Chemotaxis is the movement of bacteria up a concentration gradient of chemoattractants (e.g. malate in our project) and away from repellents (e.g. poisons). E.coli is too small to detect any concentration gradient between its two ends. These bacteria resample their surroundings every 3-4 seconds, causing to either tumble or to run^[1]. Chemoattractants transiently increase the probability of ‘tumbling’ (or bias). This is followed by a sensory adaptation process that returns the bias to baseline, enabling the cell to detect and respond to further concentration changes. A small step change in chemoattractant concentration in a spatially uniform environment increases the response time from 1 second to 2-4 seconds ^[2]. Changes to saturating levels of chemoattractant can increase the response time to several minutes.

Each chemoreceptor on the bacterium has a periplasmic binding domain and a cytoplasmic signalling domain that communicate with the flagellar motors via a phospho-relay sequence involving the CheA, CheY, and CheZ proteins. The results of modelling the chemotaxis pathway will determine the threshold chemoattractant concentration for bacterial detection and the level at which the bacterial chemoattractant detection becomes saturated, decreasing the efficiency of the bacterial response to the chemoattractant.

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2.1 Objective

Use MATLAB to model the chemotaxis pathway and thereby determine the detection threshold and saturation level of chemoattractant for bacteria with 8 μM chemoreceptor. As it is believed that the auxin should be placed close to the seed (< 0.25 cm ^[4]) for optimal growth. It is therefore crucial to determine whether our bacteria will be able to stay close to the seed.

2.2 Description

Figure 1^[1]: Chemotaxis signaling components and pathways for E.coli.

The chemotaxis pathway in E. coli is demonstrated in Figure 1. Chemoreceptors form stable ternary complexes with the CheA and CheW proteins to generate signals that control the direction of rotation of the flagellar motors ^[5]. The signalling group's currency is in the form of phosphoryl (p), which is made available to the CheY and CheB effector proteins through autophosphorylation of CheA ^[1]. CheY_p initiates flagellar responses by interacting with the motor to enhance the probability of ‘run’ ^[1]. CheB_p is part of a sensory adaptation circuit that terminates motor responses ^[1]. Therefore, studying the methylation and phosphorylation levels of CheB and CheY is important to understanding chemotaxis of a single cell. The model based on Spiro et al. (1997) ^[1] was used to study how the concentrations of proteins in the chemotaxis pathway change over time.

In addition, the quantity that links the CheY_p concentration with the type of motion (run vs. tumble) is called bias. It is defined as the fraction of time spent on the run with respect to the total movement time. The relative concentration of CheY_p is converted into motor bias using a Hill function (Equation 1).

2.3 Results and discussion

Based on the Spiro model^[1], the methylation levels of receptors and phosphorylation levels of CheY and CheB were studied (Figure 2).

Figure 2(a): The phosphorylation level of CheY. Figure 2(b): Phosphorylation of CheB. Figure 2(c): Methylation level of Chemoreceptor. Figure 2(d): The probability of of bacteria in the running state at different levels of CheY_p. Figure 2 (a)(b)(c) show that the threshold detection concentration is 10^-8 M, and the saturation concentration is 10^-5 M.

The results from our modelling of the chemotaxis pathway show that the threshold chemoattractant concentration for bacterial detection is 1x10^-8 M and the level where bacterial chemoattractant detection becomes saturated is 1x10^-5 M, causing the bacterial response to the chemoattractant to becomes less efficient. In addition, the modelling of bacterial population chemotaxis in next the next segment will further inform us about the whether our bacteria will be able to move close to the root and at which actual distance the bacteria should be placed.

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3.1 Objective

Model the chemotaxis of bacterial population dynamics under experimental and natural conditions. The model of laboratory conditions will aid the wet lab team in designing their experiments. The model of real soil conditions will further inform our project about how and where we can place our bacteria.

Under experimental conditions, the chemoattractant diffuses constantly from the source. However, in real soil, the root produces malate all the time. Therefore, we assume that the distribution of chemoattractant outside the seed is steady and time-independent. Hence, the modelling of bacterial population chemotaxis will be built with different patterns of chemoattractant distribution.

3.2 Description

This part of the modelling focused on creating a movement model of a bacterial population for chemotaxis. In chemotaxis, chemoreceptors sense an increase in the concentration of a chemoattractant and then send a signal that suppresses tumbling. Simultaneously, the receptor becomes increasingly methylated. Conversely, a decrease in the chemoattractant concentration increases the tumbling frequency and causes receptor demethylation. The tumbling frequency is approximately 1 Hz when flagellar movement is unbiased, and decreases to almost zero as the bacteria move up a chemotatic gradient [5]. In order to accurately built this model, the following assumptions were made based on literature:

1) During the directed movement phase, the mean velocity at which E. coli moves is 24.1 μm/s [7]. Whereas during the tumbling phase, the speed is insignificantly slower and can be neglected.

2) E. coli usually takes the previous second as their basis on deciding whether the concentration has increased or not. Therefore, in our model the bacteria will be able to compare the concentration of chemoattractant at t seconds to that at t-1 seconds.

3) In our model, we ignored that E. coli do not travel in a straight line during a run, but take curved paths due to unequal firing of flagella.

