PHYSICS OF FLUIDS 18, 113102 ͑2006͒ Tuning gastropod locomotion: Modeling the influence of mucus rheology on the cost of crawling ͒ Eric Laugaa Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139 A. E. Hosoi Hatsopoulos Microfluids Laboratory, Department of Mechanical Engineering, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139 ͑Received 17 August 2006; accepted 10 October 2006; published online 22 November 2006͒ Common gastropods such as snails crawl on a solid substrate by propagating muscular waves of shear stress on a viscoelastic mucus. Producing the mucus accounts for the largest component in the gastropod’s energy budget, more than 20 times the amount of mechanical work used in crawling. Using a simple mechanical model, we show that the shear-thinning properties of the mucus favor a decrease in the amount of mucus necessary for crawling, thereby decreasing the overall energetic cost of locomotion. © 2006 American Institute of Physics. ͓DOI: 10.1063/1.2382591͔ I. INTRODUCTION with crawling velocities typically between 1 mm/s and 1 cm/s. The muscular waves are said to be direct if they Common gastropods crawl on land by propagating propagate in the same direction as the direction of locomo- waves of shear stress, driven by alternating regions of mus- tion, and retrograde otherwise. Direct waves are waves of cular contraction and expansion, on top of a thin film of contraction, retrograde waves are waves of extension. Simi- viscoelastic mucus released from their foot.1–4 The only in- larly, waves are classified as monotaxic when a single wave teractions between the organism and the substrate are spans the foot of the animal and ditaxic if two alternating through this thin mucus layer, hence locomotion is made waves span the foot. In the three large classes of gastropods, possible solely via the fluid dynamics within this thin film. pulmonates are ͑mostly͒ terrestrial crawlers and use direct The nonlinear rheological properties of the mucus are re- monotaxic waves. Prosobranchs represent the largest class of sponsible for the remarkable ability of these mollusks to gastropods and are the most varied in structure; most of them walk on solid ground without detaching the foot from the are marine gastropods using ditaxic retrograde waves. Fi- substrate.5,6 Mucus production accounts for about a third of nally, opisthobranchs do not crawl, but burrow or swim.17 the total energy budget of the animal, and is an order of Lissman was the first to study in detail the kinematics magnitude larger than the mechanical work required for lo- and dynamics of gastropod motion.18,19 He analyzed the lo- comotion, making slug crawling the most energetically ex- comotion of three pulmonates, Helix aspera ͑monotaxic di- pensive mode of locomotion known among vertebrates and rect͒, Haliotis tuberculata, and Pomatias elegans ͑both di- invertebrates.7–10 In this paper, we address the relationship taxic direct͒. Jones and Trueman20,21 were the first to study between the cost of locomotion and the mechanical proper- retrograde wave locomotion, on Patella vulgata ͑ditaxic ret- ties of the mucus. rograte͒. However, it was not until 1980 with the work of Locomotion strategies employed by gastropods have Denny and coauthors that a complete mechanical picture for been of interest to the scientific community for more than gastropod locomotion emerged. The key feature of gastropod 100 years, starting with the works of Simroth,11 Vlès,12 and locomotion lies in the mechanical properties of the mucus Parker;13 the reader is referred to Refs. 1–4 and 14–16 for layer. This was first discovered for the Pumonate slug reviews. Gastropods possess a single flexible foot enhanced Ariolimax columbianus,5,6 but is generally valid for all with arrays of cilia. In many cases, the motion of the cilia is gastropods22,23 ͑see also Refs. 24–27͒. The mucus is com- responsible for movement of the mucus underneath the gas- posed of more than 95% water, some dissolved salts, and the tropod and the resulting motion of the animal. However, remaining 3–4% is a high molecular weight glycoprotein many gastropods, including all snails that crawl on hard sub- ͑mucin͒ which, in solution, forms a cross-linked gel network. strates, use a second mechanism to crawl, namely the actua- This network is responsible for the elastic component of the tion of muscular pedal waves. Using a series of foot muscles, mucus. At small shear strains, the mucus behaves like an gastropods are able to stress the thin layer of mucus ͑which elastic solid, but at large shear strains ͑on the order of 5͒, the is typically on the order of tens of micrometers thick͒ with mucus yields and behaves like a viscous liquid, with the alternating regions of muscular contraction and expansion. shear viscosity dropping by about three orders of magnitude These regions oscillate in time, leading to traveling waves to a value of approximately 20 times that of water. After that shear the mucus, resulting in translation of the animal yielding, and when shear is stopped, the mucus locally heals back into a gel network on a time scale of less than 0.1 s.36 ͒ a Electronic mail: [email protected] Underneath a crawling snail, shear strains vary in time be- 1070-6631/2006/18͑11͒/113102/9/$23.0018, 113102-1 © 2006 American Institute of Physics Downloaded 22 Nov 2006 to 18.74.6.133. Redistribution subject to AIP license or copyright, see http://pof.aip.org/pof/copyright.jsp 113102-2 E. Lauga and A. E. Hosoi Phys. Fluids 18, 113102 ͑2006͒ tween 0, in the so-called “interwave” regions in which the foot does not move relative to the ground, and approximately 50 in the regions of largest relative displacement. It is this combination of yield-stress properties with judicious peri- odic shearing of the mucus that allows parts of the foot to adhere to the ground while other portions are displaced, lead- ing to an overall translation of the organism. The use of a viscoelastic mucus for locomotion has a second interesting consequence. Measurements of the net increase in oxygen consumption due to locomotion for Ariolimax columbianus showed that adhesive crawling is the most expensive known mode of locomotion,7 12 times as costly as running, for example. Perhaps surprisingly, the largest component of this energy budget does not arise from the mechanical work of the muscles, but is due to the chemi- cal cost of mucus production ͑that is, the chemical cost of FIG. 1. ͑Color online͒ Schematic representation of the model used to rep- creating the glycoprotein based on its sequence of amino resent a slightly non-Newtonian mucus in this paper. The straight line rep- ͒ resents a Newtonian mucus; when the stress-shear rate curve is above the acids . For Ariolimax columbianus, mucus production is Newtonian one ͑⑀Ͼ0͒, the mucus is shear-thickening, and when it is below about 20 times more costly than the mechanical work of ͑⑀Ͻ0͒ the mucus is shear-thinning. locomotion.7 Since Denny’s seminal work, the cost of loco- motion has been analyzed for other gastropods, including Patella vulgata,8 Buccimum undatum,9 and Haliotis kamtschatkana,28 and similar results have been obtained ͑see mucus, consider an organism mechanically similar to a snail ͑ ͒ also Ref. 10͒. that secretes a simple lubricating fluid e.g., water . This mu- In this paper, using a simple model for the nonlinear cus becomes more complex as its chemical composition rheology of the mucus, we show that its shear-thinning prop- evolves. Hence, as its rheology is perturbed away from erties allow the gastropod to decrease the aforementioned Newtonian, it is necessary to elucidate which changes to the energetic cost of locomotion by minimizing the amount of rheology are beneficial to the organism. This is the problem fluid dispensed. The paper is organized as follows. We intro- we propose to address, perhaps shedding some light on the duce our model for the nonlinear rheological properties of importance of material properties in an evolutionary context. the mucus in Sec. II, set up the geometry and notation for the Guided by this evolutionary picture, we make the fol- particular mathematical problem to be solved in Sec. II A, lowing modeling assumption: Let us consider the mucus of and present its asymptotic solution in Sec. II B. We apply it this primitive snail-like organism as slightly non-Newtonian, to the typical stress distribution underneath a crawling slug as illustrated in Fig. 1, with a quadratic relationship between ␴ ͒ ץ ץ ͑␥ in Sec. III and show that choosing the mucus to be shear- shear rate, ˙ = u/ y , and shear stress, , given by thinning leads to a decrease in the amount of required fluid. ␴ ͉␴͉ ␥˙ = ͩ1−⑀ ͪ. ͑1͒ ␮ ␴ * II. MODEL As a first approach to the problem, we model the mucus as As described in the Introduction, adhesive locomotion, purely viscous ͑generalized Newtonian͒ and neglect its elas- the crawling strategy employed by gastropods, has two dis- tic properties. Our empirical quadratic model ͓Eq. ͑1͔͒,a tinctive features: ͑i͒ The nonlinear rheology of the mucus, Cross model of order 1, is equivalent to a Taylor expansion solid-like at low stresses and liquid-like for larger stresses, at small stress of the inverse of the shear viscosity, and ͑ii͒ the high cost of locomotion due to the internal ␩͑=␴/␥˙ ͒, as a function of the absolute value of the shear chemical production of the proteins that compose the mucus. stress, and is arguably the simplest model that captures the 37 ␴ Ͼ In this paper, we address the possible link between these two effects of the nonlinear rheology.