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)

Glycolysis

Fructose-1,6-bisphosphate (FBP)

ATP

ADP

Glucose

Fructose-6-phosphate (F6P)

Glucose-6-phosphate (G6P)

ATP

ADPHexokinase (HK), Mg2+

Phosphoglucose isomerase (PGI)

Phosphofructokinase (PFK), Mg2+

Dihydroxyacetone-phosphate (DHAP)Glyceraldehyde-phosphate (GAP)

Aldolase

Triose phosphate isomerasePi + NAD+

NADH + H+

1,3-Bisphosphoglycerate

Glyceraldehyde-3-phosphate dehydrogenase (GAPDH)

ADP

ATP

3-Phosphoglycerate (3PG)

Phosphoglycerate kinase (PGK), Mg2+

2-Phosphoglycerate (2PG)

Phosphoenolpyruvate (PEP)

Pyruvate

ADP

ATP

Phosphoglycerate mutase (PGM)

Enolase, Mg2+ H2O

Pyruvate kinase (PK), Mg2+, K+

Control of glycolysis by metabolite-modulated dynamic

enzyme associations

• Phosphofructokinase/Aldolase

• Aldolase/Glyceraldehyde-3-phosphate dehydrogenase

• Aldolase/Triose phosphate isomerase

Ovadi J (1988) TIBS 13, 486-490.

Functional advantages of FDSs

• Channelling

• increased resistance to hydrolytic enzymes

• protection against toxic or very reactive intermediates

• protection of labile intermediates

-1,2

-0,6

0

-10 40 90

[S1]

v

-Km

Vm

0

0,1

0,2

0,3

0,4

0,5

0 20 40 60 80 100

[S1]

v

00,010,020,030,040,05

0 0,2 0,4 0,6

v

v/[S

1]

0

0,05

0,1

-0,5 0 0,5

v

v/[S

1]

k1f

k1r

k2f

k2r

E1 + S1 E1S1 E1 + S2

=

v = k2f [E1S1] =

1t1f2 SEk

1

f1

f2r1 Sk

kk

1m

1m

SK

SV

The case of a Michaelis-Menten enzyme under steady-state conditions

0

0,1

0,2

0,3

0,4

0,5

0 1 2 3 4 5

S1

v

0

0,01

0,02

0,03

0,04

0,05

0,06

0 0,1 0,2 0,3 0,4 0,5

v

v/[

S1

]^n

v =

n1m

n1m

SK

SV

The case of an allosteric enzyme under steady-state conditions

The rate equation becomes a Hill function :

S1S2

E1

E1S1 E1S2

S2S3

E2

E2S2 E2S3

A two-enzyme system made of free enzymes

S1

E1S1E2S3

E1E2S3

S3

S1

E1

E1S1

E2

E1S1E2

E1S2E2

E1E2S2

An example of a two-enzyme system in which the enzymes are assembled in a FDS

28

S1 E1 S2

E1S1 E1S2

E1E2S1

E1E2S2

E1E2S1S2

E1E2S2S2

1 2

5 622 24

14 15 18

S3E2

E2S2

E1E2 S2

E1E2S1S3

43

872325

2019

16

21

E1E2S2S3

17

9 E2S310

13 E1E2 S31211

2926

27

29 reactions acting on 17 different chemical species

The reactions of the system(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)

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

Dimensionless quantitiesx = [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.

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 +

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.

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

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)

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

(K5K11)/(K6K9) = 1

Hence (since K5, K9 and K11 belong to the base)

K6 = (K5K11)/K9

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

Reaction circuits with a zero balance Derivation of the remaining 15 equilibrium constantsReference Expression

L1 5f 6r 9r 11f K6 = (K5∙K11)/K9

L2 3f 6r 15r 24f K24 = (K5∙K11∙K15)/(K3∙K9)

L3 12r 15f 18r K18 = K15/K12

L4 2f 7r 18r 24f K7 = (K2∙K5∙K11∙K12)/(K3∙K9)

L5 1f 7r 17r 22f K22 = (K2∙K5∙K11∙K12∙K17)/(K1∙K3∙K9)

L6 9r 22f 24r 26f K26 = (K1∙K9∙K15)/(K2∙K12∙K17)

L7 10r 22f 23r 27f K23 = (K2∙K5∙K11∙K12∙K17∙K27)/(K1∙K3∙K9∙K10)

L8 4f 5r 19r 23f K19 = (K11∙K12∙K17∙K27)/K

L9 11r 14f 15r 26f K14 = (K2∙K11∙K12∙K17)/(K1∙K9)

L10 9r 23f 25r 29f K25 = (K2∙K5∙K11∙K12∙K17∙K27∙K29)/(K1∙K3∙(K9)2∙K10)

L11 11f 16f 19r 29r K16 = (K12∙K17∙K27∙K29)/K

L12 26f 27r 28f 29r K28 = (K2∙K12∙K17∙K27∙K29)/(K1∙K9∙K15)

L13 7f 8r 10r 13f K8 = (K2∙K5∙K11∙K12∙K13)/(K3∙K9∙K10)

L14 13r 17f 20r 27f K20 = (K17∙K27)/K13

L15 13r 18f 21r 28f K21 = (K2∙K17∙K27∙K29)/(K1∙K9∙K13)

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

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

Modification of k9f

0

0,02

0,04

0,06

0,08

0,1

0,12

0,14

0,16

0,18

0 0,01 0,02 0,03 0,04 0,05 0,06 0,07 0,08 0,09 0,1

K9f=100

K9f=1000

K9f=10000

K9f=100000

Hyperbole k9f=1000000

A

B

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.

Main references of our team concerning FDSs