Simple toy models

(An experimentalist’s perspective)

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Lattice Polymers

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Lattice Polymers

Do they predict absolute folding rates?

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Lattice Polymers

Do they predict relative folding rates?

Two-state folding rates

kf = 2 x 105 s-1 kf = 2 x 10-1 s-1

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Landscape Roughness

Energy Gap

Collapse Cooperativity

Putative rate-defining criterion

Bryngelson & Wolynes (1987) PNAS, 84, 7524

Landscape Roughness

Kinetics switch from single exponential:

A(t) = A0 exp(-t·kf)1/h

To stretched exponential:

A(t) = A0 exp(-t·kf)1/h

When Landscape Roughness Dominates Kinetics

Socci, Onuchic & Wolynes (1998) Prot. Struc. Func. Gen. 32, 136Nymeyer, García & Onuchic (1998) PNAS, 95, 5921Skorobogatiy, Guo & Zuckermann (1998) JCP, 109, 2528Onuchic (1998) PNAS, 95, 5921

The energy landscape of protein L

Gillespie & Plaxco (2000) PNAS, 97, 12014

0.0

0.5

1.0

0.00 0.05 0.10

37˚C

Fraction folded

Time (s)

h = 0.98 0.08

0.0

0.5

1.0

0.0 0.1 0.2

-15˚C

Fraction folded

Time (s)

h = 1.04±0.07

The pI3K SH3 domain

Gillespie & Plaxco (2004) Ann. Rev. Bioch. Biophy, In press

0

25

50

75

100

0 500 1000 1500 2000

Relative Fluorescence (-100%)

Time (s)

h = 1.004±0.008

The Energy Gap

“The necessary and sufficient condition for [rapid] folding in this

model is that the native state be a pronounced

global minimum [relative to other

maximally compact structures].”

Sali, Shakhnovich & Karplus (1994) Nature, 369, 248

Gap Size Correlates with theFolding Rates of Simple Models

Dinner, Abkevich, Shakhnovich & Karplus (1999) Proteins, 35, 34

The uniqueness of the native state indicates that it is significantly more stable than any other

compact state: the energy gap is generally too large to measure experimentally.

An Indirect Test

For many simple models,

Tm correlates with Energy Gap size

15-mers (B0 = -2.0) r = 0.73

15-mers (B0 = -0.1) r = 0.92

27-mers (B0 = -2.0) r = 0.89

27-mers (B0 = -0.1) r = 0.97

Dinner, Abkevich, Shakhnovich & Karplus (1999) Proteins, 35, 34Dinner & Karplus (2001) NSB, 7, 321

Gillespie & Plaxco (2004) Ann. Rev. Bioch. Biophy., In press

Collapse cooperativity

“The key factor that determines the foldability of

sequences is the single, dimensionless parameter

…folding rates are determined

by ””””

Thirumalai & Klimov (1999) Curr. Op. Struc. Biol., 9, 197

Thirumalai & Klimov (1999) Curr. Op. Struc. Biol., 9, 197

101

102

103

104

105

106

107

108

0 0.2 0.4 0.6 0.8

101

102

103

104

105

106

107

108

Cytochrome C

10

20

30

40

-30

-20

-10

0 2 4 6

R

g

(Å)

[GdnHCl] (M)

Ellipticity 288 nm (m

o

)

Protein Rate Reference

Cytochrome C 6400 s-1 Gray & Winkler, pers com.

Ubiquitin 1530 s-1 Khorasanizadeh et al., 1993

Protein L 62 s-1 Scalley et al., 1997

Lysozyme 37 s-1 Townsley & Plaxco, unpublished

Acylphosphatase 0.2 s-1 Chiti et al., 1997

See also: Jaby et al., (2004) JMB, in press

101

102

103

104

105

106

107

108

0 0.2 0.4 0.6 0.8

101

102

103

104

105

106

107

108

101

102

103

104

105

106

107

108

0 0.2 0.4 0.6 0.8

101

102

103

104

105

106

107

108

Millet, Townsley, Chiti, Doniach & Plaxco (2002) Biochemistry, 41, 321

All “foldability” criterion optimal

1. Energy landscapes unmeasurably smooth

2. Energy gaps unmeasurably large

3. All within error of zero

Plaxco, Simons & Baker (1998) JMB, 277, 985

-2

0

2

4

6

5 10 15 20 25

log(k)

Relative Contact Order (%)

When the energy gap dominates folding kinetics, none

of a long list of putatively important parameters,

including the “number of short- versus long-range

contacts in the native state*”, plays any measurable

role in defining lattice polymers folding rates.

