### Introduction: Questions and outline

Looking at synthetic biology and the iGEM-competition as a team participating for the first time, we were astonished by the sheer amount of parts and devices already existing. After forming an idea of how a part can look like and coming up with possible functions we can let *C. reinhardtii* perform, there was one question that arose: What other characteristics does a part - no matter if DNA-sequence or other hardware - need to perform its function optimally? This general engineering question comes up early in the Design phase, where ‘other characteristics’ is defined by parameters which influence or are assumed to influence the performance of the part.

Because we slowly started to realize some of the important aspects of degrading PET and protein expression in general, there was one question coming up: How long would it take to degrade PET after transformation of *C. reinhardtii* ? To provide us with an upper estimate, we started modeling the degradation of PET.

### Model of PET degradation by *Chlamydomonas reinhardtii*

A *C. reinhardtii* which expresses and secretes the enzymes PETase and MHETase could pose as a solution for the problem of micro-plastic polluted water. Nevertheless, the viability of PET degradation by *C. reinhardtii* at a larger scale is yet unknown. Models of biological systems allow us to design experiments in silico that are difficult to reproduce in vivo and give us special insights into the role that parameters might play in the given biological system. Therefore, to assess the efficiency of PET degradation by *C. reinhardtii*, a model of PET degradation in continuous culture of *C. reinhardtii* was designed.

The overall goal of the model is to determine the time needed to degrade 1 mg of PET. As we are building a bioreactor for *C. reinhardtii*, it is imperative to know the best parameters that have to be fulfilled by our bioreactor and our algae to achieve the successful degradation of PET. The expression rate, secretion rate and kinetics of the enzymes, such as also the cultivation density, influence the degradation rate of PET in the bioreactor. Based on this assumption, the model was designed to take these factors into account. The model was programmed in Tellurium (Choi et al., 2018) and encompasses six reactions. The reactions are as listed on Fig. 1 and Tab. 1.

"The overall goal of the model is to determine the time needed to degrade 1 mg of PET."

Reaction | Rate | Value |
---|---|---|

R1: \( \rightarrow \) PETase_in | \( k1\) | \(0.01 \frac{\mu M}{s} \) |

R2: \( \rightarrow \) MHETase_in | \( k2\) | \(0.01 \frac{\mu M}{s} \) |

R3: PETase_in \( \rightarrow \) PETase_out | \(k3 \) | \( 1\) |

R4: MHETase_in \( \rightarrow \) MHETase_out | \( k4\) | \( 1\) |

R5: PET + PETase_out \( \rightarrow \) MHET + PETase_out | \( k5 \) | \( 25 \frac{mg}{\mu M \, Enzyme \cdot day}\) |

R6: MHET + MHETase_out \( \rightarrow \) TPA + EG + PETase_out | \( k6 \) | \( kcat \cdot [E] \cdot \frac{[S]}{(km*(1+\frac{TPA}{ki})+[S])} \) |

There are six reaction rates in the model, one for each reaction. There is the expression rates k1 and k2 of the enzymes and the secretion rates k3 and k4. The kinetics k5 of reaction number five are the kinetics of the PETase enzyme. The exact kinetics of the PETase has yet to be described in detail. Nevertheless, approximations can be found in literature. According to Ma et al., the optimized PETase I179F has a reaction rate of 25 mg per µM enzyme per day, which we used in this model (Ma et al., 2019). In contrast to the PETase, the kinetics of the MHETase are better studied. TPA seems to inhibit the functionality of the MHETase and its activity and inhibition can be described by michaelis menten kinetics (Palm et al., 2019). The reaction rates and their values are listed on Tab. 1.

### Assumptions and Hypothesis

The reaction rate of the PETase enzyme is known to be one of the main limiting factors in PET degradation. It is a slow enzyme and because of this reason there have been efforts to optimize it (Ma et al., 2019). This was one of our main concerns while designing and programming our model. Our hypothesis was, that the process of PET degradation would be a slow process that is yet unviable for industrial application and that its speed could eventually be regulated to a certain degree by biological parameters and cultivation parameters. We decided to focus our simulations on two main parameters, one of them external to the biology of the cell, and the other of biological nature. The external parameter that we assumed could influence the degradation of PET was the cultivation density of *C. reinhardtii*. A higher cultivation density would lead to a higher concentration of the secreted enzymes PETase and MHETase and thus to a faster PET degradation. The biological parameter that we chose to variate was the enzyme kinetics of the PETase. By increasing the kinetics of the enzyme PETase by a factor of 1000, a significantly faster degradation of PET is expected. The arbitrary value of 1000 was chosen as an extremely optimistic optimization of the PETase to examine the effect of such a substantial change to the kinetic parameters.

