Information about Penrose Lecture, November 2007 by R.D. James

Published on November 8, 2007

Author: rdjames

Source: slideshare.net

James discusses what he calls “objective structures,” structures like carbon nanotubes, buckyballs and viral capsids that occur frequently in organic and inorganic materials. James has given a precise definition of these structures and has developed a methodology to compute all of them. This could lead to the discovery of new nanostructures with unusual collective properties.

Bacteriophage T4: a virus that attacks bacteria November 1, 2007 AEM Bacteriophage T-4 attacking a bacterium: phage at the right is injecting its DNA Wakefield, Julie (2000) The return of the phage. Smithsonian 31:42-6 F. Eiserling (with permission)

Mechanism of infection November 1, 2007 AEM A 100nm bioactuator We focus on the tail sheath (joint work with Wayne Falk) Thomasson and Raaij

Structure of T4 sheath November 1, 2007 AEM 1) Approximation of molecules using electron density maps Gives orientation and position of one molecule in extended and contracted sheath one molecule of extended sheath Data from Leiman et al., 2005

Structure of T4 sheath November 1, 2007 AEM 3) Helices II: formulas for the helices Let 2) Helices I: the 8/3 rule 3 consecutive molecules on the lowest annulus 8 consecutive molecules on the main helix For contracted sheath there is a similar 12/1 rule

Structure of T4 sheath November 1, 2007 AEM where , Parameters:

Objective structures M = 1: objective atomic structure November 1, 2007 AEM is an objective molecular structure if there are orthogonal transformations such that

M = 1: objective atomic structure

is an objective molecular structure if there are orthogonal transformations such that

Preservation of species An objective molecular structure preserves species if Only discrete structures are of interest. November 1, 2007 AEM Can write the definition using a permutation: where is a permutation. is the species of atom j (any molecule)

An objective molecular structure preserves species if

Only discrete structures are of interest.

Examples Bravais lattice November 1, 2007 AEM Multilattice (or, an arbitrary periodic structure)

Bravais lattice

Multilattice (or, an arbitrary periodic structure)

Bacteriophage T4 tail sheath (extended to infinity) November 1, 2007 AEM describes the molecule

C 60 and most viral capsids November 1, 2007 AEM Icosahedral rotation group: choose

Torsion-tension-bending of a beam November 1, 2007 AEM

Periodic Table of the Elements November 1, 2007 AEM 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 1 H He Hex Hex 2 Li Be B C N O F Ne Cub Hex Rhom Hex Hex Cub Cub Cub 3 Na Mg Al Si P S Cl Ar Cub Hex Cub Cub Mono Ortho Ortho Cub 4 K Ca Sc Ti V Cr Mn Fe Co Ni Cu Zn Ga Ge As Se Br Kr Cub Cub Hex Hex Cub Cub Cub Cub Hex Cub Cub Hex Ortho Cub Rhom Hex Ortho Cub 5 Rb Sr Y Zr Nb Mo Tc Ru Rh Pd Ag Cd In Sn Sb Te I Xe Cub Cub Hex Hex Cub Cub Hex Hex Cub Cub Cub Hex Tet Tet Rhom Hex Ortho Cub 6 Cs Ba * Hf Ta W Re Os Ir Pt Au Hg Tl Pb Bi Po At Rn Cub Cub Hex Cub Cub Hex Hex Cub Cub Cub Rhom Hex Cub Rhom Mono ? Cub

Bravais lattice November 1, 2007 AEM FCC e 1 e 3 e 2

Periodic Table: Bravais lattices November 1, 2007 AEM = not a Bravais lattice 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 1 H He Hex Hex 2 Li Be B C N O F Ne Cub Hex Rhom Hex Hex Cub Cub Cub 3 Na Mg Al Si P S Cl Ar Cub Hex Cub Cub Mono Ortho Ortho Cub 4 K Ca Sc Ti V Cr Mn Fe Co Ni Cu Zn Ga Ge As Se Br Kr Cub Cub Hex Hex Cub Cub Cub Cub Hex Cub Cub Hex Ortho Cub Rhom Hex Ortho Cub 5 Rb Sr Y Zr Nb Mo Tc Ru Rh Pd Ag Cd In Sn Sb Te I Xe Cub Cub Hex Hex Cub Cub Hex Hex Cub Cub Cub Hex Tet Tet Rhom Hex Ortho Cub 6 Cs Ba * Hf Ta W Re Os Ir Pt Au Hg Tl Pb Bi Po At Rn Cub Cub Hex Cub Cub Hex Hex Cub Cub Cub Rhom Hex Cub Rhom Mono ? Cub

Objective atomic structure November 1, 2007 AEM

Objective atomic structures November 1, 2007 AEM ? ? 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 1 H He Hex Hex 2 Li Be B C N O F Ne Cub Hex Rhom Hex Hex Cub Cub Cub 3 Na Mg Al Si P S Cl Ar Cub Hex Cub Cub Mono Ortho Ortho Cub 4 K Ca Sc Ti V Cr Mn Fe Co Ni Cu Zn Ga Ge As Se Br Kr Cub Cub Hex Hex Cub Cub Cub Cub Hex Cub Cub Hex Ortho Cub Rhom Hex Ortho Cub 5 Rb Sr Y Zr Nb Mo Tc Ru Rh Pd Ag Cd In Sn Sb Te I Xe Cub Cub Hex Hex Cub Cub Hex Hex Cub Cub Cub Hex Tet Tet Rhom Hex Ortho Cub 6 Cs Ba * Hf Ta W Re Os Ir Pt Au Hg Tl Pb Bi Po At Rn Cub Cub Hex Cub Cub Hex Hex Cub Cub Cub Rhom Hex Cub Rhom Mono ? Cub

