# Introduction

One application of the project would be the direct use in wastewater treatment plants with bioreactors. In order to better visualize the process, we have developed a model for the flow. This consists of the kinetic reactions in the reactor and the flows of wastewater and phosphate out of the reactor. Subsequently, we have also developed a program to optimize the optimal flow rate and the ideal time to turn off the light.

# General Procedure in the Bioreactor

We have a bioreactor with the modified phosphate binding proteins (PBP) separated from the effluent by a membrane. Wastewater containing phosphate runs into the reactor at the flow rate F1. There, the PBPs bind the phosphate and thus separate it from the wastewater residues. The phosphate and the wastewater can pass the membrane and the liquid is homogeneous due to the use of the luminous stirrer. Finally, the wastewater flows back out of the reactor. When all the wastewater has left the reactor, the reactor is filled with fresh media and the light in the stirrer is turned off. Now phosphate dissolves from the PBP through the light-dependent switch and is released into the solution. The reactor is then emptied with flow rate F2 and the phosphate is recovered from the solution and further processed to polyphosphate. After the regeneration time of the PBP, the reactor is ready for reuse and the process can be repeated.

# Bioreactor Animation

**Figure 1**: Animation of the schematic visualization of the process.

First, the wastewater that transports phosphate is filled in the reactor with flowrate F1, or flows through the reactor continuely. While the light is on, the PBP binds the phosphate and therefore seperates it from the wastewater particles. Since they are retained by a membrane, the PBPs cannot leave the reactor. It can happen that wastewater particles stuck at the membrane, but this can be avoided by cleaning the reactor occasionally.

When most of the PBPs have bounded some phosphate, the inflow is stopped and the remaining wastewater leaves the reactor. Now we fill it with a fresh media and switch off the light inside the reactor. The phosphate is then released into the media and leaves the reactor with flow rate F2. From this solution you can extract the phosphate and process it to polyphosphate.

The light in the reactor is then activated again and after some time the PBPs can be used again.

# Kinetics

For the processes in the reactor, independent of the inflow and outflow rates, we established a kinetic reaction model and derived a system of differential equations (ODEs).

**Figure 2**: Kinetic model of the phosphate binding protein to bind and release phosphate.

Here k1 indicates the rate of how fast phosphate is bound by the modified PBP. k2 gives the information about the change from active PBP to inactive PBP induced by light and k3 stands for the phosphate release rate. The regeneration of the inactive PBP back to the active state is shown by k4. Finally the back reactions k−1, k−2 and k−4 depend on the strength of the forward reactions. The back reaction of k3 is not needed, since inactive PBP is not able to bind phosphate.

The black and orange arrows indicate reactions that happen in light or in darkness, and are later modeled by a binary switch.

First, some notation:

$PhA:=PhA(t)$ Phosphate in Wastewater at Time t

$Ph:=Ph(t)$ Released Phosphate at Time t

$Ep:=Ep(t)$ Active PBP at Time t

$Epp:=Epp(t)$ Active PBP with bounded Phosphate at Time t

$Enp:=Enp(t)$ Inactive PBP with bounded Phosphate at Time t

$En:=En(t)$ Inactive PBP at Time t

$L:=L(t)$ Light at Time t

By using Michaelis-Menten kinetics we can write reactions in a more compact form, where v1,…,v−4 denote the speed of the reaction.

$$\frac{\partial}{\partial t} \begin{pmatrix} PhA \ Ph \ Ep \ En \ Epp \ Enp \end{pmatrix}= \begin{pmatrix} -L & L & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & (1-L) & 0 & 0 \\ -L& L& 0& 0& 0& L & -(1-L) \\ 0& 0& 0& 0& (1-L)& -L & (1-L) \\

L& -L& -(1-L)& L&0 &0 &0 \\ 0&0 & (1-L)& -L&-(1-L) & 0 &0 \end{pmatrix} \cdot \begin{pmatrix} v_{1} \ v_{-1} \ v_{2} \ v_{-2} \ v_3 \ v_4 \ v_{-4} \end{pmatrix} $$

Now the speed $v_i$ can be specified with the rate of change of concentration $k_i$.

