Team:USTC/Modeling
Overview
This part starts with the bacteria movement ability, analyse the interaction between bacteria and the special film, and obtain the requests of the material properties eventually.
Force of Single Bacterium
The interaction between bacteria and film is essential for our work. To characterize the additional pressure exerted by bacteria, we need to calculate the force produced by single bacterium at beginning.
Assume the driving force of the movement of single bacteria is \(F_{0}\) in horizontal plain.
When bacteria move without extra environmental impact, let the speed as V1,
\(F_{0}=f_{1}=kV_{1}\)
Where f1 means the resistant force bacteria containing in water solution, k is a constant called drag coefficient at this conditions.
Besides, in the gravitational field, if we consider bacteria dragged by gravity in solution, then the static equation is revised as below,
\(mg-\rho Vg=f_{2}=kV_{2}\)
Where rho represents the density of solution, V indicates the extra volume of liquid immersed with bacteria, g means the acceleration of gravity. Because of the same interaction between solution and bacteria, in gravitational field, bacteria share the same k with the horizontal movement constant.
Assume the motor ability of each bacteria does not change. Then, in accordance with the data in previous research, the speed of bacterial movement((\(V_{1}\))) is approximately ~10\(\mu m\)/s, while the speed of bacterial sedimentation((\(V_{2}\))) ~\(\mu m\)/s.
And considering the size of single bacterium, assume bacteria as cylinder, whose diameter equals 0.5um and length is 1~3\(\mu m\).
Calculating with all data, consequently, we concluded that driving force of bacteria is \(10^{-13} N\) by solving the equations.
Film Deformation Modeling
Additional Pressure Produced by Bacteria
Now let's see how to calculate additional pressure caused by bacteria.
To prepare different size of film, we prepare two clips to match film,
This is the schematic program of special film I, used for circle film.
and this is film II for square film.
Film I
Firstly, let's check the geometric size of our special film I. The film is shown in green edge and the clip that used to fix the film presents in black edge. The film I is a circle with the radius(r) of 2cm. The thickness(d) of film is 0.1mm.
Assume the numerical density(σ) of bacteria is ~10000/mm^2, which means a single bacteria occupying the area of approximately ~100 \(\sim 100 \mu m^2\).
Let's assume addition pressure:
\(\Delta P= \frac{\sum F}{S}=\sigma F_{0}\)
Calculating with these data, we concluded additional pressure ΔP is ~0.001Pa.
The wavelength of laser we used is 650nm. Thus deformation is recommended at ~um level.
Stress Analysis
This is the schematic diagram of of deformed film. Assume the radius of film is r, deformation range equals h.
The film is subject to several forces acting, including force f given by bacteria and tensile force F through the film.
Let's get some approximation on this model. As the deformation range(h) is much more smaller than the radius(r) of the film(h< \(f=F\times \frac{h}{r}\)
\(\frac{F}{2\pi rd}=\frac{\Delta r}{r}\times G\)
\(\Delta r=\frac{1}{2}\times r\times (\frac{h}{r})^{2}\)
where f is the resultant force of the bacteria, F is the tensile force in the film, h is the deformation distance, r is the radius of the film,Δr is the variation of the radius, which is r, d is the thickness of the film, G is the Young Modulus. Solving these equations, we are able to get Young Modulus, a critical physical parameters to depicting the characteristic of film candidates. Film II Similarly, we are able to calculate the characteristic of film. The schematic program of film goes like this, The film is shown in black edge and the clip that used to fix the film presents in green edge. The film II is a square with the parameter of film is 2.8cmX2.5cm. The thickness(d) of film is 0.16mm. The equations after stress analysis are, \(f=2F\times \frac{h}{b}\) \(\frac{F}{a c}=\frac{\Delta b}{b}\times G\) \(\Delta b=\frac{1}{2}\times b\times (\frac{h}{b})^{2}\) Then, we are able to get Young Modulus, a critical physical parameters to depicting the characteristic of film candidates. Film Candidates Assume that 1% of bacteria are push ahead statistically, then the additional pressure given by bacteria is ΔP=0.01xσF0. After solving these equations above, we ultimately get the Young modulus of material required G <1GPa in order to get ~um order deformation. There are some common material's Young modulus, Therefore, we selected several materials as possible films: low pressure polyethylene, rubbers, and glass as our material of the film. See our results on film candidates at Results-Film Candidate Just think about it, if every bacteria pull together, we can detect the deformation even the film is made of steel and even bigger than steel.
Material type
Young modulus(GPa)
Gray cast iron
118~126
Carbon steel
206
Roll copper
108
Brass
89~97
Roll aluminium
68
Roll zinc
82
Lead
16
Rubber
0.00008
Polyamides
0.011
High pressure polyethylene
0.015~0.025
Low pressure polyethylene
0.49~0.78
Polypropylene
1.32~1.42
Glass
50
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*Vzdt 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 (KaCVz/(Kd*σ0+KaCVz)) 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:
- Loading the image.
- Calculate a self-adapting or special threshold value in the image binay progress.
- Use mathematical morphology operations.
- Use filtering processing make the image more smooth.
- Delete the small area to reduce the error noises.
- 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 :(Add super link here)
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\propto c^{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
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=C*(A)^n.
When the movement percentage reachs a steady state, that means m=K/(K+k), we can infer that m=C(A)^n/(C(A)^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}
So f=acG(\frac{h}{b})^{3}
Now we can substitute each equations and predict the relationship between deformation 'h' and concentration 'A':
'N' is the total number of sticked bacteria.
Get the reciprocal of the upper equation:
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 ('ΔN') is: ΔN=2h/λ
So replace 'h' with 'ΔN' in equations and we will get:
Define acGλ^3/8Nb^3 as a complex constant 'A0', ackGλ^3/8NCb^3 as another complex constant 'B0', and simplify the formula as below:
So if we measure the two constants 'A0'&'B0', 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.