Hierarchical Modeling of Biomolecular
Systems: From Microscopic to Macroscopic
Simulations
VAGELIS HARMANDARIS
Department of Applied Mathematics
University of Crete, and FORTH, Heraklion, Greece
Cell Biology and Physiology: PDE models, 05/10/12
Outline
 Introduction: General Overview of Biomolecular systems. Characteristic
Length-Time Scales.
 Multi-scale Particle Approaches: Microscopic (atomistic), Mesoscopic
(coarse-grained) simulations, Macroscopic PDEs.
 Applications:
 Self-assembly of Peptides through Microscopic Simulations.
 Elasticity of Biological Membranes through Mesoscopic Simulations.
 Conclusions – Open Questions.
INTRODUCTION - MOTIVATION
 Systems
 biological macromolecules (cell membrane, DNA, lipids)
 Applications
 Nano-, bio-technology (biomaterials in nano-dimensions)
Biological processes
Time – Length Scales Involved in Biomolecular Systems
 Bond length ~ 1 Å (10-10 m)
 Radius of gyration ~ 1-10 nm (10-9 m)
 Self-assembly of biomolecules ~ 10 μm (10-5 m)
 Multi-compartment biological systems (e.g. cell) ~ 1 mm (10-3 m)
Time – Length Scales Involved in Polymer Composite Systems
 Bond vibrations: ~ 10-15 sec
 Angle rotations: ~ 10-13 sec
 Dihedral rotations: ~ 10-11 sec
 Segmental relaxation: 10-9 - 10-12 sec
 Maximum relaxation time of a biomacromolecule, τ1: ~ 1 sec (in Τ < Τm)
 Dynamics of multi-component system: ~days
THEORIES & COMPUTER SIMULATIONS:
-- probe microscopic structural features
-- organization of the adsorbed groups
-- dynamics at the interface
-- study in the molecular level
Hierarchical Modeling of Molecular Materials
D) description in macroscopic continuum level
C) description in mesoscopic
(coarse-grained) level
Β) description in microscopic
(atomistic) level
Α) description in quantum
level
 Main goal:
Built rigorous “bridges” between different simulation levels.
Quantitative prediction of properties of complex biomolecular systems.
Microscopic – Atomistic Modeling: Molecular Dynamics Simulations
Molecular Dynamics (MD) [Alder and Wainwright, J. Chem. Phys., 27, 1208 (1957)]
Classical mechanics: solve classical equations of motion in phase space (r, p).
 System of 3N PDEs (in microcanonical , NVE, ensemble):


 
iL  K , H    t ri
 Fi


r

p
i 1 
i
i
N
Liouville operator:
The evolution of system from time t=0 to time t is given by : (t )  exp iLt  (0)
pi
 t ri 
mi
U  r1 , r2 ,..., rN 
 t pi  
 Fic
ri
Hamiltonian (conserved quantity):
H NVE
pi 2
 K V  
 V (r)
i 2mi
Molecular Interaction Potential (Force Field): Atomistic Simulations
 Important question: What is the potential energy function?
U  R : U r1, r2 ,..., rN 
 Assumption - The complex quantum many-body interaction can be:
1) Described by semi-empirical functions.
2) Decomposed into various components.
Molecular model: Information for the functions describing the molecular interactions
between atoms.
U  R  Vbonded  R  Vnonbonded  R  Vext
Vbonded: Interaction between atoms connected by one or a few (3-5) chemical bonds.
Vnon-bonded: Interaction between atoms belonging in different molecules or in the
same molecule but many bonds (more than 3-5) apart.
Vext: External potential (force) acting on atoms.
Molecular Interaction Potential (Force Field): Atomistic Simulations
Vbonded r   Vstr  Vbend  Vtors
 stretching potential
1 (
2
V
l
)
o
s
t
rk
s
t
rl
2
 bending potential
1
2
V

