Transitions in Finite Systems

Michael BachmannCenter for Simulational Physics, The University of Georgia, Athens GA, USA,

UFMT, Cuiaba MT & UFMG, Belo Horizonte MG, Brazil

CompPhys13, University of Leipzig, 28-30 November 2013

Overview

Introduction

1. Crystallization of Polymers

2. Peptide Aggregation

3. Adsorption of Polymers at Solid Substrates

4. Protein Folding

Summary and Conclusions

1

Introduction

Exemplified small molecular system: Proteins

• Heterogeneous linear chains of 40...3000 amino acids• Geometric structure ⇔ Biological function• Structure formation ⇔ Structural phase transition?• But: no thermodynamic limit, no scaling, no transition points• Finite-size, surface, and disorder effects

Hemoglobin

Oxygen transport inred blood cells,

4 units, 550 atomsATP synthase

Synthesis of ATP from ADP,2 units, 40,000 atoms

Ribosome

Protein synthesis,

2 units, 200,000 atoms

Herpes

Icosahedral virus,105–106 atoms

2

Introduction

Definition of temperature

• canonical:

Theatbath ≡ T cansystem(〈E〉) =

(

∂Scan(〈E〉)

∂〈E〉

)

−1

N,V

,

Scan(〈E〉) =1

T cansystem(〈E〉)

[〈E〉 − F (T cansystem(〈E〉))]

• microcanonical:

Tmicrosystem(E) =

(

∂Smicro(E)

∂E

)

−1

, Smicro(E) = kB ln g(E)

thermodynamic limit: Tmicrosystem = T can

system

small systems: Tmicrosystem 6= T can

system, deviation due to finite-size effects

3

Introduction

Canonical analysis

Indicators of thermal activity:Peaks and “shoulders” offluctuating quantities

• specific heat,• susceptibility,• order parameter flucts.

Phases of semiflexiblepolymers ⇒

E = ELJ + EFENE + Ebend

w/ Ebend = κ∑

i[1 − cos(θi)]

θi . . . bending angle

D. T. Seaton, S. Schnabel, D. P. Landau, M.B., Phys. Rev. Lett. 110, 028103 (2013).

4

Introduction

Microcanonical analysis

• Central quantity: density of states g(E)

• Entropy: S(E) = kB ln g(E)

• Inverse temperature: β(E) = T−1(E) = ∂S(E)/∂E

S(E)

H(E)

Emin

∆Q

(a)

β(E)

βtr

EEo Etr Ed

so

sd

(b)

First-order scenario:

• Convex region:Eo < E < Ed

• Phase coexistence,latent heat ∆Q 6= 0

• Gibbs constructionHS(E) = S(Eo) + E/Td

• Transition temperatureTtr = [∂HS(E)/∂E]−1

• Entropy reduction∆S = HS(Etr) − S(Etr)

5

Introduction

Classification of transitions by inflection-point analysis

2.5

βABtr

βBCtr

3.5

4.0

4.5

-4.8 -4.7 eABtr eBC

tr-4.5 -4.3 -4.2 -4.1-6.0

-5.0

-4.0

-3.0

-2.0

-1.0

0.0

1.0

2.0

β(e

)=

dS

(e)/

de γ

(e)=

dβ(e)/d

e

e = E/N

∆qBCβ(e)

γ(e)

γ = 0

γABtr < 0

γBCtr > 0

A B C

β(E) = 1/T (E)

γ(E) = dβ(E)/dE

Transitions:

1st order:

γtr > 0 (∆qtr > 0)

2nd order:

γtr < 0 (∆qtr = 0)

S. Schnabel, D. T. Seaton, D. P. Landau, M.B., Phys. Rev. E 84, 011127 (2011).

6

Introduction

Density of states (flexible polymer with 309 monomers)

-3500

-3000

-2500

-2000

-1500

-1000

-500

0

500

-3 -2 -1 0 1 2 3 4

log

10[g

(E)]

log10(E − E0)

S. Schnabel, W. Janke, M.B., J. Comp. Phys. 230, 4454 (2011).

7

1. Crystallization of Polymers

Elastic model for flexible polymers

Vintra: Lennard-Jones type interaction between all pairs of monomers

Vbond: Finitely extensible nonlinear elastic (FENE) potential for theinteraction between bonded monomers

8

1. Crystallization of Polymers

Nucleation: Liquid-solid transitions

Basic cores:

anti-Mackay(hcp)

Mackay(fcc)

Overlayer growth:

S. Schnabel, T. Vogel, M.B., W. Janke, Chem. Phys. Lett. 476, 201 (2009).S. Schnabel, M.B., W. Janke, J. Chem. Phys. 131, 124904 (2009).

