2015, 12(6): 1303-1320. doi: 10.3934/mbe.2015.12.1303

The performance of discrete models of low reynolds number swimmers

1. 

Department of Mathematics, University of California Irvine, Irvine, CA, United States

2. 

School of Mathematics, University of Minnesota, Minneapolis, MN 55445

Received  September 2014 Revised  June 2015 Published  August 2015

Swimming by shape changes at low Reynolds number is widely used in biology and understanding how the performance of movement depends on the geometric pattern of shape changes is important to understand swimming of microorganisms and in designing low Reynolds number swimming models. The simplest models of shape changes are those that comprise a series of linked spheres that can change their separation and/or their size. Herein we compare the performance of three models in which these modes are used in different ways.
Citation: Qixuan Wang, Hans G. Othmer. The performance of discrete models of low reynolds number swimmers. Mathematical Biosciences & Engineering, 2015, 12 (6) : 1303-1320. doi: 10.3934/mbe.2015.12.1303
References:
[1]

G. Alexander, C. Pooley and J. Yeomans, Hydrodynamics of linked sphere model swimmers,, Journal of Physics: Condensed Matter, 21 (2009).

[2]

F. Alouges, A. DeSimone and A. Lefebvre, Optimal strokes for low reynolds number swimmers: An example,, Journal of Nonlinear Science, 18 (2008), 277. doi: 10.1007/s00332-007-9013-7.

[3]

F. Alouges, A. DeSimone and A. Lefebvre, Optimal strokes for axisymmetric microswimmers,, The European Physical Journal E: Soft Matter and Biological Physics, 28 (2009), 279. doi: 10.1140/epje/i2008-10406-4.

[4]

F. Alouges, A. DeSimone and L. Heltai, Numerical strategies for stroke optimization of axisymmetric microswimmers,, Mathematical Models and Methods in Applied Sciences, 21 (2011), 361. doi: 10.1142/S0218202511005088.

[5]

J. E. Avron, O. Gat and O. Kenneth, Optimal swimming at low reynolds numbers,, Phys. Rev. Lett, 93 (2004). doi: 10.1103/PhysRevLett.93.186001.

[6]

J. Avron, O. Kenneth and D. Oaknin, Pushmepullyou: An efficient micro-swimmer,, New Journal of Physics, 7 (2005). doi: 10.1088/1367-2630/7/1/234.

[7]

J. Avron and O. Raz, A geometric theory of swimming: Purcell's swimmer and its symmetrized cousin,, New Journal of Physics, 10 (2008). doi: 10.1088/1367-2630/10/6/063016.

[8]

C. Barentin, Y. Sawada and J. P. Rieu, An iterative method to calculate forces exerted by single cells and multicellular assemblies from the detection of deformations of flexible substrates,, Eur Biophys J, 35 (2006), 328. doi: 10.1007/s00249-005-0038-2.

[9]

N. P. Barry and M. S. Bretscher, Dictyostelium amoebae and neutrophils can swim,, PNAS, 107 (2010). doi: 10.1073/pnas.1006327107.

[10]

G. K. Batchelor, Brownian diffusion of particles with hydrodynamic interaction,, J. Fluid Mech., 74 (1976), 1. doi: 10.1017/S0022112076001663.

[11]

G. Batchelor, Slender-body theory for particles of arbitrary cross-section in stokes flow,, Journal of Fluid Mechanics, 44 (1970), 419. doi: 10.1017/S002211207000191X.

[12]

L. Becker, S. Koehler and H. Stone, On self-propulsion of micro-machines at low reynolds number: Purcell's three-link swimmer,, Journal of fluid mechanics, 490 (2003), 15. doi: 10.1017/S0022112003005184.

[13]

F. Binamé, G. Pawlak, P. Roux and U. Hibner, What makes cells move: Requirements and obstacles for spontaneous cell motility,, Molecular BioSystems, 6 (2010), 648.

[14]

J. P. Butler, I. M. Tolic-Norrelykke, B. Fabry and J. J. Fredberg, Traction fields, moments, and strain energy that cells exert on their surroundings,, Am J Physiol Cell Physiol, 282 (2001). doi: 10.1152/ajpcell.00270.2001.

[15]

G. T. Charras and E. Paluch, Blebs lead the way: How to migrate without lamellipodia,, Nat Rev Mol Cell Biol, 9 (2008), 730. doi: 10.1038/nrm2453.

[16]

R. Cox, The motion of long slender bodies in a viscous fluid part 1. General theory,, Journal of Fluid mechanics, 44 (1970), 791. doi: 10.1017/S002211207000215X.

