Model page
Modelling Enzymatic Reactions in Engineered Organisms
I. Introduction
Our project aim is to engineer Escherichia Coli, as a proof of concept, so that they are able to reduce polyamine concentrations in the colon to non-risky levels. We modified bacteria by introducing new genes such as putrescine aminotransferase (PatA) and a polyamine oxidase (FMS1), enzymes involved in the degradation of polyamines so that we can regulate their levels in the intestinal tract. Our approach consists on creating auxotroph bacterias on polyamines. This way, the engineered organisms will be taking polyamines from the medium in which they are (in this case the colon lumen) and thus, the concentration of this compound in the colon will be reduced. This auxotrophy is achieved by knocking-out the two main sources of polyamine producers: Ornithine Decarboxylase (ODC) and Agmatinase.
Figure 1: Polyamine pathway in wildtype E.Coli
Due to the fact that FMS1 produces hydrogen-peroxide, it was needed to include a peroxidase gene in our bacteria, with a hydrogen-peroxidase regulated promoter. So as to prevent hydrogen-peroxide from producing harm in the bacterias or in the surrounding tissue.
Figure 2: Polyamine pathway in genetically modified E.Coli
We developed 2 mathematical models: the wildtype model was aimed to obtain a valid range of values for the unknown parameters. Once all this parameters were found we could simulate the behavior of auxotroph bacterias’polyamine pathway in order to gain intuition, characterize better this dynamical system and to mathematically prove that our auxotroph bacteria would be able to reduce polyamine concentrations in human intestinal tract.
For this purpose, the next paragraph presents a brief background on enzyme kinetics.
II. Enzyme kinetics
Enzymes are proteins that can operate with other molecules accelerating their reactions. Enzyme kinetics consists on the study of the chemical reactions that are catalyzed (i.e. increased its reaction rate and decreased its requirement of activation energy) by enzymes. The most common scheme for the enzymatic mechanisms is the following:
Where $E$ stands for enzyme, $S$ for substrate, the “*” indicates a modification in the structure of the molecule and $P$ stands for product. The double directed arrow refers to the double directionality of the reaction. The production of $P$ is described by the activity along time of the enzyme $E$. This activity is described by the time derivative function of $S$, also called “Michaelis-Menten rate of an enzyme catalyzed reaction”:
Where [S] is the concentration of the substrate, $V_{max}$ is the maximum rate and $K^{S}_{m}$ is the Michaelis constant for $S$, representing the concentration of $S$ at which the enzyme activity is half of its maximum capability.
Figure 3: Typical curve of an enzyme activity depending on substrate concentration. $K_{m}$ value indicates the substrate concentration at which the enzyme activity is at half of its maximum.
The above equation can also be written as:
Where $K_{o}$ is the catalytic constant (also called $K_{cat}$, representing the maximum number of chemical conversions of substrate molecules per second that a single catalytic site will execute for a given enzyme concentration), $K_{S}$ is the specificity constant and $[E]_o$ is the total concentration of catalytic centres of the specific enzyme $E$, which corresponds to the total enzyme concentration in the case there is a single catalytic centre per molecule. The previous parameters obey the following relations:
Equations 1-3 make reference to the most simple case in which only one substrate takes part in the reaction.They can also describe reactions with two substrates, where the concentration of one of the both is so high or regular that its change over time is neglected. When this is the case, the participation of the enzyme remains enclosed in all the parameters.
When we want to model a two-substrate ($S_{1}$ and $S_{2}$) reaction in the most general case, the equation gets a little more complicated:
Taking into account Equation 4.1:
And now Equation 4.2:
where we defined $K_{i}^{S_{1}} = \frac{K_{S_{2}}}{K_{S_{1}S_{2}}}$, which represents the inhibition constant of $S_{1}$ to the enzyme (usually $K_{i}^{S_{1}}$ is neglected because$K_{S_{1}S_{2}}»K_{S_{1}}$).
Figure 4: Curve of a two substrate enzyme activity depending on substrates concentration. $K_{m}^{S_1}$ and $K_{m}^{S_2}$ value are the substrate concentration at which the enzyme activity is at half of its maximum.
