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Team:Groningen/Modelling/Arsenic
From 2009.igem.org
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!Description | !Description | ||
| - | |- | + | |-class="unknown" |
|k8 | |k8 | ||
|1/s | |1/s | ||
|Reaction rate constant representing how fast ArsB can export arsenic. | |Reaction rate constant representing how fast ArsB can export arsenic. | ||
| - | |- | + | |-class="estimate" |
| - | |K1 | + | |K1<sub>d</sub> |
|M | |M | ||
| - | |Dissociation | + | |Dissociation constant for ArsR and As(III). |
* K1<sub>d</sub> = unknown (assumed to be about an order of magnitude smaller than K2<sub>d</sub> = 60µM, the corresponding constant for the similar protein ArsD from [[Team:Groningen/Literature#Chen1997|Chen1997]]) | * K1<sub>d</sub> = unknown (assumed to be about an order of magnitude smaller than K2<sub>d</sub> = 60µM, the corresponding constant for the similar protein ArsD from [[Team:Groningen/Literature#Chen1997|Chen1997]]) | ||
| + | |-class="fromPaper" | ||
| + | |K3<sub>d</sub> | ||
| + | |M | ||
| + | |Dissociation constants for ArsR and ars. | ||
* K3<sub>d</sub>² = k3<sub>off</sub>/k3<sub>on</sub> = (0.33µM)²? ([[Team:Groningen/Literature#Chen1997|Chen1997]], suspect as the relevant reference doesn't actually seem to give any value for this) | * K3<sub>d</sub>² = k3<sub>off</sub>/k3<sub>on</sub> = (0.33µM)²? ([[Team:Groningen/Literature#Chen1997|Chen1997]], suspect as the relevant reference doesn't actually seem to give any value for this) | ||
| - | |- | + | |-class="selfDerived" |
|v5 | |v5 | ||
|mol/(s·L) | |mol/(s·L) | ||
|Maximum import rate per liter of cells (see [[Team:Groningen/Glossary#MichaelisMenten|Michaelis-Menten equation]]). Note that we have purposefully chosen to write the units as mol/(s·L) instead of M/s, to emphasize the fact that the rate is per liter of ''cells''. | |Maximum import rate per liter of cells (see [[Team:Groningen/Glossary#MichaelisMenten|Michaelis-Menten equation]]). Note that we have purposefully chosen to write the units as mol/(s·L) instead of M/s, to emphasize the fact that the rate is per liter of ''cells''. | ||
* v5 = k6 GlpFT (Vs/Vc) | * v5 = k6 GlpFT (Vs/Vc) | ||
| - | |- | + | |-class="selfDerived" |
| - | |K5 | + | |K5 |
|M | |M | ||
| - | |Concentration at which | + | |Concentration at which import reaches half its maximum import rate (see [[Team:Groningen/Glossary#MichaelisMenten|Michaelis-Menten equation]]). |
* K5 = (k5off+k6) / k5on | * K5 = (k5off+k6) / k5on | ||
| + | |-class="unknown" | ||
| + | |K7 | ||
| + | |M | ||
| + | |Concentration at which export reaches half its maximum export rate (see [[Team:Groningen/Glossary#MichaelisMenten|Michaelis-Menten equation]]). | ||
* K7 = (k7off+k8) / k7on | * K7 = (k7off+k8) / k7on | ||
| - | |- | + | |-class="unknown" |
|τB, τR, τG | |τB, τR, τG | ||
|s | |s | ||
|Half-lifes (of ArsB, ArsR and GV, respectively). Degradation rate = ln(2)/τ {{infoBox|1=If you take just the degradation into account you will have the equation dC/dt = -k*C, which leads to C(t) = C(0) e<sup>-k t</sup>. So if k = ln(2)/τ we get C(t) = C(0) e<sup>-ln(2)/τ t</sup> = C(0) 2<sup>-t/τ</sup>. In other words τ is the time it takes for the concentration to half.}} | |Half-lifes (of ArsB, ArsR and GV, respectively). Degradation rate = ln(2)/τ {{infoBox|1=If you take just the degradation into account you will have the equation dC/dt = -k*C, which leads to C(t) = C(0) e<sup>-k t</sup>. So if k = ln(2)/τ we get C(t) = C(0) e<sup>-ln(2)/τ t</sup> = C(0) 2<sup>-t/τ</sup>. In other words τ is the time it takes for the concentration to half.}} | ||
| - | |- | + | |-class="unknown" |
|β1, β2, etc. | |β1, β2, etc. | ||
|1/s | |1/s | ||
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|L | |L | ||
