Team:Groningen/Modelling/Arsenic
From 2009.igem.org
m (Marked how much we know about constants.) |
(Further simplifications of model and some cleaning up.) |
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[[Category:Team:Groningen/Disciplines/Analysis_and_Design|Modelling]] | [[Category:Team:Groningen/Disciplines/Analysis_and_Design|Modelling]] | ||
[[Category:Team:Groningen/Roles/Modeller|Modelling]] | [[Category:Team:Groningen/Roles/Modeller|Modelling]] | ||
+ | <html><style type="text/css"> | ||
+ | .fromPaper { background:#dfd; } | ||
+ | .selfDerived { background:#dfd; } | ||
+ | .experimental { background:#ddf; } | ||
+ | .estimate { background:#ffc; } | ||
+ | .unknown { background:#fee; } | ||
+ | </style></html> | ||
<div style="height:10px;"></div> | <div style="height:10px;"></div> | ||
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|colspan="4"|''Transport'' | |colspan="4"|''Transport'' | ||
|- | |- | ||
- | | ||As(III)<sub>ex</sub>T → As(III)T||Import of arsenic.||style="white-space:nowrap;"|v5<sup>†</sup> As(III)<sub>ex</sub> / (K5+As(III)<sub>ex</sub>) | + | | ||As(III)<sub>ex</sub>T → As(III)<sub>in</sub>T||Import of arsenic.||style="white-space:nowrap;"|(Vc/Vs) v5<sup>†</sup> As(III)<sub>ex</sub>T / (K5+As(III)<sub>ex</sub>T) |
|- | |- | ||
- | | ||As(III)T → As(III)<sub>ex</sub>T||Export of arsenic.|| k8 ArsB<sub>As</sub> | + | | ||As(III)<sub>in</sub>T → As(III)<sub>ex</sub>T||Export of arsenic.|| k8 ArsB<sub>As</sub> |
|- | |- | ||
| ||style="white-space:nowrap;"|ars1T → ars1T + ArsBT||Production of ArsB.|| β4 ars1 | | ||style="white-space:nowrap;"|ars1T → ars1T + ArsBT||Production of ArsB.|| β4 ars1 | ||
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|colspan="4"|''Extracellular'' | |colspan="4"|''Extracellular'' | ||
|- | |- | ||
- | | ||As(III)<sub>ex</sub>T || As(III) in the solution. || (Vc/Vs) k8 ArsB<sub>As</sub> - (Vc/Vs) v5<sup>†</sup> As(III)<sub>ex</sub> / (K5+As(III)<sub>ex</sub>) | + | | ||As(III)<sub>ex</sub>T || As(III) in the solution. || (Vc/Vs) k8 ArsB<sub>As</sub> - (Vc/Vs) v5<sup>†</sup> As(III)<sub>ex</sub>T / (K5+As(III)<sub>ex</sub>T) |
|- | |- | ||
- | |colspan="4"|''Membrane (all naturally occurring, but we plan to bring GlpF to overexpression)'' | + | |colspan="4"|''Membrane (naturally occurring<!--all naturally occurring, but we plan to bring GlpF to overexpression-->)'' |
- | + | ||
- | + | ||
|- | |- | ||
| ||ArsBT || Exporter of As(III) (concentration w.r.t. the interior of the cell). || β4 ars1 - ln(2)/τB ArsB | | ||ArsBT || Exporter of As(III) (concentration w.r.t. the interior of the cell). || β4 ars1 - ln(2)/τB ArsB | ||
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|colspan="4"|''Intracellular (ars2, pro and GV are introduced)'' | |colspan="4"|''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 ArsB<sub>As</sub> | + | | ||As(III)<sub>in</sub>T || As(III) (bound and unbound) in the cell. || v5 As(III)<sub>ex</sub>T / (K5+As(III)<sub>ex</sub>T) - k8 ArsB<sub>As</sub> |
- | |- | + | |-class="estimate" |
- | | ||ars1T || ArsR repressed promoters (bound and unbound) naturally occurring in E. coli. || (concentration is constant) | + | | ||ars1T || ArsR repressed promoters (bound and unbound) naturally occurring in E. coli. || (concentration is constant = 1.6605nM, one per cell) |
