Model

Structural Modeling

Metabolic Modeling

Overview

In the following we present our model of the Nootkatone biosynthesis pathway, to give you an insight into its behaviour and dynamics. We start with an oversimplified luminar model to get a sense for the behaviour of the enzymes in the pathway. Then we will continue with a model introducing a function penalizing high concentrations of some of the products, as they have been shown to be toxic at certain levels. As the toxicity is the main culprit of Nootkatone production, we further modeled the production inside a peroxisome, as we assume that intermediates inside the peroxisome cannot pass the membrane and thus have no toxic effect on the cells.

Basic System

The basic reactions of the Nootkatone pathway that are introduced by our team are the following.

$$\ce{FPP ->[ValS] Valencene ->[\text{HPO & CPR}][NADH + H+ + O2 -> NAD+ + H2O] Nootkatol <->[ADH][NAD+ + H+ -> NADH] Nootkatone}$$

However during research we found that using the p450-BM3 enzyme will simplify and enhance Nootkatone production, giving the following reaction pathway.

$$\ce{FPP ->[ValS] Valencene ->[\text{p450-BM3}][NADH + H+ + O2 -> NAD+ + H2O] Nootkatol <->[ADH][NAD+ + H+ -> NADH] Nootkatone}$$

We assumed Michaelis-Menten kinetics for each reaction, with the last step being reversible.

Michaelis-Menten kinetics

$$\frac{dP}{dt} = \frac{V_{Max} \cdot c_{S}}{K_{M} + c_{S}}$$

Reversible Michaelist-Menten kinetics

$$\frac{dP}{dt} = \frac{\frac{V_{M+} \cdot c_{S}}{K_{M+}} - \frac{V_{M-} \cdot c_{P}}{K_{M-}}}{1 + \frac{c_{S}}{K_{M+}} + \frac{c_{P}}{K_{M-}}}$$

We further assumend a permanent FPP production proportional to the need, but with an upper boundary and a factor controlling the production speed. This behaviour is similar to an unlimited pool and diffusion.

$$\frac{dFPP}{dt} = \mu_{FPP} \cdot (Max_{FPP} - c_{FPP}) - \frac{V_{Max,ValS} \cdot c_{FPP}}{K_{M, ValS} + c_{FPP}} $$

This gives us the following system of differential equations.

$$\frac{dFPP}{dt} = \mu_{FPP} \cdot (Max_{FPP} - c_{FPP}) - \frac{V_{Max,ValS} \cdot c_{FPP}}{K_{M, ValS} + c_{FPP}} $$

$$\frac{dValencene}{dt} = \frac{V_{Max,ValS} \cdot c_{FPP}}{K_{M, ValS} + c_{FPP}} -\frac{V_{Max,p450\_BM3} \cdot c_{Valencene}}{K_{M, p450\_BM3} + c_{Valencene}}$$

$$\frac{dNootkatol}{dt} = \frac{V_{Max,p450\_BM3} \cdot c_{Valencene}}{K_{M, p450\_BM3} + c_{Valencene}} - \frac{\frac{V_{M,ADH+} \cdot c_{Nootkatol}}{K_{M,ADH+}} - \frac{V_{M,ADH-} \cdot c_{Nootkatone}}{K_{M,ADH-}}}{1 + \frac{c_{Nootkatol}}{K_{M,ADH+}} + \frac{c_{Nootkatone}}{K_{M,ADH-}}}$$

$$\frac{dNootkatone}{dt} = \frac{\frac{V_{M,ADH+} \cdot c_{Nootkatol}}{K_{M,ADH+}} - \frac{V_{M,ADH-} \cdot c_{Nootkatone}}{K_{M,ADH-}}}{1 + \frac{c_{Nootkatol}}{K_{M,ADH+}} + \frac{c_{Nootkatone}}{K_{M,ADH-}}}$$

Parameters

As we could not find information about the physiological FPP concentration in yeast cells, we deduced from Tong 2004 (Typical fibroblast FPP concentration = $0.125 \frac{\text{pmol}}{10^6 \ \text{cells}}$) and Bionumbers (Volume of a typical fibroblast = $2 \cdot 10^{-12} \text{L}$) that the FPP concentration in a fibroblast is around $\frac{c_{\text{FPP}}}{V_{\text{Fibroblast}}} = \frac{0.125 \frac{\text{pmol}}{\text{L}}}{2 \ \text{pL}} = 0.0625 \frac{\text{mol}}{\text{L}}$, which we used as the maximal FPP concentration in our model.

