Flagellated micro-swimmers in a viscous fluid

Sookkyung Lim Department of Mathematical Sciences

University of Cincinnati, USA IMA Workshop for Women in Mathematical Biology

University of Minnesota May 29-31, 2018

Acknowledgement

Howard Berg, Harvard U Charles S. Peskin, NYU Boyce Griffith, UNC Chapel Hill William Ko, University of Cincinnati Wanho Lee, NIMS, Korea Yongsam Kim, Choong-Ang U, Korea Yunyoung Park, Choong-Ang U, Korea

Bacterial flagella types

Monotrichous: Vibrio cholerae, Vibrio alginolyticus Amphitrichous: Alkaligens faecalis Lophotrichous: Spirillum Peritrichous: E. coli & Salmonella Typhimurium

Swimming bacteria, E. coli

E. Coli is a single-celled organism with a rod-shaped cell body and is equipped with a set of rotary motors. (peritrichous bacterium)

Number of flagella : typically 5-10 flagella Cell body : 2000nm long and 1000nm wide Flagellar length : typically 5000-10000nm long Helical wavelength (pitch) : 2000-2500nm Helical diameter : 400-600nm Diameter of flagella : about 20nm Rotation rate: 100Hz Hook length: 50-80nm

Howard Berg, Motile behavior of bacteria, Physics Today, 2000

Schematic of a swimming bacteria

Lauga (2016), Turner et al (2000), Darton et al (2007) From H. Berg, Harvard U

Modeling a flagellum: Kirchhoff rod theory

Left-handed Right-handed

D 1

D 3D 2

X

Unconstrained Kirchhoff rod model

Unconstrained Kirchhoff rod model

Double-well:

Rela%on  between  helical  geometry  and  intrinsic  property  

Equations of motion

Fluid equations

Structure equations

Interaction equations

Lim et al (SISC 2008), Olson, Lim, & Cortez (JCP 2013), Lim et al (PRE2017)

A left-handed helix (Lim & Peskin, PRE 2012)

CCW CW

Twsit  energy  single  well  poten%al  

right left

CC

W

CW

up

up

down

down

Two left-handed helices bundling unbundling

A bistable flagellum

Polymorphic transformations

Flagellar unbundling with bistable flagella

Flagellar unbundling with single-stable flagella

Bacterial flagellar unbundling

Various polymorphic forms

Turner et al (2000), Darton et al (2007)

A typical sequence of polymorphic transformations

Bundling process of four rotating flagella

Hook’s role: duration of bundle formulation with two flagella

Hook’s role

Flagellar bundling and unbundling caused by a single flagellum

Ongoing projects

•  A free swimmer: •  Bacterial chemotaxis (with Xue, OSU) •  Microswimmers in complex fluids (with Griffith, NC Chapel Hill) •  Microswimmers near wall (with Y. Kim, Chungang U) •  Lubrication theory for flagellar bundling •  (with T. Fai & C. Rycroft, Harvard)