- | |- | + | |-class="estimate" |
- | | ||ars2T || ArsR repressed promoters in front of gas vesicle genes. || (concentration is constant) | + | | ||ars2T || ArsR repressed promoters in front of gas vesicle genes. || (concentration is constant = 16.605nM, ten per cell) |
- | |- | + | |-class="estimate" |
- | | ||pro || Constitutive promoters in front of arsR. || (concentration is constant) | + | | ||pro || Constitutive promoters in front of arsR. || (concentration is constant = 16.605nM, ten per cell) |
|- | |- | ||
| ||ArsRT || ArsR in the cell. || β1 ars1 + β3 pro - (ln(2)/τR) ArsR | | ||ArsRT || ArsR in the cell. || β1 ars1 + β3 pro - (ln(2)/τR) ArsR | ||
|- | |- | ||
| ||GV || Concentration of gas vesicles. || β5 ars2 - ln(2)/τG GV | | ||GV || Concentration of gas vesicles. || β5 ars2 - ln(2)/τG GV | ||
+ | |-style="border:none;" | ||
+ | |colspan="4"| | ||
+ | {|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. | ||
+ | |} | ||
|} | |} | ||
<div style="text-align:right;font-size:smaller;"><sup>†</sup> 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.</div> | <div style="text-align:right;font-size:smaller;"><sup>†</sup> 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.</div> | ||
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!Component | !Component | ||
!Relative abundance | !Relative abundance | ||
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|- | |- | ||
|rowspan="2"|ArsBT | |rowspan="2"|ArsBT | ||
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|- | |- | ||
|ArsB<sub>As</sub> | |ArsB<sub>As</sub> | ||
- | |As(III) | + | |As(III)in |
|- | |- | ||
- | |rowspan=" | + | |rowspan="2"|As(III)inT |
- | |style="padding-left:0;"|As(III) | + | |style="padding-left:0;"|As(III)in |
- | | | + | |K1<sub>d</sub> |
|- | |- | ||
|ArsR<sub>As</sub> | |ArsR<sub>As</sub> | ||
- | |ArsR | + | |ArsR |
- | + | ||
- | + | ||
- | + | ||
|- | |- | ||
|rowspan="2"|arsT | |rowspan="2"|arsT | ||
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|ars2T | |ars2T | ||
|- | |- | ||
- | |rowspan=" | + | |rowspan="2"|ArsRT |
|style="padding-left:0;"|ArsR | |style="padding-left:0;"|ArsR | ||
- | | | + | |K1<sub>d</sub> |
|- | |- | ||
|ArsR<sub>As</sub> | |ArsR<sub>As</sub> | ||
- | |As(III) | + | |As(III)<sub>in</sub> |
- | + | ||
- | + | ||
- | + | ||
|} | |} | ||
- | |[[Image:Arsenic Model - Substances.png|frame|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.]] | + | |[[Image:Arsenic Model - Substances.png|frame|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. Also, the complexes formed with ars, GlpF and ArsB are considered to have such a low concentration that they are of no importance to the concentrations of As(III)in/-ex and ArsR.]] |
|} | |} | ||
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{|class="ourtable" | {|class="ourtable" | ||
|+ Constants | |+ Constants | ||
!Name | !Name | ||
!Units | !Units | ||
+ | !Value | ||
!Description | !Description | ||
|-class="unknown" | |-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" | |-class="estimate" | ||
|K1<sub>d</sub> | |K1<sub>d</sub> | ||
|M | |M | ||
