hierarchical modeling of biomolecular systems: from microscopic to macroscopic simulations
Post on 19-Mar-2016
42 Views
Preview:
DESCRIPTION
TRANSCRIPT
Hierarchical Modeling of Biomolecular Systems: From Microscopic to Macroscopic
Simulations
VAGELIS HARMANDARIS
Department of Applied MathematicsUniversity 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.
Conclusions – Open Questions.
Applications:
Self-assembly of Peptides through Microscopic Simulations.
Elasticity of Biological Membranes through Mesoscopic Simulations.
INTRODUCTION - MOTIVATION
Systems biological macromolecules (cell membrane, DNA, lipids)
Applications
Nano-, bio-technology (biomaterials in nano-dimensions)
Biological processes
Radius of gyration ~ 1-10 nm (10-9 m)
Time – Length Scales Involved in Biomolecular Systems
Self-assembly of biomolecules ~ 10 μm (10-5 m)
Bond length ~ 1 Å (10-10 m)
Multi-compartment biological systems (e.g. cell) ~ 1 mm (10-3 m)
Bond vibrations: ~ 10-15 sec
Segmental relaxation: 10-9 - 10-12 sec
Maximum relaxation time of a biomacromolecule, τ1: ~ 1 sec (in Τ < Τm)
Angle rotations: ~ 10-13 sec
Dihedral rotations: ~ 10-11 sec
Time – Length Scales Involved in Polymer Composite Systems
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
Α) description in quantum level
Β) description in microscopic (atomistic) level
C) description in mesoscopic (coarse-grained) level
D) description in macroscopic - continuum level
Hierarchical Modeling of Molecular Materials
Main goal: Built rigorous “bridges” between different simulation levels.
Quantitative prediction of properties of complex biomolecular systems.
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):
Liouville operator:
The evolution of system from time t=0 to time t is given by : ( ) exp (0)t iLt
1
, t
N
i ii i i
iL H
r Fr p
K
it i
im
pr 1 2, ,..., N c
t i ii
U
r r r
p Fr
2
( )2
iNVE
i i
pH K V Vm
rHamiltonian (conserved quantity):
Microscopic – Atomistic Modeling: Molecular Dynamics Simulations
Molecular model: Information for the functions describing the molecular interactions between atoms.
bonded non bonded extU V V V R R R
Molecular Interaction Potential (Force Field): Atomistic Simulations
Important question: What is the potential energy function?
Assumption - The complex quantum many-body interaction can be:
1) Described by semi-empirical functions.
2) Decomposed into various components.
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.
1 2: , ,..., NU UR r r r
Potential parameters are obtained from more detailed simulations or fitting to experimental data.
21 ( )2 obend bendV k bending potential
21 ( )2 ostr strV k l l stretching potential
dihedral potential5
0cos ( )i
tors ii
V c
non-bonded potential12 6
4LJV r r
Van der Waals (LJ) Coulomb
ri j
qij
qqV
ε
bonded str bend torsV V V V r
Molecular Interaction Potential (Force Field): Atomistic Simulations
non bonded LJ q hybridV V V V r
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.
2. Microscopic (atomistic) simulations of short chains (oligomers) for short times.
-- number of atoms that correspond to a ‘super-atom’ (coarse grained bead)
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.
r
BONDED POTENTIAL Degrees of freedom: bond lengths (r), bond angles (θ), dihedral angles ()
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: ( , , ), , exp
CGCG U rP r
kT
Assumption:
( , ) ln , , ( , , )CG CGBU x T k T P x T x r
, ,CG CG CG CGP r P r P P
Finally:
( ) ( ) ( )CG CG CGtotal bonded non bondedU U U Q Q Q
DEVELOP THE EFFECTIVE MESOSCOPIC CG POLYMER FORCE FIELD
NONBONDED INTERACTION PARAMETERS: REVERSIBLE WORK
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:
exp ,( , ) lnCG ATnb UU T rq
,
,AT ATij
i j
U U r r
Average < > over all degrees of freedom Γ that are integrated out (here orientational ) keeping the two center-of-masses fixed at distance r.
