Difference between revisions of "Team:USTC/Modeling"

Overview

This part introduces the data of adhesion experiment, makes exploration and interpretation on bacteria adhesion dynamics. Through the modeling of bacteria adhesion, we can get some characteristics important but difficult to directly measure on bacteria adhesion and reaction to antibiotics. Also give some irreplaceable constructive opinions and key indicators on the operation of detecting the concentration of antibiotics.

Bacterial Absorption Simulation

Let's assume variables firstly,

[C]: Concentration of bacteria.(/\(m^{3}\))

S: Area of film.(\(m^{2}\))

V: Average swiming speed of bacteria.(m/s)

Vz: Average swiming speed component in the z axis, which is perpendicular to S.(m/s)

σ: Density of the adhesive bacteria.(/\(m^{2}\))

N: Total number of sticked bacteria.

m: Movement percentage of bacteria.(%)

M: Exact movement number of bacteria.

Bacterial Absorption Dynamics Fits Langmuir Equation

Assuming velocity of bacteria in any direction is the same, let's set it V,

and irrespective of contacting with each other when swimming, we are able to conclude bacterial movement is free.

Then we calculate the average velocity in z axis,

\(V_{z}=\int_{0}^{\frac{\pi }{2}}V\times cos\theta \times \frac{dS}{S}=\int_{0}^{\frac{\pi }{2}}\frac{V}{2}cos\theta sin\theta d\theta =\frac{V}{4}\)

Consider during the interval dt, in area dS, there are dN bacteria in tiny volume \(dS \times V_{z}dt\) hitting the wall whose area is S. Schematic image illustrates the process mentioned above,

Consequently we can know that

\(dN=C\times dS\times V_{z} dt\)

Because the number of bacteria in solution is much more larger than it on the glass surface whose area is S. So the concentration of bacteria (C) can be regarded as steady during the measurement period.

Assuming the number of hitting wall bacteria is stable, the surface can only adhere one layer of bacteria, that is the place which has already adhered bacteria can not stick any more bacteria. This means we could use Langmuir Adsorption Isotherm to solve this problem!

Let's see how we figure out this issue using Langmuir Equation. Consider in a current area S, the density of bacteria on surface is σ, and during a interval dt, the change of bacteria density is dσ

\(Sd\sigma =K_{a}\times (1-\frac{\sigma }{\sigma _{0}})\times dN-K_{d}\times S\sigma dt\)

Where Ka is the adhesive rate of each hit, Kd is the drop rate of the adhered bacteria.

Then, after solving this Ordinary Differentiate Equation, ODE, we got the equation shown below,

\(\sigma (t)=\frac{K_{a}CV_{z}\sigma _{0}}{KCV_{z}+K_{a}\sigma _{0}}\times (1-e^{-\frac{K_{a}CV_{z}+K_{d}\sigma _{0}}{\sigma _{0}}\times t})\)

According to this result, the density of bacteria at time t is related to adhesive rate of bacteria Ka, concentration of bacteria c and velocity of bacteria Vz. And we can let \(\frac{K_{a}CV_{z}}{K_{d}\sigma _{0}+K_{a}CV_{z}}\) equals an integrative constant, K. In addition, we cannot start recording the image data the moment we put the bacteria on the cover glass in reality, so there should be a time delay in our modeling. Thus we got the accurate equation depicting the density of bacteria time t as,

\(\sigma (t)=K\sigma _{0}\times (1-e^{-\frac{K_{a}CV_{z}}{K\sigma _{0}}\times (t-t_{0})})\)

In order to fit the data conveniently, we change the equation form into a more general one,

\(\sigma (t)=ae^{-bt}+c\)

Bacteria Density Simulation

With assuming a constant value group:

Ka	Kd	C	Vz	σ0
0.5	0.01	10^9/m^3	5um/s	10^10/m^2

we got the plot expressing the bacteria density variate through time,

which is quite similar to the real data.

Bacteria Counting Program

This program is used to calculate the amount of bacteria and get the percentage of moving bacteria, which is essential for our further research. More details on our coding using Matlab please refer to 2015 USTC in Github.

