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All-or-none protein-like folding of a homopolymer chain
Mark P. Taylor1, Wolfgang Paul2, and Kurt Binder2
1Dept. of Physics, Hiram College, Hiram, OH 2Institut für Physik, Johannes Gutenberg Universität,
Mainz, Germany
native
molten globule
denatured
native
denatured
CompPhys09: New Developments in Computational Physics, Leipzig, Nov. 26, 2009
Can a simple homopolymer model capture some essentials of protein folding?
Single Chain Conformational Transitions
Protein Native ⇔ Denatured
Homopolymer Globule ⇔ Coil
Energy
Entropy
compact ← expanded (both disordered)
continuous transition order ← disorder discontinuous transition E
ntropy
Energy
limited number of low E conformations
coil
"molten globule"
Finkelstein & Ptitsyn, "Protein Physics" (2002)
Multiple Transitions in a Simple Chain Model
Zhou, Hall, and Karplus found a "first-order" transition in a
model homopolymer
Model: Flexible Square-Well Chain Length: N = 64
Well Diameter: λ = 1.5 MC and DMD simulations
PRL 77, 2822 (1996)
Reduced Temperature
chain collapse
chain freezing
Specific H
eat E
nergy
Size
Analogy with Bulk Fluid: Solid ⇔ Liquid ⇔ Gas
Similar to Protein: Native⇔Globule⇔Denatured
freezing
collapse
peaks merge as N→∞
Protein-like "all-or-none" transition in a simple model?
Europhys Lett. 70, 628 (2005); J. Polym. Sci. B 44, 2542 (2006);
PRE 75, 060801(R) (2007)
Lattice Bond-Fluctuation Model Chain Length: N = 32-512
Interaction Range: λ = 1.225 Wang-Landau Simulation Method
Ten years later ... a new result: Rampf, Paul, and Binder find
evidence for a "direct freezing" transition in a simple model system.
Analogous to simple liquid: Solid ⇔ Gas
and single-step protein folding?: Native ⇔ Denatured
Question: can a finite-length homopolymer exhibit such an all-or-none transition?
SW Chain Model
Model has a discrete energy spectrum: En = nε (n = number of monomer-monomer interactions)
σ
1 2 3
4 5 6
L
Polymer:
built from simple monomers:
Model Parameters: ε = well depth (sets energy scale) σ = hard-sphere diameter L = fixed bond length (L = σ) λ = interaction range/σ T* = kBT/ε = reduced temperature
Can study this model for a continuous range of λ
For SW monomers: liquid phase is unstable if λ ≤ 1.25 Question: Does the SW chain exhibit similar behavior? Note: chain connectivity places an upper limit on entropy of "gas" phase
-1
0
1
2
1 2 3r/σ
Square-WellPotential
bond length L=σ
welldiameter: λ
u(r)/ε
monomer-monomer interaction
Density of States and Wang-Landau Sampling I
Starting w/ g(En)=1, H(En)=0 ∀ n, ƒ0 =e
Generate sequence of chain conformations using acceptance criteria:
Update DOS: g(En) → ƒm g(En)
Update visitation histogram: H(En) → H(En)+1
When histogram ~flat ... reduce modification factor: ƒm+1 = (ƒm )1/2 reset histogram to zero: H(En) = 0 ∀ n
€
Pacc (a→ b) = min 1, g(Ea )g(Eb )
iterate m levels
m=20 is standard
we need m>25
Density of States:
g(En) = volume of configurational phase space for energy state En
Thermodynamics:
microcanonical entropy: S(E) = kB lng(E)
canonical partition function: Z(T) = ∑ g(E)exp(-E/kBT)
*Wang & Landau, PRL 86, 2050 (2001); PRE 64, 056101 (2001).
Wang-Landau algorithm* ... an iterative simulation method to compute g(En):
Wang-Landau Sampling II
Success of the WL methods depends critically on underlying MC move set These "standard" moves easily sample most of configuration space:
Reptation Single bead crankshaft Pivot
Escobedo & de Pablo, JCP 102, 2636 (1995)
End-bridging ... However, we need this move to access the lowest energy regions of phase space:
This move requires weight factors in the acceptance criteria:
wa = naJa
a b # of bridgable sites in state a
Jacobian factor for state a
Single Chain DOS and Canonical Analysis
Canonical Analysis
Partition Function: Z = ∑ g(E) e–E/kT
Probability: P(E,T) = g(E) e–E/kT / Z
Average Energy: 〈E(T)〉 = ∑EP(E,T)
Specific Heat: C(T) = d〈E(T)〉/dT
-1500
-1000
-500
0
-1000 -500 0
SW ChainDensity of States
λ = 1.10
E / ε
ln[g
(E)/g
(0)]
N = 256128
6432
1
10
100
1000
0.0 0.2 0.4 0.6 0.8 1.0
Cv /
Nk B
T *
SW ChainSpecific Heatλ = 1.10
N = 32
64
128
256
freezing
collapse
Taylor, Paul, & Binder, PRE 79, 050801(R) (2009)
For N = 256: g(E) spans ~700 orders of magnitude!
