Discussion topic for week 5 : Enzyme reactions

• The lock in key hypothesis (Emil Fischer) asserts that both the

enzyme and the substrate possess specific complementary

geometric shapes that fit exactly into one another.

What are the problems associated with this hypothesis?

Enzymes and Molecular Machines (Nelson, chap. 10)

Enzymes are biological catalysts that enhance the rate of chem. reactions.

Machines use free energy from an external source (e.g. ATP,

concentration or potential difference) to do useful work. Examples:

• Motors: transduce free energy into linear or rotary motion

– myosin on actin in muscles, kinesin on microtubules in cells.

• Pumps: create concentration differences across membranes

– sodium-potassium pump transports 3 Na+ ions out of the cell and

2 K+ ions into the cell in one cycle.

• Synthases: drive chemical reactions to synthesize biomolecules

– ATP synthase synthesizes the ATP molecules that are used by

most of the molecular machines in the cells.

Enzymes

An extreme example: catalese

Consider the decomposition of hydrogen peroxide: H2O2 H2O + ½ O2

DG0 = -41 kT so the reaction is highly favoured but due to a high

activation barrier it proceeds very slowly:

for 1 M solution the rate is 10-8 M/s (reaction velocity)

Adding 1 mM catalese into the solution increases the rate by 1012 !

10-3 NA catalese molecules perform 104 NA hydrolisis reactions per sec.

So 1 catalese molecule catalyses 107 reactions per sec. (rate: 10-7 s)

H2O2 is produced in cells while eliminating free radicals. Because it is

toxic, its rapid breakdown is important.

More typical rates for enzymes are around 103 s-1

Simple model of enzyme reactions:

Chemical reactions involving biomolecules are extremely complex.

Free energy surface typically involves thousands of coordinates.

Nevertheless a reaction usually proceeds along the path of least

resistance (called reaction coordinate) which allows a simple description.

A simple reaction: H + H2 H2 + H

transition

state

An enzyme facilitates a chemical reaction by binding to the transition

state and thereby reducing the activation energy, DG‡ (but not DG)

Free energy surface along the reaction coordinate

kTEkTEkSkTSTEkTG eeeee D-D--D-D- )(rate

Substrate Enzyme + substrate

DG‡

DG‡

DG DG

Direction of the reaction is controlled by DG. By changing DG, we can

reverse the direction. The reverse reaction does not necessarily follow

the same reaction coordinate.

reaction reversefumarate malate-Lfumarase

A schematic picture of an enzyme E binding to a substrate S:

E + S ES EP E + P

E+S: The enzyme has a binding site that is a good match for the subst. S

ES: In order to bind, S must deform which stretches a bond to breaking pt.

EP: Thermal fluctuations break the bond producing an EP complex

E+P: The P state is not a good match to the binding site, hence it unbinds,

leaving the enzyme free for binding of the next substrate.

Corresponding

free energy

surface

Enzyme Kinetics:

Consider an enzyme reaction with rate constants k1, k2 and k3

Assume:

For a single enzyme, the reaction simplifies to

Let probability of E unoccupied be PE and occupied PES = (1- PE)

The rate of change of PE is

},{,,},,{ 2133322 kkkkkkkccc PES --

ESESE PkkPck

dt

dP211 - -

PEEPESSEk

k

k

k

k

k

--- 3

3

2

2

1

1

PEESSEk

k

ck S

-

2

1

1

Assuming quasi-steady state, the time derivative vanishes, yielding

Rate of production of P per enzyme:

Reaction velocity for a concentration cE of enzymes

1

212max

max

121

122

,k

kkKkcv

cK

cvv

ckkk

ckkcPkcv

ME

SM

S

S

SEESE

-

-

S

SES

ESEsS

ckkk

ckP

PkkPck

121

1

211 0)1(

--

-

-

ESPk2

Michaelis-Menten (MM) rule

Experimental data for pancreatic carboxypeptidase

S

M

SM

S

c

K

vvcK

cvv 1

11

maxmax

vmax=0.085 mM/s

KM=6.4 mM

MM rule displays saturation kinetics, which has very general validity

The key idea is the processing time for S P

At low substrate concentrations, there are more enzymes than S so that

there is no waiting and hence v is proportional to cS

As cS is increased beyond KM, there is competition among S for access

to an enzyme, and they have to queue for processing.

Maximum velocity of the reaction is determined by the number of

enzymes available and the processing rate (the rate limiting step)

Modulation of enzyme activity:

• Regulate the rate of enzyme production

• Competitive inhibition: direct binding of another molecule

• Noncompetitive inhibition: binding of a molecule to a second site

kTGekD-2

Recent developments (Adenylate kinase)

We know very little about the

actual dynamical processes

occurring in enzymes.

There are only a few simple

cases where the physical

mechanism is understood, e.g.

oxygen binding in myoglobin.

Adenylate kinase catalyzes:

ADP + ADP ATP + AMP

Recent work indicates that the

rate limiting step is the

enzyme conformation, and not

the chemistry.

Molecular motors in muscles: myosin and actin

For structure of the myosin and actin filaments in a myofibril, see

http://distance.stcc.edu/AandP/AP/AP1pages/Units5to9/Unit7/myofibri.htm

Experiment with optical tweezers demonstrates how myosin pulls an actin

filament when 1 mM of ATP is added to the system.

From Finer et al. “Single myosin molecule mechanics” Nature, 1994.

Translocation of proteins across membrane:

Proteins produced in the cell are exported outside through proteins in the

membrane that form pores. To pass through the pore, the protein has to

unfold. The reverse motion is suppressed because the chemical

asymmetries between inside and outside of the cell leads to a more

stable protein structure outside.

Factors contributing to

asymmetry:

• pH

• ion concentration

• disulfide bonding

• binding of sugars

protein catalyzes translocation

Macroscopic machines are deterministic, there are no random fluctuations

But molecular machines operate in a noisy environment with lots of

random fluctuations.

Consider the ratchets below as possible models for molecular machines.

In G-ratchet the spring retracts during the passage but pops back after

In S-ratchet a latch releases the spring after the passage, which stays up

Can either ratchet pull a load f towards right doing useful work?

Unloaded G-ratchet makes no net motion, the loaded one moves to the left

S-ratchet moves to the right if e f.L, and to the left if e < f.L

(no net motion if e f.L)

Simple model for a perfect Brownian ratchet: (e kT)

In the absence of any forces, the ratchet diffuses freely until it travels a

distance L. From

Thus the average speed is:

Next we introduce a load f that pulls the ratchet to the left.

The potential energy increases as

in the interval [0, L]

From Boltzmann distribution, the

equilibrium probability will be like

DLtDtx step 22 22

LDtLv step 2

fxU

0)( - kTfxexP

We need an equation to describe the nonequilibrium probability distribution

of the ratchet’s position (cf. Fick’s law and Nerst-Planck Eq.)

| | | | | x Dx L

a-Dx/2 a a+Dx/2 a+Dx

The net flux from a a+Dx depends on (1) the probabilities at those points

and (2) the external forces. If there are N ratchets in our ensemble,

the bins at a and a+Dx have ratchets.

Assuming they move randomly, the net migration from left to right is

xxaNPxaNP DDD )()( and

axax

RL

dx

xdPtND

dx

xdPxN

xxaPaPNN

D-D-

DD-D

)()(

2

1

)()(2

1

2

)1(

Next consider the flux due to an external force,

Drift velocity due to this force:

The number of ratchets moving from left to right:

Adding the two contributions and dividing by Dt, we obtain for the flux

Steady state: flux is constant, and from continuity eq. it is also uniform

)( kTDdx

dU

kT

D

kT

Dffvd -

axdRL

dx

dUt

kT

NDPtvaNPN

D-DD )()2(

dx

dUf -

-

dx

dU

kT

P

dx

dPNDj

00

dx

dU

kT

P

dx

dP

dx

d

dx

dj(Smoluchowski eq.)

Since the potential is periodic, the solutions must be

periodic too.

First consider the equilibrium case:

A possible nonequilibrium solution for the perfect ratchet is

1. vanishes at x = L

2. yields a constant flux

3. hence solves the Smoluchowski eq.

kT

NDCfe

kT

fe

kT

fNDC kTfLxkTfLx

--- ---- 1)()(

1)( )( - -- kTfLxeCxP

)()( xULxU

)( fxU

kTUCexPdx

dU

kT

P

dx

dPj - )(00 (Boltzmann dist.)

Average speed of the perfect ratchet:

The average number of ratchets in the interval [0, L]

The time it takes for these ratchets move is

and the speed is

12

1

1

1

-

-

--

--

D

kT

fLe

kT

fL

L

D

kT

fLe

NCkT

f

kT

NCDfL

N

Lj

t

Lv

kTfL

kTfL

jNt D

--

--

-

-

--

kT

fLe

f

kTNCxe

f

kTNC

dxeNCdxxNPN

kTfLL

kTfxL

LkTfLx

L

1

1)(

0

)(

0

)(

0

Too complicated to make sense, so consider the limits:

Plot of the ratchet

speed / (2D/L)

as a function of

z = fL/kT

Activation barrier kicks

in around fL = 5 kT

kTfLekT

fL

L

DvkTfL

L

Dz

zzz

L

Dv

kT

fLz

-

-

--<<

2

122

.2

21

211.1

e

→ activation barrier

Estimate the speed for a typical molecular machine

For small molecules, ions etc.: R 1-3 Å, D 10-9 m2/s

For macromolecules, proteins: R 1-3 nm, D 10-10 m2/s

Typical length scale: L = 1 nm

Average speed: v = 2D/L = 0.2 m/s, (e.g. to move 200 steps takes 1 ms)

The perfect ratchet assumption is that backward rate vanishes

When the forward and backward rates become equal and no

net motion is possible.

In summary:

1. Molecular machines move by random walk over free energy surface

2. Their speed is determined by the activation energy barrier (but not e)

)( fLe

fLe

R

kTkTD

6

Molecular Recognition

Cells contain thousands of different proteins.

Each protein performs a specific task that may require its interaction with

a specific biomolecule, e.g. DNA, another protein or a ligand.

How does a protein distinguish that biomolecule from the thousands of

others that are floating around the cell?

The lock and key hypothesis of Fischer (1894), namely, shape

complementarity of the interacting parts, provided the first clues.

Going beyond the descriptive accounts of protein interactions using

cartoons to a quantitative accounts that can make predictions has only

become possible in the last decade thanks to the advances in

• Structure determination of complexes and single molecule exp’s

• Computer power and simulation methods

Molecular recognition covers a vast area of research

• Enzyme function

• Protein-ligand interactions: binding of a ligand changes the

conformation of a protein enabling its function, e.g. ligand-gated ion

channels, oxygen binding to hemoglobin.

• Protein-protein interactions: e.g. formation of protein complexes

(tertiary structure), signal transduction across membrane, protein transport

and modification

• Protein-DNA (or RNA) interactions: reading and duplication of DNA,

protein manufacturing

• Protein interactions with non-native peptides: e.g. toxins from the

venomous animals (spiders, snakes, scorpions, snails)

• Protein interactions with chemical compounds: e.g. drugs

Experimental methods:

• Structure determination of complexes using x-ray diffraction or NMR

• Measurement of dissociation (or binding) constants.

(mM range: weak binding, μM range: intermediate, nM range: strong)

• High-throughput screening (automated testing of large number of

compounds to discover new drugs)

Theoretical methods:

• Docking methods (popular in “in silico” drug design)

• Monte Carlo methods: search for the free energy minimum using the

Metropolis algorithm

• Brownian dynamics simulations: water is treated as continuum and

protein is rigid, but simulations are fast enough to observe docking

• Molecular dynamics simulations: realistic representation but too slow

to observe docking

Crystal structure of the barnase (blue) - barstar (green) complex

The unbound conformations are superimposed in light blue and orange.

Close up view showing the side chain pairs in the hot spot.

In the complex: barnase (blue) - barstar (green)

Comparison of

the two structures

shows the

importance of side

chain flexibility

Docking methods

There are various docking methods that search for the free energy

minimum of a protein-macromolecule system. The basic ingredients are:

• A phenomenological energy functional. Typically consists of:

electrostatic, Lennard-Jones, hydrogen bond, solvation and entropic

terms. It is parametrized using a training set.

• A search algorithm. Two common methods employed:

1. Random search using the Monte Carlo method

2. Systematic search using a grid over the active site

In the current docking methods, ligand flexibility (mainly torsion angles) is

also taken into account (target protein is still rigid). Here genetic

algorithms provide a very efficient tool (different conformations correspond

to mutations). AutoDock is the most popular method at present.

Computer simulation of protein interactions

Protein association can be broadly divided into two stages:

1. Diffusional motion until they form an encounter complex

2. Non-diffusional rearrangement process leading to the final bound

complex.

The first stage could take quite a long time (ms), so it is neither possible

nor desirable to use molecular dynamics. Brownian dynamics (BD) is the

natural tool for this stage.

The second stage involves conformational changes in the protein, and

also dehydration and rehydration of water molecules. Thus a microscopic

description that treats all the atoms in the system is necessary at this

stage, which is provided by molecular dynamics (MD).

The focus is, however, on the binding. Can we avoid the BD stage?

Molecular dynamics combined with docking

Test study in

gramicidin channel:

1. Find the initial

gramicidin channel-

organic cation

configuration

from AutoDock

2. Then employ

this in MD

simulations

Organic cations that bind to the gramicidin channel

Methylammonium

Ethylammonium

Formamidinum

Guanidinum

TMA

TEA

Calculation of free energy profiles for ions

• Potential of Mean Force (PMF) of a molecule is calculated using the

channel axis (z) as the reaction coordinate

• The PMF is obtained from the Boltzmann factor by measuring the z

coordinates of the molecule

• Umbrella sampling

A harmonic potential is used to constrain the molecule at various points

on the channel axis (typical interval, fraction of an Å), and its z

coordinate is sampled during MD simulations

The z distributions are unbiased and combined to obtain the PMF

profile along the z axis.

-

)ρ(z

ρ(z)kTzWzW

00 ln)()(

Free energy profiles (potential of mean force, PMF) of cations determined

from umbrella sampling calculations

Binding constants

Binding constant is obtained by integrating the free energy of the ligand

in a volume around the binding site

where we have approximated the volume with a cylinder of radius R.

Using the PMF’s, we can estimate the binding constants:

Methylammonium: K = 4.1 M-1 (exp: 4.4 M-1)

Ethylammonium : K = 0.2 M-1 (exp: ~ 0)

Formamidinium: K = 0.6 M-1 (exp: 23 M-1) (there is a deeper site)

-

--

0

0

0

/)(2

3/])([

zkTzW

V

kTWW

dzeR

rdeK

r

Drugs from toxins

Development of new drugs is at an all time low.

Major problem: finding new compounds with high specificity and affinity.

High hopes from “in silico drug design” methods.

Example: Conotoxins as drug leads

Conotoxins are small peptides found

in the venom of cone snails that

selectively bind to specific ion

channels with high affinity.

It is estimated that there are over

50,000 different conotoxins.

Already a few new drugs have been

developed from conotoxins.

The potential for development of

further drugs is enormous.

k-conotoxin bound to K+ channel

Exp. structure of the KcsA*- charybdotoxin complex (NMR)

Important pairs:

Y78 (ABCD) – K27

D80 (D) – R34

D64, D80 (C) - R25

D64 (B) - K11

K27 is the pore

inserting lysine –

a common thread in

scorpion and other

toxins.

K11 R34

Developing drugs from ShK toxin for autoimmune diseases

ShK toxin has three

disulfide bonds and

three other bonds:

D5 – K30

K18 – R24

T6 – F27

These bonds confer

ShK toxin an

extraordinary stability

not seen in other toxins

NMR structure of ShK toxin

ShK toxin binds to Kv1.3 channels with picomolar affinity, hence a good

candidate for treatment of autoimmune diseases.

Kv1.3-ShK complex (Docking + MD)

Monomers A and C Monomers B and D

Pair distances in the Kv1.3-ShK complex (in A)

Kv1.3 ShK HADDOCK MD aver. Exp.

D376–O1(C) R1–N1 5.0 4.5

S378–O(B) H19–N 3.2 3.0 **

Y400–O(ABD) K22–N1 2.9 2.7 **

G401–O(B) S20–OH 2.9 2.7 **

G401–O(A) Y23–OH 3.5 3.5 **

D402–O(A) R11–N2 3.2 3.5 *

H404-C(C) F27-C"1 9.7 3.6 *

V406–C1(B) M21–C" 9.4 4.7 *

D376–O1(C) R29–N1 12.2 10.2 *

** strong, * intermediate ints. (from alanine scanning Raucher, 1998)

R24 (**) and T13 and L25 (*) are not seen in the complex (allosteric)

Convergence of the PMF for the Kv1.3-ShK complex

PMF of ShK for Kv1.1, Kv1.2, and Kv1.3

Comparison of binding free energies of ShK to Kv1.x

Binding free energies are obtained from the PMF by integrating it

along the z-axis.

Complex DGwell DGb(PMF) DGb(exp)

Kv1.1–ShK 18.0 14.3 ± 1.1 14.7 ± 0.1

Kv1.2–ShK 13.8 10.1 ± 1.1 11.0 ± 0.1

Kv1.3–ShK 17.8 14.2 ± 1.2 14.9 ± 0.1

Excellent agreement with experiment for all three channels, which

provides an independent test for the accuracy of the complex

models.

Average pair distance as a function of window position

** **

**

**

* * *

** denotes strong coupling and * intermediate coupling