Team:Groningen/Modelling/Arsenic
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- | {{Team:Groningen/Header}} | + | {{Team:Groningen/Modelling/Header}} |
+ | <div style="clear:both;"></div> | ||
+ | <div style="float:left" >{{linkedImage|GroningenPrevious.png|Team:Groningen/Modelling}}</div> | ||
+ | <div title="Arsie Says UP TO ACCUMULATION" style="float:right" >{{linkedImage|Next.JPG|Team:Groningen/Modelling/Characterization}}</div> | ||
[[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]] | ||
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.experimental { background:#ddf; } | .experimental { background:#ddf; } | ||
.estimate { background:#ffc; } | .estimate { background:#ffc; } | ||
- | .unknown { background:#fee; } | + | .unknown { background:#fee; } |
</style></html> | </style></html> | ||
- | + | <html><style type="text/css"> | |
- | < | + | .intro { margin-left:0px; margin-top:10px; padding:10px; border-left:solid 5px #FFF6D5; border-right:solid 5px #FFF6D5; text-align:justify;background:#FFFFE5; } |
+ | </style></html> | ||
+ | <div class="intro introduction"> | ||
+ | ==Detailed Model== | ||
Based on the [[#QuasiSteadyState|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"): | Based on the [[#QuasiSteadyState|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). | *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, and | + | *The concentration of transporters is insignificant compared to the concentration of arsenic inside and outside the cell. |
+ | |||
+ | This leads to the [[Team:Groningen/Glossary#MichaelisMenten|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. | ||
+ | </div> | ||
+ | |||
+ | 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 [[Team:Groningen/Glossary#MichaelisMenten|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. | |
[[Image:Arsenic_filtering.png|frame|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.<!-- Also, ArsD is not shown here, as it is [[Team:Groningen/BLAST|not present in our E. coli]] and has a role analogous to ArsR.-->]] | [[Image:Arsenic_filtering.png|frame|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.<!-- Also, ArsD is not shown here, as it is [[Team:Groningen/BLAST|not present in our E. coli]] and has a role analogous to ArsR.-->]] | ||
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| ||ArsRT || ArsR in the cell. || βRN ars1 + βR proR - (ln(2)/τR) ArsR | | ||ArsRT || ArsR in the cell. || βRN ars1 + βR proR - (ln(2)/τR) ArsR | ||
|- | |- | ||
- | | ||MBPArsRT || MBPArsR in the cell. || βM proM - (ln(2)/ | + | | ||MBPArsRT || MBPArsR in the cell. || βM proM - (ln(2)/τM) MBPArsR |
|- | |- | ||
- | | ||fMTT || fMT in the cell. || βF proF - (ln(2)/ | + | | ||fMTT || fMT in the cell. || βF proF - (ln(2)/τF) fMT |
|- | |- | ||
| ||GV || Concentration of gas vesicles. || βG ars2 - (ln(2)/τG) GV | | ||GV || Concentration of gas vesicles. || βG ars2 - (ln(2)/τG) GV | ||
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|- | |- | ||
|ArsR<sub>As</sub> | |ArsR<sub>As</sub> | ||
- | | | + | |ArsR / KR<sub>d</sub> |
|- | |- | ||
|MBPArsR<sub>As</sub> | |MBPArsR<sub>As</sub> | ||
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|- | |- | ||
|fMT<sub>As</sub> | |fMT<sub>As</sub> | ||
- | |fMTT As(III)<sub>in</sub><sup>n<sub>f</sub>-1</sup> / (KF<sub>d</sub><sup>n<sub>f</sub></sup> + As(III)<sub>in</sub><sup>n<sub>f</sub></sup>) | + | |n<sub>f</sub> fMTT As(III)<sub>in</sub><sup>n<sub>f</sub>-1</sup> / (KF<sub>d</sub><sup>n<sub>f</sub></sup> + As(III)<sub>in</sub><sup>n<sub>f</sub></sup>) |
|- | |- | ||
|rowspan="2"|arsT | |rowspan="2"|arsT | ||
|style="padding-left:0;"|ars | |style="padding-left:0;"|ars | ||
- | | | + | |KA<sub>d</sub>² |
|- | |- | ||
|ArsR<sub>ars</sub> | |ArsR<sub>ars</sub> | ||
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|ars2T | |ars2T | ||
