lecture of graduiertenkolleg - helms
DESCRIPTION
Lecture of Graduiertenkolleg - Helms. V1:Reaction rate theory V2:Kramer‘s theory V3:Reaction rates for electronic transitions V4:Potential and free energy landscapes V5:Lattice optimization V6:Optimization methods in protein folding V7:Protein folding with molecular dynamics - PowerPoint PPT PresentationTRANSCRIPT
1. Lecture SS 2006
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Lecture of Graduiertenkolleg - Helms
V1: Reaction rate theory
V2: Kramer‘s theory
V3: Reaction rates for electronic transitions
V4: Potential and free energy landscapes
V5: Lattice optimization
V6: Optimization methods in protein folding
V7: Protein folding with molecular dynamics
V8: Manipulating potential energy landscapes with forces
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V1: Chemical Kinetics & Transition States
see chapter 19 in book of K. Dill
Aim: describe kinetics of processes on energy landscapes
(e.g. chemical reactions).
- detailed balance- mass action law- temperature effect, Arrhenius law- concept of transition state/activation barrier transition state theory- -value analysis- effect of catalysts
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Rate theory
Rate theory provides the relevant information on the long-time behavior of
systems with different metastable states important for understanding of many
different physical, chemical, biological, and technical processes.
Arrhenius (1889)
Wigner (1932), Eyring (1935) Transition State Theory (TST)
Pechukas (1976): proper definition of the transition state
Chandler (1978): the activation energy is a free energy
Kramers (1940): effect of friction on reaction rates
Pollak (1986): link of Kramers‘ expression to TST
Pollak, Talkner, Chaos 15, 026116 (2005)
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Reaction rates are proportional to concentrations
B
k
k
A
r
f
Lets consider a simple kinetic process, the interconversion between 2 states,
kf and kr : forward and reverse rate coefficients.
How do the amounts of A and B change with time t,
given the initial amounts at time t = 0 ?
tBktAkdt
tBd
tBktAkdt
tAd
rf
rf
The two equations are coupled.
One can solve them by matrix algebra …
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Excursion: coupled differential equations
yxy
yxx
2
4
y
x
y
x
12
14
Matrix diagonalisation can be used to solve coupled ordinary differential equations.
For example, let x(t) and y(t) be differentiable functions and x' and y' their time derivatives.
The differential equations are relatively difficult to solve:
By diagonalizing the square matrix, we get
but u' = ku for k = const is easy to solve.
The solution is u = Aekx where A = const.
translate the ODEs into matrix form
y
x
y
x1
21
11
20
03
21
11
www.algebra.com
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Excursion: coupled differential equations
y
x
y
x1
21
11
20
03
21
11
y
x
v
u1
21
11
y
x
v
u1
21
11
By diagonalizing the square matrix, we get
We then put It follows that
Thus
v
u
v
u
20
03
The solutions of this system are found easily: t
t
Dev
Ceu2
3
with some constants C and D.
y
x
v
u1
21
11
y
x
v
u
21
11With
tt
tt
DeCey
DeCex23
23
2
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Reaction rates are proportional to concentrations
tAkdt
tAdf
tkA
tAdtk
A
dAf
t
f 0ln
0
tk feAtA 0
With this technique, we could solve our system of coupled diff. equations.
In the special case that kr << kf, the first equation simplies to
If [A(t)] + [B(t)] = constant, then
tk feAtB 0constant
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[A]eq and [B]eq : equilibrium concentrations.
To see that this is a condition of equilibrium follows from inserting
into
resulting in
At equilibrium, rates obey detailed balance
eqreqf BkAk
eqreqf BkAk
tBktAkdt
tBd
tBktAkdt
tAd
rf
rf
The principle of detailed balance says that the forward and reverse transitions must
be identical for an elementary reaction at equilibrium:
Taken from Dill book
0,0
dt
tBd
dt
tAd
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At equilibrium, rates obey detailed balance
r
f
eq
eq
k
k
A
BK
The detailed balance condition relates the rate coefficients kf and kr to the
equilibrium constant K:
Taken from Dill book
For more complex systems, the principle of detailed balance gives more information beyond
the statement of equilibrium. For a system having more than one elementary reaction, the
forward and reverse rates must be equal for every elementary reaction.
