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 $ 01 mo save balance borrow

Aggregate Variables: primer on “pooling” Glu HK ATPADP G6PF6P PGIPFK ATPADP 1,6FDP “slow”“fast”“slow” HK ATP Glu HP PFK ATP Time scale separation (TSS) Temporal decomposition Aggregate pool HP= G6P+F6P

Transitions Transition homeostatic or steady Transient response: 1 “smooth” landing 2 overshoot 3 damped oscillation 4 sustained oscillation 5 chaos The subject of non-linear dynamics

Representing the Solution fastslow Glu G6P F6P HP Example:

THE DYNAMIC MASS BALANCES

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

Chemical Reactions vs. Fluxes Through Them The columns of the stoichiometric matrix represent the reactions (n in number) The actual reaction rates, or the fluxes that take place through these reaction are denoted by v i The assignment of a flux through a reaction can be performed by a simple matrix multiplication S 11S 1n S m1S mn v 1 v 2 v ni b j b n S v

Matrix Multiplication: refresher ()()() + = s 11 v 1 + s 12 v 2 = dx 1 /dt =

SOME KINETICS

Kinetics/rate laws =Sv(x;k) dx dt Two fundamental types of reactions: 1)Linear 2)Bi-linear x v x+y v Example: Hemoglobin Actual  Lumped 2  +2   2  2 2222 Special case x+x dimerization  +   2 x,y ≥ 0, v ≥ 0 fluxes and concentrations are non-negative quantities

Mass Action Kinetics rate of reaction ( )  collision frequency v=kx a a<1if collision frequency is hampered by geometry v=kx a y b 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 Biological Design 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 (  ) G6PF6P PGI K eq = [F6P] eq [G6P] eq == [F6P] ss [G6P] ss closed system open system  K eq x1x1 x2x2 v+v+ v-v- v net =v + -v - v net >0 v net <0 v net =0 equil Reversible reactions Equilibrium constant, K eq, is a physico-chemical quantity Convert a reaction mechanism into a rate law: S+Ex v1v1 v -1 P+E qssa or qea v(s)= VmsVms K m +s v2v2 mechanism assumption rate law

MULTI-SCALE DYNAMIC MODELS

PA P + + Capacity: =2(ATP+ADP+AMP) Occupancy:2ATP+1ADP+0AMP EC= ~ [ ] occupancy capacity Example: ATP=10, ADP=5, AMP=2 Occupancy=210+5=25 Capacity=2(10+5+2)= EC= Pbase PA PP High energy phosphate bond trafficking in cells

Kinetic Description ATP+ADP+AMP=A tot 2ATP+ADP= total inventory of ~P Slow Intermediate Fast pooling:

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=  -1 x ’, pooling matrix p=Px ’ log(x’(t)) Total ResponseDecoupled Response time m i m i0 log m 3 ; “slow” m 2 ; “intermediate” m 1 ; “fast” Example: x’: deviation variable ( )

Dynamic Simplification ( ( Reduction in Dimensionality ( ( = Column A xy y=Ax Row Null left Null In general: J x’ RowCol x’ full rank J; r=m x’ 1 1 m-1 eliminate a time scale rank (J)=m-1 p Null l Left Null Jp=0 i.e., one qssa or qea l J=0 l x’=0 conservation=pool Jacobian x’ =Jx’

STOICHIOMETRY VS. DYNAMICS The fundamental ‘structure vs. function’ theme from molecular biology

Dominant Effects of Stoichiometry on Network Dynamics Steady states on ATP

Dynamic Balance at Steady State Key Concepts: 1.Stability of steady state 2.Capacity limitations 3.Robustness Plateau Extinction point ATP SS v load v generation =v stst load can be met load too high: collapse typical small range

IMPORTANT ASSUMPTIONS

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

Osmotic balance:  in =  out ;  in =RT  i X i Electro-neutrality:  i  Z i X Z i =0 Fundamental physical constraints Gluc 2lac ATP ADP 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 the process of cell division but volume assumed to be constant 2. Nuclear translocation NF  c VNVN ANAN VCVC dNF  c dt =…-(A N /V c )v translocation dNF  n dt =…+(A N /V N )v translocation 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:

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 ~ -G T Understand the assumptions that lead to dt dx =Sv(x;k)