[17]

O. Fackler and R. Grosse, Cell motility through plasma membrane blebbing,, The Journal of Cell Biology, 181 (2008), 879. doi: 10.1083/jcb.200802081.

[18]

R. Golestanian and A. Ajdari, Analytic results for the three-sphere swimmer at low reynolds number,, , (). doi: 10.1103/PhysRevE.77.036308.

[19]

G. Hancock, The self-propulsion of microscopic organisms through liquids,, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences, 217 (1953), 96. doi: 10.1098/rspa.1953.0048.

[20]

R. Johnson, An improved slender-body theory for stokes flow,, Journal of Fluid Mechanics, 99 (1980), 411. doi: 10.1017/S0022112080000687.

[21]

R. Johnson and C. Brokaw, Flagellar hydrodynamics. A comparison between resistive-force theory and slender-body theory,, Biophysical journal, 25 (1979), 113. doi: 10.1016/S0006-3495(79)85281-9.

[22]

J. Keller and S. Rubinow, Slender-body theory for slow viscous flow,, J. Fluid Mech, 75 (1976), 705. doi: 10.1017/S0022112076000475.

[23]

S. Kim and S. J. Karrila, Microhydrodynamics: Principles and Selected Applications, vol. 507,, Butterworth-Heinemann Boston, (1991).

[24]

T. Lämmermann, B. L. Bader, S. J. Monkley, T. Worbs, R. Wedlich-Söldner, K. Hirsch, M. Keller, R. Förster, D. R. Critchley, R. Fässler et al., Rapid leukocyte migration by integrin-independent flowing and squeezing,, Nature, 453 (2008), 51.

[25]

E. Lauga and T. R. Powers, The hydrodynamics of swimming microorganisms,, Repts. on Prog. in Phys., 72 (2009). doi: 10.1088/0034-4885/72/9/096601.

[26]

J. Lighthill, Flagellar hydrodynamics: The john von neumann lecture,, SIAM Review, (1975), 161. doi: 10.1137/1018040.

[27]

M. Lighthill, On the squirming motion of nearly spherical deformable bodies through liquids at very small reynolds numbers,, Communications on Pure and Applied Mathematics, 5 (1952), 109. doi: 10.1002/cpa.3160050201.

[28]

S. Michelin and E. Lauga, Efficiency optimization and symmetry-breaking in a model of ciliary locomotion,, Physics of Fluids (1994-present), 22 (2010). doi: 10.1063/1.3507951.

[29]

A. Najafi and R. Golestanian, A simplest swimmer at low reynolds number: Three linked spheres,, , ().

[30]

N. Osterman and A. Vilfan, Finding the ciliary beating pattern with optimal efficiency,, Proceedings of the National Academy of Sciences, 108 (2011), 15727. doi: 10.1073/pnas.1107889108.

[31]

E. Paluch, M. Piel, J. Prost, M. Bornens and C. Sykes, Cortical actomyosin breakage triggers shape oscillations in cells and cell fragments,, Biophysical journal, 89 (2005), 724. doi: 10.1529/biophysj.105.060590.

[32]

C. Pozrikidis, Boundary Integral and Singularity Methods for Linearized Viscous Flow,, Cambridge Univ Pr, (1992). doi: 10.1017/CBO9780511624124.

[33]

E. Purcell, Life at low Reynolds number,, Amer. J. Physics, 45 (1977), 3. doi: 10.1063/1.30370.

[34]

E. Purcell, Life at low reynolds number,, Am. J. Phys, 45 (1977), 3. doi: 10.1063/1.30370.

[35]

J. Renkawitz and M. Sixt, Mechanisms of force generation and force transmission during interstitial leukocyte migration,, EMBO reports, 11 (2010), 744. doi: 10.1038/embor.2010.147.

[36]

A. Shapere and F. Wilczek, Geometry of self-propulsion at low Reynolds number,, J. Fluid Mech., 198 (1989), 557. doi: 10.1017/S002211208900025X.

[37]

D. Tam and A. E. Hosoi, Optimal stroke patterns for purcell's three-link swimmer,, Physical Review Letters, 98 (2007). doi: 10.1103/PhysRevLett.98.068105.

[38]

P. J. M. Van Haastert, Amoeboid cells use protrusions for walking, gliding and swimming,, PloS one, 6 (2011).

[39]

Q. Wang, Modeling of Amoeboid Swimming at Low Reynolds Number,, PhD thesis, (2012).

[40]

Q. Wang, J. Hu and H. G. Othmer, Natural Locomotion in Fluids and on Surfaces: Swimming, Flying, and Sliding, chapter Models of low Reynolds number swimmers inspired by cell blebbing, 197-206,, Frontiers in Applications of Mathematics, (2012).

