lectures 9-13: kinetic approaches

what do time scales of biological processes tell us

how fast a molecule moves

why are kinetic methods used

to understand mechanisms and allows you to piece together bits to get the whole speed frame

rate law

the experimental rate of reaction is often found to be proportional to the concentration of reactants raised to some power e.g. rate of reaction = k [A]^α [B]^β

rate of reaction = k [A]^α [B]^β what does everything represent

• [A]/[B] is the molar concentration of the reactants

• k is the rate constant that tells us how fast a reaction is

• the exponents are the orders which tell you how much a reactant affects the speed

0 order

changing the concentration has no effect on rate

1st order

doubling the concentration, doubles the rate

2nd order

doubling the concentration, quadruples the rate

how do you find the overall order

the sum of all the orders

when do transitions take place

when (through randomly bouncing around) molecules gain enough energy to pass over an energy barrier

irreversible unimolecular reactions

when a single molecule transforms into a product and cannot return to its original state, following first-order reactions

differential rate law:

d[A]/dt = -k[A], given by concentration over time

when would you use d[A]/dt

for an analytical approach → exact numbers to describe [A], k, t (needs calculus) but only works in simple cases

when would you use -kA

for approximates, can be used with complex cases

what is the integration of the analytical equation

[A]_t = A₀e^-kt gives an exponential function

ln([A]/[A]₀) = -kt

the slope of the straight line would be the rate

unimolecular reversible reactions

a process where a single molecule transforms into a product but the product has enough energy to transform back into the reactant

what is the equation for a unimolecular reversible reaction that is at equilibrium

K+1/K-1 = [B]_eq/[A]_eq = K_eq

For example: [A] = 100 µM, [B] = 0, and K = 10. Find [A]_eq and [B]_eq

• [A]_eq = proportion of [total] that is A at equilibrium

  = [total] 1 / (1 + K)  = 100 µM 1/(10 + 1) = 9.1 µM

• [B]_eq = proportion of [total] that is B at equilibrium

  = [total] K / (1 + K)  = 100 µM 10/(10 + 1) = 90.9 µM

what is K_obs

When one reactant is present in large excess compared to others, the reaction can be approximated as pseudo-first-order. In this case, k_obs is derived from the concentration of the limiting reactant while the excess reactant’s concentration is effectively constant

analytical approach for unimolecular reversible reactions involves the integration of the rate equation to give:

• [A] = [A]₀(Ae^-k_obst + B)

• K_obs = k+1 + k-1

bimolecular irreversible reactions

2 distinct molecules (or 2 same) collide and react to form a product. the reaction is heavily dependent on the probability of 2 things hitting each other in the right orientation and with enough force (activation energy)

what kind of order reaction is bimolecular irreversible

second-order, d[A]/dt = -k+1[A][B]

analytical approaches to solving biomolecular rate laws part 1: [A]=[B]

since they react in a 1:1 ratio, their concentrations will stay equal throughout the entire reaction, so if you replace [B] with [ A], the equation becomes d[A]/dt = -k+1[A]²

what is the problem with [A]=[B] in analytical approaches to solve bimolecular rate laws

it is difficult to physically measure out exact equal moles of 2 different substances down to the last molecule

analytical approaches to solving bimolecular rate laws part 2: [A]«[B]

if you have 1,000,000 molecules of B but only 10 of A, even after all of A has reacted, you would still have 999,990 of B, so we say [B] is constant → giving a first-order reaction (integrating gives K_obs)

how would pseudo-first order kinetics work in real life

• keep one reagent at a low concentration

• vary the concentration of the other reagent (always above a 10-fold excess)

• record and plot observed rate constants against concentration

bimolecular reversible reactions

where 2 molecules collide to form a product but the product can also break apart/ react to reform the original molecules (realistic binding)

what qualities can we make a quantitative assessment of in bimolecular reversible reactions?

• interaction strength

• on mutation

• structural variation

• predict binding under a given set of experiments

bimolecular reversible equation

d[A]/dt = -k₊₁[A][B] + k_-1[C]

kd

[A][B]/[C] = k-1/k+1

lower the Kd, the…

higher the binding affinity

whats the equation to find the fraction of protein bound to ligand

[PL]/[P]_T =[L]/[L] + K_d

simple enzymes

the enzyme-substrate complex can either revert back into the substrate and enzyme or can go forward in the reaction

the pre-equilibrium assumption

k+1 » k + 2, this means that the complex falls apart back into the reactants much faster than it creates product

because of this we can assume the first step in the reaction is at equilibrium, this allows us to determine the Michaelis constant (Km)

Km =

[E][S]/[ES]= k-1/k+1

Kcat

K+2

rate of simple enzymes

(Kcat[E]_T[S])/[S]+K_m

the Briggs-Haldane approach

if k-1 is not » k+2 → Kcat/Km = k+1 (k+2/k-1+k+2)