|Total volume of cells (in solution) (so Vs+Vc is the total volume). | |Total volume of cells (in solution) (so Vs+Vc is the total volume). | ||
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| + | {|class="ourtable" style="width:100%" | ||
| + | !colspan="5"| | ||
| + | |- style="text-align:center;" | ||
| + | |class="fromPaper" style="padding:0;"|Directly from paper. | ||
| + | |class="selfDerived" style="padding:0;"|Based on data from paper. | ||
| + | |class="experimental" style="padding:0;"|Based on experiment. | ||
| + | |class="estimate" style="padding:0;"|Rough estimate. | ||
| + | |class="unknown" style="padding:0;"|Totally unknown. | ||
| + | |} | ||
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Revision as of 12:31, 16 September 2009
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Based on the quasi-steady-state derivation below we have defined the following simplified model:
A schematic representation of the processes involved in arsenic filtering (keep in mind that ArsR represses the expression of the genes behind ars). Note that ArsD is not shown here, as it is not present in our E. coli and has a role analogous to ArsR.
| Reaction | Description | Rate | |
|---|---|---|---|
| Transport | |||
| As(III)exT → As(III)T | Import of arsenic. | v5† As(III)ex / (K5+As(III)ex) | |
| As(III)T → As(III)exT | Export of arsenic. | k8 ArsBAs | |
| ars1T → ars1T + ArsBT | Production of ArsB. | β4 ars1 | |
| ArsBT → null | Degradation of ArsB | ln(2)/τB ArsB | |
| Accumulation | |||
| ars1T → ars1T + ArsRT | Transcription + translation from the chromosomal operon. | β1 ars1 | |
| pro → pro + ArsRT | Transcription + translation from a constitutive promoter. | β3 pro | |
| ArsRT → null | Degradation of ArsR. | (ln(2)/τR) ArsR | |
| Gas vesicles | |||
| ars2T → ars2T + GV | Transcription + translation. | β5 ars2 | |
| GV → null | Degradation of gas vesicles. | ln(2)/τG GV | |
| Name | Description | Derivative to time | |
|---|---|---|---|
| Extracellular | |||
| As(III)exT | As(III) in the solution. | (Vc/Vs) k8 ArsBAs - (Vc/Vs) v5† As(III)ex / (K5+As(III)ex) | |
| Membrane (all naturally occurring, but we plan to bring GlpF to overexpression) | |||
| GlpFT | Importer of As(III) (concentration w.r.t. the exterior of the cell). | (concentration is constant) | |
| ArsBT | Exporter of As(III) (concentration w.r.t. the interior of the cell). | β4 ars1 - ln(2)/τB ArsB | |
| Intracellular (ars2, pro and GV are introduced) | |||
| As(III)T | As(III) (bound and unbound) in the cell. | v5 As(III)ex / (K5+As(III)ex) - k8 ArsBAs | |
| ars1T | ArsR repressed promoters (bound and unbound) naturally occurring in E. coli. | (concentration is constant) | |
| ars2T | ArsR repressed promoters in front of gas vesicle genes. | (concentration is constant) | |
| pro | Constitutive promoters in front of arsR. | (concentration is constant) | |
| ArsRT | ArsR in the cell. | β1 ars1 + β3 pro - (ln(2)/τR) ArsR | |
| GV | Concentration of gas vesicles. | β5 ars2 - ln(2)/τG GV | |
† Note that the "constant" v5 depends on the concentration of GlpF transporters in the cell, and this can depend on whether we bring GlpF to overexpression or not. For simplicity the production/degradation of GlpF is not included explicitly in the model, instead we can vary the constant v5 relative to the value found for wild-type E. coli.
|  Circles correspond to core substances. We consider the reactions between the overlapping substances so fast that we model them by determining the ratios between the substances when the reactions between them are in equilibrium. |
| Name | Units | Description | ||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| k8 | 1/s | Reaction rate constant representing how fast ArsB can export arsenic. | ||||||||||
| K1d | M | Dissociation constant for ArsR and As(III).
