# Team:Groningen/Modelling/Arsenic

(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)  ## Detailed Model

Based on the quasi-steady-state derivation below we have made the simplified version of our model shown below. The simplification is based on two key assumptions (which are also illustrated below, next to the table "Breakdown of core substances"):

• Binding and unbinding of arsenic to/from the transporters occurs on a much smaller time scale than changes in the concentration of arsenic inside and outside the cell. And similarly, we assume that (un)binding of ArsR to/from the ars promoter is much faster than the production of ArsR (for example).
• The concentration of transporters is insignificant compared to the concentration of arsenic inside and outside the cell.

This leads to the Michaelis-Menten equation for import, but also some more general equations for export using ArsB and accumulation with ArsR (for example, the Hill equation can be recognized in the activity of the ars promoter). We explicitly state relative abundances instead of substituting them into the differential equations. This leads to clearer and more insightful equations and gives more freedom to define complicated, interdependent ratios between substances.

The inexperienced viewer may find the following tables and formulas baffling. We would recommend that one would look at the raw model first to gain an understanding of the basic reactions involved then have a look at the steady-state and the quasi steady-state model. It is not mandatory, but it is probably the the best route to get a better understanding of the model as a whole. Also, perhaps first have a look at Michaelis-Menten kinetics before proceeding.

In contrast to how the quasi-steady-state assumption is normally used we mostly leave the specific states (bound/unbound) of substances intact in the differential equations and explicitly state the relative abundances. This keeps the differential equations shorter and gives more insight into what is actually happening, clearly mapping the "fast" reactions to ratios between substances. This also makes it possible to use quite complicated equations (the Asin and ArsR interdependency is virtually impossible to define using normal methods for example) that would otherwise be unwieldy to handle. 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 MBPArsR and fMT are not shown for clarity.
Reactions
Reaction Description Rate
Transport
As(III)exT → As(III)inTImport of arsenic.(Vc/Vs) v5 As(III)exT / (K5+As(III)exT)
As(III)inT → As(III)exTExport of arsenic. k8 ArsBAs
ars1T → ars1T + ArsBTProduction of ArsB. βB ars1
ArsBT → nullDegradation of ArsB (ln(2)/τB) ArsB
Accumulation
ars1T → ars1T + ArsRTFrom chromosomal operon. βRN ars1
proR → proR + ArsRTProduction of ArsR. βR pro
proM → proM + MBPArsRTProduction of MBPArsR. βM pro
proF → proF + fMTTProduction of fMT. βF pro
ArsRT → nullDegradation of ArsR. (ln(2)/τR) ArsR
MBPArsRT → nullDegradation of MBPArsR. (ln(2)/τM) MBPArsR
fMTT → nullDegradation of fMT. (ln(2)/τF) fMT
Gas vesicles
ars2T → ars2T + GVTranscription + translation. βG ars2
GV → nullDegradation of gas vesicles. (ln(2)/τG) GV
Core Substances
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 (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). (not used directly in model, assumed to be constant)
ArsBT Exporter of As(III) (concentration w.r.t. the interior of the cell). βB 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 = 0-166.05nM)
proR Constitutive promoters in front of arsR. (concentration is constant = 0-166.05nM)
proM Constitutive promoters in front of mbp-arsR. (concentration is constant = 0-166.05nM)
proF Constitutive promoters in front of fMT. (concentration is constant = 0-166.05nM)
ArsRT ArsR in the cell. βRN ars1 + βR proR - (ln(2)/τR) ArsR
MBPArsRT MBPArsR in the cell. βM proM - (ln(2)/τM) MBPArsR
fMTT fMT in the cell. βF proF - (ln(2)/τF) fMT
GV Concentration of gas vesicles. βG ars2 - (ln(2)/τG) GV
Directly from paper. Based on data from paper. Based on experiment. Rough estimate. Totally unknown.
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.
Breakdown of core substances
Core substance Component Relative abundance
ArsBT ArsB K7
ArsBAs As(III)in
As(III)inT As(III)in 1
ArsRAs ArsR / KRd
MBPArsRAs MBPArsRT / (KMd + As(III)in)
fMTAs nf fMTT As(III)innf-1 / (KFdnf + As(III)innf)
ArsRars ArsR²
ars ars1 ars1T
ars2 ars2T
ArsRT ArsR 1
ArsRAs As(III)in / KRd
ArsRars 2 ArsR ars / KAd²
MBPArsRT MBPArsR KMd
MBPArsRAs As(III)in
fMTT fMT KFdnf
fMTAs As(III)innf 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 (the small circles) are considered to have such a low concentration that they are of no importance to the concentrations of As(III)in/-ex and ArsR (the large circles).
Constants
Name Units Value Description
k8 1/s Reaction rate constant representing how fast ArsB can export arsenic.
KRd M 6µM Dissociation constant for ArsR and As(III). Assumed to be about an order of magnitude smaller than KDd = 60µM, the corresponding constant for the similar protein ArsD from Chen1997.
KMd M 6µM Dissociation constant for MBPArsR and As(III). We assume this to be roughly equal to KRd.
KFd M Dissociation constant for fMT and As(III).
nf Hill coefficient for the formation of the complex fMTAs. This is related to the number of arsenic ions that bind to fMT.
KAd M 0.33µM Dissociation constants for ArsR and ars.
• KAd² = kAoff/kAon = (0.33µM)²? (Chen1997, suspect as the relevant reference doesn't actually seem to give any value for this)
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.
• v5 = k6 GlpFT (Vs/Vc)
K5 M 27.718µM Concentration at which import reaches half its maximum import rate (see Michaelis-Menten equation).
• K5 = (k5off+k6) / k5on
K7 M Concentration at which export reaches half its maximum export rate (see Michaelis-Menten equation).
• K7 = (k7off+k8) / k7on
τB, τR, τG, etc. 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
βB, βR, etc. 1/s Production rates.
• βRN = the production rate for ArsR behind the ars1 promoter
• βB = the production rate for ArsB behind the ars1 promoter
• βG = the production rate for GV behind the ars2 promoter
• βR = the production rate for ArsR behind a constitutive promoter
• βM = the production rate for MBPArsR behind a constitutive promoter
• βF = the production rate for fMT behind a constitutive promoter
Vs L Volume of solution (excluding cells).
Vc L Total volume of cells (in solution) (so Vs+Vc is the total volume).
Directly from paper. Based on data from paper. Based on experiment. Rough estimate. Totally unknown.

