Abstract
Mathematical modeling and quantitative study of biological motility is producing new biophysical insight and opportunities for discoveries at the level of both fundamental science and technology. One example is the elucidation of how complex behavior of simple organisms emerges from specific (and sophisticated) body architectures, and how this is affected by environmental cues. Moreover, the two-directional interaction between biology and mechanics is promoting new approaches to problems in engineering and in the life sciences: understand biology by constructing bio-inspired machines, build new machines thanks to bio-inspiration.
This article contains an introduction to the mathematical study of swimming locomotion of unicellular organisms (e.g., unicellular algae). We use the tools of geometric control theory to identify some general principles governing life at low Reynolds numbers, that can guide the design of engineered devices trying to replicate the successes of their biological counterparts. Locomotion strategies employed by biological organism are, in fact, a rich source of inspiration for studying mechanisms for shape control. We focus on morphing mechanisms based on Gauss’ theorema egregium, which shows that the curvature of a thin shell can be controlled through lateral modulations of stretches induced in its mid-surface. We discuss some examples of this Gaussian morphing principle both in nature and technology.
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References
D. Agostinelli, F. Alouges, A. De Simone, Peristaltic waves as optimal gaits in metameric bio-inspired robots. Front. Robot. and AI 5, 99 (2018)
D. Agostinelli, A. Lucantonio, G. Noselli, A. DeSimone, Nutations in growing plant shoots: the role of elastic deformations due to gravity loading. J. Mech. Phys. Solids 136, 103702 (2019)
D. Agostinelli, R. Cerbino, J. Del Alamo, A. DeSimone, A. Hoehn, C. Micheletti, G. Noselli, E. Sharon, J. Yeomans, Micromotility: state of the art, recent accomplishments and perspectives on the mathematical modeling of bio-motility at microscopic scales. Math. Eng. (2020) https://doi.org/10.3934/mine.2020011
V. Agostiniani, A. DeSimone, Γ-convergence of energies for nematic elastomers in the small strain limit. Contin. Mech. Thermodyn. 23(3), 257–274 (2011)
V. Agostiniani, A. DeSimone, A. Lucantonio, D. Lucic, Foldable structures made of hydrogel bilayers. Math. Eng. 1, 204–223 (2018). https://doi.org/10.3934/mine.2018.1.204
H. Aharoni, Y. Abraham, R. Elbaum, E. Sharon, R. Kupferman, Emergence of spontaneous twist and curvature in non-Euclidean rods: application to Erodium plant cells. Phys. Rev. Lett. 108, 238106 (2012)
H. Aharoni, E. Sharon, R. Kupferman, Geometry of thin nematic elastomer sheets. Phys. Rev. Lett. 113, 257801 (2014)
B. Alberts, A. Johnson, J. Lewis, D. Morgan, M. Raff, K. Roberts, P. Walter, Molecular Biology of the Cell, 6th ed. (Garland Science, New York, 2014)
F. Alouges, A. DeSimone, A. Lefebvre, Optimal strokes for low Reynolds number swimmers: an example. J. Nonlinear Sci. 18(3), 277–302 (2008)
F. Alouges, A. DeSimone, A. Lefebvre, Optimal strokes for axisymmetric microswimmers. Eur. Phys. J. E 28(3), 279–284 (2009)
F. Alouges, A. Desimone, L. Heltai, Numerical strategies for stroke optimization of axisymmetric microswimmers. Math. Models Methods Appl. Sci. 21(2), 361–387 (2011)
F. Alouges, A. DeSimone, L. Giraldi, M. Zoppello, Self-propulsion of slender micro-swimmers by curvature control: N-link swimmers. Int. J. Non Linear Mech. 56, 132–141 (2013)