| ||||||||||
| K3d | M | Dissociation constants for ArsR and ars.
| ||||||||||
| v5 | mol/(s·L) | Maximum import rate per liter of cells (see Michaelis-Menten equation). Note that we have purposefully chosen to write the units as mol/(s·L) instead of M/s, to emphasize the fact that the rate is per liter of cells.
| ||||||||||
| K5 | M | Concentration at which import reaches half its maximum import rate (see Michaelis-Menten equation).
| ||||||||||
| K7 | M | Concentration at which export reaches half its maximum export rate (see Michaelis-Menten equation).
| ||||||||||
| τB, τR, τG | s | Half-lifes (of ArsB, ArsR and GV, respectively). Degradation rate = ln(2)/τ If you take just the degradation into account you will have the equation dC/dt = -k*C, which leads to C(t) = C(0) e-k t. So if k = ln(2)/τ we get C(t) = C(0) e-ln(2)/τ t = C(0) 2-t/τ. In other words τ is the time it takes for the concentration to half. i | ||||||||||
| β1, β2, etc. | 1/s | Production rates.
| ||||||||||
| Vs | L | Volume of solution (excluding cells). | ||||||||||
| Vc | L | Total volume of cells (in solution) (so Vs+Vc is the total volume). | ||||||||||
| ||||||||||||
The raw model
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)ex
- Membrane (all naturally occurring, but we plan to bring GlpF to overexpression):
- GlpF (concentration w.r.t. the exterior of the cell)
- GlpFAs (concentration w.r.t. the exterior of the cell)
- ArsB (concentration w.r.t. the interior of the cell)
- ArsBAs (concentration w.r.t. the interior of the cell)
- Intracellular (ars2, pro and GV are introduced):
- As(III)
- ars1 (concentration of unbound promoters naturally occurring in E. coli)
- ars2 (concentration of unbound promoters in front of gas vesicle genes)
- pro (concentration of constitutive promoters in front of arsR)
- ArsR ArsR binds to ars to repress production of the genes they regulate, and binds to As(III) to make it less of a problem for the cell.i
- ArsRAs (bound to As(III))
- ArsRars1 (bound to ars1)
- ArsRars2 (bound to ars2)
- GV (concentration of gas vesicles)
The variables above can be related to each other through the following "reactions" (color coding is continued below to show which parts of the differential equations refer to which groups of reactions):
- Transport (based on Rosen1996, Meng2004 and Rosen2009)
- As(III)ex + GlpF ↔ GlpFAs
- GlpFAs → GlpF + As(III)
- As(III) + ArsB ↔ ArsBAs
- ArsBAs → ArsB + As(III)ex
- ArsB → null (degradation)
- Accumulation (based on Chen1997)
- As(III) + ArsR ↔ ArsRAs
- ars1 + 2 ArsR ↔ ArsRars1
- ars2 + 2 ArsR ↔ ArsRars2
- ars1 → ars1 + ArsR + ArsB (transcription + translation)
- ars2 → ars2 + GV (transcription + translation)
- pro → pro + ArsR (transcription + translation)
- ArsR → null (degradation)
- GV → null (degradation)
Resulting in the following differential equations (please note that some can be formed by linear combinations of the others), using color coding to show the correspondence to the reactions above:
- (d/dt) As(III)ex = - (d/dt) GlpFAs - k6 GlpFAs + (Vc/Vs) k8 ArsBAs
- (d/dt) GlpF = - (d/dt) GlpFAs
- (d/dt) GlpFAs = k5on As(III)ex GlpF - (k5off+k6) GlpFAs
- (d/dt) ArsB = - (d/dt) ArsBAs + β4 ars1 - ln(2)/τB ArsB
- (d/dt) ArsBAs = k7on As(III) ArsB - (k7off+k8) ArsBAs
- (d/dt) As(III) = - (d/dt) ArsRAs - (d/dt) ArsBAs - k8 ArsBAs + (Vs/Vc) k6 GlpFAs
- (d/dt) ars1 = - (d/dt) ArsRars1
- (d/dt) ars2 = - (d/dt) ArsRars2
- (d/dt) ArsR = β1 ars1 + β3 pro - (ln(2)/τR) ArsR - (d/dt) ArsRAs - 2 (d/dt) ArsRars1 - 2 (d/dt) ArsRars2
- (d/dt) ArsRAs = k1on ArsR As(III) - k1off ArsRAs
- (d/dt) ArsRars1 = k3on ArsR² ars1 - k3off ArsRars1
- (d/dt) ArsRars2 = k3on ArsR² ars2 - k3off ArsRars2
- (d/dt) GV = β5 ars2 - ln(2)/τG GV
Using the following constants/definitions:
| Name | Units | Description |
|---|---|---|
| k1on, k5on, etc. | 1/(M·s) | Reaction rate constants for reactions to a complex. |
| k3on | 1/(M²·s) | Reaction rate constants for reactions to a complex. |
| k1off, k3off, k5off, etc. | 1/s | Reaction rate constants for reactions from a complex. |
| k6, k8 | 1/s | Reaction rate constants representing how fast transporters transport their cargo to "the other side". |
| τB, τR, τG | s | Half-lifes (of ArsB, ArsR and GV, respectively). Degradation rate = ln(2)/τ If you take just the degradation into account you will have the equation dC/dt = -k*C, which leads to C(t) = C(0) e-k t. So if k = ln(2)/τ we get C(t) = C(0) e-ln(2)/τ t = C(0) 2-t/τ. In other words τ is the time it takes for the concentration to half. i |
| β1, β2, etc. | 1/s | Production rates.
|
| Vs | L | Volume of solution (excluding cells). |
| Vc | L | Total volume of cells (in solution) (so Vs+Vc is the total volume). |
See Chen1997 for the interplay between ArsR and ArsD (the latter has a role similar to ArsR, but we do not treat it, as it is not present in our system).
Quasi steady state
When there are many molecules "waiting" to be transported and/or the concentrations in the cell and outside the cell are relatively slow changing compared to it is not unreasonable to assume that the amount of bound transporters is constant. Similarly, within the cell we assume diffusion processes are very fast and binding/unbinding of substances is quite fast compared to the production of proteins. This leads us to assume that the following ratios between substances are constantly in equilibrium:
As(III)ex : GlpFAs GlpF : GlpFAs ArsB : ArsBAs As(III) : ArsRAs : ArsBAs ArsR : ArsRAs : 2 ArsRars ars : ArsRars
Here we use the following to simplify the equations involving the ars promoters:
arsT = ars + ArsRars ars1 / ars1T = ars2 / ars2T ars = ars1 + ars2 ars = ars1 (1 + ars2T / ars1T) ars1 = ars / (1 + ars2T / ars1T) ars1 = ars ars1T / arsT ars2 = ars ars2T / arsT (d/dt) ArsR = β1 ars ars1T/arsT + β3 pro - (ln(2)/τR) ArsR - (d/dt) ArsRAs - 2 (d/dt) ArsRars (d/dt) GV = β5 ars ars2T/arsT - ln(2)/τG GV
To determine what these ratios are we can set the following derivatives to zero (these are the derivatives of the complexes corresponding to the four overlapping regions in the diagram):
