Guillaume Legent 1 , Patrick Amar 2,3 , Vic Norris 1,3 , Camille Ripoll 1,3 and Michel Thellier 1,3 1 Laboratoire AMMIS (Assemblages Moléculaires: Modélisation et Imagerie SIMS), Faculté des Sciences et Techniques, Université de Rouen, F-76821 Mont-Saint-Aignan Cedex, France 2 Laboratoire de Recherche en Informatique, UMR CNRS 8623, Université de Paris Sud, F-91405 0rsay Cedex, France 3 Epigenomics Project, Génopole, 93 rue Henri Rochefort, F- 91000 Evry, France Functioning-dependent structures Functioning-dependent structures and the coordination between and within and the coordination between and within metabolic and signalling pathways metabolic and signalling pathways
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Guillaume Legent 1, Patrick Amar 2,3, Vic Norris 1,3, Camille Ripoll 1,3 and Michel Thellier 1,3 1 Laboratoire AMMIS (Assemblages Moléculaires: Modélisation.
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Guillaume Legent1, Patrick Amar2,3, Vic Norris1,3, Camille Ripoll1,3 and Michel Thellier1,3
1Laboratoire AMMIS (Assemblages Moléculaires: Modélisation et Imagerie SIMS), Faculté des Sciences et Techniques, Université de Rouen, F-76821 Mont-Saint-
Aignan Cedex, France2 Laboratoire de Recherche en Informatique, UMR CNRS 8623, Université de Paris
Sud, F-91405 0rsay Cedex, France3 Epigenomics Project, Génopole, 93 rue Henri Rochefort, F-91000 Evry, France
Functioning-dependent structuresFunctioning-dependent structuresand the coordination between and within metabolic and and the coordination between and within metabolic and
signalling pathwayssignalling pathways
E2E1
C
S1e
S3
E2
E1
C
S1e
S1
S2
S2S1
Metabolic pathways
With or without a carrier “C”
Antigens Hormones
Other signals (e.g. electric
depolarisation)
Signalling pathways
The concept of FDS(Functioning-dependent structure)
For those rate constants that are expressed in s−1:k9f = k’9f/k’1r k9r = k’9r/k’1r k5r = k’5r/k’1r etc.
k1r = k’1r/k’1r ≡ 1
For those rate constants that are expressed in mol−1 s−1 m3:k1f = ([E1]t + [E2]t)·k’1f/k’1r k5f = ([E1]t + [E2]t)·k’5f/k’1r etc.
The steady-state
• External mechanisms are assumed to supply S1 and remove S3 as and when they are consumed and produced, respectively, in such a way as to maintain S1 at a constant concentration (s1 = constant) and S3 at a zero concentration (s3 = 0).
• The equations of the system are obtained by
writing down the mass balance of the other 15 chemical species involved.
The equations of the systemde1/dτ = k1r∙e1s1 − k1f∙e1∙s1 + k2r∙e1s2 − k2f∙e1∙s2 + k7r∙e1e2s2 − k7f∙e1∙e2s2 +
For each of the 29 reactions, j, the equilibrium constant, Kj, is written
Kj = kjf/kjr
Using the MAPLE software, the rank of the 29×17 matrix of stoichiometric coefficients is shown to be equal to 14. This means that, to solve the equations of the system, the values of 14 equilibrium constants (or linear combinations of these constants) can be chosen arbitrarily, while the other 15 equilibrium constants will be calculated by appropriate linear combinations of the 14 basic ones. We have chosen the base
K1, K2, K3, K5, K9, K10, K11, K12, K13, K15, K17, K27, K29 and K
in which K is the equilibrium constant of the overall reaction S1 → S3
Numerical simulations• The list of the reactions (here 29) acting upon the involved chemical species (here
17) is written down.• The rank (here 14) of the matrix (here 29×17) of the stoichiometric coefficients is
determined using the MAPLE software. • A base of independent equilibrium constants (here 14 equilibrium constants), K
included, is chosen and their values are also chosen. • The other equilibrium constants (here 15) are calculated i) along a reaction pathway
S1 → S3 and ii) along the reaction circuits with a zero balance of an arbitrarily chosen base of such circuits (here 15 circuits).