|- | |- | ||
- | |rowspan=" | + | |rowspan="3"|ArsRT |
|style="padding-left:0;"|ArsR | |style="padding-left:0;"|ArsR | ||
- | | | + | |1 |
|- | |- | ||
|ArsR<sub>As</sub> | |ArsR<sub>As</sub> | ||
- | |As(III)<sub>in</sub> | + | |As(III)<sub>in</sub> / KR<sub>d</sub> |
+ | |- | ||
+ | |ArsR<sub>ars</sub> | ||
+ | |2 ArsR ars / KA<sub>d</sub>² | ||
|- | |- | ||
|rowspan="2"|MBPArsRT | |rowspan="2"|MBPArsRT | ||
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|As(III)<sub>in</sub><sup>n<sub>f</sub></sup> | |As(III)<sub>in</sub><sup>n<sub>f</sub></sup> | ||
|} | |} | ||
- | |[[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 (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).]] | + | |[[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 <strike>ars,</strike> 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).]] |
|} | |} | ||
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|Hill coefficient for the formation of the complex fMTAs. This is related to the number of arsenic ions that bind to fMT. | |Hill coefficient for the formation of the complex fMTAs. This is related to the number of arsenic ions that bind to fMT. | ||
|-class="fromPaper" | |-class="fromPaper" | ||
- | | | + | |KA<sub>d</sub> |
|M | |M | ||
|0.33µM | |0.33µM | ||
|Dissociation constants for ArsR and ars. | |Dissociation constants for ArsR and ars. | ||
- | * | + | * KA<sub>d</sub>² = kA<sub>off</sub>/kA<sub>on</sub> = (0.33µM)²? ([[Team:Groningen/Literature#Chen1997|Chen1997]], suspect as the relevant reference doesn't actually seem to give any value for this) |
|-class="selfDerived" | |-class="selfDerived" | ||
|v5 | |v5 | ||
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|} | |} | ||
|} | |} | ||
+ | |||
==The raw model== | ==The raw model== | ||
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<html></style></html> | <html></style></html> | ||
- | The following | + | 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. |
- | + | [[Image:Arsenic_filtering.png|frame|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.<!-- Also, ArsD is not shown here, as it is [[Team:Groningen/BLAST|not present in our E. coli]] and has a role analogous to ArsR.-->]] | |
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- | + | {|class="ourtable" | |
+ | |+ Reactions | ||
+ | !colspan="2"|Reaction | ||
+ | !Description | ||
+ | |- | ||
+ | |colspan="3"|''Transport''{{infoBox|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. }}(based on [[Team:Groningen/Literature#Rosen1996|Rosen1996]], [[Team:Groningen/Literature#Meng2004|Meng2004]] and [[Team:Groningen/Literature#Rosen2009|Rosen2009]]) | ||
+ | |- | ||
+ | | ||<span class="import">As(III)<sub>ex</sub> + GlpF ↔ GlpF<sub>As</sub></span>||The binding and detachment of arsenic to GlpF on the outside of the cell.|| | ||
+ | |- | ||
+ | | ||<span class="import">GlpF<sub>As</sub> → GlpF + As(III)</span>||The release of arsenic on the inside of the cell by GlpF|| | ||
+ | |- | ||
+ | | ||<span class="export">As(III)<sub>in</sub> + ArsB ↔ ArsB<sub>As</sub></span>||The binding and detachment of arsenic to the Exporter ArsB|| | ||
+ | |- | ||
+ | | ||<span class="export">ArsB<sub>As</sub> → ArsB + As(III)<sub>ex</sub></span>||The release of the bound arsenic by ArsB on the outside of the cell.|| | ||
+ | |- | ||
+ | | ||<span class="export">ArsB → null</span> ||The degradation of Ars B|| | ||
+ | |- | ||
+ | |colspan="3"|''Accumulation''{{infoBox|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}}(mostly based on [[Team:Groningen/Literature#Chen1997|Chen1997]]) | ||
+ | |- | ||
+ | | ||As(III)<sub>in</sub> + ArsR ↔ ArsR<sub>As</sub>||The binding and detachment of arsenic to ArsR|| | ||
+ | |- | ||
+ | | ||As(III)<sub>in</sub> + MBPArsR ↔ MBPArsR<sub>As</sub>||The binding and detachment of arsenic to MBPArsR || | ||