A reaction $$S_1 + S_2 \overset{k_1}{\rightarrow} P$$ is written as:

$$\frac{\partial P}{\partial t} = v = k_1\cdot S_1 \cdot S_2 $$

In conclusion we get:

$$\frac{\partial}{\partial t} \begin{pmatrix} PhA \ Ph \ Ep \ En \ Epp \ Enp \end{pmatrix}= \begin{pmatrix} -L*k_1*Ep*PhA & L*k_{-1}*Epp & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & k_3*Enp*(1-L) & 0 & 0 \\ -L*k_1*PhA*Ep & L*k_{-1}*Epp & 0 & 0 & 0 & L*k_4*En & -(1-L)*k_{-4}*Ep \\ 0 & 0 & 0 & 0 & k_3*Enp*(1-L) & -L*k_4*En & (1-L)*k_{-4}*Ep \\ L*k_1*Ep*PhA & -L*k_{-1}*Epp & -(1-L)*k_2*Epp & L*k_{-2}*Enp & 0 & 0 & 0 \\ 0 & 0 & (1-L)*k_2*Epp & -L*k_{-2}*Enp & -k_3*Enp*(1-L) & 0 & 0 \end{pmatrix} $$

Therefore, we obtain a system of ordinary differential equations.

$$\partial PhA=-L\cdot k1 \cdot Ep \cdot PhA+L\cdot k1n\cdot Epp $$

$$\partial Ph=k3\cdot Enp\cdot (1-L) $$

$$\partial Ep=-L\cdot k1\cdot Ep\cdot PhA+L\cdot k1n\cdot Epp+k4\cdot En\cdot L-k4n\cdot Ep\cdot (1-L) $$

$$\partial En=k3\cdot Enp\cdot (1-L)-k4\cdot En\cdot L+k4n\cdot Ep\cdot (1-L) $$

$$\partial Epp=k1\cdot Ep\cdot PhA\cdot L-k2\cdot Epp\cdot (1-L)+k2n\cdot Enp\cdot L-k1n\cdot Epp\cdot L $$

$$\partial Enp=k2\cdot Epp\cdot (1-L)-k2n\cdot Enp\cdot L-k3\cdot Enp\cdot (1-L) $$

Next, we want to include the flows. Since only phosphate wastewater and phosphate are affected by the inflows and outflow, these are the only ones we have to adapt.

The mass bilance is determined by the change of concentration. Since it is difficult to add up concentrations, you can formulate it with flows:

$$(C\cdot V)'=V\cdot C' = f_{In}\cdot C_{In}- f_{Out}\cdot C$$

With the assumption that the inflow $f_{In}$ equals the outflow $f_{Out}$ and is denoted by $f$, you get the following formula:

$$C'=\frac{f}{V}\cdot (C_{In}-C)$$

Here $C_{In} $ denotes the starting concentration of the product, that get filled in the reactor.

$F1$ is now the flow of the wastewater through the reactor and $F2$ is the flow of the solution, that transports the released phosphate after the light is off.

We obtain the following system of ODEs:

$$\partial PhA=-L\cdot k1\cdot Ep\cdot PhA+L\cdot k1n\cdot Epp + \frac{F1}{V}(PhA_{In}-PhA)\cdot L$$

$$\partial Ph=k3\cdot Enp\cdot (1-L) - \frac{F2}{V}Ph\cdot (1-L) $$

$$\partial Ep=-L\cdot k1\cdot Ep\cdot PhA+L\cdot k1n\cdot Epp+k4\cdot En\cdot L-k4n\cdot Ep\cdot (1-L) $$

$$\partial En=k3\cdot Enp\cdot (1-L)-k4\cdot En\cdot L+k4n\cdot Ep\cdot (1-L) $$

$$\partial Epp=k1\cdot Ep\cdot PhA\cdot L-k2\cdot Epp\cdot (1-L)+k2n\cdot Enp\cdot L-k1n\cdot Epp\cdot L $$

$$\partial Enp=k2\cdot Epp\cdot (1-L)-k2n\cdot Enp\cdot L-k3\cdot Enp\cdot (1-L) $$

For simplicity we set $V=1$ since this is just the volume scale of the reactor and does not influence the reaction scheme.

