1. Overall rotation due to internal motions in the N-body dynamics of protein molecules F. J. Lin University of Southern California, Department of Mathematics, KAP 108, 3620 S. Vermont Ave., Los Angeles, CA 90089-2532 Background (cont’d.) Abstract Results (cont’d.) This approach separates the coordinates as well as the energies of overall rotation of a generalized Eckart frame in the center-of-mass frame and internal motion. This agrees with Jellinek and Li (1989) when J tot = 0. This approach applies to other Eckart generalized coordinates as well, e.g., internuclear distances. Examples of physical consequences of the coupling of overall rotation and internal motion are: (i)The recoil angle  R in triatomic dissociation for arbitrary total rotational angular momentum including internal motions is which agrees with Demyanenko et al. (1999) when J tot = 0. (ii) The scattering angle  for atom-diatomic molecule collisions with internal motions is which agrees with Cross and Herschbach (1965) when r = 0. Presented at the Institute for Mathematics and Its Applications Workshop on Protein Folding in Minneapolis, MN, held January 14 - 18, 2008; © 2008 F. J. Lin. This work describes the relationship between “internal motions” expressed in Eckart generalized coordinates q λ and overall rotation ΔΘ of a molecular frame in the center-of-mass frame. It shows that the net overall rotation does not necessarily vanish, i.e., ΔΘ ≠ 0, under methods (i) – (iii) due to the contribution arising from nonzero internal angular momentum. Application (ii): Minimizing RMSD through finite rotations with respect to a rigid initial configuration Total angular momentum for a rigid molecule  = angle of rotation =  rig q = Eckart generalized coordinate (constant for a rigid molecule) A = moment of inertia in terms of its coefficients = A(t i ) B = Eckart coefficient in terms of its coefficients =B (t i ) (  ) geom  is nonzero unless the molecule is rigid or has zero internal angular momentum. Application (iii): Setting total angular momentum to zero The total angular momentum for a nonrigid molecule is  = angle of rotation q = Eckart generalized coordinate A = moment of inertia in terms of its coefficients B = Eckart coefficient in terms of its coefficients (  ) dyn  when J tot = 0. (  ) geom  is nonzero when the internal angular momentum is nonzero. Eckart generalized coordinates The total angular momentum is The net rotation ΔΘ of a generalized Eckart frame is The molecular connection is (Lin, 2007). 1. R. J. Cross and D. R. Herschbach, Classical scattering of an atom from a diatomic rigid rotor, J. Chem. Phys. 43, 3530 – 3540 (1965). 2. A. V. Demyanenko, V. Dribinski, H. Reisler, H. Meyer, and C. X. W. Qian, Product quantum-state dependent anisotropies in photoinitiated unimolecular decomposition, J. Chem. Phys. 111, 7383 – 7396 (1999). 3. C. Eckart, Some studies concerning rotating axes and polyatomic molecules, Phys. Rev. 47, 552 – 558 (1935). 4. A. Guichardet, On rotation and vibration motions of molecules, Ann. Inst. Henri Poincaré, Phys. Théor. 40, 329 – 342 (1984). 5. J. Jellinek and D. H. Li, Separation of the energy of overall rotation in any N-body system, Phys. Rev. Lett. 62, 241 – 244 (1989). 6. F. J. Lin, Symplectic reduction, geometric phase, and internal dynamics in three-body molecular dynamics, Phys. Lett. A 234, 291 – 300 (1997). 7. F. J. Lin, Hamiltonian dynamics of atom-diatomic molecule complexes and collisions, Discrete Contin. Dyn. Syst., Suppl. 2007, 655 – 666 (2007). 8. F. J. Lin, Dynamics of the N-body problem. I: Molecular rotation due to internal motions, 2006/2007a. 9. F. J. Lin, Separation of overall rotation and internal motion in the N-body dynamics of protein molecules, 2007a. 10. J. E. Marsden, R. Montgomery, and T. Ratiu, Reduction, symmetry, and phases in mechanics, Mem. Am. Math. Soc. 88, No. 436, (American Mathematical Society, Providence, RI, 1990). 11. E. B. Wilson, Jr., J. C. Decius, and P. C. Cross, Molecular Vibrations: The Theory of Infrared and Raman Vibrational Spectra, (Dover, New York, 1980; republication of McGraw-Hill edition of 1955). 