Numerical Simulation Of Spirochete Motility Alexei Medovikov, Ricardo Cortez, Lisa Fauci Tulane University Professor Stuart Goldstein (Department of Genetics

Numerical Simulation Of Spirochete Numerical Simulation Of Spirochete

MotilityMotility

Alexei Medovikov, Ricardo Cortez, Lisa Fauci Tulane University

Professor Stuart Goldstein (Department of Genetics and Cell Biology at the

University of Minnesota)

E-coli flagella

Introduction

Figure 1. Typical shape of spirochete L. illini

Introduction

Axial filament involvement in the motility of Leptospira interrogans. DB Bromley and N W Charon

Introduction

Dynamics of spirochete L. illini (Professor Stuart Goldstein Department of Genetics and Cell Biology at the University of Minnesota)

Introduction

SummarySummary

Model of the geometryModel of the mechanical motion Fluid dynamics of the spirocheteNumerical resultsDynamical simulations

Geometrical modelGeometrical model

)u(L)u( i

m

1ii

rr

ijji

ijj

i )uu(

)uu(

)u(L

)1u3u3u(61

)u(B

)4u6u3(61

)u(B

)1u3u3u3(61

)u(B

u61

)u(B

2333i

2332i

2331i

33i

)i)1k(u(B)u(m

1i

ki

rr

1m

1i

1ki )iku(B

du)u(d

Qr

i1ii rrQ

s)s(z),wssin()s(R)s(y),wscos()s(R)s(x

Step 1: Flagella along the whole body length

Step 2: Superhelix on top of the flagella


Torsion:

11 s

0

111

s

0

111 ds)s(swsinrds)s(swcosr)u()u( bnrR

Radius of the super helix:

dudu

)u(ds

u

0

1 r

1

1

11

1

dsd

)s(

kdsd

)u(duds

du)u(d

ds)u(d

bn

nτb

nτ

τrr


2/0.176r1

Arc length of the flagella:

Radius of the cell body:

2/0.147r2

Tangent:

Normal:

Binormal:

Tangent

Normal

Binormal

Mechanical model Mechanical model

z11 eωω τωω 22

))t()(t()t( 2n1 XARAX

XωXRωv 2n1 )()t(

2sin~

2cos

2sin~2

xIxIA

Reduce number of parameters describing the system (DoF) totwo rotations and translation

Rotation about vertical center line (0,0,1):

Rotation about tangent vector:

Rodrigues rotation matrix:

0xx

x0x

xx0~

12

13

23

xzor eτx where

Coordinate and velocity of a point on the surface:

X - coordinate in the “moving frame” coordinate system

))t()(t()t( 2n1 XARAX Mechanical model Mechanical model

02 ω11 ω 12 ω01 ω

82 ω11 ω 42 ω11 ω

No fluid yet

)s()s( 22 τωω


XωXRωv 2n1 )()t(

Velocity distribution due to rotation about tangent vectors of the flagellum:

z11 eωω

12 ω

01 ω

)s(τ

0L~VL~L~0F~VF~F~

3z

2z2

1z1

3z

2z2

1z1


Fluid mechanics of the swimming spirochete Fluid mechanics of the swimming spirochete

j

j

j

iijij x

)(u

x)(u

)(p)(xx

xx

3

1iiijj )(n)()(f xxx

D),()()()(

0

p

321

xxvxvxvxu

u

0σuStokes equation for the velocity of the fluid is LINEAR equation!

Hydrodynamical forces

where

- hydrostatic pressurep

- stress tensor

)( nz11 XRev

Xτv 22

z3 Vev

D

)(ds)( xxfF

D

)(ds)( xxfxL

0L~VL~L~0VFF~F~

3z

2z2

1z1

3z

2z2

1z1


We compute distribution of hydrodynamical forces over the surface for each boundary condition,and compute total force and moment

If motion is steady state – sum of forces and moments equal to zero:

Because Stokes equation is linear

)1(V)V(),1()(

)1(V)V(),1()(

2,12,1

2,12,1

LLLL

FFFF

Stokes equations can be resolved in terms of Stokeslets

3

1i

2i,0i

3j,0ji,0iij

0ij

)xx(||||r

,r

)xx)(xx(

r),(G

r

xx

3

1i

-iiD 0ij0j )(S))d(f-)(f(),(G

81

)(u xxxxxx


R. Cortez, L. Fauci, A. Medovikov The method of regularized stokeslets in three dimensions: analysis, validation, and application to helical swimming. Physics of Fluids 17, 031504 2005(also March 1, Volume 9, Issue 5, 2005 of Virtual Journal of Biological Physics Research)

