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Team:Groningen/Modelling/Arsenic
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Our initial ideas on how and what to model can be found at Brainstorm/Modelling.
Usage of graphs in wiki: Graphs
The raw model
Note: Math support is currently not enabled on this Wiki... (I've asked hq if they can enable it.)
The following variables play an important role in our system (these can be concentrations of substances, the density of the cell, etc.):
- Extracellular:
-
As(III) -
As(V)
-
- Intracellular:
- As(III)
- Operator (concentration of unbound operators)
-
As(V) -
ArsC - ArsR
- ArsD
- ArsRAs (bound to As(III))
- At equilibrium: ArsR As(III) = (k1off/k1on) ArsRAs
- ArsDAs (bound to As(III))
- ArsRop (bound to operator)
- ArsDop (bound to operator)
The variables above can be related to each other through the following "reactions" and/or equations:
-
As(V)ex → As(V), using phosphate transporters? (Summers2009) -
As(V)ex → As(III), using ArsC (Summers2009) -
As(III) → As(III)ex, using ArsAB (helped by ArsD) (Summers2009) - As(III)in + ArsR ↔ ArsRAs
- As(III)in + ArsD ↔ ArsDAs
- Operator + ArsR ↔ ArsRop
- Operator + ArsD ↔ ArsDop
- Operator → Operator + ArsR + ArsD (transcription + translation)
- ArsR → null (degradation)
- ArsD → null (degradation)
Resulting in the following differential equations (please note that the first two can be formed by linear combinations of the other six):
- (d/dt) As(III) = - (k1on ArsR+k2on ArsD) As(III) + k1off ArsRAs + k2off ArsDAs
- (d/dt) Operator = - (k3on ArsR+k4on ArsD) Operator + k3off ArsRop + k3off ArsDop
- (d/dt) ArsR = β1 Operator - (ln(2)/τ1+k1on As(III)+k3on Operator) ArsR + k1off ArsRAs + k3off ArsRop
- (d/dt) ArsD = β2 Operator - (ln(2)/τ2+k2on As(III)+k4on Operator) ArsD + k2off ArsDAs + k4off ArsDop
- (d/dt) ArsRAs = k1on ArsR As(III) - k1off ArsRAs
- (d/dt) ArsDAs = k2on ArsD As(III) - k2off ArsDAs
- (d/dt) ArsRop = k3on ArsR Operator - k3off ArsRop
- (d/dt) ArsDop = k4on ArsD Operator - k4off ArsDop
Using the following constants/definitions:
- K1d = k1off/k1on
- K2d = k2off/k2on = 60µM (Chen1997)
- K3d = k3off/k3on = 0.33µM (Chen1997, suspect as the relevant reference doesn't actually seem to give any value for this)
- K4d = k4off/k4on = 65µM (Chen1997)
- degradation rate = ln(2)/τ
- ArsR half-life time = τ1
- ArsD half-life time = τ2
- β1 = β2 ??? (and either value is unknown)
See Chen1997 for the interplay between ArsR and ArsD.
TODO Figure out relevant equations for metallochaperone function of ArsD?
TODO Make sure all the multiplicities are correct (and/or taken care of in constants). E.g. does 1 mol ArsR (if it is bound) bind 1 mol As(III)?
Equilibrium
For many purposes, like determining the total amount of accumulated arsenic, it can be quite useful to consider the system at equilibrium. That is, when the derivatives of all variables to time are zero.
The following derives an equation for the concentration of free Operators (which is directly related to the amount of ArsR and ArsD produced) based on ArsR and ArsD (utilizing the same kind of equilibrium assumption as in the derivation of equation A.1.4 in Alon2007):
ArsR*Op=Kd3*ArsROp ArsD*Op=Kd4*ArsDOp OpT = Op + ArsROp + ArsDOp ArsD*Op=Kd4*(OpT-Op-ArsROp) ArsD*Op/Kd4=OpT-Op-ArsR*Op/Kd3 (1+ArsR/Kd3+ArsD/Kd4)*Op=OpT Op/Opt = 1/(1+ArsR/Kd3+ArsD/Kd4)
And similarly for As(III):
ArsR*As3=Kd1*ArsRAs3 ArsD*As3=Kd2*ArsDAs3 As3T = As3 + ArsRAs3 + ArsDAs3 As3/As3T = 1/(1+ArsR/Kd1+ArsD/Kd2)
Finally the bound forms of ArsR and ArsD can be eliminated from the equations completely:
0 = β1*Op - (ln(2)/τ1+k1on*As3+k3on*Op)*ArsR + k1off*ArsRAs3 + k3off*ArsROp = β1*Op - (ln(2)/τ1+k1on*As3+k3on*Op)*ArsR + k1off*ArsR*As3/Kd1 + k3off*ArsR*Op/Kd3 = β1*Op - (ln(2)/τ1+k1on*As3+k3on*Op)*ArsR + (k1on*As3+k3on*Op)*ArsR = β1*Op - ln(2)/τ1*ArsR 0 = β2*Op - ln(2)/τ2*ArsD
Kinetic Laws
TODO Add references.
TODO Find out how to determine experimentally which is applicable (and if you know, what the parameters are).
- Mass Action
- Molecules randomly interact, the reaction rate is simply the product of the concentrations of the reactants (multiplied by a constant).
- Michaelis-Menten
- Applicable to situations where there is a maximum reaction rate (due to needing a catalyst/transporter/binding site of which there is only a limited amount for example) under the assumption that there is much more of the "main" reactant than of the catalyst/transporter. Has two constants, the maximum reaction rate and the concentration at which the reaction rate is half the maximum reaction rate.
- Michaelis-Menten reversible
- TODO
- Hill
- Generalization of Michaelis-Menten. More detail.
