Protein Folding and Structure Prediction

A Statistician's View

Ingo Ruczinski Department of Biostatistics, Johns Hopkins University

Proteins

Amino acids without peptide bonds. Amino acids with peptide bonds. ¡ Amino acids are the building blocks of proteins. Proteins

Both ﬁgures show the same protein (the bacterial protein L). The right ﬁgure also highlights the secondary structure elements.

Space

Resolution limit of a light microscope

Glucose Ribosome Red blood cell

C−C bond Hemoglobin Bacterium

1 10 100 1000 10000 100000

1nm 1µm

Distance [ A° ] Energy

C−C bond

Green Noncovalent bond light Glucose

Thermal ATP

0.1 1 10 100 1000

Energy [ kcal/mol ]

Non-Bonding Interactions

Amino acids of a protein are joined by covalent bonding interactions. The polypep- tide is folded in three dimension by non-bonding interactions. These interactions, which can easily be disrupted by extreme pH, temperature, pressure, and denatu- rants, are:

Electrostatic Interactions (5 kcal/mol)

Hydrogen-bond Interactions (3-7 kcal/mol)

Van Der Waals Interactions (1 kcal/mol) ¡ Hydrophobic Interactions ( 10 kcal/mol)

The total inter-atomic force acting between two atoms is the sum of all the forces they exert on each other. Energy Proﬁle

Transition State

Denatured State

Native State

Radius of Gyration of Denatured Proteins

Do chemically denatured proteins behave as random coils?

The radius of gyration Rg of a protein is deﬁned as the root mean square distance from each atom of the protein to their centroid.

For an ideal (inﬁnitely thin) random-coil chain in a solvent, the average radius

0.5 ¡

of gyration of a random coil is a simple function of its length n: Rg n

For an excluded volume polymer (a polymer with non-zero thickness and non- trivial interactions between monomers) in a solvent, the average radius of gyra- 0.588

tion, we have Rg n (Flory 1953). ¡ The radius of gyration can be measured using small angle x-ray scattering. Radius of Gyration of Denatured Proteins

90 80 70 60 50

Creatine Kinase 30 ] ° A [ g

R 20

Angiotensin II

10 Confidence interval for the slope: [ 0.579 ; 0.635 ]

10 50 100 500 Length [residues]

Deviations from Random Coil Behaviour

Are there site-speciﬁc deviations from random coil dimensions?

Forster¨ Resonance Energy Transfer enables us to measure the distance between two dye molecules within a certain range. This can be used to study site-speciﬁc deviations from random coil dimensions in highly denatured peptides. Deviations from Random Coil Behaviour

20 number of photons

0 200 400 600 800 1000

time

30 25 20 15 10

number of photons 5 0

0 10 20 30 40 50

time

Deviations from Random Coil Behaviour

200 We have two underlying distributions for the green and red photons:

150 One stemming from a peptide only having a donor dye.

100 One stemming from a peptide being properly tagged with a donor and an number of green acceptor dye.

Assume a photon has probability ¢¡ of be- £ ing red in the former situation, and in 0 the latter. 0 20 40 60 80

number of red photons

Deviations from Random Coil Behaviour £

Assume we observe ¢¡ photons at time point . Then the number of red photons

¤¡¦¥ ¡ ¡

¢¡ £ is simply Bernoulli( ), where is either or . Assume that the probability

of observing photons from a peptide without an acceptor dye at any time is ,

independent of the total number of photons observed. Let § be the number of red

photons. Then

¨ © ¨ © ¨ © ©

§ § ¥ § ¥

¡ ¡ ¡ ¡ ¡ ¡

¡ £

¡ ¡

¡!¦" & £("

$#% )#*

¡ £

¡ '¡

and hence

¢¡ 4¡

¥ ¥ -

¡!¦" 2 £(" 76 ¡ £

$#% 5#%

¡ £

¡3 '¡

¡0/

£21

Deviations from Random Coil Behaviour

40 number of red photons

50 60 70 80 90 100

total number of photons Deviations from Random Coil Behaviour

^ p1 = 0.431

0.0 0.2 0.4 0.6 0.8 1.0 nred

nred + ngreen

Energy Proﬁle

Tmutant

∆∆GT−D

Twildtype

Dmutant

Dwildtype

Nmutant

∆∆GN−D

Nwildtype

¡¢¡ ¡¢¡ The -value is deﬁned as the ratio GT-D GN-D. Energy Proﬁle

Tmutant

∆∆GT−D = ∆∆GT−D

Tmutant ∆∆GT−D = 0 Twildtype Twildtype

Dmutant Dmutant

Dwildtype Dwildtype

Nmutant Nmutant

∆∆GN−D ∆∆GN−D

Nwildtype Nwildtype

If the part of the protein that contains the mutant amino acid is fully structured

¡¢¡ in the transition state, we have ¡¢¡ GT-D = GN-D and hence = 1.

If the part of the protein that contains the mutant amino acid is equal in dena-

tured and the transition state, we have ¡¢¡ GT-D = 0, and hence = 0.

