michael bachmann - physik.uni-leipzig.de
TRANSCRIPT
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
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