Team:Groningen/Modelling/Arsenic
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Detailed Model
Based on the quasisteadystate 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 MichaelisMenten 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 steadystate and the quasi steadystate 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 MichaelisMenten kinetics before proceeding.
In contrast to how the quasisteadystate 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.
Reaction  Description  Rate  

Transport  
As(III)_{ex}T → As(III)_{in}T  Import of arsenic.  (Vc/Vs) v5^{†} As(III)_{ex}T / (K5+As(III)_{ex}T)  
As(III)_{in}T → As(III)_{ex}T  Export of arsenic.  k8 ArsB_{As}  
ars1T → ars1T + ArsBT  Production of ArsB.  βB ars1  
ArsBT → null  Degradation of ArsB  (ln(2)/τB) ArsB  
Accumulation  
ars1T → ars1T + ArsRT  From chromosomal operon.  βRN ars1  
proR → proR + ArsRT  Production of ArsR.  βR pro  
proM → proM + MBPArsRT  Production of MBPArsR.  βM pro  
proF → proF + fMTT  Production of fMT.  βF pro  
ArsRT → null  Degradation of ArsR.  (ln(2)/τR) ArsR  
MBPArsRT → null  Degradation of MBPArsR.  (ln(2)/τM) MBPArsR  
fMTT → null  Degradation of fMT.  (ln(2)/τF) fMT  
Gas vesicles  
ars2T → ars2T + GV  Transcription + translation.  βG ars2  
GV → null  Degradation of gas vesicles.  (ln(2)/τG) GV 
Name  Description  Derivative to time  

Extracellular  
As(III)_{ex}T  As(III) in the solution.  (Vc/Vs) k8 ArsB_{As}  (Vc/Vs) v5^{†} As(III)_{ex}T / (K5+As(III)_{ex}T)  
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)_{in}T  As(III) (bound and unbound) in the cell.  v5 As(III)_{ex}T / (K5+As(III)_{ex}T)  k8 ArsB_{As}  
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 = 0166.05nM)  
proR  Constitutive promoters in front of arsR.  (concentration is constant = 0166.05nM)  
proM  Constitutive promoters in front of mbparsR.  (concentration is constant = 0166.05nM)  
proF  Constitutive promoters in front of fMT.  (concentration is constant = 0166.05nM)  
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  


Name  Units  Value  Description  

k8  1/s  Reaction rate constant representing how fast ArsB can export arsenic.  
KR_{d}  M  6µM  Dissociation constant for ArsR and As(III). Assumed to be about an order of magnitude smaller than KD_{d} = 60µM, the corresponding constant for the similar protein ArsD from Chen1997.  
KM_{d}  M  6µM  Dissociation constant for MBPArsR and As(III). We assume this to be roughly equal to KR_{d}.  
KF_{d}  M  Dissociation constant for fMT and As(III).  
n_{f}  Hill coefficient for the formation of the complex fMTAs. This is related to the number of arsenic ions that bind to fMT.  
KA_{d}  M  0.33µM  Dissociation constants for ArsR and ars.
 
v5  mol/(s·L)  3.1863µmol/(s·L)  Maximum import rate per liter of cells (see MichaelisMenten 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 MichaelisMenten equation).
 
K7  M  Concentration at which export reaches half its maximum export rate (see MichaelisMenten equation).
 
τB, τR, τG, etc.  s  Halflifes (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.
 
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 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.
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  
As(III)_{ex} + GlpF ↔ GlpF_{As}  The binding and detachment of arsenic to GlpF on the outside of the cell.  
GlpF_{As} → GlpF + As(III)  The release of arsenic on the inside of the cell by GlpF  
As(III)_{in} + ArsB ↔ ArsB_{As}  The binding and detachment of arsenic to the Exporter ArsB  
ArsB_{As} → ArsB + As(III)_{ex}  The 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  
As(III)_{in} + ArsR ↔ ArsR_{As}  The binding and detachment of arsenic to ArsR  
As(III)_{in} + MBPArsR ↔ MBPArsR_{As}  The binding and detachment of arsenic to MBPArsR  
n_{f} As(III)_{in} + fMT ↔ fMT_{As}  The binding and detachment of arsenic to fMT  
ars1 + 2 ArsR ↔ ArsR_{ars1}  the repression of the promotor of the ars1 operon by 2 arsR molecules  
ars2 + 2 ArsR ↔ ArsR_{ars2}  the 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  

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 steadystate model and the steadystate model.
substance  Description  Derivative to time  

