Our Model for our project started out with the sole objective of proving through simulation the viability of our idea. After the realization of the possibilities that we could achieve through the tools that we had available we wanted to use our model to be able to make our whole system more efficient. All our modelling was made possible through Mathwork’s Matlab, specifically with the SimBiology toolbox.

To prove if our project was plausible we started out by considering the kinetics of the channels and transporters which we wanted to transform into our organism Saccharomyces cerevisiae. As we had different types of integrated proteins also different types of kinetic mechanism had to be considered. This were specifically: channels, antiporters, transporters and even not characterized efflux und influx from the organism itself. In the following section, we will go into detail how the mechanisms work.

GEF1-Cl^-

The native channel GEF1 was characterized as a chloride voltage-gated channel (Flis et al. 2002) with a conductivity of 42 ± 4 pS. With a high voltage tolerance, the channel remains mostly in the open state. As a channel, a direct Michaelis–Menten kinetic was assumed. The definition of its constants v_max and K_m are defined for the Simulation as it follows:

The equation that represents the chloride current can be taken from Ohm’s law:

where V represents the voltage varying from -50 to – 120 mV (Vacata et al. 1981) G is the conductivity of the channel and I the current in C/s. The current still must be converted into a reaction rate. To this end the charge of chloride was assumed as it’s one valence electron. Through the next equation the rate was then calculated and incorporated into the model:

Here two natural constants describe the relationship between the electron current and the molecular current. N_A represents the Avogadro’s Number, which states the relationship between molecules per mole. e the elementary charge of an electron, in this case Cl^-.

As no other membrane protein was found to be strongly responsible for the intake of Chloride and GEF1 was found to be ubiquitous to all compartments in the cell (López-Rodríguez et al. 2007). The same kinetics of the GEF1 channel were then assumed from the influx of Cl^- although with a different v_max (Roberts et al. 2001) as the surface area is increased and the protein number is reduced.

This lead to the voltage dependant Michaelis-Menten kinetic equation:

K_m is the substrate concentration by which the reaction achieves the half of its maximal reaction rate, while [Cl^- out] is the chloride concentration and v the reaction rate. The value of K_m is 93 ± 32 mM (Roberts et al. 2001).

halleluia

AVP1 – H⁺, GmSULTr1;2 – SO₃^2-, AKT1 – Na⁺

AVP1, GmSULTr1;2 and AKT1 have been characterized as Transporters from H⁺ , SO₃^2- and Na⁺ respectively
(Drozdowicz et al. 2000), (Ding et al. 2016), (Pyo et al. 2010).

Their kinetics can be easily explained through the following model (Beard, Qian 2008; Pradhan et al. 2010):

For the Uniport reaction this can be sketched on the following flow chart:

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The overall reaction can be stated in the following chemical reaction:

For the Uniport reaction this can be sketched on the following flow chart:

Throughout the reaction the enzyme goes through the 4 different states 1 -> 2 -> 3 -> 4 -> 1. On each state, there is a specific concentration of enzymes, this can be represented through es₁, e₂, e₃, e₄. This leads to the following relation in equilibrium state:

To calculate the equilibrium constant K_eq, the free Gibbs Energy formula can be used:

As the educts and products are thermodynamically equal it is feasible to assume that ΔG₀ = 0. As all our uniporters transport one ion per transport process the equation can be simplified with the Reaction Quotient Q to:

In equilibrium is ΔG = 0 and Q = K_eq. Therefore, the final equation for our equilibrium constant can be calculated for cation uniporters through the following equation:

It is due to note that for electroneutral transporters, the equation would equal 1, thus simplifying the kinetic process. We assume that the change of potential is through the inclusion of ions in the vacuole near 0, because the influx of opposite charged ions follows directly.

Using the newly found equilibrium constant equation 4 can be represented through the mass-action rate constants as follows:

Through the assumption of rapid equilibrium, in which both the unbinding and binding of the substrates are rapid thus making the constricting states of the enzymes both e₂ and e₃. This relationship can be expressed through the following equations:

Whereby K_d = k₁₂/k₂₁ = k₂₃/k₃₂ is the dissociation constant from the enzyme at both the binding and unbinding site. Through this assumption equation 8 can be simplified to:

Next, we introduce the assumption that the protein does not undergo any conformational transformations in the process of the transport of the bounded protein. Thus, to fulfil equation 9 k₂₃ = k₁₄ = k₊ and k₃₂ = k₁₄ = k_- must follow. Through the conservation of total protein and the flux quasi-steady approximation:

As we assumed rapid equilibrium it also follows that K_d = K_m therefore we can summarize our flux equation as follows:

Equation 11 is the one which we used in Matlab’s tool symbiology to define our flux rates.

NHX1 - Na⁺/H⁺, AtNHXS1 - Na⁺/H⁺