Protein Folding and Structure Predictioniruczins/presentations/ruczinski.05.03.rutgers.pdfNon-Bonding Interactions Amino acids of a protein are joined by covalent bonding interactions.

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 alsohighlights the secondary structure elements.

Space

1 10 100 1000 10000 100000

1nm 1µm

Distance [ A° ]

C−C bond

Glucose

Hemoglobin

Ribosome

Resolution limit of a light microscope

Bacterium

Red blood cell

Distance [ A° ]

Energy

0.1 1 10 100 1000

Energy [ kcal/mol ]

Thermal

Noncovalent bond

ATP

Green light

C−C bond

Glucose

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 forcesthey exert on each other.

Energy Profile

DenaturedState

TransitionState

NativeState

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 dis-tance from each atom of the protein to their centroid.

� For an ideal (infinitely thin) random-coil chain in a solvent, the average radiusof gyration of a random coil is a simple function of its length n: Rg � n0.5 �

� 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-tion, we have Rg � n0.588 (Flory 1953).

� � The radius of gyration can be measured using small angle x-ray scattering.

Radius of Gyration of Denatured Proteins

Length [residues]

Rg

[A°]

10 50 100100 500

10

20

30

40

50

60

708090

Angiotensin II

Creatine Kinase

Confidence interval for the slope: [ 0.579 ; 0.635 ]

Deviations from Random Coil Behaviour

Are there site-specific deviations from random coil dimensions?

Forster Resonance Energy Transfer enables us to measure the distance betweentwo dye molecules within a certain range. This can be used to study site-specificdeviations from random coil dimensions in highly denatured peptides.

Deviations from Random Coil Behaviour

0 200 400 600 800 1000

0

20

40

60

80

time

num

ber

of p

hoto

ns

0 10 20 30 40 50

0

5

10

15

20

25

30

time

num

ber

of p

hoto

ns

Deviations from Random Coil Behaviour

0 20 40 60 80

0

50

100

150

200

number of red photons

num

ber

of g

reen

gre

en

We have two underlying distributions forthe green and red photons:

� One stemming from a peptide onlyhaving a donor dye.

� One stemming from a peptide beingproperly tagged with a donor and anacceptor dye.

Assume a photon has probability ��� of be-ing red in the former situation, and � � inthe latter.

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 probabilityof 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 redphotons. Then� � � � ��� � ��� � � � ��� ��� � � � � � ��� ��� � � � � ��� � � � � � ��� ��

� ���� � � ������ ��� �� ��

� � �!��" �$#%� � � �&� � � ��'��� ��� �� ��� � ���(" �)#*� � � ��

� ��� �and hence

+,� � � � � � ��� � -. �0/ �21

� ������3� ��� �� ��� � �!��" �$#%� � � �2� � �4��'��� ��� �� ��

� � ���(" �5#%� � � ��� ���76 �

Deviations from Random Coil Behaviour

50 60 70 80 90 100

0

20

40

60

80

total number of photons

num

ber

of r

ed p

hoto

ns

Deviations from Random Coil Behaviour

0.0 0.2 0.4 0.6 0.8 1.0nred

nred + ngreen

p1 = 0.431

Energy Profile

Dmutant

Dwildtype

Tmutant

Twildtype

Nmutant

Nwildtype

∆∆GN−D

∆∆GT−D

� � The � -value is defined as the ratio ��� GT-D���� GN-D.

Energy Profile

Dmutant

Dwildtype

Tmutant

Twildtype

Nmutant

Nwildtype

∆∆GN−D

∆∆GT−D = ∆∆GT−D

Dmutant

Dwildtype

Tmutant

Twildtype

Nmutant

Nwildtype

∆∆GN−D

∆∆GT−D = 0

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

��� GT-D = RT ��� log(kwildtypef ) – log(kmutant

f ) ���� GN-D = RT ��� log(kwildtype

f ) – log(kwildtypeu ) – log(kmutant

f ) + log(kmutantu ) �

−2

0

2

4

6

Denaturant concentration ( GuHCl [M] )

log(

k obs

)