[41]

K. Yoshida and T. Soldati, Dissection of amoeboid movement into two mechanically distinct modes,, J. Cell Sci., 119 (2006), 3833. doi: 10.1242/jcs.03152.

show all references

References:
[1]

G. Alexander, C. Pooley and J. Yeomans, Hydrodynamics of linked sphere model swimmers,, Journal of Physics: Condensed Matter, 21 (2009).

[2]

F. Alouges, A. DeSimone and A. Lefebvre, Optimal strokes for low reynolds number swimmers: An example,, Journal of Nonlinear Science, 18 (2008), 277. doi: 10.1007/s00332-007-9013-7.

[3]

F. Alouges, A. DeSimone and A. Lefebvre, Optimal strokes for axisymmetric microswimmers,, The European Physical Journal E: Soft Matter and Biological Physics, 28 (2009), 279. doi: 10.1140/epje/i2008-10406-4.

[4]

F. Alouges, A. DeSimone and L. Heltai, Numerical strategies for stroke optimization of axisymmetric microswimmers,, Mathematical Models and Methods in Applied Sciences, 21 (2011), 361. doi: 10.1142/S0218202511005088.

[5]

J. E. Avron, O. Gat and O. Kenneth, Optimal swimming at low reynolds numbers,, Phys. Rev. Lett, 93 (2004). doi: 10.1103/PhysRevLett.93.186001.

[6]

J. Avron, O. Kenneth and D. Oaknin, Pushmepullyou: An efficient micro-swimmer,, New Journal of Physics, 7 (2005). doi: 10.1088/1367-2630/7/1/234.

[7]

J. Avron and O. Raz, A geometric theory of swimming: Purcell's swimmer and its symmetrized cousin,, New Journal of Physics, 10 (2008). doi: 10.1088/1367-2630/10/6/063016.

[8]

C. Barentin, Y. Sawada and J. P. Rieu, An iterative method to calculate forces exerted by single cells and multicellular assemblies from the detection of deformations of flexible substrates,, Eur Biophys J, 35 (2006), 328. doi: 10.1007/s00249-005-0038-2.

[9]

N. P. Barry and M. S. Bretscher, Dictyostelium amoebae and neutrophils can swim,, PNAS, 107 (2010). doi: 10.1073/pnas.1006327107.

[10]

G. K. Batchelor, Brownian diffusion of particles with hydrodynamic interaction,, J. Fluid Mech., 74 (1976), 1. doi: 10.1017/S0022112076001663.

[11]

G. Batchelor, Slender-body theory for particles of arbitrary cross-section in stokes flow,, Journal of Fluid Mechanics, 44 (1970), 419. doi: 10.1017/S002211207000191X.

[12]

L. Becker, S. Koehler and H. Stone, On self-propulsion of micro-machines at low reynolds number: Purcell's three-link swimmer,, Journal of fluid mechanics, 490 (2003), 15. doi: 10.1017/S0022112003005184.

[13]

F. Binamé, G. Pawlak, P. Roux and U. Hibner, What makes cells move: Requirements and obstacles for spontaneous cell motility,, Molecular BioSystems, 6 (2010), 648.

[14]

J. P. Butler, I. M. Tolic-Norrelykke, B. Fabry and J. J. Fredberg, Traction fields, moments, and strain energy that cells exert on their surroundings,, Am J Physiol Cell Physiol, 282 (2001). doi: 10.1152/ajpcell.00270.2001.

[15]

G. T. Charras and E. Paluch, Blebs lead the way: How to migrate without lamellipodia,, Nat Rev Mol Cell Biol, 9 (2008), 730. doi: 10.1038/nrm2453.

[16]

R. Cox, The motion of long slender bodies in a viscous fluid part 1. General theory,, Journal of Fluid mechanics, 44 (1970), 791. doi: 10.1017/S002211207000215X.

[17]

O. Fackler and R. Grosse, Cell motility through plasma membrane blebbing,, The Journal of Cell Biology, 181 (2008), 879. doi: 10.1083/jcb.200802081.

[18]

R. Golestanian and A. Ajdari, Analytic results for the three-sphere swimmer at low reynolds number,, , (). doi: 10.1103/PhysRevE.77.036308.

