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Functioning-dependent structures

Functioning-dependent structures and the coordination between and within metabolic and signalling pathways. Guillaume Legent 1 , Patrick Amar 2,3 , Vic Norris 1,3 , Camille Ripoll 1,3 and Michel Thellier 1,3.

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Functioning-dependent structures

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  1. Functioning-dependent structures and the coordination between and within metabolic and signalling pathways 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, France 2Laboratoire 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

  2. C E1 E2 E1 E2 C The concept of FDS(Functioning-dependent structure) Metabolic pathways Signalling pathways S1e Antigens Hormones Other signals (e.g. electric depolarisation) With or without a carrier “C” S1 S2 S2 S1 S3 S1e

  3. Glycolysis Glucose ATP Hexokinase (HK), Mg2+ ADP Glucose-6-phosphate (G6P) Phosphoglucose isomerase (PGI) Fructose-6-phosphate (F6P) ATP Phosphofructokinase (PFK), Mg2+ ADP Fructose-1,6-bisphosphate (FBP) Aldolase Dihydroxyacetone-phosphate (DHAP) Glyceraldehyde-phosphate (GAP) Triose phosphate isomerase Pi + NAD+ Glyceraldehyde-3-phosphate dehydrogenase (GAPDH) NADH + H+ 1,3-Bisphosphoglycerate ADP Phosphoglycerate kinase (PGK), Mg2+ ATP 3-Phosphoglycerate (3PG) Phosphoglycerate mutase (PGM) 2-Phosphoglycerate (2PG) Enolase, Mg2+ H2O Phosphoenolpyruvate (PEP) ADP Pyruvate kinase (PK), Mg2+, K+ ATP Pyruvate

  4. Control of glycolysis by metabolite-modulated dynamic enzyme associations • Phosphofructokinase/Aldolase • Aldolase/Glyceraldehyde-3-phosphate dehydrogenase • Aldolase/Triose phosphate isomerase Ovadi J (1988) TIBS13, 486-490.

  5. Functional advantages of FDSs • Channelling • increased resistance to hydrolytic enzymes • protection against toxic or very reactive intermediates • protection of labile intermediates

  6. k1f k2f E1 + S1 E1S1 E1 + S2 k1r k2r [ ] V S m 1 [ ] + + K S k k [ ] m 1 + S 1 r 2 f k 1 1 f The case of a Michaelis-Menten enzyme under steady-state conditions [ ] [ ] k E S = v = k2f [E1S1] = 2 f 1 t 1

  7. [ ] n V S v = m 1 [ ] n + K S m 1 The case of an allosteric enzyme under steady-state conditions The rate equation becomes a Hill function :

  8. A two-enzyme system made of free enzymes S1 S2 E1 E2 S2 S3 E1S1 E1S2 E2S2 E2S3

  9. E1 S1 E2 E1S1 E1S1E2 S3 E1S1E2S3 E1S2E2 S1 E1E2S2 E1E2S3 An example of a two-enzyme system in which the enzymes are assembled in a FDS

  10. S1 E1 S2 E2 S3 1 2 3 4 E1S1 E1S2 E2S2 E2S3 10 9 5 6 7 8 22 24 25 23 E1E2 S1 E1E2 S2 E1E2 S2 E1E2 S3 11 12 13 14 18 19 20 15 21 17 16 E1E2 S1S2 E1E2 S2S2 E1E2 S1S3 E1E2 S2S3 26 28 29 27 29 reactions acting on 17 different chemical species

  11. (1) E1 + S1 = E1S1 k’1f k’1r (2) E1 + S2 = E1S2 k’2f k’2r (3) E2 + S2 = E2S2 k’3f k’3r (4) E2 + S3 = E2S3 k’4f k’4r (5) E1S1 + E2 = E1S1E2 k’5f k’5r (6) E1S2 + E2 = E1S2E2 k’6f k’6r (7) E2S2 + E1 = E1E2S2 k’7f k’7r (8) E2S3 + E1 = E1E2S3 k’8f k’8r (9) E1S1 = E1S2 k’9f k’9r (10) E2S2 = E2S3 k’10f k’10r (11) E1S1E2 = E1S2E2 k’11f k’11r (12) E1S2E2 = E1E2S2 k’12f k’12r (13) E1E2S2 = E1E2S3 k’13f k’13r (14) E1S1E2 + S2 = E1S1E2S2 k’14f k’14r (15) E1S2E2 + S2 = E1S2E2S2 k’15f k’15r (16) E1S2E2 + S3 = E1S2E2S3 k’16f k’16r (17) E1E2S2 + S1 = E1S1E2S2 k’17f k’17r (18) E1E2S2 + S2 = E1S2E2S2 k’18f k’18r (19) E1S1E2 + S3 = E1S2E2S3 k’19f k’19r (20) E1E2S3 + S1 = E1S1E2S3 k’20f k’20r (21) E1E2S3 + S2 = E1S2E2S3 k’21f k’21r (22) E1S1 + E2S2 = E1S1E2S2 k’22f k’22r (23) E1S1 + E2S3 = E1S1E2S3 k’23f k’23r (24) E1S2 + E2S2 = E1S2E2S2 k’24f k’24r (25) E1S2 + E2S3 = E1S2E2S3 k’25f k’25r (26) E1S1E2S2 = E1S2E2S2 k’26f k’26r (27) E1S1E2S2 = E1S2E2S3 k’27f k’27r (28) E1S2E2S2 = E1S2E2S3 k’28f k’28r (29) E1S1E2S3 = E1S2E2S3 k’29f k’29r (globale) S1 → S3 (Cte d’équilibre = K) The reactions of the system k’1f to k’8f and k’14f to k’25f in mol−1 s−1 m3, k’9f to k’13f, k’26f to k’29f and all the k’jr in s−1

  12. Dimensionless quantities x = [X]/([E1]t + [E2]t) e.g. e1 = [E1]/([E1]t + [E2]t) e1s1e2s3 = [E1S1E2S3]/([E1]t + [E2]t) etc. Time (k’1r in s−1): τ = k’1r∙t 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.

