Energy Landscapes: From Molecules to Nanodevices

Objective: to exploit stationary points (minima and transition states) of the

PES as a computational framework (J. Phys. Chem. B, 110, 20765, 2006):

• Basin-hopping for global optimisation (J. Phys. Chem. A, 101, 5111 1997)

• Basin-sampling for global thermodynamics (J. Chem. Phys., 124, 044102, 2006)

• Discrete path sampling for global kinetics (Mol. Phys., 100, 3285, 2002)

For small molecules (left), all the relevant stationary points and pathways can

be located. Free energy surfaces for larger systems (right) require sampling.

Disconnectivity Graphs of ‘Funnelled’ Landscapes

The nonrandom searches that result in magic number clusters, crystallisation,

self-assembly, and protein folding are associated with a ‘palm tree’ organisa-

tion of the potential energy landscape (Phil. Trans. Roy. Soc. A, 363, 357, 2005).

This ‘funnelling’ pattern has been verified for various structure-seeking sys-

tems, including the LJ13 cluster, icosahedral shells composed of pentago-

nal and hexagonal pyramids, crystalline (Stillinger-Weber) silicon, and the

polyalanine ala16. Large systems can exhibit relatively simple phenomenology.

Landscapes for the P46 Model Protein (JCP, 111, 6610, 1999)

ǫ

ǫ

The global minimum of the off-lattice bead model B9N3(LB)4N3B9N3(LB)5L

is a four-stranded β-barrel, where B=hydrophobic, L=hydrophilic, and

N=neutral. The original system exhibits frustration, which is eliminated in the

corresponding Go model [and reduced by salt bridges (JCP, 121, 10284, 2004)].

Glassy Landscapes (J. Chem. Phys., 129, 164507, 2008)10ǫA

A10ǫA

A

Disconnectivity graphs for BLJ60 including only transition states for noncage-

breaking (top) and cage-breaking (bottom) paths. Changes in colour indicate

disjoint sets of minima. Cage-breaking transitions, defined by two nearest-

neighbour changes, define a higher order metabasin structure.

Basin-Hopping Global Optimisation (J. Phys. Chem. A, 101, 5111, 1997)

(NaCl)18Na+HYPA/FBP11

38 75 76

77

98

102 103 104

Non-icosahedral Lennard-Jones Clusters Binary LJ unit cell

corronene10 (H2O)20 H3O+(H2O)20 Eigen H3O

+(H2O)20 Zundel

Stockmayer

Thomson problem

Influenza Virus Hemagglutinin (J. Virology , 84, 11802, 2010)

Influenza virus is a variable pathogen, causing problems for vaccination.

Antigenic shift of the hemagglutinin glycoprotein, which binds sialic acid at

the cell surface, produces pandemics. Antigenic drift gives rise to epidemics.

Crystal structure is known for HK68 (H3N2). The single TY155 mutation,

from threonine to tyrosine led to a cluster transition from HK68.

Discrete Path Sampling (Mol. Phys., 100, 3285, 2002).

A AB B

I

aa bb

i

no intervening minima pi(t) = 0pa(t)pa′(t)

=peq

a

peq

a′

pb(t)pb′(t)

=p

eq

b

peq

b′

Phenomenological A ↔ B rate constants can be formulated as sums over

discrete paths, defined as sequences of local minima and the transition states

that link them, weighted by equilibrium occupation probabilities, peqb :

kSSAB =

1

peqB

∑

a←b

Pai1Pi1i2 · · ·Pin−1inPinbτ−1b p

eqb =

1

peqB

∑

b∈B

CAb p

eqb

τb

,

where Pαβ is a branching probability and CAb is the committor probability that

the system will visit an A minimum before it returns to the B region.

Kinetic Analysis by Graph Transformation (JCP, 124, 234110, 2006)

The graph transformation procedure is non-stochastic and non-iterative. Min-

ima, x, are progressively removed, while the branching probabilities and wait-

ing times in adjacent minima, β ∈ Γ, are renormalised:

P ′

γβ = Pγβ + PγxPxβ

∞∑

m=0

Pmxx = Pγβ +

PγxPxβ

1 − Pxx

, τ ′

β = τβ +Pxβτx

1 − Pxx

.