The principle of programming is told below:

1. Loading the image.
2. Calculate a self-adapting or special threshold value in the image binay progress.
3. Use mathematical morphology operations.
4. Use filtering processing make the image more smooth.
5. Delete the small area to reduce the error noises.
6. Auto-counting the number of objects.

Now we are able to count bacteria using this program, see what we got!

Adhesive Assay Analysis-Bacteria Number

Now we got the theoretical bacteria density variation formula and an efficient program to demonstrate the real number of bacteria on film. We are trying to explain everything we gained from our experiment. Results are posted in Results-Adhesion assay.

Using MATLAB® to simulate these data with the function 'f(x)=a exp(-b*x)+c', here is our analysis,

HCB1-PLL(+)-no antibiotics number-time

Fitting result are shown below

the constants value and details are,

HCB1-PLL(+)-0.1ug/ml Cl number-time

Fitting result:

Constant value and details:

HCB1-PLL(+)-0.5ug/ml Cl number-time

Fitting result:

Constant value and details:

HCB1-PLL(+)-1ug/ml Cl number-time

Fitting result,

Constant value in detail,

PAO1-PLL(+)-no antibiotics number-time

Fitting plot is

Constants value and details,

PAO1-PLL(-)-no antibiotics number-time

All these perfect fitting results illustrate that our hypothesis of adhesion mechanism is totally correct. All Modelings and image analysis programs hit the truth.

Adhesive Assay Analysis-Movement Percentage

Now we know that bacteria can move straight because of its flagellum rotating counterclockwise according to previous research. However, due to the stickiness of PLL, some flagella may stick to film when spinning. Assuming the rate of stick (P) is constant, the movement percentage of bacteria will present in an exponential form changing through time.

Assume the function of movement percentage ( M) related to time is,

m=m0exp(-kt)

Note: the data is fixed by the previous analysis result t0.

PAO1-PLL-no antibiotics

Fitting plot

Constant value and details

We can see that the raw data perfectly match to this model.

Important Parameters in Adhesion Equation

According to the fitting results shown above and equations, we could get some useful information when operating NDM, such as,

• Initiation Moment (t0)
• Adhesion Ability

Initiation Moment(t0)

Because we can not start recording the image data the moment we put the bacteria solution on the cover glass, so there is a starting time delay in the equation. According to our model, we are able to retrospect the exact the moment when we conduct experiment,

\(\because a=-c\times e^{-bt_{0}} \therefore t_{0}=\frac{ln(-\frac{a}{c})}{-b}\)

Substituting this function into fitting result, we can get the real starting time of each test. Results are delivered below:

Conditions	PAO1-PLL(-)-0	PAO1-PLL(+)-0	HCB1-PLL(+)-0	HCB1-PLL(+)-0.1	HCB1-PLL(+)-0.5	HCB1-PLL(+)-1
t0	60.3s	33.3s	24.4s	109.2s	39.5s	60.3s

It is interesting that we could know the "Initiation moment" through our data analysis, which is very cool.

Adhesion Ability

Another intriguing and important property we got through our data analysis is the adhesion ability(Bacteria to Cover Glass)(AA-B-CS in brief).

According to the fitting results and modeling equation, the derivative of the fitting function at the beginning(t0) is the bacteria number growth rate in its maximum. So we defined this derivative value as the adhesion ability of the bacteria solution to cover glass, or call it AA-B-CS for short.

Refering to the modeling result, we know that:

AA-B-CS=dσ/dt|(t=0)=KaCVz=c*b

where, c and b are the constant value in fitting result.

This equation told us AA-B-CS relates to Ka, the adhesion rate of bacteria on every hit, C, the concentration of bacteria solution, Vz, the average swim speed of the bacteria. This explains why we emphasize both bacteria and film material when we defined this parameter.

The AA-B-CS of HCB1 shown in table,

Conditions	HCB1-PLL(+)-0	HCB1-PLL(+)-0.1	HCB1-PLL(+)-0.5	HCB1-PLL(+)-1
AA-B-CS	2.01	6.69	12.64	8.23

Note: The bacteria solution in 'HCB1-PLL(+)-0' sample was not the same with other samples, so the AA-B-CS value is much different with other's.

