Multiscale Modeling and Simulation of Soft Matter Materials

Transcription

1 Multiscale Modeling and Simulation of Soft Matter Materials IMA : Development and Analysis of Multiscale Methods November 2008 Paul J. Atzberger Department of Mathematics University of California Santa Barbara (UCSB) Support: NSF DMS In collaboration with : Frank Brown (UCSB), Peter Kramer (RPI), Charles Peskin (NYU)

2 Outline Motivation for Studying Soft Matter Materials Our Modeling and Simulation Framework Incorporating the Role of Thermal Fluctuations Important Issues with Numerical Approximation Stochastic Time Step Integrators (Stiffness) Physical Validity of the Framework Applications in Rheology of Soft Matter Materials

3 Outline Motivation for Studying Soft Matter Materials Our Modeling and Simulation Framework Incorporating the Role of Thermal Fluctuations Important Issues with Numerical Approximation Stochastic Time Step Integrators (Stiffness) Physical Validity of the Framework Applications in Rheology of Soft Matter Materials

4 Examples of Soft-Matter Materials Lipid Bilayer Membranes Polymeric Fluids Gels (Actin) Colloids Energy scale of interactions / deformations comparable to thermal energy (K B T) Thermal fluctuations play an important role in structure of material. Hydrodynamic coupling of components often important. Material properties depend on multiple temporal and spatial scales. Presents challenges for modeling and simulating soft-matter systems.

5 Molecular Structure of Typical Lipid Amphiphiles Phosphogylceride air water micelle bilayer Molecules consist of polar head group and non-polar tail group. Polar head groups have an affinity to be in contact with water molecules. Non-polar tails are favored to be out of contact with water molecules. Molecules form aggregates with orientational (not nec. positional order). Phases consist of rich structures with sometimes subtle dependence on on hydrophilic vs hydrophobic group size and overall molecular shape.

6 Phases of Lipid Systems (Surfactants / Amphiphiles) Fig. Source: D. Cleaver Sheffield Hallam University Long Hydrophilic Head Group Short Hydrophilic Head Group Biological systems typically have relatively short hydrophilic head group. head group ~0.2nm hydrocarbon tails ~1-3nm. Lamellar and Micellar phases appear to be most common in biology.

7 Biological Lipid Systems (Fluid Mosaic) Many proteins anchor in lipid bilayer membranes and serve many roles, for example enzymatic or mechanical roles. Fluidity appears important to biological function: mobility of surface proteins, resistance to rupture, vesicle budding and fusion. Species of artic fish actually modify lipid mixture composition (increase number double carbon bonds in lipid tails) to lower the critical temperature of the fluid to gel phase transition.

8 Role of Membranes in Cellular Processes Cell organelles such as the Golgi apparatus make extensive use of lipid bilayer membranes. Proteins are sorted and packaged into vesicles. Vesicles are transported and fuse with membranes of other organelles or the plasma membrane. Understanding the mechanisms responsible for organelle function will likely require insight into the fundamental physics of membrane bilayers.

9 Importance of Hydrodynamics and Thermal Fluctuations: Some Illustrative Problems Membrane Deformation Protein-Protein Interactions & Diffusion Deformations are expected to induce flows both of the lipids and of the bulk solvent. Diffusivity of membrane embedded proteins may be effected by these flows (via thermal fluctuations of the membrane shape / or active deformations). Exerting a force on a protein within a membrane induces motions at large distances mediated by the hydrodynamics of both the lipids of the membrane and bulk solvent fluid (if treated as only 2D fluid Stokes paradox ) (Saffman- Delbruck Theory). Hydrodynamics and membrane geometry may also play a role in collective diffusion of proteins (aggregation phenomena / observed bleaching in exp.).

