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Viscoelastic Properties of the Nematode Caenorhabditis Elegans, a Self-Similar, Shear-Thinning Worm

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Viscoelastic Properties of the Nematode Caenorhabditis Elegans, a Self-Similar, Shear-Thinning Worm Viscoelastic properties of the nematode Caenorhabditis elegans, a self-similar, shear-thinning worm Matilda Backholma, William S. Ryub, and Kari Dalnoki-Veressa,c,1 aDepartment of Physics and Astronomy and the Brockhouse Institute for Materials Research, McMaster University, Hamilton, ON, Canada L8S 4M1; bDepartment of Physics and the Donnelly Centre, University of Toronto, Toronto, ON, Canada M5S 1A7; and cLaboratoire de Physico-Chimie Théorique, Gulliver, Unité Mixte de Recherche 7083, Centre National de la Recherche Scientifique-École Supérieure de Physique et de Chimie Industrielles, 75005 Paris, France Edited by David A. Weitz, Harvard University, Cambridge, MA, and approved February 11, 2013 (received for review November 16, 2012) Undulatory motion is common to many creatures across many scales, bending of the entire worm. As the nematode is known to consist from sperm to snakes. These organisms must push off against their of anisotropic materials (21), the two stiffnesses should not be external environment, such as a viscous medium, grains of sand, or considered the same. Additionally, the elasticity related to un- a high-friction surface; additionally they must work to bend their own dulatory motion is the longitudinal stiffness, as the worm needs body. A full understanding of undulatory motion, and locomotion in to bend its entire body to swim or crawl. There exist experimental general, requires the characterization of the material properties of limitations in directly measuring the longitudinal stiffness, and the animal itself. The material properties of the model organism Cae- many measurements have been made indirectly through model- norhabditis elegans were studied with a micromechanical experiment ing assumptions (18, 19). Models used to elucidate the mechanics used to carry out a three-point bending measurement of the worm. of undulatory motion typically involve assumptions of the ma- Worms at various developmental stages (including dauer) were mea- terial properties of C. elegans that are yet to be proven experi- sured and different positions along the worm were probed. From mentally (22, 23). these experiments we calculated the viscoelastic properties of the Here, we present a method used to probe the dynamic visco- worm, including the effective spring constant and damping coeffi- elastic properties of C. elegans at a biologically, physically, and cient of bending. C. elegans moves by propagating sinusoidal waves structurally relevant length scale. Direct micromechanical mea- PHYSICS along its body. Whereas previous viscoelastic approaches to describe surements were performed, and a simple elastic model was used to the undulatory motion have used a Kelvin–Voigt model, where the gather results for the bending stiffness of C. elegans at all of its life elastic and viscous components are connected in parallel, our mea- stages. Furthermore, we have measured the viscoelastic response to surements show that the Maxwell model, where the elastic and vis- bending, and show that commonly used models do not adequately cous components are in series, is more appropriate. The viscous describe the measured material properties of C. elegans.Bymod- component of the worm was shown to be consistent with a non- eling the viscoelasticity of the worm, our dynamic experiments re- Newtonian, shear-thinning fluid. We find that as the worm matures veal unexpected viscous properties. The Young’s modulus of the it is well described as a self-similar elastic object with a shear-thinning worm as a whole is reported, and an attempt to decouple the damping term and a stiffness that becomes smaller as one approaches contributions from cuticle and muscles to the total stiffness is made. BIOPHYSICS AND the tail. COMPUTATIONAL BIOLOGY Results and Discussion biomechanics | viscoelasticity Micropipette deflection (MD) (24) was used to perform three- point bending measurements on anesthetized C. elegans nemat- he undulatory motion of snakes and fish as they crawl or swim odes to probe their force–deflection response. The experiment is Tthrough a medium is considered a superior form of locomo- illustrated in Fig. 1A, and described in more detail at the end of tion in terms of its adoption across a broad range of length scales this paper. In short, the worm was held with a flexible force- and efficiency (1). Several attempts have been made to achieve calibrated pipette through suction, and bent by moving a simple the same level of performance artificially (2), but the agility seen support toward the “holding” pipette (from left to right in the in nature is far from being reproduced in manmade systems. A figure) with a constant speed vu. The deflection x of