Team:Groningen/Modelling/Arsenic

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==Transport==
==Transport==
 +
===Quasi-steady-state===
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When there are many molecules "waiting" to be transported and/or the concentrations in the cell and outside the cell are relatively slow changing compared to  it is not unreasonable to assume that the amount of bound transporters is constant. Together with the assumption that the total amount of transporters is constant this leads to the well-known [[Team:Groningen/Glossary#MichaelisMenten|Michaelis-Menten equation]] (focussing just on import):
 +
 +
<pre>
 +
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
 +
 +
GlpFT = GlpF + GlpFAs
 +
GlpFT = GlpFAs (K5 / As(III)ex + 1)
 +
GlpFAs = GlpFT / (K5 / As(III)ex + 1)
 +
GlpFAs = GlpFT As(III)ex / (K5 + As(III)ex)
 +
 +
(d/dt) As(III) = (Vs/Vc) k6 GlpFAs
 +
              = (Vs/Vc) k6 GlpFT As(III)ex / (K5 + As(III)ex)
 +
              = v5 As(III)ex / (K5 + As(III)ex)
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</pre>
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 +
Obviously any increase in concentration (of arsenic) in the cell is accompanied by a proportional decrease in concentration outside the cell (and vice versa). So:
 +
 +
<pre>
 +
(d/dt) As(III)ex = (Vc/Vs) v5 As(III)ex / (K5 + As(III)ex)
 +
</pre>
 +
 +
{{todo}} Work in progress.
 +
 +
===Equilibrium===
By looking at the system in equilibrium we can more easily assess the impact of parameters and derive formulas for obtaining them. To this end we consider the system when all derivatives are zero (just taking the equations relevant for transport):
By looking at the system in equilibrium we can more easily assess the impact of parameters and derive formulas for obtaining them. To this end we consider the system when all derivatives are zero (just taking the equations relevant for transport):
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</pre>
</pre>
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===Without ArsB regulation===
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====Without ArsB regulation====
If we disregard ArsB regulation ArsBAs can be determined similarly to GlpFAs:
If we disregard ArsB regulation ArsBAs can be determined similarly to GlpFAs:
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An important flaw in this model is that the production of ArsB is dependent on the concentration of arsenic in the cell (via regulation by ArsR, see [[Team:Groningen/Project/Transport|our transport page]]). This could be one of the reasons that this model is unable to fit the curve shown in figure 3A in [[Team:Groningen/Literature#Kostal2004|Kostal2004]] (if we try a least squares fit with the equation above, filling in v5 and K5 from what we computed for figure 1B in [[Team:Groningen/Literature#Meng2004|Meng2004]], we get negative values for v7 and K7). (And something similar is true of figure 1 from [[Team:Groningen/Literature#Singh2008|Singh2008]].) This led us to consider the regulation of ArsB, as is described in the following section.
An important flaw in this model is that the production of ArsB is dependent on the concentration of arsenic in the cell (via regulation by ArsR, see [[Team:Groningen/Project/Transport|our transport page]]). This could be one of the reasons that this model is unable to fit the curve shown in figure 3A in [[Team:Groningen/Literature#Kostal2004|Kostal2004]] (if we try a least squares fit with the equation above, filling in v5 and K5 from what we computed for figure 1B in [[Team:Groningen/Literature#Meng2004|Meng2004]], we get negative values for v7 and K7). (And something similar is true of figure 1 from [[Team:Groningen/Literature#Singh2008|Singh2008]].) This led us to consider the regulation of ArsB, as is described in the following section.
-
===With ArsB regulation===
+
====With ArsB regulation====
As explained above ArsBAs needs a slightly different approach, as ArsB can be produced in response to arsenic ArsBT is not a (known) constant and we have a formula dependent on OpG instead:
As explained above ArsBAs needs a slightly different approach, as ArsB can be produced in response to arsenic ArsBT is not a (known) constant and we have a formula dependent on OpG instead:

Revision as of 12:54, 18 August 2009

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The raw model

A schematic representation of the processes involved in arsenic filtering (keep in mind that ArsR represses the expression of the genes behind OpG). Note that ArsD is not shown here, as it is not present in our E. coli and has a role analogous to ArsR.