2/722

4

)r(815

)(

x

)(ds)ff(),(G81

V)()(uV)()(u

3

1iiiD 0ij,

0D j0D\V j

xxx

dxxxdxxx

2/322j,0ji,0i

2/322

22

ij0ij, )r(

)xx)(xx(

)r()2r(

),(G

xx

where

is regularized Stokeslet

The method of regularized Stokeslets in three dimensionsThe method of regularized Stokeslets in three dimensions

Approximating of the regularized integral equation obtain local error estimate

10j0D j0D\V j Err)(uV)()(uV)()(u xdxxxdxxx

)(Oss

ErrErr),s,(Err3

3

3

3

21

)(ds)ff(),(G81 3

1iiiD 0ij, xxx

r

1

3

1i2

ppppppppi00ppij,00j Errws),s(J),s(f),s,,s(G

81

),s(u

2/5)x,D(distif,2

2/5)x,D(distif,1where

0

0

The method of regularized Stokeslets in three dimensionsThe method of regularized Stokeslets in three dimensions

r

1

3

1i ppppppppi00ppij,00j ws),s(J),s(f),s,,s(G

81

),s(u

FU

A


Given velocities of the boundary - compute hydrodynamical forces on the boundary

1U,0U,0U 2i31i3i3

For example for translational motion into z direction, velocity vector is:

and forces can be calculated by solving the linear algebra system (1)

(1)

Numerical ResultsNumerical Results

Velocity field of the liquid

3

1iiD 0ij0j )(S)d(f),(G

81

)(u xxxxx


1z

3z1

3z

1z1

F~F~

V

0F~VF~

0L~VL~L~0F~VF~F~

3z

2z2

1z1

3z

2z2

1z1

0L~VL~L~0F~VF~F~

3z

2z2

1z1

3z

2z2

1z1


Balance of forces and moments along z direction for steady-state motion


0L~VL~L~0F~VF~F~

3z

2z2

1z1

3z

2z2

1z1

1z

2z

2z

1z

1z

3z

3z

1z

1

1z

2z

2z

1z

3z

2z

2z

3z

LFLFLFLF

v

LFLFLFLF

v

1LFLFLFLF3z

2z

2z

3z

1z

3z

3z

1z1

Ratio of angular velocities (vertical axis) for different length of the spirochete (horizontal axis) : experiment vs. computations (blue)

Ratio of angular velocities of the cell body and anterior helix

Dynamical ModelDynamical Model

We approximate surface by network of connected springs

+reconfigure()+MotorSpring()+MotorSpring()+force()+getNumberOfSwitches()+getTotalAdditionalEnergy()

-m_reconfigurator : Reconfigurator *-m_numberOfSwitches : long-m_totalAdditionalEnergy : double-observer : IObserver *-switchForceValue : double

MotorSpring

+addSpring()+removeSpring()+getNumberOfAttachedSprings()+KineticEnergy()+getListOfSprings()+setParentFibre()+setArea()+getArea()+setJacob()+getJacob()+totalForce()+getWeight()+Node()+Node()+Node()+Node()

-listOfAttachedSprings : ListOfSprings-m_weight : double-m_jacob : double+ID : long+m_parentFibre : Fibre *+mass : double+m_area : double+q : Point+p : Point

Node

+Point()+Point()+Point()+Point()+operator =()+operator *()+crossProduct()+operator ==()+norm()

+x : double+y : double+z : double

Point

+Spring()+Spring()+attach()+detach()+removeSpring()+PotentialEnergy()+force()+getStrengthCoefficient()

+node1 : Node *+node2 : Node *+m_strengthCoefficient : double+restLength : double

Spring

-End3

1

-End4

1

-End5

1

-End6

*

+Fibre()+Fibre()+attach()+getNumOfSprings()+getNumOfNodes()+getNextNode()+getPreviouseNode()

+m_numOfSprings : long+m_numOfNodes : long+isMotor : bool+m_nodes : <unspecified> **+m_springs : <unspecified> **

Fibre

-End7

1-End8

*

Spring

Node

Fibre

1

UML Model (more about computational geometry)

+CCreatureBase()+~CCreatureBase()

CCreatureBase

+RingCreature()+RingCreature()+~RingCreature()+CollectNodes()+CollectVelocityNodes()+RotateRing()

+m_numOfSprings : int+m_numOfNodes : int+m_numOfVelocityNodes : int+m_velocityNodes : Node **+m_nodes : Node **+m_SprCoeffForIntRg : double+m_TotalMassOfRing : double+m_centerInfos : vector<RingCenterInfo>+forceRings : vector<Ring*>+velicityRings : vector<Ring*>+totalNumOfNodes : long+totalNumOfVelocityNodes : long

RingCreature

+MotorRing()+attachMotors()+reAttachMotors()+setStrengthCoefficientOfMotor()+getMotorSprings()+~MotorRing()

-m_motorSprings : MotorSpring **-motorOn : bool-strengthCoefficientOfMotor : double+numOfSwitches : long

