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)

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

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

Geometrical model

Step 1: Flagella along the whole body length Step 2: Superhelix on top of the flagella Geometrical model

Torsion: Radius of the super helix: Geometrical model Arc length of the flagella: Radius of the cell body: Tangent: Normal: Binormal: Tangent Normal Binormal

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

Mechanical model No fluid yet

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

Mechanical model

Fluid mechanics of the swimming spirochete Stokes equation for the velocity of the fluid is LINEAR equation! Hydrodynamical forces where - hydrostatic pressure - stress tensor

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 If motion is steady state – sum of forces and moments equal to zero: Because Stokes equation is linear

Stokes equations can be resolved in terms of Stokeslets Fluid mechanics of the swimming spirochete

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, (also March 1, Volume 9, Issue 5, 2005 of Virtual Journal of Biological Physics Research) where is regularized Stokeslet The method of regularized Stokeslets in three dimensions

Approximating of the regularized integral equation obtain local error estimate The method of regularized Stokeslets in three dimensions

Fluid mechanics of the swimming spirochete Given velocities of the boundary - compute hydrodynamical forces on the boundary 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 Results Velocity field of the liquid

Numerical Results

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

Numerical Results 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 Model We approximate surface by network of connected springs

1 UML Model (more about computational geometry)

UML Model

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

Because Dynamical Model Dynamical problem is a system of ODEs: Where is calculated from elastic and geometrical properties of the surface

To solve system of ODE we use fast explicit DUMKA3 -a fast explicit solver for stiff ordinary differential equations as large as possible

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

RF

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)).

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

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.