Surprisingly, we found that the concentration of antibiotics doesn't effect the AA-B-CS in our assay, which suggests that we are able to use the same type of bacteria treated with antibiotic in different concentrations, which is another important theoretical base of our NDM.

Experiment Guidance

In Adhesion Assay we finally could know several important things,

• Film-coating time
• Bacteria-film interaction time(Ti)
• Concentration of the bacteria solution,
• Observation Moment

All of these can be known through the pre-test results analysis.

Film-coating time

Through the pre-test and data in paper, we use 20ug/ml poly-L-lysine coating the film over 4 hours or overnight at the temperature 4℃ eventualy.

Bacteria-film interaction time(Ti)

Because the motility of bacteria will decrease when we do not administrate antibiotics. So we need to balance the total number of bacteria and motility. If we implement the data from "PAO1-PLL-0" assay as sample to analyse the best time of bacteria-film interaction time. We got the best interaction time is the moment when the number of movement bacteria reach the maximum value.

The total movement bacteria is,

\(M=S\sigma m\)

Getting S, σ and m in detail, we got,

\(M=c(1-e^{-bt})\times m_{0}e^{-kt}\)

Now we can simulate those data!

Thus we highly recommend the Bacteria-film intraction time(Ti) is approximately 100s after treating with bacteria solution, which means since you inoculate the bacteria about 100s, you should put the film into your sample solution and determine its antibiotics concentration.

Concentration of bacteria solution

In test "PAO1-PLL-0", the bacteria solution was culture overnight in 37℃(which means the bacteria was in platform stage). diluted bacteria solution 50 times.

Observation Moment

If we want to observe the deformation of the film, the bacterial additional pressure on film should reach a stable stage. Assume there are K% of inactivated bacteria move when administrating with antibiotics. Then the movement percentage differential equation will be changed to this:

\(\frac{dm}{dt}=(1-m)K-bm\)

After solving this differential equation, we received a m to t function:

\(m(t)=\frac{K+(m_{0}k-(1-m_{0})K)e^{-(K+k)t}}{K+k}\)

Let \(m_{0}\)=0.5, and choose different 'K'. We are able to acquire different simulating curve.(According to previous analysis, 'K'=0.0065.)

Very luckily, that's very similar to our experiment data,

If we want to limit the error probability less than 5%, we need the movement percentage reach over 95% of the maximum or minimun value. According to the simulation result, when 'm' reach 95% of its maximum or minimum value(K/(K+k)), t~100s.

Consequently, we recommend user to get the results in NDM, you should wait approximately 100s after putting film into optical path.

Overview

This part explains the fundamental principle of our detecting method - interference. Develop a pre-experiment to verifiy the modeling in advance. Give a method to process the interference image and calculate the deformation of the film in a high precision.

Pre-experiment

Our experiment is initiated from interference fringes pattern formed naturally. One thing need to know is that the different clip we implement in our pre-experiment will get different fringes.

(I) If we use clip I (the round clip),

we will get Newton's rings like fringes.

In the pre-experiment1 (you can read protocol of our pre-experiment in annex showed below).

As a matter of fact, we captured a picture like this,

That's a typical Newton ring interference image.

(2) If we use clip II (the square clip),

we will get an equal thickness interference fringe pattern.

In the pre-experiment2 (you can read protocol of our pre-experiment in annex showed below).

As well, we captured a picture like this,

That's a typical equal thickness interference fringes image.

Modeling Prinicple

(1) In method 1, we choose clip I.

Let's concentrate on the deformation of film.

As the deformation range(h) is much more smaller than the radius(r) of the film (h<<r),

we can consider the light is approximately paraxial spherical.

Because the area of CCD camera is relatively small, whose size is ~cm x cm, the interference pattern is approximately paraxial spherical as well.

Ideally, light path sketch should be as follows. (Light shown in blue line is the reflected light from holophote, light shown in red line is the reflected light from film).

Where L is the distance form the virtual image to the CCD camera.

However, in reality, because the virtual image of holophote and the film can not be set strict parallel in actual situation, the light path sketch becomes like below, where light shown in blue line is the reflected light from holophote, light shown in red line is the reflected light from film.