k
(



)
o
b
e
n
d 2
b
e
n
d
 dihedral potential
i()
V
c
o
s


t
o
r
s
ic
5
i

0
Vnonbonded r   VLJ  Vq  Vhybrid
Van der Waals (LJ)
 non-bonded potential
 1

2
6





V
4






L
J
r
 
r








Coulomb
Vq 
qi q j
εrij
 Potential parameters are obtained from more detailed simulations or fitting to
experimental data.
MULTISCALE – HIERARCHICAL MODELING OF BIOMOLECULAR SYSTEMS
Limits of Atomistic Molecular Dynamics Simulations (with usual computer
power):
-- Length scale: few (4-5) Å - (10 nm)
-- Time scale:
few fs - (0.5 μs)
-- Molecular Length scale (concerning the global dynamics):
up to ~ 10.000 – 100.000 atoms
Need:
 Study phenomena in broader range of time-length scales
 Study more complicated systems.
 COARSE-GRAINED MESOSCOPIC MODELS
 Integrate out some degrees of freedom as one moves from finer to coarser
scales.
GENERAL PROCEDURE FOR DEVELOPING MESOSCOPIC PARTICLE MODELS
DIRECTLY FROM THE CHEMISTRY
1. Choice of the proper mesoscopic description.
-- number of atoms that correspond to a ‘super-atom’
(coarse grained bead)
2. Microscopic (atomistic) simulations of short chains (oligomers) for short
times.
3. Develop the effective mesoscopic force field using the atomistic data.
4. CG (MD or MC) simulations with the new CG model.
Re-introduction (back-mapping) of the atomistic detail if needed.
DEVELOP THE EFFECTIVE MESOSCOPIC CG POLYMER FORCE FIELD
CG
CG
CG
Utotal
(Q)  Ubonded
(Q)  Unon
bonded (Q)
BONDED POTENTIAL
 Degrees of freedom: bond lengths (r), bond angles (θ),
dihedral angles ()
r
PROCEDURE:
 From the microscopic simulations we calculate the distribution functions of the
degrees of freedom in the mesoscopic representation, PCG(r,θ,).
 PCG(r,θ, ) follow a Boltzmann distribution:
P
 Assumption:
 Finally:
CG
 U CG (r , ,  ) 
 r , ,   exp 

kT


PCG  r, ,   PCG  r  PCG   PCG  
U CG ( x, T )  kBT ln PCG  x, T  ,
(x  r, , )
NONBONDED INTERACTION PARAMETERS: REVERSIBLE WORK
 CG Hamiltonian – Renormalization Group Map:
e
  U nbCG ( q ,T )
 e
  U AT ( r ,T )
PN  dr | q 
q
Reversible work method [McCoy and Curro, Macromolecules, 31, 9362 (1998)]
 By calculating the reversible work (potential of mean force) between the centers of
mass of two isolated molecules as a function of distance:
e  U
CG
nb
( q ,T )

AT

....
exp


U
 r, T  dr1 ,...rN
  
ZN
U nbCG (q, T )   ln exp  U AT  r,   
U AT  r,    U AT rij 
i, j
 Average < > over all degrees of freedom Γ that are integrated out (here orientational )
keeping the two center-of-masses fixed at distance r.
APPLICATION I: SELF – ASSEMBLY OF PEPTIDES THROUGH
ATOMISTIC MOLECULAR SIMULATIONS
Experimental Motivation
 Peptides can assemble into various structures (fibrillar, or
spherical) depending on conditions such as solvent.
 The diphenylalanine core motif of the Alzheimer’s disease
b-amyloid
 E. Gazit et al, 2003, 2005, 2007
Diphenylalanine FF
Simulation Method and Model
Atomistic Molecular Dynamics (MD) NPT Simulations.
P=1atm (Berendsen barostat)
 U (r1 , r2 ,..., rN ) 
d 2ri
m

F



i
i
T=300K (velocity rescaling thermostat)
2
d t
r

Periodic boundary conditions were used in all three dimensions.
Gromos53a6 Atomistic Force Field was used
Di-alanine (AA) / Di-phenylalanine (FF) molecule in explicit solvent
i

