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

WET LAB

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.

MODELING:

Hill Kinetics:

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:

Estimating the amount of AiiA per cell:

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.

WET LAB

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.

WET LAB

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 .

WET LAB

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.

WET LAB

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.

WET LAB

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.

MODELING:

Stoichiometric Matrix:

 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

 

Image from Systems Biology: Properties of Reconstructed Networks by Bernhard O. Palsson

The vector produced by a linear transformation is in two orthogonal spaces (the column and left null spaces), called the domain, and the vector being mapped is also in two orthogonal spaces (the row and null spaces), called the codomain or the range of the transformation.

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.

MODELING:

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

       Null space. The null space of S contains all the steady-state flux distributions allowable in the network. The steady state is of much interest since most homeostatic states are close to being steady states.

       Row space. The row space of S contains all the dynamic flux distributions of a network and thus the thermodynamic driving forces that change the rate of reaction activity.

       Left null space. The left null space of S contains all the conservation relationships, or time invariants, that a network contains. The sum of conserved metabolites or conserved metabolic pools do not change with time and are combinations of concentration variables.

       Column space. The column space of S contains all the possible time derivatives of the concentration vector and thus shows how the thermodynamic driving forces move the concentration state of the network.

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.

Image from Systems Biology: Properties of Reconstructed Networks by Bernhard O. Palsson Image from Systems Biology: Properties of Reconstructed Networks by Bernhard O. Palsson

 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.

MODELING:

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 null space represents all the possible functional, or phenotypic, states of a network.

       A particular point in the polytope represents one network function or one particular phenotypic state.

       As we will see in Chapter 16, there are equivalent points in the cone that lead to the same overall functional state of a network. Biologically, such conditions are called silent phenotypes.

       The edges of the flux cone are the unique extreme pathways. Any flux state in the cone can be decomposed into the extreme pathways. The unique set of extreme pathways thus gives a mathematical description of the range of flux levels that are allowed.

- 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.