Analysis of nematode mechanics by piezoresistive displacement clamp

Abstract

Studying animal mechanics is critical for understanding how signals in the neuromuscular system give rise to behavior and how force-sensing organs and sensory neurons work. Few techniques exist to provide forces and displacements appropriate for such studies. To address this technological gap, we developed a metrology using piezoresistive cantilevers as force–displacement sensors coupled to a feedback system to apply and maintain defined load profiles to micrometer-scale animals. We show that this system can deliver forces between 10⁻⁸ and 10⁻³ N across distances of up to 100 μm with a resolution of 12 nN between 0.1 Hz and 100 kHz. We use this new metrology to show that force–displacement curves of wild-type nematodes (Caenorhabditis elegans) are linear. Because nematodes have approximately cylindrical bodies, this finding demonstrates that nematode body mechanics can be modeled as a cylindrical shell under pressure. Little is known about the relative importance of hydrostatic pressure and shell mechanics, however. We show that dissipating pressure by cuticle puncture or decreasing it by hyperosmotic shock has only a modest effect on stiffness, whereas defects in the dpy-5 and lon-2 genes, which alter body shape and cuticle proteins, decrease and increase stiffness by 25% and 50%, respectively. This initial analysis of C. elegans body mechanics suggests that shell mechanics dominates stiffness and is a first step in understanding how body mechanics affect locomotion and force sensing.

Analyses of the nematode Caenorhabditis elegans and its mutants enable systematic study of the interplay between genes and behaviors that range from simple movement to successful mating. C. elegans and other nematodes move in a sinusoidal fashion by generating waves of alternating dorsal and ventral muscle contraction. These waves of muscle contraction produce local bending in the cuticle, which is opposed by a high hydrostatic pressure (1, 2). In the laboratory and in the natural soil environment, C. elegans crawls across surfaces in a thin layer of moisture. The transformation of signals in the neuromuscular plant into behavior is constrained by body mechanics. Similarly, body mechanics determines how loads applied to the outer body surface are conveyed to mechanosensory neurons. To learn more, we developed a piezoresistive (PR)-based system with force and displacement ranges that are unavailable with existing methods.

The nematode body plan consists of an outer tube separated from an inner tube by a fluid-filled pseudocoelom (Fig. 1). The cuticle, hypodermis, excretory system, neurons, and longitudinal muscles comprise the outer tube or shell, and the pharynx, intestine, and gonad form the inner tube (3). Internal tissues are under pressures on the order of 2–30 kPa (1), suggesting that nematodes have a shell-type hydrostatic skeleton. Very little is known about the relative importance of hydrostatic pressure and cuticle structure and elasticity to overall body stiffness. The multilayered cuticle is formed primarily from collagen proteins (4). Mutations that disrupt genes encoding collagen proteins dramatically alter body shape (4), indicating that cuticle structure plays a significant role in determining the body shape. Some of these mutants are long and narrow (called “Lon”), whereas others are short and wide (called “Dpy”). Hydrostatic pressure also contributes to body shape, because acute osmotic shock changes body shape (5). In addition to their role in body shape, cuticle elasticity and hydrostatic pressure may limit the sensitivity of touch receptor neurons, although such neurons can detect point loads as small as 100 nN in the absence of hydrostatic pressure (6).

Fig. 1.

C. elegans structure. (A) C. elegans body plan, showing a side view (left) and 3D cross-section (right). The cuticle has circumferential furrows and annuli, is synthesized by the hypodermis and encloses a pressurized pseudocoelomic fluid (adapted from www.wormatlas.org). (B) Transmission electron micrograph (left) of a cross-section taken near the terminal bulb of the pharynx and a magnified view (right) (electron micrograph courtesy of J. Cueva). (C) Model of nematode body mechanics.

