Biology Reference
In-Depth Information
If each metabolite x i is controlled by a set of m microRNAs and let M ij denote the
control strength of the microRNA j with concentration m j on x i , i.e., we have:
M ij ¼
CM is a
matrix whose the sum of coefficients on a line is 1, summarizing the effects of the
microRNAs on the metabolic network. If we calculate the stationary flux increment
ΔΦ k resulting from a perturbation
Log
Δ
x i /
Log
Δ
m j , then we get that the product of matrices P
¼
Δ
m from the value m
¼
ΔΦ k (
Δ
m )
¼ Φ k (0)
0, as:
m j , then the general term of P , i.e., the control
strength P kj exerted by the microRNA j on the flux
Σ j v kj Δ
m j )
Φ k (0)
Σ j v kj Δ
[exp(
1]
Φ k , can be calculated, if all
Δ
m j
are small as:
P kj ¼ @
Log
ΔΦ k =@
Log
Δ
m j
v kj Δ
m j j v kj Δ
m j ;
where v kj ½@ΔΦ k =@Δ
m j k ð
0
Þ
The directed signed graph associated to the incidence (or adjacency) matrix P is the
same as the up-tree part of the interaction graph of a genetic network corresponding
to inhibitions by microRNAs, where the interaction weight of the microRNA j on
the gene expressing the enzyme k is equal to v kj (cf. Mathematical Annex), which
justifies the use of similar tools like the entropy of the network and the subdominant
eigenvalue of the Markovian matrix P , for characterizing the stability and the
robustness of the metabolic network. The molecules controlled by microRNAs
through the expression of their genes are proteins, enzymes, carriers, or membrane
receptors, and the control strength equation can be used to prove that the most
regulated molecules in glycolysis are enzymes like hexokinase, PFK, GPDH, and
ADK, which rule the pool of the energetic molecules, ATP, ADP, and AMP, which
conversely are mainly produced by the glycolysis (Wolf and Heinrich 2000 ; Ruoff
et al. 2003 ; Bier et al. 1996 ; Mourier et al. 2010 ). In Fig. 4.12 , we see that the most
sensitive steps of glycolysis are hexokinase, PFK, and GPDH, inhibited in human
by the microRNAs hsa-miR-19, 4659/320, and 142, respectively.
When oscillations occur, we can use the variables T ki (resp A ki ) to quantify the
control by
τ k (resp. intensity amplitude I k ) of the k th
Δ
x i of the period
step flux
x i being the perturbation of the concentration of the i th
“pacemaker” effector (i.e., parameter causing oscillations), from its bifurcation
values x i :
(Baconnier et al. 1993 ),
Δ
T ki ¼ @
Log
τ k =@
Log
Δ
x i and A ki ¼ @
Log I k =@
Log
Δ
x i :
is the eigenvalue of the Jacobian matrix of the differential system for
which the stationary state has bifurcated in a limit cycle (Hopf bifurcation), then
τ k ¼
If
ξ
π
/ I m
ξ
2
, and if we consider a 2D potential-Hamiltonian example like:
d x 1 =
d t
¼@
P
=@
x 1 þ @
H
=@
x 2 ;
d x 2 =
d t
¼@
P
=@
x 2 @
H
=@
x 1 ;
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