We could use the method of coordinate transformation to simplify them as follows, where light shown in blue line is the reflected light from holophote, light shown in red line is the reflected light from film.

These are the parameters we preset shown in table,

r	h	a	θ
0.02m	5e-6m	0.02m	5e-4rad

where r is the radius of the film, h is the deformation length of the film, a is the length of each side of the CCD camera, θ is the slip angle between the film and the holophote which we estimate.

Using these parameters, we simulated the interference fringes image as following,

How amazing our model is ! Simulational result just hit the row image we got before!

(2) In method 2, we choose clip II.

Consider the reflect surface was not strictly parallel with the CCD, and the angle between the surface and x/y axis is α/β.

So we can know the optical path differences and light intensity distribution on CCD are

\(\Delta L=2L+2(xtan\alpha +ytan\beta )\)

\(\delta =\frac{2\pi }{\lambda }\Delta L\)

Use matlab to simulate the interference pattern, and the result showed as following,

When bacteria pushing the film and making film deform, the angle between film and axis will change. So the number of fringes on y axis will increase or decrease. And obviously the change of number (ΔN) will be linear to deformation (h) in equal thickness interference case, that is: ΔN=2h/λ.

Fringe Analysis Protocol

Methods

(1)

1.Take a series of photos at the same position in a short time.

2.Superpose these photos to sharp the edge of every object.

3.Choose two points in multi-image, the point must be on the black fringes.

4.Scaning these two fringes to find the shortest distance between them.

5.Calculate the radius and rank of every fringes.

6.Calculate the deformation of film.

(2)

1.Take a series of photos at the same position in a short time.

2.Superpose these photos to sharp the edge of every object.

3.Calculate the number of fringes on vertical direction(y axis).

4.Transform the number to deformation of the film.

More details on coding please refer to Github:2015USTCiGEM.

Annex

Pre-experiment Protocol

Optical path image in pre-experiment is as follows

Light shown in red line is the light from laser, light shown in green is reflected by film, light shown in purple is reflected by holophote.

The wave length of our laser is 650nm.

The distance between 50% reflector and film is about 10cm.

In method I:

The film was covered tightly on a tube's hole, and we give a perturbation on temperature to change the pressure in tube. So the film would deform too, then we could simulate the bacteria force in a physical way.

In method II:

Because the interference fringes are similar to the situation that the film does not deform, we can observe the interference with the film directly sticked on the clip.

Overview

This part wil give the basic method of establish a bio-calibration of the concentration of antibiotics. Using modeling prediction and results developed in previous work, we propose a transform formula between concentration and deformation. We also get the deformation of film from interference fringes by using the interference analyzing program.

Basic hypotheses

Hypotheses

Let's assume that bacteria response on antibiotic is linear to antibiotic concentration to a supposing n power. That is \(K=CA^{n}\)

And let's show the variable lists before we get started:

[A]: Concentration of antibiotics.

m: Percentage of moving bacteria.(%)

M: Exact movement number of bacteria.

h: Deformation length of the film.

m0:Percentage of moving bacteria at the time we start the test.

Concentration→Motility

The motility of bacteria can be divided into two parts: original motility and induced motility (by antibiotics). According to our previous work, Adhesion Dynamics, we conclude the relationship between the motility and time. That is

\(m(t)=\frac{K+(m_{0}k-(1-m_{0})K)\ e^{ -(K+k)t}}{K+k}\)

More information about the formula please refer to Github:2015USTCiGEM

Because we assume the effect of antibiotics on motility is n power, we can define \(K=CA^{n}\).

When the movement percentage reachs a steady state, that means \(m=\frac{K}{K+k}\), we can infer that \(m=\frac{CA^{n}}{CA^{n}+k}\).

And due to the concentration of bacteria solutions we used are almost the same(OD~0.3), the AA-B-CS will be almost the same as well. That means the total number 'N' and the 'k' will be almost the same, which is very important in our standardized operation.