Existing tools used to analyze biomechanics at the micrometer scale include optical tweezers (OT) (7), atomic force microscopy (AFM) (8), and micropipette aspiration (MA) (9). These tools operate in the range of 10⁻¹² to 10⁻⁹ N with displacements typically less than a few 10⁻⁶ m (Fig. 2); each has provided significant insights into fundamental aspects of biology. For example, OT has been critical in advancing our understanding of how molecular motors function (10), and AFM has advanced understanding of protein folding and unfolding (11). However, few are appropriate for mechanical studies of multicellular systems such as tissues, organs, and small organisms. Here, we describe a force–displacement (F–D) measurement system matched to the dimensions and material properties of tissues and small organisms and discuss its application to the analysis of the mechanics of C. elegans. Our custom-designed microelectromechanical system (MEMS)-based tool uses a silicon PR sensor to apply forces in the 10⁻⁸ to 10⁻³ N range over distances of up to 100 μm.

Fig. 2.

Force and displacement range of common instruments for biomechanics (A) and biological materials (B). The color in A indicates the bandwidth of the measurement method.

Principles of Operation

MEMS-based PR cantilevers offer several advantages for mechanical studies of multicellular biological systems, including small animals. Because displacement is measured by monitoring displacement-induced changes in the resistance of the PR region at the cantilever root with a simple Wheatstone bridge (12), PR cantilevers are free of complications imposed by laser-based optical detection methods used in OT and AFM, which include laser-induced thermal and optical disturbances of samples being studied and constraints on the relative geometry between the probe and sample. Four additional advantages favor PR cantilevers over OT and AFM. (i) PR cantilevers have force and displacement resolutions on the order of nanonewtons and nanometers, respectively. This resolution is sufficient for studies of organelles, cells, tissues, and multicellular animals. (ii) The working displacement range of a given PR cantilever is limited primarily by its stiffness, whereas that of an AFM is limited by the optical detector. (iii) PR cantilevers can be integrated into high-bandwidth feedback control systems incorporating high-resolution actuators. (iv) PR-cantilever-based feedback systems are amenable to integration with other cellular measurements, such as patch-clamp electrophysiology, and live-cell imaging of fluorescent proteins and intracellular Ca²⁺, whereas the position control and optical measurements needed for OT and AFM complicate integration with such measurements (13, 14).

Here, we describe a PR-cantilever-based system for applying user-defined force profiles and measuring the applied force and deformed indentation depth of samples. We use this system to evaluate the relative importance of cuticle stiffness and internal hydrostatic pressure to body shape in the nematode C. elegans. To our knowledge, this is the first study to measure the stiffness of C. elegans. We propose that, to a first approximation, C. elegans body mechanics is consistent with a shell-like model in which the cuticle is a major determinant of stiffness.

Results and Discussion

Characterization of PR Cantilevers and the Measurement System.

The resistance of a typical PR cantilever (Fig. 3 A and B) was 1,188 Ω, in close agreement with values predicted by TSUPREM4 for sheet resistance (53 Ω/square) and number of squares, 21. Current–voltage (I–V) relations were linear over the relevant voltage range, ±5 V. We determined effective spring constants (k_c), force sensitivities (S_f), and first-mode resonant frequencies (ω₀) of PR cantilevers using a laser doppler vibrometer (Polytec OFV3001) and resonance technique described previously (15, 16). For cantilevers with the geometries described above, effective spring constants (k_c) were between 0.123 and 2.07 N/m, with force sensitivities (S_f) between 21,400 and 3,144 μN/V and first-mode resonant frequencies (ω₀) between 0.56 and 4.92 kHz. The maximum load was defined to maintain <0.1% nonlinearity and was between 170 and 850 μN.

Fig. 3.

PR-cantilever-based metrology. (A) Schematic of PR cantilever. (B) PR cantilever with 10-μm-diameter glass bead at tip. (C) Schematic of feedback control system. ΔR ∝ stress ∝ tip displacement. (D) Noise power spectral density (PSD) of piezoresistor and associated conditioning circuitry and integrated noise (Inset). (E) Six F–D curves of cantilever-on-glass loading profiles. Markers in the Inset correspond to different measurements.