- | |Dissociation constant for ArsR and As(III). | + | |6µM |
- | + | |Dissociation constant for ArsR and As(III). 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" | |-class="fromPaper" | ||
|K3<sub>d</sub> | |K3<sub>d</sub> | ||
|M | |M | ||
+ | |(0.33µM)² | ||
|Dissociation constants for ArsR and ars. | |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) | ||
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|v5 | |v5 | ||
|mol/(s·L) | |mol/(s·L) | ||
+ | |3.1863µmol/(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) | ||
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|K5 | |K5 | ||
|M | |M | ||
+ | |27.718µM | ||
|Concentration at which import reaches half its maximum import rate (see [[Team:Groningen/Glossary#MichaelisMenten|Michaelis-Menten equation]]). | |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 | ||
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|K7 | |K7 | ||
|M | |M | ||
+ | | | ||
|Concentration at which export reaches half its maximum export rate (see [[Team:Groningen/Glossary#MichaelisMenten|Michaelis-Menten equation]]). | |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 | ||
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|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" | |-class="unknown" | ||
|β1, β2, etc. | |β1, β2, etc. | ||
|1/s | |1/s | ||
+ | | | ||
|Production rates. | |Production rates. | ||
* β1 = the production rate for ArsR behind the ars1 promoter | * β1 = the production rate for ArsR behind the ars1 promoter | ||
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|Vs | |Vs | ||
|L | |L | ||
+ | | | ||
|Volume of solution (excluding cells). | |Volume of solution (excluding cells). | ||
|- | |- | ||
|Vc | |Vc | ||
|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). | ||
|-style="border:none;" | |-style="border:none;" | ||
- | |colspan=" | + | |colspan="4"| |
{|class="ourtable" style="width:100%" | {|class="ourtable" style="width:100%" | ||
!colspan="5"| | !colspan="5"| | ||
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** ArsB<sub>As</sub> (concentration w.r.t. the interior of the cell) | ** ArsB<sub>As</sub> (concentration w.r.t. the interior of the cell) | ||
* Intracellular (ars2, pro and GV are introduced): | * Intracellular (ars2, pro and GV are introduced): | ||
- | ** As(III) | + | ** As(III)<sub>ex</sub> |
** ars1 (concentration of unbound promoters naturally occurring in E. coli) | ** ars1 (concentration of unbound promoters naturally occurring in E. coli) | ||
** ars2 (concentration of unbound promoters in front of gas vesicle genes) | ** ars2 (concentration of unbound promoters in front of gas vesicle genes) | ||
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** <span class="import">As(III)<sub>ex</sub> + GlpF ↔ GlpF<sub>As</sub></span> | ** <span class="import">As(III)<sub>ex</sub> + GlpF ↔ GlpF<sub>As</sub></span> | ||
** <span class="import">GlpF<sub>As</sub> → GlpF + As(III)</span> | ** <span class="import">GlpF<sub>As</sub> → GlpF + As(III)</span> | ||
- | ** <span class="export">As(III) + ArsB ↔ ArsB<sub>As</sub></span> | + | ** <span class="export">As(III)<sub>in</sub> + ArsB ↔ ArsB<sub>As</sub></span> |
