Multiscale Modeling and Simulation of Soft Matter Materials · IMA : Development and Analysis of...
Transcript of Multiscale Modeling and Simulation of Soft Matter Materials · IMA : Development and Analysis of...
Multiscale Modeling and Simulation of
Soft Matter Materials
IMA : Development and Analysis of Multiscale Methods
November 2008
Paul J. AtzbergerDepartment of Mathematics
University of California Santa Barbara (UCSB)
In collaboration with :
Frank Brown (UCSB), Peter Kramer (RPI), Charles Peskin (NYU)
Support: NSF DMS – 0635535.
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
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
Examples of Soft-Matter Materials
• Energy scale of interactions / deformations comparable to thermal energy (KBT)
• 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.
ColloidsGels (Actin)
Lipid Bilayer Membranes Polymeric Fluids
Molecular Structure of Typical Lipid Amphiphiles
Phosphogylceride
• 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.
micelle
water
air
bilayer
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.
• 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.
Biological Lipid Systems (Fluid Mosaic)
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.
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.).
Importance of Hydrodynamics and Thermal Fluctuations:
Some Illustrative Problems
Resolution of Modelingan
gstr
om
nan
om
ete
rm
icro
ns
mil
lem
ete
r
femtosecond picosecond nanoseconds milleseconds
Quantum
chemistry
Molecular
Dynamics
Coarse-Grained
Modeling I
Continuum
Mechanics
Coarse-Grained
Modeling II
time-scale
len
gth
-sc
ale
Resolution of Modelingan
gstr
om
nan
om
ete
rm
icro
ns
mil
lem
ete
r
femtosecond picosecond nanoseconds milleseconds
Quantum
chemistry
Molecular
Dynamics
Coarse-Grained
Modeling I
Continuum
Mechanics
Coarse-Grained
Modeling II
time-scale
len
gth
-sc
ale
hydrodynamics and thermal
fluctuations
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
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.
• How should these descriptions be coupled?
(conserve energy and momentum)
• How can thermal fluctuations be
incorporated consistently?
Polymeric Fluid
Euler-Lagrange Framework (IB)
Stokes’ Fluid Equations (R << 1):
Euler-Lagrange Coupling (IB Approximation):
How can we use this approach in practice to model soft matter systems?
Polymer / Membrane
particle
polymer
particle
Peskin, Immersed Boundary Method, Acta Numerica, 11, 2002.
Polymer / Membrane
particleparticle
structure ! particles
Coarse-Grained Lipid Model (Mechanics)
Interface
Head
Tail
• U_core repulsion between all beads.
• U_tail attractions between all interface and tail beads.
• U_interface attraction between only interface beads (hydrophobic effect).
Interactions:
Lipid Mechanics:
• U_bend acts on all sequential triples.
• bond length constraint on all sequential beads.
Interactions: Potentials:
G. Brannigan, P. F. Philips and F. L. H. Brown, Phys. Rev. E, 72, 011915 (2005).
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, 011915 (2005).
Control Point Energy:
Control Point Dynamics:
Control Point Forcing of Fluid:
Representation within the SIB Framework
Hydrodynamic Coupling
Control
Points:
curvature stretching
stretching
Polymers
• Rouse and Worm-like Chains can
within the SIB framework.
• Rouse Chain:
(very flexible polymers)
• Worm-like Chain:
(DNA, unstructured RNA, proteins, filaments)
• Modeled by discretization into n nodes {X[j]}.
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
Euler-Lagrange Framework
Stokes’ Fluid Equations (R << 1):
Euler-Lagrange Coupling (IB Approximation):
How should thermal effects be taken into account?
Polymer / Membrane
particle
polymer
particle
Atzberger, Kramer, & Peskin JCP 2007.
Euler-Lagrange Framework
Stokes’ Fluid Equations (R << 1):
Euler-Lagrange Coupling (IB Approximation):
How should these terms be chosen to account for the physics?
Polymer / Membrane
particle
polymer
particle
Euler-Lagrange Framework
Stokes’ Fluid Equations (R << 1):
Euler-Lagrange Coupling (IB Approximation):
How should these terms be chosen to account for the physics?
Polymer / Membrane
particle
polymer
particle
• Equilibrium fluctuations should be Boltzmann distributed:
• 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)
Determining the Thermal Fluctuations
Z is the normalization constant.
(fluid kinetic energy)
(“probability density”)
How should thermal forcing be introduced into these equations? (need to ensure physically meaningful / statistical mechanics)
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:
Determining the Thermal Fluctuations
.
Fluctuation-dissipation relations will be sufficient to determine
forcing.
(for reference):
By Ito Calculus the covariance of the fluid satisfies:
At statistical steady-state we have for the equilibrium fluctuations:
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?
