The Old Well 10/27/2003 Duke Applied Math Seminar 1 Continuum-molecular computation of biological microfluidics Sorin Mitran Electron micrograph of cilium cross section Cilium beat pattern Computational grid Applied Mathematics Program The University of North Carolina at Chapel Hill

The Old Well 10/27/2003 Duke Applied Math Seminar 2 Background Bring together an interdisciplinary team to build a complete model of a lung including: Models of molecular behavior (molecular motors) Microfluidics, microelasticity Biochemical networks Viscoelastic fluids Physiology, pathology to guide models UNC Virtual Lung Project Physics: Rich Superfine, Sean Washburn Applied Mathematics: Greg Forest, Sorin Mitran, Jingfang Huang, Rich McLaughlin, Roberto Camassa, Mike Minion Computer Science: David Stotts Chemistry: Michael Rubinstein, Sergei Sheiko Cystic Fibrosis Center: Richard Boucher, Bill Davis Biochemistry and Biophysics John Sheehan

The Old Well 10/27/2003 Duke Applied Math Seminar 3 Overview 1.Cilium structure 2.Ciliary flow 3.Open issues 4.A continuum-microscopic model for the single cilium 1.A fluid-structure interaction model 2.Adaptive mesh, model refinement 3.Continuum-microscopic interaction 5.A toy problem in continuum-microscopic interaction 6.Lessons learned and applications to cilium microfluidics and microelasticity 7.Conclusions

The Old Well 10/27/2003 Duke Applied Math Seminar 4 Cilium structure Dimensions: Cilium diameter: 225 nm Cilium length: 7000 nm Microtubule exterior diameter: 25 nm Microtubule interior diameter: 14 nm Microtubule centers diameter: 170 nm Microtubules form supporting structure with flexural rigidity: 10 ⁻ ² ⁴ Nm² ≲ EI ≲ 10 ⁻ ²³ Nm² Dynein molecules “walk” between adjacent microtubule pairs and exert a force: F≈6 × 10 ⁻ ¹² N Collective effect of dynein molecules (~4000 per cilium) leads to beat pattern Collective effect of thousands of cilia per cell lead to fluid entrainment cilia.htmhttp://cellbio.utmb.edu/cellbio/ cilia.htm Gwen V. Childs

The Old Well 10/27/2003 Duke Applied Math Seminar 5 Ciliary Flow Cilia maintain motion of PCL (periciliaryliquid) and mucus layer with role in filtration and elimination of harmful agents Cilia beat times/sec: ~2000 nm/sec peak velocity PCL flows at ~40 nm/sec PCL thickness: 7000 nm PCL rheoleogical behavior is uncertain, commonly assumed to be saline solution Re PCL = 3 × 10 ⁻⁹ (very slow, viscous flow)

The Old Well 10/27/2003 Duke Applied Math Seminar 6 Open issues 1.Exact mechanism of cilium motion; how do dynein molecules determine beat patterns? 2.Cilium elastic structure; how do dynein forces on microtubules lead to overall deformation? 3.Fluid dynamics around cilia – is Stokes flow of a Newtonian fluid a good working hypothesis? 4.Are we in a continuum setting?

The Old Well 10/27/2003 Duke Applied Math Seminar 7 Fluid-structure interaction model: The structure 1.Measure cilium structure dimensions 2.Construct large-deflection beam model of each microtubule: 1.Geometric description of microtubule mean fiber 2.Use local Frenet triad at each beam element cross section 3.12 DOF per beam element

The Old Well 10/27/2003 Duke Applied Math Seminar 8 Fluid-structure interaction model: The structure 4. Large deflection leads to geometric nonlinearity in the finite element model 5. Series expansion 6. Options: 1. Use higher number of terms; maintain the reference state as the zero-displacement state 2. Use a linear truncation with an averaged stiffness matrix 3. Use a linear truncation with respect to a reference state that does not necessarily correspond to zero-displacements but is close to the expected deformed state

The Old Well 10/27/2003 Duke Applied Math Seminar 9 Fluid-structure interaction model: The structure 3. Use a linear truncation with respect to a reference state that does not necessarily correspond to zero-displacements but is close to the expected deformed state 7. Assemble elemental rigidity matrix to obtain overall description of cilium structure

The Old Well 10/27/2003 Duke Applied Math Seminar 10 Fluid-structure interaction model: Dynein forces Tim Elston, John Fricks – Stochastic model of “stepping” dynein molecules Given microtubule geometry return distribution of forces along microtubules Rich Superfine – experimental measurements

The Old Well 10/27/2003 Duke Applied Math Seminar 11 Fluid-structure interaction model: Fluid forces Simultaneously solve unsteady Stokes equations to provide full fluid stress tensor at cilium membrane surface

The Old Well 10/27/2003 Duke Applied Math Seminar 12 Fluid computation Hybrid moving mesh – overlapping mesh technique Moving mesh is generated outward from cilium surface to (10-40) cilium diameters; mesh is orthogonal in the two polar directions. Moving mesh overlaps with fixed Cartesian mesh spanning computational domain Interpolation of fixed mesh grid data provides boundary conditions for moving mesh Moving mesh data updates overlapping fixed mesh grid points transmitting influence of cilium flow to far field