III. Modeling of Polyamine Pathway in Wild Type E.Coli
The following section is aimed to describe mathematically the whole pathway shown in Figure 1. The main purpose of modeling this system consists on obtaining a valid range of values for the unknown parameters. This can be done by following a phenomenological approach in which it is assumed stability of certain molecular concentrations found in literature and parameters are tuned in order to satisfy stability conditions. First of all, the equations governing the system and their parameters are presented here:
SpeA: Arginine Decarboxylase $\hspace{0.1cm}$ (EC 4.1.1.19)
| Parameter | Value | |
|---|---|---|
| $K_{m}^{L-ARG}$ | 0.65 [mM] | |
| $V_{max}^{SpeA}$ | Unknown |
SpeB: Agmatinase $\hspace{0.1cm}$ (EC 3.5.3.1.1)
| Parameter | Value | |
|---|---|---|
| $K_{m}^{AGM}$ | 1.1 [mM] | |
| $V_{max}^{SpeB}$ | Unknown |
SpeC: Ornithine Decarboxylase $\hspace{0.1cm}$ (EC 4.1.1.1.7)
\begin{equation} \nu_{SpeC} = \frac{V_{max}^{SpeC}}{1+\frac{[Antz]}{K_{i}^{Antz}}}\frac{[L-ORN]}{K_{m}^{L-ORN}+[L-ORN]} \end{equation}
Note: Antizyme (Antz) is a non-competitive inhibitor. Almost complete inhibition of SpeC occurs at high concentrations.
| Parameter | Value | |
|---|---|---|
| $K_{m}^{L-ORN}$ | 3.3 [mM] | |
| $K_{cat}$ | 3.4 $[s^{-1}]$ | |
| $\frac{K_{cat}}{K_m}$ | 1 $[mM^{-1}s^{-1}]$ | |
| $K_{i}^{Antz}$ | Unknown |
SpeD: Adenosynmethinine Decarboxylase $\hspace{0.1cm}$ (EC 4.1.1.50)
\begin{equation} \nu_{SpeD} = \frac{V_{max}^{SpeD}[SAM]}{K_{m}^{SAM}+[SAM]} \end{equation}
Note: SAM refers to S-adenosylmethionine and SAMdc refers to this molecule decarboxylated.
| Parameter | Value | |
|---|---|---|
| $K_{m}^{SAM}$ | 0.06-0.1 [mM] | |
| Specific activity | 0.88 $[\frac{\mu mol}{min*mg}]$ |
SpeE: Spermidine Synthase $\hspace{0.1cm}$ (EC 2.5.1.16)
\begin{equation} \nu_{SpeE} = \frac{V_{max}^{SpeE}[A][P]}{[A][P]+K_{m}^{A}[P]+K_{m}^{P}[A]} \end{equation}
Note: [A] or S-Ad3MPA refers to S-Adenosyl-3-Methylthio-Propylamine, [P] refers to Putrescine and S-MT5TAd refers to S-Methyl-5’-thioadenosine.
| Parameter | Value | |
|---|---|---|
| $K_{m}^{P}$ | 0.0778 [mM] | |
| $K_{m}^{A}$ | 0.32 [mM] | |
| $K_{cat}^{SpeE}$ | 0.128 [$s^{-1}]$ | |
| $\frac{K_{cat}}{K_{m}}$ | 4.4 $[mM^{-1}s^{-1}]$ | |
| Specific activity | 1.83 $[\frac{\mu mol}{min*mg}]$ |
SpeG: Spermidine Acetyl Transferase $\hspace{0.1cm}$ (EC 2.3.1.57)
\begin{equation} \nu_{SpeG} = \frac{V_{max}^{SpeG}[acCoA][SPD]}{[acCoa][SPD]+K_{m}^{acCoa}[SPD]+K_{m}^{SPD}[acCoA]} \end{equation}
Note: [SPD] refers to spermidine and “ac” refers to acetyl.
III.I Time-dependent variables
| Variable | Differential equation | |
|---|---|---|
| $[P]$ | $\frac{d[P]}{dt} = \nu_{SpeB}+\nu_{SpeC}-\nu_{SpeE}-\nu_{N1}$ | |
| $[A]$ | $\frac{d[A]}{dt} = \nu_{SpeD}-\nu_{SpeE}$ | |
| $[AGM]$ | $\frac{d[AGM]}{dt} = \nu_{SpeA}-\nu_{SpeB}$ | |
| $[aSPD]$ | $\frac{d[aSPD]}{dt} = \nu_{SpeG}-\nu_{N2}$ | |
| $[SPD]$ | $\frac{d[SPD]}{dt} = \nu_{SpeE}-\nu_{SpeG}$ | |
| $[Antz]$ | $\frac{d[Antz]}{dt}=K_{s}^{Antz}\left(1-\frac{1}{1+K_{eq}^{Antz}[SPD]}\right)-K_{d}^{Antz}[Antz]$ |
Note: $\nu_{N1}$ and $\nu_{N2}$ refers to the entrance of Putrescine and acetyl-Spermidine into the cell coming from the extracellular media, respectively.
III.II Parameter analysis
Having this information, stability conditions can be obtained from the partial derivatives of all variables evaluated at their steady states, i.e., the Jacobian matrix of the state vector. By simply computing the eigenvalues of the Jacobian matrix and forcing them to be negative, we obtained specific conditions for having a stable system which did not explode.
