Transcript Slide 1

Lecture #2
Basics of Kinetic Analysis
Outline
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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
save
balance
borrow
$
0
1 mo
Aggregate Variables:
primer on “pooling”
Glu
“slow”
“fast”
“slow”
HK
PGI
PFK
G6P
F6P
ATP ADP
ATP ADP
Time scale separation (TSS)
Temporal decomposition
Aggregate pool
HP= G6P+F6P
PFK
HK
Glu
HP
ATP
1,6FDP
ATP
Transitions
Transition
homeostatic
or
steady
Transient response:
1 “smooth” landing
2 overshoot
3 damped oscillation
4 sustained oscillation
5 chaos
1 2 3 4
The subject of
non-linear dynamics
Representing the Solution
Example:
Glu
G6P
HP
F6P
fast
slow
THE DYNAMIC MASS BALANCES
Units on Key Quantities
Dynamic Mass Balance
Dimensionless
mol/mol
Example:
1 mol ATP/
1 mol glucose
dx
= S•v(x;k)
dt
Mass (or moles)
per volume
per time
mM/sec
mM/sec
Mass (or
moles) per
volume
1/time, or
1/time • conc.
mM
mM
sec-1
sec-1 mM-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
dx =Sv(x;k)
dt
Two fundamental types of reactions:
x
1) Linear
2) Bi-linear
v
fluxes and concentrations
are non-negative quantities
x,y ≥ 0, v ≥ 0
v
x+y
Example: Hemoglobin
Special case
Actual
a+b
a+b
x+x
Lumped
2a+2b
ab
ab
a2b2
a2b2
dimerization
a+a
a2
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
v=kxayb
a<1
a>1,
or b>1
if collision frequency is
hampered by geometry
opposite case
Kinetic Constants are Biological
Design Variables
Angle of Collision
reaction
no 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
Combining Elementary Reactions
Reversible reactions
vnet >0
v+
x1
x2
v-
vnet=v+-v-
vnet <0
equil
vnet =0
Equilibrium constant, Keq, is a physico-chemical quantity
Convert a reaction mechanism into a rate law:
v1
S+E
x
v2
P+E
qssa
or qea
v-1
v(s)=
Vms
Km+s
assumption
mechanism
rate law
Mass action ratio (G)
closed system
PGI
G6P
F6P
Keq=
[F6P]eq
[G6P]eq
open system
G=
[F6P]ss
[G6P]ss
G
Keq
MULTI-SCALE DYNAMIC MODELS
High energy
phosphate bond
trafficking in cells
P
P
P
A
+
P
P
A
P
+
Capacity:
=2(ATP+ADP+AMP)
Occupancy:
2ATP+1ADP+0AMP
EC=
occupancy
capacity ~ [0.85-0.90]
Example:
ATP=10, ADP=5, AMP=2
Occupancy =2•10+5=25
Capacity =2(10+5+2)=34
EC=
25
34
base
Kinetic Description
2ATP+ADP=
total
inventory
of ~P
ATP+ADP+AMP=Atot
pooling:
Slow
Intermediate
Fast
Time Scale Hierarchy
•Observation
•Physiological process
Examples:
sec
ATP
binding
min
energy
metabolism
days
adenosine carrier:
blood storage in RBC
Untangling dynamic response:
modal analysis m=M-1x’, pooling matrix p=Px ’
Total Response
Decoupled Response
m3; “slow”
x’: deviation variable
mi
log ( m
i0
)
log(x’(t))
m2; “intermediate”
m1; “fast”
time
Example:
IMPORTANT ASSUMPTIONS
The Constant Volume Assumption
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
Fundamental physical constraints
Osmotic balance: Pin=Pout; Pin=RTfPiXi
Si ZiXZi=0
Electro-neutrality:
Albuminred blood cell
Gluc
HbATP
ADP
2lac
3K+
2Na+
membranes:
typically permeable to anions
not permeable to cations
Two Historical Examples of Bad
Assumptions
1. Cell volume doubling
during division
modeling the
process of cell
division
but
volume
assumed to
be constant
2. Nuclear translocation
NFkBc
AN
VN
dNFkBc
dt
dNFkBn
dt
VC
=…-(AN/Vc)vtranslocation
=…+(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
• 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