Lecture 1 for BIOS 203 Mini-course at Stanford University taught by Heather J. Kulik. http://bios203.stanford.edu for more info or email bios203.course@gmail.com

1. Introduction to simulations:
empirical minimizations
and potentials!
Heather J Kulik!
hkulik@stanford.edu!
2/25/13!

2. Welcome to BIOS 203!
• Objective: to introduce students to the theory and application of a
variety of computational simulations methods.!
• Instructors: Dr. Heather Kulik, Dr. Lee-Ping Wang, Professor Todd
Martinez, Professor Tom Markland, Professor Vijay Pande.!
• TAs: Martinez group members: Sofia Ismailov, Liguo Kong, Sara
Kokkila, Aaron Sisto, Fang Liu, Brendan Mar.!
• When: Monday, Wednesday, Friday 9am-11:50am (lecture
9am-10am, lab work 10am-11:50am).!
• Grade (letter students): two lab assignments + one presentation
(credit students): presentation is optional.!

3. Mini-course schedule!
Mon, Feb 25! Course introduction,
empirical potentials and
minimizations
(Dr. Heather Kulik)!
Wed, Feb 27! Electronic structure basics Fri, Mar 8! Guest lecture: Markov
(Dr. Heather Kulik)! state models (Prof. Vijay
Pande)!
Fri, Mar 1! Classical molecular Mon, Mar 11! Transition state theory and
dynamics (Dr. Lee-Ping rare event techniques
Wang)! (Dr. Heather Kulik)!
!
Mon, Mar 4! Ab initio molecular Wed, Mar 13! Guest lecture: Free energy
dynamics (Dr. Lee-Ping methods (Prof. Tom
Wang) ! Markland)!
Wed, Mar 6! Guest lecture: Excited Fri, Mar 15! Methods for bioinorganic
states, mixed methods chemistry, presentations
(Prof. Todd Martinez)! (Dr. Heather Kulik)!
Lab 1 due! Lab 2 due!
!

4. Optional background texts!
• F. Jensen “Introduction to Computational Chemistry” Wiley !
• W. Koch and M. C. Holthausen “A Chemistʼs Guide to Density Functional
Theory” Wiley-VCH!
• A. Szabo and N. S. Ostlund, “Modern Quantum Chemistry: Introduction to
Advanced Electronic Structure Theory” Dover!
!
• C. J. Cramer “Essentials of Computational Chemistry: Theories and Models”
Wiley!
• M. P. Allen and Tildesley “Computer simulations of Liquids” Oxford Science
Publishers!
• T. Schlick “Molecular Modeling and Simulation” Springer!
• D. A. McQuarrie and J. D. Simon “Physical Chemistry: A molecular
approach” University Science Books!
• D. Frenkel and B. Smit “Understanding Molecular Simulations: From
Algorithms to Applications” Academic Press!
!
Specific literature will be provided along with each lecture.!

5. Why simulations?!
Protein folding: how proteins fold and
misfold (Prof. Vijay Pande)!
Voelz, Bowman, Beauchamp, Pande. JACS (2010).!

6. Why simulations?!
Drug design: 2nd generation HIV protease inhibitor Kaletra!
See Cobb “Biomedical Computational Review” 2007 and references therein.!

7. Why simulations?!
Photochemistry for RNA bases: mechanisms for
alternative proton transfer between RNA bases.!
Golan et al. Nature Chem. (2012).!

8. Simulations beyond
biochemistry!
Materials Genome Project: Identifying elements that
substitute for each other, chemical trends…!
Hautier, G… Ceder, G. Chemistry of Materials (2010).!

9. Choosing a computational
model!
Empirical models – functional form with parameters
from experimental or other calculated data:!
!Pair potentials!
!Many body potentials!
More transferable!
More efficient!
! Best tool for the
Semi-empirical models – model Hamiltonians:! job? Depends on
!Tight binding! the job!!
!MNDO, AM1…!
!
Quantum mechanical models – approximations to the
Schrödinger equation:!
!Hartree-Fock!
!Density functional theory!
!Post-Hartree-Fock (Configuration interaction, MP2)!

10. Exercise: Choose a
computational journal article.!
• Letter grade students: select a journal
article that uses computation (at least 50% of
the article) that interests you.!
• Throughout the course, or consulting with the
lecturers, build up an understanding of the
material.!
• Last class: present and explain the methods
used in the article and the major findings.!
• Need inspiration? See us for a list of
suggested journal articles.!

