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Lecture #9

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  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 ‘-’ • activationinhibition • 2. Active concentration range • Gain • Local vs. distant active range rate i ∂rate/∂i i y x x

  5. 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 x x - + regulated rates - +

  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. Inhibition

  11. The Steady State

  12. Parametric Sensitivity steady state concentration increases response is faster

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

  14. Activation

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

  16. Key Quantities

  17. Multiple Steady States =fn(a) one to three one = fn(a)

  18. Eigenvalues and their location in the complex plane Im 1 2 3 4 Re Transient response: “smooth” landing overshoot damped oscillation sustained oscillation chaos

  19. 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)

  20. Simulating regulation ENZYMES AS MOLECULES

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

  22. The Dynamic Equations Time derivative Kinetic expressions Fluxes

  23. The Steady-State Equations

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

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

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

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

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

  29. Allosteric Regulation of Enzyme Activity dimer tetramer

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

  31. 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

  32. 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.

  33. 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

  34. Key Regulatory Step in Glycolysis (Advanced)

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

  36. Kinetic Description (Advanced)

  37. Scaling the Equations (Advanced)

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

  39. Computation of Multiple Steady States (Advanced)

  40. Additions • Compute the fluxes across the multiple steady state region