Electronic noise limits the force resolution of this PR-based measurement system. We determined these limits by analyzing the spectral density of noise (0.1 Hz to 100 kHz bandwidth) generated at the output of the instrumentation amplifier (Fig. 3D). Such signals include noise from the piezoresistor and the associated conditioning circuitry. In our system, the instrumentation amplifier (AD620; 10 nV/Hz^1/2) was the dominant source of Johnson noise, whereas the PR cantilever (4 nV/Hz^1/2) dominated 1/f noise. To extract the relationship between noise and measurement bandwidth, we integrated the noise density over the band 0.1 Hz to 100 kHz (Fig. 3E). Based on this analysis, our system has a force resolution of 12 nN in this band.

Displacement Feedback System.

We integrated the PR cantilever and circuit with a “real-time” control system (Fig. 3C) and verified robust displacement stabilization for a 100-kHz control loop. To validate performance for F–D measurements, we used displacement control to apply the cantilever to a hard surface (glass). Fig. 3E overlays six F–D curves from this measurement and demonstrates both excellent linearity and repeatability; the slopes of fits to the data were distributed narrowly around a mean value of 0.710 N/m (SD = 0.005). Snap-in of the cantilever is apparent, although this was reduced in later experiments by coating cantilevers with Parylene C.

Poly(dimethyl siloxane) (PDMS) and Agarose F–D Measurements.

We validated our system and indentation method by evaluating the elastic modulus of compliant materials: PDMS and agarose gels. Consistent with the Hertz model, F–D curves of thick PDMS and agarose gel samples were nonlinear (Fig. 4 A and B). We fit F–D curves with the Hertz model to derive an estimate of the elastic moduli of both materials. The effective elastic modulus of PDMS was comparable to values obtained by using other methods (17–20) (Table 1). The effective modulus of agarose gels increased with agarose concentration (Fig. 4B). Our elastic moduli were smaller than those derived by Normand et al. (21) from compression testing. However, their samples may have hardened during storage before testing. Consistent with this idea, the effective modulus of agar increases with drying time (22).

Fig. 4.

System validation. F–D curves of PDMS (A) and agarose gel (B). Dashed lines are fits to the data according to a nonlinear Hertz model. Effective moduli of PDMS and agarose gels were 1.12 ± 0.12 MPa, 132 ± 20 kPa, 213 ± 32 kPa, and 301 ± 32 kPa for PDMS and 2%, 5%, and 8% for agarose gels, respectively. Values are the mean (±SD) of 8 and 14 measurements of PDMS and agarose gels, respectively.

Table 1.

Young's modulus of PDMS determined by various methods

Method	E, MPa
PDMS cantilever (17)	0.75
Microtensile test (18)	1
Compression (19)	1.5∼2.5
Texture analyzer (20)	1.5∼2
PR cantilever (this work)	1.12 ± 0.12

C. elegans Body Mechanics.

Like other biological samples that have cylindrical, shell-type structures, including microtubules (23–25), hair cells (26), and certain bacteria (27), the nematode F–D relation is linear (Fig. 5B). This observation demonstrates that C. elegans can be modeled as a homogeneous cylindrical shell with internal hydrostatic pressure (see below). In support of this assumption of homogeneity, there was no detectable difference in mean stiffness between the head, center, and tail of the worm (Fig. 5C). The increased variance in measurements taken near the head and tail could reflect variability in the effects of gluing the head and tail.

Fig. 5.

PR-cantilever-based analysis of C. elegans stiffness. (A) Photomicrographs of immobilized wild-type, dpy-5, and lon-2 animals. Anterior is left. (B) Linear F–D curve of a wild-type animal. (C) Dependence on placement of point load (no significant effect, Wilcoxon test, P = 0.85). (D) Puncture decreases stiffness (matched-pair Wilcoxon test, P = 0.063). On average, stiffness decreased by only 18%. (E) Cuticle defects alter stiffness (Wilcoxon test, P < 0.0001). Shaded band indicates the 25% and 75% percentile range of wild-type stiffness; 22–27 animals were tested in each condition.

