Hierarchical Modeling of Biomolecular Systems: From Microscopic to Macroscopic Simulations

1. 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

2. 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.

3. INTRODUCTION - MOTIVATION • Systems • biological macromolecules (cell membrane, DNA, lipids) • Applications • Nano-, bio-technology (biomaterials in nano-dimensions) • Biological processes

4. Time – Length Scales Involved in Biomolecular Systems • Bond length ~ 1 Å (10-10m) • Radius of gyration~ 1-10 nm (10-9m) • Self-assembly of biomolecules ~ 10 μm (10-5m) • Multi-compartment biological systems (e.g. cell) ~ 1mm (10-3m)

5. Time – Length Scales Involved in Polymer Composite Systems • Bond vibrations: ~ 10-15sec • Angle rotations: ~ 10-13sec • Dihedral rotations: ~ 10-11sec • Segmental relaxation: 10-9- 10-12sec • 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

6. D) description in macroscopic - continuum level C) description in mesoscopic (coarse-grained) level Β) description in microscopic (atomistic) level Α) description in quantumlevel Hierarchical Modeling of Molecular Materials • Main goal: Built rigorous “bridges” between different simulation levels. Quantitative prediction of properties of complex biomolecular systems.

7. Hamiltonian (conserved quantity): 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): Liouville operator: • The evolution of system from time t=0 to time t is given by :

8. Molecular Interaction Potential (Force Field): Atomistic Simulations • Important question: What is the potential energy function? • Assumption - The complex quantum many-body interaction can be: • Described by semi-empirical functions. • Decomposed into various components. Molecular model: Information for the functions describing the molecular interactions between atoms. • 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.

9. stretching potential • bending potential • dihedral potential Van der Waals (LJ) Coulomb • non-bonded potential Molecular Interaction Potential (Force Field): Atomistic Simulations • Potential parameters are obtained from more detailed simulations or fitting to experimental data.

10. MULTISCALE – HIERARCHICAL MODELING OF BIOMOLECULAR SYSTEMS Limits of AtomisticMolecular Dynamics Simulations (with usual computer power): -- Length scale: few (4-5) Å - (10 nm) -- Time scale: few fs - (0.5μs) -- 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.

11. GENERAL PROCEDURE FOR DEVELOPING MESOSCOPIC PARTICLE MODELS DIRECTLY FROM THE CHEMISTRY • Choice of the proper mesoscopic description. -- number of atoms that correspond to a ‘super-atom’ (coarse grained bead) • Microscopic (atomistic) simulations of short chains (oligomers) for short times. • Develop the effective mesoscopic force field using the atomistic data. • CG (MD or MC) simulations with the new CG model. Re-introduction (back-mapping) of the atomistic detail if needed.

12. r DEVELOP THE EFFECTIVE MESOSCOPIC CG POLYMER FORCE FIELD • 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: • Assumption: • Finally:

13. q NONBONDED INTERACTION PARAMETERS: REVERSIBLE WORK • CG Hamiltonian – Renormalization Group Map: • 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: • Average < > over all degrees of freedom Γ that are integrated out (here orientational ) keeping the two center-of-masses fixed at distance r.

14. 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

15. Simulation Method and Model • Atomistic 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

16. Simulated Systems

17. Potential of Mean Force (PMF): Alanine • Effect of solvent: • Slight attraction of Alanine in Water. • No attraction in Methanol.

18. Potential of Mean Force (PMF): Diphenylalanine • Attraction is apparent only in Water. • Phenyl groups are responsible for strong attraction between FF molecules.

19. 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. • choose a reference atom and look for its neighbors:

20. Structure – Self Assembly of Peptides • Strong tendency for self assembly of FF in water in contrast to its behavior in methanol.

22. 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.

23. 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

24. 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

25. Dynamics of Peptides • Dynamics can be directly quantified through mean square displacements of molecules

26. Dynamics of Peptides • Slower Dynamics in Water • Phenyl groups retard motion

27. Temperature Dependence at the same concentration: c= 0.0385gr/cm3 FF in Water FF in Methanol • Temperature increase reduces structure in water. • Aggregates do not exist at any temperature in methanol.

28. Temperature Dependence at the same concentration: c= 0.0385gr/cm3 CM - radius of 2nm • Number of FF in the aggregates decreases with temperature for water solutions.

29. 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 CELL MEMBRANE -- 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.

30. (MC, MD, …) (CG, DPD, Triangulated surfaces, …) (continuum) Atomistic ------------------> Mesoscopic ------------------> Macroscopic • 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

31. : hydrophilic group, “head” particle : hydrophobic group, “tail” particles : no solvent (water) particles h t1 t2 COARSE-GRAINED LIPID MODEL (SOLVENT FREE MODEL): [I.R. Cooke, M. Deserno, K. Kremer, J. Chem. Phys. 2005] Real Lipid molecule: Lipid model: • 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): • Integrated with a DPD (pairwise) thermostat using ESPResSO package

32. unstable fluid gel like PARAMETERIZING CG PHENOMENOLOGICAL MODEL -- length unit: σ -- energy unit: ε -- wc : model parameter that control the ¨hydrophobic effect¨. Phase Diagram: Select wcso as to simulate a stable liquid phase.

33. 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

34. Fluid Membranes: Free Energy (Continuous Approach) Definitions: two principal radius R1 andR2 Mean curvature: Gaussian curvature: • 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 Funder constant areaA and volumeV[Seifert, 1997; Lipowsky 1999; …] Question: how can someone calculate κ, κGfrom simulations?

35. 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).

36. R L STUDYING THE CURVATURE ELASTICITY: CALCULATION OF ELASTIC CONSTANTS FROM DEFORMED VESICLES. Cylinder with fixed area: (one principal curvature radius R). • Helfrich theory: w Tensile force: Bending rigidity:

37. 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). • The smaller the radius R, the higher the bending of the cylinder

38. Result from Thermal fluctuations BENDING RIGIDITY [V. Harmandaris and M. Deserno, J. Chem. Phys., 125, 204905, 2006] • Helfrich theory holds even for very small curvatures !!

39. 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

40. 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.

41. 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.

42. 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)]

43. 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:capsidstilt towards each other thus reducing local curvature.

44. 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.

45. 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-Thr-Ile-Gly) and polyethylene-oxide (PEO) and/or poly(N-isopropylacrylamide) (PNIPAM).

46. 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 [Max Planck Institute for Polymer Research, Mainz] Prof. M. Deserno [Carnegie Mellon] Dr. I. Cooke [Department of Zoology, Cambridge] Dr. B. Reynolds [MPIP] Funding: DFG [SPP 1369 “Interphases and Interfaces ”, Germany] ACMAC UOC [Greece] MPIP [Germany]

47. Thank you !!!