10 angstrom nanometer microns millemeter length-scale Resolution of Modeling Continuum Mechanics Coarse-Grained Modeling II Coarse-Grained Modeling I Molecular Dynamics Quantum chemistry time-scale femtosecond picosecond nanoseconds milleseconds

11 angstrom nanometer microns millemeter length-scale Resolution of Modeling Continuum Mechanics Coarse-Grained Modeling II Coarse-Grained Modeling I Molecular Dynamics hydrodynamics and thermal fluctuations Quantum chemistry time-scale femtosecond picosecond nanoseconds milleseconds

12 Outline Motivation for Studying Soft Matter Materials Our Modeling and Simulation Framework Incorporating the Role of Thermal Fluctuations Important Issues with Numerical Approximation Stochastic Time Step Integrators (Stiffness) Physical Validity of the Framework Applications in Rheology of Soft Matter Materials

13 Soft Matter Materials Modeling Approach Soft Matter Material Model: (Newtonian Solvent Fluid + Microstructures) Membranes. Polymers. Lipid molecules. Particles: Colloids, Proteins, Ions, etc... Fluid flow accounted using hydrodynamic description in an Eulerian reference frame. Microstructure accounted in a Lagrangian reference frame moving with the structures. Polymeric Fluid How should these descriptions be coupled? (conserve energy and momentum) How can thermal fluctuations be incorporated consistently?

14 Euler-Lagrange Framework (IB) Stokes Fluid Equations (R << 1): particle Polymer / Membrane polymer structure! particles Euler-Lagrange Coupling (IB Approximation): Peskin, Immersed Boundary Method, Acta Numerica, 11, How can we use this approach in practice to model soft matter systems?

15 Coarse-Grained Lipid Model (Mechanics) Interactions: Potentials: Head Interface Tail Interactions: U_core repulsion between all beads. U_tail attractions between all interface and tail beads. U_interface attraction between only interface beads (hydrophobic effect). Lipid Mechanics: U_bend acts on all sequential triples. bond length constraint on all sequential beads. G. Brannigan, P. F. Philips and F. L. H. Brown, Phys. Rev. E, 72, (2005).

16 Coarse-Grained Lipid Model (Self-Assembly) Lipid Model Forms Bilayer Membranes G. Brannigan, P. F. Philips and F. L. H. Brown, Phys. Rev. E, 72, (2005).

17 Representation within the SIB Framework Control Points: Control Point Energy: Hydrodynamic Coupling Control Point Dynamics: Control Point Forcing of Fluid:

18 Polymers Rouse and Worm-like Chains can within the SIB framework. Rouse Chain: (very flexible polymers) Worm-like Chain: stretching (DNA, unstructured RNA, proteins, filaments) curvature stretching Modeled by discretization into n nodes {X [j] }.

19 Outline Motivation for Studying Soft Matter Materials Our Modeling and Simulation Framework Incorporating the Role of Thermal Fluctuations Important Issues with Numerical Approximation Stochastic Time Step Integrators (Stiffness) Physical Validity of the Framework Applications in Rheology of Soft Matter Materials

20 Euler-Lagrange Framework Stokes Fluid Equations (R << 1): particle Polymer / Membrane polymer Euler-Lagrange Coupling (IB Approximation): How should thermal effects be taken into account? Atzberger, Kramer, & Peskin JCP 2007.

21 Euler-Lagrange Framework Stokes Fluid Equations (R << 1): particle Polymer / Membrane polymer Euler-Lagrange Coupling (IB Approximation): How should these terms be chosen to account for the physics?

22 Euler-Lagrange Framework Stokes Fluid Equations (R << 1): particle Polymer / Membrane polymer Euler-Lagrange Coupling (IB Approximation): How should these terms be chosen to account for the physics?

23 Determining the Thermal Fluctuations How should thermal forcing be introduced into these equations? (need to ensure physically meaningful / statistical mechanics) Equilibrium fluctuations should be Boltzmann distributed: (fluid kinetic energy) Z is the normalization constant. ( probability density ) Thermal forcing will be modeled by Gaussian stochastic processes -correlated in time. The thermal forcing should account appropriately for numerical approximation of the differential operators. (ex: under-damping of high-freq modes by numerical Laplacian)

24 Determining the Thermal Fluctuations Fluctuation-dissipation relations will be sufficient to determine forcing. Let the thermal forcing be of the form (ansantz): then using Ito Calculus the covariance is: The fluid equations can be written as: The covariance of the fluid velocity is:

25 Determining the Thermal Forcing (for reference): By Ito Calculus the covariance of the fluid satisfies: At statistical steady-state we have for the equilibrium fluctuations: (Fluctuation-Dissipation Principle) This determines the thermal forcing via covariance factor Q:, where. Important Issue: Given C and operator L can Q be derived and its action numerically computed efficiently in practice?

