Protein folding and structure
Reading
Dobson CM. Experimental investigation of protein folding and misfolding.
Methods. 2004 Sep;34(1):4-14. Review.
Dobson CM. Protein folding and misfolding.
Nature. 2003 Dec 18;426(6968):884-90. Review.
Lindorff-Larsen K, Best RB, Depristo MA, Dobson CM, Vendruscolo M. Simultaneous determination of protein structure
and dynamics. Nature. 2005 Jan 13;433(7022):128-32.
Lindorff-Larsen K, Vendruscolo M, Paci E, Dobson CM. Transition states for protein folding have native topologies
despite high structural variability. Nat Struct Mol Biol. 2004 May;11(5):443-9.
Dedmon MM, Lindorff-Larsen K, Christodoulou J, Vendruscolo M, Dobson CM. Mapping long-range interactions in
alpha-synuclein using spin-label NMR and ensemble molecular dynamics simulations.
J Am Chem Soc. 2005 Jan 19;127(2):476-7.
Klein-Seetharaman J, Oikawa M, Grimshaw SB, Wirmer J, Duchardt E, Ueda T, Imoto T, Smith LJ, Dobson CM,
Schwalbe H. Long-range interactions within a nonnative protein.
Science. 2002 Mar 1;295(5560):1719-22.
Chan HS, Dill KA. Protein folding in the landscape perspective: chevron plots and non-Arrhenius kinetics.
Proteins. 1998 Jan;30(1):2-33. Review.
Schonbrun J, Dill KA. Fast protein folding kinetics.
Proc Natl Acad Sci U S A. 2003 Oct 28;100(22):12678-82.
Liwo A, Khalili M, Scheraga HA Ab initio simulations of protein-folding pathways by molecular dynamics with the
united-residue model of polypeptide chains.
Proc Natl Acad Sci U S A. 2005
Questions
1.
2.
Explain
a.
b.
c.
d.
e.
f.
g.
h.
i.
j.
k.
l.
Native state, unfolded state, denatured state, metastable state, denaturation, renaturation
Why has a folded protein lower free energy G in spite of unfavorable conformational entropy S.
Levinthal paradox
-value analysis
contact order and folding rate
Chevron-plot
Transition state ensemble
Free energy funnel of folding
Chaperones
Difference between a prion disease and amyloid disease
Thermodynamic and kinetic hypothesis of protein folding
Why it is difficult to observe directly transition state
What experimental techniques there are to study protein folding?
a. Real time methods, (H-D exchange using stop-flow techniques)
1
b.
3.
Steady-state, denaturant, chemical modification
Folding can be described as transition between the native and unfolded state. In increasing concentrations of
denaturant, a fold disintegrates with a rate
ln ku  ln kuH 2O  mku denat 
when the concentration of denaturant is above that needed to unfold 50 % of protein. Conversely folding takes place
with rate
ln k f  ln k Hf 2O  mku denat 
when the concentration of denaturant is below that needed to unfold 50 % of protein.
Write an equation for the observed rate constant ln kobs and plot it (Chevron plot) as a function of denaturant
concentration.
How do you determine from the figure kfH2O?
How will the Chevron plot change when a mutation will affect the transition state?
4.
Near the transition temperature Tm the Gibbsin
free energy is

T 
T
G(T )  1  H (Tm )  T  Tm C P  TC P ln
Tm
 Tm 
Show that there will be another transition
temperature Tm’ < Tm.
Provide an estimate of Tm’ using an approximation
5.
6.
T
T  T 1  Tm  T 
ln m  m
 

T
T
2 T 
2
Describe experiments that allow you to follow the unfolding of a globular protein
.
All proteins are increasingly unstable with increasing temperature, but only few regain full activity on cooling back to
room temperature. Why?
7.
Why is the unfolding curve of a protein often sigmoidal as a function of denaturant concentration.
8.
Why many protein display rollover in Chevron plots? K. Dill
9.
Draw Kratky-plot I(S)*S2 vs. S, where I is the intensity as a function of scattering vector, for a fully unfolded protein.
You may consider the unfolded polypeptide as a chain that scatters as
.
I (S ) 


