1)Abstract

It is difficult to apply our system to real condition with optimizing initial yeast amount to targeted salt reduction because of non-obviousness of growth of initial yeast and Na⁺ influx which depends on medium Na⁺ concentration which is changing. So we made model which gives initial value for targeted salt reduction about SseNHXS1 yeast cell which had the best performance in our assay by describing the salt absorption kinetics of yeast cells constitutively. We calculated some parameters of kinetics by our result of assay and we simulated salt reduction expected to grow the initial input yeast. As a result, it turned out to be inefficient for large Na⁺ absorption even when initial yeast amount took the value of upper of growth population size. From this modeling, it is suggested that Need is introducing population which had size exceeding the upper limit by preculture for our design, and model of adding by larger population reproduced the assay results. By using the results of this modeling it is possible to estimate the optimal initial yeast input when using the device of this study.

2)Transporter Kinetics Model

We have to describe influx of AVP1 which is vacuolar PPase on vacuole membrane and NHX1 which is Na⁺/H⁺ Antiporter SseNHXS1 yeast cell which had the best performance in our assay. In this model, these transporter was dealed with state transition model and number of parameters was decreased by Rapid Equilibrium and structural symmetry.[1]

2.1)4 state transition model:NHX1

One substrate Uniporter and Two substrate Antiporter which has assumption of simultaneous binding of internal and external substrates are described by 4 state transition model.(Fig 2.1.1)

Fig 2.1.1: 4 state transition model of Uniporter. $S^o$ is outer medium, $S^i$ is inner medium transporting substrate S. E is Uniporter and transitions between four different states of it are illustrated. When the cycle progresses clockwise, S is transported from outer medium into the cell.

Here, differential equations expressing the time change of each state are

$$ \begin{split} \frac{d[1]}{dt}&=-k^*_{2\leftarrow1}[S^o][1]-k_{4\leftarrow1}[1]+k_{1\leftarrow2}[2]+k_{1\leftarrow4}[4]\\ \frac{d[2]}{dt}&=-k_{1\leftarrow2}[2]-k_{3\leftarrow2}[2]+k^*_{2\leftarrow1}[S^o][1]+k_{2\leftarrow3}[3]\\ \frac{d[3]}{dt}&=-k_{2\leftarrow3}[3]-k_{4\leftarrow3}[3]+k_{3\leftarrow2}[2]+k^*_{3\leftarrow4}[S^i][4]\\ \frac{d[4]}{dt}&=-k_{1\leftarrow4}[4]-k^*_{3\leftarrow4}[S^i][4]+k_{4\leftarrow1}[1]+k_{4\leftarrow3}[3] \end{split} \tag{2.1.1} $$

Here, each $ k_ {i \leftarrow j}, k ^ * _ {i \leftarrow j} $ is the rate constant of the state transition to $ i \leftarrow j $, (subscript $*$ is the first order rate constant),$[1],[2],[3],[4]$are the concentration of each state, $ [S ^ o], [S ^ i] $ are the concentrations of transporting substances of each region. This system can be regarded as Non-equilibrium steady state(For each state X, $d[X]/dt = 0$) finally,Non-equilibrium steady state solution of the concentration of each state ( can be completely solved as a solution to the quaternary linear equation which combines three formulas selected from eq (2.1.1) and conservation law concerning the enzyme concentration. However, the number of parameters is too large to handle, so this time we describe the model by two general approximations. First, in the non-equilibrium steady state, it is assumed that the rate-determining step in the state transition is the structural transformation of the transporter and the equilibrium state is established for the bond with the substrate. That is introducing constant $K_d^o,K_d^i$ about Fig 2.1.1. \[ \begin{split} K_d^o&=\dfrac{[1][S^o]}{[2]}\\ K_d^i&=\dfrac{[3][S^i]}{[4]} \end{split} \tag{2.1.2} \] Second,if we ignore the effect of membrane potential,from the structural symmetry that sandwiches the cell membrane of the transporter, the following equation can be assumed. $$ \begin{split} k_{1\leftarrow4} &= k_{4\leftarrow1} = k_{2\leftarrow3} = k_{3\leftarrow2} = k\\ K_d^o&=K_d^i = K \end{split} \tag{2.1.3} $$ In these assumption, influx of one uniporter was described by two parameters. Non-equilibrium steady state solution is given by coordinate 4 formulas which are eq(2.1.2) and preservation formula of enzyme amount $[1]+[2]+[3]+[4] = E_0$, sum of 2 formulas of eq(2.1.1) in Non-equilibrium steady state, and next equation is given by calculated crock-wise flux of state transition from appropriate two enzymes. $$ {J_{ss}}_4 = \frac{E_{0}\,\mathrm{K}\,k\,\left(\mathrm{[S^o]}-\mathrm{[S^i]}\right)}{2\,\left(\mathrm{K}+\mathrm{[S^i]}\right)\,\left(\mathrm{K}+\mathrm{[S^o]}\right)} \tag{2.1.4} $$

2.2)3 state transition model:AVP1

AVP1 is a transporter that binds to pyrophosphate on the cytoplasmic side, hydrolyzes it with structural transformation, releases hydrogen ions to the vacuolar side, and releases phosphate to the cytoplasmic side. This kinetics is described by 3 state transition model(Fig 2.2.1).