Another assumption we made is a five-fold reduction in the speed of the reversible reaction of the ADH-21, based on the knowledge, that the forward reaction is favored. All enzymes were assumed to have a constant concentration of $1 \ \text{µM}$.

Simple model

A simulation in python using scipy's integrate.ode function gave the following results:

Bioreactor simulation

In order to check the validity of our model we took the results Wriessnegger 2014, $208 \ \frac{\text{mg}}{\text{L}}$ Nootkatone production after 108 h, as a point of reference. For that we changed our modeling approach from a single cell model to a population-based model and assumed the maximal yeast density in a bioreactor, $200 \frac{\text{g dry weight}}{\text{L}}$, (Source) and simulated the yield:

The yield of $154.9 \ \frac{\text{mg}}{\text{L}}$ Nootkatol was lower than the published results of Wriessnegger 2014. Our first guess was that the assumption of all enzymes being equally concentrated was probably false. We therefore varied the enzyme concentrations and found that overexpression of valencene synthase increased the yield dramatically by converting way more FPP than before, while overexpressing the other enzymes had little to no effect at all:

What can be seen in the plot above is that the system is mostly influenced by concentration changes of valencene synthase. Since the Nootkatone production did not seem to increase further after increasing the valencene synthase concentration by 20-fold, we stuck to that number and simulated our model under the changed conditions.

The maximal yield of this simulation was $2965.0 \ \frac{\text{mg}}{\text{L}}$ Nootkatone and $670.0 \frac{\text{mg}}{\text{L}}$ Nootkatol, greatly exceeding the maximal concentration achieved by Wriessnegger 2014. But our $\frac{\text{Nootkatone}}{\text{Nootkatol}}$ ratio was quite similar and we therefore deduced that the reaction mechanism we assumed seemed to be quite accurate. We found two possible explanations for the overly high yield. On the one hand our assumption for the maximal FPP concentration could have been false. The total yield of the model scales with FPP supply, so that could be the error. On the other hand, a known fact about Nootkatone production is the the toxicity of the Nootkatone precursor Nootkatol. According to Gavira 2013 the toxic nootkatol concentration for yeast is around $ 100 \frac{\text{mg}}{\text{L}}$.

Nootkatol penalty

We therefore expanded our model using a Hill function alike penalty function for increasing nootkatol concentration, which we applied to the FPP production representative for the whole yeast cell biomass production: $$\frac{dFPP}{dt} = \mu_{FPP} \cdot (Max_{FPP} - c_{FPP}) \cdot \frac{c_{Nootkatol,Toxic} \cdot K_M^n}{c_{Nootkatol}+ c_{Nootkatol,Toxic} \cdot K_M^n} - \frac{V_{Max,ValS} \cdot c_{FPP}}{K_{M, ValS} + c_{FPP}} $$ The system reacted in the following way:

The yield of $438.5 \ \frac{\text{mg}}{\text{L}}$ Nootkatone and $88.7 \frac{\text{mg}}{\text{L}}$ Nootkatol with a $\frac{\text{Nootkatone}}{\text{Nootkatol}}$ ratio of $\approx 4.9$ is way closer to the publication of Wriessnegger 2014 ($208 \ \frac{\text{mg}}{\text{L}}$), which led us to the conclusion that our model is already a quite accurate description of the pathway.

Extended Nootkatol penalty

The assumption of penalizing only the FPP influx representative for the whole activity of the cell is rather crude and we therefore wanted to check whether penalizing every reaction in the pathway with increased Nootkatol concentration would yield different results.

$$\frac{dFPP}{dt} = \mu_{FPP} \cdot (Max_{FPP} - c_{FPP}) \cdot \frac{c_{Nootkatol,Toxic} \cdot K_M^n}{c_{Nootkatol}+ c_{Nootkatol,Toxic} \cdot K_M^n} - \frac{V_{Max,ValS} \cdot c_{FPP}}{K_{M, ValS} + c_{FPP}}$$

