Modeling

Magnetosome can be used in different perspectives, thus a great potential to apply in multiple instances. Three applications has been modeled as follows:

(1) Protein Extraction

Magnetosome can be used in microscopic point of view. We tried to model the efficiency to bind with (a) different proteins and (b) use GFP-nanobody for immunoprecipitation. The main purpose of this modelling is to stimulate the binding dynamics of a fixed concentration of magnetosome and GFP-nanobody in different initial concentration of antigens.

Various conditions and parameters:

Fixed Quantity	Quantity
Volume of the mixture	1000 μl

Parameters	Quantity
Molar Mass of Magnetosome	6.89 × 108
Number of GFP-nanobody per Magnetosome	362
Association Rate Constant (kon; Marta H. Kubala†, 2010)	8.84 × 104 M-1 s-1
Dissociation Rate Constant (koff; Marta H. Kubala†, 2010)	1.24 × 10-4 s-1

Condition	Quantity
Amount of Magnetosome	1.5 mg
Weight of GFP-nanobody	Negligible
Initial Molarity of Antigen (GFP)	Varying from 0 to 1.6 μM
Initial Amount of GFP:GFP-nanobody complex	0

First, the molarity of magnetosomes is calculated since the amount of magnetosome and its molecular weight are known,

Molarity of Magnetosome = (1.5 mg / 6.89 × 10⁸ g) / (1 ml) = 2.18 nM

There are 362 GFP-nanobody per each magnetosome, so the molarity of GFP-nanobody is:

Molarity of GFP-nanobody = 2.18 nM × 362 = 7.78 × 10^-7 M

After that, a software called Simbiology in MATLAB is used to model and stimulate the dynamics of the association and dissociation between the molecules. By constructing a model about the mathematical relationship between molecules, reaction process can be stimulated.

Figure 1: Binding activity

For Forward Reaction (Association) rate:

k_on × [GFP-nanobody] × [Antigen]

For Reverse Reaction (Dissociation) rate:

k_off × [GFP-nanobody-antigen Complex]

Net Reaction Rate:

k_on × [GFP-nanobody] × [Antigen] − k_off × [GFP-nanobody-antigen Complex]

Note: k_on, k_off are the reaction rate constants described in the parameters table above.

By using SimBiology, we stimulated the dynamic of the system with the initial concentration of antigen from 0 to 1.6 μM with an interval of 0.2 μM.

Figure 2

From Figure 2, we can see that when the molarity of antigen below that of GFP-nanobody (7.78 × 10^-7 M), it becomes the limiting reagent, and the final molarity of the nanobody-antigen complex equals the initial molarity of antigen, vice versa.

Another observation is that, as the molarity of antigen increase, the reaction (i.e. the formation of nanobody-antigen complex) goes equilibrium more quickly. This can be explained by the increased forward reaction rate, which depends on the molarity of GFP-nanobody and antigen as well.

(2) Microbial Fuel Cell

In a microbial fuel cell (MFC), chemical energy is transformed into electrical energy through a cascade of electrochemical reaction. The mutated nitrogenase in Azotobacter will produce hydrogen gas by the side reaction and break down into hydrogen ion due to the existence of hydrogenase. Alternatively, the electrons can be transferred to the electrode via the oxidized mediator molecules. By using magnetosome, the distance between the bacteria and electrode can be decreased, thus reduces the diffusion distance of the oxidized mediator. This approach would increase the current density and efficiency of electricity generation in MFC.

In this model, current density distribution in hydrogen-oxygen fuel cell was studied. It included the fuel coupling between the mass balances at the anode and cathode; the momentum balances in the gas channel; the gas flow in the porous electrodes; the balance of the ionic current carried by the mediator; and an electronic current balance.

Figure 3

The fuel cell in the cathode and anode is counter-flow and it shows that the hydrogen-rich anode gas is entering from the left. The electrochemical reaction in the cell are give below:

Anode: H₂ + 2 e⁻ → 2 H⁺

Cathode: 1/2 O₂ + 2 e⁻ → O^2-

This model includes different processes:

• Electronic charge balance (Ohm’s law)

• Ionic charge balance (Ohm’s law)

• Butler-Volmer charge transfer kinetics

• Flow distribution in gas channels (Navier-Stokes)

• Flow in the porous GDEs (Brinkman equations)

• Mass balances in gas phase in both gas channels and porous electrodes (Maxwell-Stefan Diffusion and Convection)

Assuming the Butler-Volmer charge transfer kinetics describes the charge transfer current density and the first electron transfer is used to be rate determining step at the anode, hydrogen is oxidized to form hydrogen ion.

i₀,a = the anode exchange current density (A m^-2)

c_h2 is the molar concentration of hydrogen

c_h+ is the molar concentration of water

c_t the total concentration of species (mol m^-3)

c_h2,ref and c_h+,ref is the reference concentrations (mol m^-3)

F is Faraday’s constant (C mol^-1)

R is the gas constant (J mol^-1 K^-1))

T is the temperature (K)

η is the overvoltage (V)

For the cathode:

At the anode’s inlet boundary, the potential is fixed at a reference potential of zero. At the cathode’s inlet boundary, set the potential to the cell voltage, V_cell. The latter is given by

where V_pol is the polarization.

In this model, Δ φ_eq,c = 1 V and Δ φ_eq,a = 0 V ,and the fuel cell over the range 0.2 ≤ V_cell ≤ 0.95 V is simulated by using V_pol in the range 0.05 V through 0.8 V as the parameter for the parametric solver.

Results: The following figure shows the hydrogen mole fraction in the anode at a cell polarization of 0.5 V

The following figure shows the oxygen mole fraction in the cathode:

For the following figure, it shows power output as a function of cell voltage. The maximum power-output for this unit cell is about 940 W m^-2