To investigate the contribution of hydrostatic pressure to stiffness, we collected F–D curves before and after puncturing animals with a sharp probe. F–D curves were linear before and after puncture (data not shown), suggesting that, at least in the short term, animals retain their cylindrical shape in the absence of hydrostatic pressure. Although puncture decreased stiffness by ≈20% (Fig. 5D), final values were not significantly different from those of intact animals. These findings suggest that although hydrostatic pressure contributes to stiffness in wild type, it is not a dominant factor.

Next, we analyzed the effect of mutations in the cuticle and alterations in body shape. Two cuticle mutants were studied: dpy-5, which encodes a cuticle procollagen (28), and lon-2, which encodes a member of the glypican family known to regulate growth factor signaling (29). dpy-5 mutants are shorter and wider than wild-type animals, whereas lon-2 mutants are longer and skinnier than wild-type animals (Fig. 5A). In principle, dpy-5 and lon-2 could alter the surface tension of the cuticle network, its elastic modulus, or its thickness. We assumed that the pressure of dpy-5 and lon-2 worms was similar to wild type because neither gene affects transport proteins likely to regulate osmotic pressure. In support of this assumption, unlike other mutants that confer a Dpy phenotype, dpy-5 does not affect the internal concentration of glycerol (30), a major osmoticant in terrestrial invertebrates. Given the assumption of constant pressure, Laplace's law (T = PR) predicts that the wider mutant (dpy-5) should have a larger surface tension than the narrower mutant (lon-2). Thus, if surface tension is dominant, then dpy-5 should be stiffer than wild type and lon-2 because overall stiffness is proportional to membrane tension under these conditions, k ∝ T (27). In contrast to this prediction, we found that dpy-5 was less stiff than wild type, which was less stiff than lon-2 (Fig. 5 C and D). Consistent with the idea that body stiffness is related to cuticle stiffness and efficiency of force transfer to sensory neurons, touch sensitivity is decreased in lon-2 mutants (I. Chin, M. Chalfie, and M.B.G., unpublished data).

The foregoing analysis indicates that hydrostatic pressure is not the sole factor regulating stiffness in nematodes and suggests that cuticle elasticity is an important determinant of body stiffness. We probed this idea further by determining the stiffness of a mutant defective in osmotic regulation: clh-1. Defects in the clh-1 gene, which encodes a putative Cl⁻ channel expressed by seam cells in the cuticle (31), increase body width under standard growth conditions, a phenotype that is reversed by exposing clh-1 mutants to hyperosmotic shock (31) (Fig. 6B). Because clh-1 mutants have no cuticle defects, these findings suggest that, under standard conditions, clh-1 has a higher internal hydrostatic pressure than wild type. The stiffness of clh-1 mutants did not differ from wild type (Fig. 6B), however, providing support for the idea that pressure alone does not determine hydrostatic skeleton stiffness.

Fig. 6.

Effect of acute and chronic exposure to hyperosmotic media on stiffness. (A) Schematic of hyperosmotic treatments. Worms (n = 23–25 for each treatment) were grown on either standard (i–iv) or hyperosmotic (v) plates for at least 4 weeks and tested on standard (i and iii) or hypertonic (ii, iv, and v) plates. (B) Stiffness of animals treated as schematized in A. The Wilcoxon test indicates dependence on osmotic conditions (wild type, P = 0.0095; clh-1, P = 0.071) but no dependence on genotype (P = 0.36). Hyperosmotic shock (ii) induced longitudinal shrinkage. For comparison, the shaded band is repeated from Fig. 5 and shows the 25–75% percentile range of wild-type stiffness.

Acute hyperosmotic shock should decrease hydrostatic pressure and cause animals to shrink. Consistent with this prediction and previous reports (5, 31), we found that hyperosmotic shock induced shrinkage, although body length decreased to a larger extent than body width (Fig. 6B). Such anisotropy could be due to compression of furrows in the cuticle and suggests that C. elegans is more compliant along its body axis than in the circumferential direction. We reasoned that if pressure and surface tension were the primary determinants of stiffness, then hyperosmotic shock, like animal puncture, should decrease pressure, surface tension, and stiffness. Unexpectedly, we found that hyperosmotic shock increased stiffness (Fig. 6B). Taken together, these observations suggest that acute hyperosmotic shock evokes hypercontraction of body wall muscles, which are oriented longitudinally. Support for this interpretation comes from prior work showing that hyperosmotic saline increases neurotransmitter release at the C. elegans neuromuscular junction (32), which would, in turn, lead to increased muscle contraction. In any case, such responses to hyperosmotic shock are inconsistent with the idea that hydrostatic pressure is the principal determinant of stiffness.