F. Alouges, A. DeSimone, L. Heltai, A. Lefebvre, B. Merlet, Optimally swimming stokesian robots. Discrete Contin. Dynam. Systems B 18, 1189–1215 (2013)
F. Alouges, A. DeSimone, L. Giraldi, M. Zoppello, Can magnetic multilayers propel artificial microswimmers mimicking sperm cells? Soft Rob. 2, 117–128 (2015)
F. Alouges, A. Desimone, L. Giraldi, Y. Or, O. Wiezel, Energy-optimal strokes for multi-link microswimmers: Purcell’s loops and Taylor’s waves reconciled. New J. Phys. 21(4), 043050 (2019)
D. Ambrosi, M.B. Amar, C.J. Cyron, A. DeSimone, A. Goriely, J.D. Humphrey, E. Kuhl, Growth and remodelling of living tissues: perspectives, challenges and opportunities. J. R. Soc. Interface 16(157), 20190233 (2019)
M. Arroyo, A. DeSimone, Shape control of active surfaces inspired by the movement of euglenids. J. Mech. Physics Solids 62, 99–112 (2014)
M. Arroyo, D. Milan, L. Heltai, A. DeSimone, Reverse engineering the euglenoid movement. Proc. Nat. Acad. Sci. USA 109, 17874–17879 (2012)
M. Barchiesi, A. DeSimone, Frank energy for nematic elastomers: a nonlinear model. ESAIM Control Optim. Calc. Var. 21(2), 372–377 (2015)
H.C. Berg, D.A. Brown, Chemotaxis in escherichia coli analysed by three-dimensional tracking. Nature 239(5374), 500–504 (1972)
M. Bergert, A. Erzberger, R.A. Desai, I.M. Aspalter, A.C. Oates, G. Charras, G. Salbreux, E.K. Paluch, Force transmission during adhesion-independent migration. Nat. Cell Biol. 17(4), 524 (2015)
L. Berti, L. Giraldi, C. Prud’Homme, Swimming at Low Reynolds Number. ESAIM: Proceedings and Surveys, EDP Sciences, 2019, pp. 1–10
P. Bladon, E.M. Terentjev, M. Warner, Transitions and instabilities in liquid crystal elastomers. Phys. Rev. E 47, R3838–R3840 (1993)
A. Bressan, Impulsive control of lagrangian systems and locomotion in fluids. Discrete Contin. Dynam. Systems 20(1), 1 (2008)
A.B.C. Buskermolen, H. Suresh, S.S. Shishvan, A. Vigliotti, A. DeSimone, N.A. Kurniawan, C.V.C. Bouten, V.S. Deshpande, Entropic forces drive cellular contact guidance. Biophys. J. 116(10), 1994–2008 (2019)
M. Camacho-Lopez, H. Finkelmann, P. Palffy-Muhoray, M. Shelley, Fast liquid-crystal elastomer swims into the dark. Nat. Mater. 3(5), 307 (2004)
L. Cardamone, A. Laio, V. Torre, R. Shahapure, A. DeSimone, Cytoskeletal actin networks in motile cells are critically self-organized systems synchronized by mechanical interactions. Proc. Natl. Acad. Sci. 108(34), 13978–13983 (2011)
D. Carroll, E. Hankins, E. Kose, I. Sterling, A survey of the differential geometry of discrete curves. Math. Intell. 36(4), 28–35 (2014)
A.N. Caruso, A. Cvetkovic, A. Lucantonio, G. Noselli, A. DeSimone, Spontaneous morphing of equibiaxially pre-stretched elastic bilayers: the role of sample geometry. Int. J. Mech. Sci. 149, 481–486 (2018)
B. Chan, N.J. Balmforth, A.E. Hosoi, Building a better snail: lubrication and adhesive locomotion. Phys. Fluids 17(11), 113101 (2005)
G. Charras, E. Paluch, Blebs lead the way: how to migrate without lamellipodia. Nat. Rev. Mol. Cell Biol. 9(9), 730 (2008)
S. Childress, Mechanics of Swimming and Flying, vol. 2 (Cambridge University Press, Cambridge, 1981)
G. Cicconofri, A. DeSimone, A study of snake-like locomotion through the analysis of a flexible robot model. Proc. R. Soc. London, Ser. A Math. Phys. Eng. Sci. 471(2184), 20150054 (2015)
G. Cicconofri, A. DeSimone, Motion planning and motility maps for flagellar microswimmers. Eur. Phys. J. E 39, 72 (2016)
G. Cicconofri, A. DeSimone, Modelling biological and bio-inspired swimming at microscopic scales: recent results and perspectives. Comput. Fluids 179, 799–805 (2019)
