Nano based Cancer drug delivery
systems: Molecular Perspective
Muhammad Shadman (PhD)
Assistant Professor of Physical Chemistry
Department of Chemistry, Faculty of
Science,
University of Zanjan.
Molecular Dynamics Simulation
References:
1) A.R. Leach, “Molecular Modeling”, second edition, Prentice Hall
pp. 353-406.
2) D. Frenkel and B. Smit, “Understanding Molecular Simulations from
Algorithms to Applications”, second edition, Academic Press.
Chapters 4 and 6.
3) M.P. Allen and D. J. Tildesley, “Computer Simulation of Liquids”, 1991,
Oxford. Chapter 3.
4) The Art of Molecular Dynamics Simulation, D.C. Rapaport, Camb.
Univ. Press (2004)
5) Computer Simulation of Liquids, M.P. Allen and D.J. Tildesley, Oxford
(1989).
6) Theory of Simple Liquids, J.-P. Hansen and I.R. McDonald, Academic
Press (1986).
What is Molecular Dynamics?
• MD is the solution of the classical equations of motion for
atoms and molecules to obtain the time evolution of the
system.
• Applied to many-particle systems - a general analytical
solution not possible. Must resort to numerical methods
and computers
• Classical mechanics only - fully fledged many-particle timedependent quantum method not yet available
• Maxwell-Boltzmann averaging process for thermodynamic
properties (time averaging).
Introduction
Biological processes are commonly studied by experimental
techniques (X-ray, NMR, etc.). However, to gain deeper
insights in terms of atomic interactions we try to model
biological macromolecules (proteins, DNA, carbohydrates,
etc.) and simulate their behavior by Monte Carlo (MC)
methods or molecular dynamics (MD) techniques that obey
the rules of physics.
Before discussing MD let us refresh some basic notions from
high school physics.
Forces
Examples of F (F vector; F=|F|) :
1) Stretching a spring by a distance x: F= -fx, Hook’s Law
f- spring constant. F negative because it is opposite to x.
2) Gravitation force: F= kMm/r2 - m and M masses with
distance r; k - constant. On earth (R,M large), g=kM/R2
F=mg
3) Coulomb law: F=kq1q2/r2 , q1 and q2 charges.
Newton’s second law (F - resultant force):
dv
d 2x
F  ma  m
m 2
dt
dt
a, v, and x vectors; t time

Mechanical work W:
If a constant force is applied along distance d, W=Fd (F=|F|). More general, W=! F.dx.
Potential energy:
If mass m is raised to height h negative work is done, W = –mgh and the mass gains
potential energy,Ep= -W = +mgh - the ability to do
mechanical work: when m falls dawn, Ep is converted into
kinetic energy,
Ek = mv2/2, where v2/2=gh (at floor).
A spring stretched by d: Ep= -W = f! xdx = fd2/2
Two charges: Ep = kq1q2/r
In a closed system the total energy, Etot = Ep+ Ek is constant but Ep/Ek can change; e.g.,
oscillation of a mass hung on a spring.
11
Linear momentum
p=mv
The total momentum of a system of particles is equal to the momentum of a single
particle with the total mass of the system moving with the velocity of the center of
mass of the system, vCM
vCM = Σmivi / Σmi
In a system with no external forces the total momentum is conserved - the center
of mass moves in a straight line with constant speed.
Notice, the force exerted on a particle is the negative derivative of the potential
energy with respect to the coordinates.
F = dp/dt = - dEp/dx
MD simulations
Pair Potential:
rcut
  12   6 
V (r )  4      
 r   r  
Lagrangian:
N 1
1N
 
L(ri , vi )   mi vi 2    V (rij )
2 i
i j i
Lennard -Jones Potential
  12   6 
V (r )  4      
 r   r  
V(r)


r
rcut
Equations of Motion
d L   L
   
dt   vi   ri


mi ai  Fi
N 

Fi   f ij
Lagrange
Newton
j i
12
6








        
f ij   iV (rij )  24 2 2

rij

r 
rij   rij 
 ij  


LennardJones
Periodic Boundary Conditions
Minimum Image Convention
Use rij’ not rij
L
xij = xij - L* Nint(xij/L)
rcut
j’
i
j
Nint(a)=nearest integer to a
rcut < L/2
Integration Algorithms: Essential Idea
r’ (t+Dt)
r (t+Dt)
v (t)Dt
r (t)
f(t)Dt2/m
[r (t), v(t), f(t)]
[r (t+Dt), v(t+Dt), f(t+Dt)]
Time step Dt chosen to balance efficiency
and accuracy of energy conservation
Integration Algorithms (i)
2
 n 1
 n  n 1 t  n
ri
 2 ri  ri