[19]

G. Hancock, The self-propulsion of microscopic organisms through liquids,, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences, 217 (1953), 96. doi: 10.1098/rspa.1953.0048.

[20]

R. Johnson, An improved slender-body theory for stokes flow,, Journal of Fluid Mechanics, 99 (1980), 411. doi: 10.1017/S0022112080000687.

[21]

R. Johnson and C. Brokaw, Flagellar hydrodynamics. A comparison between resistive-force theory and slender-body theory,, Biophysical journal, 25 (1979), 113. doi: 10.1016/S0006-3495(79)85281-9.

[22]

J. Keller and S. Rubinow, Slender-body theory for slow viscous flow,, J. Fluid Mech, 75 (1976), 705. doi: 10.1017/S0022112076000475.

[23]

S. Kim and S. J. Karrila, Microhydrodynamics: Principles and Selected Applications, vol. 507,, Butterworth-Heinemann Boston, (1991).

[24]

T. Lämmermann, B. L. Bader, S. J. Monkley, T. Worbs, R. Wedlich-Söldner, K. Hirsch, M. Keller, R. Förster, D. R. Critchley, R. Fässler et al., Rapid leukocyte migration by integrin-independent flowing and squeezing,, Nature, 453 (2008), 51.

[25]

E. Lauga and T. R. Powers, The hydrodynamics of swimming microorganisms,, Repts. on Prog. in Phys., 72 (2009). doi: 10.1088/0034-4885/72/9/096601.

[26]

J. Lighthill, Flagellar hydrodynamics: The john von neumann lecture,, SIAM Review, (1975), 161. doi: 10.1137/1018040.

[27]

M. Lighthill, On the squirming motion of nearly spherical deformable bodies through liquids at very small reynolds numbers,, Communications on Pure and Applied Mathematics, 5 (1952), 109. doi: 10.1002/cpa.3160050201.

[28]

S. Michelin and E. Lauga, Efficiency optimization and symmetry-breaking in a model of ciliary locomotion,, Physics of Fluids (1994-present), 22 (2010). doi: 10.1063/1.3507951.

[29]

A. Najafi and R. Golestanian, A simplest swimmer at low reynolds number: Three linked spheres,, , ().

[30]

N. Osterman and A. Vilfan, Finding the ciliary beating pattern with optimal efficiency,, Proceedings of the National Academy of Sciences, 108 (2011), 15727. doi: 10.1073/pnas.1107889108.

[31]

E. Paluch, M. Piel, J. Prost, M. Bornens and C. Sykes, Cortical actomyosin breakage triggers shape oscillations in cells and cell fragments,, Biophysical journal, 89 (2005), 724. doi: 10.1529/biophysj.105.060590.

[32]

C. Pozrikidis, Boundary Integral and Singularity Methods for Linearized Viscous Flow,, Cambridge Univ Pr, (1992). doi: 10.1017/CBO9780511624124.

[33]

E. Purcell, Life at low Reynolds number,, Amer. J. Physics, 45 (1977), 3. doi: 10.1063/1.30370.

[34]

E. Purcell, Life at low reynolds number,, Am. J. Phys, 45 (1977), 3. doi: 10.1063/1.30370.

[35]

J. Renkawitz and M. Sixt, Mechanisms of force generation and force transmission during interstitial leukocyte migration,, EMBO reports, 11 (2010), 744. doi: 10.1038/embor.2010.147.

[36]

A. Shapere and F. Wilczek, Geometry of self-propulsion at low Reynolds number,, J. Fluid Mech., 198 (1989), 557. doi: 10.1017/S002211208900025X.

[37]

D. Tam and A. E. Hosoi, Optimal stroke patterns for purcell's three-link swimmer,, Physical Review Letters, 98 (2007). doi: 10.1103/PhysRevLett.98.068105.

[38]

P. J. M. Van Haastert, Amoeboid cells use protrusions for walking, gliding and swimming,, PloS one, 6 (2011).

[39]

Q. Wang, Modeling of Amoeboid Swimming at Low Reynolds Number,, PhD thesis, (2012).

[40]

Q. Wang, J. Hu and H. G. Othmer, Natural Locomotion in Fluids and on Surfaces: Swimming, Flying, and Sliding, chapter Models of low Reynolds number swimmers inspired by cell blebbing, 197-206,, Frontiers in Applications of Mathematics, (2012).

[41]

K. Yoshida and T. Soldati, Dissection of amoeboid movement into two mechanically distinct modes,, J. Cell Sci., 119 (2006), 3833. doi: 10.1242/jcs.03152.

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