# Team:Utrecht/Model

### Background & Motivation

The chemotaxis pathway, which directs the movement of a bacterium towards attractants and away from repellents, is a complicated system involving receptors. This can be challenging, but it also gives us possibilities when it comes to biosensors. An important characteristic of methyl-accepting chemotaxis receptors is that the sensitivity is regulated by the methylation and demethylation of several methylation sites. This regulation mechanism facilitates the spatio-temporal adaptation necessary for chemotaxis by adjusting the sensitivity of the receptor over time. This inspired us to engineer these methylation sites to create several variants of the receptors with different sensitivities, allowing quantification over a broader range. Determining the level of contamination is considered very innovative and useful according to the experts we consulted.

The Tar-receptor has 4 methylation sites (Q295, E302, Q309, E491), all adhering to the consensus sequence Glu-Glu-X-X-Ala-Ser/Thr (Perez et al., 2004). The more astute readers will notice that the 1nd and 3th sites are not glutamates. However, the cell post-translationally deamidates these positions, resulting in all glutamates. The methylation of these positions can be mimicked through site-directed-mutagenesis (E/Q -> A) fixating the methylation. In future we will denote all artificial methylation variants with a four-letter code: EEEE through AAAA.

To explore the feasibility of this idea a simple ODE-model was constructed, illustrating how the methylation of the different sites relates to the activity of the chemotaxis receptor and consequently the signal we can measure. Using this model, we engineered methylation sites and created receptor mutants that varied several orders of magnitude in sensitivity. It also offered us an indication of which concentrations to use for our assays to be able to measure clear signals. An important prerequisite of this model is that we need to distinguish between the methylation sites, in contrast to most models where the regulation of chemotaxis is assumed to have an equal effect for the sake of simplicity (Deshpande et al., 2016; Miller et al., 2010; Morton-Firth et al., 1999). Krembel et al. showed that it matters greatly which positions are methylated, as the effect of the artificial methylation of different position proved significantly different.

### Parameters

Table 1: Parameters
Internal Concentrations
RT 17.8 µMTotal Tar receptor (Bitbol and Wingreen, 2015)
AT 2.97 µM Total CheA (Bitbol and Wingreen, 2015)
YT 9.73 µM Total CheY (Bitbol and Wingreen, 2015)
ZT 3.80 µM Total CheZ (Bitbol and Wingreen, 2015)
Michaelis-Menten
KcatA 400 Turnover CheA (Stewart, 1997)
KmA 6.5 Michaelis constant CheA (Stewart, 1997)
HillA 1 Hill coefficient CheA (Stewart, 1997)
KcatZ 4.9 Turnover CheZ (Silversmith et al., 2008)
KmZ 0.8 Michaelis constant CheZ (Silversmith et al., 2008)
HillZ 2 Hill coefficient CheZ (Silversmith et al., 2008)

An important part of every ODE-model is the parametrization. Fortunately, the chemotaxis pathway is relatively well studied, allowing us to retrieve the relevant parameters from literature. For our purposes we focused on the Tar-receptor as a model receptor because the efficacy of the hybrid receptors had yet to be determined and proved difficult to predict. It will, of course, be very interesting to model the custom receptors once more data is obtained.

### Model Description

Ligand sensitivity of the Tar receptor will be measured using our BRET-pair, which is linked to the level of phosphorylated CheY (CheY-p). Using michaelis-menten kinetics and literature (Silversmith et al., 2008, Stewart, 1997) we can form the differential equation expressing the change in CheY-p which consists of two parts: The forward flux (CheY -> CheY-P) catalysed by CheA (1) and the backward flux (CheY-P -> CheY) catalysed by CheZ (2).

$$V1= \frac{V_{maxA}*Y}{(K_{mA}+Y)}=\frac{V_{maxA}*(YT-Yp)}{K_{mA}+(YT-Yp)}$$ $$V2 =\frac{V_{maxZ}*Yp^2}{K_{mZ}^2+Yp^2}$$

Together these lead to equation (3)

$$\frac{dYp}{dt}=V1-V2=\frac{V_{maxA}*(YT-Yp)}{K_{mA}+(YT-Yp)}-\frac{V_{maxZ}*Yp^2)}{K_{mZ}^2+Yp^2}$$

VmaxZ can be easily determined:

$$V_{maxZ}= k_{catZ} * ZT$$

Where we assume that all CheZ is active and therefore equal to the total CheZ concentration.