For this system:
Let‘s consider a 3-state mechanism with
kIA 0, kBI 0, kAB 0.
AB
BA
eq
eq
BI
IB
eq
eq
IA
AI
eq
eq
k
k
B
A
k
k
I
B
k
k
A
I ,,
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These are two independent equations for 3
unknown concentrations
the system has an infinite number of
solutions.
In mechanism (b), all rates of the
Denominator in
are zero mechanism (b) is impossible.
At equilibrium, rates obey detailed balance
BAIB
IBAI
kBkI
kIkA
This results in the mechanism shown right.
The only conditions for equilibrium are:
BIABIA
IBBAAI
IBBAAI
BIABIA
BI
IB
eqAI
IA
BA
ABeq
BA
ABeqeqeq
IA
AIeq
AB
BA
eq
eq
BI
IB
eq
eq
IA
AI
eq
eq
kkk
kkk
kkk
kkk
k
k
Ak
k
k
kA
k
kABA
k
kI
k
k
B
A
k
k
I
B
k
k
A
I
1or,1
or)2(
)3(,)1(
,,
Taken from Dill book
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At equilibrium, rates obey detailed balance
The principle of detailed balance says that forward and backward reactions at
equilibrium cannot have different intermediate states.
That is, if the forward reaction is A I B,
the backward reaction cannot be B A.
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The mass action laws describe mechanisms in chemical kinetics
Suppose the following reaction leading from reactants A, B, and C to product P:
PcCbBaA In general, the initial reaction rate depends on
- the concentrations of the reactants
- the temperature and pressure
- and on the coefficients a, b, and c.
Kinetic law of mass action (CM Guldberg & P Waage, 1864):
„the reactants should depend on the stoichiometry in the
same way that equilibrium constants do“.
cbaf CBAk
dt
Pd
Although mass action is in agreement with many experiments,
there are exceptions.
These require a quantum mechanical description.
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Energy Barriers
Where do energy barriers come from?
Why do reactions have activation barriers?
Different processes are characterized by similar energy barriers that are due to
very different mechanisms.
Effects on energy barriers- Chemical reactions - temperature- protein:ligand association - pH
- protein:protein association - D2O vs. H2O
- protein:membrane association - viscosity- protein:DNA association- during protein folding- time scales of protein dynamics- vesicle budding- virus assembly
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History of Energy Barriers of Chemical Reactions
1834 Faraday: chemical reactions are not instantaneous because there is an
electrical barrier to reaction
1889 Arrhenius: reactions follow Arrhenius law
with an activation barrier Ea.
Bodenstein: reactions occur via a series of elementary steps where bonds break
and form. Bodenstein showed that Arrhenius‘ law is applicable only to
elementary reactions. Overall reactions often show deviations.
1935 Polanyi & Evans: bonds need to stretch during elementary reactions.
The stretching causes a barrier. Bonds also break.
Physical causes of barriers to chemical reactions
bond stretching and distortion
orbital distortion due to Pauli repulsions (not more than 2 electrons may occupy one orbital)
quantum effects
special reactivity of excited states
kT
Ea
erateRate
0
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Energy Barriers of Chemical Reactions
When chemical bonds need to be „broken“,
- nuclei need to move only over small distances,
- electrons need to redistribute „settle down“ in different orbitals intermediate state has high energy
One can compute the energy barriers by electronic structure methods.
However, these calculations do not explain why the barriers arises.
Chemists like to think in concepts and rules and like to separate these.
How fast can a reaction proceed? Even within one bond vibration.
Such processes need to be activated.
E.g. bond length should stretch far beyond equilibrium distance.
This is possible by statistical fluctuations and by coupling with other modes
(large energy becomes concentrated in this mode).
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Energy Barriers of Protein:Ligand InteractionStep 1: Protein and Ligand are independently solvated (left picture below)
Step 2: The Ligand may preorganize into its binding conformation (costs usually 3 kcal/mol)
Step 3: The Ligand approaches the binding pocket of the protein.
System partly looses 6 degrees of freedom (CMS of ligand: 3 translation, 3 rotation)
Step 4: The Ligand enters the binding pocket of the protein Waters are displaced from binding pocket.
Sometimes: simultaneous conformational changes of protein/receptor
Receptor
Ligand
Bound and associated H2O
Displaced H2O
No collective modes!