Phase Behavior for Finite Length Chain?
1
10
100
1000
0.3 0.4 0.5 0.6 0.7 0.8
C(T
) / N
k B
T *
1.061.08
1.10
SW ChainN = 128
λ = 1.04
collapse
freezing
In the "canonical analysis", collapse and freezing specific heat peaks merge for small λ ...
... a "microcanonical analysis" can be used to distinguish these transitions
Microcanonical Analysis I
Phase transitions in a finite system determined from curvature of
the microcaonical entropy: S(E) = kB ln g(E)
Gross, "Microcanonical Thermodynamics" (2001) Junghans, Bachmann, & Janke, PRL 97, 218103 (2006)
Taylor, Paul, & Binder, PRE 79, 050801(R) (2009)
-1200
-900
-600
-300
0
-500 -250 0
SW ChainN = 128
E / ε
ln[g
(E)/g
(0)]
λ = 1.15
1.02
1.05
1.10
Density of States
-900
-600
-300
0 SW ChainN = 128, λ = 1.10
ln[g
(E)/g
(0)]
Entropy: S(E) = kB ln g(E)
0.5
1.0
1.5
2.0
2.5
3.0
-500 -250 0E / ε
T –1 (E
) = d
S(E
) / dE
"convex intruder" signals discontinuous transition
(chain freezing)
inflection point in T(E) signals continuous
transition (chain collapse)
Phase boundaries obtained via "equal areas" construction
Microcanonical Analysis II
Phase transitions in a finite system determined from curvature of
the microcaonical entropy: S(E) = kB ln g(E)
Gross, "Microcanonical Thermodynamics" (2001) Junghans, Bachmann, & Janke, PRL 97, 218103 (2006)
Taylor, Paul, & Binder, JCP 131, 114907 (2009)
-1200
-900
-600
-300
0
-500 -250 0
SW ChainN = 128
E / ε
ln[g
(E)/g
(0)]
λ = 1.15
1.02
1.05
1.10
Density of States
-10
0
10
20
-400 -200 0E / ε
c(E
)/Nk B
0.5
1.0
1.5
2.0
2.5
3.0 SW Chain
N = 128λ = 1.15
[ T *(
E) ]
–1 =
dS
(E)/d
E
µ-canonical specific heat
isolated peak in c(E): continuous transition
poles in c(E): stability limits of discontinuous transition
0.5
1.0
1.5
2.0
2.5
3.0
-500 -250 0E / ε
T –1 (E
) = d
S(E
) / d
E
T –1 (E
) = d
S(E
) / dE
1.5
2.0
2.5
3.0
-500 -400 -300 -200 -100 0E / ε
SW ChainN = 128
λ = 1.05
1.06
1.08
shifted upby 0.25
shifted downby 0.25
Phase transitions in a finite system determined from curvature of S(E)
chain collapse
Microcanonical Analysis III
Collapse transition is preempted by freezing for short-range interaction!
-900
-600
-300
0 SW ChainN = 128, λ = 1.10
ln[g
(E)/g
(0)]
Entropy: S(E) = kB ln g(E)
Freezing Transition Phase Boundaries
Single Chain Phase Diagram: T-λ Version
Protein-like all-or-none "folding"
0.2
0.4
0.6
0.8
1.0
1.2
1.00 1.10 1.20 1.30Interaction Range λ
T *
expandedcoil
collapsedglobule
frozencrystallite
SW ChainN = 128
Taylor, Paul, & Binder, J. Chem. Phys. 131, 114907 (2009)
0.2
0.4
0.6
0.8
1.0
1.2
1.00 1.10 1.20 1.30Interaction Range λ
T *
expandedcoil
collapsedglobule
frozencrystallite
SW ChainN = 128
Calculation:
range of interaction
Prediction:
Binder and Paul Macromolecules 41, 4537 (2008)
Taylor, Paul, & Binder J. Chem. Phys. 131, 114907 (2009)
triple point moves to λ = 1.15 in N → ∞ limit
Single Chain Phase Diagram: E-λ Version
-500
-400
-300
-200
-100
0
1.00 1.05 1.10 1.15 1.20 1.25Interaction Range: λ
Ene
rgy:
E / ε
Egs
crystallite
collapsedglobule
expandedcoil
SW ChainN = 128
unstable
metastable
meta- stable
Defining a physical folding temperature sets the model energy scale
-50
-40
-30
-20
-10
0
-500 -400 -300 -200 -100 0E / ε
(G –
Go)
/ ε 60 ºC
"unfolded"states
37 ºC
75 ºC
"folded"states
stability
barrier
Free Energy Profiles: G = E–TS(E)
transition state (nucleation-condensation)
Protein Thermodynamics: Free Energy Landscape
Here we take Tfold = 333 K (60 ºC) ⇒ ε = 1.5 kcal/mol
All-or-none folding of a N = 128 SW Chain
with λ = 1.05
See Chan and Kaya: Proteins 40, 543 & 637 (2000); 52, 510 (2003).