MotorRing

-setTransformationMatrix()+Ring()+~Ring()+getRadius()+getStrengthCoefficient()+getNormal()+getCenter()+setCenter()+setNormal()+refreshCenter()+attach()+moveTo()+getNextNode()+getPreviouseNode()

-m_angle : double-m_radius : double-m_strengthCoefficient : double-m_normal : Point-m_center : Point-m_mass : double-m_transformationMatrix[3] : Point

Ring

-End1

1

-End2

*

+CenterLine()+CenterLine()+GetCentralLine()+lagrange()+interpolation()+tangent()+BSplineBasis()+BSplineDerivative()+BSpline2Derivative()+BSpline3Derivative()+dRds()+d2Rd2s()+d3Rd3s()+getRecursiveBSpline()+Flagella()

#radius : double+m_order : int

CenterLine

+GetCentralLine()+StaticCenterLine()+getFrameRotationalMatrix()

+m_RadiusOfFlagella : double+m_numOfPoints : int+flagellaCenters : vector<Point>+spiroCenters : vector<Point>+flagellaTangents : vector<Point>+spiroTangents : vector<Point>

StaticCenterLine

+Fibre()+Fibre(in numOfNodes : long, in numOfSprings : long)+attach(in fibre : Fibre*, in connector : <unspecified>*) : <unspecified> *+getNumOfSprings() : long+getNumOfNodes() : long+getNextNode(in ID : long) : <unspecified> *+getPreviouseNode(in ID : long) : <unspecified> *

+m_numOfSprings+m_numOfNodes+isMotor+m_nodes+m_springs

Fibre

1

Ring

N+1 N+1

N

UML Model


We approximate the initial intrinsic shape of the spirochete by a network of points and springs.

We use the boundary integral equations to calculate the surface velocities of elastic structures in Stokes fluid from surface elastic forces

The regularized Stokeslet method allows us to overcome difficulties related to the weak singularities of the boundary integral formulation

Numerical approximation leads to system of stiff ordinary differential equations, which we solve by DUMKA3 -a fast explicit solver for stiff ordinary differential equations

Because

ux

dt

)t(d

r

1

3

1i ppi0ij,

j w~)(f),(G8

1

dt

dx

xxx


)X(F)X(dtXd

elastic

A

Dynamical problem is a system of ODEs:

Where is calculated from elastic and geometrical properties of the surfaceelasticF

To solve system of ODE we use fast explicit DUMKA3 -a fast explicit solver for stiff

ordinary differential equations

n

ii

n

iin

n

ii

n

iin

nPnh

ASpP

hyy

1

'

1

110

)0(/

)(1|)1(||)(|

)1(

as large as possible

D,vvv)(u

,0x

)(u

,0x

)(

x)(p

)(u

3i

2i

1ii

3

1i i

i

3

1j j

ij

ii

xx

ux

xxx

r

1 ppppppppij ws),s(J),s(fF

)i)1k(u(B)u(m

1i

ki

rr

1m

1i

1ki )iku(B

du)u(d

Qr

i1ii rrQ

1)u(Bi

)y(max)u(y)y(min

)x(max)u(x)x(min

jij3i

jij3i

jij3i

jij3i

)1u3u3u(61

)u(B

)4u6u3(61

)u(B

)1u3u3u3(61

)u(B

u61

)u(B

2333i

2332i

2331i

33i

From: Mark DePristo's notes on biology: raven.bioc.cam.ac.uk/~mdepristo/

R F

0.01 0.201135

0.1 0.168359

0.2 0.119337

0.3 -0.137456

0.4 -0.128172

0.5 -0.37992

Rotation of a fragment of the spirochete about flagellum tangent vectors with angular velocity (a), rotation of the spirochete about with angular velocity (b) (view is taken from the point (0.8,0,13)).

)s( it 11 ze

1

Combination of rotations: , (a); , (b); , (c); (view is taken from the point (-3,3,14)).

11 1 01 111 0

A spirochete is a bacterium with a characteristic helical, elastic body. Because of its unique structure, a spirochete can swim in highly viscous, gel-like media, such as collagen within the mammal, and mucosal surfaces. Several species of spirochetes cause medically important diseases, some with grave consequences: Weil's disease, syphilis, yaws, bejel, pinta, Lyme disease (which is the most prevalent vector-borne disease in the United States), relapsing fever, leptospirosis and more. The spirochete is composed of different connected parts that have complicated shapes (several flagella, elastic spirochete's body, outer sheath and motors). We consider three aspects of the model: model of the geometry, model of the mechanical motion and the regularized Stokeslet method for simulation of fluid dynamics of the spirochete. We investigate the role of the geometry for the swimming and how we can compute global measurable characteristics of the motion.

Documents

Numerical Simulation Of Spirochete Motility Alexei Medovikov, Ricardo Cortez, Lisa Fauci Tulane University Professor Stuart Goldstein (Department of Genetics