Team:LCG-UNAM-Mexico/Notebook/2008-October

We  used 18 tubes with 5ml of liquid  LBAm100 and  one control to grew the 18 samples that we streaked yesterday from the pJET/rcnA clones.
At the afternoon  we extracted the pJET/rcnA plasmids from the 18 clones using the  Alkaline Lysis protocol.
We load the plasmidic DNA from the 18 pJET/rcnA samples in a 1% agarose gel and ran them 1hr at 100volts.

REFERENCE: Irwin H. Segel's Enzyme kinetics: Behaviour and Analysis of rapid equilibrium and Steady-state Enzyme systems.

Multiple Inhibition Analysis

v = kp[ES]          ->         v/[Et]=kp[ES]/([E]+[IE]+[EI]+[IEI]+[ES])

Ks=[E][S]/[ES]   ->         [ES]=[E][S]/Ks  
Ki=[I][E]/[IE]     ->         [IE]=[I][E]/Ki    
Ki=[EI][I]/[IEI]   ->         [IEI]=[EI][I]/Ki=[I]2[E]/Ki2

v/[Et]=([E][S]/Ks)kp/([E]+([I][E]/Ki)+([E][I]/Ki)+([I]2[E]/Ki2)+([E][S]/Ks)) 
v/[Et]=([S]/Ks)kp/(1+2([I]/Ki)+([I]2/Ki2)+([S]/Ks))                       
v=([S]/Ks)kp[Et]/(1+2([I]/Ki)+([I]2/Ki2)+([S]/Ks))=([S]/Ks)Vmax/(1+2([I]/Ki)+([I]2/Ki2)+([S]/Ks))

With cooperativity:    
v=([S]/Ks)Vmax/(1+2([I]/Ki)+([I]2/aKi2)+([S]/Ks)) 
*a factor

It can be written in Hill's terms (if the cooperativity is strong). 

System: cI repression

Inhibitor:        cI:cI      (I)       
“Enzyme”:       ρ          (ρ)       
Substrate:          -          
Product:        RcnA     (P)

Binding sites:         OR2 & OR1         
I in OR1             ->         ρI        
I in OR2             ->         Iρ

K5-1=[I][ρ]/[ρI] 
K5-2=[I][ρ]/[Iρ] 
a & b cooperativity factor

* K5=[ρ][I]2/[IρI]

K5=K5-1·K5-2·a

ΔGº=ΔGº1+ ΔGº2+ ΔGº12

1/K5=exp(-ΔGº/RT)=exp(ΔGº1/RT)+ exp(ΔGº2/RT)+ exp(ΔGº12/RT)

ρI         ->         ΔGº1=-11.7 kcal/mol    
Iρ         ->         ΔGº2=-10.1 kcal/mol    
Coop.   ->         ΔGº12=-2 kcal/mol        
ΔGº =-23.8 kcal/mol

v=k6·ρ  
v/[ρt]=kp[ρ]/([ρ]+[Iρ]+[ρI]+[IρI])          
v/[ρt]=[ρ]kp/([ρ]+([ρ][I]/K5-1)+([I][ρ]/K5-2)+([I]2[ρ]/K52))
v/[ρt]= kp/(1+([I]/K5-1)+([I]/K5-2)+([I]2/K5))        
v= kp[ρt]/(1+([I]/K5-1)+([I]/K5-2)+([I]2/K5))

NOTE: We are not considering the fact that cI:cI will be sequestered by the promotor. This doesn't seem important since we only have 10 molecules of the promoter per cell, compared with 150 cI:cI molecules (without the AHL signal).

WET LAB

From the 18 samples we ran on the gel only 8 were useful because they were between 3.5 and 4Kbp. (pJET/rcnA should have 3874bp)..
We used the samples : 2,4,6,9,10, 12,14,18 and double-digested them all night at 37°C using XbaI and HindIII restriction enzymes; we also double-digested the pBBR1MCS-5  vector using the same restriction enzymes mentioned before.

MODELING:

AiiA is under the control of the lac promoter. The transcription and mRNA degradation rates help us estimate the amount of mRNA present on the cell.

“The half-life of protein A is assumed to be around 10 minutes which is similar to what is used in Elowitz’s repressilator model [1]. Furthermore, we assume that a more aggressive degradation tail can enable half-times on the order of two minutes for protein B.”

Modeling the Lux/AiiA Relaxation Oscillator by Christopher Batten

In the paper AiiA is called protein B. Therefore the degradation rate for AiiA with an aggressive degradation tail is 0.0058/s. This would give us a lower limit.