L is here the light function. It is 1, if the light is on and 0 if the light is off.

$$L(t):= \begin{cases} 1 & \text{if Light on} \ 0 & \text{if Light off} \end{cases} $$

For the solution we used an implicit ODE solver, since our system is stiff. The stiffness is shown in the Jupyter Live Script in A1. Explicit we used the scipy.integrate solver with the Radau method, which is a implicit Runge-Kutta method of Radau IIA family of order 5.

We switch off the light at the time tOff given in minutes.

Let's look at the solution of this model for some parameters. The parameters are orientated at the process time of the enzymes, that we used (further informations in A2 of the Live Script). But they can change by using different PBP or light switches.

We can exchange the parameters easily and look at the change in the solution.

Parameter:

k1=0.024

k1n=0.002

k2=0.5

k2n=0.2

k3=0.024

k4=0.0017

k4n=0.0084

The time the light is switched off: tOff=30 Minutes

And the initial conditions:

[PhA,Ph,Ep,En,Epp,Enp]=[100,0,100,0,0,0]

**Figure 3**: Plot of the model solution with the given parameters and initial conditions.

Now we want to include the flows and take a look what happens for different flows and tOFFs.

**Figure 4**: Plots of the model solutions for different flows and tOffs.

We see that when the flow is high, a lot of phosphate is recovered, but a lot of phosphate is also lost in the process. However, if the flow is too low, not enough phosphate is bound, and you only get a small yield.

The time at which the light is turned off and directed into the harvesting process is also important. If this time is too short, not enough phosphate is bound and the yield is small. If you wait too long, all the enzymes are utilized and you waste unnecessary time.

The time while the light is turned off and directed into the harvesting process is also important. If this time is too short, not enough phosphate is bound and the yield is small. If you wait too long, all the enzymes will be used up and you will lose time unnecessarily.

It is necessary to find the optimal flow and time.

# Optimization

We want to optimize the function to obtain the maximum of phosphate concentration. Therefore, we wrote a program that calculates the best flow and best time for switching off the light, so that the maximum amount off phosphate is bound by PBP, but is not satisfied to the maximum.

First, we defined an objective function that returns the phosphate production at the end of the interval for a given flow at the optimal time to turn off the light.

The best switch-off time is determined by a program that you can find in the Jupyter Live Script. Here we return the time where the bound active PBP does not change significantly anymore. Therefore, $$\frac{\partial Epp}{\partial t} < TOL \approx 0$$

So the change is smaller than a tolerance TOL which is choosen close to zero.

Now we can use Nelder-Mead optimization or a different one to gain the optimum flow and the best time to turn off the light at this flow.

**Figure 5**: Plot of the model solution for the optimal flow and tOff.

Finally, lets have a look what happens after we turn the light back on:

**Figure 6**: Plot of the kinetic model solution with optimal tOff and light turned on after 200 minutes.

When the light is back on, it will take some time to reactivate the PBP. This depends on how fast the regeneration time of the used PBP with the optogenetic switch is.

This could be a problem in the industry. Therefore, we provide a solution to avoid this problem.

# Industrial Outlook

In order to avoid waiting periods you could use three or more reactors parallel.

The wastewater flows inside reactor 1 (R1), where phosphate binds under the influence of light. After about a certain time, the flow is directed to the reactor 2 (R2) and in reactor 1 flows fresh media. While in R2 phosphate is bound, phosphate is released in R1 in the dark state and therefore provides usable phosphate. After the next time interval, R1 is emptied and the light is turned on again. Now the PBPs can return to the active state, while in R2 phosphate is released and in reactor 3 (R3) phosphate is bound. After the regeneration time in R1, the fusion protein can be used again and we obtain a cycle that gives us a maximum amount of phosphate.

**Figure 7**: Animation of the bioreactor chain.

# Jupyter Notebook

The whole modeling is also available as Jupyter Notebook. There the parameters can be freely adjusted and with an existing python installation the model can be executed with different values. The notebook is available for download in our repository under `/modeling`

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We hope that in this way the method of light-driven phosphate recovery will experience a rapid development.