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)