1. Metabolic Control Theory
Change of activity
of an enzyme, e.g. PFK
? Change of steady-state
fluxes eg. within TCA cycle ?
? Change of concentration
of metabolites, e.g. pyruvat ?

2. Example: Flux Control

3. Metabolic Control Theory
Quantification of metabolic control
Quantification of the impact of small parameter changes on the variables
of a metabolic system.
Problem:
Relations between steady-state variables and parameters are usually non-linear
and can not be expressed analytically.
There exists no theorie, which permits quantitative prediction of the effect of
large changes of enzyme activity on fluxes.
Restriction to small (infinitesimal) changes.
(Linearisation of the system in vicinity of steady state).
Controlling parameters: kinetic constants, enzyme concentrations,...
Controlled variables: fluxes, substrate concentrations   
Wanted: mathematical function quantifying control.

4. Metabolic Control Theory
Relevant questions:
Many mechanisms and regulatory properties of isolated enzyme reactions are known
 what is their quantitative meaning for metabolism in vivo?
Which step of a metabolic systems controls a given flux?
(Is there a rate-limiting step?)
Which effectors or modifiers have the most influence on the reaction rate?
Example: biotechnological production of a substance, Increase of turnover rate
Question: which enzyme to activate in order to yield the most effect?
Example: disease of metabolism, overproduction of a substance
Question: which reaction to modify to reduce overproduction in a predictable way?

5. Coefficients used in Control Theory
Metabolic systems are networks; their behavior depends on the structure
of the network and the properties of the individual components.
There are two types of coefficients : local and global ones
Elasticity coefficients
Control coefficients
Response coefficients
quantify the sensitivity of a rate
for the change of a concentration
Or a parameter value
directly, immediate
(no steady state)
Quantitative measure for change
of steady-state variables
Assume reaching new steady states
Depends on network structure

6. Locally: Elasticity Coefficients
Question: How sensitive is a rate
of an enzyme reaction with respect
to small changes of a metabolite
concentration? v1 v2 v3 S1 S2 ? ? ?
Consider enzyme as isolated,
Wanted: immediate effect
Elasiticity coefficient of
reaction rate k with respect to
metabolite concentration Si

7. Parameter Elasticity v1 v2(Km, Vmax) v3 S1 S2 ?
-elasticities comprise derivatives with respect to
a metabolite concentration (a variable!).
-elasticities comprise derivatives with respect to
parameter values (kinetic constants, enzyme concentrations,...) v1
v2(Km, Vmax) v3 S1 S2 ?

8. Globally: Control Coefficients
1. The system of metabolic reactions is in steady state.
J = v(S(p),p) S = S(p)
2. A small perturbation of a reaction is performed
(Addition of enzyme, addition of metabolite,....)
3. The system approaches a new (nearby) steady state.
J J+DJ S S+DS
What is the change of steady state-variables (fluxes,
concentrations) due to the perturbation of a single reaction?

9. Definition of Control Coefficients
Flux control coefficient
- Change of rate of the k-th reaction under isolated fixed conditions
- Resulting change of steady state flux through the j-th reaction
- Normalization factor v1 v2 v3 S1 S2 ? ? ?

10. Definition of Control Coefficients
Concentration
control coefficient
- Change of rate of the k-th reaction under isolated fixed conditions
- Resulting change of steady state concentration of Si
- Normalization factor v1 v2 v3 S1 S2 ? ?

11. Choice of Perturbation Parameter
The change of vk is based on a change of some parameters pk, which influences only this k-th reaction.
(Enzymkonzentration, Inhibitoren, Aktivatoren, ....)
Extended expression of flux control
coefficient:
Important:
Perturbation of pk influences directly
only vk and no further reaction
The flux control coefficients are then
independent of the choice of the perturbed parameters pk.
The can be interpreted as measure for the degree to which reaction k
controls a given flux in steady state.

12. Response Coefficients, global
Consider: complete system in steady state.
This state is determined by the values of the parameters;
Parameter changes influence the steady state.
The sensitivity of steady-state variables with respect to parameter
Perturbations is expressed by response coefficients. v1
v2(Km, Vmax, I) v3 S1 S2 ? ? ?

13. Response Coefficients, Additivity
If several reactions are sensitive
for this parameter v1
v2(p)
v3(p) S1 S2
if parameter m influences
Only one reaction j :

14. Example 1 Add inhibitor to a biochemical reaction
Experimentelly measurable quantities:
Flux control coefficient

15. Non-normalized Coefficients
Non-normalized flux and
Concentration control coefficients
Non-normalized elasticities
Examples
Glycolysis:
Glucose
2 Lactat Be
Control of second reaction?
Non-normalized flux control coeff.
Normalized coefficients

16. Matrix Notation

17. Theorems of Control Theory
The problem:
The fluxes J usually cannot be expressed as mathematical functions
Of the reaction rates. How can one calculate the global control
coefficients from the local (measurable) changes ??
The solution:
Use of theoremes.
Here, the theoremes are only given and described with examples.
The mathematical derivation is given for your information on the
following pages.

18. The Summation Theorems
Thought experiment: What happens, if we induce by experimental manipulation the same fractional change  in the local rate of all steps of the system?
Result: Flux J must also increase by factor . Since all rates increase in the same ratio, remain the concentration of the variable metabolites S1 and S2 unchanged.
The combined effect of all changes in the local rates on the systems variables J, S1 and S2 can be described
As the sum of all individual effects caused by the change of each local rate. For flux J holds:
Thus holds
Analog for S1 and S2
It follows:

19. The Summation Theorems
The flux control coefficients of a metabolic
pathway add up to 1.
The enzymes share the control over flux.
The concentration control coefficients for a
substance add up to zero.
Some enzymes increase a metabolite
concentration, others decrease it.
Matrix notation:

20. Connectivity Theorems – General Relations
Connectivity between flux control coefficients and elasticities
Connectivity between concentration control coefficients and elasticities

21. Example: Calculate flux control coefficients
Summation theorem:
Connectivity theorem:
Result:
Since in general:
and
follows (i.a.!):
Both reaction exert positive control over the flux.

22. Example: Calculate concentration control coefficients
Summation theorem:
Connectivity theorem:
Result:
It holds:
and
Producing reactions have positive control,
consuming reactions have negative control.

23. Example: Linear pathway1 2 S
i-1 r - v
i+1 i .
... S1 S5
Concentration
Control
Coefficients S9
Reaction
Producing reactions have positive control,
consuming reactions have negative control.

24. Linear Metabolic Pathway1 2 S
i-1 r - v
i+1 i .
...
Each rate is a function of the concentrations of
substrates and productes
Assuming mass action kinetics
With the equilibrium constants
One can derive an equation for the
Steady state flux

25. Linear Metabolic Pathway – Flux Control1 2 S
i-1 r - v
i+1 i .
...
General Expression for flux control coefficients
(if )
r flux control coefficients
1 summation theorem
r-1 connectivity theorems

26. Linear Pathway - Properties1 2 S
i-1 r - v
i+1 i .
...
Ratio of two successive flux control coeff.:
Flux control coefficients: Summation theorem
Since sum of all flux control coeff is 1,
and the ratio of two successive flux control coefficients is positiv,
all flux control coefficients in an unbranched pathway are positiv.

27. Linear Pathway - Properties1 2 S
i-1 r - v
i+1 i .
...
Case 1:
Be the kinetic constants of all
involved enzymes equal and the
equilibrium constants larger than 1
Ratio of two successive flux control coeff.:
Flux control coefficients tend to be larger at the beginning than at the end.

28. Linear Pathway - Properties1 2 S
i-1 r - v
i+1 i .
...
Case 2:
with
holds:
Using Relaxation time  as measure for the velocity of an enzyme:
with or holds and therefore
All enzymes are involved in control.
Slow enzymes exert more control.
There is no „rate-limiting step“

29. Flux increase – how? Simple case: E1  E1 + 1%v v v 1 2 3 v P S S S 4 P 1 1 2 3 2
Flux control coefficients
Simple case:
0.5
0.4
0.3
0.2
0.1 1 2 3 4
Reaction
E1  E1 + 1%
J  J + C1 * 1% 

30. Flux increase P S S S P P S P S P S v v v v 1 1 2 3 2 1 2 v 3 1 2 v 34 P S S S P 1 1 2 3 2 P 1 2 S v 4 3 P 1 2 S v 4 3 P 1 2 S v 4 3

31. Irreversibility and Feedback1 v 2 v 3 P S S S v 4 P 1 1 2 3 2 v 1 v 2 v 3 S S v 4 P S P 1 1 2 3 2 v 1 v 2 v 3 v 4 P S S S P 1 1 2 3 2

32. Bis hier am 7. Juni 2010

33. Branching System v1 v2 v3 P0 S1 S2 P3 v4 v5 P4 P5

34. Branching System with ATP/ADP-Exchange1 S 3 2 v
ATP
ADP
Branching System with ATP/ADP-Exchange
Rang(N) = 2 < r
Conservation relation ATP + ADP = const. G
Reduced stoichiometric matrix
Basis vector for admissible steady state fluxes

35. Mathematical Derivation of the Theorems
Start with equation
Implicite Differentiation w.r.t.
parameter vector p
regular Jacobi matrix M
Rearrange to
Rearrange to

36. Mathematical Derivation of Theorems, 2
Start with equation
Implicite differentiation w.r.t.
parameter vector p
Non-normalized
Flux CC
Rearrange to
Both non-normalized CC are independent of the choice of perturbed parameter.
They depend only on stoichiometry (N) and kinetics (dv/dS) !!

37. Theorems, Normalized Control Coefficients

38. Reaction System with Conservation Relations
Problem: Jacobi-Matrix is not regular
Rearrange rows of N and S,
Such that dependent rows are at bottom.
Implicite Differentiation of independent
steady-state equations
w.r.t parameter vector p
The non-singular Jacobi matrix
Of the reduced systems :
Non-normalized concentrations cc
Non-normalized flux cc

39. Real System – Glycolysis Concentration control coefficients1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
AMP
NADH
NAD+
ADP
ATP
CNx
ACAx
Glycx
Glyc
EtOHx
EtOH
ACA
Pyr
PEP
BGP
DHAP
GAP
FBP
F6P
G6P
Glc
Glcx

40. „Real System“ – Glycolysis Flux control coefficients

41. Concentration control in the MAP K cascade
MAPKKK 1
MAPKKK-P 3
MAPKKK-PP 2 4
MAPKK 5
MAPKK-P 7
MAPKK-PP 6 8
MAPK 9
MAPK-P 11
MAPK-PP 10 12
All parameter values: k, p=1

42. Concentration control in Wnt signaling
2003

43. Non-Steady State Trajectories
What is the effect of parameter
perturbations on time courses ?
S1[0] = 0
S2[0] = 0
S3[0] = 0
S4[0] = 1
S[t] S2 S1 S2 S4 S3 v1 v2 v3 v4 v5 S3 S4 p1 p2
p1 = 1
p2 = 1
p3 = 1
p4 = 0.5
p5 = 0.5
S2[0]
S4[0]
S1[0] p3
p1,3
RS2
S3[0] p5 p4 p2 p4 p5 p5
S3[0]
RS3
S4[0]
S2[0]
p1,3
S1[0] p2
B.P. Ingalls, H.M. Sauro, JTB, 222 (2003) 23–36 p4
Time

44. Time-dependent Response:
B.P. Ingalls, H.M. Sauro, JTB, 222 (2003) 23–36

45. Experimental Methods to Determine Control Coefficients
- Titration with purified enzyme
- Addition of specific inhibitors
- Overexpression of an enzyme using
genetic techniques
- Downregulation of individual genes / Reduction of
enzyme amount

46. Metabolic Control Analysis - History/People
1973 Kacser /Burns
1974 Heinrich /Rapoport - Definition of coefficients
about 1980 Discovery by Experimentalists (Westerhoff)
1988 Reder – Matrix Formulation
BTK - Models and Experiments
(Fell, Cornish-Bowden, Hofmeyr, Bakker, Schuster,….)