** <span class="export">ArsB<sub>As</sub> → ArsB + As(III)<sub>ex</sub></span> | ** <span class="export">ArsB<sub>As</sub> → ArsB + As(III)<sub>ex</sub></span> | ||
** <span class="export">ArsB → null</span> (degradation) | ** <span class="export">ArsB → null</span> (degradation) | ||
* Accumulation (based on [[Team:Groningen/Literature#Chen1997|Chen1997]]) | * Accumulation (based on [[Team:Groningen/Literature#Chen1997|Chen1997]]) | ||
- | ** As(III) + ArsR ↔ ArsR<sub>As</sub> | + | ** As(III)<sub>in</sub> + ArsR ↔ ArsR<sub>As</sub> |
** ars1 + 2 ArsR ↔ ArsR<sub>ars1</sub> | ** ars1 + 2 ArsR ↔ ArsR<sub>ars1</sub> | ||
** <span class="production">ars2 + 2 ArsR ↔ ArsR<sub>ars2</sub></span> | ** <span class="production">ars2 + 2 ArsR ↔ ArsR<sub>ars2</sub></span> | ||
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* (d/dt) GlpF<sub>As</sub> = <span class="import">k5<sub>on</sub> As(III)<sub>ex</sub> GlpF - (k5<sub>off</sub>+k6) GlpF<sub>As</sub></span> | * (d/dt) GlpF<sub>As</sub> = <span class="import">k5<sub>on</sub> As(III)<sub>ex</sub> GlpF - (k5<sub>off</sub>+k6) GlpF<sub>As</sub></span> | ||
* (d/dt) ArsB = <span class="export">- (d/dt) ArsB<sub>As</sub> + β4 ars1 - ln(2)/τB ArsB</span> | * (d/dt) ArsB = <span class="export">- (d/dt) ArsB<sub>As</sub> + β4 ars1 - ln(2)/τB ArsB</span> | ||
- | * (d/dt) ArsB<sub>As</sub> = <span class="export">k7<sub>on</sub> As(III) ArsB - (k7<sub>off</sub>+k8) ArsB<sub>As</sub></span> | + | * (d/dt) ArsB<sub>As</sub> = <span class="export">k7<sub>on</sub> As(III)<sub>in</sub> ArsB - (k7<sub>off</sub>+k8) ArsB<sub>As</sub></span> |
- | * (d/dt) As(III) = - (d/dt) ArsR<sub>As</sub><span class="export"> - (d/dt) ArsB<sub>As</sub> - k8 ArsB<sub>As</sub></span><span class="import"> + (Vs/Vc) k6 GlpF<sub>As</sub></span> | + | * (d/dt) As(III)<sub>in</sub> = - (d/dt) ArsR<sub>As</sub><span class="export"> - (d/dt) ArsB<sub>As</sub> - k8 ArsB<sub>As</sub></span><span class="import"> + (Vs/Vc) k6 GlpF<sub>As</sub></span> |
* (d/dt) ars1 = - (d/dt) ArsR<sub>ars1</sub> | * (d/dt) ars1 = - (d/dt) ArsR<sub>ars1</sub> | ||
* (d/dt) ars2 = <span class="production">- (d/dt) ArsR<sub>ars2</sub></span> | * (d/dt) ars2 = <span class="production">- (d/dt) ArsR<sub>ars2</sub></span> | ||
* (d/dt) ArsR = β1 ars1 + β3 pro - (ln(2)/τR) ArsR - (d/dt) ArsR<sub>As</sub> - 2 (d/dt) ArsR<sub>ars1</sub><span class="production"> - 2 (d/dt) ArsR<sub>ars2</sub></span> | * (d/dt) ArsR = β1 ars1 + β3 pro - (ln(2)/τR) ArsR - (d/dt) ArsR<sub>As</sub> - 2 (d/dt) ArsR<sub>ars1</sub><span class="production"> - 2 (d/dt) ArsR<sub>ars2</sub></span> | ||
- | * (d/dt) ArsR<sub>As</sub> = k1<sub>on</sub> ArsR As(III) - k1<sub>off</sub> ArsR<sub>As</sub> | + | * (d/dt) ArsR<sub>As</sub> = k1<sub>on</sub> ArsR As(III)<sub>in</sub> - k1<sub>off</sub> ArsR<sub>As</sub> |
* (d/dt) ArsR<sub>ars1</sub> = k3<sub>on</sub> ArsR² ars1 - k3<sub>off</sub> ArsR<sub>ars1</sub> | * (d/dt) ArsR<sub>ars1</sub> = k3<sub>on</sub> ArsR² ars1 - k3<sub>off</sub> ArsR<sub>ars1</sub> | ||
* (d/dt) ArsR<sub>ars2</sub> = <span class="production">k3<sub>on</sub> ArsR² ars2 - k3<sub>off</sub> ArsR<sub>ars2</sub></span> | * (d/dt) ArsR<sub>ars2</sub> = <span class="production">k3<sub>on</sub> ArsR² ars2 - k3<sub>off</sub> ArsR<sub>ars2</sub></span> | ||