9

1. Crystallization of Polymers

Length-dependence of caloric temperatures T (E)(N = 13, . . . , 309)

0.1

0.2

0.3

0.4

0.5

-5.5 -5.0 -4.5 -4.0 -3.5 -3.0 -2.5

T(e

)

e = E/N• inflection points

black lines: N = 13, 55, 147, 309 (“magic” chain lengths)

10

1. Crystallization of Polymers

Size dependence of transition temperatures

0.05

0.20

0.30

0.40

0.50

0.10

13 20 30 40 55 80 100 147 200 309100

Ttr

(N)

N

0.64 −1.42

N1/3

• 1st-order, × 2nd-order transitions

⇒ Unique identification of transitions points and order possible!

S. Schnabel, D. T. Seaton, D. P. Landau, M.B., Phys. Rev. E 84, 011127 (2011).

11

1. Crystallization of Polymers

Interaction range dependence

(90-mer)

-1.0

−ǫsq

0.0

0.5

1.0

1.5

2.0

0.5 0.6 r1 r2 0.9 1.0 1.1 1.2 1.3 1.4 1.5

Vmod,r

LJ

(r),Vsq(r)

r

δ

−420 −360 −300 −240 −180 −120 −60

1.0

1.5

2.0

2.5

3.0

3.5

4.0

β(E

)

δ ≈ 0.220

δ ≈ 0.015

−420 −360 −300 −240 −180 −120 −60

E

−0.06

−0.04

−0.02

0.00

0.02

0.04

γ(E

)

δ ≈ 0.220 δ ≈ 0.015

Microcanonical indicators for collapse, freezing, and solid-solid transitions

12

1. Crystallization of Polymers

Phase diagram0.00 0.05 0.10 0.15 0.20 0.25

0.2

0.4

0.6

0.8

1.0

T

G

L

Sfcc/deca Sico−M

Sico−aM

0.01 0.02 0.03

0.25

0.28

0.31

0.34

0.00 0.05 0.10 0.15 0.20 0.25

δ

0.0

0.5

1.0

pnic

T = 0.2

nic = 0

nic ≥ 1

Merger of collapse and liquid-solid transition for short-range interaction

J. Gross, T. Neuhaus, T.Vogel, M.B., J. Chem. Phys. 138, 074905 (2013).

G = “gas”

L = “liquid”

S = “solid”

13

2. Peptide Aggregation

Coarse-grained model for the aggregation of proteins

• Heteropolymer chains of a sequence of amino acids (disorder!)

• Simple hydrophobic-polar protein aggregation model:

Vintra: interaction betweennon-bonded amino acids ofthe same chain

Vinter: interaction betweenamino acids of differentchains

Vbend: bending energy

Bond length is constant(stiff bonds)

14

2. Peptide Aggregation

Microcanonical analysis (4 chains)

0.16

0.18

0.20

Tagg

0.24

0.26

0.28

-0.6 -0.5 eagg -0.3 -0.2 -0.1 0.0 efrag 0.2

T(e

)

e = E/N

nucleation transition region

∆qag

greg

ate

phas

e

frag

men

tphas

e

subphase1

subphase2

subphase3

fragment

subphase 1

subphase 2

subphase 3

aggregate

• hierarchy of subphase transitions (→ finite-size effects)

• 1st-order phase transition: infinite number of subphase transitionsC. Junghans, W. Janke, M.B., Comp. Phys. Commun. 182, 1937 (2011).

15

2. Peptide Aggregation

Special example: GNNQQNY (4 chains)

1

1.5

2

2.5

3

3.5

100 120 140 160 180 200-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04

β(E

) γ(E

)

E

β(E)

γ(E)

γ = 0

• same behavior: hierarchy of subphase transitions

• O(10) CPU years using parallel tempering MC (Lund model)

M.B., unpublished.K. L. Osborne, M.B., B. Strodel, Proteins 81, 1141 (2013).