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Steps involved in protein-protein association:
- random diffusion (1) + hydrodynamic interaction
- electrostatic steering (2)
- formation of encounter complex (3)
(Possible: large-scale conformational changes of
one or two proteins)
- dissociation or formation of final complex via TS (4)
Origin of Barrier (4): System partly looses 6 degrees of freedom (CMS: 3 translation, 3 rotation) Desolvation: large surface patches need to be partially cleared from water Induced fit of side chains at interface potential entropy loss
Energy Barriers of Protein:Protein Interaction
Effects of hydrodynamicInteractions:(left) effect of translation(right) effect of rotation
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Energy Barriers of Protein:Membrane Interaction
Membrane surface either carries a net negative charge (mixture of neutral and
anionic lipids) or has a partially negative character.
Cloud of positive counter ions accumulates near membrane to compensate
membrane charge.
Membrane surface is not well defined and quite dynamic, ondulations.
wikipedia.org
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Energy Barriers of Protein:Membrane Interaction
Neutron scattering: four layers of ordered water molecules
above membrane – also found by MD simulation.
Water layers significantly weaken membrane potential.
Lin, Baker, McCammon, Biophys J, 83, 1374 (2002)
Interaction potential of protein:membranesystems is largely unknown.
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Energy Barriers of Protein:DNA Interaction
DNA backbone carries strong permanent negative charge.
is surrounded by cloud of positive ions and coordinating water molecules
Protein must displace this cloud and must form very polar interactions with DNA
backbone.
Spatial distribution functions of water, polyamine atoms and Na+ ions around a CA/GT fragment. View of the minor groove. Data are for systems with 30 diaminopropane2+ (A), 30 putrescine2+ (B), 20 spermidine3+ (C) and 60 Na+ (D), averaging the MD trajectories over 6 ns, with three decamers with three repeated CA/GT fragments in each decamer. Water (oxygen, red; hydrogen, gray) is shown for a particle density >40 p/nm3 (except for the Na/15 system, where this value is 50 p/nm3); Spherical distribution function of the polyamine N+ atoms (blue) and Na+ (yellow) ions are drawn for a density >10 p/nm3; polyamine carbon and hydrogen atoms not shown.
Korolev et al. Nucl Acid Res 31, 5971 (2003)
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Energy Barriers of Protein:DNA Interaction
Recognition shows Faster-than-diffusion paradox (similar to Anfinsen paradox
for protein folding).
Maximal rate achieveable by 3D diffusion: 108 M-1s-1
This would correspond to target location in vivo on a timescale of only a few
seconds, when each cell contains several tens of TFs.
However:
Experimentally found (LacI repressor and its operator on DNA): 1010 M-1s-1.
Suggests that dimensionality of the problem changes during the search
process. While searching for its target site, the protein periodically scans the
DNA by sliding along it. This is best done if the TF is only partially folded and
only adopts its folded state when it recognizes its binding site.
Slutsky, Mirny, Biophys J 87, 4021 (2004)
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Time scales of protein dynamics
Motion Spatial extent (nm)Log10 of characteristic
time (s)
Relative vibration of bonded atoms 0.02 to 0.05 -14 to –13
Elastic vibration of globular region 1 to 2 -12 to –11
Rotation of side chains at surface 0.5 to 1 -11 to –10
Torsional libration of buried groups 0.5 to 1 -14 to –13
Relative motion of different globular regions (hinge bending)
1 to 2 -11 to –7
Rotation of medium-sized side chains in interior
0.5 -4 to 0
Allosteric transitions 0.5 to 4 -5 to 0
Local denaturation 0.5 to 1 -5 to 1
Protein folding 3 to 5 -5 to 2
Adapted from http://www.dbbm.fiocruz.br/class/Lecture/d22/kolaskar/ask-11june-4.ppt
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Energy Barriers during protein folding
Peptide chain must organize into particular 3D fold: large entropy loss.
Formation of secondary structure elements formation of hydrogen bonds.
This is almost cancelled by loss of hydrogen bonds with solvent molecules.
Burial of hydrophobic surface free energy gain due to hydrophobic effect.
Charged active site residues must be buried in hydrophobic protein interior
often electrostatically unfavorable
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Vesicle budding (right, above) does not occur
spontaneously: would be too dangerous for cell.
