Lecture #4 Chemical Reactions

Basic Properties of Chemical Reactions Stoichiometry -- chemistry Relative rates – thermodynamics; K eq = f(P,T) Absolute rates -- kinetics Bi-linearity X + Y X:Y DNA sequence dependent Example: Hb assembly  +    Pomposiello, P.J. and Demple, B. Tibtech, 19: , (2001). TF binding Example:

Cases Studied Reversible linear reaction Reversible bi-linear reaction Connected reversible linear reactions Connected reversible bi-linear reactions

The Reversible Reaction: basic equations dynamic mass balances

The Reversible Reaction: a simulated response k 1 =1

The Reversible Reaction: a simulated response k 1 =1

The Reversible Reaction: forming aggregate variables conservation variable dis-equilibrium variable the fraction of the pool p 2 that is in the form of x 1

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

The Reversible Reaction: a simulated response k 1 =1

The Bi-linear Reaction: basic equations =-v 1,net bi-linear term ; conservation variables x1+x2x1+x2 v1=k1x1x2v1=k1x1x2 v -1 =k -1 x 3 x3x3

The Bi-linear Reaction: forming aggregate variables 2) conservation variables1) dis-equilibrium variable C + A CA p3p3 p2p2 schematic:

The Bi-linear Reaction: a simulated response x 1 (0) x 2 (0) x 3 (0) disequilibrium variable conservation variables k -1 =1/t linearized disequilibrium variable k 1 =1/t/conc

The Bi-linear Reaction: linearization constant linear higher-order terms truncated combine solve linearized equation slope of curve at x 0

The Bi-linear Reaction: basic equations in terms of pools x(t) p(t) P x 1 +x 2 x 3 bi-linear reaction linearized dis-equilibrium variable conservation quantities

dynamic coupling Connected Linear Reactions: basic equations fastslow fast simulate

Connected Linear Reactions: the origin of dynamic coupling one-way dynamic coupling block-diagonal terms ;

Connected Linear Reactions: a simulated response k 1 =k 2 =k 3 =1 K 1 =K 3 =1 x 10 =1 x 20 =x 30 =x 40 =0

Connected Linear Reactions: forming aggregate variables If reactions are decoupled; k 2 =0 If reactions are dynamically coupled; k 2 ≠0 p 1 &p 3 are the same dis-equilibrium variables p 2 changed from a conservation variable to the coupling variable p 4 changed from an individual reaction conservation to a total conservation variable for the system

Connected Linear Reactions: simulated dynamic response not coupled P coupled P K 1 =K 3 k 1 =5k 2 =k 3

PS=0 explain

Connected Linear Reactions: basic equations in terms of pools, conservation quantity, has no effect

The Bi-linear Reaction: basic equations simulate block diagonal terms coupling variable if x 2 =x 4 this is the reversible form of the MM mechanism

Coupled Bi-linear Reactions: a simulated response ConcentrationsPools x 1 and x 2 identical x3x3 x 4 and x 5 identical p1p1 p2p2 p 3, p 4, p equilibrium: IC

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

Coupled Bi-linear Reactions: forming aggregate variables x(t) p(t) P 3 conservation variables—not unique 2 coupled dis-equilibrium variables

Coupled Bi-linear Reactions: linearization ; conservation variables dis-equilibrium variables

Simulation Results

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

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.