Extracellular  
As(III)_{ex}  As(III) in the solution  (d/dt) As(III)_{ex} =  (d/dt) GlpF_{As}  k6 GlpF_{As} + (Vc/Vs) k8 ArsB_{As}  
Membrane (all naturally occurring, but we plan to bring GlpF to overexpression)  
GlpF  concentration w.r.t. the exterior of the cell  (d/dt) GlpF =  (d/dt) GlpF_{As}  
GlpF_{As}  concentration w.r.t. the exterior of the cell  (d/dt) GlpF_{As} = k5_{on} As(III)_{ex} GlpF  (k5_{off}+k6) GlpF_{As}  
ArsB  concentration w.r.t. the interior of the cell  (d/dt) ArsB =  (d/dt) ArsB_{As} + β4 ars1  ln(2)/τB ArsB  
ArsB_{As}  concentration w.r.t. the interior of the cell  (d/dt) ArsB_{As} = k7_{on} As(III)_{in} ArsB  (k7_{off}+k8) ArsB_{As}  
Intracellular (ars2, pro and GV are introduced)  
As(III)_{in}  concentration of As(III) inside the cell  (d/dt) As(III)_{in} =  (d/dt) ArsR_{As}  (d/dt) MBPArsR_{As}  n_{f} (d/dt) fMT_{As}  (d/dt) ArsB_{As}  k8 ArsB_{As} + (Vs/Vc) k6 GlpF_{As}  
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) ArsR_{ars1}  
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) ArsR_{ars2}  
proR  concentration of constitutive promoters in front of arsR  (d/dt)proR = 0 in our model  
proM  concentration of constitutive promoters in front of mbparsR  (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) ArsR_{As}  2 (d/dt) ArsR_{ars1}  2 (d/dt) ArsR_{ars2}  
ArsR_{As}  the concentration of ArsR bound to As(III)  (d/dt) ArsR_{As} = kR_{on} ArsR As(III)_{in}  kR_{off} ArsR_{As}  
ArsR_{ars1}  the concentration of ArsR bound to ars1  (d/dt) ArsR_{ars1} = kA_{on} ArsR² ars1  kA_{off} ArsR_{ars1}  
ArsR_{ars2}  the concentration of ArsR bound to ars2  (d/dt) ArsR_{ars2} = kA_{on} ArsR² ars2  kA_{off} ArsR_{ars2}  
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) MBPArsR_{As}  
MBPArsR_{As}  bound to As(III)  (d/dt) MBPArsR_{As} = kM_{on} MBPArsR As(III)_{in}  kM_{off} MBPArsR_{As}  
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) fMT_{As}  
fMT_{As}  bound to multiple As(III)  fMT_{As} = kF_{on} fMT As(III)_{in}^{nf}  kF_{off} fMT_{As}  
ArsR_{As}  bound to As(III)  
GV  concentration of gas vesicles  (d/dt) GV = βG ars2  ln(2)/τG GV  

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/(M^{nf}·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  Halflifes (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.

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). 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 timescale 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 : 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 KAd² : 2 ArsR ars ars : ArsRars ars : ArsR² ars / KAd² KAd² : ArsR²
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^(nf1) / (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^(nf1) / (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
Steady state
By looking at the steady state of the system we can say something about its longterm 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
This directly leads to:
0 = βB ars1  ln(2)/τB ArsB ArsB = βB (τB/ln(2)) ars1 ArsB = βB (τB/ln(2)) ars1T KAd²/(KAd²+ArsR²) 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 GV = βG (τB/ln(2)) ars2T KAd²/(KAd²+ArsR²)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)x/(cx)
is nonnegative and nondecreasing for x∈[0,c⟩.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^(nf1)/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^(nf1)/KFd^nf)  As(III)T
As the function on the righthand side is nondecreasing 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 thirdorder equation:
0 = βRN ars1 + βR pro  (ln(2)/τR) ArsR 0 = βRN ars1T KAd²/(KAd²+ArsR²) + βR pro  (ln(2)/τR) ArsR 0 = βRN ars1T KAd² + βR pro (KAd²+ArsR²)  (ln(2)/τR) ArsR (KAd²+ArsR²) 0 = βRN ars1T KAd² + βR pro KAd² + βR pro ArsR²  (ln(2)/τR) ArsR KAd²  (ln(2)/τR) ArsR³ 0 = (βRN ars1T + βR pro) KAd²  (ln(2)/τR) KAd² ArsR + βR pro ArsR²  (ln(2)/τR) ArsR³ 0 = (βRN ars1T + βR pro) (τR/ln(2)) KAd²  KAd² ArsR + βR (τR/ln(2)) pro ArsR²  ArsR³
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.