G. Cicconofri, M. Arroyo, G. Noselli, A. DeSimone, Morphable structures from unicellular organisms with active, shape-shifting envelopes: variations on a theme by Gauss. Int. J. Non Linear Mech. 118, 103278 (2020)
S. Conti, A. DeSimone, G. Dolzmann, Soft elastic response of stretched sheets of nematic elastomers: a numerical study. J. Mech. Phys. Solids 50(7), 1431–1451 (2002)
G. Corsi, A. DeSimone, C. Maurini, S. Vidoli, A neutrally-stable shell in a Stokes flow: a rotational Taylor sheet. Proc. R. Soc. A 475, 20190178 (2019)
S.M. Coyle, E.M. Flaum, H. Li, D. Krishnamurthy, M. Prakash, Coupled active systems encode emergent behavioral dynamics of the unicellular predator Lacrymaria olor. Curr. Biol. 29(22), 3838–3850. e3 (2019)
G. Dal Maso, A. DeSimone, M. Morandotti, An existence and uniqueness result for the motion of self-propelled micro-swimmers. SIAM J. Math. Anal. 43, 1345–1368 (2011)
C. Darwin, The Power of Movement in Plants (John Murray, London, 1880)
A. DeSimone, Hysteresis and imperfection sensitivity in small ferromagnetic particles. Meccanica 30(5), 591–603 (1995)
A. DeSimone, Energetics of fine domain structures. Ferroelectrics 222(1–4), 275–284 (1999)
A. DeSimone, Spontaneous bending of pre-stretched bilayers. Meccanica 53, 511–518 (2018)
A. DeSimone, G. Dolzmann, Macroscopic response of nematic elastomers via relaxation of a class of SO(3)-invariant energies. Arch. Ration. Mech. Anal. 161(3), 181–204 (2002)
A. DeSimone, L. Teresi, Elastic energies for nematic elastomers. Eur. Phys. J. E 29(2), 191–204 (2009)
A. DeSimone, A. Tatone, Crawling motility through the analysis of model locomotors: two case studies. Eur. Phys. J. E 35(9), 85 (2012)
A. DeSimone, F. Guarnieri, G. Noselli, A. Tatone, Crawlers in viscous environments: linear vs non-linear rheology. Int. J. Non Linear Mech. 56, 142–147 (2013)
A. DeSimone, P. Gidoni, G. Noselli, Liquid crystal elastomer strips as soft crawlers. J. Mech. Phys. Solids 84, 254–272 (2015)
M.P. do Carmo, Differential Geom. of Curves and Surfaces. Paperback (Prentice-Hall, Inc, Englewood Cliffs, 1976)
K. Drescher, R.E. Goldstein, N. Michel, M. Polin, I. Duval, Direct measurement of the flow field around swimming microorganisms. Phys. Rev. Lett. 105, 168101 (2010)
A. Fukunaga, K. Urayama, T. Takigawa, A. DeSimone, L. Teresi, Dynamics of electro-opto-mechanical effects in swollen nematic elastomers. Macromolecules 41(23), 9389–9396 (2008)
E.A. Gaffney, H. Gadelha, D.J. Smith, J.R. Blake, J.C. Kirkman-Brown, Mammalian sperm motility: observation and theory. Annu. Rev. Fluid Mech. 43, 501–528 (2011)
P. Gidoni, G. Noselli, A. DeSimone, Crawling on directional surfaces. Int. J. Non Linear Mech. 61, 65–73 (2014)
N. Giuliani, N. Heltai, A. DeSimone, Predicting and optimizing micro-swimmer performance from the hydrodynamics of its components: the relevance of interactions. Soft Rob. 5(4) (2018)
R.E. Goldstein, Green algae as model organisms for biological fluid dynamics. Annu. Rev. Fluid Mech. 47, 343–375 (2015)
H. Gu, L. Bumke, C. Chluba, E. Quandt, R.D. James, Phase engineering and supercompatibility of shape memory alloys. Mater. Today 21(3), 265–277 (2018)
J.S. Guasto, K.A. Johnson, J.P. Gollub, Oscillatory flows induced by microorganisms swimming in two dimensions. Phys. Rev. Lett. 105, 168102 (2010)
Z.V. Guo, L. Mahadevan, Limbless undulatory propulsion on land. Proc. Natl. Acad. Sci. 105(9), 3179–3184 (2008)
P. Holmes, R.J. Full, D. Koditschek, J. Guckenheimer, The dynamics of legged locomotion: models, analyses, and challenges. SIAM Rev. 48(2), 207–304 (2006)
D.L. Hu, J. Nirody, T. Scott, M.J. Shelley, The mechanics of slithering locomotion. Proc. Natl. Acad. Sci. 106(25), 10081–10085 (2009)
