Team:SydneyUni Australia/Modelling Principles

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System of ODEs used to model the intracellular concentration of the metabolic intermediates inside a single cell. The concentrations of each metabolite are determined through the associated MM equation of the introduced enzymes.  
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<div align="justify">System of ODEs used to model the intracellular concentration of the metabolic intermediates inside a single cell. The concentrations of each metabolite are determined through the associated MM equation of the introduced enzymes.  
The symbols A, X, D & E represent the intracellular enzyme concentrations, where X = B or C, and the symbols α<sub>in</sub>,α<sub>out</sub>, β, γ & δ represent the concentrations of the metabolic intermediates.  
The symbols A, X, D & E represent the intracellular enzyme concentrations, where X = B or C, and the symbols α<sub>in</sub>,α<sub>out</sub>, β, γ & δ represent the concentrations of the metabolic intermediates.  
The function J(α<sub>out</sub>,α<sub>in</sub> ) represents the flux of extracellular DCA across the plasma membrane and is a function of the extra and intracellular concentrations of DCA. The value S represents the average surface area of the cellular membrane of ''E. coli''. See below for the derivation of J.
The function J(α<sub>out</sub>,α<sub>in</sub> ) represents the flux of extracellular DCA across the plasma membrane and is a function of the extra and intracellular concentrations of DCA. The value S represents the average surface area of the cellular membrane of ''E. coli''. See below for the derivation of J.
The initial conditions for the system are α<sub>in</sub>=β=γ=δ=0 M at t=0 min, ε=ε<sub>phys</sub> &  α<sub>out</sub>=α<sub>out</sub>(t=0) . Where  is the physiological concentration of non-DCA-derived glycolate and α<sub>out</sub>(t=0) is the initial concentration of DCA outside of the cellular membrane.
The initial conditions for the system are α<sub>in</sub>=β=γ=δ=0 M at t=0 min, ε=ε<sub>phys</sub> &  α<sub>out</sub>=α<sub>out</sub>(t=0) . Where  is the physiological concentration of non-DCA-derived glycolate and α<sub>out</sub>(t=0) is the initial concentration of DCA outside of the cellular membrane.
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The intracellular enzyme concentrations will be determined experimentally.
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The intracellular enzyme concentrations will be determined experimentally.</div>
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[[File:j=p.png|center]]
[[File:j=p.png|center]]
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Fick’s first law of diffusion is appropriate since 1,2-DCA is non-polar. 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 has units L<sup>2</sup> T<sup>-1</sup>. The permeability coefficient of DCA is not reported in the literature, but can be estimated through the relation:
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<div align="justify"></div>
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<div align="justify">Fick’s first law of diffusion is appropriate since 1,2-DCA is non-polar. 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 has units L<sup>2</sup> T<sup>-1</sup>. The permeability coefficient of DCA is not reported in the literature, but can be estimated through the relation:
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</div>
[[File:p=kow.png|center]]
[[File:p=kow.png|center]]
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Where D is the diffusion constant of DCA 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 the octanal-water partition coefficient (K<sub>ow</sub>). This constant states that the equilibrium ratio of DCA in octanol and water:
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<div align="justify">Where D is the diffusion constant of DCA 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 the octanal-water partition coefficient (K<sub>ow</sub>). This constant states that the equilibrium ratio of DCA in octanol and water:
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The diffusion constant, D, is not documented in the literature but can be estimated through the Stokes-Einstein equation based on an estimation of the radius of DCA and viscosity of the cellular membrane.
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The diffusion constant, D, is not documented in the literature but can be estimated through the Stokes-Einstein equation based on an estimation of the radius of DCA and viscosity of the cellular membrane.</div>
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[[File:D=kb.png|center]]
[[File:D=kb.png|center]]
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The Stokes-Einstein equation for determining the diffusion constant D. k<sub>b</sub>, T, η & r represent the Boltzmann constant, temperature, viscosity of the membrane and radius of DCA respectively. Here we must assume DCA is spherical.
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<div align="justify">The Stokes-Einstein equation for determining the diffusion constant D. k<sub>b</sub>, T, η & r represent the Boltzmann constant, 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<sub>W</sub>, and hence the van der Waal radius, r<sub>W</sub><sup>3</sup>. Note DCA is not spherical, and this method is used to calculate atoms.</div>
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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<sub>W</sub>, and hence the van der Waal radius, r<sub>W</sub><sup>3</sup>. Note DCA is not spherical, and this method is used to calculate atoms.
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[[File:Igem b=.png|center]]
[[File:Igem b=.png|center]]

Revision as of 14:00, 27 September 2013


SydneyUniversity Top Banner.jpg SydneyUniversity Bottom Banner.jpg

A Brief Background on Michaelis-Menton Kinetics

The majority of single-substrate metabolic enzymes follow MM kinetics, for which the equations accurately describe enzyme activity. 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:

Igem pathway 1.png
Igem Re.jpg

The rate of catalysis for an enzyme is a function of the enzyme and substrate concentration and incorporates the catalytic constant Kcat and the Michaelis constant Km, which are given in most of the literature that involves enzyme kinetics. The symbol denotes the reaction rate: the rate of product formation, i.e.

