Difference between revisions of "Team:Goettingen/Model"

B. subtilis as a glyphosate detector

As described in the Background section, glyphosate specifically inhibits enolpyruvyl-shikimate-3-phosphate (EPSP) synthase, an enzyme present in plants, fungi, bacteria and archaea. The EPSP synthase converts shikimate-3-phosphate (S3P) and phosphoenolpyruvate (PEP) to EPSP, an intermediate in the shikimate pathway and required for synthesis of aromatic amino acids. Interaction of glyphosate with EPSP leads to depletion of cellular levels of aromatic amino acids resulting in growth inhibition. We have observed that growth of B. subtilis is strongly inhibited by glyphosate (50% reduction of the growth rate at 1 mM glyphosate, see Results section). To generate a glyphosate detection system, we wanted to make use of the glyphosate sensitivity of B. subtilis. However, the wild type allows to detect glyphosate only in the range between 0 mM and 2 mM. Fortunately, we have discovered the first glyphosate transporters of a living organism (see Results section). So we could make use of the B. subtilis double mutant strain (BP235, trpC2 gltT::spc gltP::cat) lacking both glyphosate transporters. The growth rate of this strain is reduced by 50% at a glyphosate concentration of about 8 mM! Therefore, the transporter mutant allows us develop a system with an increased detection range between 8 mM and 30 mM. To employ B. subtilis as a detector for glyphosate, we had to quantify the glyphosate-dependent reduction of the growth rate of the different B. subtilis strains. The growth rate seems to be ideal for this application because it is easy to reproduce. The relation between the growth rate and the glyphosate concentration can be obtained by modeling our growth curves with the bacterial growth model.

The bacterial growth model

Under lab conditions, the right environmental parameters and nutrients provided, bacterial growth can be observed in form of growth curves, plotting the bacterial mass as a function of time. A convenient way to observe those growth curves is provided by measuring the optical density also referred to as the turbidity.
During the growth of bacteria three distinct phases can be differentiated (Figure 1), all of which can be associated with physiological changes in the cell cultures:

1. During lag phase the bacteria are suggested to adapt to physiological changes in the medium. Adaption can be achieved by e.g. changes in the proteome of the bacteria in order to meet the needs for the new culture requirements. Bacterial growth is strongly reduced
2. After adaption bacterial growth enters the log phase alternatively named exponential phase. During this phase the bacteria show exponential growth, the most rapid growth possible under given conditions. The actual double-timing of a certain strain is determined by the growth rate, the increase of cell mass per time.
3. In the stationary phase no net growth can be observed. Nutrients and carbon sources are used up and increase in cell mass is no longer possible. Dead cells can lyse and provide another source of nutrients, explaining a small increase in cell mass that sometimes still occurs.

To be scientifically correct a fourth phase can be observed in the bacterial growth curve namely the death phase. As we measured our bacterial mass in terms of optical density this phase couldn't be observed in our experiments and has no relevance for our modeling approaches.

Correlation between growth rate and glyphosate concentration

Obtaining the growth rate μ

During their growth bacteria reach a phase of exponential growth as described above. In this phase every bacteria in the culture divides at the fastest rate possible, therefore the growth of the colony is directly dependend on the amount of cells already grown. This is also true for the optical density as it's proportional to cell mass. To put this growth into a context that can be calculated, a constant of proportionality is needed. This constant can be described as the growth rate of the colony, it is specific to strains and also dependend on environmental influences like glyphosate. These dependencies can be described by the following formula:

This short differential equation describes the growth behaviour of the bacteria during the log phase. The equation can be solved by first separating the derivates:

Next the equation has to be integrated at both sides yielding to the following form:

Inserting the initial condition t=0 and OD_t=0 results in:

From this solution a very easy and convenient way to obtain the growth rate for every growth curve arrives. As it is probably known by everyone working in the field of microbiology plotting the logarithm of cell mass (or OD) against time leads to a linear section in the affiliated graph. This linear section is exactly what is described by the last formula, the logarithm of the exponential phase of bacterial growth with the growth rate as its slope. To reproducibly calculate the slope/growth rate we wrote a python script, that enables us to select the linear section of the graph by hand. After that a linear fit is calculated and the resulting growth rate is safed. In Figure 2 usage of the program is shown, for further experiments we used two mutant strains expressing fluorescent proteins, namely iGEM23 (trpC2 amyE::(GFP-lacZ aphA3)), resembling the wild type strain and iGEM36 (trpC2 amyE::(mOrange-lacZ aphA3) gltT::spc gltP::cat), resembling the transporter double mutant. Why the fluorescent labeling is important for our experiments is laid out in detail later in this article.