+ | |- | ||
+ | | ||n<sub>f</sub> As(III)<sub>in</sub> + fMT ↔ fMT<sub>As</sub>||The binding and detachment of arsenic to fMT || | ||
+ | |- | ||
+ | | ||ars1 + 2 ArsR ↔ ArsR<sub>ars1</sub>||the repression of the promotor of the ars1 operon by 2 arsR molecules|| | ||
+ | |- | ||
+ | | ||<span class="production">ars2 + 2 ArsR ↔ ArsR<sub>ars2</sub></span>||the repression of the promotor of the ars1 operon by 2 arsR molecules|| | ||
+ | |- | ||
+ | | ||ars1 → ars1 + ArsR<span class="export"> + ArsB</span> ||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|| | ||
+ | |- | ||
+ | |colspan="3"|''Gas vesicles''{{infoBox|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.}} | ||
+ | |- | ||
+ | | ||ars2 → ars2<span class="production"> + GV</span> ||The transcription and translation of the ars2 operon to produce GVP clusters wich will make the cell float|| | ||
+ | |- | ||
+ | | ||<span class="production">GV → null</span> ||The degradation of GVP|| | ||
+ | |- | ||
+ | |colspan="3"| | ||
+ | {|class="ourtable" style="width:100%" | ||
+ | !colspan="5"| | ||
+ | |- style="text-align:center;" | ||
+ | |class="fromPaper" style="padding:0;"|Import related. | ||
+ | |class="fromPaper" style="padding:0;"|Import related. | ||
+ | |class="experimental" style="padding:0;"|Export related. | ||
+ | |class="experimental" style="padding:0;"|Export related. | ||
+ | |class="estimate" style="padding:0;"|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. | |
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- | + | {|class="ourtable" | |
+ | |+ Core substances | ||
+ | !colspan="2"|substance | ||
+ | !Description | ||
+ | !Derivative to time | ||
+ | |- | ||
+ | |colspan="4"|''Extracellular'' | ||
+ | |- | ||
+ | | ||As(III)<sub>ex</sub>||As(III) in the solution||(d/dt) As(III)<sub>ex</sub> = <span class="import">- (d/dt) GlpF<sub>As</sub> - k6 GlpF<sub>As</sub></span><span class="export"> + (Vc/Vs) k8 ArsB<sub>As</sub></span> | ||
+ | |- | ||
+ | |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 = <span class="import">- (d/dt) GlpF<sub>As</sub></span> | ||
+ | |- | ||
+ | | ||GlpF<sub>As</sub>||concentration w.r.t. the exterior of the cell||(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> | ||
+ | |- | ||
+ | | ||ArsB||concentration w.r.t. the interior of the cell||(d/dt) ArsB = <span class="export">- (d/dt) ArsB<sub>As</sub> + β4 ars1 - ln(2)/τB ArsB</span> | ||
+ | |- | ||
+ | | ||ArsB<sub>As</sub> ||concentration w.r.t. the interior of the cell||(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> | ||
+ | |- | ||
+ | |colspan="4"|''Intracellular'' (ars2, pro and GV are introduced) | ||
+ | |- | ||
+ | | ||As(III)<sub>in</sub>||concentration of As(III) inside the cell||(d/dt) As(III)<sub>in</sub> = - (d/dt) ArsR<sub>As</sub> - (d/dt) MBPArsR<sub>As</sub> - n<sub>f</sub> (d/dt) fMT<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> | ||
+ | |- | ||
+ | | ||ars1 {{infoBox|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}} ||concentration of unbound promoters naturally occurring in <i>E. coli</i>||(d/dt) ars1 = - (d/dt) ArsR<sub>ars1</sub> | ||
+ | |- | ||
+ | | ||ars2 {{infoBox|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}}||concentration of unbound promoters in front of gas vesicle genes||(d/dt) ars2 = <span class="production">- (d/dt) ArsR<sub>ars2</sub></span> | ||
+ | |- | ||
+ | | ||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 {{infoBox|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.}}||concentration of the accumulation protein ArsR||(d/dt) ArsR = βRN ars1 + βR proR - (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> | ||
+ | |- | ||
+ | | ||ArsR<sub>As</sub> || the concentration of ArsR bound to As(III)||(d/dt) ArsR<sub>As</sub> = kR<sub>on</sub> ArsR As(III)<sub>in</sub> - kR<sub>off</sub> ArsR<sub>As</sub> | ||