Each transformation conserves the MFPT from every reactant state to the

set of product states with an execution time independent of temperature:

kT/K ∆Fbarrier Nmin Nts NGT/s SOR/s KMC/s

298 5.0 272 287 8 13 85,138

298 4.5 2,344 2,462 8 217,830

1007 - 40,000 58,410 35 281 1,020,540

1690 - 40,000 58,410 39 122,242

Geometry Optimisation

Minimisation: Nocedal’s algorithm, LBFGS, with line searches removed.

Transition states: single-ended searches use hybrid eigenvector-following

(Phys. Rev. B, 59, 3969, 1999; Chem. Phys. Lett., 341, 185, 2001), double-ended searches

use the doubly-nudged elastic band approach (J. Chem. Phys., 120, 2082, 2004).

The GMIN (global optimisation), OPTIM (transition states and pathways) and

PATHSAMPLE (discrete path sampling) programs are available under the Gnu

General Public License. Access to the svn source can be arranged for devel-

opers. Current svn tarball image: http://www-wales.ch.cam.ac.uk.

Interfaces to many electronic structure codes are included. Example:

• Ammonia synthesis and dissociation on Fe{211}: the Haber-Bosch process.

We predict that atomic nitrogen can be hydrogenated above around 340 K,

with ammonia being evolved at temperatures above 570-670 K.

A Modified Superposition Approach (Chem. Phys. Lett., 466, 105, 2008)

Here the partition function is broken down into contributions from local min-

ima and pathways as a function of order parameter a, with terms like

Zi(a, T ) =

(

kT

hνi

)3N−6exp (−Vi/kT )√

2πkTAi

exp

[

−(a − ai)

2

2kTAi

]

,

where Ai is a weighted sum of order parameter derivatives.

Free energy surfaces for alanine dipeptide (CHARMM22/vacuum) from su-

perposition, replica exchange, and reaction path Hamiltonian superposition:

The ‘filling in’ problem for barrier regions in low-dimensional projections due

to overlapping distributions can be avoided using disconnectivity graphs.

The effect of regrouping for a barrier threshold of 3 kcal/mol is shown below

for AMBER(ff03)/GBOCB (left) and compared with the CHARMM22/vacuum

surface (right). Free energy of group J : FJ(T ) = −kT ln∑

j∈J Zj(T ) with

F†LJ(T ) = −kT ln

∑

l←j

Z†lj(T ), and kLJ(T ) =

kT

he−[F

†LJ

(T )−FJ (T )]/kT .

Folding of Beta3s

−620

−630

−640

−650

−660

−670

−680

−460

−480

−500

−520

−5400 500 1000 1500 2000

s/A

Etot

Enb

−686

−682

−678

−674

−670

−666

−662

−658

−654

−650

−646

−642

−638

−634

−630

−626

−622

V(k

cal/

mol)

Beta3s is a designed 20-residue peptide with a three-stranded antiparallel

β-sheet. Folding with CHARMM19/EEF1 involves early formation of the

C-terminal hairpin followed by docking of the N-terminal strand.

Mean first passage time is 300 ns at 298 K, consistent with other calculations

and the experimental upper bound of 4000 ns (J. Phys. Chem. B, 112, 8760, 2008).

A Knotted Protein (PLoS Comput. Biol., 6, e1000835, 2010)−136

−138

−140

−142

−144

−146

−148

−150

−152

−154

−156

−158

−160

−162

−164

−166

−168

−170

−172

−174

−110

−120

−130

−140

−150

−160

−170

−180

0 500 1000 1500 2000 2500 3000 3500

The tRNA methyltransferase protein 1UAM contains a deep trefoil knot.

The folding pathway has two slipknot-type steps for a truncated (residues

78–135) Go model representation using an associated memory Hamiltonian.

The estimated rate constant is between 0.04 and 0.4 s−1.

