Team:ETH Zurich/motilityModel

Our first biosensor approach is based on bacterial chemotaxis. During this process, bacteria sense a change in ligand concentration which induces a change in their swimming period. However, this change is only temporary, as an adaptation process within the chemotactic pathway restores the original movement of the bacteria. The exact timescale of the adaptation depends on the ligand concentration change in the surrounding of the bacteria. Therefore, the adaptation time represents an indicator of concentration changes in the environment. In order to gain more insight in the timescales of the adaptation process, we constructed a model that allows us to analyze these timescales in detail. Using experimental data, several parameters in our model were re-estimated to achieve a model that reflects the behaviour of our bacteria in the sensor. This showed us the range of concentrations that we are able to sense within this system and allowed us to model the behavior of AROMA in silico.

Model

Over the years, a lot of chemotactic models have been proposed. For our project, we decided to explore the Monod-Wyman-Changeux (MWC) model as this model has been shown to be in good agreement with experimental data [1]. Moreover, this model is relative simple while still representing all the key signalling pathway processes, as can be seen in figure 1.

Schematic overview of the chemotactic process in e.coli (for detailed biology information visit the Approach A webpage). When CheA is active, it autophosphorylates itself. Afterwards it can transfer it's phosphate group to the response regulators CheB and CheY. CheY-P binds to the motor of the flagellum to change the swimming pattern of the cell. CheZ can dephosphorylate CheYp, returning CheY to its initial value. Upon ligand binding CheA becomes less active, leading to a temporary decrease in CheYp. CheR and CheB are important for adaptation. Where CheR enhances the receptors signaling activity by methylation, CheB performs the opposite reaction. representation

Reactions

The reactions below represent the chemotactic model. As chemotaxis happens on a post-translational level, we can assume that we don't have to take any protein production and degradation into account.

$CheA \rightarrow CheAp$
$CheAp + CheY \rightarrow CheA + CheYp$
$CheAp + CheB \rightarrow CheA + CheBp$
$CheYp + CheZ \rightarrow CheY + CheZ$
$CheYp \rightarrow CheY$
$CheBp \rightarrow CheB$
Receptor$_{methylation}$ $_{level}$ + CheR $\rightarrow$ Receptor$_{increased}$ $_{methylation}$ $_{level}$ + CheR
Receptor$_{methylation}$ $_{level}$ + CheBp $\rightarrow$ Receptor$_{decreased}$ $_{methylation}$ $_{level}$ + CheBp

Equations

An important feature of the MWC model is that receptors form a cluster (with size N) in which the receptors are either all simultaneously active or inactive. Importantly, these clusters do not interact with each other. When assuming also no spatial heterogeneity, we can use the reactions above to obtain the following set of non-linear ODE's. [2]

\begin{equation} \frac{dCheA_p}{dt} = A*k_a*([CheATot] - [CheA_p]) - k_y*[CheA_p]*([CheYTot] - [CheY_p]) - k_b*[CheA_p]*([CheBTot] - [CheB_p]) \end{equation}

\begin{equation} \frac{dCheY_p}{dt} = k_y*[CheA_p]*([CheYTot] - [CheY_p]) - k_1*[CheZTot] - [CheY_p CheZ]*[CheY_p] + k_2* [CheY_p CheZ] \end{equation}

\begin{equation} \frac{dCheY_p CheZ}{dt} = k_1*[CheZTot] - [CheY_p CheZ]*[CheY_p] - (k_2+k_3) * [CheY_p CheZ] \end{equation}

\begin{equation} \frac{dCheB_p}{dt} = k_{-b}*[CheA_p]*([]CheBTot] - [CheB_p) - k_b*[CheB_p] \end{equation}

Moreover, to describe the receptor methylation kinetics, we use:

\begin{equation} \frac{dm}{dt} = g_R*[R_T]*(1-\Phi) - g_B*[CheB_P]^2*\Phi \end{equation}

with the activity of the receptor complex describes as:

\begin{equation} \ \Phi = \frac{1}{1+e^F} \end{equation}

Hereby, F stands for the 'free energy of the receptor cluster' and is defined as

\begin{equation} \ F = N * (1 - \frac{m}{2} + ln \frac{1+\frac{L}{koff}}{1+\frac{L}{kon}}) \end{equation}

Importantly, in this model we only include the Tar receptor, but neglect the contribution of the Tsr receptor which can also respond to aspartate in the environment.