Transcription initiation rate, km

Malan et al. (1984) measured the transcription initiation rate at P1 and report the following value: km ≈ 0.18min-1

mRNA degradation rate, jM

Kennell and Riezman (1977), measured a lacZ mRNA half-life of 1.5 min: ξM = 0.46/min

lacZ mRNA translation initiation rate, кB

From Kennell and Riezman (1977), translation starts every 3.2 s at the lacZ mRNA. This leads to the following translation initiation rate: кB ≈ 18.8/min

Santillán M. and Mackey M. C. (2004). Influence of Catabolite Repression and Inducer Exclusion on the Bistable Behavior of the lac Operon. Biophys J 86:1282–1292

Simulating with simbiology, AiiA reaches stationary state at almost 3500 molecules per cell.

We inactivated the enzymes by putting them  at 65°C for 20min, Then we loaded them into an 1% agarose gel and ran them for about 1hr  at 100volts.

We prepared a 1% low melting point agarose Gel in order to cut from it the rcnA bands that we obtained by double-digesting the pJET/rcnA plasmid.

pJET/rcnA(sample 6) 0.2418g
pJET/rcnA(sample 12) 0.1872g
We purified the gel band using the QIAquick Gel Extraction Kit.
After that we proceeded to ligate the pBBR1MCS-5 vector with the purified rcnA fragment using the Fermentas Rapid DNA ligation kit.
We left the ligation reaction at the 20°C room the complete weekend.

We use the  ligated samples of pBBR1MCS-5/rcnA to transform E.coli TOP10 by electroporation.
We grew for 1hr the electroporated cells in  1ml of LB at 37°C. We plated the growth cells on a LB Gm20 agar plate .

We streaked 4 LBGm20 agar plates with different TOP10/pBBR1MCS-5-rcnA colonies, which were taken from the plates that we made yesterday and let them grow all night.

We took 3 samples from the TOP10/pBBR1MCS-5-rcnA plates and put them into 3 tubes with 3ml of Liquid LBGm20.
We used the  Roche Pasmid DNA extraction Kit  in order to obtain the pBBR1MCs-5 vector with rcnA.

We ran a 1% agarose gel to verify the extraction.

The vector with rcnA was double-digested in order to verify the correct ligation of the fragments.
A 1% agarose gel was made to see the digested vector.
Gel NENE VI
The pBBR1MCs-5/rcnA was used to transform W3110/YohM- competent cells by Heat shock. The transformed cells were plated on LBGm20 agar plates and incubated at 37°C the whole night.

WET LAB

The  W3110/YohM-/pBBR1MCs-5/rcnA colonies were streaked on LBGm20 agar plates and incubated the whole night at 37°C.
The planning for the measurements was completed and ready to start.  
The propose for the determination of nickel internalization considerates to assume that when the medium has certain nickel concentration, the presence of cells from different strains will cause a variation in the resistivity(which, by the way will affect conductivity). For the different strains that we are using, we expect:

The two known critical concentrations of NiSO4 are 2.2 mM, the minimum inhibitory, and 500uM the minimun needed to repress RcnR activity.
Planned methodology:
-Establishment of a place to do the measurements.
  -We expect to have a constant temperature during the measurement process.
-Test the sensitivity of the measurement  dispositive. To achieve this, measurements will be done in agitation of:

• Just LB
• LB with NiSO4
• LB with cells

• (,001) Dilución de 10 microlitros de 0.1 M en 1 mililitro
• (,00001) Dilución de 10 microlitros de 0.001M en 1 mililitro
• (0,0000001) Dilución de 10 microlitros de 0.00001 M en un mililitro

Internalization

-Grow the strain YohM- in LB medium until reach an OD(optical density) of 0.4 to ensure that the cells are in exponential phase. (Later this was changed to an OD of 0.5 at a lambda of 600nm).

-Plot concentration vs  time and calculate the slope
slope= Internalization rate of Ni2+

Extrusion
The following steps were performed in order to prove the RcnA activity, and to get the enough data to calculate conductivity  from the resistivity measurements. These conductivity data will be useful to get the extrusion rate of RcnA.

-          Grow the strains YohM-, YohM- + pBBMICS-5+RcnA, YohM- + pBBMICS-5 and YohM+  in LB medium until reach an OD(optical density) of 0.4 to ensure that the cells are in exponential phase. (Later this was changed to an OD of 0.5 at a lambda of 600nm).
-          Take 1ml  from each culture for each of the next nickel concentrations:

- The measurements will be performed during three minutes for each one strain

WET LAB
Some measurements were done in LB with and without nickel in order to calibrate and prove the apparatus. Initially the measurement dispositive included a gold electrode which seemed to be working properly.

