Lecture 5: Chemical Reactions Outline: basic concepts Nonlinearities: saturation: Michaelis-Menten kinetics switching: Goldbeter-Koshland

Basics Simple reaction

Basics Simple reaction Number of AB pairs in volume v :

Basics Simple reaction Number of AB pairs in volume v : Reaction equation:

Basics Simple reaction Number of AB pairs in volume v : Reaction equation:

Basics Simple reaction Number of AB pairs in volume v : Reaction equation: Note: r has units 1/t, k has units volume/t

Reversible reactions, stoichiometry Reaction can go both ways:

Reversible reactions, stoichiometry Reaction can go both ways:

Reversible reactions, stoichiometry Reaction can go both ways: Equilibrium:

Reversible reactions, stoichiometry Reaction can go both ways: Equilibrium: Stoichiometry:

Reversible reactions, stoichiometry Reaction can go both ways: Equilibrium: Stoichiometry:

Reversible reactions, stoichiometry Reaction can go both ways: Equilibrium: Stoichiometry:

Michaelis-Menten Enzyme + substrate complex -> enzyme + product reversible  rates a,d

Michaelis-Menten Enzyme + substrate complex -> enzyme + product

Michaelis-Menten Enzyme + substrate complex -> enzyme + product

Michaelis-Menten Enzyme + substrate complex -> enzyme + product reversible  rates a,d

Michaelis-Menten Enzyme + substrate complex -> enzyme + product reversible   irreversible rates a,d rate k

Michaelis-Menten Enzyme + substrate complex -> enzyme + product reversible   irreversible rates a,d rate k Rate equations:

Michaelis-Menten Enzyme + substrate complex -> enzyme + product reversible   irreversible rates a,d rate k Rate equations:

Michaelis-Menten Enzyme + substrate complex -> enzyme + product reversible   irreversible rates a,d rate k Rate equations:

Michaelis-Menten Enzyme + substrate complex -> enzyme + product reversible   irreversible rates a,d rate k Rate equations:

Reduction: Eliminate E :

Reduction: Eliminate E : To solve:

Reduction: Eliminate E : To solve: Initial conditions:

Initial regime: Lots of S, [S] hardly changes from S 0

Initial regime: Lots of S, [S] hardly changes from S 0

Initial regime: Lots of S, [S] hardly changes from S 0

Initial regime: Lots of S, [S] hardly changes from S 0

Initial regime: Lots of S, [S] hardly changes from S 0

Initial regime: Lots of S, [S] hardly changes from S 0

Initial regime: Lots of S, [S] hardly changes from S 0 Michaelis constant

Slow dynamics If [S] changes slowly compared with τ fast,

Slow dynamics If [S] changes slowly compared with τ fast,

Slow dynamics If [S] changes slowly compared with τ fast,

Slow dynamics If [S] changes slowly compared with τ fast,

Slow dynamics If [S] changes slowly compared with τ fast,

Slow dynamics If [S] changes slowly compared with τ fast,

Slow dynamics If [S] changes slowly compared with τ fast, consistent

Result: Reaction rate

Result: Reaction rate saturation

Cooperative binding

Equations ->

Cooperative binding After fast transient: Equations ->

Cooperative binding Hill coefficient n After fast transient: Equations ->

Goldbeter-Koshland switching 2 MM reactions, 1 in each direction:

Goldbeter-Koshland switching 2 MM reactions, 1 in each direction:

Goldbeter-Koshland switching 2 MM reactions, 1 in each direction:

Goldbeter-Koshland switching 2 MM reactions, 1 in each direction:

Goldbeter-Koshland switching 2 MM reactions, 1 in each direction:

Goldbeter-Koshland switching 2 MM reactions, 1 in each direction:

Goldbeter-Koshland switching 2 MM reactions, 1 in each direction:

Goldbeter-Koshland switching 2 MM reactions, 1 in each direction: Steady state (add 1 st 2 or 2 nd 2 eqns):

Steady state:

Quadratic equation for S 1

Solution: where

Solution: where S 1, 1-S 1

Solution: where S 1, 1-S 1 Sharp switching for small κ