Penrose Lecture, November 2007 by R.D. James

Slide 1: Lessons on structure from the structure of viruses Richard James Department of Aerospace Engineering and Mechanics james@umn.edu Thanks: Kaushik Dayal, Traian Dumitrica, Ryan Elliott, Wayne Falk, Felix Hildebrand, Peter Kuchment, John Maddocks, Stefan Müller, Rob Phillips, Egon Schulte, Ellad Tadmor, Giovanni Zanzotto. Welcome: Amartya Sankar Banerjee November 1, 2007

Slide 2: Bacteriophage T4: a virus that attacks bacteria 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) November 1, 2007 AEM

Slide 3: Mechanism of infection Thomasson and Raaij We focus on the tail sheath A 100nm bioactuator (joint work with Wayne Falk) November 1, 2007 AEM

Slide 4: Structure of T4 sheath 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 November 1, 2007 AEM

Slide 5: Structure of T4 sheath 2) Helices I: the 8/3 rule 3 consecutive molecules on the 8 consecutive molecules on the lowest annulus main helix For contracted sheath there is a similar 12/1 rule 3) Helices II: formulas for the helices Let November 1, 2007 AEM

Slide 6: Structure of T4 sheath , where Parameters: November 1, 2007 AEM

Slide 7: Objective structures is an objective s molecular structure if there are orthogonal transformations such that M = 1: objective atomic structure s November 1, 2007 AEM

Slide 8: Preservation of species 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 s Only discrete structures are of interest. s November 1, 2007 AEM

Slide 9: Examples Bravais lattice s Multilattice (or, an arbitrary periodic structure) s November 1, 2007 AEM

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

Slide 11: C60 and most viral capsids Icosahedral rotation group: choose November 1, 2007 AEM

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

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

Slide 14: Bravais lattice e3 FCC e2 e1 November 1, 2007 AEM

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

Slide 16: Objective atomic structure November 1, 2007 AEM

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

Slide 18: Quantum mechanical significance of objective structures: the atomic case where November 1, 2007 AEM

Slide 19: The molecular case where November 1, 2007 AEM

Slide 20: Equilibrium equations (atomic case) If one atom is in equilibrium then all atoms are in equilibrium Objective structures have free parameters: atomic case structural equilibrium structural equilibrium implies atomic equilibrium if November 1, 2007 AEM

Slide 21: Explicit formulas for all objective molecular structures Dayal, Elliott, James Iterate g1: More generally, November 1, 2007 AEM

Slide 22: Main theorem Dayal, Elliott, James November 1, 2007 AEM

Slide 23: Bacteriophage T4 tail sheath, revisited describes the molecule parameters Experimental values of the parameters satisfy, for both short or tall forms, (structural parameters: ) November 1, 2007 AEM

Slide 24: 3-term formula for objective molecular structures, abelian case describes the molecule Some structures generated by this formula November 1, 2007 AEM

Slide 25: Four molecule arrays, eight molecule arrays November 1, 2007 AEM

Slide 26: Pairs of rings unstaggered staggered November 1, 2007 AEM

Slide 27: Bilayers November 1, 2007 AEM

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

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

Slide 30: 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 s problem For simpler atomic models, e.g., Density Functional Theory (DFT), s we do, and this is what underlies the success of DFT: periodic BC for the density The same simplifications are possible for objective structures s – Use density functional theory – Replace periodic boundary conditions by objective boundary conditions November 1, 2007 AEM

Slide 31: Finding equilibria by first principles Find the energy as a function of the structural parameters and seek local s minima energy every atom is in equilibrium structural parameters, 3M dimensions Objective structures are the natural structures in which to seek collective s properites – Ferromagnetism – Ferroelectricity – Superconductivity November 1, 2007 AEM

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

Slide 33: Objective MD study of a carbon nanotube under torsion Three-body Tersoff potentials for carbon s Twist was controlled by controlling the group parameters s (interesting open question: what generalized forces answer to variations of group parameters?) The groups chosen were various subgroups associated to the s two-term Abelian formula. For each subgroup a fundamental domain was found. November 1, 2007 AEM

Slide 34: Objective MD: study of buckling of C nanotube under torsion t1 a b (12, 12) CNT b t2 a b 3 deg/Angstrom twist November 1, 2007 AEM

Slide 35: Effect of different choices of the fundamental domain bifurcation diagram November 1, 2007 AEM

Slide 36: Objective MD simulation of bending of a carbon nanotube Is there a St. Venant’s principle at atomic level? November 1, 2007 AEM

Slide 37: Large-scale transient mode November 1, 2007 AEM

Slide 38: Example 3: The measurement of structure Friesecke) (joint work with Gero Structure of matter as we Analog for objective know it = structure of crystals structures The function ? s The function s – Eigenfunction of symmetry group – Eigenfunction of the translation of an objective structure group of a crystal – Eigenfunction of the Laplacian, i.e., – Eigenfunction of the Laplacian, i.e., steady solution of Maxwell’s steady solution of Maxwell’s equations equations ? - transform Fourier transform s s Specially prepared incoming radiation Plane wave source s s (not plane waves) Bragg Law s Analog of the Bragg Law s Peaks in the spectrum of emitted s Peaks in the spectrum of emitted radiation s radiation Procedures of x-ray crystallography s New kind of x-ray machine s In-vivo x-ray diffraction? s November 1, 2007 AEM