Lessons on structure   from the  structure of viruses Richard James Department of Aerospace Engineering and Mechanics [ema...
Bacteriophage T4: a virus that attacks bacteria November 1, 2007 AEM Bacteriophage T-4 attacking a bacterium:  phage at th...
Mechanism of infection November 1, 2007 AEM A 100nm bioactuator We focus on  the tail sheath (joint work with Wayne Falk) ...
Structure of T4 sheath November 1, 2007 AEM 1) Approximation of molecules using electron density maps Gives orientation an...
Structure of T4 sheath November 1, 2007 AEM 3) Helices II: formulas for the helices Let 2) Helices I: the 8/3 rule 3 conse...
Structure of T4 sheath November 1, 2007 AEM where , Parameters:
Objective structures <ul><li>M = 1:  objective atomic structure </li></ul>November 1, 2007 AEM <ul><li>is an  objective mo...
Preservation of species <ul><li>An objective molecular structure  preserves species  if </li></ul><ul><li>Only discrete st...
Examples <ul><li>Bravais lattice </li></ul>November 1, 2007 AEM <ul><li>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 ...
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...
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...
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  equilibriu...
Explicit formulas for all objective molecular structures November 1, 2007 AEM Iterate g 1 : More generally, Dayal, Elliott...
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 ...
3-term formula for objective molecular structures, abelian case  November 1, 2007 AEM Some structures generated by this fo...
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 <ul><li>For full quantum mechanics we d...
Finding equilibria by first principles <ul><li>Find the energy as a function of the  structural parameters  and seek local...
Example 2: Molecular dynamics November 1, 2007 AEM Proof: invariant solutions of MD using (joint work with Traian Dumitric...
Objective MD study of a carbon nanotube under torsion <ul><li>Three-body Tersoff potentials for carbon </li></ul><ul><li>T...
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 ...
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...
Large-scale transient mode November 1, 2007 AEM
Example 3: The  measurement of structure November 1, 2007 AEM <ul><li>The function  </li></ul><ul><ul><li>Eigenfunction of...
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Penrose Lecture, November 2007 by R.D. James

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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.

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  • Transcript of "Penrose Lecture, November 2007 by R.D. James"

