Model

Goal

The aim of this model is to analyze iron uptake by the introduced Pft ferritin within the statolith. The analysis would demonstrate (i) the expected rate of uptake and (ii) the expected amount of loaded iron at equilibrium (ie. once steady state is reached).

Assumptions

1. Plant iron uptake is limited by diffusion through the ferritin iron channel. No other diffusion-governed transport step is limiting.
• The diffusion constant from the xylem to the cell was calculated to be D = 3.43E-9 m*s-2
• Notably, iron uptake from the roots depends on the soil iron concentration, which is variable. For our experimental conditions, the medium has a high concentration of iron, so we considered this to not be limiting, but this may not be the case in other mediums.
• Statolith has fluid-like behavior: iron that enters the cell should be uptaken quickly (source)
2. Convection is not limiting
• Ion channels mean iron particles must move through a finite amount of pores.
3. The catalytic oxidation reactions of the iron are non-limiting. -> just diffusion
• There are more catalytic sites than ion channels (source)
4. Iron outside ferritin is a constant high concentration
• This assumption is made because the plants are to be grown in a medium with excess iron.
5. Unsteady-state
• We would like to find the time it takes to equilibrate
6. Single-pore diffusion analysis assumes 1D diffusion in the z-direction
• Cylindrical coordinates were used, but cylindrical or cartesian coordinates are both acceptable.

Parameters

1. D (iron to cell) = 3.43 x 10-9 m2/s
This parameter is used to demonstrate that diffusion into the cell is negligible. We know that for a lower diffusion coefficient, the concentration drop in this region is high (since fluxes are equal at the interface by Fick’s First Law).
2. Number of pores = 1
3. Entry channels = 15 angstroms

Equations

Using chemical kinetics equations for Langmuir adsorption adapted for single-pore diffusion,

Total intra-pore sites = Vacant sites + Sum of sites occupied by iron

[Iron]external + Intra-pore site → [Iron]internal

The rate laws are given as first order

Forward rate = k[Iron][Vacant Sites]

Reverse rate = k[Internalized iron]

Using these equations, we can develop steady-state profiles for internalized iron concentration as a concentration of bulk iron concentrations.

Preliminary Results

We see from the following plot that as external iron concentration increases, internalized iron concentration increases. The maximum iron load is approximately 80 M Fe2+ (4000 atoms in an ~8nm diameter spherical cage). Although the iron loaded within the ferritin at any given time has a higher concentration than the bulk, this is somewhat expected since biologically, ferritin serves as an iron storage molecule. The large disparity is due to a limitation in our current preliminary model, where bulk ferritin concentration is considered constant.

We classify this as chemisoprtion. Technically, the overall process includes physisorption for the limiting diffusion step and chemisorption is the negligible redox step that follows.

Next Steps and Improvements

The parameters could be improved through experimental determinations of iron loading rate . Iron uptake and clearance kinetics profiles could indicate whether our system follows the kinetic trend described by the model or not.

The parameters could also be improved through analysis of plants grown in various concentrations of iron: this would yield the equilibrium iron stored within the plant as a function of iron media concentration

Goal

The aim of this model is to analyze the diffusion of essential nutrients and iron into arabidopsis roots from various plant growth mediums. The goal of this model is to compare the transport efficacy of various mediums. Specifically, our goal was to compare diffusion rates from the medium to arabidopsis roots in recyclable mediums, which we learned from our discussion with NASA scientist Dr. Gioia Massa are preferred for plant growth in space.