Describing the dependency of growth rate from glyphosate concentration

After assessing the growth rates by doing fits of the growth curves, we plotted them against the glyphosate concentration in the environment of every growth experiment respectively. The graphs can be observed in Figure 3 and 4:

As stated in the Results section of our project the growth rate is strongly dependend on glyphosate concentration. While low concentrations of glyphosate (relative to the concentration of 50% growth reduction for every strain) do not strongly affect the growth behavior , increasing the concentration yields to very strong reduction. Moreover, higher doses of glyphosate do not change the growth rate very strongly anymore. The graphs strongly resemble the well-known Hill equation (see below) suggesting cooperativity in either the transport of glyphosate or interaction with the EPSP synthase.

We fitted the Hill equation to the data using a python script and the leastsq method of the scipy library, the fitted data are shown in Figure 5 and 6.

The hill equation can be transposed to show growth rate as dependency for glyphosate concentration:

With this transposition it is possible to determine the glyphosate concentration by evaluating the growth behaviour of the strains. For every strain these modelling parameters are different, for iGEM23 and the iGEM36 we already determined the fitting parameters (the values for the parameters can be found in the two diagrams above), enabling us to detect glyphosate from 0.5 up to 50 mM.

HIER FEHLT NOCH EIN EINSETZEN DER WACHSTUMSGESCHWINDIGKEIT IN DIE GLEICHUNG UM DAMIT EINEN GLYPHOSAT-WERT AUSZURECHNEN!

But there is one catch, that made us try to further extend our consideration. We can't be exactly sure, that there are no other environmental conditions that influence our growth experiments except for glyphosate. At last correlation must not mean causality. To ensure that glyphosate is the dominating factor in our experiments, we set out a competition assay between the iGEM23 and iGEM36, gradually increasing the concentration of the herbicide in the media.

Modeling the competition for different glyphosate concentrations

Modelling the optical density

During a competition assay both strains should be influenced by glyphosate in the same way they would while growing isolated from each other. The growth rate and it's reduction by glyphosate can be assumed to follow the hill equations we already determined for both strains. The difference to a normal growing experiment can be found in the competition between both strains for nutrients. Like in an isolated growing experiment, the growth curve can be plotted as logistic growth, but the limiting threshhold for the growth is not only influenced by the cell mass of one single strain but by both strains together. On top of this theory two dependencies can be written out in the following form:

This set of differential equations can't be solved elegantly in an analytical way, therefore we used numerical integration to model the optical density (representing the cell mass) of both of our strains in competition. This is quite conveniently done by employing the NDSolve method of the computing system mathematica.

Results of modelling

Now we had all the necessary information to model the cell mass of our both strains in competition for different glyphosate concentrations over the course of several hours. The outcome of this simulation is best depicted as ratio of the optical densitiy of iGEM23 and iGEM36 representing the ratio of cell mass between the two (Figure 7):

At low concentrations of glyphosate, the ratio BP235 and iGEM23 is near 1, both strains grow at the same speed and have similar cell mass. But increasing the dose of glyphosate by only as little as 2 mM, the ratio changes in favor of the mutant strain. As it can be seen in the hill plot (Figure 9), iGEM36 shows a much lower reduction in growth rate, therefore it should grow much faster than the wild type shifting the ratio in cell mass.

As laid out before, we are only able to measure the OD of both strains together. To evaluate the quality of our modelling we could therefore only compare the sum of the OD calculated with our modelled to the experimental data. The modelled graph and the experimental data are depicted in the following 3D graphs (Figure 9 and 10):

Comparing Figure 9 and 10 it can be observed, that the model is predicting the sum of ODs right in most areas. While the experimental data are of course a little rougher both model and experimental data show the same sigmoidal increase of OD. With the model the reduced growth of OD for high glyphosate concentrations is predicted right. While the experimental data show a decrease in maximum OD for high glyphosate concentrations, the model doesn't express a dependency of maximum OD and glyphosate concentration. This is expected by the nature of the model, as it doesn't take into account the death of cells, that also could be influenced by glyphosate concentration. Maximum OD in the model is 0.8, while in the competition assay only a maximum OD of around 0.5 is reached, this can be easily adapted by changing the parameter S in the modelled equations (see above).

