Team:Groningen/Modelling/Arsenic

 .fromPaper { background:#dfd; } .selfDerived { background:#dfd; } .experimental { background:#ddf; } .estimate { background:#ffc; } .unknown { background:#fee; }  .intro { margin-left:0px; margin-top:10px; padding:10px; border-left:solid 5px #FFF6D5; border-right:solid 5px #FFF6D5; text-align:justify;background:#FFFFE5; }

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.



&dagger; 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.

{|class="ourtable" !Name !Units !Value !Description
 * + Constants
 * -class="unknown"
 * k8
 * 1/s
 * Reaction rate constant representing how fast ArsB can export arsenic.
 * -class="estimate"
 * KRd
 * M
 * 6&micro;M
 * Dissociation constant for ArsR and As(III). Assumed to be about an order of magnitude smaller than KDd = 60&micro;M, the corresponding constant for the similar protein ArsD from Chen1997.
 * -class="estimate"
 * KMd
 * M
 * 6&micro;M
 * Dissociation constant for MBPArsR and As(III). We assume this to be roughly equal to KRd.
 * -class="unknown"
 * KFd
 * M
 * Dissociation constant for fMT and As(III).
 * -class="unknown"
 * nf
 * Hill coefficient for the formation of the complex fMTAs. This is related to the number of arsenic ions that bind to fMT.
 * -class="fromPaper"
 * KAd
 * M
 * 0.33&micro;M
 * Dissociation constants for ArsR and ars.
 * KAd² = kAoff/kAon = (0.33&micro;M)²? (Chen1997, suspect as the relevant reference doesn't actually seem to give any value for this)
 * -class="selfDerived"
 * v5
 * mol/(s&middot;L)
 * 3.1863&micro;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&middot;L) instead of M/s, to emphasize the fact that the rate is per liter of cells.
 * v5 = k6 GlpFT (Vs/Vc)
 * -class="selfDerived"
 * K5
 * M
 * 27.718&micro;M
 * Concentration at which import reaches half its maximum import rate (see Michaelis-Menten equation).
 * K5 = (k5off+k6) / k5on
 * -class="unknown"
 * K7
 * M
 * Concentration at which export reaches half its maximum export rate (see Michaelis-Menten equation).
 * K7 = (k7off+k8) / k7on
 * -class="unknown"
 * &tau;B, &tau;R, &tau;G, etc.
 * s
 * Half-lifes (of ArsB, ArsR and GV, respectively). Degradation rate = ln(2)/&tau;
 * -class="unknown"
 * &beta;B, &beta;R, etc.
 * 1/s
 * Production rates.
 * &beta;RN = the production rate for ArsR behind the ars1 promoter
 * &beta;B = the production rate for ArsB behind the ars1 promoter
 * &beta;G = the production rate for GV behind the ars2 promoter
 * &beta;R = the production rate for ArsR behind a constitutive promoter
 * &beta;M = the production rate for MBPArsR behind a constitutive promoter
 * &beta;F = 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).
 * -style="border:none;"
 * colspan="4"|
 * Vs
 * L
 * Volume of solution (excluding cells).
 * Vc
 * L
 * Total volume of cells (in solution) (so Vs+Vc is the total volume).
 * -style="border:none;"
 * colspan="4"|
 * Total volume of cells (in solution) (so Vs+Vc is the total volume).
 * -style="border:none;"
 * colspan="4"|


 * }

The raw model
 .import { background: LightGreen; } .export { background: LightBlue; } .accumulation { background: LightPink; } .production { background: LightGoldenRodYellow; }

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.



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.