Simulated Systems
System
Name
N-peptide
(# molec.)
N-solvent
(# molec.)
#atoms
c
(g pep./cm3
solv.)
T(K)
1
AA in Water
16
3696
11328
0.0385
300
2
AA in
Methanol
16
1632
5120
0.0385
300
3
FF in Water
16
6840
21112
0.0385
300
4
FF in
Methanol
16
3024
9648
0.0385
300
5
FF in Water
16
25452
76948
0.0103
300
6
FF in
Methanol
16
11648
35520
0.0103
300
7 RE
FF in Water
16
6840
21112
0.0385
395343
8 RE
FF in
Methanol
16
3024
9648
0.0385
385332
Potential of Mean Force (PMF): Alanine
20
AA in Water
AA in Methanol
kBT
15
V(r)(kJ/mol)
10
5
0
-5
0.0
0.3
0.6
0.9
1.2
r(rm)
Effect of solvent:
Slight attraction of Alanine in Water.
No attraction in Methanol.
1.5
Potential of Mean Force (PMF): Diphenylalanine
20
FF in Water
FF in Methanol
kBT
V(r)(kJ/mol)
15
10
5
0
-5
0.0
0.5
1.0
1.5
r(rm)
Attraction is apparent only in Water.
Phenyl groups are responsible for strong attraction between FF molecules.
STATIC PROPERTIES : LOCAL STRUCTURE
 radial distribution function gn(r): describe how the density of surrounding matter
varies as a distance from a reference point.
V N N !  .... exp   U N  drn11 ,...rN
g (r1 , r2 )  n
N ( N  n)!
ZN
n
 pair radial distribution function g(r)=g2(r): gives the joint probability to find 2
particles at distance r. Easy to be calculated in experiments (like X-ray diffraction) and
simulations.

1 N
g (r )  2   rij  r
N i , j 1

 choose a reference atom and
look for its neighbors:
G(r)
Structure – Self Assembly of Peptides
12
10
8
6
4
2
0
0.0
FF-FF in H2O
0.5
1.0
1.5
2.0
G(r)
1.5
2.5
3.0
FF-FF in CH3OH
1.0
0.5
0.0
0.0
0.5
1.0
1.5
2.0
2.5
3.0
r(nm)
 Strong tendency for self assembly of FF in water in contrast to its behavior in
methanol.
Self Assembly of Peptides: Experimental Data
 Self-assembly of Peptides
in water.
Vials A: Peptide is dissolved in water, vials labelled as B: Peptide is dissolved
in methanol.
Self Assembly of Peptides: More Experimental Data
 SEM Pictures (A. Mitraki, Dr. E. Kasotakis, E. Georgilis, Department of
Material Science, University of Crete)
 Peptide in water
 Peptide in methanol
Self Assembly of Peptides: More Experimental Data
 SEM Pictures (A. Mitraki, Dr. E. Kasotakis, E. Georgilis, Department of
Material Science, University of Crete)
 Peptide in water
 Peptide in methanol
Dynamics of Peptides
Dynamics can be directly quantified through mean square
displacements of molecules
r  t  
2
FF in Methanol
FF in Water
2
<r >/N (nm )
100
2
10
1
10
100
1000
t(ps)
10000
100000
 Rcm (t )  Rcm (0) 
2
Dynamics of Peptides
D  lim
 Rcm (t )  Rcm (0) 
t 
2
6t
Systems
D (cm2/sec)
stdev
AA in Water
1.1567
+/- 0.4352
FF in Water
0.5370
+/- 0.2897
AA in Methanol
2.3904
+/- 0.5372
FF in Methanol
0.8252
+/- 0.2190
Slower Dynamics in Water
 Phenyl groups retard motion
Temperature Dependence at the same concentration:
c= 0.0385gr/cm3
FF in Water
FF in Methanol
1.2
T=295K
T=316.39
T=342.74K
12
1.0
0.8
g(r)
g(r)
8
0.6
0.4
T=285K
T=311.84
T=331.12K
0.2
4
0.0
0.0
0.5
1.0
1.5
r(nm)
0
0.0
0.5
1.0
1.5
2.0
2.5
3.0
r(nm)
 Temperature increase reduces structure in water.
 Aggregates do not exist at any temperature in methanol.
2.0
2.5
3.0
Mean number of FF molecules in an aggregate
Temperature Dependence at the same concentration:
c= 0.0385gr/cm3
16
14
12
FF in H2O
10
FF in CH3OH
8
6
4
280
290
300
310
320
330
340
350
T(K)
CM - radius of 2nm
Number of FF in the aggregates decreases with temperature for water
solutions.
MULTI-SCALE MODELING OF BIOLOGICAL MEMBRANES
CELL MEMBRANE
Formation of a membrane: Selfaggregation of amphiphilic molecules
-- Molecules try to reduce contacts with
water. They form various structures:
• micelles
-- An amphiphilic - lipid membrane: • bilayer membranes
one water-loving (hydrophilic) and
one fat-loving (hydrophobic) group.
-- Works as a selective filter which
controls transfer of ions, molecules, • closed bilayers (vesicles)
large particles (viruses, bacteria, ..) • …... etc
between extracellular and
cytoplasm.
Motivation to Study Biomembranes:
• “Biophysical” reasons: -- 2D systems with novel physical properties,
-- their composition involves many components,
self-organization of multi-component systems,
-- specified membrane function can be studied on the
molecular level,
-- possible role of universal physical properties,
-- ………………. etc
• “Biotechnical” reasons: -- drug delivery (directly connected with the vesicles),
-- biosensors (combinations of membranes + electronics),
-- ………………. etc
SIMULATIONS OF BIOMEMBRANES
Atomistic ------------------> Mesoscopic ------------------> Macroscopic
(MC, MD, …)
(CG, DPD, Triangulated surfaces, …)
(continuum)
COARSE-GRAINED LIPID MODEL (SOLVENT FREE MODEL):
[I.R. Cooke, M. Deserno, K. Kremer, J. Chem. Phys. 2005]
Real Lipid molecule:
Lipid model:
h
t1
: hydrophilic group, “head” particle
: hydrophobic group, “tail” particles
: no solvent (water) particles
t2
Interactions:
• Bonded Interactions: FENE bonds (h-t1, t1-t2), harmonic
bending angle (h-t1-t2)
• Excluded volume potential: (Repulsive, WCA potential (fix
size of the lipid)
• Attractive (t – t):