A.J. Ijspeert, Central pattern generators for locomotion control in animals and robots: a review. Neural Netw. 21(4), 642–653 (2008)
A.Y. Khapalov, Local controllability for a swimming model. SIAM J. Control. Optim. 46(2), 655–682 (2007)
A. Khapalov, P. Cannarsa, F.S. Priuli, G. Floridia, Well-posedness of 2-d and 3-d swimming models in incompressible fluids governed by navier–stokes equations. J. Math. Anal. Appl. 429(2), 1059–1085 (2015)
J. Kim, J.A. Hanna, M. Byun, C.D. Santangelo, R.C. Hayward, Designing responsive buckled surfaces by halftone gel lithography. Science 335(6073), 1201–1205 (2012)
S. Kim, C. Laschi, B. Trimmer, Soft robotics: a bioinspired evolution in robotics. Trends Biotechnol. 31(5), 287–294 (2013)
Y. Klein, E. Efrati, E. Sharon, Shaping of elastic sheets by prescription of non-Euclidean metrics. Science 315, 1116–1120 (2007)
J.H. Lai, J.C. del Alamo, J. Rodríguez-Rodríguez, J.C. Lasheras, The mechanics of the adhesive locomotion of terrestrial gastropods. J. Exp. Biol. 213(22), 3920–3933 (2010)
E. Lauga, A.E. Hosoi, Tuning gastropod locomotion: modeling the influence of mucus rheology on the cost of crawling. Phys. Fluid. 18(11), 113102 (2006)
E. Lauga, T.R. Powers, The hydrodynamics of swimming microorganisms. Rep. Prog. Phys. 72(9), 096601 (2009)
B.S Leander, G. Lax, A. Karnkowska, A.G.B. Simpson, Euglenida. In: Handbook of the Protists (Springer, Cham, 2017)
Sir J. Lighthill. Mathematical Biofluiddynamics (SIAM, Philadelphia, 1975)
J. Lin, D. Nicastro, Asymmetric distribution and spatial switching of dynein activity generates ciliary motility. Science 360(6387), eaar1968 (2018)
C.B. Lindemann, K.A. Lesich, Flagellar and ciliary beating: the proven and the possible. J. Cell Sci. 123(4), 519–528 (2010)
J. Lohéac, J.-F. Scheid, M. Tucsnak, Controllability and time optimal control for low Reynolds numbers swimmers. Acta Appl. Math. 123(1), 175–200 (2013)
K.E. Machin, Wave propagation along flagella. J. Exp. Biol. 35(4), 796–806 (1958)
R. Marchello, M. Morandotti, H. Shum, M. Zoppello, The n-link swimmer in three dimensions: controllability and optimality results. arXiv preprint arXiv:1912.04998 (2019)
C.D. Modes, M. Warner, Negative Gaussian curvature from induced metric changes. Phys. Rev. E 92, 010401 (2015)
F. Montenegro-Johnson, E. Lauga, Optimal swimming of a sheet. Phys. Rev. E 89, 060701(R) (2014)
R. Montgomery, A Tour of Subriemannian Geometries, Their Geodesics and Applications, vol. 91 (American Mathematical Society, Providence, 2002)
A. Montino, A. DeSimone, Three-sphere low-Reynolds-number swimmer with a passive elastic arm. Eur. Phys. J. E 38(42), 1–10 (2015)
C. Mostajeran, Curvature generation in nematic surfaces. Phys. Rev. E 91, 062405 (2015)
A. Najafi, R. Golestanian, Simple swimmer at low Reynolds number: three linked spheres. Phys. Rev. E 69(6), 062901 (2004)
G. Noselli, M. Arroyo, A. DeSimone, Smart helical structures inspired by the pellicle of euglenids. J. Mech. Phys. Solids 123, 234–246 (2019)
G. Noselli, A. Beran, M. Arroyo, A. DeSimone, Swimming Euglena respond to confinement with a behavioral change enabling effective crawling. Nat. Phys. 15, 496–502 (2019)
C. Pehlevan, P. Paoletti, L. Mahadevan, Integrative neuromechanics of crawling in D. melanogaster larvae. Elife 5, e11031 (2016)
E.M. Purcell, Life at low Reynolds number. Am. J. Phys. 45(1), 3–11 (1977)
P. Recho, T. Putelat, L. Truskinovsky, Contraction-driven cell motility. Phys. Rev. Lett. 111(10), 108102 (2013)