Distorting plane membrane costs deformation energy
binding of coat proteins reduces energy cost and gives
natural membrane curvature (right, below).
SNARE proteins help to overcome energy cost for fusion of
membranes.
Energy Barriers during vesicle budding
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Energy Barriers during formation of virus capsid
Many individual particles combine into one larger particle
big loss of translational and rotational degrees of freedom
Much hydrophobic surface gets buried between assembling proteins free energy gain according to hydrophobic effect (primarily solvent entropy)
Electrostatic attraction: probably not very significant for binding affinity but
important for specificity.
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What is the effect of pH on energy barriers?
At different pH, titratable groups will adopt different protonation states.
e.g. at low pH, Asp and Glu residues will become protonated
salt-bridges (Asp – Lys pairs) in which residues were involved will break up.
proteins unfold at low and high pH.
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What is the effect of D2O on energy barriers?
It is a common strategy to compare the speed of chemical reactions in H2O and
in D2O.
The electronic energy profile of the barrier is the same.
But the deuteriums of D2O have a higher mass than the hydrogens of H2O.
Their zero-point energies are lower
they need to overcome a higher effective energy barrier
all chemical reactions involving proton transfer will be slowed down,
typically by a factor of 1.4
This is called the „kinetic isotope effect“ (KIE).
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What is the effect of viscosity on energy barriers?
Adding co-solvents in the solvent to increase the viscosity should, in principle,
slow down conformational transitions.
It is often problematic that the co-solvent will also change the equilibrium, e.g.
between folded and unfolded states of a protein.
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What is the effect of temperature on energy barriers?
In general, higher temperature will enormously speed up activated processes.
However, we often need to consider free energy barriers instead of energy
barriers. The free energy barriers often change considerably with temperature.
E.g. in a MD simulation of a protein at 500K, the protein residues will more
easily overcome individual torsional energy barriers.
But, after a certain time, the whole protein will unfold.
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Reaction rates depend on temperatureConsider a binary reaction in the gas phase:
PBAk2
Suppose that
By definition, the rate coefficient k2 is independent of [A] and [B].
But k2 can depend strongly on temperature.
BAkdt
Pd2
The observed dependence of the reaction rate on the temperature is much
greater than one would expect from just the enhanced thermal motions of the
molecules.
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1889 Arrhenius: found temperature dependence of
the rates of inversion of sugar in the presence of acids.
Arrhenius cites van‘t Hoff (1884) for suggesting
e–A/T dependence.
Ea: activation energy
Arrhenius postulated that this relationship indicates the existence of an
„activated sugar“ whose concentration is proportional to the total concentration
of sugar, but is exponentially temperature dependent.
Arrhenius is the father of rate theory
Arrhenius and activated molecules
kT
Ea
erateRate
0
Svante Arrhenius 1859 – 1927Noble price 1903
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Arrhenius equation
1889, S. Arrhenius started from the
van‘t Hoff equation for the strong
dependence of the equilibrium constant
K on temperature:
2
ln
kT
h
dT
Kd
and proposed that kf and kr also have van‘t Hoff form
22
lnand
ln
kT
E
dT
kd
kT
E
dT
kdaraf
where Ea and E‘a have units of energy that are chosen to fit exp. data.
Ea and E‘a are called activation energies.
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Activation energy diagram
According to Arrhenius, it is not the average energy of the reactants that
determines the reaction rates but only the high energies of the ‚activated‘
molecules.
Taken from Dill book
There are two plateaus, one for the reactants and one for the products.
In between lies an energy maximum (also: transition state or activation barrier)
which is the energy that activated molecules must have to overcome on their
way from reactants to products.
Measuring kf as a function of temperature, and using eq. (1) gives Ea.
Measuring the reverse rate gives E‘a.
Measuring the equilibrium constant versus temperature gives h°.
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Population at different temperatures
From
Taken from Dill book
r
f
k
kK it follows
aa EEh '
The figure shows how activation is interpreted according to the Boltzmann
distribution law: a small increase in temperature can lead to a relatively
large increase in the population of high-energy molecules.
