Team:Oxford/biosensor characterisation
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+ | <h1>Stochastic Modelling</h1> | ||
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+ | Stochastic modelling uses probability theory to predict the behaviour of a system. For our project, we used it to model the expression of GFP from bacteria. | ||
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+ | We started with the Gillespie Algorithm, which considers the expression of GFP to be binary; a molecule of GFP is either produced or degraded. Before we determined which reaction happened, we had to work out when the reaction happened. Using the random number r1 (taken from a uniform distribution between 0 and 1), we produced another random number τ, which determined the time until the next reaction. | ||
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+ | <img src="https://static.igem.org/mediawiki/2014/8/89/Oxford_Matt_equations_1.jpg" style="float:left;position:relative; height:8%; width:20%;" /> | ||
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+ | <br><br><br><br><br><br> | ||
+ | Where α0 represents the probability that any reaction will happen, given by the following equation: | ||
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+ | <img src="https://static.igem.org/mediawiki/2014/3/37/Oxford_Matt_equations_2.jpg" style="float:left;position:relative; height:4%; width:47%;" /> | ||
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+ | We modelled the probability of a molecule of GFP being created using the Michaelis-Menten model (α1), incorporating a basal transcription rate (b1). For the degradation, we assumed a simple proportional relationship: the more GFP you have, the more likely it is that a molecule degrades (δ1). The constant of proportionality will be a function of the intrinsic life time of the protein in the cell. We considered there to be no DCM originally, then a large step in DCM at time=0. This is similar to placing the detector in a DCM polluted source, to make the model more realistic the level of DCM would go down as it is degraded but we had no time to obtain data for this rate. | ||
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+ | To decide if GFP was produced, we looked at the percentage of “reactions” which were productive, and then we compared this to a second random number r2 (again taken from a uniform distribution from 0 to 1). If the random number was lower, then GFP was created. If it was higher, then GFP was degraded. In this way we make a weighted random choice about whether GFP was created or degraded. We only stored the time and amount of GFP when there was a reaction, to save on computation time. | ||
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+ | <img src="https://static.igem.org/mediawiki/2014/e/e1/Oxford_Matt_equations_3.jpg" style="float:left;position:relative; height:8%; width:30%;" /> | ||
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+ | Stochastic modelling is useful because it can show us the stochastic effects which are often observed in individual bacteria. By calculating the variation of the mean of multiple GFP producing bacteria, we can also work out the standard deviation. Then, if we assume that the system varies with respect to the normal distribution, we can produce error bounds for the production of GFP, such that we can say that 90% of the time we can expect the production of GFP from a single bacterium to be within these two curves. This could be useful for seeing if results are unexpected, or, if there are multiple outliers, that our model is incorrect. If we average an increasing number bacteria, then the mean curve tends towards the deterministic response. This is to be expected, as we are now looking at the system as a whole and fluctuations in the production from individual bacteria are averaged out. In terms of their use, when looking at small amounts of bacterium the stochastic model would be better, because real random fluctuations can be seen. For larger bacterial populations, the deterministic response models the growth very well. The stochastic model can also model large groups but requires large number of realisations which causes simulations to take a lot longer to run. | ||
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+ | When running the models, we picked arbitrary constants to view the general response. If we had more time we would have attempted to work out the basal rate, transcription rate and degradation rate of the GFP from DCM. | ||
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Revision as of 21:31, 17 October 2014
![](https://static.igem.org/mediawiki/2014/4/4d/Oxford_plus-sign-clip-art.png)
Wetlab data showing response in level of mCherry expressed with different concs of ATC
By making a translational fusion of mCherry at the C terminus of the dcmR gene under the tet promoter and tet operator system (see our Construction page for details) we could measure mCherry fluorescence to gain information about dcmR induction by ATC. Expression was induced with various amounts of ATC and the following fluorescence data acquired. Exposure time was 0.2 seconds. As no calibration data was obtained using purified mCherry, the results have been left in fluorescence arbitrary units. Images were analysed using imageJ software.mCherry fluorescence increases with amount of ATC used confirming that the dcmR gene was expressed under the control of the tet promoter and operator system.