P. Recho, A. Jerusalem, A. Goriely, Growth, collapse, and stalling in a mechanical model for neurite motility. Phys. Rev. E 93(3), 032410 (2016)
M. Rossi, G. Cicconofri, A. Beran, G. Noselli, A. DeSimone, Kinematics of flagellar swimming in Euglena gracilis: Helical trajectories and flagellar shapes. Proc. Natl. Acad. Sci. U. S. A. 114(50), 13085–13090 (2017)
C. Santangelo, Buckling thin disks and ribbons with non-Euclidean metrics. Europhys. Lett. 86, 34003 (2011)
P. Sartori, V.F. Geyer, A. Scholich, F. Jülicher, J. Howard, Dynamic curvature regulation accounts for the symmetric and asymmetric beats of Chlamydomonas flagella. Elife 5, e13258 (2016)
Y. Sawa, K. Urayama, T. Takigawa, A. DeSimone, L. Teresi, Thermally driven giant bending of liquid crystal elastomer films with hybrid alignment. Macromolecules 43, 4362–4369 (2010)
P. Sens, Rigidity sensing by stochastic sliding friction. Europhys. Lett. 104(3), 38003 (2013)
A. Shahaf, E. Efrati, R. Kupferman, E. Sharon, Geometry and mechanics in the opening of chiral seed pods. Science 333(6050), 1726–1730 (2011)
D. Tam, A.E. Hosoi, Optimal stroke patterns for Purcell’s three-link swimmer. Phys. Rev. Lett. 98, 068105 (2007)
D. Tam, A.E. Hosoi, Optimal kinematics and morphologies for spermatozoa. Phys. Rev. E 83, 045303(R) (2011)
G.I. Taylor, Analysis of the swimming of microscopic organisms. Proc. R. Soc. Lond. A Math. Phys. Eng. Sci. 209, 447–461 (1951)
G.I. Taylor, Low Reynolds number flows, movie for The National Committee for Fluid Mechanics Films (1966). http://web.mit.edu/hml/ncfmf.html. Online accessed 6 Nov 2019
S. Timoshenko, Analysis of bi-metal thermostats. J. Optical Soc. Am. 11, 233–255 (1925)
B. Tondu, Modelling of the McKibben artificial muscle: a review. J. Intell. Mater. Syst. Struct. 23, 225–253 (2012)
Q. Wang, H. Othmer, The performance of discrete models of low Reynolds number swimmers. Math. Biosci. Eng. 12, 1303 (2015)
Q. Wang, H.G. Othmer, Computational analysis of amoeboid swimming at low Reynolds number. J. Math. Biol. 72(7), 1893–1926 (2016)
M. Warner, E.M. Terentjev, Liquid Crystal Elastomers, vol. 120 (Oxford University Press, Oxford, 2007)
M. Warner, C.D. Modes, D. Corbett, Curvature in nematic elastica responding to light and heat. Proc. R. Soc. Lond. A Math. Phys. Eng. Sci. 466(2122), 2975–2989 (2010)
T.J. White, D.J. Broer, Programmable and adaptive mechanics with liquid crystal polymer networks and elastomers. Nat. Mat. 14(11), 1087 (2015)
H. Wu, A. Farutin, W.-F. Hu, M. Thiébaud, S. Rafaï, P. Peyla, M.-C. Lai, C. Misbah, Amoeboid swimming in a channel. Soft Matt. 12(36), 7470–7484 (2016)
S. Zhang, R.D. Guy, J.C. Lasheras, J.C. del Álamo, Self-organized mechano-chemical dynamics in amoeboid locomotion of physarum fragments. J. Phys. D. Appl. Phys. 50(20), 204004 (2017)
J. Zhu, A. Mogilner, Mesoscopic model of actin-based propulsion. PLoS Comput. Biol. 8(11), e1002764 (2012)
Acknowledgements
We gratefully acknowledge the support by the European Research Council through the ERC Advanced Grant 340685-MicroMotility. These lecture notes draw freely from results obtained with several co-authors over the last 10 years, and published in the papers referenced in the bibliography (in particular, references [35, 36]). The collaboration with them has been a source of endless joy and inspiration.
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DeSimone, A. (2020). Cell Motility and Locomotion by Shape Control. In: Ambrosi, D., Ciarletta, P. (eds) The Mathematics of Mechanobiology. Lecture Notes in Mathematics(), vol 2260. Springer, Cham. https://doi.org/10.1007/978-3-030-45197-4_1
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