, r  rc


   r  rc  

Vatt ( r )    cos2 
 , rc  r  rc  wc
2
w
c




0
, r  rc  wc

 Integrated with a DPD (pairwise) thermostat using ESPResSO
package
PARAMETERIZING CG PHENOMENOLOGICAL MODEL
-- length unit: σ
-- energy unit: ε
-- wc : model parameter that control the ¨hydrophobic effect¨.
Phase Diagram:
Select wc so as to simulate a stable liquid phase.
unstable
fluid
gel like
Application 1: Studying The Curvature Elasticity Of Biomembranes
Through Numerical Simulations
[V. Harmandaris, M. Deserno, J. Chem. Phys. 125, 204905 (2006)]
OUR GOAL: Study the curvature elasticity (predict the elastic
constants) through simulation methods
Fluid Membranes: Free Energy (Continuous Approach)
Definitions: two principal radius R1 and R2
Mean curvature:
K  1/ R1 1/ R2  / 2
Gaussian curvature:
KG  1/(R1R2 
Bending Elasticity Theory:
[Helfrich, 1973]
-- κ: bending rigidity
-- κG: Gaussian bending rigidity
E

2
 dA  2K 
2
  G  dAK G
Assumptions:
 fluidity of the membrane, 2D representation, insolubility (constant number of lipids)
Membrane shape can be calculated by minimizing F under constant area A and
volume V [Seifert, 1997; Lipowsky 1999; …]
Question: how can someone calculate κ, κG from simulations?
STUDYING THE CURVATURE ELASTICITY – AN ALTERNATIVE WAY:
CALCULATION OF ELASTIC CONSTANTS FROM DEFORMED VESICLES
[V. Harmandaris, M. Deserno, J. Chem. Phys. 125, 204905 (2006)]
-- Main idea: impose a deformation on the membrane and measure the
force required to hold it in the deformed state.
 Simple Method: Stretch a Membrane !
(a well-controlled bending deformation is created by the periodic boundary
conditions).
STUDYING THE CURVATURE ELASTICITY: CALCULATION OF ELASTIC CONSTANTS
FROM DEFORMED VESICLES.
Cylinder with fixed area:
(one principal curvature radius R).
A  2 RL
R
 Helfrich theory:
Tensile force:
Bending rigidity:

1
E  2A
2 R
 E 
2
Fz  
  ... 
R
 Lz  A

Fz R
2
w
L
Coarse-graining MD simulations:
(5000 lipids, kBT = 1.1 ε, radius R = 6 – 24 σ)
Tensile Force (due to the deformation), Fz
-- Stress tensor, τ, can be calculated directly in the simulation (using the Virial theorem).
16
   V1   r