Assumptions

1. Concentrations of the nutrients are uniform in the medium
2. The composition of the arabidopsis roots are uniform
• Although cell densities at the root tip might vary, affecting the diffusion coefficient, such discrepancies were considered to be negligible. Further, since the surface composition of the roots are made up of cells with extracellular components of the plant roots being further inside the root, the diffusion coefficient was approximated as the diffusion coefficient of the cells, which was measured by Kramer et. al 2.5+/-0.7x10(-12) m(2) s(-1) [2]
3. Convection is negligible
• Since we are assuming a stationary medium, it was assumed that convective flow of nutrients through the medium is negligible and that concentrations of nutrients of interest throughout the medium are uniform
4. The arabidopsis roots can be approximated by a cylinder
• Only one root was analyzed in this model, but the results can be extended for more complex systems. However, because this would make the dimensions very difficult to analyze and cause partial differential equations to arise due to multidimensional diffusion, this simplification to the system was made. This assumption is useful to us because we are less interested in the exact uptake of a specific root arrangement (which varies for each plant and during a single plant’s life) and are instead interested in comparing different mediums.
5. Diffusion is 1-dimensional in the radial direction
• The radial direction is set to be the direction radial to the root. Variation in the z-direction (parallel to the root) and the theta direction were assumed to be negligible.
6. Any surface reactions are negligible
• Since only uptake transport phenomena were analyzed, reactions that would affect nutrient concentrations at the root tip were considered to be negligible. This simplifying assumption is made because there is no common species evolution surface reactions on the plant root.
7. Pressure and gravitation terms are considered to be negligible
• Nutrient transport is not driven by a pressure gradient or gravitational field.
8. Steady-state
• For these concentration profiles, we are assuming a steady-state has been reached, and that the concentration profile is not time-dependent. This is valid for systems where the bulk concentration of nutrients varies by a negligible amount.

Parameters

1. Calculated diffusion coefficients using particle sizes, viscosities

• 

2. Bulk Concentrations (in the medium)
• Ammonium nitrate (NH4NO3) 1650 mg/L
• Calcium chloride (CaCl2 · 2H2O) 440 mg/L
• Magnesium sulfate (MgSO4 · 7H2O) 370 mg/L
• Monopotassium phosphate (KH2PO4) 170 mg/L
• Potassium nitrate (KNO3) 1900 mg/lL
• FeSO4 (We assume that this concentration is significantly higher so the FeCl concentration is negligible): 100 umol/L (152 g/mol)
3. Concentrations (in Plant)
• Root meristem radius: Using ImageJ software and a fluorescence microscopy image of the arabidopsis root tip meristem from Nicol et. al, the radius of the root was determined to be approximately 85 microns. Note that this value varies for each root, so this is an approximation.

Equations

We can calculate concentration profiles using Fick’s Second Law. Ignoring all terms aside from radial diffusion, the concentration profile in each medium is given by:

Ca = Aln(r) + B

At the medium-root interface, we know that the fluxes are equal. By Fick’s First Law:

D*grad(C) = D*grad(C)

Next Steps and Improvements

As in the prior model, experimental work to determine diffusion coefficients would improve our existing model significantly. We could also seek to consider a more diverse set of media and nutrients.

Goal

The aim of this model is to take the ideal parameter for current determined by solving an ODE and find the combination of passives and NPN devices that will work for generation of a constant current in that range. Ideally, the output current of the device should be tunable via a 5 KOhm rheostat.

Assumptions:

1. Transistor nonlinearity and current gain are ideally modeled with CircuitLab’s Gummell-Poon like model
2. Resistor tolerances are negligible in comparison with their values
3. The rheostat can assume a K value anywhere between 0 and 1 (0 ohm and 5K ohm).
4. The load resistance is close enough to 200 mOhm
5. The zener breakdown voltage and current are ideal
6. PCB trace and wire terminal resistances are negligible
7. Parasitic inductances are negligible
8. The ambient and transistor temperatures are close enough to room temperature
9. Capacitor leakage current is negligible in the DC circuit.
10. The influence of power supply noise is negligible.
11. Load inductance does not influence the system during DC performance.
12. The load can be modeled as an ideal resistor.

Parameters:

1. Transistors:
• D44VH10 SPICE model from Maxim
• TIP122 model from CircuitLab
2. Zener:
• 1N4732 model from CircuitLab
3. Zener Voltage Reference:
• Generic Zener Diode in parallel with series resistors. Zener diode reverse voltage was set to the voltage at the node between resistors.
4. Resistors:
• PCB mount
• 30 Ohm
• MCP4642 rheostat modeled as ideal rheostat
• Chassis mount
• Primary coil resistor: 200 mOhm
• Secondary coil resistor: 3.3 Ohm
5. Load
• 200 mOhm resistor

Equations:

Unfortunately, CircuitLab is closed-source software but does state that it uses “SPICE-like component models” to give “accurate results for nonlinear circuit effects”. This is most important for modeling of the transistors, where nonlinear behavior is responsible for changes in gain current. SPICE uses a Gummel-Poon model to more accurately model changes in transistor gain over different base currents.