Igem Re= dproduct.png


A Schematic of the Engineered Metabolic Pathway:

Pathway 2.png

In the above figure the symbols represent the intracellular concentration of the associated metabolite or enzyme.More information is given below


General Information regarding the Enzymes Involved in the Metabolic Pathway

Enzyme Gene Symbol Constants Substrate Product Ref
1,2-Dichloroethane Dechlorinase dhlA A KM A=0.53mM, kcat A = 3.3 s-1 DCA 2-chloroethanol [1]
Alcohol Dehydrogenase Adhlb 1/2* B KM B=0.94 mM, kcat B = 0.0871 s-1 2-chloroethanol chloroacetaldehyde [2]
p450  ? C KM C = 7.2 mM, kcat C = 89.8 s-1 2-chloroethanol chloroacetaldehyde [3]
Chloroacetaldehyde Dehydrogenase aldA D KM D = 0.0.6mM, kcat D = 0.60 s-1 chloroacetaldehyde chloroacetate [4]
Haloacetate Dehydrogenase dhlB E KM E= 20 mM, kcat E = 25.4 s-1 chloroacetate glycolate [5]
* values based on ethanol as a substrate

ODE Model

Igem ode 1.png


System of ODEs used to model the intracellular concentration of the metabolic intermediates inside a single cell. The concentrations of each metabolite are determined through the associated MM equation of the introduced enzymes.

The symbols A, X, D & E represent the intracellular enzyme concentrations, where X = B or C, and the symbols αinout, β, γ & δ represent the concentrations of the metabolic intermediates. The function J(αoutin ) represents the flux of extracellular DCA across the plasma membrane and is a function of the extra and intracellular concentrations of DCA. The value S represents the average surface area of the cellular membrane of E. coli. See below for the derivation of J. The initial conditions for the system are αin=β=γ=δ=0 M at t=0 min, ε=εphys & αoutout(t=0) . Where is the physiological concentration of non-DCA-derived glycolate and αout(t=0) is the initial concentration of DCA outside of the cellular membrane.

The intracellular enzyme concentrations will be determined experimentally.


DCA Diffusion Across the Plasma Membrane

The diffusion of DCA across the cell membrane was modelled based on Fick’s first law of diffusion:

J=p.png
Fick’s first law of diffusion is appropriate since 1,2-DCA is non-polar. 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 has units L2 T-1. The permeability coefficient of DCA is not reported in the literature, but can be estimated through the relation:
P=kow.png
Where D is the diffusion constant of DCA 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 the octanal-water partition coefficient (Kow). This constant states that the equilibrium ratio of DCA in octanol and water: The diffusion constant, D, is not documented in the literature but can be estimated through the Stokes-Einstein equation based on an estimation of the radius of DCA and viscosity of the cellular membrane.


D=kb.png


The Stokes-Einstein equation for determining the diffusion constant D. kb, T, η & r represent the Boltzmann constant, 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, VW, and hence the van der Waal radius, rW3. Note DCA is not spherical, and this method is used to calculate atoms.


Igem b=.png

In Summary, the flux across the membrane is:

Igem summary J=.png


Summary of Constants
Symbol Name Value (units) Ref
η Cellular Membrane Viscosity 1.9 kg/m/s [6]
S E. coli Membrane Surface Area 6E-12 m2 [7]
r DCA radius 0.3498 nm [8]
Kow Octanol-water Partition Coefficient for DCA 28.2 [9]
d Length of Cellular Membrane  ??? [10]

Extending the System to Cell Cultures

Extending the System to Cell Cultures: The model can be extended to a homogeneous solution/culture of cells. Let Ψ represent the concentration of the cells in solution. The overall rate of decrease of DCA (αtotal) in solution is

Igem datot=daout.png

By applying this change, the resulting values of αinout, β, γ & δ represent the metabolite concentration across all cells. Furthermore the cells are expected to grow due to the production of glycolate (which can be used as a carbon source for growth).

Igem dpsi=f.png

The rate at which E. coli can grow based on the amount of glycolate is not described in the literature. However one will be able to infer the growth as a function of amount of glycolate, ε, in the cell through experimentation.


With thanks to: