Lecture #4

1. Lecture #4 Chemical Reactions

2. Basic Properties of Chemical Reactions • Stoichiometry -- chemistry • Relative rates – thermodynamics; • Keq= f(P,T) • Absolute rates -- kinetics • Bi-linearity • X + Y <=> X:Y DNA sequence dependent Example: Hb assembly Example: TF binding a+bab a+bab a2b2 Pomposiello, P.J. and Demple, B. Tibtech, 19: 109-114, (2001).

3. Cases Studied • Reversible linear reaction • Reversible bi-linear reaction • Connected reversible linear reactions • Connected reversible bi-linear reactions

4. The Reversible Reaction: basic equations dynamic mass balances

5. k1=1 The Reversible Reaction: a simulated response

6. k1=1 The Reversible Reaction: a simulated response

7. The Reversible Reaction: forming aggregate variables dis-equilibrium variable conservation variable the fraction of the pool p2 that is in the form of x1

8. P maps x p The Reversible Reaction: basic equations in terms of pools concentrations, x pooled variables, p p(t) = Px(t)

9. k1=1 The Reversible Reaction: a simulated response

10. v1=k1x1x2 x1+x2 x3 v-1=k-1x3 The Bi-linear Reaction: basic equations ; bi-linear term =-v1,net conservation variables

11. The Bi-linear Reaction: forming aggregate variables 1) dis-equilibrium variable 2) conservation variables p3 schematic: C + A CA p2

12. k1=1/t/conc The Bi-linear Reaction: a simulated response disequilibrium variable x1(0) k-1=1/t conservation variables x2(0) x3(0) linearized disequilibrium variable

13. Connected Linear Reactions: basic equations fast slow fast simulate dynamic coupling

14. x10=1 k1=k2=k3=1 Connected Linear Reactions: a simulated response K1=K3=1 x20=x30=x40=0

15. If reactions are decoupled; k2=0 If reactions are dynamically coupled; k2≠0 Connected Linear Reactions: forming aggregate variables • p1&p3 are the same dis-equilibrium variables • p2 changed from a conservation variable to the coupling variable • p4 changed from an individual reaction conservation to a total • conservation variable for the system

16. not coupled P Connected Linear Reactions: simulated dynamic response coupled P K1=K3 k1=5k2=k3

17. if x2=x4 this is the reversible form of the MM mechanism The Bi-linear Reaction: basic equations coupling variable block diagonal terms simulate

18. Concentrations Pools Coupled Bi-linear Reactions: a simulated response IC x1 and x2 identical p1 x3 p2 p3, p4, p5 x4 and x5 identical IC @ equilibrium:

19. Conservation Pools Conservation of A Conservation of B Conservation of C The 3 conservation quantities

20. Summary • Dynamics of chemical reactions can be represented in terms of aggregate variables • Fast dis-equilibrium pools can be relaxed • Removing a dynamic variable reduces the dynamic dimension of the description • Linearizing bi-linear kinetics does not create much error

21. Summary • Each net reaction can be described by pooled variables that represent a dis-equilibrium quantity and a mass conservation quantity that is associated with the reaction. • If a reaction is fast compared to its network environment, its dis-equilibrium variable can be relaxed and then described by the conservation quantity associated with the reaction. • Removing a time scale from a model corresponds to reducing the dynamic dimension of the transient response by one. • As the number of reactions grow, the number of conservation quantities may change. • Irreversibility of reactions does not change the number of conservation quantities for a system. • Linearizing bi-linear rate laws does not create much error.