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 Vnonbonded 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 rk 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 Vnonbonded 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 drn11 ,...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 2A 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]