Results:

Initially, an adjustable Zener voltage reference was modeled as a zener diode with a set reverse voltage and a resistor pair in parallel as described in Texas Instruments’ typical application schematic for the LM4041 adjustable Zener voltage reference. A single NPN power transistor (D44VH) was used to control the output current.

Simulations were performed with a 200 mOhm load as an overestimate in order to supply the requisite 10A.

The circuit utilizing the Zener voltage reference was a valid result, but was very susceptible to changes in the load resistance and had rigid requirements for passive component values to prevent overcurrenting the Zener diode. Furthermore, due to the low-resistance series resistors lots of power was lost as heat in the constant current generation side of the circuit.

A Zener diode was chosen to replace the Zener voltage reference, and the resistor pair in parallel with it was removed. Less power was lost as heat when an NPN darlington was used to drive the D44VH. This means that the overall gain is very significantly amplified, and even with small currents of 10 mA through the MCP4642 we can modulate the output current between:

• Primary coil: 7.6 to 11.4 A
• Secondary coil: 1.038 to 1.079A

The circuits for the primary and secondary coil are identical except that the desired output current is set by a chassis mount resistor.

The resistor values are as follows:

• Primary coil: 220 mOhm
• Secondary coil: 3.3 Ohm

The result of these changes is that the circuit is much more efficient (it takes advantage of the additional gain provided by a darlington pair) and more adjustable. The only device that has to be changed to grossly alter the coil current is the chassis mount resistor, and lots of fine adjustment is achievable by altering the tap fraction of the MCP4642. We went on to perform DC sweep simulations for this tap fraction (between 0 and 1) and plotted the resulting load current for the primary and secondary coils.

Circuit for constant current driver with 220 mOhm chassis resistor (R4) installed.

Dependency of load current (Amperes) on tap fraction of the MCP4642 (R10.K) for a 220 mOhm chassis resistor. The relationship is approximately linear (y = 3.7646x + 6.8596, R2 = 0.9848) and the current can vary between 7.117 to 10.928 A. This provides evidence that our revised circuit could meet the calculated theoretical current (9.87 A) and can still be adjusted to a wider range (+ 1.058A, -2.753A). This would allow us to compensate for real-world inaccuracies when building the constant current driver and still achieve the desired 9.87A.

Circuit for constant current driver with 3.3 Ohm chassis resistor (R4) installed.

Dependency of load current (Amperes) on tap fraction of the MCP4642 (R10.K) for a 220 mOhm chassis resistor. The relationship is still approximately linear (y = 0.0146x + 1.0347, R2 = 0.9723) but a second order polynomial fit the data better (y = 0.009x2 + 0.0056x + 1.0362, R² = 0.9974). This displays that for lower currents about 1A, the constant current driver’s rheostat-based current adjustment may not be able to adjust the current quite effectively. For our application, the accuracy and modulability of the current used to drive the secondary coil is less consequential than the primary. However, a separate selection of transistors may be desired for more accurate low-current operation in other projects.

Goal

The goal of this model is to get a better understanding of the mechanisms of iron loading in ferritin, as well as to estimate values such as diffusion constants. We intend to use molecular simulation packages such as Pyrosetta along with tools such as AlphaFold2. This model is still in very early stages as these simulations are very resource and time intensive.

Assumptions

We intend to continue developing this model with the following assumptions:

1. Mutated ferritins will behave similarly to existing ferritin
2. Ferritin behaves similarly to other proteins
This is an important assumption to make, as it allows us to use machine-learning based tools such as Alphafold2
3. All ferritin subunits behave similar to similar subunits (i.e., one ion channel has the same behavior as other ion channels for the same ferritin
We must make this assumption as it is too computationally demanding to simulate the entirety of ferritin
4. Solvent effects are negligable
We do not intend on including solvents in molecular models, as it would also increase the computational cost of the model beyond reasonable limits

Parameters and Equations

There are no explicit equations we are directly solving. We intend to use the pre-existing packages of pyRosetta, which are primarily Monte Carlo in nature, and AlphaFold2, which is a machine-learning algorithm

For the purpose of calculating diffusion through ferritin's ion channels, we intend on simplifying the system to a single ion-channel and it's immediate surroundings. Simulating the entire ferritin is unfeasible due to it's molecular size.

Results

As this model is still in early development, we do not have significant results beyond evidence that different ferritin subunits are able to dock with one another

The following images are acquired using the OVITO software