*Sali, Shacknovich & Karplus (1994) “How does a protein fold?” Nature, 369, 248

Do subtle, topology-dependent kinetic

effects appear only in the absence of

confounding energy landscape issues?

Go Polymers

• Native-centric energy potential

• Extremely smooth energy landscape

• Topologically complex

Topology-dependence of Go folding

5.0

5.5

6.0

0.3 0.4 0.5

log(MFPT)

Relative Contact Order

r = 0.2; p = 0.06

The topomer search model

1. The chain is covalent

2. Rates largely defined by native topology

3. Local structure formation is rapid

4. Equilibrium folding is highly two-state

Local structure formationis not the rate-limiting step

-1 10

-2

0 10

0

1 10

-2

2 10

-2

3 10

-2

4 10

-2

5 10

-2

6 10

-2

7 10

-2

-100 0 100 200

∆ Absorbance

Time (ns)

Closure of10-residue loop

Oh, Heeger & Plaxco, unpublished

Protein folding is highly two-state

Fyn SH3 domain

∆Gu = -3 kcal/mol

55 residue protein

Kohn, Gillespie & Plaxco, unpublished

-10123

Chemical Shift

4 residue truncation

Kohn, Gillespie & Plaxco, unpublished

∆Gu ~ 2 kcal/mol

-10123

Chemical Shift

The Topomer Search Model

Makarov & Plaxco (2003) Prot. Sci., 12, 17

P(QD) <K>QD

kf = QD<K>QD

Testing the topomer search model

We can test the model if we assume that all sequence-

distant residues in contact in the NATIVE STATE

must be in proximity in the TRANSITION STATE

Sequence-distant: > 4-12 residues

Native contact: CCÅ

-2

0

2

4

0 25 50 75 100

( k

f

/Q

D

)

Q

D

log

kf QD<K>QD

r = 0.88

Crowding Effects

Real Polymers

Gaussian Chains

Persistence lengthExcluded volume

-2

0

2

4

0.0 0.5 1.0

( k

f

/Q

D

)

Q

D

N

-1

log

kf QD<K>QD/N

r = 0.92

Makarov & Plaxco (2003) Prot. Sci., 12, 17

“It is also a good rule not to put

overmuch confidence in

observational results that are put

forward until they have been

confirmed by theory.”

Paraphrasing Sir Arthur Eddington

theoretical

simulation

Minimum requirements for topology-dependent kinetics

1. Connectivity

2. Rapid local structure formation

3. Smooth landscapes

4. Cooperativity

Go polymers are not cooperative

-10

-5

0

5

10

15

0.70.80.91

Contacts (Fraction Native)

Experimental

SimulationGF

(kB

T)

Δ

-20

-10

0

0 10 20

Energy ( )

Number of Native Contacts

ε

=1 = =3 =5s s s s

-εQ

Q(1 - s)/QN + sQ

E =

-10

-5

0

5

10

15

0.70.80.91

GF

(kB

T)

Δ

( )Contacts Fraction Native

=3s

=s

=1s

Experimental

s = 2

6.5

7.0

7.5

0.3 0.4 0.5

log(MFPT)

Relative Contact Order

r = 0.71; p = 10-16

s = 3

7.5

8.0

8.5

9.0

0.3 0.4 0.5

log(MFPT)

Relative Contact Order

Jewett, Pande & Plaxco (2003) JMB, 326, 247

See also: Kaya & Chan (2003) Proteins, 52, 524

r = 0.76; p = 10-18

AcknowledgementsUCSBBlake GillespieLara TownsleyJonathan Kohn Andrew JewettHoria Metiu

UT AustinDima Makarov

StanfordSeb DoniachIan MilletVijay Pande

Universita di FirenzeFabrizio Chiti

NIH, UC BioSTAR, ONR

Acknowledgements

Dziekowac!