26 Euler-Lagrange SIB Framework Stokes Fluid Equations (R << 1): Complex fluid particle Polymer / Membrane polymer Euler-Lagrange Coupling (IB Approximation): Atzberger, Kramer, & Peskin, Stochastic Immersed Boundary Method, JCP 2007.

27 Outline Motivation for Studying Soft Matter Materials Our Modeling and Simulation Framework Incorporating the Role of Thermal Fluctuations Important Issues with Numerical Approximation Stochastic Time Step Integrators (Stiffness) Physical Validity of the Framework Applications in Rheology of Soft Matter Materials

28 Thermal Fluctuations for Discretized Equations Must be careful in how thermal fluctuations are introduced when discretizing the equations. Fluctuations should be considered in relation to the approximate dissipative operator of the discrete system. Central Difference Laplacian on Uniform Periodic Mesh Eigenvalue Magnitude (Damping Strength) Eigenvalues agree well for small wavenumbers. eigenvalue k Continuum Laplacian Discretized Laplacian Higher frequency modes are underdamped by the discretized Laplacian. Using thermal forcing from continuum model gives incorrect fluctuations (too large for high freq. modes). wavenumber k

29 Central Difference Approximation on Uniform Periodic Mesh Stokes Fluid Equations (3D periodic lattice): (projection method is used) Will be helpful to work in Fourier space (diagonalize Laplacian). Discrete Fourier Transform (DFT) will be used:

30 Determining Thermal Forcing for Central Difference Approximation on Periodic Mesh Stokesian Fluid Equations (DFT on N 3 lattice points): (incompressibility) (real-valuedness) where, Represent by Fourier Modes This gives the numerical Laplacian (projection not included): (diagonal matrix)

31 Determining Thermal Fluctuations for Central Difference Approximation on Periodic Mesh The energy can be expressed as: (fluid kinetic energy) This gives a Boltzmann distribution of the form: (Parseval s Lemma) Gaussian with mean 0 and covariance: Fluctuation-dissipation relations (under the constraints): ; (incompressibility) (real-valuedness) (Kramer and Peskin 2003) (Atzberger et al. 2007) (self-conjugate modes)

32 Determining Thermal Fluctuations for Central Difference Approximation on Periodic Mesh Thermal forcing for the fluid (Fourier space): where, : Brownian motion path (complex valued) : Strength of forcing of the k th mode. Represent by Fourier Modes Constraints are handled by operations on the generated thermal force.

33 Summary: Euler-Lagrange SIB Formalism (Uniform Periodic Mesh) Fluid Equations (Fourier-Space): Uniform Mesh Periodic Boundary Conditions (particle force) (thermal force) (incompressibility) (real-valuedness) (viscous damping) particle Polymer / Membrane polymer Euler-Lagrange Coupling:

34 Euler-Lagrange SIB Formalism (Adaptive Mesh) Fluid Equations (MAC Laplacian): Adaptive Mesh Dirichlet Boundary Conditions (viscous damping) (particle force) (thermal force) particle Euler-Lagrange Coupling: polymer For MAC Laplacian we have method to generate thermal forces directly 2D/3D: (challenge to generate efficiently)

35 Sampler for Adaptive Meshes Gauss-Siedel vs Multigrid for stochastic sampling. Multigrid only requires a few iterations to generate nearly decorrelated variates. Can be used for uniform or adaptive non-periodic meshes with imposed boundary conditions.

36 Time Scales of a Typical System Water at physiological temperature. L = 1 m (length-scale), N = 128 (number of modes) Relaxation Time Scale (k th mode) Diffusion Time Scale Time scales indicate a stiff set of stochastic equations. Resolving fastest time-scales of the fluid is expensive. One Solution: Treat fluid modes as relaxed on the time scales of the particles and track just the particles. (Brownian-Stokesian Dynamics). May be of interest to resolve fluctuations of hydrodynamic modes. (Brownian-Stokesian Dynamics pre-computes Green s functions, presents challenges when imposing boundary conditions.)