$$\frac{dValencene}{dt} = \frac{V_{Max,ValS} \cdot c_{FPP}}{K_{M, ValS} + c_{FPP}} \cdot \frac{c_{Nootkatol,Toxic} \cdot K_M^n}{c_{Nootkatol}+ c_{Nootkatol,Toxic} \cdot K_M^n} - \frac{V_{Max,p450\_BM3} \cdot c_{Valencene}}{K_{M, p450\_BM3} + c_{Valencene}} \cdot \frac{c_{Nootkatol,Toxic} \cdot K_M^n}{c_{Nootkatol}+ c_{Nootkatol,Toxic} \cdot K_M^n} $$

$$\frac{dNootkatol}{dt} = \frac{V_{Max,p450\_BM3} \cdot c_{Valencene}}{K_{M, p450\_BM3} + c_{Valencene}} \cdot \frac{c_{Nootkatol,Toxic} \cdot K_M^n}{c_{Nootkatol}+ c_{Nootkatol,Toxic} \cdot K_M^n} - \frac{\frac{V_{M,ADH+} \cdot c_{Nootkatol}}{K_{M,ADH+}} - \frac{V_{M,ADH-} \cdot c_{Nootkatone}}{K_{M,ADH-}}}{1 + \frac{c_{Nootkatol}}{K_{M,ADH+}} + \frac{c_{Nootkatone}}{K_{M,ADH-}}} \cdot \frac{c_{Nootkatol,Toxic} \cdot K_M^n}{c_{Nootkatol}+ c_{Nootkatol,Toxic} \cdot K_M^n} $$

$$\frac{dNootkatone}{dt} = \frac{\frac{V_{M,ADH+} \cdot c_{Nootkatol}}{K_{M,ADH+}} - \frac{V_{M,ADH-} \cdot c_{Nootkatone}}{K_{M,ADH-}}}{1 + \frac{c_{Nootkatol}}{K_{M,ADH+}} + \frac{c_{Nootkatone}}{K_{M,ADH-}}} \cdot \frac{c_{Nootkatol,Toxic} \cdot K_M^n}{c_{Nootkatol}+ c_{Nootkatol,Toxic} \cdot K_M^n}$$

This model yielded a maximal yield of $398.4 \ \frac{\text{mg}}{\text{L}}$ Nootkatone, $90.4 \frac{\text{mg}}{\text{L}}$ Nootkatol and a $\frac{\text{Nootkatone}}{\text{Nootkatol}}$ ratio of $ \approx 4.4$.

Reversibility

Since we assumed that Nootkatone is not degraded and that the reaction is reversible both substances accumulate in our model. This challenges the assumption that only the reaction catalysed by the alcohol dehydrogenase is reversible. We therefore set up a model in which every reaction is reversible and varied the speed of the back reaction to get a feeling of how the system might react to the overaccumulation. We kept the penalty on FPP.

$$\ce{FPP <->[ValS] Valencene <->[HPO] ValenceneO <->[CPR] Nootkatol <->[ADH] Nootkatone}$$

$$\frac{dFPP}{dt} = \mu_{FPP} \cdot (Max_{FPP} - c_{FPP}) \cdot \frac{c_{Nootkatol,Toxic} \cdot K_M^n}{c_{Nootkatol}+ c_{Nootkatol,Toxic} \cdot K_M^n} - \frac{\frac{V_{M,ValS+} \cdot c_{FPP}}{K_{M,ValS+}} - \frac{V_{M,ValS-} \cdot c_{Valencene}}{K_{M,ADH-}}}{1 + \frac{c_{FPP}}{K_{M,ValS+}} + \frac{c_{Valencene}}{K_{M,ValS-}}}$$

$$\frac{dValencene}{dt} = \frac{\frac{V_{M,ValS+} \cdot c_{FPP}}{K_{M,ValS+}} - \frac{V_{M,ValS-} \cdot c_{Valencene}}{K_{M,ADH-}}}{1 + \frac{c_{FPP}}{K_{M,ValS+}} + \frac{c_{Valencene}}{K_{M,ValS-}}} - \frac{\frac{V_{M,p450+} \cdot c_{Valencene}}{K_{M,p450+}} - \frac{V_{M,p450-} \cdot c_{Nootkatol}}{K_{M,p450-}}}{1 + \frac{c_{Valencene}}{K_{M,p450+}} + \frac{c_{Nootkatol}}{K_{M,p450-}}}$$