11. Empirical model potentials!
1 N ! !
E = E 0 + # V ( Ri ! R j )
2 i, j"i
V(ΔR)!
• Species are repulsive for
small distances!
• Attractive for longer distances!
• Only need to calculate ΔR!
potentials for atoms within a
certain distance.!

12. Lennard-Jones Potential!
200
150 Argon! (! " $12 ! ! $6 +
VLJ = 4! *# & ' # & -
100 *" r % " r % -
) ,
50
VLJ! (K)
E/kB
0 σ!
-50
Ar parameters:!
-100 σ=3.345 Å!
ε! ε/KB=125.7 K! J. A. White,
-150 JCP (1999).!
2.0 4.0 6.0 8.0 10.0
rm! r (Å)

13. Morse Potential!
5
De!
4 H2!
H2 parameters:!
De=4.75 eV!
3
V(r) (eV)
β=1.93 1/Å
re=0.741 Å!
2
! ! (r!re ) 2
1
re! V (r) = De(1! e ) adapted from
McQuarrie
and Simon!
0
0.0 1.0 2.0 3.0 4.0
r (Å)

14. Morse potential: CH4 stretch!
adapted from
Jensen!

15. Limitations of pair potentials!
Counts only bonds, not organization:!
!
=!
Nbonds = 3! Nbonds = 3!

16. Limitations of pair potentials!
Counts only bond length, no orientation or
angular effects (e.g. ethylene):!
Pair potentials:!
C-H, C-C bonds.!
!
θ
θ
No treatment: !
H-C-H angle, !
H-C-C-H dihedral.!

17. Limitations of pair potentials!
Preference for high number/high density of
bonds formed to lower total energy:!
!
!
!
Bond energy for blue atom on left is four times the right.!
In real systems: more bonds means lower energy/bond.!

18. Adding angular terms!
H2O Bending! Harmonic
works well!
!
Note:
derivative
should be set
to zero at 180o!
adapted from
Jensen!

19. Adding torsional terms!
Ethane eclipsed! Ethane staggered! Torsional potential is
steric, non-bonded
electrostatics.!
!
!
!
!
!
!
!
Periodicity needs
Energy
to be enforced,
e.g. ethane.!
0 60 120 180 240 300 360
Dihedral (degrees)

20. AMBER!
Assisted Model Building with Energy Refinement:
AMBER is a force field and a software package.
http://www.ambermd.org!
1) AMBER Force field: ffXX (year) peptides and nucleic
acids, some ions (Mg2+).!
2) GAFF (Generalized Amber Force Field): generalized
scheme for force field for any organic molecule based
upon topology.!
3) Parameter sets available on the web:
http://www.pharmacy.manchester.ac.uk/bryce/amber/!

21. The AMBER force field!
Vn
E= " kb (l ! l0 )2 + " ka (! ! ! 0 )2 + " 2 [1+ cos(n" ! "0 )]+
bonds angles torsions
!(1)! ! ! !(2)! ! ! ! ! !(3)!
1, 2, 3: Harmonic oscillator-like bonding, angular, torsional terms!
(! $12 ! r $ + N'1 N q q
6
N'1 N (
N'1 N
r0ij Cij Dij +
. . !i, j *# r & ' 2 # r & - + . . 4"! r + . . * r12 ' r10 -
*# ij &
0ij
# &-
i j
j=1 i= j+1 )" % " ij % , j=1 i= j+1 0 ij j=1 i= j+1* ij
) ij -,
! ! !(4)! ! ! ! !(5)! ! ! ! !(6)!
van der Waals ! !electrostatic ! !hydrogen bonding!

22. Things that are missing…!
More advanced force fields:!
– Cross terms: e.g. bend affects stretch
in a water molecule. !
bend!
– Polarizability from point charges or
multipoles: e.g. AMOEBA.!
– Bond breaking and formation: e.g.
ReaxFF.!
!
Not all advanced methods are
feasible for very large simulations. !

23. Generalized Amber FF!
Same functional form as AMBER but more general
(http://ambermd.org/antechamber/gaff.html):!
!

24. GAFF parameters!
• Bond constants, force constants, torsional
dependence: based on assigned bonding
topology (you or a code decides).!
• Across large test set: MP2/6-31g*,
MP4/6-31g* and Cambridge Structural
Database training set.!
• Charges for electrostatics from semi-
empirical methods or HF/6-31g*. Can
generate custom charges for each molecule.!

25. Optimizing on a PES!
N atoms, 3N-6 degree
potential energy
hypersurface. Simplify!!
!
For classical MD: biggest
bottleneck is counting all
the bonds, need neighbor
lists to limit cost of
summation.!
Minimizations/optimizations: finding our way to local or
global minima!
from H. B. Schlegel, Wayne State U.!