Taylor, Paul, & Binder, PRE 79, 050801(R) (2009)
Protein Thermodynamics: Calorimetry
-50
-40
-30
-20
-10
0
-500 -400 -300 -200 -100 0E / ε
(G –
Go)
/ ε 60 ºC
"unfolded"states
37 ºC
75 ºC
"folded"states
stability
barrier
Free Energy Profiles: G = E–TS(E)
transition state (nucleation condensation)
Experimental test for "two-state" folding: Equality of calorimetric and van't Hoff* enthalpies
0
200
400
600
800
1000
0.0
0.2
0.4
0.6
0.8
1.0
50 55 60 65 70T (ºC)
Cv /
Nk B
[U]ΔHcal /ε = (area) N = 306 ΔHvH /ε
= –T2 dK/dT = 307
F ⇔ U Keq = [U]/[F]
@ T = T1/2:
See Chan and Kaya: Proteins 40, 543 & 637 (2000); 52, 510 (2003).
Transition enthalpy gives "distance" between folded/unfolded states.
*experimentally this is determined from the height of Cv
-20
-15
-10
-5
0
2.12.22.32.42.5
unfolding folding
1/T*
–ΔG
‡/k
BT
= ln
(kra
te/a
)
increasingtemperature/denaturant
slope ≈ –ΔH ‡/ε
ΔHu‡/ε ≈ –183
ΔHƒ‡/ε ≈ 113
Chevronrollover
Protein Thermodynamics: Chevron Plot
-50
-40
-30
-20
-10
0
-500 -400 -300 -200 -100 0E / ε
(G –
Go)
/ ε 60 ºC
"unfolded"states
37 ºC
75 ºC
"folded"states
stability
barrier
Free Energy Profiles: G = E–TS(E)
transition state (nucleation condensation)
Experimental test for "two-state" folding: Barrier height via kinetics ...
Anti-Arrhenius/Arrhenius behavior for folding/unfolding.
barrier
room temp
Chevron slopes give "distance" from folded/unfolded states
to transition state.
Experimental Data for Chymotrypsin Inhibitor 2 (CI2)
-20
-15
-10
-5
0
2.12.22.32.42.5
unfolding folding
1/T*
–ΔG
‡/k
BT
= ln
(kra
te/a
)
increasingtemperature/denaturant
slope ≈ –ΔH ‡/ε
ΔHu‡/ε ≈ –183
ΔHƒ‡/ε ≈ 113
Chevronrollover
0
200
400
600
800
1000
0.0
0.2
0.4
0.6
0.8
1.0
50 55 60 65 70T (ºC)
Cv /
Nk B
[U]ΔHcal /ε = (area) N = 306 ΔHvH /ε
= –T2 dK/dT = 307
F ⇔ U Keq = [U]/[F]
@ T = T1/2:
128 bead SW chain
Heat Capacity
∆Hcal/∆HvH
=1.04
Jackson et al, Biochemistry 42,
11259 (1993)
83 residue protein
Folding/Unfolding Rates
Maxwell et al, Protein Sci. 14, 602 (2006)
[GuHCl] (M)
fast
er
slower
ln(k
obs)
T (ºC)
Exc
ess
Cp (
cal/d
eg/m
ol)
∆Hcal/∆HvH
=1.00
← folding unfolding →
Summary and Outlook
Funding: DFG (SFB 625-A3) NSF (DMR-0804370) Hiram College
Special thanks to the Binder group for their hospitality!
Flexible SW Chain Model
Findings: Short range interaction results in "all-or-none" folding. see: PRE 79, 050801(R) (2009); JCP 131, 114907 (2009) Reproduces many key aspects of protein thermodynamics.
To do: Heteropolymer model (such as HP type). [difficulty: bridging moves change sequence] Explore free energy landscapes.
Happy "American" Thanksgiving