Motility→Deformation

According to the analysis in Film Candidate we can know the relation between deformation and bacteria's movement. That is

\(f=MF_{0}\)

\(f=2F\times \frac{h}{b}\)

\(\frac{F}{ac}=\frac{\Delta b}{b}G\)

\(\Delta b=\frac{b}{2}\times (\frac{h}{b})^{2}\)

So \(f=acG(\frac{h}{b})^{3}\)

Now we can substitute each equations and predict the relationship between deformation 'h' and concentration 'A':

\(\frac{CA^{^{n}}}{CA^{n}+k}N=acG(\frac{h}{b})^{3}\)

'N' is the total number of sticked bacteria.

Get the reciprocal of the upper equation:

\(\frac{acG}{Nb^{3}}(1+\frac{k}{CA^{n}})=\frac{1}{h^{3}}\)

That means 1/h^3 is linear to 1/A^n.

Deformation→Fringes

According to the modeling of fringes analysis we conclude the relationship between deformation 'h' and the changes of number of fringes on y axis (\(\Delta N\)) is: \(\Delta N=\frac{2h}{\lambda }\)

So replace 'h' with 'ΔN' in equations and we will get:

\(\frac{acG\lambda ^{3}}{8Nb^{3}}(1+\frac{k}{CA^{n}})=\frac{1}{\Delta N^{3}}\)

Define \(\frac{acG\lambda ^{3}}{8Nb^{3}}\) as a complex constant \(A_{0}\), \(\frac{ackG\lambda ^{3}}{8NCb^{3}}\) as another complex constant \(B_{0}\), and simplify the formula as below:

\(A_{0}+\frac{B_{0}}{A^{n}}=\frac{1}{\Delta N^{3}}\)

So if we measure the two constants \(A_{0}\)&\(B_{0}\), we can build a calibration on concentration detecting. And when we get a solution with unknown concentration, we can test it and find the concentration in calibration!

Fringes analysis

Methods

1.Use matlab recognize the number of fringes in each image.

2.Collect the output information from program and calculate the change fringe numbers in average.

3.Find the concentration in calibration.

More details on coding please refer to Github:2015USTCiGEM

This part introduce a genetically engineered bacteria with inhibition circuit to detect the trace antibiotics. Giving the modeling result of inhibitive circuit and tradition positive control circuit, and show the advantage of inhibitive circuit in trace detection.

Basic Hypothesis

• Gene expression under regulation has a linear relation with inducer or inhibitor at an appropriate concentration range.
• Modeling on quorum sensing is based on steady-state model.

Variables containing:

S: Concentration of antibiotics, such as sulfamonamide or tetracycline.
A: Concentration of AHL
R: Concentration of LuxR
RA: AHL-LuxR complex
cI： Concentration of cI
G: Relative fluorescence internsity
F: micF transcription initiation effciency
C: Lac transcription initiation effciency
X: Promoter Lux efficiency
Λ: Promoter λP efficiency

In antibiotic sensing part:

\(J=-D\nabla C(1)\)

\(\frac{d}{dt}[AHL]=k_{2}F(2)\)

About AHL diffusion modeling:

Assume bacteria expressing AHL are uniform distributed, and consider as a single bacteria, AHL production speed is:

\(\frac{d}{dt}A(3)\)

At the distance r, the concentration contribution of this bacteria is a. Let the diffusion constant as D. According to Fick's Law:

\(J=-D\nabla C(4)\)

Owing to the uniform distribution of producer, the concentration of AHL is uniform as well, thus:

\(A=k[AHL](5)\)

In Bacteria II

Due to the promoter lac is induced by IPTG, the concentration of R is stable and maximum:

\(R+A\rightarrow RA(6)\)

\([RA]=k_{3}[R][A] (7)\)

\([X]=k_{4}[RA] (8)\)

\(\frac{d}{dt}[cI]=k_{5}[X] (9)\)

\(\lambda =1-k_{6}[cI] (10)\)

\(\frac{d}{dt}G=k_{7}\lambda -k_{8}G (11)\)

Consequently, we could get our exact modeling result using Matlab:

Time consumption compared to traditional reporter system:

Concentration resolution response compared to traditional reporter system:

More information on our code please refer to Github:2015USTCiGEM.