Determining the Thermal Forcing
.
(Fluctuation-Dissipation Principle)
Euler-Lagrange SIB Framework
Stokes’ Fluid Equations (R << 1):
Euler-Lagrange Coupling (IB Approximation):
Polymer / Membrane
particle
polymer
particle
Complex fluid
Atzberger, Kramer, & Peskin, Stochastic Immersed Boundary Method, JCP 2007.
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
Thermal Fluctuations for Discretized Equations
Continuum Laplacian
Eigenvalue Magnitude (Damping Strength)
Discretized Laplacian
wavenumber k
eig
en
va
lue®k
• Eigenvalues agree well for
small wavenumbers.
• Higher frequency modes are
“underdamped” by the
discretized Laplacian.
• Using thermal forcing from
continuum model gives
incorrect fluctuations
(too large for high freq. modes).
• 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
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:
Determining Thermal Forcing for
Central Difference Approximation on Periodic Mesh
.
Stokesian Fluid Equations (DFT on N3 lattice points):
where,
(incompressibility)
(real-valuedness)
This gives the numerical Laplacian (projection not included):
(diagonal matrix)
Represent
by
Fourier Modes
Determining Thermal Fluctuations for
Central Difference Approximation on Periodic Mesh
.
• The energy can be expressed as:
• This gives a Boltzmann distribution of the form:
Gaussian with mean 0 and covariance:
• Fluctuation-dissipation relations (under the constraints):
(fluid kinetic energy)
(Parseval’s Lemma)
(incompressibility)
(real-valuedness);
(Kramer and Peskin 2003)
(Atzberger et al. 2007)
(self-conjugate modes)
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 kth mode.
• Constraints are handled by operations on the generated thermal force.
Represent
by
Fourier Modes
Summary: Euler-Lagrange SIB Formalism (Uniform Periodic Mesh)
Fluid Equations (Fourier-Space):
Euler-Lagrange Coupling:
(particle force)
(thermal force)
(incompressibility)
(real-valuedness)
(viscous damping)
Uniform Mesh
Periodic Boundary Conditions
Polymer / Membrane
particle
polymer
particle
Euler-Lagrange SIB Formalism (Adaptive Mesh)
Fluid Equations (MAC Laplacian):
Euler-Lagrange Coupling:
(particle force)
(thermal force)
(viscous damping)
Adaptive Mesh
Dirichlet Boundary Conditions
• For MAC Laplacian we have method to generate thermal forces directly 2D/3D:
(challenge to generate efficiently)
polymer
particle
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.
Time Scales of a Typical System
• Water at physiological temperature.
• L = 1 m (length-scale), N = 128 (number of modes)
• 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.)
Relaxation Time Scale (kth mode) Diffusion Time Scale
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
Numerical Methods for Dynamics Over
Long Time Steps (Basic Problem)
t
X
t
XConsequences of non-smoothness of
the system dynamics:
• 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.
Integrators for the Stochastic Dynamics
Fluid Equations (Stokes Flow):
Euler-Lagrange Coupling:
(incompressibility)
(particle force)
(thermal force)
(viscous damping)
Uniform Mesh
Adaptive Mesh
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:
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.
Stochastic Integrator for the Fluid Dynamics
(Semigroup Approach)
Uniform Periodic Mesh: (Laplacian diagonalizable via FFT)
This gives the integrator:
;
Integrator for the Lagrangian Dynamics
Fluid Equations (Stokes Flow):
Euler-Lagrange Coupling:
(incompressibility)
(particle force)
(thermal force)
(viscous damping)
Uniform Mesh
Adaptive Mesh
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 .
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:
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)
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
Physical Validity of the Formalism
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, …)
• 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)
.
Does the SIB formalism reproduce well-known physics of microscopic systems?
±a
Diffusion of Immersed Boundary Particles
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.
Diffusion Coefficient:Diffusion Coefficient
Resolution of the fluid potentially allows for subtle features of the
diffusive dynamics of particles to be considered. [hydrodyamic memory]
Diffusion of SIB Particles(hydrodynamic memory effects)
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.
Velocity Autocorrelation Function
Equilibrium Fluctuations of 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.
soft-wall potential
Statistical mechanics requires Gibbs-Boltzmann distributed
fluctuations of fluid and particles.
particleparticle
Equilibrium Fluctuations of Polymer (WC)
Longitudinal & Transverse Fluctuations
, : variances of kth mode (Boltzmann Statistics)
Periodic
Do polymers exhibit Gibbs-
Boltzmann fluctuations?
Results: Variance of Fourier
modes indicate fluctuations are
accurately captured over scales
larger than Eulerian mesh
spacing.