## The raw model

The following table gives all the reactions that take place inside the cell. You can look at the schematic representation of the processes involved to get a good grasp as how every reaction works to the other. Note that proR, ProM and MBPArsR, ProF and Fmt are not displayed in the figure. This has been done for clarity. These reactions are simple constituative promotor reactions. Once you have an insight in the reactions involved you can have a look at the next table. 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 MBPArsR and fMT are not shown for clarity.
Reactions
Reaction Description
Transport
In the reactions below you can see the import of arsenic by GlpF and the export of arsenic by ArsB. Only the degradation of ArsB is taken into acount because the ars operon also produces ArsB, as can be seen in the accumulation section. We assume a constant number of GlpF importers.
i
(based on Rosen1996, Meng2004 and Rosen2009)
As(III)ex + GlpF ↔ GlpFAsThe binding and detachment of arsenic to GlpF on the outside of the cell.
GlpFAs → GlpF + As(III)The release of arsenic on the inside of the cell by GlpF
As(III)in + ArsB ↔ ArsBAsThe binding and detachment of arsenic to the Exporter ArsB
ArsBAs → ArsB + As(III)exThe release of the bound arsenic by ArsB on the outside of the cell.
ArsB → null The degradation of Ars B
Accumulation
In the reactions below you can see the production and degradation of all our accumulation proteins. Two things should be noticed: ArsR represses it's own production and that of the GVP clusters and the ars1 operon does not only produce ArsR but also the exporter ArsB
i
(mostly based on Chen1997)
As(III)in + ArsR ↔ ArsRAsThe binding and detachment of arsenic to ArsR
As(III)in + MBPArsR ↔ MBPArsRAsThe binding and detachment of arsenic to MBPArsR
nf As(III)in + fMT ↔ fMTAsThe binding and detachment of arsenic to fMT
ars1 + 2 ArsR ↔ ArsRars1the repression of the promotor of the ars1 operon by 2 arsR molecules
ars2 + 2 ArsR ↔ ArsRars2the repression of the promotor of the ars1 operon by 2 arsR molecules
ars1 → ars1 + ArsR + ArsB The transcription and translation of the ars1 operon to produce ArsR and ArsB
proR → proR + ArsR The transcription and translation of the proR operon to produce ArsR
proM → proM + MBPArsR The transcription and translation of the proM operon to produce MBPArsR
proF → proF + fMT The transcription and translation of the proF operon to produce fMT
ArsR → null The degradation of ArsR
MBPArsR → null The degradation of MBPArsR
fMT → null The degradation of fMT
Gas vesicles
These two reactions give the production and degradation rate of the GVP clusters. Keep in mind that ars2 is repressed by the accumulation protein ArsR. This reaction can be found under accumulation part.
i
ars2 → ars2 + GV The transcription and translation of the ars2 operon to produce GVP clusters wich will make the cell float
GV → null The degradation of GVP
Import related. Import related. Export related. Export related. GVP Production related.