2 ex  x 1
, x  4 2 S 2 RG2
2
x
where RG is the radius of gyration. At high values of scattering vector you may treat the elements of a chain as a needle
I (S ) 
1
2

2 2 2
2 SL 4 S L
where L is the length of the needle.
How does the scattering differ from that of a folded protein at small values of S
And at high values of S
I (S )  1
ln
I (S )
I ( 0)

  4 2 S 2 RG2 / 3

S4
10. Rearrange equation H - TS = RTln[D]/[N], where [D] is the concentration of the denatured and [N] the native
protein, to [D]/([D]+[N]) that is determined by measurement (spectroscopy). Draw the curve [D]/([D]+[N]) vs. T and
identify the midpoint as Tm.
2
11. Below are given thermodynamic parameters for folding of few proteins at 25 ºC. What is the most stable protein?
What are the melting temperatures assuming that enthalpy and entropy do not change with temperature?
Protein
Ribonuclease
Chymotrypsin
Lysozyme
Cytochrome C
G (kJ/mol)
-46
-55
-62
-44
H (kJ/mol)
-280
-270
-220
-52
S (kJ/Kmol)
-790
-720
-530
-27
12. Ribonuclease structure is stabilized by G = -7.1 kJ/mol at pH 2.5 and T = 25 ºC. What is the ratio between folded and
denatured molecules? The enthalphy for folding is 238.6 kJ/mol at 25 ºC. Assume that enthalpy and entropy do not
change when temperature is raised to 37 ºC. What is G at 37 ºC and what is the ratio between folded and denatured
molecules? Finally what is the Tm for ribonuclease at pH 2.5?
13. The action of a denaturant such as urea or guanidinium chloride is to solubilize all parts of the polypeptide. To a good
approximation the free energy change is proportional to the denaturant concentration G = GH2O – m[denaturant]
where m is a constant (gradient of G). Rewrite the equation in 2 for [D]/([D]+[N]) vs. [denaturant] and draw the
curve.
14. Show that there is a temperature Ts where the protein structure is most stable from the expression of free energy
G(T )
H (T )
S (T )
G(T )  G(Tm )  
dT  H (Tm )  
dT  TS (Tm )  T 
dT
T
T
T
Tm
Tm
Tm
T
T
T
where Tm is the melting temperature. Assume constant pressure.
15. Based on the previous result, show first that Ts < Tm and then that there is another temperature Tm´ below Ts where
protein denatures anew. Express Tm´.
16. Use your previous results and draw curves for free energy, enthalpy and entropy as function of temperature for a
protein with Tm = 60 ºC, H(Tm) = 500 kJ/mol and specific heat Cp = 10 kJ/(Kmol).
17. Pressure and temperature will affect together protein stability according to
G ( P, T )  G (To , Po )  Vo ( P  Po ) 
1
 ( P  Po ) 2   ( P  Po )(T  To )
2
  T


 S o (T  To )  C P T  ln
 1  To 

  To

where V is the change in the specific volume,  is the change in the isothermal compressibility (∂V/∂P)T and  is
the change in the thermal expansivity (∂V/∂T)P = -(∂S/∂P)T. Make an approximation (T - To)/To << 1 to cast G(P,T)
to a quadratic form. What is the condition in terms of  Cp and To for the quadratic equation to be an ellipse in
the P,T-plane?
18. Find an expression for the center of the G(P,T)-ellipse and write the condition to determine whether pressure will
simply destabilize the protein or the most stable state is at an elevated pressure.
3
19. Paramagnetic relaxation of nuclear spins is governed by
I para
I

C exp(  R2 P t )
C  R2 P
where t is time and paramagnetic relaxation rate is
R2 P 
K
r6