VmaxA is the interesting part as we know its activity depends on the receptor activity.This means that to determine the level of active CheA we need to know how this level depends on the ligand concentration (basically the main question we are trying to figure out with our project). To achieve this, we need to relate receptor activity to CheA inactivation for the different methylation states. To accomplish this, the EC50s of the receptors are needed, allowing us to determine the activity of CheA as a function of ligand concentration. Giving us the last piece of the puzzle (equation (6)).

$$V_{maxA} = k_{catA} * A_{act}$$
$$A_{act} = \frac{AT}{1+ (\frac{L}{EC50})^2}$$
Table 2: Receptor EC50 values
ReceptorΔCheBΔCheRWT
AAAA385.5 303.3
EAAA57.9 78.9
AEAA108.8 118.5
AAEA43.3 58.6
AAAE68.7 117.6
AAEE0.8 1.3
AEAE1.3 3.5
AEEA- 0.9
EAAE- 4
EAEA- 1.3
EEAA52.8 53.2
AEEE- 0.6
EAEE- 1.3
EEAE- 3.0
EEEA- 0.6
EEEE- 0.8

This data was obtained by Krembel et al., who determined the EC50s for different methylation states (table 2). They separated their data in two categories: one with a CheB/CheR double knock-out, and the other with wildtype CheB/CheR. Under the double knock-out circumstance only artificial methylation is possible, allowing investigation of the effect of different methylation states under static circumstances. The dynamics of the other set of data measured in E. coli with a functioning methylation regulation system is less static. These cells start adapting upon change of the ligand concentration. However, methylation is a slow process (in the order of minutes to hours (Tu et al., 2008) in comparison to the other processes. This allows us to assume that if the measurement is performed within minutes, the methylation state can be considered constant. On the long term the system will start to adapt itself slowly lowering the sensitivity, something that was accounted for in our experimental design.

The other dynamics, such as ligand-receptor binding and the phosphorylation of regulation factors, take place on the timescale of seconds to milliseconds (Meir et al., 2010). We can therefore assume that the system reaches equilibrium rapidly, allowing us to study the system in equilibrium. This is done by equating differential equation (3) to 0, providing us with an equilibrium value for CheYp. Since this value depends on the fraction of active CheA, which in turns depends on the ligand concentration, we can express this equilibrium value as a function of ligand and plot for the different methylation states. However, because of the non-linearity of equation 3, solving this algebraically proved challenging and was therefore done with Mathematica.

### Results

In figure 1 we can see the projected concentration of CheYp as a function of ligand concentration under native conditions. We need to keep in mind that because of the peculiarity of the chemotaxis system, our assay works ‘backwards’. This means that at some point all CheY-p gets removed rather than being built up, resulting in the absence of the BRET-signal. We should also note that the x-axis is logarithmic. The EEEE variant corresponds to the wildtype, where no methylation site has been mimicked (blue line). Interestingly, we have one artificially altered variant that has even higher sensitivity than this wildtype receptor. This is most likely caused by the methylation at that site influencing the methylation rates of the other sites. We can now identify which mutations are the most impactful. Mutating the third position (Q309, EEAE) lowers sensitivity by almost an order of magnitude making it the most influential single mutation. If the fourth position is mutated as well (E941, EEAA) we can alter the sensitivity another order of magnitude. This means that with only two mutations we can drastically alter the sensitivity of the receptor, something we decided to test experimentally. We can also identify a transition region where the signal goes from a full signal to no signal. This means that theoretically you can determine the concentration of ligand if you knew exactly how CheY-p concentrations relate to signal strength. With current techniques this might not yet be feasible, but it should be possible in the future

#### References

Bitbol, A.-F., and Wingreen, N.S. (2015). Fundamental Constraints on the Abundances of Chemotaxis Proteins. Biophys. J. 108, 1293–1305.
Deshpande, A., Samanta, S., Das, H., and Layek, R.K. (2016). A Boolean approach to bacterial chemotaxis. Conf. Proc. Annu. Int. Conf. IEEE Eng. Med. Biol. Soc. IEEE Eng. Med. Biol. Soc. Annu. Conf. 2016, 6125–6129.
Krembel, A., Colin, R., and Sourjik, V. (2015). Importance of Multiple Methylation Sites in Escherichia coli Chemotaxis. PLOS ONE 10, e0145582.
Meir, Y., Jakovljevic, V., Oleksiuk, O., Sourjik, V., and Wingreen, N.S. (2010). Precision and Kinetics of Adaptation in Bacterial Chemotaxis. Biophys. J. 99, 2766–2774.
Morton-Firth, C.J., Shimizu, T.S., and Bray, D. (1999). A free-energy-based stochastic simulation of the tar receptor complex 1 1Edited by I. B. Holland. J. Mol. Biol. 286, 1059–1074.
Perez, E., West, A.H., Stock, A.M., and Djordjevic, S. (2004). Discrimination Between Different Methylation States of Chemotaxis Receptor Tar by Receptor Methyltransferase CheR. Biochemistry 43, 953–961.
Silversmith, R.E., Levin, M.D., Schilling, E., and Bourret, R.B. (2008). Kinetic Characterization of Catalysis by the Chemotaxis Phosphatase CheZ MODULATION OF ACTIVITY BY THE PHOSPHORYLATED CheY SUBSTRATE. J. Biol. Chem. 283, 756–765.
Stewart, R.C. (1997). Kinetic Characterization of Phosphotransfer between CheA and CheY in the Bacterial Chemotaxis Signal Transduction Pathway. Biochemistry 36, 2030–2040.
Tu, Y., Shimizu, T.S., and Berg, H.C. (2008). Modeling the chemotactic response of Escherichia coli to time-varying stimuli. Proc. Natl. Acad. Sci. U. S. A. 105, 14855–14860.