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

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Example: Hb assembly + +

Pomposiello, P.J. and Demple, B. Tibtech, 19: 109-114, (2001).

TF bindingExample:

Cases Studied

• Reversible linear reaction

• Reversible bi-linear reaction

• Connected reversible linear reactions

• Connected reversible bi-linear reactions

The Reversible Reaction: basic equations

dynamic massbalances

The Reversible Reaction: a simulated response

k1=1

The Reversible Reaction: a simulated response

k1=1

The Reversible Reaction: forming aggregate variables

conservation variable

dis-equilibrium variable

the fraction of the pool p2 that is inthe form of x1

The Reversible Reaction: basic equations in terms of pools

concentrations, x pooled variables, p

P maps x p

p(t) = Px(t)

The Reversible Reaction: a simulated response

k1=1

The Bi-linear Reaction: basic equations

=-v1,net

bi-linear term

;

conservation variables

x1+x2

v1=k1x1x2

v-1=k-1x3

x3

The Bi-linear Reaction: forming aggregate variables

2) conservation variables

1) dis-equilibrium variable

C + A CA

p3

p2

schematic:

The Bi-linear Reaction: a simulated response

x1(0)

x2(0)

x3(0)

disequilibrium variable

conservation variablesk-1=1/t

linearized disequilibriumvariable

k1=1/t/conc

dynamic coupling

Connected Linear Reactions: basic equations

fast slow fast

simulate

Connected Linear Reactions: a simulated response

k1=k2=k3=1

K1=K3=1

x10=1

x20=x30=x40=0

Connected Linear Reactions: forming aggregate variables

If reactions are decoupled; k2=0

If reactions are dynamically coupled; k2≠0

•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

Connected Linear Reactions: simulated dynamic response

not coupled P

coupled P

K1=K3

k1=5k2=k3

The Bi-linear Reaction: basic equations

simulateblock

diagonalterms

coupling variable

if x2=x4

this is the reversible form of the MM

mechanism

Coupled Bi-linear Reactions: a simulated response

Concentrations Pools

x1 and x2 identical

x3

x4 and x5 identical

p1 p2

p3, p4, p5

@ equilibrium:

IC

IC

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.