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==Quasi steady state{{anchor|QuasiSteadyState}}== | ==Quasi steady state{{anchor|QuasiSteadyState}}== | ||
- | + | First of all, we assume the concentration of transporters is quite low compared to the concentration of the transported substances. After all, if this were not the case the transporters would act more like "storage" proteins than transporters (note that this can be even more rigorously justified if, for example, GlpFT<<K5). Similarly, there will generally only be a few ars promoters, compared to very many ArsR molecules. This leads to: | |
<pre> | <pre> | ||
- | As(III)ex | + | As(III)exT ≈ As(III)ex |
- | + | As(III)inT ≈ As(III)in + ArsRAs | |
- | + | ArsRT ≈ ArsR : ArsRAs | |
- | As(III) | + | |
- | ArsR : ArsRAs | + | |
- | + | ||
</pre> | </pre> | ||
- | + | Also, we assume the binding and unbinding of molecules to the transporters occurs on a much finer time-scale than any actual changes to the concentrations inside and outside the cell. 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: | |
+ | {{frame|1= | ||
+ | <div style="text-align:left;"> | ||
+ | We use the following when grouping the ars promoters: | ||
<pre> | <pre> | ||
arsT = ars + ArsRars | arsT = ars + ArsRars | ||
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ars2 = ars ars2T / arsT | ars2 = ars ars2T / arsT | ||
+ | </pre> | ||
+ | </div> | ||
+ | }} | ||
- | ( | + | <pre> |
- | + | As(III)ex : GlpFAs ≈ As(III)ex : 0 | |
+ | GlpF : GlpFAs | ||
+ | ArsB : ArsBAs | ||
+ | As(III)in : ArsRAs : ArsBAs ≈ As(III)in : ArsRAs : 0 | ||
+ | ArsR : ArsRAs : 2 ArsRars ≈ ArsR : ArsRAs : 0 | ||
+ | ars : ArsRars | ||
</pre> | </pre> | ||
- | To determine what | + | To determine what the unknown 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): |
<pre> | <pre> | ||
0 = (d/dt) GlpFAs = k5on As(III)ex GlpF - (k5off+k6) GlpFAs | 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) ArsBAs = k7on As(III)in ArsB - (k7off+k8) ArsBAs |
0 = (d/dt) ArsRars = k3on ArsR² ars - k3off ArsRars | 0 = (d/dt) ArsRars = k3on ArsR² ars - k3off ArsRars | ||
- | 0 = (d/dt) ArsRAs = k1on ArsR As(III) - k1off ArsRAs | + | 0 = (d/dt) ArsRAs = k1on ArsR As(III)in - k1off ArsRAs |
</pre> | </pre> | ||
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K5 : As(III)ex | K5 : As(III)ex | ||
- | + | 0 = (d/dt) ArsBAs = k7on As(III)in ArsB - (k7off+k8) ArsBAs | |
- | As(III) | + | |
- | + | ||
- | + | k7on As(III)in ArsB = (k7off+k8) ArsBAs | |
+ | ArsB = (k7off+k8)/k7on ArsBAs / As(III)in | ||
+ | ArsB = K7 ArsBAs / As(III)in | ||
- | + | ArsB : ArsBAs | |
- | + | K7 ArsBAs / As(III)in : ArsBAs | |
- | + | K7 : As(III)in | |
- | + | ||
- | + | ||
- | K7 ArsBAs / As(III) : ArsBAs | + | |
- | + | ||
</pre> | </pre> | ||
- | The other two differential equations can be used to determine the relative abundances of ArsR | + | The other two differential equations can be used to determine the relative abundances of ArsR and ArsRAs, and ars and ArsRars: |
<pre> | <pre> | ||
- | 0 = (d/dt) ArsRAs = k1on ArsR As(III) - k1off ArsRAs | + | 0 = (d/dt) ArsRAs = k1on ArsR As(III)in - k1off ArsRAs |
- | k1on ArsR As(III) = k1off ArsRAs | + | k1on ArsR As(III)in = k1off ArsRAs |