26. Features of a PES!
from H. B. Schlegel, Wayne State U.!

27. Energy minimizations!
Minimize:! Force = !"(Energy)
Gradient on our PES in # "f "f &
terms of all coordinates !f = % ,!, (
(internal, cartesian):!
$ "x1 "xn '
Any stationary point
For !f != ! 0
! ! !:! (a) local minimum,
(b) global minimum,
What we usually
want!!
(c) saddle point.!
!

28. Steepest descent!
Optimization direction:! g = !"E
!i g i
Update coordinates:! ri+1 = ri +
! gi
Reduce λ near minimum.!
Pros: Fast when far from minimum,
local minimization guaranteed.!
!
Cons: slow descent for certain
PESs, can oscillate near minimum.!
From E. Eliav,
Tel Aviv U!

29. Conjugate gradient!
Initial direction:! h 0 = g 0 = !"E
Update coordinates:! ri+1 = ri + ! hi
Next steps:! hi+1 = g i+1 + ! i+1hi
!
where:! (g ! g )g
! i+1 = i+1 2i i+1
! gi
!
Pro: Using history, faster
convergence near minimum.!
From E. Eliav,
Tel Aviv U!

30. The Hessian/Force matrix!
(3N-6)
" %
!2 f !2 f !2 f '
(3N-6)x(3N-6) matrix $ !
with elements:! $ !x12 !x1!x2 !x1!xn '
$ '
(3N-6)
$ !2 f !2 f 2
! f '
2
! f 2
!
H ij ( f ) = H ( f ) = $ !x2!x1 !x2 !x2!xn '
!xi!x j $ '
$ " " # " '
$ !2 f !2 f !2 f '
$ ! 2 '
$ !xn!x1
# !xn!x2 !xn ' &
Approximate PES around stationary point by harmonic potentials.!

31. The Hessian/Force matrix!
Diagonalize Hessian Eigenvalues:! Eigenvectors:!
(eigenvalue problem)! 2 normal
n ! k = m" k coordinates!
!l (k )
k j = !H l
ij i
(k )
Harmonic
i=1 frequencies!
minimum! maximum! saddle point!
εk>0! εk<0! εk>0, except one εj<0!
From E. Eliav, ωk
all real! ωk
all imaginary! one imaginary ωj
on RC!
Tel Aviv U!

32. Minimizations… in practice!
• Explicit Hessian methods accurate but
too expensive for large molecules (9N2,
27N3). Approximations, updating.
!
• In AMBER: SD,CG, BFGS, following
low freq modes for struct. change.!
• May use only CG (gradient history ~
implicit treatment of Hessian)
!
• Combinatorial explosion: large
molecules (eg.rotatable bonds), many
possible conformers. Need better ways
to seek out global minimum: genetic
algorithms, Monte Carlo, etc.!

33. Follow-up reading!
• Force fields for biological systems:!
– J. W. Ponder and D. A. Case, “Force fields for protein simulations” Adv. Prot. Chem.
(2003).!
– A. D. Mackerell “Empirical force fields for biological macromolecules: Overview and
issues” J. Comput. Chem. (2004).!
• Non-biological force fields:!
– H. Balamane, T. Halicioglu, W. A. Tiller “Comparative study of silicon empirical
interatomic potentials” Phys. Rev. B. (1992).!
– M. Finnis, Interatomic Forces in Condensed Matter, Oxford University Press (2003).!
– M. J. Buehler, A. C. T. van Duin, W. A. Goddard, III “Multiparadigm modeling of
dynamical crack propagation in silicon using a reactive force field” Phys. Rev. Lett.
(2006).!
• Optimization for biological systems:!
– P. M. Pardalos, D. Shalloway, and G. Xue “Optimization methods for computing global
minima of nonconvex potential energy functions” J. Global Opt. (1994).!
– I. Kolossvary and W. C. Guida “Low mode search. An efficient, automated
computational method for conformational analysis: application to cyclic and acyclic
alkanes and cyclic peptides.” JACS (1996).!
– B. Das, H. Meirovitch, and I. M. Navon “Performance of hybrid methods for large-scale
unconstrained optimization as applied to models of proteins” J. Comput. Chem.
(2003).!
– K. Zhu, M. R. Shirts, R. A. Friesner, and M. P. Jacobson “Multiscale optimization of a
truncated Newton minimization algorithm and application to proteins and protein-ligand
complexes.” JCTC (2007).!