Difference between revisions of "Team:Cologne-Duesseldorf/Model"

Metabolic Modeling

In the following we modeled the Nootkatone biosynthesis pathway, to get an insight into its behaviour and dynamics. We started with an oversimplified model to get a sense for the behaviour of the enzymes in the pathway. The basic reactions without cofactors are the following: $$\ce{FPP ->[ValS] Valencene ->[HPO] ValenceneO ->[CPR] Nootkatol <->[ADH] Nootkatone}$$ Which gives us this system of differential equations:

$$\frac{dFPP}{dt} = \mu_{FPP} \cdot (Max_{FPP} - c_{FPP}) - \frac{V_{Max,ValS} \cdot c_{FPP}}{K_{M, ValS} + c_{FPP}} $$ $$\frac{dValencene}{dt} = \frac{V_{Max,ValS} \cdot c_{FPP}}{K_{M, ValS} + c_{FPP}} -\frac{V_{Max,HPO} \cdot c_{Valencene}}{K_{M, HPO} + c_{Valencene}}$$ $$\frac{dValenceneO}{dt} = \frac{V_{Max,HPO} \cdot c_{Valencene}}{K_{M, HPO} + c_{Valencene}} -\frac{V_{Max,CPR} \cdot c_{ValenceneO}}{K_{M, CPR} + c_{ValenceneO}}$$ $$\frac{dNootkatol}{dt} = \frac{V_{Max,CPR} \cdot c_{ValenceneO}}{K_{M, CPR} + c_{ValenceneO}} - \frac{\frac{V_{M,ADH+} \cdot c_{Nootkatol}}{K_{M,ADH+}} - \frac{V_{M,ADH-} \cdot c_{Nootkatone}}{K_{M,ADH-}}}{1 + \frac{c_{Nootkatol}}{K_{M,ADH+}} + \frac{c_{Nootkatone}}{K_{M,ADH-}}}$$ $$\frac{dNootkatone}{dt} = \frac{\frac{V_{M,ADH+} \cdot c_{Nootkatol}}{K_{M,ADH+}} - \frac{V_{M,ADH-} \cdot c_{Nootkatone}}{K_{M,ADH-}}}{1 + \frac{c_{Nootkatol}}{K_{M,ADH+}} + \frac{c_{Nootkatone}}{K_{M,ADH-}}}$$

With these kinetic parameters:

We assumend a permanent FPP production proportional to the need, but with an upper boundary. 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 $\frac{c_{\text{FPP}}}{V_{\text{Fibroblast}}} = \frac{0.125 \frac{\text{pmol}}{\text{L}}}{2 \ \text{pL}} = 0.0625 \frac{\text{mol}}{\text{L}}$. 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 $1 \ \text{µM}$. The model further makes the assumption that the Michaelis-Menten kinetic assumption ([S] >> [E]) is met and that the enzyme concentration is constant, due to permanent production and recycling of the enzymes.

Simple model

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

p450-BM3

During research we found that using the p450-BM3 enzyme will simplify and enhance Nootkatone production, giving the following reactions: $$\ce{FPP ->[ValS] Valencene ->[\text{p450-BM3}] Nootkatol <->[ADH] Nootkatone}$$ And the reduced set of differential equations: $$\frac{dFPP}{dt} = \mu_{FPP} \cdot (Max_{FPP} - c_{FPP}) - \frac{V_{Max,ValS} \cdot c_{FPP}}{K_{M, ValS} + c_{FPP}} $$ $$\frac{dValencene}{dt} = \frac{V_{Max,ValS} \cdot c_{FPP}}{K_{M, ValS} + c_{FPP}} -\frac{V_{Max,p450\_BM3} \cdot c_{Valencene}}{K_{M, p450\_BM3} + c_{Valencene}}$$ $$\frac{dNootkatol}{dt} = \frac{V_{Max,p450\_BM3} \cdot c_{Valencene}}{K_{M, p450\_BM3} + c_{Valencene}} - \frac{\frac{V_{M,ADH+} \cdot c_{Nootkatol}}{K_{M,ADH+}} - \frac{V_{M,ADH-} \cdot c_{Nootkatone}}{K_{M,ADH-}}}{1 + \frac{c_{Nootkatol}}{K_{M,ADH+}} + \frac{c_{Nootkatone}}{K_{M,ADH-}}}$$ $$\frac{dNootkatone}{dt} = \frac{\frac{V_{M,ADH+} \cdot c_{Nootkatol}}{K_{M,ADH+}} - \frac{V_{M,ADH-} \cdot c_{Nootkatone}}{K_{M,ADH-}}}{1 + \frac{c_{Nootkatol}}{K_{M,ADH+}} + \frac{c_{Nootkatone}}{K_{M,ADH-}}}$$

There sadly were no data available on this particular enzyme, so we modeled the pathway using the P450 enzyme from Rhodococcus ruber (EC 1.14.14.1) with the following kinetics:

Using this enzyme the model reacted in the following way:

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 expected and the published results of Wriessnegger 2014. 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:

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 Wriessnegger 2014 was $208 \ \frac{\text{mg}}{\text{L}}$ with a $\frac{\text{Nootkatone}}{\text{Nootkatol}}$ ratio of $\frac{208}{44} \approx 4.7$ Our maximal yield was $2965.0 \ \frac{\text{mg}}{\text{L}}$ Nootkatone and $670.0 \frac{\text{mg}}{\text{L}}$ Nootkatol with a $\frac{\text{Nootkatone}}{\text{Nootkatol}}$ ratio of $\approx 4.4$. While our yield was way higher the $\frac{\text{Nootkatone}}{\text{Nootkatol}}$ ratio was quite similar and we therefore deduced that the reaction mechanism we assumed seemed to be quite accurate. The overly high yield was probably based on a lack of the model to implement 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.