Team:BNU-China/modeling.html

Abstract

The dynamic analysis of the E.coli’s chemotaxis and the effectiveness estimation of the Prometheus are hard to do in experiments. It requires a high accuracy to finish. A virtual Peanut-Root-and-E.coli system was built in computer (by c++ and OpenGL) to quantify the process that “Prometheus”, our engineered bacteria, carrying Mo as a poter move towards peanut roots. We at last estimate the best “memory” time (See here) of Prometheus and its effectiveness (as shown below). The code (in c++) of the project is available all here.

Our model requires the acknowledge to the mechanism of E.coli’s motivation and chemotaxis and the distribution of the peanut root and its [分泌物]. We then put both into code to make a complete system to [进行我们想要的分析]. A [思路] to introduce or work is shown below. You can [分别了解] them by click the icons.

E.coli model

Mechanism

Escherichia coli’s movement is divided into two parts: “running” for straight line movement, and “tumbling” for reorientation (See [1] ). In a uniform environment, E.coli’s motivation can be describe as “random walk” (See [4]) and the running duration fits a normal distribution. In a environment with spatial concentration gradients of chemical attractance,when bacteria sense a higher attractant concentration this time compared to last time (in a positive temporal gradients environment),the average running duration time increases.Otherwise, their duration times keep the same as that in the uniform environment, no matter how high or low the concentration is(See [4]).

Running

The running of E.Coli can be descibed as a straight line. The distribution of running duration time fits a normal curve, with average τ_run0 and standard division σ_run.

An paper by Brown and Berg (See [4]) suggested that the mean run duration time τ has a functional relationship with the current attractant concentration and the time rate of change of concentration.. The chemical attractant in his experiments is glutamate. From experiment data and analysis, he concluded the relationship (See formula (1),(2)).

K_D is the dissociation constant of the complex formed by glutamate and its receptor. Pb is the fractional amount of receptor (protein) bound with the attractance (See [4]).

The mean velocity during a run, however, is constant while the duration time grow bigger. It keeps the value that in a uniform environment (See [5,6])

Obtain K_D

Brown and Berg found the curve K_DC/(K_D + C)² fits the points of sensitivity assays for taxis toward glutamate well. (See a picture from their paper below)

Fig. 2

The dissociation constants K_D of coplexes in our project is no way to obtain by experiment. However, similar results (points) (See our experiment) were got and we tried to make the constant by fitting it well. The result of Least Square Straight-Line Fit of the data from McfR-Succinate shows the K_D of Succinate-Receptor bound is 3.5mM with the formula.

Fig. the result from experiment.

Fig. The result fitted by a( K_DC / (K_D + C)²) + d where a = 389 and d = 260 . This work is made by c++ and OpenGL. The code to get K_D and a and to make sketch is here.

Thus the value of τrun can be calculate now iff there is a concrete constant number of α in function (1). This one was made by a huge amount of experiments in [4]. We lend his number to fit here* with α = 660 seconds.

Tumbling

E. coli reorient between two runs.

Fig.3 A schematic drawing of a tumble, from [2], page 45

The angle dθ is not random strictly (See [1], [5], [6]), and the angle distribution is shown in Fig.4. However, the distribution of the angle is not formularily clear and the process to build a random number fit this distribution should take too much time, so we roughly regard it as a random process.

Fig.4 the distribution of change in direction ( A figure from [1] ).

The statement of the model

The E.coli model is base on the following statements (it will be shown in two parts).

In runs

1. The velocity during a run is constant (always equal to the mean)
2. E.coli moves straightly during a run
3. An E.coli’s running duration time trun is got by a random number fit N(<τrun>,σrun)
4. The mean duration time <τrun> fit formula (1) and (2)
5. The specific duration_time of a single bacterium changes in the same pace with the changes of the statistical distribution’s mean (relying on C and dC/dt). ( See Fig. 2)

Fig. 2 For a single bacteria, its surrounding environment decides the statistical distribution of its own run duration time. The change of a real duration time is equal to the change of the statistical distribution’s mean change. ( The variance doesn’t change. See above or [])

That can be described in a equation

where t_run2 is the specified duration time that a bacteria should run for (a timer records the time has passed), and the t_run1 is the one at the last second.

τ_run2 is the mean duration time now which is calculate from the function (1) and (2) on the value of C and dC/dt at this moment; τrun1 is the mean duration time at last second which is calculated from the value C and dC/dt at the last second.

NOTE
The statement is made to calculate the process more easily. We suppose it is the run duration time, not the difference of velocity, is the main variant of the whole chemotaxis system.

In tumbles

1. E. coli ’s positions do not change as they are tumbling;
2. the direction after reorientation is randomly chosen.

Coefficients

• Velocity = 100μm
• run duration time mean in uniform environment = 1.3
• run duration standard division = 3
• mean tumble duration time = 0.14 sec
• alpha (See equation (1)) = 660 sec
• KD = 3500 (μmol/L)

Root model

Virtual Root made by Laser scanning

We use laser scanner to get the root surface points position and their topological information. The scanner is offered by the College of Information Science and Technology of Beijing Normal University. The scanner 型号。

We gain 142735 vertexes and 253404 faces from it. The result can be seen in Fig.

Fig. The virtual peanut from the scanning

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The Story of E.coli Prometheus

BNU-China