### Results: Variation of the Cultivation Density

As a first approach, the cultivation density of the algae was varied to examine its effect on the degradation rate of PET. For the simulation presented in Fig. 2, the cell volume to culture volume ratio of 1:10 was examined. This means that of 10 ml culture, you would have 1 ml of cells. The time needed to degrade 1000 µg (or 1 mg) of PET was extracted from this simulation, leading to a total degradation time of 8200 s, or 2,27 hours. This is a very optimistic simulation for the cultivation of the alga *C. reinhardtii* because the cultivation density of 1:10 is very difficult to achieve and sustain. Even with this optimistic calculation, a PET plastic bottle weighing 40 g would need approximately 10 years to be completely degraded. According to Klein et al. (2105), the water of the Rhein river in Germany is polluted with micro-plastic up to 1 g per kg. This would mean that to clean one liter of water, assuming that all the micro plastic is PET, 94,5 days would be needed. To analyse a more realistic cultivation density, a simulation for a cultivation ratio of 1:100 was made, as can be seen on Fig. 3. This simulation led to the degradation of 1 mg of PET in 82000 s, or 22,7 h. According to this simulation, it would take approximately 100 years for a 40 g PET bottle to be completely degraded. These simulations show that even after altering the cultivation density parameter, an efficient degradation of PET on industrial scale is not in sight.

### Results: Improving the PETase Kinetics by Factor 1000

The results for the first simulations, shown in Fig. 2 and Fig. 3, confirmed that the PETase kinetics are slow, especially in comparison to the kinetics of the MHETase. On the aforementioned graphs, the curve describing the concentration of MHET stays constant at a value converging to zero. This means that the MHETase immediately converts all MHET to terephthalate (TPA) and ethylene glycol (EG), demonstrating a big difference between the kinetics of the PETase and MHETase. Because the enzyme PETase is much slower than MHETase, MHET can not be accumulated but is rather immediately degraded to TPA and EG. Advanced methods to improve enzymes, for example by directed evolution, have shown that it is possible to drastically improve the kinetics of an enzyme. To assess the effect of an optimistically enhanced and improved PETase on the degradation rate of PET, two simulations were made, where the kinetics of the PETase were improved by a factor of 1000. We are aware that such an optimization of the PETase would be very difficult, if not impossible to achieve. Nevertheless, the drastic improvement of the enzyme for these simulations is expected to serve as an exaggerated case that shows under what conditions, if any, PET degradation is applicable at an industrial scale.

One simulation was made for the cultivation density ratio of 1:10 and the other one for a ratio of 1:100. The results can be seen on Fig. 4 and Fig. 5 respectively. For the simulation shown in Fig. 5, the time needed to degrade 1mg of PET was 260 s, or 4,3 min, which is a substantial improvement to the simulations discussed in the previous section. According to this simulation, the time needed for the degradation of a 40 mg plastic bottle would be approximately 119 days. Still, the problem of achieving and sustaining a cultivation ratio of 1:10 prevails. Therefore, the simulation was repeated for a cultivation ratio of 1:100 with the optimized PETase by a factor of 1000 (Fig. 5). The results of this simulation show that for this cultivation setup a time of 2600 s, or 43 min, is needed to degrade 1 mg of PET. The time needed to degrade a 40 mg PET bottle would be approximately 3,3 years.

### Conclusion: Model of PET degradation by *Chlamydomonas reinhardtii*

The inevitable conclusion of our model is that the PET degradation process by *C. reinhardtii* is for the moment a slow process which is difficult to optimize for industrial application. To be able to use a PET degrading alga at an industrial scale, there are several factors that need optimization. What became clear to us after the results of these simulations is the fact that we needed to improve all possible factors that might have an impact on the PET degradation rate. To achieve this, we took the following decisions regarding the improvement of PET degradation. First, we chose the light inducible promoter PsaD for the design of our parts and constructs. This promoter is the promoter for an abundant chloroplast protein of the Photosystem 1 complex (Fischer et al. 2001). By choosing the promoter of this abundantly expressed protein we expect to boost the expression of our parts and therefore a higher PET degradation. Our second improvement was the introduction of the Sp20 glycomodule to our secretion constructs, due to the fact that this module seems to drastically improve secretion yield in *C. reinhardtii* (Ramos-Martinez et al. 2017). By having a higher secretion rate of the enzymes, we expect a higher PET degradation rate. Third, we chose the *C. reinhardtii* strain UVM4 which was designed to allow efficient expression of transgenes (Neupert et al. 2009). Because the results of the model made clear that culture parameters are a decisive factor, we seeked help from experts on algae cultivation to improve the cultivation of our algae and the design of our own bioreactor. For this we visited the company MINT Engineering, focused on the construction of big scale algae bioreactors, and the algae farm Roquette Klötze.

