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