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Lecture #2

Lecture #2. Basics of Kinetic Analysis. Outline. Fundamental concepts The dynamic mass balances Some kinetics Multi-scale dynamic models Important assumptions. FUNDAMENTAL CONCEPTS. Fundamental Concepts. Time constants: measures of characteristic time periods Aggregate variables:

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Lecture #2

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  1. Lecture #2 Basics of Kinetic Analysis

  2. Outline • Fundamental concepts • The dynamic mass balances • Some kinetics • Multi-scale dynamic models • Important assumptions

  3. FUNDAMENTAL CONCEPTS

  4. Fundamental Concepts • Time constants: • measures of characteristic time periods • Aggregate variables: • ‘pooling’ variables as time constants relax • Transitions: • the trajectories from one state to the next • Graphical representation: • visualizing data

  5. Time Constants • A measure of the time it takes to observe a significant change in a variable or process of interest save balance borrow $ 0 1 mo

  6. Aggregate Variables: primer on “pooling” “slow” “fast” “slow” HK PGI PFK Glu 1,6FDP G6P F6P ATP ADP ATP ADP Time scale separation (TSS) Temporal decomposition Aggregate pool HP= G6P+F6P PFK HK Glu HP ATP ATP

  7. Transitions Transition homeostatic or steady Transient response: “smooth” landing overshoot damped oscillation sustained oscillation chaos 1 2 3 4 The subject of non-linear dynamics

  8. Representing the Solution Example: Glu G6P F6P HP fast slow

  9. THE DYNAMIC MASS BALANCES

  10. Units on Key Quantities dx dt Dynamic Mass Balance = S•v(x;k) Mass (or moles) per volume per time Mass (or moles) per volume Dimensionless mol/mol 1/time, or 1/time • conc. 1 mol ATP/ 1 mol glucose mM/sec mM/sec mM mM sec-1 sec-1mM-1 Example: Need to know ODEs and Linear Algebra for this class

  11. Matrix Multiplication: refresher + = ( ) ( ) ( ) = s11•v1 + s12•v2 = dx1/dt

  12. SOME KINETICS

  13. dx dt Kinetics/rate laws =Sv(x;k) Two fundamental types of reactions: v fluxes and concentrations are non-negative quantities x • Linear • Bi-linear v x,y ≥ 0, v ≥ 0 x+y Special case x+x dimerization a+aa2 Example: Hemoglobin Actual a+b ab a+b ab Lumped 2a+2ba2b2 a2b2

  14. Mass Action Kinetics ( ) a rate of reaction collision frequency Continuum assumption: Collision frequency a concentration Linear: v=kx; Bi-linear: v = kxy Restricted Geometry (rarely used) v=kxa a<1 if collision frequency is hampered by geometry v=kxayb a>1, opposite case or b>1

  15. Kinetic Constants are Biological Design Variables Angle of Collision reaction • What determines the numerical value of a rate constant? • Right collision; enzymes are templates for the “right” orientation • k is a biologically determined variable. Genetic basis, evolutionary origin • Some notable protein properties: • Only cysteine is chemically reactive (di-sulfur, S-S, bonds), • Proteins work mostly through hydrogen bonds and their shape, • Aromatic acids and arginine active (p orbitals) • Proteins stick to everything except themselves • Phosporylation influences protein-protein binding • Prostetic groups and cofactors confer chemical properties no reaction

  16. Combining Elementary Reactions Reversible reactions vnet >0 vnet <0 vnet =0 v+ x1 x2 vnet=v+-v- v- equil Equilibrium constant, Keq, is a physico-chemical quantity Convert a reaction mechanism into a rate law: v1 v2 qssa Vms S+E x P+E v(s)= Km+s or qea v-1 assumption mechanism rate law Mass action ratio (G) closed system open system PGI [F6P]eq [F6P]ss G Keq Keq= G= G6P F6P [G6P]eq [G6P]ss

  17. MULTI-SCALE DYNAMIC MODELS

  18. High energy phosphate bond trafficking in cells occupancy capacity base A A P P P P P P + + Capacity: =2(ATP+ADP+AMP) Occupancy: 2ATP+1ADP+0AMP EC= 25 34 EC= ~ [0.85-0.90] Example: ATP=10, ADP=5, AMP=2 Occupancy =2•10+5=25 Capacity =2(10+5+2)=34

  19. total inventory of ~P 2ATP+ADP= Kinetic Description ATP+ADP+AMP=Atot pooling: Slow Fast Intermediate

  20. Time Scale Hierarchy • Observation • Physiological process sec ATP binding min energy metabolism days adenosine carrier: blood storage in RBC Examples:

  21. Untangling dynamic response: modal analysis m=M-1x’, pooling matrix p=Px’ Total Response Decoupled Response m3; “slow” x’: deviation variable mi mi0 ( ) log log(x’(t)) m2; “intermediate” m1; “fast” time Example:

  22. Important Assumptions

  23. The Constant Volume Assumption concentration volume M = V • x mol/cell vol/cell mol/vol Total mass balance mol/cell/time f = formation, d = degradation =0 if V(t)=const mol/vol/time

  24. Fundamental physical constraints Osmotic balance: Pin=Pout; Pin=RTfPiXi Electro-neutrality: SiZiXZi=0 Albumin- red blood cell Gluc Hb- ATP membranes: typically permeable to anions not permeable to cations ADP 2lac 3K+ 2Na+

  25. Two Historical Examples of Bad Assumptions • Cell volume doubling • during division 2. Nuclear translocation AN NFkBc modeling the process of cell division but volume assumed to be constant VN VC dNFkBc =…-(AN/Vc)vtranslocation dt dNFkBn =…+(AN/VN)vtranslocation dt Missing (A/V) parameters make mass lost during translocation

  26. Hypotheses/Theories can be right or wrong… Models have a third possibility; they can be irrelevant Manfred Eigen Also see: http://www.numberwatch.co.uk/computer_modelling.htm

  27. Summary • ti is a key quantity • Spectrum of ti time scale separation temporal decomposition • Multi-scale analysis leads to aggregate variables • Elimination of a ti reduction in dim from m  m-1 • one aggregate or pooled variable, • one simplifying assumption (qssa or qea) applied • Elementary reactions; v=kx, v=kxy, v≥0, x≥0, y≥0 • S can dominate J; J=SG S ~ -GT • Understand the assumptions that lead to dx =Sv(x;k) dt

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