Once the sytem behaved under a regime close to the concentrations found experimentally in the literature, we performed a parameter sensibility analysis in order to understand better the influence power of each unknown parameter and determine more its viability range. We obtained the following figures by tuning those parameters and looking at the concentrations of the implicated parts after a simulation:
*Figures 5,6,7,8,9,10 and 11:These heatmaps show the concentration after the transient time of each substance modeled under different values for specific parameters. $V_{max}$ is considered as the unknown parameter although it is the multiplication of two parameters (enzyme concentration and turnover number). *
IV. Modeling of Polyamine Pathway in Auxotroph E.Coli
The following model is based on the previous Wild Type model but with the addition of three knock-in’s (KatG, FMS1 and PatA genes) and two knock-out’s (SpeB and SpeC genes). This section was intended to give predictions about the GMO (genetically modified organism) given the validation of the above parameters. As this validation was not achieved, the new parameters are studied under the assumption that educated guesses of WT organism unknown parameters lay down within the range of plausability.
In the following lines, the equations modeling the behavior of the knock-in genes and their parameters are presented:
PatA: Putrescine Aminotransferase $\hspace{0.1cm}$ (EC 2.6.1.82)
\begin{equation} \nu_{PatA} = \frac{V_{max}^{PatA}[P]}{K_{m}^{P}+[P]} \end{equation}
| Parameter | Value | |
|---|---|---|
| $K_{m}^{P}$ | 0.1 [mM] | |
| $K_{cat}^{PatA}$ | 0.05 [$s^{-1}$] |
KatG: Hydroperoxidase $\hspace{0.1cm}$ (EC 1.11.1.21)
\begin{equation} \nu_{KatG} = \frac{V_{max}^{KatG}[H_{2}O_{2}]}{K_{m}^{H_{2}O_{2}}+[H_{2}O_{2}]} \end{equation}
| Parameter | Value | |
|---|---|---|
| $K_{m}^{H_{2}O_{2}}$ | 3.8 [mM] | |
| $K_{cat}^{KatG}$ | 11000 [$s^{-1}$] |
FMS1: Polyamine Oxidase $\hspace{0.1cm}$ (EC 1.5.3.17)
\begin{equation} \nu_{FMS1}= \frac{V_{max}^{FMS1}[H_{2}O_{2}]}{K_{m}^{N^1acSPD}+[H_{2}O_{2}]} \end{equation}
| Parameter | Value | |
|---|---|---|
| $K_{m}^{N1aSPD}$ | 0.011 [mM] | |
| $K_{cat}^{N1aSPD}$ | 15.1 [$s^{-1}$] | |
| $K_{m}^{SPM}$ | 0.118 [mM] | |
| $K_{cat}^{FMS1}$ | 39 [$s^{-1}$] |
IV.I Time-dependent variables
| Variable | Differential equation | |
|---|---|---|
| $[P]$ | $\frac{d[P]}{dt} = \nu_{FMS1}-\nu_{SpeE}-\nu_{PatA}-\nu_{N1}$ | |
| $[SPD]$ | $\frac{d[SPD]}{dt} = \nu_{SpeE}-\nu_{SpeG}$ | |
| $[aSPD]$ | $\frac{d[aSPD]}{dt} = \nu_{SpeG}-\nu_{FMS1}-\nu_{N2}$ | |
| $[A]$ | $\frac{d[SPD]}{dt} = \nu_{SpeD}-\nu_{SpeE}$ | |
| $[H_{2}O_{2}]$ | $\frac{d[H_{2}O_{2}]}{dt}=\nu_{FMS1}^{N^{1}aSPD}-\nu_{KatG}$ |
It was analyzed the stable fixed solutions after the transient behavior for a long range of values for the different unknowns (in this case the unknowns were the enzyme concentrations, and thus the $V_{max}$ values).
*Figures 12,13 and 14: These heatmaps show the steady state concentration of each molecule modeled. *
The above figures give an intuition on how the modified bacteria should degrade putrescine and the metabolite hydrogen peroxide to non hazardous concentrations. These figures also are suggesting the use of specific biobricks according to the intended tuning.
Flux Balance Analysis (FBA)
In order to check the viability of the knock-outed GMO (polyamine auxotroph E. Coli), we also carried out a simple stochiometric flux analysis. This can be used to obtain an intuition on how the metabolic flux is changed when the organism is knocked out. When doing a FBA, one must optimize a function in order to derive the flux balancing. In our case, we optimized the growth rate of the organism, which is understood as the amount of biomass.
From these one can observe that the knock-outs would derive the metabolic charge to the fatty acids beta-oxidation system mostly due to the usage of Acetyl-Conzyme-A as a precursor of $N^{1}$acetyl-Spermidine.
Figures 15 and 16: These are the most significant changes in metabolic flux between the WT organism and the modified strain. Red arrows indicate an increased flux in the modified organism while the narrow blue arrow shows the paths where the flux has decreased with respect to the WT.
