Difference between revisions of "Team:USTC/Modeling"

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 bacteria

Assume the force of single bacteria is \(F_{0}\).

When bacteria move without outer condition with the speed of \(V_{1}\),

\(F_{0}=f_{1}=kV_{1}\)

When bacteria drag by gravity in solution,

\(mg-\rho Vg=f_{2}=kV_{2}\)

Because the motor ability of each bacteria does not change.

According to the data in literature, the speed of movement (\(V_{1}\))is about ~10\(\mu m\)/s, the speed of sedimentation (\(V_{1}\)) ~\(\mu m\)/s, the size of bacteria ~\(\mu m\).

So we could solve the equations and get \(F_{0}\sim 10^{-13} N\).

Modeling of deformation

The geometric size of film

The film is a circle with the radius(r) of 2cm.

(The film shows in green edge. The clip that used to fix the film shows in yellow edge.)

The thickness(d) of film is 0.1mm.

Assume the numerical density(σ) of bacteria is \(\sim 10000/mm^2\).(That means a single bacteria occupying ~100 sq.\(\mu m\).)

Addition pressure: \(\Delta P= \sum \frac{F}{S}=\sigma F_{0}\)

So the addition pressure ΔP is ~0.001Pa.

The wave length we used is 650nm. We need ~ \(\mu m\)deformation.

As the deformation range(h) is much more smaller than the radius(r) of the film, so we can get equations through mechanical equilibrium and geometry constraint:

\(f=F\times \frac{h}{r}\)

\(\frac{F}{2\pi rd}=\frac{\Delta r}{r}\times G\)

\(\Delta r=\frac{1}{2}\times r\times (\frac{h}{r})^{2}\)

('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(r), 'd' is the thickness of the film. )

Material requests

Assume that 1% of bacteria are push ahead statistically.

Then \(\Delta P=0.01 \times \sigma F_{0}\), and solve these equations.

Thus we require the Young modulus of material 'G' <1GPa 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

We choose low pressure polyethylene as our material of the film.

Just think about it, if every bacteria pull together, we can detect the deformation even the film is made of steel.

This part analyse the data of adhesion experiment, make 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. And give some irreplaceable constructive opinions and key indicators on the operation of detecting the concentration of antibiotics.

Modeling

Variable List
[C]: Concentration of bacteria.(\(m^{3}\))
S: Area of the place we consider.(\(m^{2}\))
V: Average swiming speed of bacteria.(m/s)
Vz: Average swiming speed component in the z axis(perpendicular to S).(m/s)
σ: Density of the cohered bacteria.(\(m^{2}\))
N: Total number of sticked bacteria.
m: Movement percentage.(%)
M: Movement number of bacteria.

Adhesion modeling

Assuming that the velocity of bacteria in any direction is the same (V).

And we believe that the bacteria has very less contact with each other when they swim, so we could consider their movement is free.

Then we can get 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 in during the interval 'dt', in area 'dS', those bacteria in tiny volume \('dS\times V_{z}dt'\)(with the amount 'dN' ) will hit the wall(S).

So we can know that

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

Because the amount of bacteria in solution is much more lager than it on the glass surface(S). So the concentration of bacteria (C) remains unchanged during the whole time.

So the hit-wall-bacteria number is stable, but the surface can only adhere one layer of bacteria, and the area that already adhere bacteria can not stick more bacteria. That means we could use Langmuir adsorption isotherm to solve this problem!

Consider a current area ('S'), the density of bacteria on surface is 'σ', and during the interval 'dt', the change of σ is 'dσ'.

Then

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

'Ka' is the success adhere rate of each hit, 'Kd' is the drop rate of the adhered bacteria.

Solve this ODEs and get the equation shows below

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

In fact we can't start to record the image data as soon as we put the bacteria on the cover glass, so there is a time delay in the real situation equation. And make '\(\frac{K_{a}CV_{z}}{K_{d}\sigma _{0}+K^{a}CV_{z}}\)'.That means

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

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

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

Simulation

And we could show some simulation results.