![](https://static.igem.org/mediawiki/parts/5/51/Oxford_DcmR-mCherry_expression_induced_by_0ng_ATC.png)
![](https://static.igem.org/mediawiki/parts/1/19/Oxford_DcmR-mCherry_expression_induced_by_10ng_ATC.png)
![](https://static.igem.org/mediawiki/parts/6/63/Oxford_DcmR-mCherry_expression_induced_by_50ng_ATC.png)
![](https://static.igem.org/mediawiki/parts/c/cb/Oxford_DcmR-mCherry_expression_induced_by_100ng_ATC.png)
![](https://static.igem.org/mediawiki/parts/3/37/Oxford_DcmR-mCherry_expression_induced_by_200ng_ATC.png)
![](https://static.igem.org/mediawiki/parts/e/e6/Oxford_Comparison_of_mean_flouresence_intensity_from_DcmR-mCherry_expression_induced_at_varying_amounts_of_ATC.png)
This data was then used to refine and test our models (see below).
![](https://static.igem.org/mediawiki/2014/4/4d/Oxford_plus-sign-clip-art.png)
![](https://static.igem.org/mediawiki/2014/4/4d/Oxford_Just_data.jpg)
Introduction
As you can see from the biochemistry bubble above, our team was only able to obtain fluorescence data for the first half of the genetic circuit (ATC-induced mCherry response). On top of this, the wet lab team were unable to obtain data that measured how the fluorescence of a single culture changed with time, again because of time constraints. This is slightly limiting because it means that we don’t have any dynamic data for any part of our system, and therefore can’t test the modelling predictions of the speed of the biosensor’s response.The original data is shown on the right with error bars showing the standard error of the measurements.
Standard error is calculated as the average standard deviation divided by the square root of the total number of readings.
How we used the model
However, to demonstrate the power of the computer models that we’ve built, we made our model simulate the same graph (mean fluorescence against ATC concentration added). To build this, we started from the graph shown in the first modelling bubble on this page, shown here with a small basal rate (see where did these equations come from? ). This graph shows how the predicted fluorescence of the cells changes with time in response to an addition of ATC halfway along the time scale. At this stage, all input values, model parameters and therefore results are arbitrary.![](https://static.igem.org/mediawiki/2014/4/46/Oxford_data2.png)
We then ran the model for the correct amount of time (2 hours 20mins incubation with ATC) and ran it for lots of different concentrations of ATC over the range that the wet-lab team tested. The parameters are still arbitrary at this point (the same as above) and the results of the graphs are therefore arbitrary are as well, but the input values are now correct. The graphic below shows how we used the existing model to obtain the same graphs as the wet-lab team had obtained.
The numerical inputs that were used to model this data set were therefore:
![](https://static.igem.org/mediawiki/2014/f/f6/Oxford_data3.png)
Oxford iGEM 2014
![](https://static.igem.org/mediawiki/2014/4/4d/Oxford_plus-sign-clip-art.png)
Parameters
As all results are arbitrary up to this point, it is now time to calculate the parameters that will make the model’s response match up with the wet-lab data. The purpose of doing this is that the model will be able to give relatively accurate predictions of the response of the bacteria to further testing, therefore making the development of the biosensor much more efficient. The amount of data here will not allow us to calculate the parameters to a high level of accuracy, but it should be able to give us some very good approximations of what we can expect.![](https://static.igem.org/mediawiki/2014/f/ff/Oxford_mCherry_circuit2.png)
These parameters are:
Remember that because the mCherry gene is tagged (translational fusion) onto the end of the dcmR gene, the mCherry fluorescence will be the same as the amount of DcmR protein present. However, there is not very comprehensive data in the literature about the values that we can expect from the behaviour of the dcmR gene and its stability in vivo.
Degradation constant
The initial steady state of the system (before ATC has been added) is determined by two constants in the model. These constants are the degradation constant of DcmR and the basal transcription rate of the system. Due to the lack of numerical information in the literature on the behaviour of the dcmR gene, the way of calculating these two parameters is by using the single basal rate data point from the wet-lab data (fluorescence value when 0ng of ATC has been added).If we assume that the half-life of the dcmR protein is 3 hours [1] (180 minutes), we can calculate the degradation constant for our model. The exponential protein decay is therefore described by:
![](https://static.igem.org/mediawiki/2014/0/0d/Oxford_data6.png)
Basal transcription rate
At the basal steady state, the rate of change of DcmR (and therefore fluorescence) is zero. As no ATC has been added to the system yet, the value of [ATC] is also zero.![](https://static.igem.org/mediawiki/2014/5/55/Oxford_data7.png)
![](https://static.igem.org/mediawiki/2014/e/e0/Oxford_data8.png)
![](https://static.igem.org/mediawiki/2014/f/f6/Oxford_data9.png)
![](https://static.igem.org/mediawiki/2014/0/0a/Oxford_data10.png)
Expression rate constant and Michaelis - Menten constant
After the ATC has been added to the system, the value of [ATC] becomes non-zero. This means that the expression constant and the Michaelis – Menten constant start to affect the system.To find the parameters that make the model’s output match the data values, we turned to the code that we had developed for parameter calibration.