12. Y. Zhou, M. Cook, and M. Karplus, Protein motions at zero-total angular momentum: The importance of long-range correlations, Biophys. J. 79, 2902 – 2908 (2000). Overall rotation in a simulation of three- helix bundle protein dynamics At zero total angular momentum, the flexible protein molecule rotates by 42 degrees in 10 5 reduced time steps due to internal motions described in terms of bond lengths, bond angles, … (Zhou, Cook, and Karplus, 2000; see Fig. 3). Overall rotation of a triatomic molecule in a differential geometric study Under zero total angular momentum a purely vibrational motion, i.e., an internal motion, can take a molecule with a specified initial shape to a final configuration with the same shape but differing from the initial configuration by a net rotation (Guichardet, 1984). Results The condition of zero overall rotation due to internal motions is equivalent to separation of overall rotation and internal motion. (i)In terms of normal modes when J tot = 0, the net overall rotation vanishes, i.e., Δθ = 0, when the internal angular momentum vanishes. (ii) In terms of rotation with respect to a rigid initial configuration when J tot = 0, the net overall rotation vanishes when the internal angular momentum vanishes. (iii) In general when J tot = 0, the net overall rotation vanishes when the internal angular momentum vanishes. Traditionally, molecular motions are separated into translation, rotation, and internal motions (Wilson, Decius, and Cross, 1955). However, here are three examples where internal motions and overall rotation are coupled. Physical approach : A net overall rotation (of a generalized Eckart frame) in the center-of-mass frame may arise as a consequence of the conservation of total rotational angular momentum. (1) Express the total rotational angular momentum as the sum of orbital plus internal angular momenta. (2) Express the orbital angular momentum as the product of a moment of inertia and an angular velocity of overall rotation. (3) Integrate the angular velocity to obtain the net overall rotation due to internal motions. Differential geometrical approach : The net rotation is expressed as the holonomy of a connection (Marsden et al., 1990). Background Methods Objective Summary Discussion References For a protein molecule in vacuo, the net overall rotation due to flexibility is expressed in internal coordinates by using Eckart's decomposition of the total rotational angular momentum. Regardless of whether the total (rotational) angular momentum vanishes, the condition for zero overall rotation is zero orbital angular momentum. Previous approaches toward the elimination of overall rotation included (i) using normal modes, (ii) minimizing the root-mean-squared deviation (RMSD) through finite rotations with respect to an initial configuration, and (iii) setting the total angular momentum to zero. These three approaches neglected the contribution of nonzero internal angular momentum. While this approach (Lin, 2007, 2007a) is motivated by results in geometric mechanics (Marsden et al., 1990), the results agree with an experimental observation of a rotation of 20 degrees in triatomic photodissociation (Demyanenko et al., 1999) and a computational observation of an overall rotation of 42 degrees in the dynamics of protein molecules (Zhou et al., 2000). Rotation of the recoil angle in a NO 2 photodissociation experiment At zero total angular momentum, the rotation of the vector R (the change in angle of the recoil velocity of the O atom) due to the rotation of the vector r of the remaining diatomic NO fragment is described by Jacobi coordinates and is (Demyanenko,Dribinski,Reisler, et al., 1999; see Fig. 9). An example: Jacobi coordinates The total rotational angular momentum is with The net overall rotation Δθ R is (Lin, 2007). Application (i): Using normal modes The total angular momentum for a nonrigid molecule is  = angle of rotation q = Eckart generalized coordinate ≈ constant for normal mode A = moment of inertia in terms of its coefficients B = Eckart coefficient in terms of its coefficients (  ) geom  is nonzero when the internal angular momentum is nonzero.