Modeling

Abstract

Analyzing dynamics of E.coli’s chemotaxis and estimating effectiveness of the Prometheus require high accuracy, and thus hard to realize with experiments. Here, we construct a virtual peanut root-E.coli system (by c++ and OpenGL), storing the 3-D environmental information by k-d tree. We quantify the process in which “Promethei”, our engineered bacteria, carrying Mo, move towards peanut roots. We at last estimate around 20-40% of bacteria could work in reality, and that the best “memory” time be around 40 min (See here). All codes (in c++) of the project are available here.

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E.coli model

Mechanism

We divide Escherichia coli’s behavior into two parts: “running” straight lines, and “tumbling” for reorientation[1]. When attractant concentration uniformed, E.coli can be described as doing “random walk”[4], and the running time fits a normal distribution. With spatial concentration gradients of chemical attractance,when bacteria sense a higher attractant concentration than before,the mean of the statistical distribution of running time increases. Otherwise, the time keeps the same as that in the uniform environment, no matter how high or low the concentration is [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.

Brown and Berg[4] suggested that the mean τ has a functional relationship, shown as follows, with the current attractant (in their case, glutamate) concentration and the change of concentration over time. (See formula (1),(2)).

K_D is the dissociation constant of the complex formed by glutamate and its receptor. Pb is the fraction of receptors (protein) bound with the attractant.

In their model, the mean velocity during a run, however, is independent from C and dC/dt [5,6].

Brown and Berg found the curve K_DC/(K_D + C)₂ fits the chemotaxis sensitivity assays well (Fig. 2).

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

Build a concentration gradient

The concentration gradient of the root’s secreta can be regarded as stable in the soil. The concentration decrease as it go farther from the root surface ( A sketch is below, see Fig. ).

Fig. A sketch of the root and its secreta’s concentration gradient shown in different colors. The green points are E.coli imagined run up the gradient towards the root.

The concentration value of the layers are estimate from [8]. We found no paper talks directly on the spatial distribution of peanut’s root exudates, but it shows that the spatial distribution could be described by the equation Y = A₁ * X^-B, where Y is the C14 activity representing the amount of exudates at the distance X from the root surface. ( An example is shown in Fig. from [8].) Thus, we imagine that the spacial distribution attractance (Succinate) exudated by peanut fit the equation above similar with maize and wheat.

Fig. This figure from [8] is to show the fir results of maize and wheat ( the y axis represent the percentage(%) of C14 activity. The curves seem good to fit the points with x >= 1 mm, but it absolutely can’t be used to get a percentage if 0<= x <1. Thus we just estimate that from 0 to 1 mm, the percentage decrease linearly, and the value is 70% in x = 1 mm to a peanut root.

Thus, the function of the concentration C on the distance to the root surface x is

C = 100 - 0.3 * x if x < 100 and

C = 70 * x^(-1.2) if x > 100.

NOTE:
The number is estimated by calculate the order of magnitudes from the total mass of Succinate a peanut root contains.

Use k-d tree to store these points

We make the concentration of the point in concentration gradient nearest to the bacteria be the value of concentration in the bacteria’s position.

To find a nearest concentration point to a bacteria from about one million will be a big jog. We implement k-d tree ( k-dimension tree, a useful and smart data structure to store information of points in k-dimension space; it is very fast, typically requiring O(logN) time, to find the nearest point with a given position) to store all the 100 thousand points of concentration and find the nearest to the given bacteria. The knowledge of k-dtree is learned from the course algorithms developed by R. Sedgewick And K. Wayne in Coursera.org and [7].

Programming structure

The attractance is specified to be Succinate here.

The program running in computer simulate the movement per second as the following pseudo-code.

move (a bunch of E.coli , the concentration gradient)

each E.coli has [state, timer, position, direction, duration_time]

*The points of concentration gradient are stored in a 3-d tree It will expounded clearly in the part below.
** The mean value is calculate by the formula(1), (2) and (4).
*** This step is by calculating duration_time (this moment) - timer, and if the result > 0, then it should go on this behavior.

Usage of the model

Part One: estimation

e implement this model to estimate the best time interval to kill the engineered E.coli at large in soil.

After run it by computer , we get that it is best to ....

Two

We tried to estimate the effectiveness of the whole system.

From the runs of the program several times, we can clearly compare with the difference.

Compared with the imaginary situation that E.coli are not attracted by anything of the root, the effectiveness of

Results and analysis

Review and forecast

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The Story of E.coli Prometheus

BNU-China