The following variables play an important role in our system (these can be concentrations of substances, the density of the cell, etc.):

  • Extracellular:
    • As(III)ex
  • Membrane:
    • GlpF (concentration w.r.t. the exterior of the cell)
    • GlpFAs (concentration w.r.t. the exterior of the cell)
    • ArsB (concentration w.r.t. the interior of the cell)
    • ArsBAs (concentration w.r.t. the interior of the cell)
  • Intracellular:
    • As(III)
    • OpG (concentration of unbound operators)
    • OpH (concentration of ArsR producing operators that are always on)
    • ArsR
      ArsR binds to OpG to repress production of the genes they regulate, and binds to As(III) to make it less of a problem for the cell.
       i 
    • ArsRAs (bound to As(III))
      • At equilibrium: ArsR As(III) = (k1off/k1on) ArsRAs
    • ArsRopg (bound to opg)

The variables above can be related to each other through the following "reactions" (color coding is continued below to show which parts of the differential equations refer to which groups of reactions):

  • Transport (based on Rosen1996, Meng2004 and Rosen2009)
    • As(III)ex + GlpF ↔ GlpFAs
    • GlpFAs → GlpF + As(III)
    • As(III) + ArsB ↔ ArsBAs
    • ArsBAs → ArsB + As(III)ex
    • ArsB → null (degradation)
  • Accumulation (based on Chen1997)
    • As(III) + ArsR ↔ ArsRAs
    • OpG + 2 ArsR ↔ ArsRopg
    • OpG → OpG + ArsR + ArsB (transcription + translation)
    • OpH → OpH + ArsR (transcription + translation)
    • ArsR → null (degradation)

Resulting in the following differential equations (please note that some can be formed by linear combinations of the others), using color coding to show the correspondence to the reactions above:

  • (d/dt) As(III)ex = -k5on As(III)ex GlpF + k5off GlpFAs + (Vc/Vs) k8 ArsBAs
  • (d/dt) GlpF = -k5on As(III)ex GlpF + (k5off+k6) GlpFAs
  • (d/dt) GlpFAs = k5on As(III)ex GlpF - (k5off+k6) GlpFAs
  • (d/dt) ArsB = -k7on As(III) ArsB + (k7off+k8) ArsBAs + β4 OpG - ln(2)/τB ArsB
  • (d/dt) ArsBAs = k7on As(III) ArsB - (k7off+k8) ArsBAs
  • (d/dt) As(III) = - (d/dt) ArsRAs - k7on As(III) ArsB + k7off ArsBAs + (Vs/Vc) k6 GlpFAs
  • (d/dt) OpG = - (d/dt) ArsRopg
  • (d/dt) ArsR = β1 OpG + β3 OpH - (ln(2)/τR) ArsR - (d/dt) ArsRAs - 2 (d/dt) ArsRopg
  • (d/dt) ArsRAs = k1on ArsR As(III) - k1off ArsRAs
  • (d/dt) ArsRopg = k3on ArsR² OpG - k3off ArsRopg

Using the following constants/definitions:

Name Units Description
k1on, k2on, etc. 1/(M·s) Reaction rate constants for reactions to a complex.
k1off, k2off, 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".
K1d - K4d M Dissociation constants.
  • K1d = k1off/k1on
  • K2d = k2off/k2on = 60µM (Chen1997)
  • K3d² = k3off/k3on = (0.33µM)²? (Chen1997, suspect as the relevant reference doesn't actually seem to give any value for this)
  • K4d = k4off/k4on = 65µM (Chen1997)
τR s ArsR half-life. Degradation rate = ln(2)/τ
If you take just the degradation into account you will have the equation dC/dt = -k*C, which leads to C(t) = C(0) e-k t. So if k = ln(2)/τ we get C(t) = C(0) e-ln(2)/τ t = C(0) 2-t/τ. In other words τ is the time it takes for the concentration to half.
 i 
β1, β2, etc. 1/s Production rates.
  • β1 = the production rate for ArsR behind the ars promoter
  • β3 = the production rate for ArsR behind our constitutive promoter
  • β4 = the production rate for ArsB behind the ars promoter
Vs L Volume of solution (excluding cells).
Vc L Total volume of cells (in solution) (so Vs+Vc is the total volume).
v5, v7 mol/(s·L) Maximum reaction rates 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 it the rates are per liter of cells.
  • v5 = k6 GlpFT (Vs/Vc)
  • v7 = k8 ArsBT (Vc/Vc) = k8 ArsBT
v7'·OpG Reaction rate when As(III) = K7
  • v7' = k8 (β4 τB/ln(2))
K5, K7 M Concentration at which the reaction reaches half its maximum reaction rate (see Michaelis-Menten equation).
  • K5 = (k5off+k6) / k5on
  • K7 = (k7off+k8) / k7on (in our model, with regulation of ArsB, this is the concentration As(III) at which the reaction rate equals v7'·OpG)

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).

Transport

Quasi-steady-state

When there are many molecules "waiting" to be transported and/or the concentrations in the cell and outside the cell are relatively slow changing compared to it is not unreasonable to assume that the amount of bound transporters is constant. Together with the assumption that the total amount of transporters is constant this leads to the well-known Michaelis-Menten equation (focussing just on import): 

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

 GlpFT = GlpF + GlpFAs
 GlpFT = GlpFAs (K5 / As(III)ex + 1)
GlpFAs = GlpFT / (K5 / As(III)ex + 1)
GlpFAs = GlpFT As(III)ex / (K5 + As(III)ex)

(d/dt) As(III) = (Vs/Vc) k6 GlpFAs
               = (Vs/Vc) k6 GlpFT As(III)ex / (K5 + As(III)ex)
               = v5 As(III)ex / (K5 + As(III)ex)

Obviously any increase in concentration (of arsenic) in the cell is accompanied by a proportional decrease in concentration outside the cell (and vice versa). So: 

(d/dt) As(III)ex = (Vc/Vs) v5 As(III)ex / (K5 + As(III)ex)

TODO Work in progress.

Equilibrium

By looking at the system in equilibrium we can more easily assess the impact of parameters and derive formulas for obtaining them. To this end we consider the system when all derivatives are zero (just taking the equations relevant for transport): 

0 = (d/dt) As(III)ex = -k5on As(III)ex GlpF + k5off GlpFAs + (Vc/Vs) k8 ArsBAs
0 = (d/dt) GlpF      = -k5on As(III)ex GlpF + (k5off+k6) GlpFAs
0 = (d/dt) GlpFAs    =  k5on As(III)ex GlpF - (k5off+k6) GlpFAs
0 = (d/dt) ArsB      = -k7on As(III) ArsB + (k7off+k8) ArsBAs + β4 OpG - ln(2)/τB ArsB
0 = (d/dt) ArsBAs    =  k7on As(III) ArsB - (k7off+k8) ArsBAs
0 = (d/dt) As(III)   = -k7on As(III) ArsB + k7off ArsBAs + (Vs/Vc) k6 GlpFAs

The third equation is equivalent to the second and the fifth can be eliminated from the fourth, leading to: 

0 = -k5on As(III)ex GlpF + k5off GlpFAs + (Vc/Vs) k8 ArsBAs
0 = -k5on As(III)ex GlpF + (k5off+k6) GlpFAs
0 = β4 OpG - ln(2)/τB ArsB
0 =  k7on As(III) ArsB - (k7off+k8) ArsBAs
0 = -k7on As(III) ArsB + k7off ArsBAs + (Vs/Vc) k6 GlpFAs

By combining the last two equations a rather obvious equation can be derived that essentially expresses that the import rate equals the export rate: 

    0 - 0 =    (-k7on As(III) ArsB + k7off ArsBAs + (Vs/Vc) k6 GlpFAs)
             + ( k7on As(III) ArsB - (k7off+k8) ArsBAs)
        0 = (Vs/Vc) k6 GlpFAs - k8 ArsBAs
k8 ArsBAs = (Vs/Vc) k6 GlpFAs

