# Lecture #9 Regulation.

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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 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

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

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

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

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

A Principle for Local Regulation

Dynamic Effects of Regulation

Inhibition

Parametric Sensitivity
steady state concentration increases response is faster

Dynamic Response a - x Mass action kinetics Hill kinetics

Activation

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

Key Quantities

to =fn(a) one three = fn(a)

Eigenvalues and their location in the complex plane
Im Re Transient response: “smooth” landing overshoot damped oscillation sustained oscillation 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(l)-> more negative) Local feedback activation/feedforeward inhibition is destabilizing (Re(l)-> more positive)

Simulating regulation
ENZYMES AS MOLECULES

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

The Dynamic Equations Time derivative Fluxes Kinetic expressions

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?

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

Regulation of Gene Expression
v7 (-) translation decay x7 x6 x5 inhibition of translation

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

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

Allosteric Regulation of Enzyme Activity
dimer tetramer

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

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)

Effective schema: x + v1(x) v2(x)