Team:SydneyUni Australia/Modelling Principles

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Revision as of 09:03, 18 October 2013

A Schematic of the Engineered Metabolic Pathway:

General Information regarding the Enzymes Involved in the Metabolic Pathway

A table that summarises the enzymes that we used in both of the engineered metabolic pathways. The constants presented were (painstakingly) obtained from the literature (referenced).

• values based on ethonal as a substrate

ODE Model

Firstly, the overview of the system of ODEs describing the 2 different pathways:

The non-monooxygenase pathway (with dhlA (A) and adh1b1*2 (B)).

The monooxygeanse pathway (with p450 (C)).

The 2 above system of ODEs were used to model 2 things:

1.The rate at which extracellular DCA is removed from solution (given the number of cells in solution) . This is considering the rate at which DCA crosses the plasma membrane (from solution to cytoplasm) of all cells in solution.

2. The rate at which the intracellular concentrations of the metabolic intermediates change over time within a single cell. This considers the relative activities of the enzymes of the metabolic pathway.

The symbols A, B, C, D & E represent the intracellular enzyme concentrations of 1,2-Dichloroethane Dechlorinase, Alcohol Dehydrogenase, Cytochrome p450, Chloroacetaldehyde Dehydrogenase and Haloacetate Dehydrogenase respectively. They have units of mM.

The symbols α_in, β, γ, δ and ε represent the intracellular concentrations of the metabolic intermediates DCA, 2-chloroethanol, chloroacetaldehyde, chloroacetate and glycolate respectively. They have units mM. α_out represents the extracellular concentration of DCA, ie the concentration of DCA in solution. It has units mM.

The symbol Ψ represents the concentration of cells in solution and has units cells/mL.

The function J(α_out,α_in) represents the rate which extracellular DCA moves across the plasma membrane and into the single cell (the flux rate). It is a function of the extra and intracellular concentrations of DCA and has a value of surface area per time m^2/s. The value S represents the average surface area of the cellular membrane of E. coli. By multiplying S by J one achieves the total amount of DCA flowing into a single cell per unit time. The total amount of DCA flowing from solution is achieved by multiplying Ψ by S and J.

Construction of a model for the metabolic pathway

Each line represents how the metabolic intermediate changes over time; this is dictated by the relative rate at which the intermediate is formed vs the rate at which it is used/removed. So, for the metabolic component of the ODE, the rate at which the intracellular concentration of metabolites (β, γ, δ and ε) change over time is purely determined through the relative activity (described by MM equations) of the enzymes creating or removing the metabolite. The rate at which once substrate is removed is directly equal to the rate at which the associated product is created, for instance, the rate at which γ is removed is the rate at which δ is formed. By observing how each intermediate simultaneously acts as a substrate and product, it is easy to see how this process gives rise to a connected system of equations.

(Luckily) the enzymes of our metabolic pathway are accurately described by MM kinetics and all constants were available in the literature. The system is based on the forward and reverse kinetic rate (k₁ and k_-1 respectively) of substrate-enzyme binding followed by the irreversible catalytic step of product formation (k₂) as depicted in the figure below:

The rate of catalysis for an enzyme is a function of the enzyme and substrate concentration and incorporates the catalytic constant kcat and the Michalies constant KM, which are given in most of the literature that involves enzyme kinetics.

The symbol R_E denotes the reaction rate: ie the rate of product formation (which is equal to the rate of substrate removal).

Estimating Protein Concentration

The accuracy of this model is strongly dependant on the intracellular enzyme concentrations, these cannot be determined theoretically was there is a weak correlation with the level of gene expression and protein abundance.

Ishihama et al ([]) profiled the protein concentration in E. coli as:

The enzyme-encoding-genes are under the regulation of an artificial promoter termed Psyn. The promoter is designed to be constituative and drive high expression of the enzymes. We estimated the protein concentration as 10,000 protein units per cell - the upper quartile limit in the ‘Highly abundant group’ box in the figure above.

By taking the cell volume as 0.65 um³ = 0.63 x 10^-18 m³ []

So the intracellular concentration for all proteins is predicted to be 25.55 mM

DCA Diffusion Across the Plasma Membrane

We were presented with a scenario where it was necessary to model how DCA would move from the solution into the cell where it’s metabolised: ie linking α_out and α_in.

This was tackled by modelling the rate of DCA diffusion across a cell membrane through Fick’s first law of diffusion:

Fick’s first law of diffusion is justified since 1,2-DCA is non-polar and the cellular membrane is thin. The law states that the flux, J, of DCA across the membrane is equal to the permeability coefficient, P, times the concentration difference of DCA across the cell membrane. The flux has units m² s^-1

The permeability coefficient of DCA is not reported in the literature, so it was estimation had to be made through the definition of the permeability coefficient:

Where D is the diffusion constant for DCA diffusion across the plasma membrane and d is the length of the membrane. The partition coefficient for DCA across a plasma membrane is not documented, but can be estimated to be similar to the octanal-water partition coefficient (K_ow). This constant determines the equilibrium ratio of DCA in octanol and water. Like many properties of DCA, the diffusion constant, D, is not documented in the literature. It was determined through the famous Stokes-Einstein equation:

Where k_B, T, η & r represent the Boltzmann constant (1.3806488 × 10-23 m2 kg s-2 K-1), temperature, viscosity of the membrane and radius of DCA respectively. Here we must assume DCA is spherical. The ‘radius’ of DCA isn’t described in the literature. But the van der Waal constant, b, can be used to calculate the van der Waal volume, V_W , and hence the van der Waal radius, r_W³. Note DCA is not spherical, and this method is used to calculate atoms.

By combining all of the above, the flux can be described as:

Extending the System to Cell Cultures

The model describes the process of DCA diffusion and metabolism of a single cell. So given a concentration of our engineered cells in solution one can calculate the rate at which the DCA concentration in solution (αα_out) decreases by simply multiply the flux rate of DCA across the membrane by the total concentration of cells in the solution. So, by letting Ψ represent the concentration of cells in solution. The overall rate of decrease of DCA in solution is:

Since the latter portion of the system of ODEs describes the intracellular concentrations within a single cell, the rate at which the DCA comes into the cell and is metabolised (dα_in/dt) is simply:

The motivation for doing this as it seems more natural to consider the intracellular metabolite concentrations of a single cell, rather than averaged across many. It also allows one to gauge whether chloroacetaldehyde reaches cytotoxic levels. The assumption made here is that all the cells in the solution are exactly the same