Team:Groningen/Modelling/Characterization

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Optimization procedure

To fit our model to experimental data from different uptake experiments and/or papers we have implemented an optimization procedure that allows for experiments with different genotypes and circumstances by letting constants be overridden per experiment. It aims to optimize the sum of the RMS errors for each experiment using Simulated Annealing. By clicking the button "Fit" the optimization is started and its progress can be followed by looking at the table of constants and the graphs shown below the table (which are updated in real-time as the best solution is improved).


best cur gradient solved
v5/K5
v5
K5
k8/K7
k8
K7
tauBbetaB
tauB
betaB
tauR
betaRN
tauFbetaF
tauF
betaF
tauKbetaK
tauK
betaK
ars2T
E

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Relative Promotor Strength

We have four kinds of parts we would like to characterize (RPS stands for Relative Promoter Strength):

     Input Output Subject
Importers
     RPS → Δvmax                                             

We can measure how much v5 (vmax for As(III) import via GlpF) is in wild-type E.coli and when we over express GlpF at a certain promoter strength S (measured in RPUs). As v5 is a constant times the amount of (active) GlpF this leads to a simple equation for Δv5, if we assume the amount of (active) GlpF produced by our construct is linearly dependent on the promoter strength (v5(0) and v5(1) would be measured):

v5(RPS) = v5wt + Δv5*RPS

v5(0) = v5wt + Δv5*0
v5(S) = v5wt + Δv5*S

Δv5 = (v5(1) - v5(0))/S

Meng 2004 was able to knock out all efflux of arsenic. If there is no efflux of arsenic the dervative of the accumulation graph is the speed at wich arsenic is pumped inside the cell. The maximum speed would be v5. In such a senario two measurements would be enough to determine the relative promotor strength. One could even determine the reaction rate k6 and GlpF with a simple calculation, since v5 = k6 GlpFT (Vs/Vc). However we do have efflux, not only do we have efflux, but we have efflux that is dependent on the total amount of arsenic inside the cell. Also a portion arsenic gets bound to ArsR.

Accumulators
     RPS → Asbound(As(III)in)

For both MBPArsR and fMT we assume the amount of bound As(III) for a given relative promoter strength S obeys (for MBPArsR n=1):

Asbound(As(III)in)
 = S Bmax As(III)in^n
    / (K^n + As(III)in^n)
The constants Bmax, K and n can be determined from uptake experiments comparing E. coli with and without fMT expression. Of course this can be done in general by fitting our model to experimental data, if enough data is provided the fit will be tight enough to allow this. However, even without fitting the full model it should be possible to make a fair estimation from equilibrium measurements.
If the total cell volume is much smaller than the volume of the solution it is reasonable to assume a constant import rate. Also, regardless of whether they feature fMT or not, in equilibrium the amount of ArsR is the same, as is the amount of ArsB, leading to the same amount of unbound arsenic being present. This means that any difference in uptake of arsenic is completely due to arsenic being bound to fMT or MBPArsR. By measuring the amount of arsenic in equilibrium in wild-type cells as well as in cells expressing fMT/MBPArsR for several different (inital) concentrations of As(III), at one or more (known) levels of expression, it is possible to determine the constants Bmax, K and n.
 i 
Sensors
     metal(t) → RPS(t)

The ars promoter is part of a feedback loop, so it is not a simple matter of defining the (instantaneous) promoter strength. Instead we suggest using the relevant equations from our model. The necessary parameters can be determined by fitting uptake measurement data to our model. Specifically, if the RPS is measured without arsenic present and with enough arsenic present to keep the promoter fully active during the experiment we can determine βRN τR as follows (under the assumption that the RPS is linearly dependent on arsT/ars and using the fact that without any arsenic present the cells will be in equilibrium):

S(max) / S(0) = ars(max) / ars(0)
S(max) / S(0) = arsT / ars(0)
S(max) / S(0) = 1 + ArsR(0)²/KAd²
      ArsR(0) = KAd √(S(max)/S(0) - 1)

0 = βRN ars1(0) - (ln(2)/τR) ArsR(0)
0 = βRN ars1T S(0)/S(max)
     - (ln(2)/τR) KAd √(S(max)/S(0) - 1)
βRN ars1T S(0)/S(max)
 = (ln(2)/τR) KAd √(S(max)/S(0) - 1)
βRN τR = (ln(2)/ars1T) KAd
   (S(max)/S(0)) √(S(max)/S(0) - 1)
GVP cluster
     RPS → GV

RPS → GV The amount of gas vesicles can be expressed in terms of buoyant density, as volume fraction, using the total mass of the vesicles, etc. No matter how it is expressed, we assume a simple linear dependency between the RPS and the amount of gas vesicles. By taking (T)EM pictures of slices the amount of gas vesicles formed under influence of different RPSes can be determined and a straightforward fit made.


Uptake measurements

To gain measurements for the optimization procedure described above we conducted ICPMS measurements on our cells containing various devices. To optimize our findings we have conducted measurements both in time and in concentration. Details on ICPMS the experiments can be found on our Protocol Page. Measurments have been conducted at at times and concentrations as indicated in the table on the right. Results can be seen below.

Sampling scheme
Time (min)
0 10 20 40 60
As(III)exT(0)
(µM)
0 x
10 x x x x x
20 x
50 x
100 x

First Uptake Measurement
In this case we looked at the arsenic uptake of our wildtype and of our wildtype with a lot of pArs promoters with RFP. The raw data can be found at our Download Section

        AsUptakeWildTypeConcentration.pngAsUptakePArsRFPConcentration.png
        AsUptakeWildTypeTime.pngAsUptakePArsRFPTime.png
        Getting (Vc/Vs) by means of the OD
Information
        Fluoresence Measurments
Information
        Combining fluoresence measurements with the uptake of arsenic
Information