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.

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

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

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

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

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

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. Objective atomic structure November 1, 2007 AEM

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

19. The molecular case November 1, 2007 AEM where

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

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

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

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

26. Pairs of rings November 1, 2007 AEM unstaggered staggered

27. Bilayers November 1, 2007 AEM

28. Molecular fibers November 1, 2007 AEM unstaggered staggered s u

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

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

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

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. Large-scale transient mode November 1, 2007 AEM