Team:USTC/Model/1
Abstract
It has been discovered that there are too many kinds of proteins which can transfer electrons in the membrane of E.coli, including the homologues of MtrCAB generated by our gene circuits and other intrinsic protein. To simplify the process of electron transfer from outer membrane to cytosol and prove the efficiency of our production, that is, MtrCAB conduits. We substitute resistances for MtrCAB and other transferring protein, which are located on the outer membrane, in the periplasm or across the internal membrane. Electrons can only transfer through the resistances. In the final achievement of this model, we can get the exact value of each kind of resistance and describe how much flux of electron conducted by MtrCAB conduits. In the end, our model has also provided a bright new method to detect the concentration of MtrCAB protein conveniently.
Description
If we simplify the protein as the resistance in some way, we can see different kinds of protein transferring the electrons one by one, which represent resistances with different value connected in series. Similarly, the same kind of protein is simplified by resistances with the same value connected in parallel. Take MtrA as an example. thousands of MtrA located in the periplasm and each electron will go through the MtrA only once. So the close analogy with resistances in parallel can be made.
We can simplify the protein conduits:
Fig. 1 | The ideal path of electrons transfer
It is substituted by electric circuit:
Fig. 2 | The electric circuit model of electrons transfer process.(We regarded every protein as a resistance)
The meanings of resistance is described as below:
Limited by the experimental condition, we cannot get the exact indispensable data from our own E.coli and MtrCAB system. We have to apply our model to the paper of Daniel E.Ross et al[1].Although the experiments are accomplished in Shewanella, but the resistance of the protein is supposed to be the same because our DNA part virtually originates from Shewanella. So we can apply the data bravely and check the identity of MtrCAB system in the same time. The data from that paper is shown as Mtr current value(excel).
Assumption
Part 1: Circuit building
Assumption 1.1: Protein can be regarded as resistance.
There are a lot of identities between MtrCAB system and electric circuit, that is the motivation for us to build this model. However the feasibility of replacement is still a hypothesis.
Assumption 1.2: The voltage of the same kind of protein is equal.
We assume the electron flux very in cell fluid, resulting in the isotropy of electron concentration to one cell. With this assumption, same resistances can be connected in parallel.
Assumption 1.3: Proteins of one kind are uniform in structure.
Due to the same structure, the resistances of the protein is the same. Suppose the number of MtrA in one cell is N0, and the value of single MtrA resistance is R0, according to the formula of resistance in parallel, all the resistance of MtrA is:
In that way, we can count for the same kind of protein in one resistance.
Part 2: Equation solving
Though the model is simplified, there are too many variables to solve the equation. To find out extra conditions, we have to make more assumptions.
Assumption 2.1: MtrC always connect to MtrB.
From the literature[2],we know that MtrB is essential to the stability of MtrC because MtrC is attached to MtrB by lipid. So that we can connect MtrC and MtrB into one resistance. Thus our varible is cut down.
Assumption 2.2: The resitivity, area density and the cross section area of the protein of one kind, including their homologues, are the same.
This assumption is based on the fact that protein MtrC, MtrA and CymA all consist of peptides and hemes, which are very critical to electron transfer. Additionally, the density of hemes in these protein is almost the same(Number of MtrC heme=10, MtrA heme=10, CymA heme=4). Cross sections of MtrCAB system are almost the same according to this illustration of size information[3]:
Fig. 3 | Precise size and position of every protein.
Assumption 2.3: MtrB has no resistance.
In fact, there is a pore in the middle of MtrB so that electron and get it through freely. We suppose that the pore has no resistance,that means the resistivity of the pore is zero.
With assumption 2, 3 we can deduce that the resistance of MtrA, MtrC, CymA including their homologues, is in proportion to the length of the protein. According to the measurement in the figure above, it is 9:8.5:1.3. Suppose the proportionality coefficient is k, we have:
Among them:
Equation and result
As we only care about the efficiency of MtrCAB system, getting the proportion among all the protein is enough. Thus we'll suppose the voltage casually as V0=1mV.
Here are the KVL equations in different strains of Shewanella.
Solve those equations with current value(excel), so far we have:
Here, the proportionality coefficient k is remained, that means, it can only be measured in the experiment. In fact, the physical meaning of k is clearly defined by the expression of resistance:
So that:
ρ represents the resistivity of protein and S is the cross section area. They are all measurable and available.
Data Processing
According to the circuit model introduced before, MtrCAB protein can be regarded as resistance in parallel or in series .In the sake of applying the model in our system, we test the electricity of our E.coli to estimate the efficiency of our system.
Here is the current measured in average in our system:
Engineered cell refers to our E.coli with plasmid transformed. We can see that WT conduct the current easier than the engineered E.coli because of metabolic pressure. Now we can abstract our system as electric circuit model.
Fig. 4 | The modified e-cricuit model of WT.
Resistance replaces the protein listed below:
In the same way, we have KVL equation:
Then we get the solution:
The efficiency is:
Conclusion
Apparently, this number is far less than MtrCAB system working in Shewanella. Electrons transferred via our system does not account for the majority, which represents a great potential to perfect the our Mtr system.
There is another possibility existing that E.coli can not adopt Mtr system as naturally as Shewanella , There may have some other mechanism shaping MtrCAB protein remained to explore.
Reference
- Jensen, H. M., et al. (2010). "Engineering of a synthetic electron conduit in living cells." Proceedings of the National Academy of Sciences of the United States of America 107(45): 19213-19218.
- Goldbeck, C. P., et al. (2013). "Tuning Promoter Strengths for Improved Synthesis and Function of Electron Conduits in Escherichia coli." Acs Synthetic Biology 2(3): 150-159.
