simple toy models (an experimentalist’s perspective)
TRANSCRIPT
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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?
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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
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Bryngelson & Wolynes (1987) PNAS, 84, 7524
Landscape Roughness
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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
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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
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0.0
0.5
1.0
0.0 0.1 0.2
-15˚C
Fraction folded
Time (s)
h = 1.04±0.07
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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
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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
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Gap Size Correlates with theFolding Rates of Simple Models
Dinner, Abkevich, Shakhnovich & Karplus (1999) Proteins, 35, 34
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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.
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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
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Gillespie & Plaxco (2004) Ann. Rev. Bioch. Biophy., In press
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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
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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
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Cytochrome C
10
20
30
40
-30
-20
-10
0 2 4 6
R
g
(Å)
[GdnHCl] (M)
Ellipticity 288 nm (m
o
)
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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
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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
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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
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All “foldability” criterion optimal
1. Energy landscapes unmeasurably smooth
2. Energy gaps unmeasurably large
3. All within error of zero
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Plaxco, Simons & Baker (1998) JMB, 277, 985
-2
0
2
4
6
5 10 15 20 25
log(k)
Relative Contact Order (%)
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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
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Do subtle, topology-dependent kinetic
effects appear only in the absence of
confounding energy landscape issues?
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Go Polymers
• Native-centric energy potential
• Extremely smooth energy landscape
• Topologically complex
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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
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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
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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
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Protein folding is highly two-state
Fyn SH3 domain
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∆Gu = -3 kcal/mol
55 residue protein
Kohn, Gillespie & Plaxco, unpublished
-10123
Chemical Shift
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4 residue truncation
Kohn, Gillespie & Plaxco, unpublished
∆Gu ~ 2 kcal/mol
-10123
Chemical Shift
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The Topomer Search Model
Makarov & Plaxco (2003) Prot. Sci., 12, 17
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P(QD) <K>QD
kf = QD<K>QD
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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
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-2
0
2
4
0 25 50 75 100
( k
f
/Q
D
)
Q
D
log
kf QD<K>QD
r = 0.88
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Crowding Effects
Real Polymers
Gaussian Chains
Persistence lengthExcluded volume
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-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
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“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
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Minimum requirements for topology-dependent kinetics
1. Connectivity
2. Rapid local structure formation
3. Smooth landscapes
4. Cooperativity
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Go polymers are not cooperative
-10
-5
0
5
10
15
0.70.80.91
Contacts (Fraction Native)
Experimental
SimulationGF
(kB
T)
Δ
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-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 =
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-10
-5
0
5
10
15
0.70.80.91
GF
(kB
T)
Δ
( )Contacts Fraction Native
=3s
=s
=1s
Experimental
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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
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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
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AcknowledgementsUCSBBlake GillespieLara TownsleyJonathan Kohn Andrew JewettHoria Metiu
UT AustinDima Makarov
StanfordSeb DoniachIan MilletVijay Pande
Universita di FirenzeFabrizio Chiti
NIH, UC BioSTAR, ONR
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Acknowledgements
Dziekowac!
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