14
 Fi ,  
i
12
Fz   zz Lx Ly
10
Fz(ε/σ)
i ,
8
6
4
2
0
4
8
12
16
20
24
28
Radius R (σ)
 The smaller the radius R, the higher the bending of the cylinder
BENDING RIGIDITY

[V. Harmandaris and M. Deserno, J. Chem. Phys., 125, 204905, 2006]
Fz Req
2
Result from
Thermal fluctuations
 Helfrich theory holds even for very small curvatures !!
Application 2: Interaction between Proteins and Biological
Membranes
 Biological problem: how do membrane proteins aggregate? Do they need
direct interactions? What is the role of the curvature-mediated interactions?
[Gottwein et al., J. Virol., 77, 9474 (2003)]
 Experimentally: very difficult to isolate curvature-mediated and direct (e.g.
specific binding) interactions.
 Modeling: needs simulations in the range of length ~ 100nm and times ~
1ms.
CG simulations
Interaction between Proteins (Colloids) and Biological Membranes
[B. Reynolds, G. Illya, V. Harmandaris, M. Müller, K. Kremer and M. Deserno, Nature, 447, 461 (2007)]
 CG modeling of proteins and biomembranes:
CG lipids
CG proteins
CG colloids
 No specific interactions: proteins are partially attracted to lipid bilayer but not
between each other.
Interaction between Proteins (Colloids) and Biological Membranes
 Evolution in time of the aggregation process:
[ System: 46080 lipids and 36 big caps. (~ 106 atoms). Time: ~ 4 ms]
 Curvature-mediated interactions: aggregation due to less
curvature energy.

2
E
2
 dA  2K 
  G  dAK G
Interaction between Proteins (Colloids) and Biological Membranes
 Colloidal spheres (model of viral capsids or nanoparticles)
 Attraction and cooperative budding: clustering in form of pairs
[Gottwein et al., J. Virol., 77, 9474 (2003)]
Interaction between Proteins (Colloids) and Biological Membranes
 Pair attraction: put two capsids on a membrane, calculate the constraint force
needed to fix them at distance d.
 Possible mechanism for attraction: capsids tilt towards each other thus
reducing local curvature.
Summary - Conclusions
 Modeling of realistic multi-component biomolecular system requires multi-scale
simulation approaches.
 Microscopic (atomistic) Molecular Dynamics can give valuable information about
the structure and the dynamics of small systems at the atomic resolution
 Effect of solvent (water or organic) is very strong on the self-assembly of
short peptides, like Di-alanine (AA) and Di-phenylalanine (FF).
 Stronger attraction between FF molecules because of phenyl groups.
 Slower Dynamics in Water. Phenyl groups retard motion.
 Mesoscopic (coarse-grained) simulations of biomembranes allows the study of
more complicated systems as well as of continuum approaches
 Interaction between colloids/proteins can lead to the rupture of membrane.
 continuum elasticity is valid even for very small distances.
Current Work – Open Questions
 Length scales: from ~ 1 Å (10-10 m) up to 100 nm (10-7 m)
 Time scales: from ~ 1 fs (10-15 sec) up to about 1 ms (10-3 sec)
 Systematic Coarse-Graining in order to study much larger systems (thousands
of peptide molecules).
 Need for efficient numerical schemes to describe complex many-body terms
 Study more complex systems:
 Boc-FF, FMoc-FF and porphyrines in water
 Bioconjugated hybrids: 8-mer peptide NSGAITIG (Asn-Ser-Gly-Ala-Ile-ThrIle-Gly) and polyethylene-oxide (PEO) and/or poly(N-isopropylacrylamide)
(PNIPAM).
ACKNOWLEDGMENTS
Modeling of Peptides
Dr. T. Rissanou [Applied Math, University of Crete, Greece]
Prof. A. Mitraki, Dr. E. Kasotakis, E. Georgilis [Department of Material Science,
University of Crete, Greece]
Biological Membranes
Prof. K. Kremer
Prof. M. Deserno
Dr. I. Cooke
Dr. B. Reynolds
[Max Planck Institute for Polymer Research, Mainz]
[Carnegie Mellon]
[Department of Zoology, Cambridge]
[MPIP]
Funding:
DFG [SPP 1369 “Interphases and Interfaces ”, Germany]
ACMAC UOC [Greece]
MPIP [Germany]