Here you can find the time derivatives for each substance we derived. The constants are explained in the next teble. After one has a full understanding of all the constants and derivatives and and reactions. One can begin the process of simplifying the model and thus one can have a look at the quasi steady-state model and the steady-state model.

Core substances
substance Description Derivative to time
Extracellular
As(III)exAs(III) in the solution(d/dt) As(III)ex = - (d/dt) GlpFAs - k6 GlpFAs + (Vc/Vs) k8 ArsBAs
Membrane (all naturally occurring, but we plan to bring GlpF to overexpression)
GlpFconcentration w.r.t. the exterior of the cell(d/dt) GlpF = - (d/dt) GlpFAs
GlpFAsconcentration w.r.t. the exterior of the cell(d/dt) GlpFAs = k5on As(III)ex GlpF - (k5off+k6) GlpFAs
ArsBconcentration w.r.t. the interior of the cell(d/dt) ArsB = - (d/dt) ArsBAs + β4 ars1 - ln(2)/τB ArsB
ArsBAs concentration w.r.t. the interior of the cell(d/dt) ArsBAs = k7on As(III)in ArsB - (k7off+k8) ArsBAs
Intracellular (ars2, pro and GV are introduced)
As(III)inconcentration of As(III) inside the cell(d/dt) As(III)in = - (d/dt) ArsRAs - (d/dt) MBPArsRAs - nf (d/dt) fMTAs - (d/dt) ArsBAs - k8 ArsBAs + (Vs/Vc) k6 GlpFAs
ars1
ars1 stands for the promotor in front of the operon which contains the information for the production of the accumulation protein ArsR and the exporter ArsB. It is selfregulatory in the sence that it produces it's own repressor in the form of ArsR
i
concentration of unbound promoters naturally occurring in E. coli(d/dt) ars1 = - (d/dt) ArsRars1
ars2
ars2 stands for the promotor in front of the operon which contains the information for the production of Gas Vesicles. Unlike ars 1 it is not selfregulatory, but the if everything goes correctly the production of gas vesicles will only start if there arsenic inside the cell
i
concentration of unbound promoters in front of gas vesicle genes(d/dt) ars2 = - (d/dt) ArsRars2
proR concentration of constitutive promoters in front of arsR (d/dt)proR = 0 in our model
proM concentration of constitutive promoters in front of mbp-arsR (d/dt)proM = 0 in our model
proF concentration of constitutive promoters in front of fMT (d/dt)proF = 0 in our model
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
concentration of the accumulation protein ArsR(d/dt) ArsR = βRN ars1 + βR proR - (ln(2)/τR) ArsR - (d/dt) ArsRAs - 2 (d/dt) ArsRars1 - 2 (d/dt) ArsRars2
ArsRAs the concentration of ArsR bound to As(III)(d/dt) ArsRAs = kRon ArsR As(III)in - kRoff ArsRAs
ArsRars1 the concentration of ArsR bound to ars1(d/dt) ArsRars1 = kAon ArsR² ars1 - kAoff ArsRars1
ArsRars2 the concentration of ArsR bound to ars2(d/dt) ArsRars2 = kAon ArsR² ars2 - kAoff ArsRars2
MBPArsR
A fusion of maltose binding protein and ArsR. It is more stable than the normal ArsR variant, but it is no longer able to act as a repressor for the ars promotor.
i
a fusion of maltose binding protein and ArsR(d/dt) MBPArsR = βM proM - (ln(2)/τM) MBPArsR - (d/dt) MBPArsRAs
MBPArsRAs bound to As(III)(d/dt) MBPArsRAs = kMon MBPArsR As(III)in - kMoff MBPArsRAs
fMT
It is another binding protein. Unlike it's counterpart it capeble of containing up to five As(III) particles or one As(V) particle
i
Arsenic binding metallotein (d/dt) fMT = βF proF - (ln(2)/τF) fMT - (d/dt) fMTAs
fMTAs bound to multiple As(III)fMTAs = kFon fMT As(III)innf - kFoff fMTAs
ArsRAs bound to As(III)
GV concentration of gas vesicles(d/dt) GV = βG ars2 - ln(2)/τG GV
Import related. Import related. Export related. Export related. GVP Production related.