• Using dimensionless quantities normalised with k’1r, k1r ≡ 1. • Values of kjf or kjr, for all reactions j other than 1, are chosen. • s1 = constant et s3 = 0 are imposed (constrained steady-state). • The set of equations of the system is solved using the MAPLE software. • The steady-state rate of functioning of the system, v corresponding to either the
consumption of S1or the production of S3) is calculated by (in our present case):v(s1) = − k1r∙e1s1 + k1f·e1·s1 − k17r·e1s1e2s2 + k17f·s1·e1e2s2 − k20r·e1s1e2s3 + k20f·s1·e1e2s3 v(s3) = − k4f·e2·s3 + k4r·e2s3 − k16f·s3·e1s2e2 + k16r·e1s2e2s3 − k19f·s3·e1s1e2 + k19r·e1s1e2s3
Classical « input/output» functions in electrical and electronic circuits
A) linear response, B) constant response, C) impulse response, Da) step response, Db) inverse step response
Kinetic behaviour of the free enzymes
Depending on the values given to the independent parameters, the following kinetic behaviours occur
Kinetic behaviour of a single free enzyme
s1
Kinetic behaviour of a set of two sequential free enzymes
• The values given to the parameters were K1 = 100, K2 variable, K3 = 10, k4f calculated, k4r = 1, K9 = 10, K10 = 1
• The dashed straight line is the slope at origin, the dashed curve is the hyperbola with the same slope at origin and the same saturation plateau as the curve K2 = 0.1.
Kinetic behaviour of FDSs
Depending on the values given to the independent parameters, the following kinetic behaviours occur
An example of a FDS with a sigmoid kinetic behaviour
Various examples of the kinetic behaviour of a FDS
A) and B) the equivalent of an impulse function impulsion, Cd) the equivalent of a constant function, D) the equivalent of an inverse step
function
modification of k10f
0
0,05
0,1
0,15
0,2
0,25
0,3
0,35
0 0,02 0,04 0,06 0,08 0,1 0,12 0,14 0,16 0,18 0,2
K10f=1
K10f=10
K10f=100
K10f=1000
Discussion1) An FDS can display kinetic properties that the individual enzymes cannot, including the full range of basic input/output characteristics found in electronic circuits such as linearity, invariance, impulse and switching
2) Hence FDSs can play a role in the control of cell metabolism and homeostasis
3) Sigmoids only are not very convincing: a role for allosteric enzymes?
4) Instead of the classical implication
Structure → Function
life involves a double implication
Structure Function
5) Via FDSs, living systems create and maintain dynamically the catalytic structures for the tasks to be carried out
What is life?
The density of entropy production
jj
jj
j JXT
1
σ = density of entropy production
j = a process under consideration (electric, transport, reaction)
T = Absolute temperature
Xj = the force acting on the process j (gradient of a potential, affinity of a reaction)
Jj = the flux of the process j (electric intensity, flux of the transport of a substance, rate of a reaction)
σ = (1/T) (XelectricJelectric + XtransportJtransport + XreactionJreaction + etc.
●Under steady-state conditions, the density of entropy production does not depend on the way how the system S1 ↔ S3 is catalysed (free enzymes or FDS) and it is written
σS1↔S3 = (1/T) AS1↔S3 vS1↔S3
●Under non steady-state conditions, i.e. if the FDS (E1E2) is in the process of associating from the free enzymes (E1 and E2), or of dissociating, according to whether the substrate concentration increases or decreases, extra terms corresponding to these modifications are going to be involved
σE1,E2↔E1E2 = (1/T) A E1,E2↔E1E2 v E1,E2↔E1E2
●More generally, if a structure in a living system is created and maintained by its own functioning (e.g. the dynamical maintaining of the cytoskeleton), this will be responsible for the presence of specific extra terms in the expression of the density of entropy production.
●In brief, when a system undergoes a transformation, independent of this system being living or inanimate, the density of entropy production associated with this transformation can be expressed as a sum of terms XjJj corresponding to the functioning of the processes j involved in the transformation, i.e., as stated above,
jj
jj
j JXT
1
●However, if some of these terms XjJj correspond to modifications of the system structure dependent on the system functioning (e.g. AE1,E2↔E1E2 vE1,E2↔E1E2 corresponding to the assembly/disassembly of free enzymes/FDSs), then this means that this system is a living one.
Thellier M, Legent G, Norris V, Baron C, Ripoll C (2004) Introduction to the concept of “functioning-dependent structures” in living cells. CR Biologies 327, 1017-1024.
Thellier M, Legent G, Amar P, Norris V, Ripoll C (2006) Steady-state kinetic behaviour of functioning-dependent structures. FEBS J. 273, 4287-4299.
Legent G, Thellier M, Norris V, Ripoll C (2006) Steady-state kinetic behaviour of two- or n-enzyme systems made of free sequential enzymes involved in a metabolic pathway. CR Biologies 329, 963-966.