Overall our modeling showed, that based on our assumptions of glyphosate as the dominating factor in our experiments, the ratio between the iGEM23 and iGEM36 should pretty drastically change in favor of the double mutant by increasing the glyphosate concentration in the media.

Evaluating growth behaviour of the strains by means of fluorescence

As the OD of both strains can't be measured separated from each other, no comparison between the strains by means of OD was possible. Therfore we created fluorescent mutant strains expressing the fluorophores green fluorescent protein (GFP, iGEM23 - wild type) and mOrange (iGEM36 - double mutant) to monitor the fluorescence of both strains over the course of the experiment. We calculated the increase in fluorescence relative to the fluorescence measured at t=0 for different glyphosate concentrations. Doing this we were able to evaluate if our proposed trend of the mutant strain being the stronger competitor at higher glyphosate concentrations is right. Although it has to be mentioned that this test is more of qualitative nature, as modelling the fluorescence signals of the strains needs further research:
To properly model the fluorescence trends of the strains a link between optical density and fluorescence has to be found. To achieve this expression, maturation and degradation rates of the proteins are needed as the model of Leveau and Lindow suggests (1).
The calculated relative fluorescence and a ratio between the strands for 0 mM and 2 mM glyphosate is shown in the following Figure (Figure 11):

Without the influence of glyphosate the fluorescence of both strains increased nearly equally (Figure 11, upper left and right), at the 10 h mark a ratio of 0.8 between the strains can be found.
Adding 2 mM glyphosate to the system strongly shifts the ratio of fluorescence. The iGEM36 is strongly expressing fluorescent proteins, nearly doubling the amount of relative fluorescence increase compared to the competition experiment without glyphosate (Figure 11, upper right). In contrast iGEM23 shows no significant increase in fluorescence over the course of the experiment, at maximum a 2-fold increase can be observed (Figure 11, upper left). The ratio of fluorescence decreases to less than 0.1 after ten hours of competition, suggesting a much stronger growing double mutant.

Even though these results can't be interpreted quantitatively they still show, that our assumptions might be right. While without glyphosate both strains show a rather equal increase in fluorescence intensity, adding glyphosate shifts the ratio strongly in favour of the double mutant strain. As higher fluorescence intensity correlates with a stronger expression of fluorescent proteins, it can be assumed, that the double mutant clearly dominates in growth rate in comparison to the wild type, as our modelled already suggested.

Evaluating cell growth with a β-galactosidase assay

Another method for distinction of two strains in one culture was an enzymatic solution. The sensitive strain expresses constitutively a β-galactosidase which leads to hydrolyzation of a chromogenic substrate. This assay is performed with the substrate o-nitrophenyll-β-D-galactopyranoside (ONPG). The hydrolyzation of ONPG leads to the formation of o-nitrophenol, which absorbs light at λ=420 nm. The amount of o-nitrophenol that is formed within a certain time is directly proportional to the activity of a promoter. Since the promoter is active in any cell from the sensitive strain, it is proportional to the growth. Measurements of the enzyme activity can be seen in Figure 12:

High concentrations of glyphosate lead to a strongly reduced enzyme activity, suggesting slower or stagnated growth of the wild type strain. These results complement what our model of the cell mass ratio suggested: For glyphosate concentrations higher than 2 mM the wild type strain is strongly inhibited in growth, reassuring that glyphosate is the dominant factor in our competition and single experiments.

This assay can also be performed with 5-bromo-4-chloro-3-indolyl-β-D-galactopyranoside (X-gal) as chromogenic substrate. The hydrolyzation leads to the formation of an insoluble indigo-dye that is located within the cells. With increasing glyphosate concentration, the intensity of the blue color decreases and is therefore an indicator for the glyphosate concentration. This method is further prove that the sensitive wild type strain is strongly reduced in growth and therefore eliminated by the glyphosate-resistant double mutant.

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References

1. Leveau & Lindow (2001) J. Bacteriol. 183: 6752-6762.