{|class="ourtable" !colspan="2"|substance !Description !Derivative to time
 * + Core substances
 * colspan="4"|Extracellular
 * ||As(III)ex||As(III) in the solution||(d/dt) As(III)ex = - (d/dt) GlpFAs - k6 GlpFAs + (Vc/Vs) k8 ArsBAs
 * colspan="4"|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) GlpFAs
 * ||GlpFAs||concentration w.r.t. the exterior of the cell||(d/dt) GlpFAs = k5on As(III)ex GlpF - (k5off+k6) GlpFAs
 * ||ArsB||concentration w.r.t. the interior of the cell||(d/dt) ArsB = - (d/dt) ArsBAs + &beta;4 ars1 - ln(2)/&tau;B ArsB
 * ||ArsBAs ||concentration w.r.t. the interior of the cell||(d/dt) ArsBAs = k7on As(III)in ArsB - (k7off+k8) ArsBAs
 * colspan="4"|Intracellular (ars2, pro and GV are introduced)
 * ||As(III)in||concentration 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 ||concentration of unbound promoters naturally occurring in E. coli||(d/dt) ars1 = - (d/dt) ArsRars1
 * ||ars2 ||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 ||concentration of the accumulation protein ArsR||(d/dt) ArsR = &beta;RN ars1 + &beta;R proR - (ln(2)/&tau;R) 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 ArsR&sup2; ars1 - kAoff ArsRars1
 * ||ArsRars2 ||the concentration of ArsR bound to ars2||(d/dt) ArsRars2 = kAon ArsR&sup2; ars2 - kAoff ArsRars2
 * ||MBPArsR || a fusion of maltose binding protein and ArsR||(d/dt) MBPArsR = &beta;M proM - (ln(2)/&tau;M) MBPArsR - (d/dt) MBPArsRAs
 * ||MBPArsRAs ||bound to As(III)||(d/dt) MBPArsRAs = kMon MBPArsR As(III)in - kMoff MBPArsRAs
 * ||fMT || Arsenic binding metallotein ||(d/dt) fMT = &beta;F proF - (ln(2)/&tau;F) 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 = &beta;G ars2 - ln(2)/&tau;G GV
 * colspan="4"|
 * ||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 ||concentration of the accumulation protein ArsR||(d/dt) ArsR = &beta;RN ars1 + &beta;R proR - (ln(2)/&tau;R) 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 ArsR&sup2; ars1 - kAoff ArsRars1
 * ||ArsRars2 ||the concentration of ArsR bound to ars2||(d/dt) ArsRars2 = kAon ArsR&sup2; ars2 - kAoff ArsRars2
 * ||MBPArsR || a fusion of maltose binding protein and ArsR||(d/dt) MBPArsR = &beta;M proM - (ln(2)/&tau;M) MBPArsR - (d/dt) MBPArsRAs
 * ||MBPArsRAs ||bound to As(III)||(d/dt) MBPArsRAs = kMon MBPArsR As(III)in - kMoff MBPArsRAs
 * ||fMT || Arsenic binding metallotein ||(d/dt) fMT = &beta;F proF - (ln(2)/&tau;F) 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 = &beta;G ars2 - ln(2)/&tau;G GV
 * colspan="4"|
 * ||MBPArsRAs ||bound to As(III)||(d/dt) MBPArsRAs = kMon MBPArsR As(III)in - kMoff MBPArsRAs
 * ||fMT || Arsenic binding metallotein ||(d/dt) fMT = &beta;F proF - (ln(2)/&tau;F) 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 = &beta;G ars2 - ln(2)/&tau;G GV
 * colspan="4"|
 * ||ArsRAs ||bound to As(III)
 * ||GV ||concentration of gas vesicles||(d/dt) GV = &beta;G ars2 - ln(2)/&tau;G GV
 * colspan="4"|
 * ||GV ||concentration of gas vesicles||(d/dt) GV = &beta;G ars2 - ln(2)/&tau;G GV
 * colspan="4"|
 * colspan="4"|


 * }

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: 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 &asymp; As(III)ex As(III)inT &asymp; 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:

As(III)ex : GlpFAs &asymp; As(III)ex : 0 GlpF : GlpFAs ArsB : ArsBAs As(III)in : ArsRAs : MBPArsRAs : nf fMTAs : ArsBAs &asymp; 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^(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 &asymp; 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 &asymp; 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 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

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  (as As(III)exT cannot be negative):

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  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 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  (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.