Team:USTC/Model/3
Introduction
In order to quantitatively analyze the electron transfer rate, a model based on the simple symmetric random walk is employed, in which the mtrA, mtrC and CymA proteins are regarded as particles on an infinite cubic lattice. It is assumed that electron transfer happens where proteins of all three kinds are in same position. To run the simulation, this system is initialized with protein positions generated randomly in a uniformly distributed manner with density measured in wet experiment. The interactions between proteins are ignored including physical collision and electrostatic force, and proteins are considered to move with equal probabilities to any one of its immediate neighbors in an arbitrary time step. The total number of times of these proteins overlapping is counted after each simulation and an average is calculated to help us analyze the electron transfer rate.
What were we modeling:
To avoid complex kinetics and dynamics process, we used Monte Carlo simulation to achieve our goal. It has been proved that Monte Carlo simulation can actually describe the Brownian motion of the motion of proteins[1], [2].
At the beginning, we consider a simple simulation of only one particle in an infinite plane to verify the feasibility of our simulation. This whole process is a Markov process. The coordinate (x,y) of the particle in every step constituted two Markov chains, one for x, the other for y. A dynamic result of a random moving simulation shown in Fig.1 below.
Fig.1 | illustration of random walking particle model
To some extent, this simulation lets us to know more about the truth of the motion of protein. So we consider this simulation can be used in our model.
In practice, we use a local coordinate on the surface of a bacteria. In this coordinate, space-time is a Minkowski Space which is a flat space-time. In this space, firstly we can see that functional MtrC is binding on MtrB which is binding on the extracellular membrane of E.coli,shown in Fig.2 (a). Because of this structure, MtrB can't move into the periplasm. We regarded MtrC can just move on a 2-D plane. As for MtrA, MtrA moves in the periplasm. Because the thickness of periplasm is very smaller than the width of the membrane, we can regard that MtrA also moves in a flat plane. At last, CymA is a globulin that binding on the intracellular membranes. Just like MtrB, we also think that the CymA moves on a flat plane.
Fig.2 (b) shows the motion of these proteins after approximation. Every protein just moves in a flat plane. Yellow and purple plane in Fig.2 (b) correspond to OM(outer membrane) and IM(inner membrane) respectively in Fig.2 (a).
a)Geometric dimensions of Mtr on E.coli[3]
b)Simplified model in our simulation
Fig.2 | illustration of random walking of MtrC, MtrA and CymA
By using MATLAB code to simulate this electron transfer process, dynamic result shown in Fig.3 below.
Fig.3 | Dynamic simulation result of random walking and electron transfer
In order to simulate the real process of the transformation of an electron, we ignore the time which the chemical reaction cost. The Brownian motion take most of the time of the whole process. So we considered that if a MtrA, MtrC, and CymA move like a line, the electron can be transferred into bacteria. Based on this simulation, we assume the current I that is a function of time t. So we get a curve of current-time and calculate the average current of each E.coli membrane.
Results:
Fortunately, we found some parameters of this model below. Fig.4 shows the E.coli concentration conversion to OD600, and the concentration of protein on membrane shown in Table 1. Geometric dimensions of protein we simulate shown in Fig.1 (a) before.
Table 1 | Concentrations of heme c, CymA, and MtrA
Fig. 2(a) | A scaled schematic of the electron conduit on E.coli
Fig. 4 | Relationship between OD600 and the number of E.coli[5]
We considered proteins’ distribution is homogenized. Once we get OD600 of our bacteria solution, we can get the number of bacteria and the number of proteins on every bacteria membrane. In this Fig.4, we can use these parameters to estimate the effective reaction area of these proteins to get a useful judgment to know whether the electrons are transferred into bacteria. By using all these constant and parameters before, we finally simulate the stable electron transfer rate or current on the membrane of E.coli, the current-time curve shown in Fig.5 below.
Fig.5 | The current-time curve of our simulation about our system
At last, we get a stable current-time diagram which means our system is feasible. This diagram also tells us that our simulation is useful to roughly describe the whole process of electron transfer.
Reference:
- Saffman, P. G., & Delbrück, M. (1975). Brownian motion in biological membranes. Proceedings of the National Academy of Sciences, 72(8), 3111-3113.
- Sonnleitner, A., Schütz, G. J., & Schmidt, T. (1999). Free Brownian motion of individual lipid molecules in biomembranes. Biophysical Journal, 77(5), 2638-2642.
- 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.
- http://www.genomics.agilent.com/biocalculators/calcODBacterial.jsp