0 = (d/dt) GlpFAs = k5on As(III)ex GlpF - (k5off+k6) GlpFAs 0 = (d/dt) ArsBAs = k7on As(III) ArsB - (k7off+k8) ArsBAs 0 = (d/dt) ArsRars = k3on ArsR² ars - k3off ArsRars 0 = (d/dt) ArsRAs = k1on ArsR As(III) - k1off ArsRAs
The first two derivates let us determine the ratios between bound and unbound transporters:
0 = (d/dt) GlpFAs = k5on As(III)ex GlpF - (k5off+k6) GlpFAs
k5on As(III)ex GlpF = (k5off+k6) GlpFAs
GlpF = (k5off+k6)/k5on GlpFAs / As(III)ex
GlpF = K5 GlpFAs / As(III)ex
GlpF : GlpFAs
K5 GlpFAs / As(III)ex : GlpFAs
K5 : As(III)ex
As(III)ex : GlpFAs
As(III)ex : GlpF As(III)ex / K5
K5 : GlpF
0 = (d/dt) ArsBAs = k7on As(III) ArsB - (k7off+k8) ArsBAs
k7on As(III) ArsB = (k7off+k8) ArsBAs
ArsB = (k7off+k8)/k7on ArsBAs / As(III)
ArsB = K7 ArsBAs / As(III)
ArsB : ArsBAs
K7 ArsBAs / As(III) : ArsBAs
K7 : As(III)
The other two differential equations can be used to determine the relative abundances of ArsR, ArsRAs and ArsRars (the latter is counted twice as ArsR binds to ars as a dimer):
0 = (d/dt) ArsRAs = k1on ArsR As(III) - k1off ArsRAs
k1on ArsR As(III) = k1off ArsRAs
ArsRAs = k1on/k1off ArsR As(III)
ArsRAs = ArsR As(III) / K1d
0 = (d/dt) ArsRars = k3on ArsR² ars - k3off ArsRars
k3on ArsR² ars = k3off ArsRars
ArsRars = k3on/k3off ArsR² ars
ArsRars = ArsR² ars / K3d²
ArsR : ArsRAs : 2 ArsRars
ArsR : ArsR As(III) / K1d : 2 ArsR² ars / K3d²
1 : As(III) / K1d : 2 ArsR ars / K3d²
ars : ArsRars
ars : ArsR² ars / K3d²
K3d² : ArsR²
And finally the relative abundances of arsenic:
ArsB = K7 ArsBAs / As(III) ArsBAs = ArsB As(III) / K7 ArsRAs = ArsR As(III) / K1d As(III) : ArsRAs : ArsBAs As(III) : ArsR As(III) / K1d : ArsB As(III) / K7 1 : ArsR / K1d : ArsB / K7
Summarizing:
As(III)ex : GlpFAs = K5 : GlpF GlpF : GlpFAs = K5 : As(III)ex ArsB : ArsBAs = K7 : As(III) As(III) : ArsRAs : ArsBAs = 1 : ArsR / K1d : ArsB / K7 ArsR : ArsRAs : 2 ArsRars = 1 : As(III) / K1d : 2 ArsR ars / K3d² ars : ArsRars = K3d² : ArsR²
Now we can look at the differential equations for the totals of ArsB (so ArsBT=ArsB+ArsBAs), ArsR, As(III) and As(III)ex (GlpFT and arsT are assumed to be constant):
(d/dt) As(III)exT = (d/dt) As(III)ex + (d/dt) GlpFAs
= - (d/dt) GlpFAs - k6 GlpFAs + (Vc/Vs) k8 ArsBAs + (d/dt) GlpFAs
= (Vc/Vs) k8 ArsBAs - k6 GlpFAs
= (Vc/Vs) k8 ArsBAs - (Vc/Vs) v5 GlpFAs / GlpFT
= (Vc/Vs) k8 ArsBAs - (Vc/Vs) v5 As(III)ex / (K5+As(III)ex)
(d/dt) ArsBT = (d/dt) ArsB + (d/dt) ArsBAs
= - (d/dt) ArsBAs + β4 ars1 - ln(2)/τB ArsB + (d/dt) ArsBAs
= β4 ars1 - ln(2)/τB ArsB
(d/dt) As(III)T = (d/dt) As(III) + (d/dt) ArsRAs + (d/dt) ArsBAs
= - (d/dt) ArsRAs - (d/dt) ArsBAs - k8 ArsBAs + (Vs/Vc) k6 GlpFAs + (d/dt) ArsRAs + (d/dt) ArsBAs
= (Vs/Vc) k6 GlpFAs - k8 ArsBAs
= v5 As(III)ex / (K5+As(III)ex) - k8 ArsBAs
(d/dt) ArsRT = (d/dt) ArsR + (d/dt) ArsRAs + 2 (d/dt) ArsRars
= β1 ars1 + β3 pro - (ln(2)/τR) ArsR - (d/dt) ArsRAs - 2 (d/dt) ArsRars + (d/dt) ArsRAs + 2 (d/dt) ArsRars