### Introduction: Modeling light limitation for *C. reinhardtii* cultivation

Because the model of PET-degradation revealed the need for high cultivation densities for degrading PET, we came to the conclusion that it would be necessary to optimize cultivation parameters. With the help of Dr. Ralf Steuer (described here) who is an experienced researcher in the field of algae biotechnology, we were able to further conclude that we should aim for light-limited cultivation, because photoautotrophic growth has a lot of potential to tackle tasks of synthetic biology in a sustainable way.

This is why we decided to use an existing model [9] of chemostat cultivation and apply it to our turbidostat cultivation setup. By summarizing key points, we want to make it easier for other teams to get familiar with chemostat cultivation.

The aim of using this model is to understand how higher cultivation densities can be achieved while defining parameters important for cultivation. Inspired by the model of PET degradation which revealed the need for high densities, it is applicable to our cultivation setup in general and serves as an overview of important culture parameters that can be used by future teams to build upon.

### Formulation of the model

To describe the interplay between vessel depth and culture density, we start from the two equations on the right. For other relevant mathematical derivations, see ([12]).
Equation (1) provides us with the light intensity gradient inside the culture vessel due to algae cell concentration \( c_X \) and is well known as the Beer-Lambert law.
Important is the minus sign on the right hand side of the equation. \(\mathrm{d}I\) is antiproportional to \(\mathrm{d}z\), meaning that the intensity is declining along the light path \( z \).
This will allow us to draw conclusions on the design of our setup pretty fast. Because we want to treat algae cells and other molecules, the Lambert-Beer law is written as a sum over all species contributing to the absorption along the z-axis.

Equation (2) can be understood by looking at the single terms being added to each other on the right hand side. It equates the temporal change of \( c_X \) to the sum of positive (growth with production rate per surface area \(P\) ) and negative (dilution \(D\) of the culture and maintenance \( \mu_e\) ) contributions. Dilution rate is the rate at which we pump out culture from the vessel. The maintenance rate is necessary because cells require energy to maintain processes such as osmoregulation, reducing growth (see [9] for details). For (2) it is also important to note that productivity per surface area \(p\) is a function of specific productivity \( \tilde p \), which expresses the amount of carbon produced per cell. By expressing \(\tilde p \) as a function of light intensity and calculating \(p\) as an integral over the light path, we can derive further conclusions already. At first, we will take a look at the Beer-Lambert law and derive some parameters important to our model.

For the sake of clarity, precise mathematical derivations for this model are given in a PDF supplement, while we still try to explain the model on this page from a physical point of view.

The solution of equation (1) requires evaluation of a well known integral (cf [11]). As a solution we get the exponential decay term on the left, describing the light intensity along the distance traveled \(z\) at a wavelength \( \lambda \). Because there are not only algae in our culture but also other compounds, we split the sum in a term dependent on algae concentration and one containing the rest, the background turbidity \( bg \). The equation shows that algae in a cultivation vessel are subject to a light gradient.

\begin{equation} \frac{\partial}{\partial z} I(\lambda, z, t) = -I(\lambda, z, t) \cdot \sum_{i=1}^{n} \epsilon_i(\lambda)_ \cdot c_i \tag{1} \end{equation} \[\newcommand\T{\Rule{0pt}{1em}{.3em}} \begin{array}{r l} I(\lambda, z,t): & \text{Light Intensity in} \: \frac{\mu mol}{m^2 \cdot s} \\ \epsilon(\lambda): & \text{specific light attenuation coefficient in} \: \frac{L}{mol \cdot cm} \\ c_i: & \text{concentration of \(i\)-th species} \: \frac{mol}{L} \\ z: & \text{light path in}\:cm \\ \lambda: & \text{wavelength in}\:nm \end{array} \]

\begin{equation} \frac{\mathrm{d}}{\mathrm{dt} } c_X= \frac{Y}{z} \cdot p(\tilde p, c_X) - D \cdot c_X - \mu_e \cdot c_X \tag{2} \end{equation} \begin{array}{r l} c_X: & \text{cell concentration in}\: \frac{mol}{L} \\ p(\tilde p, c_X): & \text{production rate per surface area}\: \frac{mol \: carbon}{m^2 \cdot h} \\ Y: & \text{cell yield per carbon fixed in}\: \frac{mol \: cells}{mol \: carbon } \\ \mu_e : & \text{maintenance rate in} \: \frac{1}{h} \\ \tilde p(I) & \text{specific productivity as a function of light intensity}\: \frac{mol \: carbon}{mol \: cells \cdot mm^3 } \end{array}