- | ArsRAs = k1on/k1off ArsR As(III) | + | ArsRAs = k1on/k1off ArsR As(III)in |
- | ArsRAs = ArsR As(III) / K1d | + | ArsRAs = ArsR As(III)in / K1d |
+ | |||
+ | ArsR : ArsRAs | ||
+ | ArsR : ArsR As(III)in / K1d | ||
+ | K1d : As(III)in | ||
0 = (d/dt) ArsRars = k3on ArsR² ars - k3off ArsRars | 0 = (d/dt) ArsRars = k3on ArsR² ars - k3off ArsRars | ||
Line 424: | Line 431: | ||
ArsRars = k3on/k3off ArsR² ars | ArsRars = k3on/k3off ArsR² ars | ||
ArsRars = ArsR² ars / K3d² | ArsRars = ArsR² ars / K3d² | ||
- | |||
- | |||
- | |||
- | |||
ars : ArsRars | ars : ArsRars | ||
Line 437: | Line 440: | ||
<pre> | <pre> | ||
- | + | ArsRAs = ArsR As(III)in / K1d | |
- | + | ||
- | + | As(III)in : ArsRAs | |
- | + | As(III)in : ArsR As(III)in / K1d | |
- | As(III) : ArsRAs | + | K1d : ArsR |
- | As(III) : ArsR As(III) / K1d | + | |
- | + | ||
</pre> | </pre> | ||
Line 450: | Line 450: | ||
<pre> | <pre> | ||
- | As(III)ex : GlpFAs | + | As(III)ex : GlpFAs ≈ As(III)ex : 0 |
GlpF : GlpFAs = K5 : As(III)ex | GlpF : GlpFAs = K5 : As(III)ex | ||
- | ArsB : ArsBAs = K7 : As(III) | + | ArsB : ArsBAs = K7 : As(III)in |
- | As(III) : ArsRAs : ArsBAs | + | As(III)in : ArsRAs : ArsBAs ≈ K1d : ArsR : 0 |
- | ArsR : ArsRAs : 2 ArsRars | + | ArsR : ArsRAs : 2 ArsRars ≈ K1d : As(III)in : 0 |
ars : ArsRars = K3d² : ArsR² | ars : ArsRars = K3d² : ArsR² | ||
</pre> | </pre> | ||
- | 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): | + | Now we can look at the differential equations for the totals of ArsB (so ArsBT=ArsB+ArsBAs), ArsR, As(III)in and As(III)ex (GlpFT and arsT are assumed to be constant): |
<pre> | <pre> | ||
Line 466: | Line 466: | ||
= (Vc/Vs) k8 ArsBAs - (Vc/Vs) v5 GlpFAs / GlpFT | = (Vc/Vs) k8 ArsBAs - (Vc/Vs) v5 GlpFAs / GlpFT | ||
= (Vc/Vs) k8 ArsBAs - (Vc/Vs) v5 As(III)ex / (K5+As(III)ex) | = (Vc/Vs) k8 ArsBAs - (Vc/Vs) v5 As(III)ex / (K5+As(III)ex) | ||
+ | = (Vc/Vs) k8 ArsBAs - (Vc/Vs) v5 As(III)exT / (K5+As(III)exT) | ||
(d/dt) ArsBT = (d/dt) ArsB + (d/dt) ArsBAs | (d/dt) ArsBT = (d/dt) ArsB + (d/dt) ArsBAs | ||
= - (d/dt) ArsBAs + β4 ars1 - ln(2)/τB ArsB + (d/dt) ArsBAs | = - (d/dt) ArsBAs + β4 ars1 - ln(2)/τB ArsB + (d/dt) ArsBAs | ||
= β4 ars1 - ln(2)/τB ArsB | = β4 ars1 - ln(2)/τB ArsB | ||
- | (d/dt) As(III) | + | (d/dt) As(III)inT = -(Vs/Vc) (d/dt) As(III)exT |
- | + | = v5 As(III)exT / (K5+As(III)exT) - k8 ArsBT As(III)in / (K7+As(III)in) | |
- | + | ||
- | + | ||
(d/dt) ArsRT = (d/dt) ArsR + (d/dt) ArsRAs + 2 (d/dt) ArsRars | (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 - (d/dt) ArsRAs - 2 (d/dt) ArsRars + (d/dt) ArsRAs + 2 (d/dt) ArsRars | ||
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<pre> | <pre> | ||
0 = (d/dt) ArsBT = β4 ars1 - ln(2)/τB ArsB | 0 = (d/dt) ArsBT = β4 ars1 - ln(2)/τB ArsB | ||
- | 0 = (d/dt) As(III) | + | 0 = (d/dt) As(III)inT = 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) ArsRT = β1 ars1 + β3 pro - (ln(2)/τR) ArsR | ||
0 = (d/dt) GV = β5 ars2 - ln(2)/τG GV | 0 = (d/dt) GV = β5 ars2 - ln(2)/τG GV | ||
Line 500: | Line 499: | ||
</pre> | </pre> | ||
+ | <!-- | ||
For the intra- and extracellular concentrations we can find the following condition: | For the intra- and extracellular concentrations we can find the following condition: | ||