4) Our model did not consider the growth of bacteria or the tendency of bacteria to congregate into a small area due to quorum sensing.

In the model, the bacteria should be able to compare the chemoattractant concentration at the current time point to the concentration at the previous second. If the concentration decreases (i.e. C_t1-C_t2 ≤0), the bacteria will tumble with a frequency of 1 Hz. If the concentration increases (C_t1-C_t2>0), the tumbling frequency decreases, hence the probability of tumbling decreases. From equation 10 in ref [6], we know that when C_t1-C_t2 >0, the probability of tumbling decreases as an exponential function of chemostatic constant, bacterial velocity , concentration difference between adjacent time points and angle between the two time points. Therefore, we can conclude the above description into the following statement [8]:

Under laboratory conditions, the chemoattractant continuously diffuses from the source, hence the distribution pattern of chemoattratctant changes with time. In this case, the error function (Equation 2) was used to describe the non-steady state chemoattractant distribution.

However, in real soil, malate is used as the chemoattractant. Malate is constantly secreted from the root tip at a concentration of 3 mM [9]. In this case, the malate source is always replenished due to continuous secretion from the root and the distribution pattern can be considered as steady (i.e. independent of time). The steady-state Keler-Segel model was used to demonstrate this distribution (Equation 3).

3.3 Results and Discussion

Under laboratory conditions, a simulation of chemotaxis of 100 bacteria placed 6 cm away from a 5 mM malate source is shown in the movie below. This MATLAB program has a graphic user interface (GUI), which can be used by the wet lab students to simply type in parameters and generate static graphs or animations.

Figure 3(a): Distribution of malate vs. distance (1D). Figure 3(b): Distribution of malate with radius (2D). Figure 3(b)shows the position of the threshold detection concentration (1e-8 M at radius = 0.028 m) and the saturation concentration (1e-5 M at radius = 0.012).

Figure 4: Chemotaxis with bacteria placed at different starting positions. Blue: 2×10⁵ seconds chemotaxis starts at radius = 0.015 m (in between 0.012 m and 0.028 m), green: 2×10⁵ seconds chemotaxis starts at radius = 0.008 m, which is less than 0.012. This shows that the chemotactic response of the bacteria is inefficient when they are placed between 0.0028 m and 0.0012 m from the root tip. The green line shows that the bacteria can be maintained close to the seed when it is placed at a distance of less than 0.012 m away from the root tip. Therefore, the model suggests that the bacteria should be placed at a distance of less than 0.012 m in our project implementation.

In conclusion, the threshold chemoattractant concentration for bacterial detection is 1x10^-8 M and the level where the bacterial chemoattractant detection becomes saturated is 1x10^-5 M and bacterial response to chemoattractant becomes less efficient. Furthermore, with malate concentration equal to 3 mM, the threshold detection concentration is at a distance of 0.028 m from the root, and the saturation concentration is reached at 0.012 m (1.2 cm) away from the root. From modelling the chemotaxis of the bacterial population, we observed that the chemotaxis to the root is slow if the bacteria are initially placed in between 0.012 m and 0.028 m, while, when placing the bacteria within 0.012 m from the root, the bacteria are likely to stabilise and remain close to the root. The distance for stable bacterial localisation (1.2 cm) is greater than 0.25 cm (the distance where nutrients can be effectively taken up by the root), therefore it is suggested that for our seed coat implementation, the bacteria should be placed at or closer than 0.25 cm away from the root.

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[1] Spiro PA, Parkinson JS and Othmer HG (1997) ‘A model of exciatation and adaptation in bacterial chemotaxis’. Proc Nat. Acd Sci USA 94: 7263-7268.

[2] Blocks SM, Segall JE and Berg HC (1982) Cell 31: 215-226.

[3] Stock JB and Surette MG (1996) ‘Escherichia coli and salmonella: cellular and molecular biology’. Am Soc Microbiol, Washington, DC.

[4] Schnepf A ‘3D simulation of nutrient uptake’

[5] Levin MD, Morton-Firth CJ, Abouhamad WN, Bourret RB, and Bray D, ‘Origins of individual swimming behavior in bacteria.’

[6] Vladimirov N, Lovdok L, Lebiedz D, Sourjik V (2008) ‘Dependence of bacterial chemotaxis on gradient shape and adaptation rate’ PloS Comput Biol 4(12): e1000242. Doi:10.1371/journal.pcb1.1000242.

[7] Liu Z and Papadopoulos K (1995) ‘Unidirectional motility of Escherichia coli’. App Env Microbiol 61: 3567–3572, 100099-2240/95/$04.0010

[8] https://2009.igem.org/Team:Aberdeen_Scotland/chemotaxis

[9] Martinoia E and Rentsch D (1994) ‘Malate compartmentation-responses to a complex metabolism’ Annual Review of Plant Physiology and Plant Molecular Biology 45: 447-467, DOI: 10.1146/annurev.pp.45.060194.002311

[10] Brokaw CJ ‘Chemotaxis of bracken spermatozoids: Implications of electrochemical orientation’.

[11] Jones DL, Prabowo AM, Kochian LV (1996) ‘Kinetics of malate transport and decomposition in acid soils and isolated bacterial populations the effect of microorganisms on root exudation of malate under Al stress.’ Plant and Soil 182: 239-247

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