WET LAB

Solutions nickel 500uM and without nickel at all were prepared from samples of LB, and the strains YohM-, YohM- + pBBIMRCS-5 and YohM- + pBBIMRCS-5 + RcnA.

WET LAB
The OD that the cells must reach was changed to 0.5 at a lambda of 600nm(This was due to the finding of the proper OD measurement in the strain W3110).
Some adjusts were done to the measurement dispositive in order to increase its sensibility.

WET LAB
Preparation of the strains YohM-, YohM- + pBBIMRCS-5, and YohM- + pBBIMRCS-5 + RcnA for the measurements. An overnight of each strain was grown in LB. Samples of YohM- + pBBIMRCS-5, and YohM- + pBBIMRCS-5 + RcnA were prepared with different concentrations of NiSO4 . That concentrations includes a gradient from 1x10-3 M – 1X10-10M. And the two critical concentrations 500uM and 2.2mM.

WET LAB
Strains were prepared in pursuit to perform some new measurements. Samples of LB, LB+ sulfate, and YohM-+RCNA+PBB were read.
Measurement intervals (100 milliseconds) semeed to be quite large, so its correction was proposed.

MODELING:

 Palsson, 2006

Flux vector                   ->         v=(v1, v2, …, vn)
Concentration vector      ->         x=(x1, x2, …, xm)          
->         δx/δt = S·v

δxi/δt=∑Sikvk

The four fundamental subspaces

The vectors in the left null space (li) represent mass conservation.

The flux vector can be decomposed into a dynamic component and a steady-state component:     
v = vdyn + vss

The steady state component satisfies Svss=0 and vss is thus in the null space of S.

WET LAB
Prosecution of correction of measurement intervals.

WET LAB
Due to the great variation observed in the data, some aspects of the samples, measurements and electrodes were reconsidered. As the variations could be caused by the ions present in the medium it was proposed to use a buffer instead of LB medium.

The higher the number of independent reaction vectors, the smaller the orthogonal left null space. The higher the number of independent reactions, the fewer the conservation quantities exist.

 FUNDAMENTAL SUBSPACES OF S

The dimensions of both the column and row space is r (rank; number of linearly independent rows and columns that the matrix contains).         
dim(Col(S)) = dim(Row(S)) = r   
Since the dimension of the concentration vector is m, we have     
dim(Left Null(S)) = m− r
Similarly, the flux vector is n-dimensional; thus, 
dim(Null(S)) = n – r

Singular Value Decomposition

SVD states that for a matrix S of dimension m× n and of rank r, there are orthonormal matrices U (of dimension m ×m) and V (of dimension n × n) and a matrix with diagonal elements ∑ = diag(σ1, σ2, ... , σr ) with σ1 ≥ σ2 ≥ ··· ≥ σr > 0 such that S = U∑VT

A non-negative real number σ is a singular value for M if and only if there exist unit-length vectors u in Km and v in Kn such that           
Mv=σu and M*u=σv
The vectors u and v are called left-singular and right-singular vectors for σ, respectively.       
In any singular value decomposition       
M=UΣV*
the diagonal entries of Σ are necessarily equal to the singular values of M. The columns of U and V are, respectively, left- and right-singular vectors for the corresponding singular values.

 The columns of U are called the left singular vectors and the columns of V are the right singular vectors. The columns of U and V give orthonormal bases for all the four fundamental subspaces of S (see Figure 8.3). The first r columns of U and V give orthonormal bases for the column and row spaces, respectively. The lastm− r columns of U give an orthonormal basis for the left null space, and the last n − r columns or V give an orthonormal basis for the null space.

WET LAB
Correction of the measurement intervals to 1 millisecond.
161008

THE (RIGTH) NULL SPACE OF S

The right null space of S is defined by    
Svss = 0
Thus, all the steady-state flux distributions, vss, are found in the null space. The null space has a dimension of n − r. Note that vss must be orthogonal to all the rows of S simultaneously and thus represents a linear combination of flux values on the reaction map that sum to zero.

Mathematics versus biology

- The stoichiometric matrix has a null space that corresponds to a linear combination of the reaction vectors that add up to zero; so-called link-neutral combinations.

- The orthonormal basis given by SVD does not yield a useful biochemical interpretation of the null space of the stoichiometric matrix.

THE LEFT NULL SPACE OF S

As with the (right) null space, the choice of basis for the left null space is important in describing its contents in biochemically and biologically meaningful terms.

…may represent mass conservation…

 THE ROW AND COLUMN SPACE OF S

The column and row spaces of the stoichiometric matrix contain the concentration time derivatives and the thermodynamic driving forces, respectively.