+ | |- | ||
+ | | ||ArsR<sub>ars1</sub> ||the concentration of ArsR bound to ars1||(d/dt) ArsR<sub>ars1</sub> = kA<sub>on</sub> ArsR² ars1 - kA<sub>off</sub> ArsR<sub>ars1</sub> | ||
+ | |- | ||
+ | | ||ArsR<sub>ars2</sub> ||the concentration of ArsR bound to ars2||(d/dt) ArsR<sub>ars2</sub> = <span class="production">kA<sub>on</sub> ArsR² ars2 - kA<sub>off</sub> ArsR<sub>ars2</sub></span> | ||
+ | |- | ||
+ | | ||MBPArsR {{infoBox|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.}}|| a fusion of maltose binding protein and ArsR||(d/dt) MBPArsR = βM proM - (ln(2)/τM) MBPArsR - (d/dt) MBPArsR<sub>As</sub> | ||
+ | |- | ||
+ | | ||MBPArsR<sub>As</sub> ||bound to As(III)||(d/dt) MBPArsR<sub>As</sub> = kM<sub>on</sub> MBPArsR As(III)<sub>in</sub> - kM<sub>off</sub> MBPArsR<sub>As</sub> | ||
+ | |- | ||
+ | | ||fMT {{infoBox|It is another binding protein. Unlike it's counterpart it capeble of containing up to five As(III) particles or one As(V) particle }} || Arsenic binding metallotein ||(d/dt) fMT = βF proF - (ln(2)/τF) fMT - (d/dt) fMT<sub>As</sub> | ||
+ | |- | ||
+ | | ||fMT<sub>As</sub> ||bound to multiple As(III)||fMT<sub>As</sub> = kF<sub>on</sub> fMT As(III)<sub>in</sub><sup>n<sub>f</sub></sup> - kF<sub>off</sub> fMT<sub>As</sub> | ||
+ | |- | ||
+ | | ||ArsR<sub>As</sub> ||bound to As(III) | ||
+ | |- | ||
+ | | ||GV ||concentration of gas vesicles||(d/dt) GV = <span class="production">βG ars2 - ln(2)/τG GV</span> | ||
+ | |- | ||
+ | |colspan="4"| | ||
+ | {|class="ourtable" style="width:100%" | ||
+ | !colspan="5"| | ||
+ | |- style="text-align:center;" | ||
+ | |class="fromPaper" style="padding:0;"|Import related. | ||
+ | |class="fromPaper" style="padding:0;"|Import related. | ||
+ | |class="experimental" style="padding:0;"|Export related. | ||
+ | |class="experimental" style="padding:0;"|Export related. | ||
+ | |class="estimate" style="padding:0;"|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): | |
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+ | Using the following constants/definitions: | ||
{|class="ourtable" | {|class="ourtable" | ||
|- | |- | ||
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|Reaction rate constants for reactions to a complex. | |Reaction rate constants for reactions to a complex. | ||
|- | |- | ||
- | | | + | |kAon |
|1/(M²·s) | |1/(M²·s) | ||
|Reaction rate constants for reactions to a complex. | |Reaction rate constants for reactions to a complex. | ||
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|Reaction rate constants for reactions to a complex. | |Reaction rate constants for reactions to a complex. | ||
|- | |- | ||
- | |kRoff, kMoff, kFoff, | + | |kRoff, kMoff, kFoff, kAoff, k5off, etc. |
|1/s | |1/s | ||
|Reaction rate constants for reactions from a complex. | |Reaction rate constants for reactions from a complex. | ||
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|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). | ||
|} | |} | ||
- | |||
See [[Team:Groningen/Literature#Chen1997|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 [[Team:Groningen/BLAST|not present in our system]]). | See [[Team:Groningen/Literature#Chen1997|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 [[Team:Groningen/BLAST|not present in our system]]). | ||
==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) | + | 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: |
<pre> | <pre> | ||
As(III)exT ≈ As(III)ex | As(III)exT ≈ As(III)ex | ||
- | As(III)inT ≈ As(III)in + ArsRAs + MBPArsRAs + fMTAs | + | As(III)inT ≈ As(III)in + ArsRAs + MBPArsRAs + nf fMTAs |
- | + | ||
</pre> | </pre> | ||
Line 432: | Line 511: | ||
GlpF : GlpFAs | GlpF : GlpFAs | ||
ArsB : ArsBAs | ArsB : ArsBAs | ||