1( , ).... exp , ,...CG
nb
ATNU T
N
U T d
Ze
qr r r
CG Hamiltonian – Renormalization Group Map:
( , ) ( , ) |CG AT
nbU T U TNP de e q r r q q
APPLICATION I: SELF – ASSEMBLY OF PEPTIDES THROUGH ATOMISTIC MOLECULAR SIMULATIONS
Experimental Motivation Diphenylalanine FF
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
Simulation Method and ModelAtomistic Molecular Dynamics (MD) NPT Simulations.P=1atm (Berendsen barostat)T=300K (velocity rescaling thermostat)
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
21 2
2
( , ,..., )i Ni i
i
d Um
d t
r r r rFr
Simulated SystemsSystem 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 395-343
8 RE FF in Methanol
16 3024 9648 0.0385 385-332
Potential of Mean Force (PMF): Alanine
0.0 0.3 0.6 0.9 1.2 1.5-5
0
5
10
15
20 AA in Water AA in Methanol k
BT
r(rm)
V(r
)(kJ
/mol
)
Effect of solvent:
Slight attraction of Alanine in Water.
No attraction in Methanol.
Potential of Mean Force (PMF): Diphenylalanine
0.0 0.5 1.0 1.5-5
0
5
10
15
20
V
(r)(
kJ/m
ol)
r(rm)
FF in Water FF in Methanol k
BT
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.
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 11 2
.... exp ,...!( , )( )!
NN n Nn
nN
U dr rV Ng r rN N n Z
2, 1
1( )N
iji j
g r rN
r
choose a reference atom and look for its neighbors:
Strong tendency for self assembly of FF in water in contrast to its behavior in methanol.
0.0 0.5 1.0 1.5 2.0 2.5 3.00.0
0.5
1.0
1.5 FF-FF in CH3OH
G(r
)
G(r
)
r(nm)
0.0 0.5 1.0 1.5 2.0 2.5 3.002468
1012
FF-FF in H2O
Structure – Self Assembly of Peptides
Self Assembly of Peptides: Experimental Data
Vials A: Peptide is dissolved in water, vials labelled as B: Peptide is dissolved in methanol.
Self-assembly of Peptides in water.
Self Assembly of Peptides: More Experimental Data
Peptide in water
SEM Pictures (A. Mitraki, Dr. E. Kasotakis, E. Georgilis, Department of Material Science, University of Crete)
Peptide in methanol
Self Assembly of Peptides: More Experimental Data
Peptide in water
SEM Pictures (A. Mitraki, Dr. E. Kasotakis, E. Georgilis, Department of Material Science, University of Crete)
Peptide in methanol
Dynamics of PeptidesDynamics can be directly quantified through mean square displacements of molecules
10 100 1000 10000 100000
1
10
100
FF in Methanol FF in Water
<r2 >/
N (n
m2 )
t(ps)
22 ( ) (0)cm cmr t R t R
Dynamics of Peptides
Systems D (cm2/sec) stdevAA 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
2( ) (0)lim
6cm cm
t
R t RD
t
Temperature Dependence at the same concentration: c= 0.0385gr/cm3
0.0 0.5 1.0 1.5 2.0 2.5 3.00
4
8
12 T=295K T=316.39 T=342.74K
g(r)
r(nm)
FF in Water
Temperature increase reduces structure in water.
Aggregates do not exist at any temperature in methanol.
FF in Methanol
0.0 0.5 1.0 1.5 2.0 2.5 3.00.0
0.2
0.4
0.6
0.8
1.0
1.2
T=285K T=311.84 T=331.12K
g(r)
r(nm)
Temperature Dependence at the same concentration: c= 0.0385gr/cm3
280 290 300 310 320 330 340 350
4
6
8
10
12
14
16
FF in H2O FF in CH3OH
Mea
n nu
mbe
r of F
F m
olec
ules
in a
n ag
greg
ate
T(K)
CM - radius of 2nm
Number of FF in the aggregates decreases with temperature for water solutions.
-- An amphiphilic - lipid membrane: one water-loving (hydrophilic) and one fat-loving (hydrophobic) group.
-- Works as a selective filter which controls transfer of ions, molecules, large particles (viruses, bacteria, ..) between extracellular and cytoplasm.