3 c 
 4 c 

2 2 
1



H c 

and r is the distance from the paramagnetic site. All other
symbols are constants
In the figure you see paramagnetic enhancement to nuclear
spin relaxation for unfolded apomyoglobin at pH 2.3. The
histograms show the intensity ratios (Ipara/I) for each residue
when the cysteine bound label is (A) E18C*, (B) K77C*, and
(C) K133C*.
Sketch in each panel the expected intensity ratios for a random
coil.
Deduce from the differences between the expected random
curve and experimental data which of the sites 18, 77 and 133
are involved in a clustering and draw a residue-residue contact
map.
20. Why it is difficult to simulate protein folding using atomic models but
instead simplified models such the one below are used?
In the simple model side chains are presented as pseudoatoms (SC) that
move in a cone given by  and  angles. The backbone is also reduced to a
single pseudoatom between two C and having  and  as the degrees of
freedom. Pseudoatoms are connected by virtual bonds d.
Construct a force field, i.e. a potential energy function U, for this simplified
model that contains potential energy terms related to the degrees of freedom
and pair wise interactions.
Assume you are running a molecular dynamics (MD) simulation using you
force field. At a given time step t the model has a set of coordinates q(t).
How do you obtain from U[q(t)] accelerations d2q/dt2?
21. Relaxation rates of nuclear spins are sensitive to the mobility i.e. to the effective size of the molecular fragment.
Below you see relaxation data from lysozyme that has been chemically modified to prevent it from folding.
4
22. Below on left you see free energy changes G of a folded WW protein introduced by mutations. Which parts of the
protein are important for its thermal stability.
On right are shown  values. What residues are important for folding?
23. Hydrogen deuterium exchange …
24. Explain how a free energy funnel devoid of major barriers may cause apparent two-state folding kinetics.
25. Observed chemical reaction kinetics is described microscopically by specific rate limiting atomic structures. Observed
protein folding kinetics resembles observed chemical reaction kinetics, however, how does the microscopic
macromolecular description differ from the chemical one?
26. How do mass-action and statistical mechanics models of protein folding differ?
27. The free energy landscape (funnel) is a high-dimensional surface E(1,2,3,…m) where s present lateral
coordinates that relate to the degrees of freedom. What is the meaning of the internal free energy E? Can one measure
it? How does temperature or denaturant affect E?
28. Simple protein folding models contain only a chain of hydrophobic and polar beads that can “walk” in a (2D) grid.
What kind of pair wise interaction you could set among the beads and how would the corresponding energy level
diagrams and landscapes look like?
29. What is the distinction between thermodynamic (Q) and kinetic () reaction coordinates? Are they correlated? Often Q
= 1/2 is taken to define the transition mid-point. How is Q biased according to the kinetic description.
30. Monte Carlo method is used in a wide range of optimization tasks. Usually the probability for the transition is obtained
using Metropolis sampling. Why this sampling may not be optimal for studies of folding but Kawasaki sampling
should be used?
31. Proteins unfold upon heating and perhaps surprisingly also upon cooling. To account for the temperature dependence
using a HP model, where there is only attractive interaction HH between the hydrophobic residues, the simplest
approximation is that  does not depend on T, the second approximation is  (T) and the third is A(T) where A is the
rate of conformational transition. Plot schematically folding and unfolding rates using the first, second and third
approximations vs 1/T.
5
32. Describe the differences between homopolymer and heteropolymer folding kinetics.
33. How does protein folding in cell (in vivo) differ from that achievable in vitro or in silico.
34. The native state, i.e. a folded and functional protein, is often the subject of primary interest. What other forms or states
a polypeptide may adopt?
35. Proteins are dynamic. Give examples of the motions that ragen over many time and length scales.
36. Protein structure determination by NMR involves minimization of penalty function


E   kr (r  ro )2  k cos 3(  o )  
to match the experimental data, i.e. distances between atoms ro, dihedrals o, etc. However the target values denoted
by subscript o are averages over the dynamic ensemble. Discuss the properties and qualities of the model that has been
obtained by minimization of E.
One may also acquire dynamic information, e.g. order parameter S2. How would you need to modify the minimization
of the penalty function to incorporate the dynamic nature.
37. Transition states are claimed to have native-like topology. What does it mean?
38. In the figure transverse relaxation rates have been measured from lysozyme that has been chemically modified to trap
a folding intermediate.
How would you build on the basis of data a model of the intermediate. Suppose the residue indicated with 5. was
mutated. What kind of data you would expect to record then?
6