- | As(III)in : ArsRAs : MBPArsRAs : fMTAs : ArsBAs ≈ As(III)in : ArsRAs : MBPArsRAs : fMTAs : 0 | + | As(III)in : ArsRAs : MBPArsRAs : nf fMTAs : ArsBAs ≈ As(III)in : ArsRAs : MBPArsRAs : nf fMTAs : 0 |
- | ArsR : ArsRAs : 2 ArsRars | + | ArsR : ArsRAs : 2 ArsRars |
ars : ArsRars | ars : ArsRars | ||
</pre> | </pre> | ||
Line 442: | Line 521: | ||
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)in ArsB - (k7off+k8) ArsBAs | 0 = (d/dt) ArsBAs = k7on As(III)in ArsB - (k7off+k8) ArsBAs | ||
- | 0 = (d/dt) ArsRars = | + | 0 = (d/dt) ArsRars = kAon ArsR² ars - kAoff ArsRars |
0 = (d/dt) ArsRAs = kRon ArsR As(III)in - kRoff ArsRAs | 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) MBPArsRAs = kMon MBPArsR As(III)in - kMoff MBPArsRAs | ||
Line 485: | Line 564: | ||
KRd : As(III)in | KRd : As(III)in | ||
- | 0 = (d/dt) ArsRars = | + | 0 = (d/dt) ArsRars = kAon ArsR² ars - kAoff ArsRars |
- | + | kAon ArsR² ars = kAoff ArsRars | |
- | ArsRars = | + | ArsRars = kAon/kAoff ArsR² ars |
- | ArsRars = ArsR² ars / | + | ArsRars = ArsR² ars / KAd² |
+ | |||
+ | ArsR : 2 ArsRars | ||
+ | ArsR : 2 ArsR² ars / KAd² | ||
+ | KAd² : 2 ArsR ars | ||
ars : ArsRars | ars : ArsRars | ||
- | ars : ArsR² ars / | + | ars : ArsR² ars / KAd² |
- | + | KAd² : ArsR² | |
</pre> | </pre> | ||
Line 513: | Line 596: | ||
ArsRAs = ArsR As(III)in / KRd | ArsRAs = ArsR As(III)in / KRd | ||
- | As(III)in : ArsRAs | + | As(III)in : ArsRAs : MBPArsRAs : n fMTAs |
- | As(III)in : | + | 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 : | + | 1 : ArsR / KRd : MBPArsRT / (KMd+As(III)in) : n fMTT As(III)in^(nf-1) / (KFd^nf+As(III)in^nf) |
</pre> | </pre> | ||
Line 523: | Line 606: | ||
GlpF : GlpFAs = K5 : As(III)ex | GlpF : GlpFAs = K5 : As(III)ex | ||
ArsB : ArsBAs = K7 : As(III)in | ArsB : ArsBAs = K7 : As(III)in | ||
- | As(III)in : ArsRAs : MBPArsRAs : fMTAs ≈ 1 : | + | 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 = | + | ars : ArsRars = KAd² : ArsR² |
- | ArsR : ArsRAs ≈ | + | ArsR : ArsRAs : 2 ArsRars ≈ 1 : As(III)in / KRd : 2 ArsR ars / KAd² |
MBPArsR : MBPArsRAs = KMd : As(III)in | MBPArsR : MBPArsRAs = KMd : As(III)in | ||
fMT : fMTAs = KFd^nf : As(III)in^nf | fMT : fMTAs = KFd^nf : As(III)in^nf | ||
Line 560: | Line 643: | ||
0 = (d/dt) As(III)inT = v5 As(III)exT / (K5+As(III)exT) - k8 ArsBAs | 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) 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 = (d/dt) GV = βG ars2 - ln(2)/τG GV | ||
</pre> | </pre> | ||
Line 568: | Line 653: | ||
0 = βB ars1 - ln(2)/τB ArsB | 0 = βB ars1 - ln(2)/τB ArsB | ||
ArsB = βB (τB/ln(2)) ars1 | ArsB = βB (τB/ln(2)) ars1 | ||
- | ArsB = βB (τB/ln(2)) ars1T | + | 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 | 0 = βG ars2 - ln(2)/τG GV | ||
GV = βG (τB/ln(2)) ars2 | GV = βG (τB/ln(2)) ars2 | ||
- | GV = βG (τB/ln(2)) ars2T | + | GV = βG (τB/ln(2)) ars2T KAd²/(KAd²+ArsR²) |
</pre> | </pre> | ||
- | For the intra- and extracellular concentrations we can find the following | + | For the intra- and extracellular concentrations we can find the following equation, giving a maximum for As(III)in of <code>K7 v5/(k8 ArsB)</code> (as As(III)exT cannot be negative){{infoBox|Conveniently the function <code>x/(c-x)</code> is non-negative and non-decreasing for x∈[0,c⟩.}}: |
<pre> | <pre> | ||
- | 0 = v5 As(III)exT / (K5+As(III)exT) - k8 ArsBAs | + | 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 / (K5+As(III)exT) - k8 ArsB As(III)in / K7 |
- | 0 = v5 As(III)exT / (K5+As(III)exT) - k8 ArsB As(III) | + | 0 = v5 As(III)exT - k8 ArsB As(III)in / K7 (K5+As(III)exT) |
- | 0 = | + | 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) | ||
</pre> | </pre> | ||
- | As we can safely assume arsenic neither disappears into nothingness nor appears from nothingness, we can use this to derive | + | 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): |
<pre> | <pre> | ||