With the constant value:

\(K_{a}\)	\(K_{d}\)	C	\(V_{z}\)	\(\sigma _{0}\)
0.5	0.01	10^9/m^3	5um/s	10^10/m^2

That's very similar to the real data.

Image recognition program

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

Programming method:
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.

Results analysis

Fitting result

Reference the experiment data.

Use MATLAB simulate these data with the function '\(f(x)=a^{-b\times x}+c\)'.

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

Fitting result:

Constants value and details:

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 and details:

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

Fitting result:

Constants value and details:

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

Fitting result:

Constants value and details:

All these perfect fitting result shows that our hypothesis of adhesion mechanism, modeling and image analysis program is just fit the truth.

Movement percentage

We know that bacteria can move straight because its flagellum can contrarotate. But due to the stickiness of PLL, some flagellum may be sticked when they spin. Assuming that the rate of stick (P) always the same all the time. So the the movement percentage will present a exponential form.

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

\(m=m_{0}^{-kt}\)

Fitting function: \(M=a^{-bt}\)

PS: the data was fixed by the previous analysis result \(t_{0}\).

PAO1-PLL-0

Fitting result

Constant value and details:

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

Result analysis

According to the Fitting result and fitting equation, we could get some useful information such as "Adhesion ability", "Starting time (\(t_{0}\))"

Starting time (\(t_{0}\))

Because we can not start record the image data as soon as we drop the bacteria solution on the cover glass, so there is a starting time in the equation. According our model, we know that:

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

Substitution this function into fitting result, we can get the starting time of each test. Results shows below:

	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 "starting time" through our data analysis, that's a big deal.

Adhesion ability

Another interesting and important properties we can get through our data analysis is the adhesion ability of the bacteria solution to cover glass. I'll explain why I called it "the adhesion ability of the bacteria solution to cover glass" later.

According to the fitting results and modeling equation, the derivative of the fitting function at the time point zero is the maximum bacteria number growth rate. So I define this derivative value as the adhesion ability of the bacteria solution to cover glass, or call it "AA" for short.

Refer to the modeling result, we know that:

\(AA=\frac{d\sigma }{dt}\mid _{t=0}=K_{a}CV_{z}=c\times b\)

c&b is the constant value in fitting result.

The AA relates to \(K_{a}\), the adhesion rate of bacteria on every hit, C, the concentration of bacteria solution,\(V_{z}\) , the average swim speed of the bacteria. So that is why I call it the adhesion ability of bacteria solution to cover glass.

The AA of HCB1 shown in table:

condition	HCB1-PLL(+)-0	HCB1-PLL(+)-0.1	HCB1-PLL(+)-0.5	HCB1-PLL(+)-1
AA	2.01	6.69	12.64	8.23

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

We can know that the concentration of antibiotics doesn't effect on AA, so we could use the same type of bacteria in different antibiotics solution.

Experiment guidance

In "antibiotics concentration detection experiment" we need to know film-coating time, bacteria-film interaction time(Ti), concentration of the bacteria solution, and observation time. All of these can be known through the pre-test result 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 not administrate antibiotics. So we need to balance the total number of bacteria and motility.

Use test "PAO1-PLL-0" data as sample to analyse the best time of bacteria-film interactintime.

The best interaction time is the time that the number of movement bacteria reach the maximum value.

\(M=S\sigma m\)

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

We can give the simulation result:

Thus we can recommend the Bacteria-film intraction time(Ti)~100s. That means since you inoculate the bacteria about 100s, you should put it into your water sample to test its antibiotics concentration.

Observation time

If we want to observe the deformation of the film, the bacteria's reaction must reach a stable stage. Assuming that K(%) of bacteria that not act at first start to act when we administrate antibiotics.

The movement percentage differential equation change to this:

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

solve this differential equation get the m~t function:

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

Make \(m_{0}\)=0.5, choose different 'K' can get different simulate curve.(According to previous analysis, 'k'=0.0065.)

That's very similar to our raw data:

If we want to limit the error probility under 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\(\frac{K}{K+k}\), t~100s.

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). And we diluted bacteria solution 50 times.

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.