the old well 10/27/2003 duke applied math seminar 1 continuum-molecular computation of biological...

10/27/2003 Duke Applied Math Seminar1

The Old Well

Continuum-molecular computation of biological microfluidics

Sorin [email protected]

http://www.amath.unc.edu/Faculty/mitran

Electron micrograph of cilium cross section

Cilium beat pattern Computational grid

Applied Mathematics Programhttp://www.amath.unc.edu

The University of North Carolina

at Chapel Hill

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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 BiophysicsJohn Sheehan


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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


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Cilium structure

• Dimensions:Cilium diameter: 225 nmCilium length: 7000 nmMicrotubule exterior diameter: 25 nmMicrotubule interior diameter: 14 nmMicrotubule centers diameter: 170 nm

• Microtubules form supporting structure with flexural rigidity:

10 ²⁴ 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

http://cellbio.utmb.edu/cellbio/cilia.htm Gwen V. Childs

http://cellbio.utmb.edu/cellbio/cilia.htm

http://cellbio.utmb.edu/cellbio/cilia.htm


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Ciliary Flow

• Cilia maintain motion of PCL (periciliaryliquid) and mucus layer with role in filtration and elimination of harmful agents

• Cilia beat 10-40 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

• RePCL = 3 × 10 (very slow, viscous flow)⁻⁹


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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?


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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

T

B

N

NV

NMBM

BV

TV

i

js u

vw

TM

R is, t x is, tî y is, t z is, tk, i 1, , 20, #

T R s , N 1

T s , B 1

N s T #

U i u i v i w i i i iT,

F i V T,i V N,i V B ,i MT,i MN,i MB ,iT.

#

# U e

U i

U j, F e

F i

F j. #


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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

T

B

N

NV

NMBM

BV

TV

i

js u

vw

TM

F e F eU e . #

F eU e F eU 0e K 0

e U e U 0e 1

2 Ue U 0

e TL0e U e U 0

e #

K eU e U 0e

U 0e

U eF e

U e UedU e #


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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

T

B

N

NV

NMBM

BV

TV

i

js u

vw

TM

F e,n 1 F e,n K e,n U e,n 1 U e,n #

K e,n F e

U e Ue,n . #

fluiddynein FFKUUM


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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


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Fluid-structure interaction model: Fluid forces

• Simultaneously solve unsteady Stokes equations to provide full fluid stress tensor at cilium membrane surface

iji

j

j

iij

A

fluid pxu

xudSF

,

uptu

21


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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

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0Z

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1

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X

Y

Z

11181),,(

),,(),,(8

1

l

llijk

N

NQq


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Fluid computation – Moving Mesh

• Use a time-dependent grid mapping between Cartesian computational space and physical space

),,(),,,(: tYYtXXT


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Fluid computation – Moving Mesh

• Solve unsteady Stokes equations using projection method in computational space

TSRGFq

trgfq

t

yyxxyxt

~

IYXYX

YXqg

qf

qF

qYXYX

YXgf

F

tt

tt

~~

~

~

~

~~~


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Fluid computation – Typical result

• Imposed dynein forces

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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)

t

ionInterpolatInject

AveragingRestrict

t

2/t 2/t

4/t 4/t 4/t 4/t


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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

A Toy problem: one-dimensional model of ductile failure in a rod

Model features

Progressive damage Bonds break due to combined thermal, mechanical effect

},...,1,0{,)(Nn

nnk

Point masses connected by multiple springs

m mcba x

)(xF

Force-deformation law


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Equations of motion

Displacement from equilibrium:

020)2(

211

11

luuuklu

uuukum

iiii

iiii

/,0 22 lkcucu xxtt

0)()( 12/112/1 iiiiiii uunuunum

0)(2 xxtt uxcu

Continuum limit

With damage

iu

Mass per lattice spacing:

lm /

No damage:

Continuum limit

/)()(2 xnxc

Presence of damage requires microscopic information, i.e. the number of broken springs

iu 1iu1iu

2/12/1 ii nk 2/12/1 ii nk

Zero temperature limit


The Old Well

Two-dimensional model of thin shell failure

cba

0)(

)(

)(

)(

,2/11,2/12/1,2/1

1,2/1,2/12/1,2/1

2/1,2/1,12/1,2/1

2/1,12/1,2/1,2/12/1,

jijiji

jijiji

jijiji

jijijiji

uun

uun

uun

uunum

i

2D lattice of oscillators

j

No damage

0)/(

0)/(2

2

ijlij

ijlij

vkv

uku

0

022

22

vcv

ucu

tt

ttContinuum limit

With damage

0),(

0),(2

2

vyxcv

uyxcu

tt

ttContinuum limit


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Failure scenarios

Bonds break under dynamic loading due to combined thermal (microscopic) and continuum motion

cba

i

Dynamic loading

j

cba

i

Melting

j


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DNS results – no damage

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time

u

Particle trajetories

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(n-n

0)/n

0

Relative number broken bonds


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DNS results – with damage

Extend rod through constant, outward velocity boundary conditions at end points

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time

u

Particle trajetories

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(n-n

0)/n

0

Relative number broken bonds


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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


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Eliminating thermal behavior Microscopic dynamics

Principal component analysis

0)2( 11 iiii uuukum contains all system information Very little of the information is relevant macroscopically Coarse graining approaches:

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Principal modes – 32 point masses

Cutoff after three decadesN

Nji

P

n

jnji

ni

ni

nii

rrC

Puuuu

C

Pntuutu

......

1))((

,...,1)},({)(

21

,11

► Spatial averaging - homogenization► Fourier mode elimination - RNG


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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

},1|),,,{( Mjivuvu nij

nij

nij

nij

Unbiased Monte Carlo simulation requires extensive sampling – too expensive► hierarchical Monte Carlo► bias sampling in accordance with principal components from immediately coarser level

-1

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1.5

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The Old Well

Continuum-Microscopic Interaction

Continuum to microscopic injection of values

frequency lowfrequencyhigh

,ij

ijijijij u

uuuu

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

TyxTt ),(

modes thermal- modes elastic - modesminor - modes principal -

,,..., 21

TEMPTEPMPR

rrrRrCr N


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Instantaneous Constitutive Relations

For simple model considered here only constitutive relation is the dependence of elastic speed upon local damage

dydxyxn

yxyxc ),(/),(2

Update of local damage: ► on each level ► after each time step ► check if displacement has increased beyond current elastic law restriction

cba x

)(xF

Force-deformation law


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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:[ 2 2 2 | 8 8 8 ]Continuum DSMC + PCA 16.7 million atoms

Animation of density of ruptured atomic bonds


The Old Well

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

the old well 10/27/2003 duke applied math seminar 1 continuum-molecular computation of biological...

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