Difference between revisions of "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,

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

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.

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:

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.

This part explain 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

In the pre-experiment(method shown in annex), we use reflector and film get interference fringes and catch picture like this

That's a typical newton's rings interference.

Modeling method

Consider 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 approximate paraxial spherical.

The area of CCD camera is small(~cm x cm), so the interference is approximate paraxial spherical as well.

In perfect situation, light path sketch shown below.(Light shows in blue line is the reflected light from holophote, light shows in red is the reflected light from film.)

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

Because the virtual image of holophote and the film can not set strict parallel in actual situation.

The light path sketch changes to this(Light shows in blue line is the reflected light from holophote, light shows in red is the reflected light from film.)

We could use the method of coordinate transformation to simplify them like that(Light shows in blue line is the reflected light from holophote, light shows in red is the reflected light from film.)

With the parameters shown in table

r	h	a	θ
0.02m	5e-6m	0.02m	5e-4rad

'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.

Simulate interference fringe result shown below

That just looks like the row image we got before!

Fringe analysis

Method

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

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

3.Choose two point in multi-image, the point must 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.

More details on our coding using Matlab please refer to 2015 USTC in Github.

Annex

Pre-experiment method

Optical path in pre-experiment shown below

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

The wave length of our laser is 650nm.

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

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