<pre> | <pre> | ||
0 = v5 As(III)ex / (K5+As(III)ex) - k8 ArsBAs | 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 / (K5+As(III)ex) - k8 ArsBT As(III)in / (K7+As(III)in) |
- | 0 = v5 As(III)ex (K7+As(III)) - k8 ArsBT As(III) (K5+As(III)ex) | + | 0 = v5 As(III)ex (K7+As(III)in) - k8 ArsBT As(III)in (K5+As(III)ex) |
- | 0 = v5 K7 As(III)ex + (v5 - k8 ArsBT) As(III) As(III)ex - k8 ArsBT K5 As(III) | + | 0 = v5 K7 As(III)ex + (v5 - k8 ArsBT) As(III)in As(III)ex - k8 ArsBT K5 As(III)in |
</pre> | </pre> | ||
- | 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){{infoBox|Note that in the final solution we can safely take just the plus (in the general ABC formula <code>(-b±√(b²-4ac))/(2a)</code>) 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) | + | 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)in{{infoBox|Note that in the final solution we can safely take just the plus (in the general ABC formula <code>(-b±√(b²-4ac))/(2a)</code>) 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)T is the total amount of arsenic): |
<pre> | <pre> | ||
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According to Mathematica's solution of <code>Reduce[eq && K3d > 0 && arsT >= 0 && pro >= 0 && β1 > 0 && β3 > 0 && τR > 0, ArsR, Reals]</code> (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. | According to Mathematica's solution of <code>Reduce[eq && K3d > 0 && arsT >= 0 && pro >= 0 && β1 > 0 && β3 > 0 && τR > 0, ArsR, Reals]</code> (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. | ||
+ | --> |
Revision as of 14:02, 17 September 2009
Based on the quasi-steady-state derivation below we have defined the following simplified model:
Reaction | Description | Rate | |
---|---|---|---|
Transport | |||
As(III)exT → As(III)inT | Import of arsenic. | (Vc/Vs) v5† As(III)exT / (K5+As(III)exT) | |
As(III)inT → 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)exT / (K5+As(III)exT) | |||||||||||
Membrane (naturally occurring) | |||||||||||||
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)inT | As(III) (bound and unbound) in the cell. | v5 As(III)exT / (K5+As(III)exT) - k8 ArsBAs | |||||||||||
ars1T | ArsR repressed promoters (bound and unbound) naturally occurring in E. coli. | (concentration is constant = 1.6605nM, one per cell) | |||||||||||
ars2T | ArsR repressed promoters in front of gas vesicle genes. | (concentration is constant = 16.605nM, ten per cell) | |||||||||||
pro | Constitutive promoters in front of arsR. | (concentration is constant = 16.605nM, ten per cell) | |||||||||||
ArsRT | ArsR in the cell. | β1 ars1 + β3 pro - (ln(2)/τR) ArsR | |||||||||||
GV | Concentration of gas vesicles. | β5 ars2 - ln(2)/τG GV | |||||||||||
|
|
Name | Units | Value | Description | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|
k8 | 1/s | Reaction rate constant representing how fast ArsB can export arsenic. | |||||||||||
K1d | M | 6µM | Dissociation constant for ArsR and As(III). Assumed to be about an order of magnitude smaller than K2d = 60µM, the corresponding constant for the similar protein ArsD from Chen1997. | ||||||||||
K3d | M | (0.33µM)² | Dissociation constants for ArsR and ars.
