was by having extra BCAM0228[P]. This suggests that an unknown compound is phosphorylating BCAM0228 in our cell.
When figure 10B is compared to figure 9B the overall expression is higher because BCAM0228 is being constitutively phosphorylated. We also infer that the presence of BDSF has very little effect on the expression of GFP. Therefore we can conclude the BDSF signal has no quantifiable effect on the production of GFP.
Conclusions
DSF
Objectives