Chevron Plots

wildtype mutant

¡ ¢ ¡¢¡ GT-D = RT log(kf ) – log(kf )

wildtype wildtype mutant mutant

¡ ¢ ¡¢¡ GN-D = RT log(kf ) – log(ku ) – log(kf ) + log(ku )

6 Wildtype Mutation I28A

4 ) s b o k

( 2 log

−2

0 1 2 3 4 5 6 7 8

Denaturant concentration ( GuHCl [M] )

CGuHCl CGuHCl

¢ ¢¥¤ log(kobs) = log exp log(kf)+ mf RT + exp log(ku)+ mu RT More Chevron Plots

6 6 6 Mutation I28A Mutation I28L Mutation I28V

4 4 4 ) s b o k 2 2 2 ( log

0 0 0

−2 −2 −2

0 1 2 3 4 5 6 7 8 0 1 2 3 4 5 6 7 8 0 1 2 3 4 5 6 7 8

6 6 6 Mutation V55A Mutation V55M Mutation V55T

4 4 4 ) s b o k 2 2 2 ( log

0 0 0

−2 −2 −2

0 1 2 3 4 5 6 7 8 0 1 2 3 4 5 6 7 8 0 1 2 3 4 5 6 7 8

Denaturant concentration ( GuHCl [M] ) Denaturant concentration ( GuHCl [M] ) 6 Wildtype

4 )

s UC Santa Barbara b o k 2 ( Rice University log UC Berkeley 0

−2

0 1 2 3 4 5 6 7 8

Denaturant concentration ( GuHCl [M] )

Variability

φ − values

0 φ

−1

UC Santa Barbara Rice University −2 UC Berkeley

∆∆GN−D − values

10 5 D − N

G 0 ∆ ∆ −5 −10 I28A−I28L I28L−I28V I28A−I28V V55A−V55T V55M−V55T V55A−V55M Wild type−I28L Wild type−I28A Wild type−I28V Wild type−V55T Wild type−V55A Wild type−V55M Variability

Between lab φ − value standard deviation

1.50

1 Wild type−I28A 6 2 Wild type−I28L 8 1.25 3 Wild type−I28V

4 I28A−I28L

5 I28A−I28V

6 I28L−I28V 1.00 11 7 Wild type−V55A

8 Wild type−V55M

9 Wild type−V55T b a l 0.75 10 V55A−V55M

^ 2 σ

11 V55A−V55T

12 V55M−V55T

0.50 3

0.25 4

10 7 5 9 12 1 0.00 0 1 2 3 4 5 6 7 8 9 10 11 12

average ∆∆GN−D values

Variability

5 D − T

G 0 ∆ ∆

−5

−10 UC Santa Barbara Rice University

UC Berkeley

−10 −5 0 5 10

∆∆GN−D Some Simulation

3.5

3.0

2.5

2.0 b a l ^ σ

1.5

1.0

0.5

0.0 2 4 6 8 10 12

average ∆∆GN−D values

Some Simulation

3.5

3.0

2.5

2.0 b a l ^ σ

1.5

1.0

0.5

0.0 2 4 6 8 10 12

average ∆∆GN−D values Some Simulation

3.5

3.0

2.5

2.0 b a l ^ σ

1.5

1.0

0.5

0.0 2 4 6 8 10 12

average ∆∆GN−D values

Some More Simulations

D ( φ^ | φ , ∆∆G )

0.37 ^

φ 0

0 1 2 3 4 5 6 7 8 9 10 11 12 13

∆∆GN−D values Some More Simulations

1 1 1 ^ φ 0.3 0.2 0.1 0 0 0

1 2 3 4 5 6 7 8 9 10 11 12 1 2 3 4 5 6 7 8 9 10 11 12 1 2 3 4 5 6 7 8 9 10 11 12

1 1 1

0.6 ^ φ 0.5 0.4

0 0 0

1 2 3 4 5 6 7 8 9 10 11 12 1 2 3 4 5 6 7 8 9 10 11 12 1 2 3 4 5 6 7 8 9 10 11 12

1 1 1 0.9 0.8 0.7 ^ φ

0 0 0

1 2 3 4 5 6 7 8 9 10 11 12 1 2 3 4 5 6 7 8 9 10 11 12 1 2 3 4 5 6 7 8 9 10 11 12

average ∆∆GN−D values average ∆∆GN−D values average ∆∆GN−D values

Phi-Value Estimation

∆∆GTD

13.5

13.0

12.5

6 8 10 12 14 D

T 12.0 G ∆ ∆

∆∆GND 11.5

11.0

10.5

6 8 10 12 14 7.0 7.5 8.0 8.5 9.0

∆∆GTD

¢¡

¡ ¡¥£ ¡¢¡¤£ £

¡ TD TD

6 ¥ 6

¡¢¡¤£

¥£ ¡¢¡ ND

ND ¦¨§ ©

1 1

with

F F

F F U U W F U M F U

F F W F U M F U

Phi-Value Estimation

¡ ¡¥£

TD ¡

¡¢¡¤£ ND ©

with

¡ ¡ ¡ ¡

¡¢¡¥£ ¡¢¡¤£ ¡ ¡¥£ ¡ ¡¥£ £

¥£ ¤£

¡¢¡ ND

0.55 0.60 0.65 0.70 0.75

Φ^

Phi-Value Estimation

§ ¦¨§

¥ ¢ ¢!

¡ ¡

£ ¡ ¡

£ ?

"*" "%"

# #

1.0

0.8

0.6

0.4

0.2

0.0 Evolution and Folding Kinetics

Are amino acids in proteins conserved because of folding kinetics? To what extent does natural selection act to optimize the details of protein folding kinetics? Is there a relationship between an amino acid’s evolutionary conservation and its role in protein folding kinetics? Some comments:

Our studies of sequence conservation among residues known to participate in the folding nuclei of all of the appropriately characterized proteins reported to date have not provided any evidence that highly conserved residues are more likely to participate in the protein folding nucleus than poorly conserved residues.

This is in contrasts to some of the beliefs stemming from theoretical considera- tions (good science, good people).

This is also in contrast to the conclusions certain people drew from experimental data (really aweful statistics).

These people do not like us.