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):

Using the following constants/definitions:

Name Units Description
kRon, kMon, k5on, etc. 1/(M·s) Reaction rate constants for reactions to a complex.
kAon 1/(M²·s) Reaction rate constants for reactions to a complex.
kFon 1/(Mnf·s) Reaction rate constants for reactions to a complex.
kRoff, kMoff, kFoff, kAoff, 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, τM, τF, τG s Half-lifes (of ArsB, ArsR, MBPArsR, fMT 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
βRN, βR, etc. 1/s Production rates.
• βRN = the production rate for ArsR behind the ars1 promoter
• βB = the production rate for ArsB behind the ars1 promoter
• βG = the production rate for GV behind the ars2 promoter
• βR = the production rate for ArsR behind a constitutive promoter
• βM = the production rate for MBPArsR behind a constitutive promoter
• βF = the production rate for fMT behind a constitutive promoter
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).

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). This leads to:

```As(III)exT ≈ As(III)ex
As(III)inT ≈ As(III)in + ArsRAs + MBPArsRAs + nf fMTAs
```

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:

We use the following when grouping 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
```
```As(III)ex : GlpFAs ≈ As(III)ex : 0
GlpF : GlpFAs
ArsB : ArsBAs
As(III)in : ArsRAs : MBPArsRAs : nf fMTAs : ArsBAs ≈ As(III)in : ArsRAs : MBPArsRAs : nf fMTAs : 0
ArsR : ArsRAs : 2 ArsRars
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 = kAon ArsR² ars - kAoff ArsRars
0 = (d/dt) ArsRAs = kRon ArsR As(III)in - kRoff ArsRAs
0 = (d/dt) MBPArsRAs = kMon MBPArsR As(III)in - kMoff MBPArsRAs
0 = (d/dt) fMTAs = kFon fMT As(III)in^nf - kFoff fMTAs
```

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 next two differential equations can be used to determine the relative abundances of ArsR and ArsRAs, and ars and ArsRars:

```0 = (d/dt) ArsRAs = kRon ArsR As(III)in - kRoff ArsRAs

kRon ArsR As(III)in = kRoff ArsRAs
ArsRAs = kRon/kRoff ArsR As(III)in
ArsRAs = ArsR As(III)in / KRd

ArsR : ArsRAs
ArsR : ArsR As(III)in / KRd
KRd  : As(III)in

0 = (d/dt) ArsRars = kAon ArsR² ars - kAoff ArsRars

kAon ArsR² ars = kAoff ArsRars
ArsRars = kAon/kAoff ArsR² ars
ArsRars = ArsR² ars / KAd²

ArsR : 2 ArsRars
ArsR : 2 ArsR² ars / KAd²

ars : ArsRars
ars : ArsR² ars / KAd²
```

For MBPArsR and fMT we find:

```0 = (d/dt) MBPArsRAs = kMon MBPArsR As(III)in - kMoff MBPArsRAs

MBPArsR : MBPArsRAs = KMd : As(III)in

0 = (d/dt) fMTAs = kFon fMT As(III)in^nf - kFoff fMTAs

fMT : fMTAs = KFd^nf : As(III)in^nf
```

And finally the relative abundances of arsenic:

```ArsRAs = ArsR As(III)in / KRd

As(III)in : ArsRAs               : MBPArsRAs                            : n fMTAs
As(III)in : ArsR As(III)in / KRd : MBPArsRT As(III)in / (KMd+As(III)in) : n fMTT As(III)in^nf / (KFd^nf+As(III)in^nf)
1         : ArsR / KRd           : MBPArsRT / (KMd+As(III)in)           : n fMTT As(III)in^(nf-1) / (KFd^nf+As(III)in^nf)
```

Summarizing:

```GlpF : GlpFAs = K5 : As(III)ex
ArsB : ArsBAs = K7 : As(III)in
As(III)in : ArsRAs : MBPArsRAs : n fMTAs ≈ 1 : ArsR / KRd : MBPArsRT / (KMd+As(III)in) : n fMTT As(III)in^(nf-1) / (KFd^nf+As(III)in^nf)
ars : ArsRars = KAd² : ArsR²
ArsR : ArsRAs : 2 ArsRars ≈ 1 : As(III)in / KRd : 2 ArsR ars / KAd²
MBPArsR : MBPArsRAs = KMd : As(III)in
fMT : fMTAs = KFd^nf : As(III)in^nf
```

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 + βB ars1 - ln(2)/τB ArsB + (d/dt) ArsBAs
= βB 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
= βRN ars1 + βR proR - (ln(2)/τR) ArsR - (d/dt) ArsRAs - 2 (d/dt) ArsRars + (d/dt) ArsRAs + 2 (d/dt) ArsRars
= βRN ars1 + βR proR - (ln(2)/τR) ArsR
(d/dt) MBPArsRT = (d/dt) MBPArsR + (d/dt) MBPArsRAs
= βM proM - (ln(2)/τM) MBPArsR
(d/dt) fMTT = (d/dt) fMT + (d/dt) fMTAs
= βF proF - (ln(2)/τF) fMT
```

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 = βB ars1 - ln(2)/τB ArsB
0 = (d/dt) As(III)inT = v5 As(III)exT / (K5+As(III)exT) - k8 ArsBAs
0 = (d/dt) ArsRT = βRN ars1 + βR pro - (ln(2)/τR) ArsR
0 = (d/dt) MBPArsRT = βM proM - (ln(2)/τM) MBPArsR
0 = (d/dt) fMTT = βF proF - (ln(2)/τF) fMT
0 = (d/dt) GV = βG ars2 - ln(2)/τG GV
```

```   0 = βB ars1 - ln(2)/τB ArsB
ArsB = βB (τB/ln(2)) ars1

0 = βM proM - (ln(2)/τM) MBPArsR
MBPArsR = βM (τM/ln(2)) proM

0 = βF proF - (ln(2)/τF) fMT
fMT = βF (τF/ln(2)) proF

0 = βG ars2 - ln(2)/τG GV
GV = βG (τB/ln(2)) ars2
```
For the intra- and extracellular concentrations we can find the following equation, giving a maximum for As(III)in of `K7 v5/(k8 ArsB)` (as As(III)exT cannot be negative)
Conveniently the function `x/(c-x)` is non-negative and non-decreasing for x∈[0,c⟩.
i
:
```         0 = v5 As(III)exT / (K5+As(III)exT) - k8 ArsBAs
0 = v5 As(III)exT / (K5+As(III)exT) - k8 ArsB As(III)in / K7
0 = v5 As(III)exT - k8 ArsB As(III)in / K7 (K5+As(III)exT)
0 = v5 As(III)exT - k8 ArsB As(III)in As(III)exT / K7 - k8 ArsB As(III)in K5 / K7
0 = As(III)exT (v5 - k8 ArsB As(III)in / K7) - k8 ArsB As(III)in K5 / K7
As(III)exT = k8 ArsB As(III)in K5 / (v5 K7 - k8 ArsB As(III)in)
As(III)exT = K5 As(III)in / (K7 v5/(k8 ArsB) - As(III)in)
```

As we can safely assume arsenic neither disappears into nothingness nor appears from nothingness, we can use this to derive (As(III)T is the total amount of arsenic):

```As(III)inT = As(III)in (1 + ArsR/KRd + MBPArsR/KMd + fMT As(III)in^(nf-1)/KFd^nf)

As(III)T = Vs As(III)exT + Vc As(III)inT
0 = Vs As(III)exT + Vc As(III)inT - As(III)T
0 = Vs K5 As(III)in / (K7 v5/(k8 ArsB) - As(III)in) + Vc As(III)in (1 + ArsR/KRd + MBPArsR/KMd + fMT As(III)in^(nf-1)/KFd^nf) - As(III)T
```

As the function on the right-hand side is non-decreasing for `As(III)in∈[0,K7 v5/(k8 ArsB)⟩` it at most has one zero on this interval (and it has one, as it starts at a negative value and gets arbitrarily large as As(III)in approaches the end of its range). So this zero can safely be found using any number of numerical methods.

Finally, for ArsR we can find the following third-order equation:

```0 = βRN ars1 + βR pro - (ln(2)/τR) ArsR
According to Mathematica's solution of `Reduce[eq && KAd > 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.