Quantum mechanical significance of objective structures: the atomic case November 1, 2007 AEM where

The molecular case November 1, 2007 AEM where

Equilibrium equations (atomic case) November 1, 2007 AEM Objective structures have free parameters: structural equilibrium structural equilibrium implies atomic equilibrium if atomic case If one atom is in equilibrium then all atoms are in equilibrium

Explicit formulas for all objective molecular structures November 1, 2007 AEM Iterate g 1 : More generally, Dayal, Elliott, James

Main theorem November 1, 2007 AEM Dayal, Elliott, James

Bacteriophage T4 tail sheath, revisited November 1, 2007 AEM describes the molecule Experimental values of the parameters satisfy, for both short or tall forms, parameters (structural parameters: )

3-term formula for objective molecular structures, abelian case November 1, 2007 AEM Some structures generated by this formula describes the molecule

Four molecule arrays, eight molecule arrays November 1, 2007 AEM

Pairs of rings November 1, 2007 AEM unstaggered staggered

Bilayers November 1, 2007 AEM

Molecular fibers November 1, 2007 AEM unstaggered staggered s u

Branden and Tooze, Introduction to protein structure November 1, 2007 AEM

Example 1: first principles computations of the energy of an objective structure For full quantum mechanics we do not know how to write a cell problem For simpler atomic models, e.g., Density Functional Theory (DFT), we do, and this is what underlies the success of DFT: periodic BC for the density November 1, 2007 AEM The same simplifications are possible for objective structures Use density functional theory Replace periodic boundary conditions by objective boundary conditions

For full quantum mechanics we do not know how to write a cell problem

For simpler atomic models, e.g., Density Functional Theory (DFT), we do, and this is what underlies the success of DFT: periodic BC for the density

The same simplifications are possible for objective structures

Use density functional theory

Replace periodic boundary conditions

by objective boundary conditions

Finding equilibria by first principles Find the energy as a function of the structural parameters and seek local minima November 1, 2007 AEM every atom is in equilibrium Objective structures are the natural structures in which to seek collective properites Ferromagnetism Ferroelectricity Superconductivity structural parameters, 3M dimensions energy

Find the energy as a function of the structural parameters and seek local minima

Objective structures are the natural structures in which to seek collective properites

Ferromagnetism

Ferroelectricity

Superconductivity

Example 2: Molecular dynamics November 1, 2007 AEM Proof: invariant solutions of MD using (joint work with Traian Dumitrica) Periodic MD Objective MD

Objective MD study of a carbon nanotube under torsion Three-body Tersoff potentials for carbon Twist was controlled by controlling the group parameters (interesting open question: what generalized forces answer to variations of group parameters?) The groups chosen were various subgroups associated to the two-term Abelian formula . For each subgroup a fundamental domain was found. November 1, 2007 AEM

Three-body Tersoff potentials for carbon

Twist was controlled by controlling the group parameters (interesting open question: what generalized forces answer to variations of group parameters?)

The groups chosen were various subgroups associated to the two-term Abelian formula . For each subgroup a fundamental domain was found.

November 1, 2007 AEM 3 deg/Angstrom twist (12, 12) CNT a b t 1 t 2 b Objective MD: study of buckling of C nanotube under torsion a b

Effect of different choices of the fundamental domain November 1, 2007 AEM bifurcation diagram

Objective MD simulation of bending of a carbon nanotube November 1, 2007 AEM Is there a St. Venant’s principle at atomic level?

Large-scale transient mode November 1, 2007 AEM

Example 3: The measurement of structure November 1, 2007 AEM The function Eigenfunction of the translation group of a crystal Eigenfunction of the Laplacian, i.e., steady solution of Maxwell’s equations Fourier transform Plane wave source Bragg Law Peaks in the spectrum of emitted radiation Procedures of x-ray crystallography Structure of matter as we know it = structure of crystals Analog for objective structures The function ? Eigenfunction of symmetry group of an objective structure Eigenfunction of the Laplacian, i.e., steady solution of Maxwell’s equations ? - transform Specially prepared incoming radiation (not plane waves) Analog of the Bragg Law Peaks in the spectrum of emitted radiation New kind of x-ray machine In-vivo x-ray diffraction? (joint work with Gero Friesecke)

The function

Eigenfunction of the translation group of a crystal

Eigenfunction of the Laplacian, i.e., steady solution of Maxwell’s equations

Fourier transform

Plane wave source

Bragg Law

Peaks in the spectrum of emitted radiation

Procedures of x-ray crystallography

The function ?

Eigenfunction of symmetry group

of an objective structure

Eigenfunction of the Laplacian, i.e., steady solution of Maxwell’s equations

? - transform

Specially prepared incoming radiation (not plane waves)

Analog of the Bragg Law

Peaks in the spectrum of emitted radiation

New kind of x-ray machine

In-vivo x-ray diffraction?

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