Fig 2.2.1: 3 state transition model of Uniporter.$PP_i$ is pyrophosphoric acid, $P_i$ is phosphoric acid, H⁺ is hydrogen ion.

This model can be dealed with by same way of 4 state transition model, and next flux ${J_{ss}}_3 $ is given. $$ {J_{ss}}_3 = \frac{ E_0 K k^2 ( [PP_i]_{cyt} - [H^+]_{vac} ) }{ ( K + [H^+]_{vac} ) ( 2[PP_i]_{cyt} + K({[P_i]_{cyt}}^2 + 1) ) } \tag{2.2.1} $$ Here, subscripts of various ion concentrations represent intracellular cytosols and vacuoles.

3)SseNHX1 yeast Transporter Kinetics Model

The introduction of AVP1 alone in this assay resulted in an influence on salt absorption performance, so it is considered that the energy conjugate transport system via H⁺ on the endoplasmic vacuolar membrane is also important. So in addition to NHX1 and AVP1 introduced by transformation, endogenous NHX1-like and AVP1-like transport systems were incorporated in this model using the transporter model described above. After all, the model used this time is a simultaneous differential equation and preservation formula of the same form as the following basic equation. $$ \begin{split} [Na^+]_{out} &= [Na^+]_{out}(0)-\frac{N}{V_r-N\overline{V}_{c}}\left([Na^+]_{cyt}(\overline{V}_{c}-\overline{V}_{vac})+\frac{\overline{V}_{vac}}{\overline{V}_{c}-\overline{V}_{vac}}[Na^+]_{vac}\right)\\ \frac{ d[Na^+]_{cyt} }{dt} &= {{J_{ss}}_4}_{If} -\frac{\overline{V}_{vac}}{\overline{V}_{c}-\overline{V}_{vac}}\frac{d[Na^+]_{vac} }{dt} \\ \frac{ d[Na^+]_{vac} }{dt} &= {{{J_{ss}}_4}_{NHX}}_{like}+{{J_{ss}}_4}_{SseNHX}\\ \frac{d[H^+]_{vac} }{dt} &= {{{J_{ss}}_3}_{AVP}}_{like}-\frac{ d[Na^+]_{vac} }{dt}\\ \frac{ dN }{dt} &= f(N) \end{split} \tag{3.1} $$ Here,$\overline{V}_{c}$,$\overline{V}_{vac}$ is average volume of yeast cells and mean vacuolar volume, $V_r$ is volume of system, and $f$ is suitable function to describe cell proliferation.${{J_{ss}}_4}_{If}$is flux of endogenous cell membrane Na⁺ uptake mechanism, and it is described by 4 state transition model. $[H^+]$,$[PP_i]$,$[P_i]$ in cytosol is supposed that constant because of intrinsic compensation mechanism.

4)Handling of model parameters

From the assay results of this study, $k,K$ was determined for ${{J_{ss}}_4}_{NHX}$, and $k$ was determined for ${{J_{ss}}_3}_{AVP}$ as a parameter for use in later simulations, and other parameters were acquired and estimated as summarized in Table 1.

Table1: Parameters used in simulation of our device.

5)Result

When actually using yeast used in this study, how much yeast should be added first is an important issue. From the viewpoint of the growth of yeast,possibility that a sufficient final salt removal amount can be expected by the growth of the initial input yeast,and possibility that a sufficient final salt removal amount can not be expected if the initial input yeast amount is not more than a certain amount are conceivable. We selected the cell growth function $f$ by these two points and analyzed the behavior of the model.

5.1)Salt reduction moodel expected to grow the initial input yeast

For models that take into account the growth of yeast initially introduced, we used a general logistic equation and the specific growth rate including growth inhibition by inhibitors according to Lenski (1986) [3] as $f$. $$ f(N)=\mu_0 e^{-\lambda[Na^+]_{cyt}}N(1-\frac{N}{N_{max}}) \tag{5.1.1} $$ Here, $\mu_0$ is the maximum specific growth rate, $\lambda$ is a parameter representing the inhibitory effect, and $N_{max}$is the maximum population size. Simulation was carried out under the same initial conditions as in the assay, and the relationship between the initial yeast input and medium Na⁺ concentration after a certain period of time was determined.(Fig 5.1.1)

Fig 5.1.1: Prediction of removal amount by growth SseNHX1 yeast

5.2)Salt reduction moodel with increased initial yeast amount

Based on the simple assumption of $f(N)=0$ in the model with the initial input yeast amount raised because of assumption that the upper limit of the population size due to proliferation was simply established by growth inhibition and the influence of death is small under the experimental conditions of this stud. Simulation was carried out under the same initial conditions as in the assay, and the relationship between the initial yeast input and medium Na⁺ concentration after a certain period of time was determined.(Fig 5.2.1)

Fig 5.2.1: Prediction of removal amount by adding SseNHX1 yeast

6)Discussion

It is suggested from the simulation results by salt reduction moodel expected to grow the initial input yeast (Fig 5.1.1), the salt absorption rate per volume defined by the maximum population group size for the purpose of salt absorption is rather inefficient, and it is necessary to populate the population size exceeding the upper limit by preculture when yeast is used in this study. The reduced model obtained by raising the initial input yeast amount constructed in this study is compatible with the assay results and it is considered that the initial yeast input amount when yeast is applied to the actual system can be predicted using this model.

7)Reference

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