Fi   ( t 4 )
mi
1  n 1  n 1

vi n 
(ri
 ri )   ( t 2 )
2 t
Verlet algorithm
Integration Algorithms (ii)
 n 1/ 2  n 1/ 2 t  n
vi
 vi
 Fi   (t 3 )
mi
 n 1  n
 n 1/ 2
ri  ri  tvi
  (t 4 )
Leapfrog Verlet Algorithm
Integration Algorithms
 n 1  n
 n t 2  n
4
ri  ri  tvi 
Fi   (t )
2mi
 n 1  n t  n  n 1
2
vi  vi 
( Fi  Fi )   (t )
2mi
Velocity Verlet
Algorithm
 n 1/ 2  n t  n
vi
 vi 
Fi
2mi
 n 1  n
 n 1/ 2
ri  ri  tvi
 n 1  n 1/ 2 t  n 1
vi  vi

Fi
2mi
As Applied
Verlet Algorithm: Derivation
Taylor' s expansions :
r (t  t )  r (t )  r(t )t  12 r(t )t 2  16 r(t )t 3  O(t 4 )
r (t  t )  r (t )  r(t )t  12 r(t )t 2  16 r(t )t 3 O(t 4 )
1
2
Add (1) and (2) :
r (t  t )  r (t  t )  2r (t )  r(t )t 2  O(t 4 )
or :
r (t  t )  2r (t )  r (t  t )  f (t )t 2 / m  O(t 4 )
Subtract (2) from (1) :
(3)
r (t  t )  r (t  t )  2r(t )t  O(t 3 )
or :
v(t)  r (t  t )  r (t  t )  / 2t  O(t 2 )
(4)
Key Stages in MD Simulation
Initialise
•Set up initial system
•Calculate atomic forces
Forces
•Calculate atomic motion
•Calculate physical properties
Motion
•Repeat !
•Produce final summary
Properties
Summarise
MD – Further Comments
Constraints and Shake
If certain motions are considered unimportant, constrained MD can be more
efficient e.g. SHAKE algorithm - bond length constraints
Rigid bodies can be used e.g. Eulers methods and quaternion algorithms
Statistical Mechanics
The prime purpose of MD is to sample the phase space of the statistical
mechanics ensemble.
Most physical properties are obtained as averages of some sort.
Structural properties obtained from spatial correlation functions e.g. radial
distribution function.
Time dependent properties (transport coefficients) obtained via temporal
correlation functions e.g. velocity autocorrelation function.
System Properties: Static (1)
• Thermodynamic Properties
• Kinetic Energy:
1 N
K . E.   mi vi2
2 i
• Temperature:
2
T
K . E.
3 Nk B
System Properties: Static (2)
• Configuration Energy:
Uc 
N
  V (rij )
i j i
• Pressure:
1
PV  NkBT 
3
• Specific Heat
 (Uc )
2
NVE
 
 rij  fij
N 1 N
i 1 j i
3 Nk B
3 2 2
 Nk B T (1 
)
2
2Cv
System Properties: Static (3)
• Structural Properties
• Pair correlation (Radial Distribution Function):
n( r )
V
g( r ) 
 2
2
4  r r N
• Structure factor:
S (k )  1  4  

0
N
   (r  rij )
i j i
sin( kr)
g (r )  1 r 2 dr
kr
Note: S(k) available from x-ray diffraction
Radial Distribution Function
 R
R
Typical RDF
g(r)
1.0
separation (r)
Free Energies?
•
All above calculable by molecular dynamics or Monte Carlo
simulation. But NOT Free Energy:
where
A(V , T )   k B T log e QN (V , T )
is the Partition Function.
But can calculate a free energy difference!
1
QN (V , T ) 
N !h3N
N N N N
exp