16

3. Adsorption of Polymers at Solid Substrates

Flexible polymer interacting with continuous substrate

E = 4

N−2∑

i=1

N∑

j=i+2

(

r−12ij − r−6

ij

)

+1

4

N−2∑

i=1

[1− cos (ϑi)]+ǫs

N∑

i=1

(

2

15z−9i − z−3

i

)

17

3. Adsorption of Polymers at Solid Substrates

Adsorption transition AE2⇔DE (20mer, εs = 5)

0.0

0.5

1.0

1.5

2.0

0.0 1.0 2.0 3.0 4.0 5.0

T

ǫs

s(e)

Hs(e)∆s(e)

∆ssu

rf

∆q

edesesepeads 1-3-4-5

e

∆s(e

)

s(e

)

0.20

0.15

0.10

0.05

0.00

0.5

0.0

-0.5

-1.0

-1.5

M. Moddel, W. Janke, M.B., PhysChemChemPhys 12, 11548 (2010); Macromolecules 44, 9013 (2011).18

3. Adsorption of Polymers at Solid Substrates

Polymer adsorption at nanowire; structural phase diagram ofground-state morphologies

B: barrellike (tube) C: clamshell

Gi: globular, included Ge: globular, excluded

T. Vogel, M.B., Phys. Rev. Lett. 104, 198302 (2010).19

3. Adsorption of Polymers at Solid Substrates

Thermodynamics of polymer–nanowire interaction

E

β(E

)

β = 0.19, T = 5.31

β = 0.23, T = 4.31

β = 0.30, T = 3.30ǫf = 1

2

3

4

ǫf = 5

N = 100, σf = 3/2

0-50-100-150-200-250

1.21

0.8

0.6

0.4

0.20

T. Vogel, M.B., Comp. Phys. Commun. 182, 1928 (2011).

1st order: adsorption; 2nd order: collapse

20

4. Protein Folding

Helix-coil transition, (AAQAA)n∆

S(E

)[k

B]

0.5

0.4

0.3

0.2

0.1

0

T−1µc (E)

T−1can(〈E〉can)

T−1c

T−

1µc,

T−

1can[k

B/E

]

0.95

0.85

0.75

Rg

[A] 9

8

7

dEsc/dE

dEhb/dE

E, 〈E〉can [E ]

dE

sc/d

E,

dE

hb/d

E

120100806040200

1.21

0.80.60.40.2

0

(a)

(b)

(c)

(d)

∆S

(E)

[kB]

0.5

0.4

0.3

0.2

0.1

0

Rg

[A]

29

25

21

17

dEsc/dE

dEhb/dE

E [E ]

dE

sc/d

E,

dE

hb/d

E

6005004003002001000

1

0.8

0.6

0.4

0.2

0tertiary

secondary

(a)

(b)

(c)

T. Bereau, M.B., M. Deserno, JACS 132, 13129 (2010).

n = 3: 1st order n = 15: 2nd order

21

4. Protein Folding

Helix-coil transition, helix bundle α3d

∆S

(E)

[kB]

0.5

0.4

0.3

0.2

0.1

0

T−1can(〈E〉can)

T−1µc (E)

T−1c

T−

1µc,

T−

1can[k

B/E

]

0.82

0.79

0.76

H(E)

θ(E)

E, 〈E〉can [E ]

θ(E

),H

(E)

350300250200150100500

32.5

21.5

10.5

0

T. Bereau, M. Deserno, M.B., Biophys. J. 100, 2764 (2011).22

Summary and Conclusions

• Goal: Understanding mechanisms of [molecular] structure formationprocesses

• Tool: Canonical and microcanonical statistical analysis

• Examples: Coarse-grained model for conformational transitions of elasticflexible homopolymers, aggregation transitions, adsorption of polymersat substrates, helix-coil transitions of peptides

• Result: Identification of structural transitions; 1st and 2nd order likebehavior ⇒ finite-size & surface effects

• Conclusion: Microcanonical entropy inflection-point analysis generalizestheory of cooperativity

• Outlook: Improving efficiency and accuracy by means of advancedsimulation and data analysis strategies; other applications

23

Advertisement

Thermodynamics andStatistical Mechanics ofMacromolecular Systems

• modern review of stat mech

• statistical data analysis

• error calculation

• Monte Carlo methods

• transitions of macromolecules:

− folding and crystallization

− aggregation

− adsorption at solid substrates

Cambridge University Press,to be released in Spring 2014

24