= β1 ars1 + β3 pro - (ln(2)/τR) ArsR
Steady state
By looking at the steady state of the system we can say something about its long-term behaviour. This also makes it easier to analyze relations between variables. To derive the steady state solution we take the quasi steady state solution and simplify it further by setting additional derivatives to zero:
0 = (d/dt) ArsBT = β4 ars1 - ln(2)/τB ArsB 0 = (d/dt) As(III)T = v5 As(III)ex / (K5+As(III)ex) - k8 ArsBAs 0 = (d/dt) ArsRT = β1 ars1 + β3 pro - (ln(2)/τR) ArsR 0 = (d/dt) GV = β5 ars2 - ln(2)/τG GV
This directly leads to:
0 = β4 ars1 - ln(2)/τB ArsB ArsB = β4 (τB/ln(2)) ars1 ArsB = β4 (τB/ln(2)) ars1T K3d²/(K3d²+ArsR²) 0 = β5 ars2 - ln(2)/τG GV GV = β5 (τB/ln(2)) ars2 GV = β5 (τB/ln(2)) ars2T K3d²/(K3d²+ArsR²)
For the intra- and extracellular concentrations we can find the following condition:
0 = v5 As(III)ex / (K5+As(III)ex) - k8 ArsBAs 0 = v5 As(III)ex / (K5+As(III)ex) - k8 ArsBT As(III) / (K7+As(III)) 0 = v5 As(III)ex (K7+As(III)) - k8 ArsBT As(III) (K5+As(III)ex) 0 = v5 K7 As(III)ex + (v5 - k8 ArsBT) As(III) As(III)ex - k8 ArsBT K5 As(III)As we can safely assume arsenic neither disappears into nothingness nor appears from nothingness, we can use this to derive a quadratic equation for As(III)
Note that in the final solution we can safely take just the plus (in the general ABC formula
i (-b±√(b²-4ac))/(2a)) instead of both plus and minus, as the square root will always yield a value that is positive and greater in magnitude than b, ensuring that the final answer is positive if and only if a plus is used.
As(III)0 = Vs As(III)ex + Vc As(III)
Vs As(III)ex = As(III)0 - Vc As(III)
0 = v5 K7 As(III)ex + (v5 - k8 ArsBT) As(III) As(III)ex - k8 ArsBT K5 As(III)
0 = v5 K7 Vs As(III)ex + (v5 - k8 ArsBT) As(III) Vs As(III)ex - k8 ArsBT K5 Vs As(III)
0 = v5 K7 As(III)0 - v5 K7 Vc As(III) + (v5 - k8 ArsBT) As(III)0 As(III) - (v5 - k8 ArsBT) Vc As(III)² - k8 ArsBT K5 Vs As(III)
0 = v5 K7 As(III)0 + (v5 (As(III)0 - Vc K7) - k8 ArsBT (As(III)0 - Vs K5)) As(III) - (v5 - k8 ArsBT) Vc As(III)²
As(III) = (√(b² + 4 a c) - b)/(2 a)
a = (v5 - k8 ArsBT) Vc
b = v5 (Vc K7 - As(III)0) - k8 ArsBT (Vs K5 - As(III)0)
c = v5 K7 As(III)0
Finally, for ArsR we can find the following third-order equation:
0 = β1 ars1 + β3 pro - (ln(2)/τR) ArsR 0 = β1 ars1T K3d²/(K3d²+ArsR²) + β3 pro - (ln(2)/τR) ArsR 0 = β1 ars1T K3d² + β3 pro (K3d²+ArsR²) - (ln(2)/τR) ArsR (K3d²+ArsR²) 0 = β1 ars1T K3d² + β3 pro K3d² + β3 pro ArsR² - (ln(2)/τR) ArsR K3d² - (ln(2)/τR) ArsR³ 0 = (β1 ars1T + β3 pro) K3d² - (ln(2)/τR) K3d² ArsR + β3 pro ArsR² - (ln(2)/τR) ArsR³ 0 = (β1 ars1T + β3 pro) (τR/ln(2)) K3d² - K3d² ArsR + β3 (τR/ln(2)) pro ArsR² - ArsR³According to Mathematica's solution of
Reduce[eq && K3d > 0 && arsT >= 0 && pro >= 0 && β1 > 0 && β3 > 0 && τR > 0, ArsR, Reals] (where eq is the equation shown above) there is only one real solution (examining the discriminant of eq confirms this), so we can solve the equation safely using Newton's (or Halley's) method.