- | + | 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 = | + | 0 = Vs As(III)exT + Vc As(III)inT - As(III)T |
- | 0 = | + | 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 |
- | + | ||
- | + | ||
- | + | ||
- | + | ||
- | + | ||
- | + | ||
- | + | ||
- | + | ||
- | + | ||
</pre> | </pre> | ||
+ | |||
+ | As the function on the right-hand side is non-decreasing for <code>As(III)in∈[0,K7 v5/(k8 ArsB)⟩</code> 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: | Finally, for ArsR we can find the following third-order equation: | ||
Line 609: | Line 694: | ||
<pre> | <pre> | ||
0 = βRN ars1 + βR pro - (ln(2)/τR) ArsR | 0 = βRN ars1 + βR pro - (ln(2)/τR) ArsR | ||
- | 0 = βRN ars1T | + | 0 = βRN ars1T KAd²/(KAd²+ArsR²) + βR pro - (ln(2)/τR) ArsR |
- | 0 = βRN ars1T | + | 0 = βRN ars1T KAd² + βR pro (KAd²+ArsR²) - (ln(2)/τR) ArsR (KAd²+ArsR²) |
- | 0 = βRN ars1T | + | 0 = βRN ars1T KAd² + βR pro KAd² + βR pro ArsR² - (ln(2)/τR) ArsR KAd² - (ln(2)/τR) ArsR³ |
- | 0 = (βRN ars1T + βR pro) | + | 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)) | + | 0 = (βRN ars1T + βR pro) (τR/ln(2)) KAd² - KAd² ArsR + βR (τR/ln(2)) pro ArsR² - ArsR³ |
</pre> | </pre> | ||
- | According to Mathematica's solution of <code>Reduce[eq && | + | According to Mathematica's solution of <code>Reduce[eq && KAd > 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. |
+ | |||
{{Team:Groningen/Footer}} | {{Team:Groningen/Footer}} |
Latest revision as of 12:26, 18 November 2009
- Modelling
- DetailedModel
- Characterization
- Downloads
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.
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. | 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)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 | |||||||||||
|
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.
| ||||||||||
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, 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.
| |||||||||||
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 ↔ GlpFAs | The 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 ↔ ArsBAs | The binding and detachment of arsenic to the Exporter ArsB | |||||||||||
ArsBAs → 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 ↔ ArsRAs | The binding and detachment of arsenic to ArsR | |||||||||||
As(III)in + MBPArsR ↔ MBPArsRAs | The binding and detachment of arsenic to MBPArsR | |||||||||||
nf As(III)in + fMT ↔ fMTAs | The binding and detachment of arsenic to fMT | |||||||||||
ars1 + 2 ArsR ↔ ArsRars1 | the repression of the promotor of the ars1 operon by 2 arsR molecules | |||||||||||
ars2 + 2 ArsR ↔ ArsRars2 | 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 steady-state model and the steady-state model.
substance | Description | Derivative to time | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Extracellular | |||||||||||||
As(III)ex | As(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) | |||||||||||||
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 + 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)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 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 | |||||||||||
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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.
|
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 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 : 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^(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
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
K7 v5/(k8 ArsB)
(as As(III)exT cannot be negative)x/(c-x)
is non-negative and non-decreasing 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^(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 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.