37 Outline Motivation for Studying Soft Matter Materials Our Modeling and Simulation Framework Incorporating the Role of Thermal Fluctuations Important Issues with Numerical Approximation Stochastic Time Step Integrators (Stiffness) Physical Validity of the Framework Applications in Rheology of Soft Matter Materials

38 X X Numerical Methods for Dynamics Over Long Time Steps (Basic Problem) Consequences of non-smoothness of the system dynamics: t Not efficient to derive schemes by simply asymptotically expanding system dynamics for small times t. Schemes must carefully account for cancellation of the fluctuating terms over the time step. t

39 Integrators for the Stochastic Dynamics Fluid Equations (Stokes Flow): (viscous damping) (particle force) (thermal force) (incompressibility) Uniform Mesh Euler-Lagrange Coupling: Adaptive Mesh

40 Stochastic Integrator for the Fluid Dynamics (Semigroup Approach) Integrating the fluid equations gives: where, Assuming forces change on slow time scale relative to fluid dynamics: Statistics of the thermal fluctuations can be computed by Ito Calculus:

41 Stochastic Integrator for the Fluid Dynamics (Semigroup Approach) [for reference]: Substituting into the above we obtain the integrator: where, is a Gaussian random variable with mean 0 and covariance Important Issue: To obtain viable numerical method we must be able to efficiently compute (or approximate) the operators and. Two cases we shall consider: 1) Operator L is easily diagonalizable (FFT). or 2) Large separation of times scales between dynamics of fluid and particles.

42 Stochastic Integrator for the Fluid Dynamics (Semigroup Approach) Uniform Periodic Mesh: (Laplacian diagonalizable via FFT) ; This gives the integrator:

43 Integrator for the Lagrangian Dynamics Fluid Equations (Stokes Flow): Uniform Mesh (viscous damping) (particle force) (thermal force) (incompressibility) Euler-Lagrange Coupling: Adaptive Mesh

44 Stochastic Integrator for the Lagrangian Dynamics (Semigroup Approach) Integrating the particle equations gives: Assuming particle positions change only a small amount over time step: Substituting above we obtain the integrator: Important Issue: To obtain a viable numerical method we must be able to efficiently compute (or approximate) the statistics of.

45 Stochastic Integrator for the Lagrangian Dynamics Semigroup Method Statistics of the fluctuations of the time integrated fluid velocity can be computed by Ito Calculus. Mean: Covariance: Fluid-Microstructure Covariance:

46 Stochastic Integrator for the Lagrangian Dynamics Semigroup Method Uniform Periodic Mesh: (Laplacian diagonalizable via FFT) ; This gives the numerical integrator: (samples time integrated fluctuations of the fluid)

47 Outline Motivation for Studying Soft Matter Materials Our Modeling and Simulation Framework Incorporating the Role of Thermal Fluctuations Important Issues with Numerical Approximation Stochastic Time Step Integrators (Stiffness) Physical Validity of the Framework Applications in Rheology of Soft Matter Materials

48 Physical Validity of the Formalism Does the SIB formalism reproduce well-known physics of microscopic systems? Some basic checks: Do particles as represented by diffuse with the appropriate dependence on the physical parameters of the system? (i.e. particle size, fluid viscosity, temperature, ) ± a Do such interacting microstructures exhibit the correct hydrodynamic correlations (i.e. Oseen hydrodynamic coupling tensor)? Does the joint fluid-microstructure system have the correct equilibrium fluctuations? (i.e. Gibbs-Boltzmann statistics)

49 Diffusion of Immersed Boundary Particles Diffusion Coefficient Diffusion Coefficient: Theory (red curve): Simulation (data points): Theory and simulation agree giving the correct scaling in the physical parameters. (particle size, fluid viscosity, temperature) Overdamped treatment of the fluid fluctuations still yields the correct diffusion of particles. Resolution of the fluid potentially allows for subtle features of the diffusive dynamics of particles to be considered. [hydrodyamic memory]

50 Diffusion of SIB Particles (hydrodynamic memory effects) Velocity Autocorrelation Function SIB Analytic Theory (green curve): Numerical Simulations (blue curve). Power law for decay of the velocity autocorrelation function predicted by MD simulations (Alder & Wainright 1950 s). SIB captures hydrodynamic memory effects contribution to the particle dynamics.