| ||||||||||
v5 | mol/(s·L) | 3.1863µmol/(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 | 27.718µM | 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)ex
- 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)in + ArsB ↔ ArsBAs
- ArsBAs → ArsB + As(III)ex
- ArsB → null (degradation)
- Accumulation (based on Chen1997)
- As(III)in + 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)in ArsB - (k7off+k8) ArsBAs
- (d/dt) As(III)in = - (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)in - 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
First of all, we assume the concentration of transporters is quite low compared to the concentration of the transported substances. After all, if this were not the case the transporters would act more like "storage" proteins than transporters (note that this can be even more rigorously justified if, for example, GlpFT<<K5). Similarly, there will generally only be a few ars promoters, compared to very many ArsR molecules. This leads to:
As(III)exT ≈ As(III)ex As(III)inT ≈ As(III)in + ArsRAs ArsRT ≈ ArsR : ArsRAs
Also, we assume the binding and unbinding of molecules to the transporters occurs on a much finer time-scale than any actual changes to the concentrations inside and outside the cell. 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 ≈ As(III)ex : 0 GlpF : GlpFAs ArsB : ArsBAs As(III)in : ArsRAs : ArsBAs ≈ As(III)in : ArsRAs : 0 ArsR : ArsRAs : 2 ArsRars ≈ ArsR : ArsRAs : 0 ars : ArsRars
To determine what the unknown 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)in ArsB - (k7off+k8) ArsBAs 0 = (d/dt) ArsRars = k3on ArsR² ars - k3off ArsRars 0 = (d/dt) ArsRAs = k1on ArsR As(III)in - 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 0 = (d/dt) ArsBAs = k7on As(III)in ArsB - (k7off+k8) ArsBAs k7on As(III)in ArsB = (k7off+k8) ArsBAs ArsB = (k7off+k8)/k7on ArsBAs / As(III)in ArsB = K7 ArsBAs / As(III)in ArsB : ArsBAs K7 ArsBAs / As(III)in : ArsBAs K7 : As(III)in
The other two differential equations can be used to determine the relative abundances of ArsR and ArsRAs, and ars and ArsRars:
0 = (d/dt) ArsRAs = k1on ArsR As(III)in - k1off ArsRAs k1on ArsR As(III)in = k1off ArsRAs ArsRAs = k1on/k1off ArsR As(III)in ArsRAs = ArsR As(III)in / K1d ArsR : ArsRAs ArsR : ArsR As(III)in / K1d K1d : As(III)in 0 = (d/dt) ArsRars = k3on ArsR² ars - k3off ArsRars k3on ArsR² ars = k3off ArsRars ArsRars = k3on/k3off ArsR² ars ArsRars = ArsR² ars / K3d² ars : ArsRars ars : ArsR² ars / K3d² K3d² : ArsR²
And finally the relative abundances of arsenic:
ArsRAs = ArsR As(III)in / K1d As(III)in : ArsRAs As(III)in : ArsR As(III)in / K1d K1d : ArsR
Summarizing:
As(III)ex : GlpFAs ≈ As(III)ex : 0 GlpF : GlpFAs = K5 : As(III)ex ArsB : ArsBAs = K7 : As(III)in As(III)in : ArsRAs : ArsBAs ≈ K1d : ArsR : 0 ArsR : ArsRAs : 2 ArsRars ≈ K1d : As(III)in : 0 ars : ArsRars = K3d² : ArsR²
Now we can look at the differential equations for the totals of ArsB (so ArsBT=ArsB+ArsBAs), ArsR, As(III)in 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) = (Vc/Vs) k8 ArsBAs - (Vc/Vs) v5 As(III)exT / (K5+As(III)exT) (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)inT = -(Vs/Vc) (d/dt) As(III)exT = v5 As(III)exT / (K5+As(III)exT) - k8 ArsBT As(III)in / (K7+As(III)in) (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)inT = 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)