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Arrhenius plots
Integrating
over temperature T
Taken from Dill book2
ln
kT
E
dT
kdaf
gives:
BkTE
f
BkTE
f
BkTE
f
af
eAAek
eek
ek
BkT
Ek
a
a
a
with
ln
H2 + I2 2HI (open circles)
2HI H2 + I2 (full circles)
Diffusion of carbon in iron
The figures show examples of
chemical systems showing
Arrhenius behavior.
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Activated processes
Arrhenius kinetics applies to many physical and chemical processes.
When should one treat a process as activated?
If a small increase in temperature gives a large increase in rate,
a good first step is to try the Arrhenius model.
E.g. breaking of bonds.
Counter example: highly reactive radicals. CNHHHCNH 223
These can be much faster than typical activated processes and
they slow down with increasing temperature.
We now describe a more microscopic approach to reaction rates,
called transition state theory.
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The energy landscape of a reaction
An energy landscape defines how the
energy of a reacting system depends
on its degrees of freedom.
E.g. A + BC AB + C
Each reaction trajectory would involve
some excursions up the walls of the
valleys.
When averaged over multiple trajectories,
the reaction process can be described as
following the lowest energy route, along
the entrance valley over the saddle point
and out of the exit valley, because the
Boltzmann populations are highest along
that average route. Taken from Dill book
Energy surface for
D + H2 HD + H
The transition (saddle) point isdenoted by the symbol ‡.It is unstable: a ball placed on thesaddle point will roll downhillalong the reaction coordinate.
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Eyring theoryThe Eyring-theory or transition state theory (Theorie des Übergangszustandes) is a
molecular reaction theory. It uses molecular descriptors like the partition function and
describes the absolute rate of chemical reactions.
The reactants are separated from the products by
an activation barrier. The reaction from the reactants
via the transition state to the products proceeds
along a trajectory = the reaction coordinate.
Transition state: point of highest potential energy along this reaction coordinate.
Activated complex: atomic arrangement in the transition state.
The main assumptions of TST are:
- the activated complex exists in an equilibrium with the reactants
- All molecules that reach the transition state from the reactant states leave it in direction of
the products. Recrossings are not allowed.
[taken from Dill book]
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where is a prefactor with the dimensions of s-1 for
unimolecular reactions and s-1cm-3 for bimolecular reactions.
Wigner and Eyring: The transition state method
kTEa
ek
In the 1930s it was well established that reaction rates k should be written in the form:
Henry Eyring (1901-1981)
[taken from Dill book]
In „The activated complex in chemical reactions“ (1935),
Eyring gave a heuristic derivation of an expression for
the prefactor based on the assumption of an equilibrium
between the activated complex and reactants.
To obtain the time constant, he postulated, that at the
saddle point, any quantum state perpendicular to the
reaction coordinate reacts with the same universal time
constant kT/2ħ.
The rate is then given by the product of this universal time
constant with the ratio of the partition function of the
activated complex to the partition function of the reactants.
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Wigner and Eyring: The transition state method
1932, Wigner also derived an estimate for the tunneling
contribution to the thermal flux of particles crossing a
barrier.
1932 Pelzer & Wigner: estimated rate of conversion of parahydrogen into normal
hydrogen.
To compute the reaction rate, they use a thermal equilibrium distribution in the vicinity of
the saddle point of the PES and estimate the unidirectional classical flux in the direction
from reactants to products
They ignore the possibility of recrossings of the saddle
point noting that their probability at room temperature
would be rather small.
How can one define the „activated complex“?
Eyring‘s definition is questionable.
Wigner‘s definition leaves no ambiguity: the best dividing
surface is that which minimizes the unidirectional flux from
reactants to products. Eugene Wigner (1902-1995)Noble price 1963
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The transition state
(left) Contour plot of a reaction pathway (- - -) on an energy landscape
for the reaction A + BC AB + C. The broken line shows the lowest-energy
path between reactants and products
(right) The transition state is an unstable point along the reaction pathway
(indicated by the arrow) and a stable point in all other directions that are
normal to the reaction coordinate.
Taken from Dill book
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Calculating rate coefficients from TST
Let us consider the reaction
by transition state theory:
Divide the reaction process into two stages:
(1) the equilibrium between the reactants ant the transition state (AB)‡ with
‚equilibrium constant‘ K‡
(2) a direct step downhill from the TS to the product with rate coefficient k‡:
PBAk2
PABBA kK ‡‡ ‡
Key assumption of TST: step (1) can be expressed as an equilibrium between
the reactants A and B and the transition state (AB)‡ , with
BAB
KA
‡‡
even though (AB)‡ is not a true equilibrium state.