51 Equilibrium Fluctuations of Particles particle soft-wall potential Statistical mechanics requires Gibbs-Boltzmann distributed fluctuations of fluid and particles. Fluctuation-dissipation consider only the fluid equations invoked to derive the thermal forcing. Must check this is sufficient to ensure Gibbs-Boltzmann is equilibrium. Results: Numerical simulations indicate correct equilibrium fluctuations of particles.

52 Equilibrium Fluctuations of Polymer (WC) At resolutions finer than the fluid mesh, polymer fluctuations are suppressed. (polymer energy), : variances of k th mode (Boltzmann Statistics) Periodic Longitudinal & Transverse Fluctuations Do polymers exhibit Gibbs- Boltzmann fluctuations? Results: Variance of Fourier modes indicate fluctuations are accurately captured over scales larger than Eulerian mesh spacing.

53 Outline Motivation for Studying Soft Matter Materials Our Modeling and Simulation Framework Incorporating the Role of Thermal Fluctuations Important Issues with Numerical Approximation Stochastic Time Step Integrators (Stiffness) Physical Validity of the Framework Applications in Rheology of Soft Matter Materials

54 Applications in Rheology of Soft Matter Materials

55 Rheological Experiments to Measure Shear Viscosity of Materials Rheomety Device

56 SIB Simulations of Sheared Materials Polymeric Fluid (effective shear viscosity as function shear rate) Membrane (directoinal shear viscosities / zero shear limit)

57 Shearing Boundary Conditions Lees-Edwards Boundary Conditions (1972). MD particle interactions periodic boundary conditions are modified: Material frame is shifted when crossing periodic boundary. Jump in velocity occurs over the boundary. We shall now discuss how to handle these conditions for the Stokes fluid flows.

58 Shearing Boundary Conditions Stokes Equations: Boundary Conditions: Using standard coordinate frame, u is no longer periodic (because of shift). We shall consider fluid flow described in the sheared coordinate frame:

59 Shearing Boundary Conditions Change of coordinates (shear in x-direction along z-axis) Stokes Equations (sheared coordinate frame): Boundary Conditions (become jump condition): Jump boundary condition can be accounted for by introducing forcing term J on RHS.

60 Shearing Boundary Conditions Stokes Equations (with jump related forcing term J): Boundary Conditions (become periodic for modified PDE): We shall discretize this PDE to obtain the fluid flow. Some important issues: Can the discrete operators be diagonalized by FFT? Can incompressibility be enforced efficiently? How should J be discretized?

61 Shearing Boundary Conditions Stokes Equations (with jump related forcing terms J): Stokes Equations (discretized in space, projection method): Central Finite Differences used for and to obtain L(t) and. J is obtained by plugging jump condition into the central difference scheme when crossing top and bottom periodic boundaries. Yields cyclic matrix (grid-translation invariant) approximation for Laplacian and Divergence (=> diagonalizable by FFT) This allows for previously discussed integration schemes to be used.

62 Shearing Boundary Conditions Stokes Equations (with jump related forcing terms J): Stokes Equations (discretized in space, projection method): Central Finite Differences used for and to obtain L(t) and. J is obtained by plugging jump condition into the central difference scheme when crossing top and bottom periodic boundaries. Yields cyclic matrix (grid-translation invariant) approximation for Laplacian and Divergence (=> diagonalizable by FFT) This allows for previously discussed integration schemes to be used.

63 Results for Lipid Bilayer Membranes

64 Coarse-Grained Lipid Model (Mechanics) Interactions: Potentials: Head Interface Tail Interactions: U_core repulsion between all beads. U_tail attractions between all interface and tail beads. U_interface attraction between only interface beads (hydrophobic effect). Lipid Mechanics: U_bend acts on all sequential triples. bond length constraint on all sequential beads. G. Brannigan, P. F. Philips and F. L. H. Brown, Phys. Rev. E, 72, (2005).

65 Representation within the SIB Framework Control Points: Control Point Energy: Hydrodynamic Coupling Control Point Dynamics: Control Point Forcing of Fluid:

66 Bending Modulus Monte-Carlo Sampling of Equilibrium Membrane Fluctuations (Metropolis). Fourier interpolation of the membrane surface. Elastic sheet model in the Monge gauge fit to Fourier modes to deduce effective k c. For constant tension simulations must be careful to average over the fluctuations of the projected area L 2. Within range of physical membranes: k c = J. Flexible k c comparable to DGDG. Stiffer k c comparable to DLPC & DMPC. Brown, Phys. Rev. E, 72, (2005).