Using the equations above (and the fact that the total amount of importers doesn't change in our model) GlpFAs can be expressed as follows (working towards a Michaelis-Menten equation): 

k5on As(III)ex GlpF = (k5off+k6) GlpFAs
               GlpF = GlpFAs (k5off+k6) / (k5on As(III)ex)

 GlpFT = GlpF + GlpFAs
 GlpFT = GlpFAs (1 + (k5off+k6) / (k5on As(III)ex))
GlpFAs = GlpFT / (1 + (k5off+k6) / (k5on As(III)ex))
GlpFAs = GlpFT As(III)ex / (As(III)ex + K5)

Without ArsB regulation

If we disregard ArsB regulation ArsBAs can be determined similarly to GlpFAs: 

ArsBAs = ArsBT As(III) / (As(III) + K7)

By substituting the above equations for GlpFAs and ArsBAs we can now derive a relation between the extracellular and intracellular concentrations of arsenic (where we recognize the constants of the well-known Michaelis-Menten equation): 

                                    k8 ArsBAs = (Vs/Vc) k6 GlpFAs
            k8 ArsBT As(III) / (As(III) + K7) = (Vs/Vc) k6 GlpFT As(III)ex / (As(III)ex + K5)
                  v7 As(III) / (As(III) + K7) = v5 As(III)ex / (As(III)ex + K5)
                  v7 As(III) (As(III)ex + K5) = v5 As(III)ex (As(III) + K7)
As(III) (v7 As(III)ex + v7 K5 - v5 As(III)ex) = v5 K7 As(III)ex
                                      As(III) = v5 K7 As(III)ex / (v7 As(III)ex + v7 K5 - v5 As(III)ex)
                                      As(III) = v5 K7 As(III)ex / ((v7 - v5) As(III)ex + v7 K5)

An important flaw in this model is that the production of ArsB is dependent on the concentration of arsenic in the cell (via regulation by ArsR, see our transport page). This could be one of the reasons that this model is unable to fit the curve shown in figure 3A in Kostal2004 (if we try a least squares fit with the equation above, filling in v5 and K5 from what we computed for figure 1B in Meng2004, we get negative values for v7 and K7). (And something similar is true of figure 1 from Singh2008.) This led us to consider the regulation of ArsB, as is described in the following section.

With ArsB regulation

As explained above ArsBAs needs a slightly different approach, as ArsB can be produced in response to arsenic ArsBT is not a (known) constant and we have a formula dependent on OpG instead: 

β4 OpG = ln(2)/τB ArsB
  ArsB = (β4 τB/ln(2)) OpG

(k7off+k8) ArsBAs = k7on As(III) ArsB
(k7off+k8) ArsBAs = (β4 τB/ln(2)) OpG k7on As(III)
           ArsBAs = (β4 τB/ln(2)) OpG (k7on/(k7off+k8)) As(III)
           ArsBAs = (β4 τB/ln(2)) OpG As(III) / K7

This now leads to the following relation between intra- and extracellular As(III), note that v7' (in contrast to v7) is not the maximum export rate (it has units of 1/second): 

                        k8 ArsBAs = (Vs/Vc) k6 GlpFAs
k8 (β4 τB/ln(2)) OpG As(III) / K7 = (Vs/Vc) k6 GlpFT As(III)ex / (As(III)ex + K5)
               v7'/K7 OpG As(III) = v5 As(III)ex / (As(III)ex + K5)
                          As(III) = v5/(v7' OpG) K7 As(III)ex / (As(III)ex + K5)

To get rid of the unknown OpG in this equation we can use two equations that are derived below for accumulation: 

              0 = β1 OpG + β3 OpH - (ln(2)/τR) ArsR
(ln(2)/τR) ArsR = β1 OpG + β3 OpH
           ArsR = (β1 OpG + β3 OpH) (τR/ln(2))