\begin{equation} p(\tilde p, c_X) = \int_0^z \tilde p(I) \cdot c_X \, \mathrm{d} z \end{equation} \begin{array}{r l} c_X: & \text{cell concentration in}\: \frac{mol}{L} \\ p(\tilde p, c_X): & \text{production rate per surface area}\: \frac{mol \: carbon}{m^2 \cdot h} \\ Y: & \text{cell yield per carbon fixed in}\: \frac{mol \: cells}{mol \: carbon } \\ \mu_e : & \text{maintenance rate in} \: \frac{1}{h} \\ \tilde p(I) & \text{specific productivity as a function of light intensity}\: \frac{mol \: carbon}{mol \: cells \cdot mm^3 } \end{array}

\begin{equation} I(\lambda, z, t) = -I_0(\lambda, t) \cdot \exp[-(\epsilon_X \cdot c_X+bg)z] \tag{1a} \end{equation} \[\newcommand\T{\Rule{0pt}{1em}{.3em}} \begin{array}{r l} I_0(\lambda, t): & \text{light intensity upon entrance into vessel}\\ \epsilon_X: & \text{light attenuation coefficent for algae } X \\ bg: & \text{background turbidity} \: \frac{1}{cm} \end{array} \] where \begin{equation} bg = \sum_{i=1}^{n} \epsilon_i(\lambda)_ \cdot c_i \end{equation}

### Further resolving the model

After arriving at this first conclusion we should step back and take a closer look at equations (1) and (2). Altough they give a basis for cultivation, we still need to give meaning to the parameters in the model. This is especially important if other people from the Synbio-community want to produce reliable results with our cultivation setup. For the steady state to be reached, one needs to finetune the system such that \( \frac{\mathrm{d}}{\mathrm{d}t} c_X \) can be equated to zero. This is important for application because it shows us we need to measure the amount of cells our pump removes from the culture. From this measurement, we then can calculate the growth rate of the cells, because overall cell number remains constant.

It is also important to mention that these equations already represent a simplification of reality; some variables are treated as constants and functional dependencies have been dropped for the sake of notation. This insight shows the importance of being able to fine tune the system. An easy example is temperature, because many 'constants' are functions of it (such as \(Y\) ).

If we take a look at equation (1a), we see that we can measure intensities \(I_{in}\) and \(I_{out}\) before and after passing the vessel to infer the course of light intensity in our vessel according to the equation. This is very important to be able to evaluate results later on: From light in- and output, one gets a first estimation of the maximum amount of photons available to the organism.

For more insights, we have to give further meaning to our equations. We will do this by calculating production rate per surface area \(p\) as a function of light intensity. To this end, we have to find a function for \( \tilde p \), describing the productivity per cell.

For calculation of \( \tilde p \)
we already gave the equation above: The productivity per surface area is given as an integral along the light path \( z \) of \( \tilde p \), multiplied by productivity.

For this approach to sufficiently describe our setup, we need to make sure that there is no intensity gradient along the other coordinates. We realized this in our setup by using an array of LEDs spaced such that light can be regarded as travelling mainly in z-direction.

To solve our question, we use a table in [9] that provides a review of possible functions for \( \tilde p \). What we already know is that for the light limited chemostat, \( \tilde p \) should only be a function of light intensity. To realize this, one also should use cultivation media tailored to this need ([9]).

For our first approach, we are choosing the Monod saturation kinetics with a characteristic asymptotic shape and a value for intensity at half maximum \(H\).

\begin{equation} p(\tilde p, c_X) = \int_0^z \tilde p(I) \cdot c_X \, \mathrm{d} \tilde z \end{equation} \begin{array}{r l} & \text{productivity per unit area as an integral over light path} \end{array}

\begin{equation} \tilde p(I) = p_{max} \cdot \frac{I(\lambda, z, t)}{I(\lambda, z, t)+H} \end{equation} \begin{array}{r l} & \text{Monod-type growth curve} \\ p_{max}: & \text{maximum specific productivity in} \: \frac{mol \, carbon}{mol \, cells \cdot h} \\ H: & \text{intensity for which} \: \tilde p(H) = \frac{p_{max}}{2} \end{array}