CELL MEMBRANE Formation of a membrane: Self-aggregation of amphiphilic molecules
-- Molecules try to reduce contacts with water. They form various structures:
• micelles
• bilayer membranes
• closed bilayers (vesicles)
• …... etc
MULTI-SCALE MODELING OF BIOLOGICAL MEMBRANES
SIMULATIONS OF BIOMEMBRANES
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
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]
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):
: hydrophilic group, “head” particle : hydrophobic group, “tail” particles : no solvent (water) particles
h
t1
t2
2
,
( ) cos , 2
0 ,
c
catt c c c
c
c c
r r
r rV r r r r w
w
r r w
Lipid model: Real Lipid molecule:
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.
gel like
fluid
unstable
Application 1: Studying The Curvature Elasticity Of Biomembranes Through Numerical Simulations
OUR GOAL: Study the curvature elasticity (predict the elastic constants) through simulation methods
[V. Harmandaris, M. Deserno, J. Chem. Phys. 125, 204905 (2006)]
Definitions: two principal radius R1 and R2
Mean curvature:
Gaussian curvature:
Fluid Membranes: Free Energy (Continuous Approach)
1 21/ 1/ / 2K R R
1 21/(GK R R
Bending Elasticity Theory: [Helfrich, 1973]
-- κ: bending rigidity-- κG: Gaussian bending rigidity
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; …]
222 G GE dA K dAK
Question: how can someone calculate κ, κG from simulations?
-- Main idea: impose a deformation on the membrane and measure the force required to hold it in the deformed state.
STUDYING THE CURVATURE ELASTICITY – AN ALTERNATIVE WAY: CALCULATION OF ELASTIC CONSTANTS FROM DEFORMED VESICLES
[V. Harmandaris, M. Deserno, J. Chem. Phys. 125, 204905 (2006)]
Simple Method: Stretch a Membrane ! (a well-controlled bending deformation is created by the periodic boundary conditions).
Cylinder with fixed area:(one principal curvature radius R).
Helfrich theory:
STUDYING THE CURVATURE ELASTICITY: CALCULATION OF ELASTIC CONSTANTS FROM DEFORMED VESICLES.
2...zz A
EFL R
2
12
E AR
Tensile force:
2A RL
Bending rigidity:2zF R
R
Lw
Coarse-graining MD simulations: (5000 lipids, kBT = 1.1 ε, radius R = 6 – 24 σ)
z zz x yF L L
The smaller the radius R, the higher the bending of the cylinder
Tensile Force (due to the deformation), Fz
-- Stress tensor, τ, can be calculated directly in the simulation (using the Virial theorem).
, ,1
i ii
r FV
BENDING RIGIDITY[V. Harmandaris and M. Deserno, J. Chem. Phys., 125, 204905, 2006]
Result fromThermal fluctuations
Helfrich theory holds even for very small curvatures !!
2z eqF R
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?
CG simulations
Modeling: needs simulations in the range of length ~ 100nm and times ~ 1ms.
Experimentally: very difficult to isolate curvature-mediated and direct (e.g. specific binding) interactions.
[Gottwein et al., J. Virol., 77, 9474 (2003)]
CG modeling of proteins and biomembranes:
[B. Reynolds, G. Illya, V. Harmandaris, M. Müller, K. Kremer and M. Deserno, Nature, 447, 461 (2007)]
CG lipids
CG proteins
No specific interactions: proteins are partially attracted to lipid bilayer but not between each other.
Interaction between Proteins (Colloids) and Biological Membranes
CG colloids
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.
222 G GE dA K dAK
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.
Interaction between Proteins (Colloids) and Biological Membranes
Summary - Conclusions
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.
Modeling of realistic multi-component biomolecular system requires multi-scale simulation approaches.
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
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-Thr-Ile-Gly) and polyethylene-oxide (PEO) and/or poly(N-isopropylacrylamide) (PNIPAM).
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)
ACKNOWLEDGMENTS
Modeling of PeptidesDr. T. Rissanou [Applied Math, University of Crete, Greece]
Prof. A. Mitraki, Dr. E. Kasotakis, E. Georgilis [Department of Material Science, University of Crete, Greece]
Funding:DFG [SPP 1369 “Interphases and Interfaces ”, Germany]ACMAC UOC [Greece]MPIP [Germany]
Biological MembranesProf. K. Kremer [Max Planck Institute for Polymer Research, Mainz]Prof. M. Deserno [Carnegie Mellon]Dr. I. Cooke [Department of Zoology, Cambridge]Dr. B. Reynolds [MPIP]
top related