67 Stresses within the Bilayer Monte-Carlo Sampling of Equilibrium Membrane Fluctuations (Metropolis-Hastings). Stresses estimated from microscopic model using the virial stress formula of Irving- Kirkwood (1950). Surface tension for slab (fixed z) given by Effective tension of membrane bilayer (z) agrees qualitatively with fully atomic MD simulations. is within range of theoretical estimates mj/m 2. Brown, Phys. Rev. E, 72, (2005).

68 Shear Viscosity [Preliminary] Hydrodynamic simulations of the membrane model using the SIB method. Stresses estimated from volume averaged microscopic stressed. Kramer-Kirkwood Estimator of Stress: Shear viscosity obtained from simulations of membrane sheared at rate _. Shear viscosity given by: m = ¾ m =_ Shear Viscosity (SIB): Sckulipa, Otter, Briels (Biophys. J. 2005): Physiological Range:

69 Head Group Diffusivity [Preliminary] Motion of individual head groups and pairs considered. Single Head Diffusion: Lipid Head Group Position Diffusion Tensor Components

70 Head Group Diffusivity [Preliminary] Diffusivity of a Pair of Head Groups: Lipid Head Group Positions Diffusion Tensor for Separation Distance D(r)

71 Polymeric Fluids

72 Polymers Rouse and Worm-like Chains can within the SIB framework. Rouse Chain: (very flexible polymers) Worm-like Chain: stretching (DNA, unstructured RNA, proteins, filaments) curvature stretching Modeled by discretization into n nodes {X [j] }.

73 Polymeric Fluid (FENE Dimers) Finitely Extensible Nonlinear Elastic Dimers: (non-zero rest length bonds) Potential Energy U(r) Force F(r)

74 Polymeric Fluid (FENE Dimers) SIB Methodology can be used to study shear thinning of a FENE dimer fluid. Preliminary studies being carried out to compute shear stress with SIB with hydrodynamic interactions. Kramers-Kirkwood Stress Tensor

75 Polymer Knots and SIB

76 Polymer Knots and Links All structures derive motion by averaging a common velocity field. Consequences: Solution map of space surrounding polymer curve is a homeomorphism. Topological invariants of curves are preserved under the stochastic flow. Knottedness or linking of polymers retained under flow (up to numerical accuracy). No need for excluded volume forces when simulating polymer knots and links to impose topological constraints.

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80 Osmotic Wall-Pressure of Polymer Knots Concentration of Monomers in Boundary Layer

81 Other Soft Matter Systems

82 Motor Proteins and Polymerization Processes Force generation in cell motility and division generated by polymerization reactions. Examples include: microtubule chromosome separation actin polymerization in leading edge of cell Motor proteins are involved in active transport and force generation within cells. Examples include: Kinesins / Dyneins : active transport along microtubules. Myosins : force generation (sliding) of actin filaments, also active transport (myosin V).

83 Brownian Ratchet (Hydrodynamic Load) diffusion The ratchet is prevented from back-slipping to the left. Brownian fluctuations drive the ratchet to the next interval. Basic model for single headed Kinesin motors and microtubule / actin (de)-polymerization. Many interesting questions about the role of the cargo (purple) and it s coupling to the motor on the rate of stochastic transport.

84 Vesicle Transport by Motor Protein

85 Brownian Ratchet (Hydrodynamic Load)

86 Acknowledgements Collaborators: Stochastic Immersed Boundary Method: Peter R. Kramer (Dept. Math, Rensselaer Polytechnic Institute) Charles S. Peskin (Dept. Math, New York University) Coarse-Grained Lipid Bilayer Membrane Modeling: Frank Brown (Chemistry and Biochemistry, University of California Santa Barbara) Evgeni Penev (Chemistry and Biochemistry, University of California Santa Barbara) Support: NSF DMS Related Papers and Codes:

87 Graduate and Postdoctoral Research in Applied Mathematics Mathematics Department Los Angeles: 90 Miles UC Santa Barbara More information about positions: