Multiscale Modeling of Physical & Biological Systems

Transcription

1 Multiscale Modeling of Physical & Biological Systems George Em Karniadakis Division of Applied Mathematics, Brown University & Department of Mechanical Engineering, MIT & Pacific Northwest National Laboratory, CM4 The CRUNCH group:

2 Multiple Scales Multiple Methods DFT Molecular Dynamics Dissipative Particle Dynamics (DPD) Continuum Equation [1] Kerson Huang Lectures on statistical mechanics and protein folding, pp75-76.

3 Methods for Multiscale Modeling Sequential Methods Separation of length and time scales Parameter passing Concurrent Methods Different length and time scales within hybrid scheme Typically DFT, MD, continuum (FE); LBM; IBM Coarse Graining Integration over fast time scales/short length scales

4 Connecting Atomistic Simulations to Continuum Theory (e.g., MD to Navier-Stokes equations) Relaxation method (O Connel & Thompson, PRE, 1995) Maxwell demon method (Hadjiconstantinou & Patera, IJ Mod. Phys. C, 1997) Flux-exchange method (Flekkoy et al., Europhys. Lett., 2000) Hybrid method (Nie et al, J. Fluid Mech., 2004) Issues: Density fluctuations at interface Slip at interface Particles are drifting Inefficient algorithms

5 CM4: Collaboratory on Mathematics for Multiscale Modeling of Materials Overview Modeling Mesoscale Processes of Scalable Synthesis Research Areas I. Coarse Graining IV. Stochastic Methods II. Particle-Based Methods V. Concurrent Coupling LEAD-PI: GEORGE EM KARNIADAKIS III. Grid-Based Methods VI. Fast Solvers VII. Scalable Algorithms and Applications 5

6 Mesoscale Phenomena and Models April 9,

7 Outline I.DPD & Triple-Decker Algorithm II.Modeling of Sickle-Cell Anemia III.Mori-Zwanzig formulation

8 Dissipative Particle Dynamics (DPD) MD DPD Navier-Stokes MICROscopic level approach atomistic approach is often problematic because larger time/length scales are involved set of point particles that move off-lattice through prescribed forces each particle is a collection of molecules MESOscopic scales momentum-conserving Brownian dynamics continuum fluid mechanics MACROscopic modeling Ref on Theory: Lei, Caswell & Karniadakis, Phys. Rev. E, 2010

9 Dissipative Particle Dynamics (DPD) Particles in DPD represent clusters of molecules and interact through simple pair-wise forces C R D Fi Fij Fij dt Fij j DPD system is thermally equilibrated through a thermostat defined by forces The time evolution equations are given by: dr vi dt i i dv i Fdt * P.J. Hoogerbrugge and J.M.V.A. Koelman, Europhys.Lett.,19: , 1992 i

10 Pairwise Interactions Forces exerted by particle J on particle I: Fluctuation-dissipation relation: σ 2 = 2 γ κ Β Τ ω D = [ ω R ] 2 Conservative fluid / system dependent Dissipative frictional force, represents viscous resistance within the fluid accounts for energy loss r ij j r i i Random stochastic part, makes up for lost degrees of freedom eliminated after the coarse-graining r j

11 Conservative Force MD a DPD From Forrest and Sutter, 1995 Soft potentials were obtained by averaging the molecular field over the rapidly fluctuating motions of atoms during short time intervals. This approach leads to an effective potential similar to one, used in DPD.

12 Intra-Polymer Forces Combinations Of the Following: Lennard-Jones Repulsion Stiff (Fraenkel) / Hookean Spring Finitely-Extensible Non-linear Elastic (FENE) Spring Marko-Siggia/WormLike Chain

13 Mixing Soft-Hard Potentials Motivation for 2 different time-steps (Δt,δt): Subcycling Symeonidis & Karniadakis, J. Comp. Phys., 2006 Solvent (soft repulsive) Polymer Lennard-Jones (hard repulsive) Forrest+Suter, (J. Chem. Phys., 1995) idea of pre-averaging - in the spirit of conservative forces in DPD solvent

14 Adaptive Boundary Conditions Locally averaged density Wall force Target density Current density Adaptive BC: layers of particles Iteratively adjust the wall repulsion force in each bounce back reflection bin based on the averaged density values. adaptive wall force I.V. Pivkin and G.E. Karniadakis, PRL, vol.96, , 2006

15 Triple-Decker Algorithm, JCP, 2008 Atomistic-Mesoscopic-Continuum Coupling Efficient time and space decoupling Subdomains are integrated independently and are coupled through the boundary conditions every time

16 CONTINUUM ATOMISTIC

17 Outline I.DPD & Triple-Decker Algorithm II.Modeling of Sickle-Cell Anemia III.Mori-Zwanzig formulation

18 Sickle cell anemia Sickle cell anemia is a genetic blood disorder affecting mainly Americans of Sub-Saharan African descent In the United States, about 1 out of 500 African-American children born will have sickle-cell anemia Life expectancy of the patients with sickle cell anemia is around 50 years [1] J. B. Herrick, Arch. Intern. Med., 6: , 1910 [2] L. Pauling, H. A. Itano, S. J. Singer, and I. C. Wells, Science, 110: , 1949.

19 Multi-scale Red Blood Cell Model Main features: Triangular mesh: 1) each vertex a DPD particle 2) each edge a viscoelastic spring 3) bending energy between faces 4) constant surface area (local or global) 5) constant volume With Subra Suresh, MIT

20 General Spectrin-level and Multi-Scale RBC Models 500 nm Pivkin & Karniadakis, PRL, 2008; Fedosov, Caswell & Karniadakis, Biophys. J, 2010

21 MS-RBC mechanics: healthy Y N m N m k J c Experiment - Suresh et al., Acta Biomaterialia, 1:15-30, 2005

22 DPD Simulations Validation (Igor Pivkin)

23 Results for healthy and malaria infected cells MIT expts & sims: Hansen & Han, Pivkin & Suresh Converging geometry Diverging geometry Flow Flow RBCs exhibited faster velocity in the channel with converging entrance geometry Ring RBCs travel with velocity approximately 50% less than that of uninfected cells from the same culture

24 Introduction: Molecular pathogenesis Packaging of hemoglobin into RBCs requires that the protein be soluble. Upon de-oxygenation: Replacement of Glu at 6 with Val results in hydrophobic interaction (HI) with another hemoglobin molecule, causing aggregation into large polymers. HI is necessary for the formation of polymers Polymerization of deoxyhemoglobin and alignment of fibers result in a distortion of the shape of the RBCs and a marked decrease in its deformability. H. F. Bunn, N. Engl. J. Med, 1997, 337, 762

25 Coarse-grained HbS model Normal RBC contains hemoglobin A that has 2 subunits denoted by and 2 subunits denoted by. hydrophilic, soluble hydrophobic, insoluble The packaging of a very high concentration of hemoglobin into RBCs requires that the protein be extraordinary soluble. A schematic of coarse-grained model for sickle hemoglobin. Change from charged to neutral hydrophobic amino acid causes aggregation upon de-oxygenation.

26 Dissipative Particle Dynamics Method The force field is usually divided into two major parts: bonded and non-bonded potential terms: Vtot Vbonded Vnonbonded Vstr Vbend Vtors Vvdw Ves Bonded interactions: Hookean spring interaction (A-B and B-B): Bond-bending interaction (A- B-B and B-B-B in same chain): FENE interaction (A-B-B in different chains): Vstr kstr r r0 Vbend kbend 0 F bend k bend max Control the chain rigidity Describe the chain chirality Non-bonded interactions: Pairwise conservative interaction: aij Vnon-bonded 1 rij r 2 c 2

27 The self-assembled microstructures Bond-bending and torsional interactions among the hydrophilic and hydrophobic particles included k bend (A - B - B) o (A - B - B) max (A - B - B) The self-assembled elongated sheet-like microstructures

28 The self-assembled microstructures Bond-bending and torsional interactions among the hydrophilic and hydrophobic particles included k bend (A - B - B) o (A - B - B) max (A - B - B) Hydrophobic particles pack more densely and form cylindrical micelles in order to minimize contact with the solvent particles. The self-assembled elongated step-like bundle microstructures

29 Shape deformation of RBC induced by HbS fibers To simulate the growth of a HbS fiber, we use a linear spring model described by Fstr kstr lref l l l ref 0 l target 1 1 l p _ target max O2 _ target O p p O Biconcave shape Sickle shape Holly leaf shape Shape deformation of RBC induced by the growth of HbS fiber in DPD simulation

30 Rheology of sickle cell suspension Shear viscosity Ht = 45% Ht = 40% [1] Usami, S., S. Chien, P. M. Scholtz, and J. F. Bertles, Microvascular Research 9:324, 1975 [2] DK. Kaul and H Xue, Blood, 77, , 1991

31 [1] D. A. Fedosov, B. Caswell, and G. E. Karniadakis, Biophysical Journal, 100, 2084, Vaso-occlusion Shear flow response Same adhesive parameters applied to the sickle cells

32 Pipe flow (SS2 + SS4) Vaso-occlusion PNAS, to appear Same adhesive parameters applied Deformable SS2 cells adherent to post capillary Trap rigid SS4 cells (mostly Irreversible sickle cells) Blood occlusion in post capillary

33 Outline I.DPD & Triple-Decker Algorithm II.Modeling of Sickle-Cell Anemia III.Mori-Zwanzig formulation

34 Dissipative Particle Dynamics (DPD) 1. Mori-Zwanzig formulation 2. Direct construction of DPD from MD

35 Equation of Coarse-Grained system CG (slower) Variables Atomistic (fast) variables

36 (Cont'd) General Equation of Coarse-Grained particles Harmonic chain Ensemble average Dissipative force Random force

37 Question Coupled term General Coarse-grained Equation? Mesoscopic Dynamics

38 Langevin Equation Markovian Approximation

39 DPD from Coarse-Grained Equation

40 (Cont'd) Markovian approximation Transverse term Standard DPD back

41

42 MD to DPD Microscopic system (Molecular Dynamics) What is Preserved? What is Lost? Conservative force Dissipative force Random force Mesoscopic system (DPD)

43 Conservative potential (force) term

44 back

45 Rg = 0.95 Dissipative force term Rg = rkovian back

46 (Cont'd)

47 Question Microscopic System Fluid Structure? Mesoscopic System

48 Static properties

49 Question Microscopic System Dynamic Properties? Mesoscopic System

50 Dynamic Properties

51 Two-Thermostats versus One Thermostat

52 (Cont'd) Strong over-damping

53 Why over-damping? Over-estimation of the dissipative force, due to pairwise decomposition of the mean force field For high density and large Rg system, the interaction between two CG clusters strongly depends on the spatial distribution of other clusters. Recall conservative force: conservative Markovian approximation of the velocity correlation term over-damps the dissipative force term. Recall random force correlation: Dissipative

54 What s left, other potential?

55 (Cont'd)

56

57 Open Issues Handshaking is still a problem (MD-DPD-NS) Coarse graining has its limits Complex dynamics requires stochastic closures. Error/Uncertainty quantification needed. New Verification & Validation procedures are required for MSM. MSM should motivate new parallel computational paradigms Special JCP issue with NIH projects.