Numerical Simulation Of Spirochete Motility

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Numerical Simulation Of Spirochete Motility

• Alexei Medovikov, Ricardo Cortez, Lisa Fauci
• Tulane University
• Professor Stuart Goldstein
• (Department of Genetics and Cell Biology at the
  University of Minnesota)

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Introduction
E-coli flagella
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Introduction

• Figure 1. Typical shape of spirochete L. illini

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Introduction

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

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

• Model of the geometry
• Model of the mechanical motion
• Fluid dynamics of the spirochete
• Numerical results
• Dynamical simulations

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Geometrical model

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Geometrical model
Step 1 Flagella along the whole body length


Step 2 Superhelix on top of the flagella
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Geometrical model
Arc length of the flagella
Tangent
Tangent
Normal
Normal
Radius of the super helix
Binormal
Binormal
Torsion
Radius of the cell body
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Mechanical model
Reduce number of parameters describing the system
(DoF) to two rotations and translation

Rotation about vertical center line (0,0,1)
Rotation about tangent vector
Rodrigues rotation matrix
where
Coordinate and velocity of a point on the
surface
- coordinate in the moving frame coordinate
system
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Mechanical model
No fluid yet
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Mechanical model
Velocity distribution due to rotation about
tangent vectors of the flagellum
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Mechanical model
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Fluid mechanics of the swimming spirochete
Stokes equation for the velocity of the fluid is
LINEAR equation!
where
- stress tensor
- hydrostatic pressure
Hydrodynamical forces
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Fluid mechanics of the swimming spirochete
We compute distribution of hydrodynamical forces
over the surface for each boundary condition, and
compute total force and moment
Because Stokes equation is linear

If motion is steady state sum of forces and
moments equal to zero
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Fluid mechanics of the swimming spirochete

• Stokes equations can be resolved in terms of
  Stokeslets

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The method of regularized Stokeslets in three
dimensions

• 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)

where
is regularized Stokeslet
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The method of regularized Stokeslets in three
dimensions

• Approximating of the regularized integral
  equation obtain local error estimate

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Fluid mechanics of the swimming spirochete

Given velocities of the boundary - compute
hydrodynamical forces on the boundary
(1)
For example for translational motion into z
direction, velocity vector is
and forces can be calculated by solving the
linear algebra system (1)
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Numerical Results
Velocity field of the liquid
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Numerical Results


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Numerical Results
Balance of forces and moments along z direction
for steady-state motion
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Numerical Results
Ratio of angular velocities of the cell body and
anterior helix

Ratio of angular velocities (vertical axis) for
different length of the spirochete (horizontal
axis) experiment vs. computations (blue)
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Dynamical Model
We approximate surface by network of connected
springs
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UML Model (more about computational geometry)
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UML Model
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Dynamical 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

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Dynamical Model
Because
Dynamical problem is a system of ODEs
Where is calculated from elastic
and geometrical properties of the surface
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• To solve system of ODE we use fast explicit
  DUMKA3 -a fast explicit solver for stiff ordinary
  differential equations

as large as possible
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From Mark DePristo's notes on biology
raven.bioc.cam.ac.uk/mdepristo/
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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
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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)).
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Combination of rotations ,
(a) , (b)
, (c)
(view is taken from the point (-3,3,14)).
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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.