lecture #2
DESCRIPTION
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: - PowerPoint PPT PresentationTRANSCRIPT
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Lecture #2
Basics of Kinetic Analysis
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Outline
• Fundamental concepts• The dynamic mass balances• Some kinetics• Multi-scale dynamic models• Important assumptions
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FUNDAMENTAL CONCEPTS
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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
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Time Constants
• A measure of the time it takes to observe a significant change in a variable or process of interest
$
0 1 mo
save
balance
borrow
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Aggregate Variables:primer on “pooling”
GluHK
ATP ADP
G6P F6PPGI PFK
ATP ADP
1,6FDP
“slow” “fast” “slow”
HK
ATP
Glu HPPFK
ATP
Time scale separation (TSS)Temporal decomposition
Aggregate poolHP= G6P+F6P
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TransitionsTransition
homeostaticor
steady
Transient response:1 “smooth” landing2 overshoot3 damped oscillation4 sustained oscillation5 chaos
The subject ofnon-linear dynamics
1 2 3 4
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Representing the Solution
fast slow
Glu
G6P
F6P
HP
Example:
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THE DYNAMIC MASS BALANCES
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Units on Key Quantities
Dynamic Mass Balance dxdt = S•v(x;k)
Dimensionlessmol/mol
Mass (or moles)per volume
per time
Mass (or moles) per
volume
1 mol ATP/1 mol glucose
mM/secM/sec
mMM
Example:
1/time, or1/time • conc.
sec-1
sec-1 M-1
Need to know ODEs and Linear Algebra for this class
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Matrix Multiplication: refresher
( )( ) ( )+
=
s11•v1 + s12•v2 = dx1/dt
=
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SOME KINETICS
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Kinetics/rate laws =Sv(x;k)dxdt
Two fundamental types of reactions:
1) Linear
2) Bi-linear
xv
x+yv
Example: Hemoglobin
Actual
Lumped2+2 22
22
Special case
x+x
dimerization
+ 2
x,y ≥ 0, v ≥ 0
fluxes and concentrations are non-negative quantities
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Mass Action Kinetics
rate ofreaction( ) collision frequency
v=kxa a<1 if collision frequency is hampered by geometry
v=kxayb a>1, opposite case or b>1
Restricted Geometry (rarely used)
Collision frequency concentration
Linear: v=kx; Bi-linear: v = kxy
Continuum assumption:
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Kinetic Constants are BiologicalDesign Variables
•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 (orbitals) •Proteins stick to everything except themselves•Phosporylation influences protein-protein binding•Prostetic groups and cofactors confer chemical properties
reaction
no reaction
Angle of Collision
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Combining Elementary Reactions
Mass action ratio ()
G6P F6PPGI
Keq=[F6P]eq
[G6P]eq
=[F6P]ss
[G6P]ss
closed system open system
Keq
x1 x2
v+
v-vnet=v+-v-
vnet >0
vnet <0
vnet =0 equil
Reversible reactions
Equilibrium constant, Keq, is a physico-chemical quantityEquilibrium constant, Keq, is a physico-chemical quantity
Convert a reaction mechanism into a rate law:
S+E xv1
v-1
P+Eqssa
or qeav(s)=
VmsKm+s
v2
mechanism assumption rate law
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MULTI-SCALE DYNAMIC MODELS
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P AP+ +
Capacity: =2(ATP+ADP+AMP)
Occupancy: 2ATP+1ADP+0AMP
EC= ~ [0.85-0.90]occupancy
capacity
Example:
ATP=10, ADP=5, AMP=2
Occupancy =2•10+5=25Capacity=2(10+5+2)=34
2534
EC=
P baseP APP
High energy phosphate bond trafficking in cells
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Kinetic Description
ATP+ADP+AMP=Atot
2ATP+ADP=total
inventoryof ~P
Slow
Intermediate Fast
pooling:
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Time Scale Hierarchy•Observation•Physiological process
Examples: secATP
binding
minenergy
metabolism
daysadenosine carrier:
blood storage in RBC
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Untangling dynamic response:modal analysis m=-1x’, pooling matrix p=Px’
log(x’(t))
Total Response Decoupled Response
time
mi
mi0log
m3; “slow”
m2; “intermediate”
m1; “fast”
Example:
x’: deviation variable
( )
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IMPORTANT ASSUMPTIONS
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The Constant Volume Assumption
M = V • xmol/cell vol/cell mol/vol
volu
me
conc
entra
tion
Total mass balancemol/cell/time
f = formation, d = degradation
=0 if V(t)=const
mol/vol/time
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Osmotic balance:in=out; in=RTiXi
Electro-neutrality: iZiXZi=0
Fundamental physical constraints
Gluc
2lac
ATPADP
3K+ 2Na+
Hb-
Albumin-
membranes:typically permeable to anions
not permeable to cations
red blood cell
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Two Historical Examples of Bad Assumptions
1. Cell volume doubling during division
modeling theprocess of cell
divisionbut
volumeassumed tobe constant
2. Nuclear translocation
NFc
VN
AN
VC
dNFc
dt=…-(AN/Vc)vtranslocation
dNFn
dt=…+(AN/VN)vtranslocation
Missing (A/V) parameters make mass lost during translocation
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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
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Summary• i is a key quantity
• Spectrum of i time scale separation temporal decomposition
• Multi-scale analysis leads to aggregate variables• Elimination of a i 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 dtdx =Sv(x;k)