Wildtype

Mutation I28A

0 1 2 3 4 5 6 7 8

log(kobs) = log � exp � log(kf)+ mf � CGuHCl

RT � + exp � log(ku)+ mu � CGuHCl

RT ���

More Chevron Plots

−2

0

2

4

6

log(

k obs

)

0 1 2 3 4 5 6 7 8

Mutation I28A

−2

0

2

4

6

0 1 2 3 4 5 6 7 8

Mutation I28L

−2

0

2

4

6

0 1 2 3 4 5 6 7 8

Mutation I28V

−2

0

2

4

6

log(

k obs

)

0 1 2 3 4 5 6 7 8

Mutation V55A

−2

0

2

4

6

Denaturant concentration ( GuHCl [M] )

0 1 2 3 4 5 6 7 8

Mutation V55M

−2

0

2

4

6

Denaturant concentration ( GuHCl [M] )

0 1 2 3 4 5 6 7 8

Mutation V55T

−2

0

2

4

6

Denaturant concentration ( GuHCl [M] )

log(

k obs

)

0 1 2 3 4 5 6 7 8

Wildtype

UC Santa Barbara

Rice University

UC Berkeley

Variability

−2

−1

0

1

2

φ

φ − values

UC Santa Barbara

Rice University

UC Berkeley

−10

−5

0

5

10

∆∆G

N−D

∆∆GN−D − values

Wild

type

−I2

8A

Wild

type

−I2

8L

Wild

type

−I2

8V

I28A

−I2

8L

I28A

−I2

8V

I28L

−I2

8V

Wild

type

−V

55A

Wild

type

−V

55M

Wild

type

−V

55T

V55

A−

V55

M

V55

A−

V55

T

V55

M−

V55

T

Variability

average ∆∆GN−D values

σ lab

Between lab φ − value standard deviation

1

2

3

4

5

6

7

8

910

11

12

1

2

3

4

5

6

7

8

9

10

11

12

Wild type−I28A

Wild type−I28L

Wild type−I28V

I28A−I28L

I28A−I28V

I28L−I28V

Wild type−V55A

Wild type−V55M

Wild type−V55T

V55A−V55M

V55A−V55T

V55M−V55T

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

0.00

0.25

0.50

0.75

1.00

1.25

1.50

Variability

−10 −5 0 5 10

−10

−5

0

5

10

∆∆GN−D

∆∆G

T−D

UC Santa Barbara

Rice University

UC Berkeley

Some Simulation

2 4 6 8 10 12

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

average ∆∆GN−D values

σ lab

Some Simulation

2 4 6 8 10 12

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

average ∆∆GN−D values

σ lab

Some Simulation

2 4 6 8 10 12

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

average ∆∆GN−D values

σ lab

Some More Simulations

∆∆GN−D values

φ

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

0.37

0

1

D ( φ | φ , ∆∆G )

Some More Simulations

φ

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

0.10

1

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

0.20

1

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

0.3

0

1

φ

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

0.4

0

1

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

0.5

0

1

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

0.6

0

1

average ∆∆GN−D values

φ

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

0.7

0

1

average ∆∆GN−D values

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

0.8

0

1

average ∆∆GN−D values

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

0.9

0

1

Phi-Value Estimation

∆∆GTD

6 8 10 12 14

∆∆GND

6 8 10 12 14 7.0 7.5 8.0 8.5 9.0

10.5

11.0

11.5

12.0

12.5

13.0

13.5

∆∆GTD

∆∆G

TD

�������� TD������ ND ��� �

1 � ��� TD

����� ND

6 � 1��� � �� �� � � 6 �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 ��� � � ������ ND � ��� � � �� ��� TD

�� ��� ND � �

������ TD � � �

Φ

0.55 0.60 0.65 0.70 0.75

Phi-Value Estimation

� � ���� � ��� ����"����%"�� # � � �� ��� �

� � ��� ����"����*"�� # � � ��� �! ?

0.0

0.2

0.4

0.6

0.8

1.0

Φ

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 foldingkinetics? Is there a relationship between an amino acid’s evolutionary conservationand 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 reportedto date have not provided any evidence that highly conserved residues aremore likely to participate in the protein folding nucleus than poorly conservedresidues.

� 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 experimentaldata (really aweful statistics).

� These people do not like us.