At resolutions finer than the
fluid mesh, polymer
fluctuations are suppressed.
(polymer energy)
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
Applications in Rheology of
Soft Matter Materials
Rheological Experiments to Measure
Shear Viscosity of Materials
Rheomety Device
SIB Simulations of Sheared Materials
Membrane (directoinal shear viscosities / zero shear limit)
Polymeric Fluid (effective shear viscosity as function shear rate)
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.
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:
Shearing Boundary Conditions
.
Stokes Equations (sheared coordinate frame):
Boundary Conditions (become jump condition):
Change of coordinates (shear in x-direction along z-axis)
• Jump boundary condition can be accounted for by introducing forcing term
J on RHS.
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?
Shearing Boundary Conditions
.
Stokes Equations (with jump related forcing terms J):
• Yields cyclic matrix (grid-translation invariant) approximation for Laplacian and
Divergence (=> diagonalizable by FFT)
• This allows for previously discussed integration schemes to be used.
• Central Finite Differences used for and to obtain L(t) and .
Stokes Equations (discretized in space, projection method):
• J is obtained by plugging jump condition into the central difference scheme
when crossing top and bottom periodic boundaries.
Shearing Boundary Conditions
.
Stokes Equations (with jump related forcing terms J):
• Yields cyclic matrix (grid-translation invariant) approximation for Laplacian and
Divergence (=> diagonalizable by FFT)
• This allows for previously discussed integration schemes to be used.
• Central Finite Differences used for and to obtain L(t) and .
Stokes Equations (discretized in space, projection method):
• J is obtained by plugging jump condition into the central difference scheme
when crossing top and bottom periodic boundaries.
Results for Lipid Bilayer Membranes
.
Coarse-Grained Lipid Model (Mechanics)
Interface
Head
Tail
• U_core repulsion between all beads.
• U_tail attractions between all interface and tail beads.
• U_interface attraction between only interface beads (hydrophobic effect).
Interactions:
Lipid Mechanics:
• U_bend acts on all sequential triples.
• bond length constraint on all sequential beads.
Interactions: Potentials:
G. Brannigan, P. F. Philips and F. L. H. Brown, Phys. Rev. E, 72, 011915 (2005).
Control Point Energy:
Control Point Dynamics:
Control Point Forcing of Fluid:
Representation within the SIB Framework
Hydrodynamic Coupling
Control
Points:
.
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 kc.
• For constant tension simulations must be careful
to average over the fluctuations of the projected
area L2.
• Within range of physical membranes: kc = 1 – 8 £ 10-20J.
• Flexible kc comparable to DGDG.
• Stiffer kc comparable to DLPC & DMPC.
Brown, Phys. Rev. E, 72, 011915 (2005).
.
Stresses within the Bilayer
Brown, Phys. Rev. E, 72, 011915 (2005).
• 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
32 – 65 mJ/m2.
• 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:
• Shear Viscosity (SIB):
• Sckulipa, Otter, Briels (Biophys. J. 2005):
• Physiological Range:
.
Shear Viscosity [Preliminary]
_°
´m= ¾m= _°
• Motion of individual head groups and pairs
considered.
Head Group Diffusivity [Preliminary]
Lipid Head Group Position
Diffusion Tensor Components
Single Head Diffusion:
Head Group Diffusivity [Preliminary]
Diffusivity of a Pair of Head Groups:
Lipid Head Group Positions
Diffusion Tensor for Separation Distance D(r)
Polymeric Fluids
.
• Rouse and Worm-like Chains can
within the SIB framework.
• Rouse Chain:
(very flexible polymers)
• Worm-like Chain:
(DNA, unstructured RNA, proteins, filaments)
• Modeled by discretization into n nodes {X[j]}.
Polymers
curvature stretching
stretching
Polymeric Fluid (FENE Dimers)
.
Potential Energy U(r) Force |F(r)|
Finitely Extensible Nonlinear Elastic Dimers:
(non-zero rest length bonds)
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
Polymer Knots and SIB
.
Polymer Knots and Links
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.
All structures derive motion by averaging a common velocity field.
Osmotic Wall-Pressure of Polymer Knots
Concentration of Monomers in Boundary Layer
Other “Soft Matter” Systems
.
• 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).
Motor Proteins and Polymerization Processes
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.
.
Vesicle Transport by Motor Protein
Brownian Ratchet (Hydrodynamic Load)
.
Related Papers and Codes: http://www.math.ucsb.edu/~atzberg/
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)
Acknowledgements
Support: NSF DMS – 0635535.
Mathematics Department
Graduate and Postdoctoral Research in Applied Mathematics
More information about positions: http://www.math.ucsb.edu/~atzberg/
Los Angeles: 90 Miles
UC Santa Barbara