1. Lecture #9
Regulation

2. Multiple levels of enzyme regulation: 1) gene expression, 2) interconversion, 3) ligand binding, 4) cofactor availability

3. Outline Phenomenology of regulation and signaling
The mathematics of regulatory coupling
Simulating regulation:
Enzymes as molecules in simulation
Fractional states of macromolecular pools
Monomers, dimers, tetramers, …

4. Phenomenology 1. Built-in bias; ‘+’ or ‘-’ activation inhibition
2. Active concentration range
Gain
Local vs. distant
active range
rate i
∂rate/∂i i x x y

5. THE MATHEMATICS OF REGULATION OF ENZYME ACTIVITY
Local regulation
THE MATHEMATICS OF REGULATION OF ENZYME ACTIVITY

6. Local Regulation: The five basic cases
mass action
kinetics x
v=kx
No regulation
Feedback inhibition
Feedback activation
Feedforward inhibition
Feedforward activation - x x +
regulated rates - x + x

7. Combination of Rate Constants
“local” regulation vs.
“distant” regulation
sign bias
gain magnitude

8. The ‘Net’ Rate Constant:
an eigenvalue or a systems time constant x + x + - x x -

9. A Principle for Local Regulation

10. Dynamic Effects of Regulation
(advanced)

11. Inhibition

12. The Steady State

13. Parametric Sensitivity
steady state
concentration increases
response is faster

14. Dynamic Response a - x
Mass action
kinetics
Hill kinetics

15. Activation

16. In a steady state the mass balance becomes:
Activation
rate x
(s)
(u) +
(s) stable
(u) unstable x
unique
mult
In a steady state the mass balance becomes:
l=0
simultaneously
satisfied

17. Key Quantities

18. Multiple Steady Statesto
=fn(a)
one
three
= fn(a)

19. Eigenvalues and their location in the complex planeIm Re
Transient response:
“smooth” landing
overshoot
damped oscillation
sustained oscillation
chaos

20. Some observations
Regulation moves the eigenvalues in the complex plane (only discussed real values here)
Eigenvalues are systemic time constants
The mathematics to analyze regulation is complex
Local feedback inhibition/feedforeward activation is stabilizing (Re(l)-> more negative)
Local feedback activation/feedforeward inhibition is destabilizing (Re(l)-> more positive)

21. Simulating regulation
ENZYMES AS MOLECULES

22. Regulation at a “Distance”
biosynthetic pathway
perturbation
primary pathway x6 x1 x2 x5 x7
regulator binding site

23. The Dynamic Equations
Time
derivative
Fluxes
Kinetic
expressions

24. The Steady-State Equations

25. Complicated to interpret the time responses: what is going on?
Simulation Results
10x
1.0
0.1
t=0 t b1 x1 x5 b1 v0 v1 v5
Complicated to interpret the time responses: what is going on?

26. Phase Portrait and Pool Interpretation
10x
1.0
0.1
t=0 t b1
flux balancing on
biosynthetic pathway
flux x1 x5 b1 v0 v1 v5
state of
the enzyme
concentration

27. Regulation of Gene Expressionv7
(-)
translation
decay x7 x6 x5
inhibition of translation

28. Simulation Results total enzyme ≠ const fast metabolic
10x
1.0
0.1
t=0 t b1 x1 x5 b1 v0 v1 v5
total enzyme ≠ const
fast metabolic
inhibitory response
slow response of
protein translation

29. Phase Portrait and Pool Interpretation
flux balancing on
biosynthetic pathway
flux
10x
1.0
0.1
t=0 t b1 x1 x5 b1 v0 v1 v5
concentration
state of
the enzyme

30. Allosteric Regulation of Enzyme Activity
dimer
tetramer

31. Simulation Results: monomer, dimer, tetramer disturbance rejection
10x
1.0
0.1
t=0 t b1 x1 x5 b1 v0 v1 v5
dimer
tetramer
monomer
disturbance rejection
tetramer > dimer > monomer

32. Some observations
Enzymes can be added as molecules into simulation models
Enzymes will have multiple functional states
The fractional state is important
Tetramers are more effective than dimers that are more effective than monomers when it comes to regulation

33. Summary The activities of gene products are often directly regulated.
Regulation can be described by:
i) its bias,
ii) the concentration range over which the regulatory molecule is active and
iii) its strength, that is how sensitive the flux is to changes in the concentration of the regulator.
In addition the `distance' in the network between the site of regulation and the formation of the regulator is an important consideration.
In general, local signals that:
support the natural mass action trend in a network are `stabilizing’
counter the mass action trend may destabilize the steady state and create multiple steady states.

34. Summary Regulation of enzyme activity comes down to:
i) the functional state of the gene product (typically fast),
ii) regulating the amount of the gene product present (typically slow); and
examining the functional state of the pool formed by the amount of the active gene product and then the total amount itself.
Regulatory mechanisms
can be build on top of the basic stoichiometric structure of a network being analyzed and its description by elementary mass action kinetics
are described by additional reactions that transform the regulated gene product from one state to the next with elementary reaction kinetics

35. Key Regulatory Step in Glycolysis (Advanced)

36. Regulatory Signals (Advanced)
Effective schema: x +
v1(x)
v2(x)

37. Kinetic Description (Advanced)

38. Scaling the Equations (Advanced)

39. Criteria for Existence of Multiple Steady States (Advanced)

40. Computation of Multiple Steady States (Advanced)

41. Additions
Compute the fluxes across the multiple steady state region