    1. 1. Lessons on structure from the structure of viruses Richard James Department of Aerospace Engineering and Mechanics [email_address] November 1, 2007 Thanks: Kaushik Dayal, Traian Dumitrica, Ryan Elliott, Wayne Falk , Felix Hildebrand, Peter Kuchment, John Maddocks, Stefan M ü ller, Rob Phillips, Egon Schulte, Ellad Tadmor, Giovanni Zanzotto. Welcome: Amartya Sankar Banerjee
    2. 2. 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)
    3. 3. Mechanism of infection November 1, 2007 AEM A 100nm bioactuator We focus on the tail sheath (joint work with Wayne Falk) Thomasson and Raaij
    4. 4. 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
    5. 5. 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
    6. 6. Structure of T4 sheath November 1, 2007 AEM where , Parameters:
    7. 7. Objective structures <ul><li>M = 1: objective atomic structure </li></ul>November 1, 2007 AEM <ul><li>is an objective molecular structure if there are orthogonal transformations such that </li></ul>
    8. 8. Preservation of species <ul><li>An objective molecular structure preserves species if </li></ul><ul><li>Only discrete structures are of interest. </li></ul>November 1, 2007 AEM Can write the definition using a permutation: where is a permutation. is the species of atom j (any molecule)
    9. 9. Examples <ul><li>Bravais lattice </li></ul>November 1, 2007 AEM <ul><li>Multilattice (or, an arbitrary periodic structure) </li></ul>
    10. 10. Bacteriophage T4 tail sheath (extended to infinity) November 1, 2007 AEM describes the molecule
    11. 11. C 60 and most viral capsids November 1, 2007 AEM Icosahedral rotation group: choose
    12. 12. Torsion-tension-bending of a beam November 1, 2007 AEM
    13. 13. 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
    14. 14. Bravais lattice November 1, 2007 AEM FCC e 1 e 3 e 2
    15. 15. 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
    16. 16. Objective atomic structure November 1, 2007 AEM
    17. 17. 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
    18. 18. Quantum mechanical significance of objective structures: the atomic case November 1, 2007 AEM where
    19. 19. The molecular case November 1, 2007 AEM where
    20. 20. 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
    21. 21. Explicit formulas for all objective molecular structures November 1, 2007 AEM Iterate g 1 : More generally, Dayal, Elliott, James
    22. 22. Main theorem November 1, 2007 AEM Dayal, Elliott, James
    23. 23. 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: )
    24. 24. 3-term formula for objective molecular structures, abelian case November 1, 2007 AEM Some structures generated by this formula describes the molecule
    25. 25. Four molecule arrays, eight molecule arrays November 1, 2007 AEM
    26. 26. Pairs of rings November 1, 2007 AEM unstaggered staggered
    27. 27. Bilayers November 1, 2007 AEM
    28. 28. Molecular fibers November 1, 2007 AEM unstaggered staggered s u
    29. 29. Branden and Tooze, Introduction to protein structure November 1, 2007 AEM
    30. 30. Example 1: first principles computations of the energy of an objective structure <ul><li>For full quantum mechanics we do not know how to write a cell problem </li></ul><ul><li>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 </li></ul>November 1, 2007 AEM <ul><li>The same simplifications are possible for objective structures </li></ul><ul><ul><li>Use density functional theory </li></ul></ul><ul><ul><li>Replace periodic boundary conditions </li></ul></ul><ul><ul><li>by objective boundary conditions </li></ul></ul>
    31. 31. Finding equilibria by first principles <ul><li>Find the energy as a function of the structural parameters and seek local minima </li></ul>November 1, 2007 AEM every atom is in equilibrium <ul><li>Objective structures are the natural structures in which to seek collective properites </li></ul><ul><ul><li>Ferromagnetism </li></ul></ul><ul><ul><li>Ferroelectricity </li></ul></ul><ul><ul><li>Superconductivity </li></ul></ul>structural parameters, 3M dimensions energy
    32. 32. Example 2: Molecular dynamics November 1, 2007 AEM Proof: invariant solutions of MD using (joint work with Traian Dumitrica) Periodic MD Objective MD
    33. 33. Objective MD study of a carbon nanotube under torsion <ul><li>Three-body Tersoff potentials for carbon </li></ul><ul><li>Twist was controlled by controlling the group parameters (interesting open question: what generalized forces answer to variations of group parameters?) </li></ul><ul><li>The groups chosen were various subgroups associated to the two-term Abelian formula . For each subgroup a fundamental domain was found. </li></ul>November 1, 2007 AEM
    34. 34. 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
    35. 35. Effect of different choices of the fundamental domain November 1, 2007 AEM bifurcation diagram
    36. 36. Objective MD simulation of bending of a carbon nanotube November 1, 2007 AEM Is there a St. Venant’s principle at atomic level?
    37. 37. Large-scale transient mode November 1, 2007 AEM
    38. 38. Example 3: The measurement of structure November 1, 2007 AEM <ul><li>The function </li></ul><ul><ul><li>Eigenfunction of the translation group of a crystal </li></ul></ul><ul><ul><li>Eigenfunction of the Laplacian, i.e., steady solution of Maxwell’s equations </li></ul></ul><ul><li>Fourier transform </li></ul><ul><li>Plane wave source </li></ul><ul><li>Bragg Law </li></ul><ul><li>Peaks in the spectrum of emitted radiation </li></ul><ul><li>Procedures of x-ray crystallography </li></ul>Structure of matter as we know it = structure of crystals Analog for objective structures <ul><li>The function ? </li></ul><ul><ul><li>Eigenfunction of symmetry group </li></ul></ul><ul><ul><li>of an objective structure </li></ul></ul><ul><ul><li>Eigenfunction of the Laplacian, i.e., steady solution of Maxwell’s equations </li></ul></ul><ul><li>? - transform </li></ul><ul><li>Specially prepared incoming radiation (not plane waves) </li></ul><ul><li>Analog of the Bragg Law </li></ul><ul><li>Peaks in the spectrum of emitted radiation </li></ul><ul><li>New kind of x-ray machine </li></ul><ul><li>In-vivo x-ray diffraction? </li></ul>(joint work with Gero Friesecke)

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