Lecture of Graduiertenkolleg Helms

1
Lecture of Graduiertenkolleg - Helms
V1 Reaction rate theory V2 Kramers
theory V3 Reaction rates for electronic
transitions V4 Potential and free energy
landscapes V5 Lattice optimization V6 Optimizati
on methods in protein folding V7 Protein folding
with molecular dynamics V8 Manipulating
potential energy landscapes with forces
2
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)
4
Reaction rates are proportional to concentrations
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 ?
The two equations are coupled. One can solve
them by matrix algebra
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Excursion coupled differential equations
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
but u' ku for k const is easy to solve. The
solution is u Aekx where A const. ?
translate the ODEs into matrix form
By diagonalizing the square matrix, we get
www.algebra.com
6
Excursion coupled differential equations
By diagonalizing the square matrix, we get
We then put
It follows that
Thus
The solutions of this system are found easily
with some constants C and D.
With
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Reaction rates are proportional to concentrations
With this technique, we could solve our system of
coupled diff. equations. In the special case
that kr ltlt kf, the first equation simplies to
If A(t) B(t) constant, then
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At equilibrium, rates obey detailed balance
The principle of detailed balance says that the
forward and reverse transitions must be identical
for an elementary reaction at equilibrium
Aeq and Beq equilibrium concentrations. To
see that this is a condition of equilibrium
follows from inserting into resulting in
Taken from Dill book
9
At equilibrium, rates obey detailed balance
The detailed balance condition relates the rate
coefficients kf and kr to the equilibrium
constant K
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
Lets consider a 3-state mechanism with kIA ? 0,
kBI ? 0, kAB ? 0.
Taken from Dill book
10
At equilibrium, rates obey detailed balance
This results in the mechanism shown right. The
only conditions for equilibrium are
Taken from Dill book
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.
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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
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.
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
• proteinligand association - pH
• proteinprotein association - D2O vs. H2O
• proteinmembrane association - viscosity
• proteinDNA 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
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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 ProteinLigand Interaction
Step 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
No collective modes!
Bound and associated H2O
Receptor
Ligand
Displaced H2O
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Energy Barriers of ProteinProtein Interaction

• 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

Effects of hydrodynamic Interactions (left)
effect of translation (right) effect of rotation
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Energy Barriers of ProteinMembrane 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 ProteinMembrane 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 proteinmembrane systems
is largely unknown.
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Energy Barriers of ProteinDNA 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
gt40 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 gt10 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 ProteinDNA 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
Adapted from http//www.dbbm.fiocruz.br/class/Lect
ure/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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Energy Barriers during vesicle budding
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.
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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 temperature
Consider a binary reaction in the gas phase
Suppose that
By definition, the rate coefficient k2 is
independent of A and B. But k2 can depend
strongly on temperature.
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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Arrhenius and activated molecules
1889 Arrhenius found temperature dependence
of the rates of inversion of sugar in the
presence of acids. Arrhenius cites vant Hoff
(1884) for suggesting eA/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
Svante Arrhenius 1859 1927 Noble price 1903
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Arrhenius equation
1889, S. Arrhenius started from the vant Hoff
equation for the strong dependence of the
equilibrium constant K on temperature
and proposed that kf and kr also have vant Hoff
form
where Ea and Ea have units of energy that are
chosen to fit exp. data. Ea and Ea are called
activation energies.
33
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.
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
Ea. Measuring the equilibrium constant versus
temperature gives ?h.
Taken from Dill book
34
Population at different temperatures
From
it follows
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.
Taken from Dill book
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Arrhenius plots
H2 I2 ? 2HI (open circles) 2HI ? H2 I2 (full
circles)
The figures show examples of chemical systems
showing Arrhenius behavior.
Taken from Dill book
Integrating over temperature T
gives
Diffusion of carbon in iron
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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.
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.
Energy surface for D H2 ? HD H
The transition (saddle) point is denoted by the
symbol . It is unstable a ball placed on
the saddle point will roll downhill along the
reaction coordinate.
Taken from Dill book
38
Eyring theory
The 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
39
Wigner and Eyring The transition state method
In the 1930s it was well established that
reaction rates k should be written in the form
where ? is a prefactor with the dimensions of s-1
for unimolecular reactions and s-1?cm-3 for
bimolecular reactions.
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/2?h. 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.
taken from Dill book
Henry Eyring (1901-1981)
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Wigner and Eyring The transition state method
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? Eyrings
definition is questionable. Wigners 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
1932, Wigner also derived an estimate for the
tunneling contribution to the thermal flux of
particles crossing a barrier.
41
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
42
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
Key assumption of TST step (1) can be expressed
as an equilibrium between the reactants A and B
and the transition state (AB) , with
even though (AB) is not a true equilibrium state.
43
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
Because the quantitiy K is regarded as an
equilibrium constant, it can be expressed in
terms of the molar partition functions
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
??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 ?Gwt and ?Gmut.
A ?-value of 1 means that ??G0 ??G for this
mutant ? the mutant has the same effect on the TS
structure as on the folded state ? this part of
the TS structure is folded as in the folded
state F.
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Catalysts speed up chemical reactions
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.
kc
ABC
Linus Pauling 1935
Free energy barrier ?G is reduced by a catalyst
C.
Taken from Dill book
46
Catalysts speed up chemical reactions
From transition theory we obtain for the
catalyzed reaction rate kc (normalized to the
uncatalyzed reaction rate k0)
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,
Paulings 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
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.
Taken from Dill book
48
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
49
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 occurs in
diffusion processes, polymer conformational change
s, and biomolecule folding. A single minimum in
the center may represent the product, but there
can be many different reactants, such as the
many open configurations of a denatured protein.
http//www.dillgroup.ucsf.edu/
50
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.