the holding number of experimental model systems have been used to study pipette produces a certain force F = k x, where k is the spring – p p undulatory motion (3 7). To achieve a deeper understanding of constant of the pipette. The bending y = xu − x of the worm is this form of motility, the biomechanics has been studied theo- defined as the difference between the motion of the support (xu) retically for snake-like systems (8, 9). Recently, computational and the deflection of the pipette. Optical microscopy images fluid dynamic models of organismal swimming have been de- from the beginning (Upper) and end (Lower) of a bending ex- veloped to simulate fluid–body interactions, including internal periment performed on an adult worm are shown in Fig. 1B. The forces and body stiffness (10, 11). However, a requirement for a total deflection of the pipette is indicated by the dashed line. successful, systems-level model is a detailed knowledge of the material properties of the crawler––insight that can only be achieved Elastic and Viscoelastic Theoretical Models. Two different theoretical experimentally. models were used to achieve an understanding of the worm ma- Caenorhabditis elegans, a millimeter-sized nematode, has been terial. A simple linearized Hookean model was applied to describe used as a model organism to study undulatory motion experi- the purely elastic properties of C. elegans. To gain deeper insight, mentally (12–15). One fundamental, unresolved question is how difficult is it for the worm to bend its own body as it moves (16). In other words, what is the bending stiffness of the model or- Author contributions: M.B., W.S.R., and K.D.-V. designed research; M.B. performed re- ganism? Efforts have been made to measure the stiffness of search; M.B., W.S.R., and K.D.-V. contributed new reagents/analytic tools; M.B. analyzed C. elegans (17–20), but a conclusive result is yet to be reached data; and M.B., W.S.R., and K.D.-V. wrote the paper. for several reasons. The authors declare no conflict of interest. Direct comparisons between transverse and longitudinal stiff- This article is a PNAS Direct Submission. ness values have caused confusion. Here, the former quantity is 1To whom correspondence should be addressed. E-mail: [email protected]. the elasticity probed by a local compression of the worm, whereas This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10. the latter corresponds to the stiffness related to a nonlocal 1073/pnas.1219965110/-/DCSupplemental. www.pnas.org/cgi/doi/10.1073/pnas.1219965110 PNAS Early Edition | 1of6 Downloaded by guest on September 24, 2021 Fig. 1. (A) Schematic diagram of the experimental micropipette deflection setup used to study the bending stiffness of C. elegans. A support (two circles) is moved from left to right with a constant speed, vu. This induces a bending y of the worm due to the force F = kpx applied by the pipette, where kp is the spring constant of the pipette. The deflection of the pipette and motion of the support are defined as x and xu, respectively, and the bending y of the worm is defined as the difference between these. (B) Optical microscopy images of an adult worm in the beginning (Upper) and end (Lower) of a bending experiment, with the total pipette deflection indicated by the dashed line. The supporting structure is a thicker, U-shaped glass pipette, into which the worm can be bent as it would be simply supported. Scale bar, 100 μm. (C) Diagram of the worm modeled as a viscoelastic Maxwell material, consisting of a spring (spring constant kw) and dashpot (damping coefficient c) connected in series. The bending y of the worm corresponds to the compression of the system due to the force F applied by the pipette. a non-Hookean viscoelastic Maxwell model was introduced. In the h i c − =½ ð + Þ limit of small deformations and short times these two models yðtÞ = v t − 1 − e kpkw c kp kw t ; [3] u k are equivalent. p Euler–Bernoulli elastic beam theory (EBT) (25) was used to where v is the speed of the support and k is the spring constant analyze the elasticity of the worm by approximating it as a spring- u p of the holding pipette. Furthermore, the force–deformation re- like beam. The bending y at the position of the applied force F is lationship can be written as given as sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiÀ Á ! k k + k 2ð − Þ2 ð Þ = kpvuc − + + 2 w p w : [4] = a L a = 1 ; [1] F y 1 1 y y F F kp + kw kpvuc 3LEI kw where L is the distance between the supports and a is the dis- The Kelvin–Voigt viscoelastic model, where a spring and a tance between the upper support and the position of the applied dashpot are connected in parallel (27), has been used by others force (pipette). The spring constant kw of the worm is a function to describe the material properties of C. elegans (18, 19), based of both the geometry of the experiment, as well as the bending on the theory of snake-like creatures in general (9).
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