In order to validate the feasibility of our design of heating the E.coli, we need to figure out what temperature the E.coli could reach when we apply the magnetic field on it.
After the former part of our modeling, we got the approximate expression quantity of ferritin molecules. Thus in this part, we would do further exploration and simulation about whole cell temperature distribution, and we had three steps to achieve our goal:
1.Calculate the heating power of single ferritin molecule and the temperature distribution around one molecule;
2.Calculate the temperature distribution around one molecule;
3.Simulate the whole cell temperature distribution with every molecule moving randomly.
SLP
After a literature review, we were able to calculate the heating power of every single ferritin molecule. And some recent studies [1][2] guided us to acquire ferritin heating formula, the specific loss power (SLP).We used this method to calculate single molecule’s heating power with specific diameter and chose parameter to adjust the magnetic field.
In order to simplify the model, we made some assumptions:
1. The medium around the ferritin would be homogeneous;
2. Ferritins are singly distributed in the medium;
3. Consider the iron core in the protein shell to be a kind of nanoparticle-like material.
Formula
As we assume ferritin molecule to be a nanoparticle-like material, thus we used the formula for magnetic nanoparticle in our model to get the SLP.SLP of magnetic nanoparticles in an electromagnetic field is given by the expression:
It is clear that the SLP is proportional to the square of magnetic field strength. At low frequencies SLP is proportional to the square of frequency, while it saturates at higher frequencies. For frequencies below the ferromagnetic resonance frequency, Brownian and Néel relaxation are the two main physical mechanisms for SLP.
Brownian relaxation is due to the physical rotation of the particle. The relaxation time is given by:
And Néel relaxation is due to the rotation of the magnetization, and its relaxation time is given by:
The anisotropy energy of Fe2O3 core would be calculated with expression given blow:
The two mechanisms operate in parallel, giving the expression for the overall relaxation time :
Except for the theoretical calculation, we analyzed the χ0 which stands for the equilibrium susceptibility with the result of magnetic analysis,and conbined with the formula shown blow.We estimated χ0 to be 1.8^(-4).
With SLP clear in mind, we moved on to the parameter choice and calculation as follow.
Parameters and Calculation
After setting all the parameters, we could initially get the result of SLP. When we set the diameter of ferritin to be 7nm, SLP would reach 8.9*10^(-3) W/g
Then we observed the SLP’s fluctuation with variety of Frequency, Electromagnetic Field Strength and Temperature respectively.
Heat Transmission
Single Molecule Level
To simulate the whole cell heating effect, we started from single molecule temperature change and the sphere around it, to observe whether its temperature can be transferred in cell.
Thus we made some assumptions to get our model clear:
1. Ferritin molecules are monodisperse;
2. Cytoplasm is well-distributed;
3. The coefficient of heat conduction and specific heat capacity of cytoplasm are the same as water.
Firstly, we hypothesized that there were several spheres outside the single ferritin molecule and temperature would be same in the same sphere. We chose heat conduction equation to describe the heat transmission between two adjoined spheres and we got the result shown as follow:
To our disappointed, the simulation result was far from the real condition in cell although we haven’t done any experiments yet. We considered that it failed because of the heat transfer areas were too small for the calculation method to conduct.
Thus we turned to the calculation method for microcosmic environment and got a better result:
Parameters
Whole Cell Level
In this part, we made five hypothesis:
1. All ferritins are randomly distributed at the initial time but there was no ferritin in nucleoid;
2. All ferritins are moving randomly in the whole cell;
3. The coefficient of heat conduction and specific heat capacity of cytoplasm are the same as water;
4. The whole cell is cuboid and divided into 200*100*100 small cuboid, every ferritin molecule would occupy one cuboid at one time;
5. One cuboid cannot contain more than one ferritin molecule at one time.
With these five assumptions above and the result of ferritin molecule quantity from the former modeling part, we were able to simulate the molecules’ stochastic moving and temperature change in cell.
Based on Smoluchowski equation for the molecule diffusion, we used MATLAB to gain the result with the condition: molecules’ diameter is 7nm, the MF strength is 6.4kA/m and the frequency is 540 kHZ.
As we can see from the result above, ferritin molecules distributed randomly could roughly raise cell’s temperature about 2-3 ℃ with in10mins. So we could drew the conclusion that our design of heating cell environment with magnetic field would work well.
Feedback on Wet Lab
After the testing about the temperature change in wet lab, we got disappointed that the temperature was hard to observe. Thus we analyzed all the parameters related to ferritin. Finally we found that the ferritin molecule’s diameter was just around 2.5nm which meant that SLP of ferritin decreased down to 4*10^(-4).
So we did simulation once again to observe the approximate temperature change with our experiment condition. And the results were shown as follow:
Except for the diameter of ferritin molecule, we also noticed that ferritins were not homogeneously distributed in the cell. So the heating effect would be decreased again.
Conclusion
From our simulation and wet lab experiment results, we could drew the conclusion that if the materials could distributed homogeneously in the cell environment, and its heating power could be at least 10^(-2), the temperature of the whole cell would be changed efficiently.
Considering the heat dissipation and environment change during applying magnetic field, we would do more detailed models to explain its mechanism deeply in the future.
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