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Calculating rate coefficients from TST
The overall rate is expressed as the number of molecules in the TS, [(AB)‡],
multiplied by the rate coefficient k‡ for the second product-forming step
BAKkABkdt
Pd ‡‡‡‡
Because the quantitiy K‡ is regarded as an equilibrium constant, it can be
expressed in terms of the molar partition functions:
kTD
BA
AB eqq
qK
‡‡
‡
where D‡ is the dissociation energy of the TS minus the dissociation energy of
the reactants.
q(AB)‡ is the partition function of the transition state.
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relation between value analysis and TST
Later we will characterize the effect of a protein mutant by its -value
0
ln
G
kkTk
wt
mutB
G0 reflects whether the mutant stabilizes the folded state F over the unfolded
state U stronger or weaker than wild-type protein.
According to TST, both wild-type and mutant folding proceed via transition
states with activation free energies G‡wt
and G‡mut.
wtFU
mutFU GGG 0
‡‡‡;‡
‡‡
‡
‡
‡
‡
wtmutkTG
kTGG
kTG
kTG
wt
mut
kTGmut
kTGwt
GGGe
ee
e
k
k
ek
ek
wtmut
wt
mut
mut
wt
A -value of 1 meansthat G0 = G‡ forthis mutant the mutant has the sameeffect on the TS structureas on the folded state this part of the TS structure is folded asin the folded state F.
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Catalysts speed up chemical reactions
Taken from Dill book
Free energy barrier G‡ is reduced by a catalyst C.
Catalysts affect the rates of chemical reactions; e.g. enzymes accelerate
biochemical reactions.
Enzymes can achieve remarkable accelerations, e.g. by a factor of
2 x 1023 for orotine 5‘-phosphate decarboxylase.
Linus Pauling proposed in 1946 that catalysts work by stabilizing the
transition state.
PBA k 0
ABC
kc
Linus Pauling 1935
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Catalysts speed up chemical reactions
From transition theory we obtain for the catalyzed reaction rate kc (normalized to
the uncatalyzed reaction rate k0)
CAB
ABC
k
kc
‡
‡
0
This ratio represents the ‚binding constant‘ of the catalyst to the transition state
the rate enhancement by the catalyst is proportional to the binding affinity of
the catalyst for the transition state.
This has two important implications:
(1) to accelerate a reaction, Pauling‘s principle says to design a catalyst that
binds tightly to the transition state (and not the reactants or product, e.g.).
(2) a catalyst that reduces the transition state free energy for the forward reaction
is also a catalyst for the backward reaction.
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Speeding up reactions by intramolecular localization or solvent preorganization
Taken from Dill book
Reactants polarize, so water reorganizes.
Two neutral reactants become charged in the transition state.
Creating this charge separation costs free energy because it orients the solvent
dipoles.
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Speeding up reactions by intramolecular localization or solvent preorganization
Enzymes can reduce the activation barrier by having a site with pre-organized
dipoles.
Taken from Dill book
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Funnel landscape describe diffusion and polymer folding
All the processes described sofar involve well-defined reactants and products, and
a well-defined reaction coordinate.
But diffusional processes and polymer conformational changes often cannot be
described in this way. The starting point of protein folding is not a single point on
an energy landscape but a broad distribution.
A bumpy energy landscape, such as occursin diffusion processes, polymer conformationalchanges, and biomolecule folding.
A single minimum in the center may representthe ‚product‘, but there can be many different‚reactants‘, such as the many openconfigurations of a denatured protein.
http://www.dillgroup.ucsf.edu/
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Summary
Chemical reactions and diffusion processes usually speed up with temperature.
This can be explained in terms of a transition state or activation barrier and an
equilibrium between reactants and a transient, unstable transition state.
For chemical reactions, the transition state involves an unstable weak vibration
along the reaction coordinate, and an equilibrium between all other degrees of
freedom.
Catalysts act by binding to the transition state structure.
They can speed up reactions by forcing the reactants into transition-state-like
configurations.