  13. 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.

  14. The equations of the system de1/dτ = k1r∙e1s1 − k1f∙e1∙s1 + k2r∙e1s2 − k2f∙e1∙s2 + k7r∙e1e2s2 − k7f∙e1∙e2s2 + k8r∙e1e2s3 − k8f∙e1∙e2s3 = 0 de2/dτ = k3r∙e2s2 − k3f∙e2∙s2 + k4r∙e2s3 − k4f∙e2∙s3 + k5r∙e1s1e2 − k5f∙e2∙e1s1 + k6r∙e1s2e2 − k6f∙e2∙e1s2 = 0 ds2/dτ = − k2f∙e1∙s2 + k2r∙e1s2 − k3f∙e2∙s2 + k3r∙e2s2 − k14f∙s2∙e1s1e2 + k14r∙e1s1e2s2 − k15f∙s2∙e1s2e2 + k15r∙e1s2e2s2 − k18f∙s2∙e1e2s2 + k18r∙e1s2e2s2 − k21f∙s2∙e1e2s3 + k21r∙e1s2e2s3 = 0 de1s1/dτ = − k1r∙e1s1 + k1f∙e1∙s1 + k5r∙e1s1e2 − k5f∙e2∙e1s1 − k9f∙e1s1 + k9r∙e1s2 − k22f∙e1s1∙e2s2 + k22r∙e1s1e2s2− k23f∙e1s1∙e2s3 + k23r∙e1s1e2s3 = 0 de1s2/dτ = k2f∙e1∙s2 − k2r∙e1s2 − k6f∙e2∙e1s2 + k6r∙e1s2e2 + k9f∙e1s1 − k9r∙e1s2 − k24f∙e1s2∙e2s2 + k24r∙e1s2e2s2− k25f∙e1s2∙e2s3 + k25r∙e1s2e2s3 =0 etc.

  15. Independent and calculated equilibrium constants 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 Then the remaining 15 constants K6, K7, K8, K14, K16, K18, K19, K20, K21, K22, K23, K24, K25, K26 et K28 are calculated from the basic 14 constants

  16. Expression of K and calculation of K4 K is calculated along any reaction pathway whose balance is s1→ s3, e.g. {1f 2r 3f 4r 9f 10f}, i.e. : (1f) E1 + S1 = E1S1 (2r) E1S2 = E1 + S2 (3f) E2 + S2 = E2S2 (4r) E2S3 = E2 + S3 (9f) E1S1 = E1S2 (10f) E2S2 = E2S3 This means that K = (k1f·k2r·k3f·k4r·k9f·k10f)/(k1r·k2f·k3r·k4f·k9r·k10r) = (K1·K3·K9·K10)/(K2·K4) or K4 = k4f/k4r = (K1∙K3∙K9∙K10)/(K2∙K)

  17. Reaction circuits with a zero balance: principle of the derivation of the remaining 15 constants Reaction circuits with a zero balance exist; this is the case of e.g. 5f 6r 9r 11f (5f) E1S1 + E2 = E1S1E2 (6r) E1S2E2 = E1S2 + E2 (9r) E1S2 = E1S1 (11f) E1S1E2 = E1S2E2 Along such a reaction circuit one may write (K5K11)/(K6K9) = 1 Hence (since K5, K9 and K11 belong to the base) K6 = (K5K11)/K9

  18. Derivation of the remaining constants from a base of 15 circuits with a zero balance

  19. 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

  20. 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

  21. Kinetic behaviour of the free enzymes Depending on the values given to the independent parameters, the following kinetic behaviours occur

  22. Kinetic behaviour of a single free enzyme s1

  23. 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.

  24. Kinetic behaviour of FDSs Depending on the values given to the independent parameters, the following kinetic behaviours occur

  25. An example of a FDS with a sigmoid kinetic behaviour K1 = 0.1 (k1r≡ 1), K2 = 0.1, K3 = 10, K5 = 1000, K10 = K11 = K12 = K13 = K15 = K17 = K29 = 1, K27 = 100 k4r = k6r = k7r = k8r = k14r = k16r = k18r = k19r = k20r = k21r = k22r = k23r = k24r = k25r = k26r = k28r = 1 It is the equivalent of a step function

  26. A B A B

  27. 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

  28. Discussion 1) 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

  29. What is life?

  30. The density of entropy production 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.

  31. ●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.

  32. ●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, ●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.

  33. Main references of our team concerning FDSs Thellier M, Legent G, Norris V, Baron C, Ripoll C (2004) Introduction to the concept of “functioning-dependent structures” in living cells. CR Biologies327, 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 Biologies329, 963-966.

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