How are we calculating the parameters?
This code gave the parameters as:
α1 = expression rate constant of dcmR = 16.5 (fluorescence/min)
k1 = Michaelis - Menten constant of dcmR = 0.015 (ml/ng)
![](https://static.igem.org/mediawiki/2014/4/4d/Oxford_plus-sign-clip-art.png)
Entering the correct parameters
When the parameters that had been calculated above were entered into the model:![](https://static.igem.org/mediawiki/2014/d/d2/Oxford_data11.png)
![](https://static.igem.org/mediawiki/2014/f/f8/Oxford_data12.png)
![](https://static.igem.org/mediawiki/2014/2/26/Final_mCherry_dots.jpg)
![](https://static.igem.org/mediawiki/2014/d/dc/Final_mCherry_line.jpg)
Sensitivity
An important part of building mathematical models is sensitivity analysis of the results. This can be basically explained as wiggling all of the input values and parameters to see how much variations in each of these values affects the system output. This is especially important for finding parameters to describe the system as it is important to know what level of accuracy the values need to be found to provide a reasonable degree of prediction accuracy.On top of this, it is possible to find what range of values the system is especially sensitive to. An example of this analysis is shown with a simple example that is relevant to our system below:
![](https://static.igem.org/mediawiki/2014/1/1a/Oxford_data13.png)
Stability
In our Engineering studies we have learnt detailed control theory. Control theory is an interdisciplinary branch of engineering and mathematics that deals with the behaviour of dynamical systems with inputs, and how their behavior is modified by feedback. The usual objective of control theory is to control a system so that its output follows a desired control signal, called the reference, which may be a fixed or changing value. This important because many dynamic systems can go unstable if they are given an unsafe set of input values and/or operating conditions.However, as there are no feedback loops in this synthetic circuit, control theory analysis of this system isn't necessary.
Future experiment ideas from an Engineering design perspective
To see details about how the wet lab team then used this model to guide their work see our optimisation page.Reference
[1] Dr George Wadhams by personal communication (14/10/2014)![](https://static.igem.org/mediawiki/2014/4/4d/Oxford_plus-sign-clip-art.png)
The first data readings that we managed to obtain from the completed genetic circuit were qualitative sfGFP fluorescence readings. The results of these are shown here:
The first system that we tested is shown here. This is the bottom level of our synthetic circuit. Testing just this plasmid allowed us to obtain some important information to allow us to characterise the genetic system.
The GFP fluorescence of this system was high and this is the basal transcription rate of the PdcmA promoter.
The second system that we tested was the whole synthetic system without any DCM added. This allowed us to analyse just the effect of DcmR on the PdcmA promoter when it was compared to the result from system 1.
The fluorescence was much lower than the basal transcription rate of the PdcmA. This strongly suggests that the DcmR protein acts as a repressor of the PdcmA.
The next step is to prove that adding DCM represses this repression and increases the sfGFP fluorescence of the system.
Therefore we know that DcmR represses PdcmA with DCM repressing this repression. This means that the system is a double repressor!
First system tested
![](https://static.igem.org/mediawiki/2014/c/c6/Oxford_bottom_line.png )
The GFP fluorescence of this system was high and this is the basal transcription rate of the PdcmA promoter.
Second system tested
![](https://static.igem.org/mediawiki/2014/c/c1/Oxford_system2.png)
The fluorescence was much lower than the basal transcription rate of the PdcmA. This strongly suggests that the DcmR protein acts as a repressor of the PdcmA.
The next step is to prove that adding DCM represses this repression and increases the sfGFP fluorescence of the system.
Third system tested
![](https://static.igem.org/mediawiki/2014/4/40/Oxford_Whole_circuit.png)