$$\frac{dNootkatol}{dt} = \frac{\frac{V_{M,p450+} \cdot c_{Valencene}}{K_{M,p450+}} - \frac{V_{M,p450-} \cdot c_{Nootkatol}}{K_{M,p450-}}}{1 + \frac{c_{Valencene}}{K_{M,p450+}} + \frac{c_{Nootkatol}}{K_{M,p450-}}} - \frac{\frac{V_{M,ADH+} \cdot c_{Nootkatol}}{K_{M,ADH+}} - \frac{V_{M,ADH-} \cdot c_{Nootkatone}}{K_{M,ADH-}}}{1 + \frac{c_{Nootkatol}}{K_{M,ADH+}} + \frac{c_{Nootkatone}}{K_{M,ADH-}}}$$

$$\frac{dNootkatone}{dt} = \frac{\frac{V_{M,ADH+} \cdot c_{Nootkatol}}{K_{M,ADH+}} - \frac{V_{M,ADH-} \cdot c_{Nootkatone}}{K_{M,ADH-}}}{1 + \frac{c_{Nootkatol}}{K_{M,ADH+}} + \frac{c_{Nootkatone}}{K_{M,ADH-}}}$$

This model yielded a maximal yield of $437.9 \ \frac{\text{mg}}{\text{L}}$ Nootkatone, $88.6 \frac{\text{mg}}{\text{L}}$ Nootkatol and a $\frac{\text{Nootkatone}}{\text{Nootkatol}}$ ratio of $ \approx 4.9$.

As with the non-reversible model we wanted to check how a penalty on all reactions would change the way the model behaved.

$$\frac{dFPP}{dt} = \mu_{FPP} \cdot (Max_{FPP} - c_{FPP}) \cdot \frac{c_{Nootkatol,Toxic} \cdot K_M^n}{c_{Nootkatol}+ c_{Nootkatol,Toxic} \cdot K_M^n} - \frac{\frac{V_{M,ValS+} \cdot c_{FPP}}{K_{M,ValS+}} - \frac{V_{M,ValS-} \cdot c_{Valencene}}{K_{M,ADH-}}}{1 + \frac{c_{FPP}}{K_{M,ValS+}} + \frac{c_{Valencene}}{K_{M,ValS-}}} \cdot \frac{c_{Nootkatol,Toxic} \cdot K_M^n}{c_{Nootkatol}+ c_{Nootkatol,Toxic} \cdot K_M^n} $$

$$\frac{dValencene}{dt} = \frac{\frac{V_{M,ValS+} \cdot c_{FPP}}{K_{M,ValS+}} - \frac{V_{M,ValS-} \cdot c_{Valencene}}{K_{M,ADH-}}}{1 + \frac{c_{FPP}}{K_{M,ValS+}} + \frac{c_{Valencene}}{K_{M,ValS-}}} \cdot \frac{c_{Nootkatol,Toxic} \cdot K_M^n}{c_{Nootkatol}+ c_{Nootkatol,Toxic} \cdot K_M^n} - \frac{\frac{V_{M,p450+} \cdot c_{Valencene}}{K_{M,p450+}} - \frac{V_{M,p450-} \cdot c_{Nootkatol}}{K_{M,p450-}}}{1 + \frac{c_{Valencene}}{K_{M,p450+}} + \frac{c_{Nootkatol}}{K_{M,p450-}}} \cdot \frac{c_{Nootkatol,Toxic} \cdot K_M^n}{c_{Nootkatol}+ c_{Nootkatol,Toxic} \cdot K_M^n} $$

$$\frac{dNootkatol}{dt} = \frac{\frac{V_{M,p450+} \cdot c_{Valencene}}{K_{M,p450+}} - \frac{V_{M,p450-} \cdot c_{Nootkatol}}{K_{M,p450-}}}{1 + \frac{c_{Valencene}}{K_{M,p450+}} + \frac{c_{Nootkatol}}{K_{M,p450-}}} \cdot \frac{c_{Nootkatol,Toxic} \cdot K_M^n}{c_{Nootkatol}+ c_{Nootkatol,Toxic} \cdot K_M^n} - \frac{\frac{V_{M,ADH+} \cdot c_{Nootkatol}}{K_{M,ADH+}} - \frac{V_{M,ADH-} \cdot c_{Nootkatone}}{K_{M,ADH-}}}{1 + \frac{c_{Nootkatol}}{K_{M,ADH+}} + \frac{c_{Nootkatone}}{K_{M,ADH-}}} \cdot \frac{c_{Nootkatol,Toxic} \cdot K_M^n}{c_{Nootkatol}+ c_{Nootkatol,Toxic} \cdot K_M^n} $$

$$\frac{dNootkatone}{dt} = \frac{\frac{V_{M,ADH+} \cdot c_{Nootkatol}}{K_{M,ADH+}} - \frac{V_{M,ADH-} \cdot c_{Nootkatone}}{K_{M,ADH-}}}{1 + \frac{c_{Nootkatol}}{K_{M,ADH+}} + \frac{c_{Nootkatone}}{K_{M,ADH-}}} \cdot \frac{c_{Nootkatol,Toxic} \cdot K_M^n}{c_{Nootkatol}+ c_{Nootkatol,Toxic} \cdot K_M^n}$$