Simplification

Comparing the time-scales of the different steps, we noticed that the CheYp-CheZ complex is formed only fast and transient [3], therefore we will replace equation (2) and (3) for the following ODE:

\begin{equation} \frac{dCheY_p}{dt} = k_y*[CheA_p]*([CheYTot] - [CheY_p]) - k_1* [CheY_p]*[CheZTot] - k_2* [CheY_p] \end{equation}

Parameter estimation and Results

The parameters used within this model can be found here (parameter page). Moreover, when using only parameters from literature, we noticed that the adaptation times deviated from our own experimental observations. We therefore refitted several parameters of the model to make it reflect our own observed adaptation times. We decided to only re-estimate parameters which were already fitted to other experimental data in the literature, while preserving biochemical obtained results. As we do not model flagellum movement directly, we model the changes in CheYp concentration to obtain the adaptation time.

We decided to split our experimental data into two sets. One set was used to do the estimation (table 1) and the another set was used to validate the model (table 2). You can read how we obtained and processed the experimental data here.

Parameter optimization

For the parameter fitting we used the following experimental data:

Experimental data obtained by reading out the adaptation time of bacteria that were exposed to different concentrations of aspartate. All values are based on a initial concentration value of 0 uM aspartate.

From zero to [uM]	Mean [s]	STD [s]	# of replicates
0.5	30	13	6
1	75	18.7	5
5	108	31.7	5
10	110	36.1	3
25	106.7	45.4	3
100	825	35.4	2

After fitting our model with these data points, we obtain the following response curves:

Graph displaying the change in CheYp concentrations in repsonse to aspartate addition and removal. The time it takes to restore the concentration of CheYp after ligand addition or removal can be read out as the adaptation time of the chemotactic process. The higher the concentration of the added aspartate, the longer it takes to restore the initial CheYp values.

Comparing figure (2) with the experimental data in table (1), we find that the lower range of aspartate induction is closely reflected by our model. Higher aspartate concentrations do not match our model. We would exclude them as outliers, but as we have no time to confirm these values experimentally we decided to include them for correctness.

Model Validation

We then obtained another set of data, shown in table 2. Here, we measured different starting concentrations of aspartate instead of starting from a concentration of 0 uM, thereby representing the situation of AROMA in the field. The adaptation times predicted by our model are displayed in figure (3).

Experimental data obtained by reading out the adaptation time of bacteria that were exposed to different concentrations of aspartate.

From-To [uM]	Mean [s]	STD [s]	Number of replicates
0,5-1	30	14.1	2
1-2,5	95.8	21.1	6
2,5 - 5	131.9	39.09	8
5-50	490	-	1
50-100	134.4	128	9
100 -150	150	-	1

Chemotactic adaptation times after adding a serie of different ligand concentrations. Number 1 to 5 represent the addition of 0.5, 1, 2.5, 5, 50 and 100 uM aspartate

Figure (3) matches correctly with the experimental data in table one. To demonstrate this even better, we will look deeper into the change of aspartate concentration from 1 to 2.5 uM.

Chemotactic adaptation process read-out from our own experiments.

Chemotactic adaptation process predicted by our model.

Measurement

Figure (4.1) displays the experimental data, which has an adaptation time of 70 seconds, and figure (4.2) shows the outcome of our model containing an adaptation time of 65 seconds. Although the y-axis of both figures are different, both curves demonstrate the adaptation time. This is due to the direct influence of CheYp on the rotational angle, as mentioned before.

Conclusion

We developed a model of the chemotaxis response of E.coli. Hereby we re-estimated parameters of the model, leading to predictions of our model that match the validation data. This showed us the range of concentrations that we are able to sense within this system. Moreover, with this model we were able to generate synthetic measurements that were used for testing and improving the path planning algorithm of our robot.

References

[1] Hansen, Clinton H., Robert G. Endres, and Ned S. Wingreen. "Chemotaxis in Escherichia coli: a molecular model for robust precise adaptation." PLoS computational biology 4.1 (2008): e1.
[2] Clausznitzer, Diana, et al. "Chemotactic response and adaptation dynamics in Escherichia coli." PLoS computational biology 6.5 (2010): e1000784.
[3] Edgington, Matthew P., and Marcus J. Tindall. "Mathematical Analysis of the Escherichia coli Chemotaxis Signalling Pathway." Bulletin of mathematical biology 80.4 (2018): 758-787.

[1] [1] Hansen, Clinton H., Robert G. Endres, and Ned S. Wingreen. "Chemotaxis in Escherichia coli: a molecular model for robust precise adaptation." PLoS computational biology 4.1 (2008): e1.

[2] [2] Clausznitzer, Diana, et al. "Chemotactic response and adaptation dynamics in Escherichia coli." PLoS computational biology 6.5 (2010): e1000784.

[3] [3] Edgington, Matthew P., and Marcus J. Tindall. "Mathematical Analysis of the Escherichia coli Chemotaxis Signalling Pathway." Bulletin of mathematical biology 80.4 (2018): 758-787.