Self-Assembly of Icosahedral Shells (PCCP, 11, 2098-2104, 2009)

R1 R2Rax

r

Palm tree disconnectivity graphs with Ih global minima are found for

T = 1 and T = 3 shells constructed from pentagonal and hexagonal pyramids.

Landscapes of this form are associated with good structure-seekers.

24 Pentagonal Pyramids

For the same parameters two T = 1 icosahedra are similar in energy to a

single shell based on a snub cube. Polyoma virus capsid protein VP1 forms a

left-handed snub cube from alkaline solution in the absence of the genome.

Emergent Behaviour from Simple Models (ACS Nano, 4, 219, 2010)

Adding two repulsive axial Lennard-Jones sites to an ellipsoidal core produces

remarkably versatile building blocks. Oblate ellipsoids favour shells, while

stronger repulsion for the longer semiaxis produces tubes and spirals.

Global minima for the oblate ellipsoids include icosahedra for N = 12, 32 and

72 (T = 1, 3 and 7), the snub cube observed for polyoma virus capsids at

N = 24, and conical, biaxial, prolate, and oblate shells at other sizes.

r1 r2

r

d h1

h2

N = 35

N = 37

N = 38

NN

V/N

−1.6−1.6

−1.7−1.7

−1.8−1.8

−1.9−1.9

−2.0−2.0

−2.1−2.1

−2.2−2.2

−2.3−2.3

−2.4−2.4

−2.5−2.51010 2020 3030 4040 5050 6060 7070 8080

Modelling Mesos opi Stru tures (ACS Nano, 4, 219, 2010)

top

side

Mixing ellipsoidal building blocks that favour shells and tubes produces struc-

tures with distinct head and tail regions (left): the Frankenphage.Particles with a Lennard-Jones site buried in the ellipsoid assemble into a

spiral structure (right) with parameters similar to tobacco mosaic virus.

V

s50 100 1500

−402

−406

−410

−414

−418

1

1

2

3

4

4 5 6

7

7 8 9

10

10 11

12

13

13

Nanodevices (Soft Matter, 7, 2325, 2011)

Coupled linear and rotary motion has been characterised for a helix composed

of 13 asymmetric dipolar dumbbells in the presence of an electric field.

The helix changes handedness as the boundary between segments propagates

along the strand via successive steps that switch the dumbbells.

Connecting Dynamics and Thermodynamics (Science, 293, 2067, 2001)

The organisation of a PES is governed by its stationary points, where Taylor

expansions provide local descriptions in terms of Hessian matrices.

The organisation of families of PES’s as a function of parameters in the

potential is determined by the stationary points that possess additional zero

Hessian eigenvalues, known as non-Morse points.

Catastrophe theory provides a local representation of the PES around non-

Morse points as a function of both atomic coordinates and parameters.

The splitting lemma reduces the dimensionality to the essential variables,

while transversality guarantees that the resulting classifications are universal.

The simplest one-parameter catastrophes are the fold, f(x) = 1

3x3 + ax, and

the symmetrical cusp, f(x) = 1

4x4 + 1

2ax2.

Geometries of the fold and cusp catastrophes.

curvature= λ curvature= λ

curvature= −λ curvature= −2λ

∆s ∆s

∆V ∆V

fold: rf =6∆V

λ∆s2

= 1 cusp: rc =4∆V

λ∆s2

= 1

ener

gy

ener

gy

distancedistance

LJ19 (NaCl)35Cl− bulk glass-formersrf

〈|rf−

1|〉

cum

ulative

probab

ility

0.001 0.01 0.1 1 100.010.01 0.10.1 11 1010

0.1

0.1

0.1

0.10.1

1 11

10 1010

0.00.0 0.50.5 1.01.0 1.51.5 2.02.00 00

1 11

0 1 2 3 4

0.0 0.00.0

0.2

0.2

0.2

0.3

0.4

0.4

0.6

0.8

path length/pair equilibrium separation

For systems with a fixed potential we effectively have a snap-shot of parameter

space. On average, rf remains close to unity for many pathways in both model

clusters and bulk, providing an explanation for Hammond’s postulate.