 OpG = OpGT/(1 + ArsR²/K3d²)
OpGT = (1 + ArsR²/K3d²) OpG
OpGT = OpG + ArsR² OpG / K3d²
   0 = K3d² OpG + ArsR² OpG - K3d² OpGT
   0 = K3d² OpG + (β1 OpG + β3 OpH)² (τR/ln(2))² OpG - K3d² OpGT
   0 = K3d² OpG + (β1 OpG + β3 OpH) (β1 OpG² + β3 OpH OpG) (τR/ln(2))² - K3d² OpGT
   0 = K3d² OpG + β1 OpG β1 OpG² (τR/ln(2))² + β1 OpG β3 OpH OpG (τR/ln(2))² + β3 OpH β1 OpG² (τR/ln(2))² + β3 OpH β3 OpH OpG (τR/ln(2))² - K3d² OpGT
   0 = β1² (τR/ln(2))² OpG³ + 2 β1 β3 (τR/ln(2))² OpH OpG² + (β3² (τR/ln(2))² OpH² + K3d²) OpG - K3d² OpGT

According to Mathematica's solution of Reduce[eq && K3d > 0 && OpGT >= 0 && OpH >= 0 && β1 > 0 && β3 > 0 && τR > 0, OpG, Reals] (where eq is the equation above) there is only one real root, so we solve the equation safely using Newton's (or Halley's) method.

Accumulation

For many purposes, like determining the total amount of accumulated arsenic, it can be quite useful to consider the system in equilibrium. That is, when the derivatives of all variables to time are zero (just taking the equations relevant for accumulation here, assuming the As(III) concentration to be constant): 

0 = (d/dt) ArsR = β1 OpG + β3 OpH - (ln(2)/τR) ArsR - (d/dt) ArsRAs - 2 (d/dt) ArsRopg
0 = (d/dt) ArsRAs = k1on ArsR As(III) - k1off ArsRAs
0 = (d/dt) ArsRopg = k3on ArsR² OpG - k3off ArsRopg

By eliminating the last two equations from the first and dividing the last two by k?on we are left with: 

0 = β1 OpG + β3 OpH - (ln(2)/τR) ArsR
0 = ArsR As(III) - K1d ArsRAs
0 = ArsR² OpG - K3d² ArsRopg

Using the fact that the "concentration" of operators remains constant the last equation can be used to derive an equation for OpG: 

      0 = ArsR² OpG - K3d² ArsRopg
ArsRopg = OpG ArsR² / K3d²

OpGT = OpG + ArsRopg
OpGT = OpG + OpG ArsR² / K3d²
OpGT = OpG (1 + ArsR²/K3d²)
 OpG = OpGT/(1 + ArsR²/K3d²)

This leads to the following: 

0 = β1 OpG + β3 OpH - (ln(2)/τR) ArsR
0 = β1 OpGT + (β3 OpH - (ln(2)/τR) ArsR) (ArsR²/K3d² + 1)
0 = β1 OpGT + β3 OpH ArsR²/K3d² + β3 OpH - (ln(2)/τR) ArsR³/K3d² - (ln(2)/τR) ArsR
0 = K3d² (τR/ln(2)) β1 OpGT + (τR/ln(2)) β3 OpH ArsR² + K3d² (τR/ln(2)) β3 OpH - ArsR³ - K3d² ArsR
0 = ArsR³ - (τR/ln(2)) β3 OpH ArsR² + K3d² ArsR - K3d² (τR/ln(2)) (β1 OpGT + β3 OpH)

According to Mathematica's solution of Reduce[eq && K3d > 0 && OpGT >= 0 && OpH >= 0 && β1 > 0 && β3 > 0 && τR > 0, ArsR, Reals] (where eq is the equation shown above) there is only one real solution, so we can find it safely using Newton's (or Halley's) method.

Analogously to the derivation of the equation for OpG an equation can be derived for the fraction of unbound arsenic in the cell: 

As(III)T = As(III) + ArsRAs
As(III)/As(III)T = 1/(ArsR/K1d + 1)
Retrieved from "http://2009.igem.org/Team:Groningen/Modelling/Arsenic"