H
(
r
, p )dr dp
 
System Properties: Dynamic (1)
●
The bulk of these are in the form of Correlation Functions :
1
C (t ) 
T
T
  f (t   )  f  f ( )  f d
av
0
or
C (t )  f (t ) f (0)  f av
2
av
System Properties: Dynamic (2)
•
Mean squared displacement (Einstein relation)
•
Velocity Autocorrelation (Green-Kubo relation)
1
2 Dt  | ri (t )  ri (0) |2
3
1 
D   v i (t )  v i (0) dt
3 0
Typical MSDs
Liquid
Solid
time (ps)
Typical VAF
<vi(t).vi(0)>
1.0
0.0
t (ps)
What is coarse-graining?
1. coarse-grained - composed of or covered
with particles resembling meal in texture or
consistency; "granular sugar"; "the
photographs were grainy and indistinct"; "it
left a mealy residue" of textures that are
rough to the touch or substances consisting
of relatively large particles; "coarse meal";
"coarse sand"; "a coarse weave"
2. coarse-grained - not having a fine texture;
"coarse-grained wood"; "large-grained
sand"- of textures that are rough to the touch
or substances consisting of relatively large
particles; "coarse meal"; "coarse sand"; "a
coarse weave"
Based on WordNet 3.0, Farlex clipart collection. © 2003-2011
Princeton University, Farlex Inc.
In Physics, Molecular Science, Molecular
Modeling:
Coarse graining: reducing the representation of
a system or a phenomenon.
Physicists prefer the term renormalization.
Coarse graining: from micro- to macro-scale
Calcite: specimen from Shullsburg
District, Lafayette County, Wisconsin
Calcite: crystallographic structure
(courtesy of the Online Mineral Museum)
Fine-grain structure does not matter in such
applications…
A cross-section of a small portion of an E. coli bacterium
Flagellar
Flagellum motor
Cell wall
Enzymes
m-RNA
Ribosomes
HU protein
(bacterial
nucleosome)
Cytop;lasm
DNA
Nucleoid
© David S. Goodsell 1999.
Molecular Grahics Laboratory
Scripps Research Institute, La Jolla, CA
QM
QM/MM
Averaging over individual components
Individual
components
Atomisticallydetailed
All-atom
Unitedatom
Description
level
Coarse-grained
System level
(Networks)
PDEs to describe
reaction/diffusion
Network graphs
Residue
level
Molecule/
domain
level
Averaging over „less important” degrees of freedom
Fully-detailed
Global optimization of the energy surface of the N-terminal portion of
the B-domain of staphylococcal protein A with all-atom ECEPP/3 force
field + SRFOPT mean-field solvation model (Vila et al., PNAS, 2003,
100, 14812–14816)
Superposition of the native fold (cyan)
and the conformation (red) with the
lowest C RMSD (2.85 Å) from the
native fold
Energy-RMSD diagram
sidechain
rotation
helix
formation
protein
folding
10-15
10-12
10-9
10-6
10-3
100
femto
pico
nano
micro
milli
seconds
bond
vibration
all atom
MD step
loop
closure
folding of
-hairpins
Time step t for some standard MD packages
MD
Package
Explicit
Solvent
Implicit
Solvent
1 fs
2 fs
3 fs
4-5 fs
1 fs
2 fs
AMBERa
CHARMMb
TINKERc
a
http://amber.scripps.edu/
b
http://www.charmm.org/
c
http:// dasher.wustl.edu/tinker/
Folding proteins at x-ray resolution using a specially designed
ANTON machine (x-ray: blue, last frame of MD) simulation
(red): villin headpiece (left), a 88 ns of simulations, WW
domain (right), 58 ms of simulations. Good symplectic
algorithm; up to 20 fs time step.
D.E. Shaw et al., Science, 2010, 330, 341-346
Folding the WW domain with the UNRES coarse-grained approach: 280 ns
instead of 58 ms
Coarse-grained approaches and accuracy
1LQ7; lowest-energy structure obtained with UNRES; Ca rmsd=2.1 Å
Ołdziej et al., J. Phys. Chem. B, 108, 16950, 2004
What is a force field?
A set of formulas (usually explicit) and parameters to
express the conformational energy of a given class of
molecules as a function of coordinates (Cartesian, internal,
etc.) that define the geometry of a molecule or a molecular
system.
Features:
• Cheap
• Fast
• Easy to program
• Restricted to conformational
analysis
• Non-transferable
• Results sometimes
unreliable
All-atom empirical force fields: a very simplified
representation of the potential energy surfaces
Vn
1
1
d
o 2

o 2
E   ki (d i  d i )   ki ( i   i )    cos( n   )
2 bonds
2 angles
dihedral n 2
angles
12
6
0
0

 rij 
 rij  
qi q j

  ij    2  
r  
 rij 
i  j rij
 ij  

Applications of force fields
•Technique
•Application
•Local minimization
•Refinement
•Global minimization
•Searching most stable (?)
structures
•Monte Carlo methods
•Molecular dynamics and
extensions
•Thermodynamical
properties, ensemble
averages
•Dynamical properties
United-residue force fields
All-atom representation
of polypeptide chain in
solution (explicit water)
United-residue
(UNRES)
representation of
polypeptide chain
Coarse-grained force fields: a general formula
U  u
local
i
i
Local terms
   u ij   u ijkl...
i
j i
Pairwise terms
ijkl...
Multibody terms
Types of coarse-grained potentials
 Physics-based potentials
 smoothing the energy surface by treating rigid objects as
single interaction sites (e.g., the Kihara potentials),
 averaging out non-essential degrees of freedom,
 reproducing thermodynamic properties of small compounds.
 Statistical potentials based on the „Boltzmann principle”
 database information implicit in the potentials,
 database information explicit in the potentials.
 Arbitrary potentials (HP, HNP).
 Structure-based potentials
 native secondary structure or other components of structure
built in the potentials
 the native structure is the global minimum (Go-like
potentials).
 Elastic network potentials.
Applications of coarse-grained potentials
 Prediction of 3D structure (proteins, nucleic acids, their
complexes)
 Large time- and size-scale molecular dynamics simulations
 Computing thermodynamic properties and ensemble
averages (thermodynamics of folding)
BN Nanotubes
MoS2 Nanotubes
Metal Organic Frameworks
Camptothecin
Paclitaxel
Doxorubicin
Machine learning applications in
cancer prognosis and prediction
Nan-Cancer: Escherichia coli
The End