Lecture #9

Regulation

Multiple levels of enzyme regulation:

1) gene expression,2) interconversion,3) ligand binding,4) cofactor availability

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, …

Phenomenology

active rangerate

∂rate/∂i

i

i

x x y

THE MATHEMATICS OF REGULATION OF ENZYME ACTIVITY

Local regulation

Local Regulation:The five basic cases

• No regulation

• Feedback inhibition

• Feedback activation

• Feedforward inhibition

• Feedforward activation

x v=kx

x-

x+

x-

x+

mass actionkinetics

regulated rates

Combination of Rate Constants

“local” regulation vs.“distant” regulation

sign biasgain magnitude

The ‘Net’ Rate Constant:an eigenvalue or a systems time constant

x+

x+

-x x

-

A Principle for Local Regulation

Inhibition

The Steady State

Parametric Sensitivity

steady state concentration increases response is faster

Dynamic Response

x-

Hill kinetics Mass actionkinetics

a

Activation

(s) stable(u) unstable x

rate

x

(s)

(u)

(s)

+

uniq

uem

ult

In a steady state the mass balance becomes:

=0

simultaneouslysatisfied

Activation

Key Quantities

Multiple Steady States

one

three

one

= fn()

=fn(a)

to

Eigenvalues and their location in the complex plane

1 2 3 4 Im

Re

Transient response:1 “smooth” landing2 overshoot3 damped oscillation4 sustained oscillation5 chaos

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()-> more negative)• Local feedback activation/feedforeward inhibition

is destabilizing (Re()-> more positive)

ENZYMES AS MOLECULESSimulating regulation

Regulation at a “Distance”pr

imar

y pa

thw

ay

perturbation

biosynthetic pathway

x6

x1 x2 x5

x5x5

x6 x7

regulator binding site

The Dynamic EquationsTime

derivative FluxesKinetic

expressions

The Steady-State Equations

Simulation Results

x1 x5

b1

v0

v1 v510x

1.0

0.1

t=0 t

b1

Complicated to interpret the time responses: what is going on?

Phase Portrait and Pool Interpretation

x1 x5

b1

v0

v1 v5

10x

1.0

0.1

t=0 t

b1

flux balancing onbiosynthetic pathway

flux

state ofthe enzyme

concentration

Regulation of Gene Expression

x6

x7

x5

v7

(-)

translation decay

inhibition of translation

Simulation Results

total enzyme ≠ const

slow response of protein translation

fast metabolicinhibitory response

x1 x5

b1

v0

v1 v5

10x

1.0

0.1

t=0 t

b1

Phase Portrait and Pool Interpretationflux balancing on

biosynthetic pathway

state ofthe enzyme

x1 x5

b1

v0

v1 v5

10x

1.0

0.1

t=0 t

b1

flux

concentration

dimer

tetramer

Allosteric Regulation of Enzyme Activity

Simulation Results:monomer, dimer, tetramer

tetramer

dimer

monomer

x1 x5

b1

v0

v1 v5

10x

1.0

0.1

t=0 t

b1

disturbance rejectiontetramer > dimer > monomer

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

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.

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

Key Regulatory Step in Glycolysis (Advanced)

Regulatory Signals (Advanced)

x+Effective schema:

v1(x) v2(x)

Kinetic Description (Advanced)

Scaling the Equations (Advanced)

Criteria for Existence of Multiple Steady States (Advanced)

Computation of Multiple Steady States (Advanced)

Additions

• Compute the fluxes across the multiple steady state region