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This lecture delves into the fundamental properties of chemical reactions, emphasizing stoichiometry, relative rates, and thermodynamics. It explores key concepts, including the equilibrium constant (Keq) and rates of reactions from both a molecular perspective (kinetics) and aggregate variables. The discussion includes examples of reversible linear and bi-linear reactions, illustrating the dynamic mass balance and conservation variables. Key insights into reaction coupling and the effects of dynamic variables on equilibrium and conservation will also be addressed to provide a clearer understanding of chemical reaction systems.
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Lecture #4 Chemical Reactions
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).
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
k1=1 The Reversible Reaction: a simulated response
k1=1 The Reversible Reaction: a simulated response
The Reversible Reaction: forming aggregate variables dis-equilibrium variable conservation variable the fraction of the pool p2 that is in the form of x1
P maps x p The Reversible Reaction: basic equations in terms of pools concentrations, x pooled variables, p p(t) = Px(t)
k1=1 The Reversible Reaction: a simulated response
v1=k1x1x2 x1+x2 x3 v-1=k-1x3 The Bi-linear Reaction: basic equations ; bi-linear term =-v1,net conservation variables
The Bi-linear Reaction: forming aggregate variables 1) dis-equilibrium variable 2) conservation variables p3 schematic: C + A CA p2
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
Connected Linear Reactions: basic equations fast slow fast simulate dynamic coupling
x10=1 k1=k2=k3=1 Connected Linear Reactions: a simulated response K1=K3=1 x20=x30=x40=0
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
not coupled P Connected Linear Reactions: simulated dynamic response coupled P K1=K3 k1=5k2=k3
if x2=x4 this is the reversible form of the MM mechanism The Bi-linear Reaction: basic equations coupling variable block diagonal terms simulate
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:
Conservation Pools Conservation of A Conservation of B Conservation of C The 3 conservation quantities
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.