### Productivity and steady state growth rate

With this set of equations, we are able to quantify the answer to many questions that arose through Human Practices and literature review: How much carbon is fixated through light \( \tilde p \) and \( p \) and which fraction of it is redirected to produce biomass \( Y \)? To link things together, we calculate: \begin{equation} p = \int_0^z \tilde{p}( I(\lambda, z, t) ) \cdot c_X \mathrm{d}\,z \end{equation} \begin{equation}\text{ } = p_{max} \cdot \int_0^z \frac{I(\lambda, z, t)}{I(\lambda, z, t)+H} \cdot c_X \mathrm{d}\,z \end{equation}

With the substitution \( \mathrm{d} \tilde z = -\frac{1}{\epsilon_{X} \cdot c_{X} + bg} \cdot \frac{1}{I} \), this integral can be solved (see supplement) to give us an expression for the productivity per surface area:
\begin{equation} p(I)= c_X \cdot \frac{p_{max}}{\epsilon_X \cdot c_X + bg} \cdot ln(\frac{I_0(\lambda,t) + H}{I(\lambda, z,t)+H}) \tag{4} \end{equation}
By substituting this equation into (1), our system is defined by the following equation and the Beer-Lambert law described above:
\begin{equation} \frac{\mathrm{d}}{\mathrm{dt} } c_X= \frac{\mu_{max}}{z \cdot \epsilon_X} \cdot \frac{c_X \cdot \epsilon}{(\epsilon_X \cdot c_X + bg)} \cdot ln(\frac{I_{in}(\lambda) + H}{I_{out}(\lambda)+H})- D \cdot c_X - \mu_e \cdot c_X \tag{2a} \end{equation}
These equations provide us with a solid framework to investigate algae growth in a chemostat under light limited conditions. In the above equation, we substituted \(\mu_{max} = Y \cdot p_{max} \), which gives us the maximum growth rate of the culture as the maximum specific productivity (How much carbon can be fixated?) weighted with the yield coefficient (How much carbon is diverted to growth?).

### Conclusions of the model

To extract first information, we see that keeping optical density constant translates to constant cell number, meaning \( \frac{\mathrm{d}}{\mathrm{d}t} c_X = 0 \).

This is equal to demanding that the positive and negative contributions in equation (2) cancel each other out. Plotting these contributions in a graph, we see that there has to be an intersection that corresponds to our steady state.

Because cell number is constant, \( I_{out}( \lambda) \) is aswell. We call this intensity after passing the vessel \( I^{\ast} \). Now we can rearrange equation (1a) to get an expression for the steady state cell concentration \(c_X^{\ast} \).

\begin{equation} c_X^{\ast}= \frac{1}{\epsilon_x} \cdot (\frac{ln(\frac{I_{in}}{I^{\ast}_{out}})}{z} - bg) \end{equation}

From this equation we see that the achievable steady state cell concentration is inversely proportional to the depth of our cultivation vessel \(z\). This is one of the reasons why we decided to build a flat panel device for cultivation. A dense culture can be more suitable for experiments that aim to produce a protein or other compounds of interest. The graph shows \(c_X\) plotted as a function of \(z\). The end of the graph is the critical light path \( z_{crit} \) showing that above a certain value, the culture can not hold itself due to the light gradient.

With these two graphs we can see that at concentrations lower than \(c_X^{\ast} \), growth exceeds dilution and maintencane, while at higher concentrations, it is vice versa. This means that \(c_X^{\ast} \) is a stable steady state to which the culture converges automatically - but only if the parameters are realized such that the two graphs have an intersection.

We are now also able to calculate the most important parameter of the culture, the specific growth rate \( \mu \), which is a factor converting concentration to temportal change of concentration. For this, we again use the Beer-lambert law and rearrange it to get an expression that is only dependable of input and output intensities:
\begin{equation} \mu(I_{in}, I_{out}) = \mu_{max} \cdot \frac{ln(\frac{I_{in}+H}{I_{out}+H})}{ln(\frac{I_{in}}{I_{out}})} - \mu_{e} \end{equation}
This equation allows us to calculate the expected growth rate solely from measurement of light intensities!

### Application and outlook

With this model, we are perfectly prepared for cultivating algae in steady state. It was very useful while designing the bioreactor, but most of it will be applied in the next iteration of building the setup.

It will further guide us in design of our cultivation setup and can help others to understand turbido- and chemostat cultivation of microalgae. Furthermore, we have a framework to implement our growth data into.

For measurement of OD, we already applied the Beer-Lambert law from the model. With this experience, we can look to further optimize our setup, for example with measurement at multiple wavelengths to get more precise measurements with equation (1).

See our Hardware page for our applications of the model.

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