This model yielded a maximal yield of $394.2 \ \frac{\text{mg}}{\text{L}}$ Nootkatone, $88.1 \frac{\text{mg}}{\text{L}}$ Nootkatol and a $\frac{\text{Nootkatone}}{\text{Nootkatol}}$ ratio of $ \approx 4.4$.

A thing we wanted to check at this point was if the introduction of reversible reactions and penalty terms changed the behaviour of our expression analysis and we thus conducted it for a second time, now with the changed model:

Apparently, less overexpression of Valencene synthase is needed, whereas now the overexpression of ADH-21 has little effect of overall Nootkatone production, compared to no effect before.

Peroxisome model

Having explored the dynamics of the reactions involved we further wanted know whether using peroxisomes to produce Nootkatone would increase the yield as expected. Since we assume that the toxic intermediate Nootkatol cannot diffuse out of the peroxisome, the production has no penalty terms, but we assume all processes to be reversible:

$$\frac{dFPP}{dt} = \mu_{FPP} \cdot (Max_{FPP} - c_{FPP}) - \frac{\frac{V_{M,ValS+} \cdot c_{FPP}}{K_{M,ValS+}} - \frac{V_{M,ValS-} \cdot c_{Valencene}}{K_{M,ADH-}}}{1 + \frac{c_{FPP}}{K_{M,ValS+}} + \frac{c_{Valencene}}{K_{M,ValS-}}}$$

$$\frac{dValencene}{dt} = \frac{\frac{V_{M,ValS+} \cdot c_{FPP}}{K_{M,ValS+}} - \frac{V_{M,ValS-} \cdot c_{Valencene}}{K_{M,ADH-}}}{1 + \frac{c_{FPP}}{K_{M,ValS+}} + \frac{c_{Valencene}}{K_{M,ValS-}}} - \frac{\frac{V_{M,p450+} \cdot c_{Valencene}}{K_{M,p450+}} - \frac{V_{M,p450-} \cdot c_{Nootkatol}}{K_{M,p450-}}}{1 + \frac{c_{Valencene}}{K_{M,p450+}} + \frac{c_{Nootkatol}}{K_{M,p450-}}}$$

$$\frac{dNootkatol}{dt} = \frac{\frac{V_{M,p450+} \cdot c_{Valencene}}{K_{M,p450+}} - \frac{V_{M,p450-} \cdot c_{Nootkatol}}{K_{M,p450-}}}{1 + \frac{c_{Valencene}}{K_{M,p450+}} + \frac{c_{Nootkatol}}{K_{M,p450-}}} - \frac{\frac{V_{M,ADH+} \cdot c_{Nootkatol}}{K_{M,ADH+}} - \frac{V_{M,ADH-} \cdot c_{Nootkatone}}{K_{M,ADH-}}}{1 + \frac{c_{Nootkatol}}{K_{M,ADH+}} + \frac{c_{Nootkatone}}{K_{M,ADH-}}}$$

$$\frac{dNootkatone}{dt} = \frac{\frac{V_{M,ADH+} \cdot c_{Nootkatol}}{K_{M,ADH+}} - \frac{V_{M,ADH-} \cdot c_{Nootkatone}}{K_{M,ADH-}}}{1 + \frac{c_{Nootkatol}}{K_{M,ADH+}} + \frac{c_{Nootkatone}}{K_{M,ADH-}}}$$

All of the above results summarized in a table.

During this investigation we noticed that under those conditions the maximal Nootkatone production is only dependent on the size of the peroxisome and therefore modeled the production depending on the peroxisomal size.

With the minimal peroxisomal diameter for equal production being $5.79 \ \text{µm}$, which we obtained by linear regression, we thus decided to create a Pex11 knockout mutant in which we can control the size of the peroxisome.