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: – ‘pooling’ variables as time constants relax
• Transitions: – the trajectories from one state to the next
• Graphical representation: – visualizing data
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
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
TransitionsTransition
homeostaticor
steady
Transient response:1 “smooth” landing2 overshoot3 damped oscillation4 sustained oscillation5 chaos
The subject ofnon-linear dynamics
1 2 3 4
Representing the Solution
fast slow
Glu
G6P
F6P
HP
Example:
THE DYNAMIC MASS BALANCES
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
Matrix Multiplication: refresher
( )( ) ( )+
=
s11•v1 + s12•v2 = dx1/dt
=
SOME KINETICS
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
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:
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
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
MULTI-SCALE DYNAMIC MODELS
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
Kinetic Description
ATP+ADP+AMP=Atot
2ATP+ADP=total
inventoryof ~P
Slow
Intermediate Fast
pooling:
Time Scale Hierarchy•Observation•Physiological process
Examples: secATP
binding
minenergy
metabolism
daysadenosine carrier:
blood storage in RBC
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
( )
IMPORTANT ASSUMPTIONS
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
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
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
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
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)