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Lecture #5 Enzyme Kinetics

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Outline The principles of enzyme catalysis Deriving rate laws for enzymes Michaelis-Menten kinetics Hill kinetics The symmetry model Scaling equations (Advanced)

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ENZYME CATALYSIS Some basic information

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Enzyme catalysis: basics http://ebooklibrary.thieme.com/SID2502958536850/ebooklibrary/flexibook/pubid1619260736/index.html

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Enzyme catalysis: basics

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EC Classification of enzymes EC # = enzyme commission # EC x.x.x.x

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Details for specific cases are available

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DERIVING RATE LAWS Mathematical description of catalytic activity

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Deriving Enzymatic Rate Laws from Postulated Reaction Mechanisms 1.Formulate mass balances on elementary reactions 2.Identify mass balances/time invariants 3.Reduce to the dynamically independent variables 4.Apply simplifying assumptions: The QSSA or the QEA 5.Use numerical integration to determine when the assumptions apply 6.Scale equations and form dimensionless numbers (optional; advanced analysis)

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MICHAELIS-MENTEN KINETICS

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Michaelis-Menten Reaction Mechanism substrate free enzyme intermediate complex product fastslow (dynamic degree of freedom) const the two time invariants

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Mass Action Kinetics: introduction of time-invariants to go from 4 variables to 2 dynamically independent variables

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The Quasi-steady State Assumption =v m KmKm choose independent variables Applying the QSSA --, ODEsAEs

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The Michaelis-Menten Rate Law vmvm vmvm 2 K m =s s (0 th order) (1 st order)

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Finding Sensitivity (Advanced) chain rule

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phase portrait fast response slow response error Michaelis-Menten Mechanism: dynamic simulation

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full and qss-solution are indistinguishable for the validity of the qssa: e 0 <<s 0 literature e 0 <<K m accurate Michaelis-Menten Mechanism: dynamic simulation

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Applicability of the QEA, QSSA When k 2 << k -1 then the QEA works When e t << K m then the QSSA works When K m << s t then the QSSA works S+E ES P+E k -1 k2k2 fast slow k 2 <<k -1 ( see Chem. Eng. Sci., 42, 447-458.)

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Key Dimensionless Groups (Advanced) a = k 2 /k -1 <<1QEA b = e t /K m <<1QSSA c = s t /K m >>1QSSA stickiness number

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Numerical simulation vs. Pooling for understanding

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Regulatory Enzymes

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HILL KINETICS Originally used to describe oxygen binding to hemoglobin

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Hill Kinetics 3. QEA on reaction (2) degree of cooperativity, rarely an integer due to lumping effect of reaction (2) Hb ~2.3-2.6, also called the Hill coefficient per site binding constant 2. Mass balance 4. Reaction rate 1. Reaction mechanism conservation quantity Inhibitor catalytically inactive form of E

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Applying Simplifying Assumptions mass balance:QEA inhibition Add e to the rate law: activation a: concentration of A

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Graphical Representation vmvm v m no sensitivity maximum sensitivity no sensitivity to effector molecule i or a inflection point activation inflection point inhibition precursor aa protein synth. example Activated form Normal form

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Sensitivity (advanced) The Hill rate law has a sigmoidal shape with sensitivity of the reaction rate to the end product concentration as: which has a maximum at the inflection point The values at the inflection point

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Dynamic Simulation of Hill Kinetics Phase portraits Dynamic responses fastslow distribution of enzyme states catalysis

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Key Dimensionless Groups (Advanced)

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THE SYMMETRY MODEL And now, chemically realistic mechanisms

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The Symmetry Model (R form) (T form)

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Deriving the Rate Law Mass balance Combine QEA

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Deriving the Rate Law (Cont) Similar equation for activators and substrates 4 4

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Dynamic Response of the Symmetry Model Phase planes Dynamic responses fastslow distribution of enzyme states catalysis

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Key Dimensionless Groups (Advanced) L:The allosteric constant t i =K i /v m time constant of inhibition b=e t /K i relative times of catalysis to inhibition

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Scaling Equations (Advanced)

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Summary Enzymes are highly specialized catalysts that accelerate reaction rates Reaction mechanisms are formulated for the chemical conversions carried out by enzymes in terms of elementary reactions. Rate laws for enzyme reaction mechanisms are derived based on simplifying assumptions. Two simplifying assumptions are commonly used: the quasi- steady state (QSSA) and the quasi-equilibrium assumptions (QEA). The validity of the simplifying assumptions can be determined using scaling of the equations followed by mathematical and numerical analysis. A number of rate laws have been developed for enzyme catalysis and for the regulation of enzymes. Only three reaction mechanisms were described in this chapter.

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