The Old Well 10/27/2003 Duke Applied Math Seminar 13 Fluid computation – Moving Mesh Use a time-dependent grid mapping between Cartesian computational space and physical space

The Old Well 10/27/2003 Duke Applied Math Seminar 14 Fluid computation – Moving Mesh Solve unsteady Stokes equations using projection method in computational space

The Old Well 10/27/2003 Duke Applied Math Seminar 15 Fluid computation – Typical result Imposed dynein forces

The Old Well 10/27/2003 Duke Applied Math Seminar 16 Adaptive mesh refinement Logically Cartesian Grids enable AMR Trial step on coarse grid determines placement of finer grids Boundary conditions for finer grids from space-time interpolation Time subcycling: more time steps (of smaller increments) are taken on fine grids Finer grid values are obtained by interpolation from coarser grid values Coarser grid values are updated by averaging over embedded fine grids Conservation ensured at coarse-fine interfaces (conservative fixups)

The Old Well 10/27/2003 Duke Applied Math Seminar 17 Adaptive mesh, model refinement Is the standard continuum fluid flow hypothesis tenable for cilia-induced flow? Unclear – try development of appropriate computational experiment Maintain idea of embedded grids Establish a cutoff length at which microscopic computation is employed Redefine injection/prolongation operators Redefine error criterion for grid refinement Redefine time subcycling

The Old Well 10/27/2003 Duke Applied Math Seminar 18 A Toy problem: one-dimensional model of ductile failure in a rod Model features Progressive damage Bonds break due to combined thermal, mechanical effect Point masses connected by multiple springs Force-deformation law

The Old Well 10/27/2003 Duke Applied Math Seminar 19 Equations of motion Displacement from equilibrium: Continuum limit With damage Mass per lattice spacing: No damage: Continuum limit Presence of damage requires microscopic information, i.e. the number of broken springs Zero temperature limit

The Old Well 10/27/2003 Duke Applied Math Seminar 20 Two-dimensional model of thin shell failure 2D lattice of oscillators No damage Continuum limit With damage Continuum limit

The Old Well 10/27/2003 Duke Applied Math Seminar 21 Failure scenarios Bonds break under dynamic loading due to combined thermal (microscopic) and continuum motion Dynamic loadingMelting

The Old Well 10/27/2003 Duke Applied Math Seminar 22 DNS results – no damage

The Old Well 10/27/2003 Duke Applied Math Seminar 23 DNS results – with damage Extend rod through constant, outward velocity boundary conditions at end points

The Old Well 10/27/2003 Duke Applied Math Seminar 24 Molecular-Continuum Interaction Prolongation operator from continuum to microscopic levels instantiates a statistical distribution of dumbbell configurations (e.g. Maxwell-Boltzmann) Prolongation operator from microscopic to microscopic levels is a finer sampling operation Restriction operator from microscopic to continuum level is a smoothing of the additional stress tensor (avoid microscopic noise in the continuum simulation) Time subcycling determined by desired statistical certainty in the stress tensor

The Old Well 10/27/2003 Duke Applied Math Seminar 25 Eliminating thermal behavior Microscopic dynamics Principal component analysis contains all system information Very little of the information is relevant macroscopically Coarse graining approaches: Principal modes – 32 point masses Cutoff after three decades ► Spatial averaging - homogenization ► Fourier mode elimination - RNG

The Old Well 10/27/2003 Duke Applied Math Seminar 26 Direct Simulation Monte Carlo Full microscopic computation too expensive We need microscopic data to evaluate elastic speed and local damage Use a Monte Carlo simulation to sample configuration space Unbiased Monte Carlo simulation requires extensive sampling – too expensive ► hierarchical Monte Carlo ► bias sampling in accordance with principal components from immediately coarser level

The Old Well 10/27/2003 Duke Applied Math Seminar 27 Continuum-Microscopic Interaction Continuum to microscopic injection of values low frequency contribution from principal components of coarser grid level high frequency contribution from Maxwell- Boltzmann distribution of unresolvable coarser grid modes Microscopic to continuum restriction: ► Subtract contribution from thermal and minor modes ► The energy of these modes defines a “temperature” valid for current grid level ► Transport coefficient from standard statistical mechanics

The Old Well 10/27/2003 Duke Applied Math Seminar 28 Instantaneous Constitutive Relations For simple model considered here only constitutive relation is the dependence of elastic speed upon local damage Update of local damage: ► on each level ► after each time step ► check if displacement has increased beyond current elastic law restriction Force-deformation law

The Old Well 10/27/2003 Duke Applied Math Seminar 29 Two-dimensional example: Rupturing membrane Simulation parameters 50 initial molecular bonds n<5 ruptured bonds reform initial Gaussian deformation along x direction with amplitude umax for x<0.2  chosen so initial umax does not cause rupture for x>0.2  chosen so initial umax causes rupture zero-displacement boundary conditions adiabatic boundary conditions initial 32x32 grid 6 refinement levels (3 visualized) refinement ratios: [ | ] Continuum DSMC + PCA 16.7 million atoms Animation of density of ruptured atomic bonds

The Old Well 10/27/2003 Duke Applied Math Seminar 30 Further Work Extend continuum-kinetic approach to fluid flow Implement realistic fluid kinetic model Verify validity of commonly used drag approximations for cilium flow Extend to multiple cilia