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Published byKorbin Chumley Modified over 2 years ago

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Lecture #9 Regulation

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

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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, …

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Phenomenology 1. Built-in bias;+or- activation inhibition 2. Active concentration range 3.Gain 4.Local vs. distant active range rate rate/i i i x x y

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THE MATHEMATICS OF REGULATION OF ENZYME ACTIVITY Local regulation

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

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

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

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A Principle for Local Regulation

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Dynamic Effects of Regulation (advanced)

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Inhibition

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The Steady State

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Parametric Sensitivity steady state concentration increases response is faster

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Dynamic Response x - Hill kinetics Mass action kinetics a

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Activation

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

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Key Quantities

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

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Eigenvalues and their location in the complex plane Im Re Transient response: 1 smooth landing 2 overshoot 3 damped oscillation 4 sustained oscillation 5 chaos

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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)

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ENZYMES AS MOLECULES Simulating regulation

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Regulation at a Distance primary pathway perturbation biosynthetic pathway x6x6 x1x1 x2x2 x5x5 x5x5 x5x5 x6x6 x7x7 regulator binding site

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The Dynamic Equations Time derivative Fluxes Kinetic expressions

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The Steady-State Equations

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Simulation Results x1x1 x5x5 b1b1 v0v0 v1v1 v5v5 10x t=0 t b1b1 Complicated to interpret the time responses: what is going on?

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Phase Portrait and Pool Interpretation x1x1 x5x5 b1b1 v0v0 v1v1 v5v5 10x t=0 t b1b1 flux balancing on biosynthetic pathway flux state of the enzyme concentration

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Regulation of Gene Expression x6x6 x7x7 x5x5 v7v7 (-) translation decay inhibition of translation

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Simulation Results total enzyme const slow response of protein translation fast metabolic inhibitory response x1x1 x5x5 b1b1 v0v0 v1v1 v5v5 10x t=0 t b1b1

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Phase Portrait and Pool Interpretation flux balancing on biosynthetic pathway state of the enzyme x1x1 x5x5 b1b1 v0v0 v1v1 v5v5 10x t=0 t b1b1 flux concentration

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dimer tetramer Allosteric Regulation of Enzyme Activity

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Simulation Results: monomer, dimer, tetramer tetramer dimer monomer x1x1 x5x5 b1b1 v0v0 v1v1 v5v5 10x t=0 t b1b1 disturbancerejection tetramer > dimer > monomer

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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

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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.

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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

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Key Regulatory Step in Glycolysis (Advanced)

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Regulatory Signals (Advanced) x + Effective schema: v 1 (x) v 2 (x)

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Kinetic Description (Advanced)

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Scaling the Equations (Advanced)

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Criteria for Existence of Multiple Steady States (Advanced)

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Computation of Multiple Steady States (Advanced)

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Additions Compute the fluxes across the multiple steady state region

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