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: – ‘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)