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

Hill Kinetics

REFERENCE: Segel; 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)

Union 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 ommiting the fact that cI:cI will be "kidnapped" by the promotor. This doesn't seem important since we only have 10 molecules of the promoter per cell, comparing with 150 basal molecules (without the AHL signal) of the dimer.

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

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-1. 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 = 0.46min-1

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 ≈ 18.8min-1”

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

We modified both transcription initiation and translation rates by multiplying both rates by 4. This due to the fact that LuxR is a four times smaller than LacZ:

LacZ has a length of 1024 aa

LuxR has a length of 250 aa

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

Stechiometric 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

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

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

 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.

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.

-