A Model of C. elegans Body Mechanics.

We adapted models developed for certain bacteria (27) and vertebrate outer hair cells (26) to estimate the elastic modulus of the C. elegans shell and the relative contributions of pressure and shell elasticity to overall stiffness. According to these models, stiffness is a function of the effective elastic modulus of the cuticle (E_c), hydrostatic pressure (P), and C. elegans dimensions (w, body width; l, body length; and t, cuticle thickness). For a very long cylinder, the flattening occurs over the characteristic length, $l_0=R\sqrt{R/t}$ (26), where R is radius of C. elegans (w/2), and t ∼ R/44 (33). The body length (≈1 mm) is much longer than the characteristic length (≈100 μm) of wild-type C. elegans, such that flattening occurs in the local area of indentation, stiffness dependence on body length may be neglected, and k_s = f(E, P, w, t). Because the stiffness decreases ≈20% after puncture, we can estimate an effective elastic modulus of wild-type C. elegans cuticle from k_s ≅ 1.37Et^5/2/R^3/2 (23) by neglecting pressure and assuming an idealized cylindrical geometry. This approach suggests that the C. elegans“shell,” which is composed of the cuticle, the hypodermis, and longitudinal muscles, has an effective elastic modulus on the order of 380 MPa.

To compare surface tension energy, U_t, and the elastic energy, U_e, we use this estimate and hydrostatic pressures reported for other nematodes [Ascaris lumbricoides, 2–30 kPa (1)] to calculate a nondimensional number, κ (27),

where the elastic energy, U_e, is comprised of bending, U_b, and stretching energy, U_s. Stretching energy is neglected because the indentation deforms a small local area of cuticle in the longitudinal direction. In this validation test, a κ ratio >1 implies pressure dominates, whereas a κ ratio <1 one implies that bending energy dominates. Assuming that C. elegans and A. lumbricoides have similar hydrostatic pressures, we estimate that κ is 0.01∼0.3, consistent with the observed 20% stiffness contribution of hydrostatic pressure. In other words, this model predicts that elastic energy is 3–100 times higher than surface tension energy and that elasticity of the “shell” dominates C. elegans body stiffness.

Conclusions

We demonstrated a PR cantilever indentation system suitable for measuring mechanical properties of C. elegans. For these studies, we fabricated a PR cantilever capable of delivering forces between 10⁻⁸ and 10⁻³ N (12 nN resolution) over 100 μm with feedback for displacement control. The useful range of forces and displacements is limited in certain applications, including analysis of C. elegans body mechanics. This system has several advantages over existing methods, including the following: (i) a large range of forces and displacements matched to the properties of biological materials; (ii) the ability to target the desired dynamic range and force resolution with PR cantilevers of varying stiffness; and (iii) the ability to measure force without optical methods. The system can be tuned to alter the sensitivity, natural frequency, and spring constant of the cantilever, as well as the characteristics of the control system. With minor alterations, for example, it is possible to operate the system in force-clamp mode (with 1 nN resolution at 1–10 kHz bandwidth) to apply and maintain user-programmed force profiles.

We used the system presented here to analyze the body mechanics of C. elegans in displacement-clamp mode. We found that C. elegans exhibits linear F–D relations, consistent with an elastic shell-type model. We also found that internal hydrostatic pressure and cuticle elasticity contribute to overall stiffness. These findings have implications for future analyses of locomotion and force sensing in nematodes. In particular, the modest contribution of hydrostatic pressure to stiffness may account for the persistence of mechanoreceptor currents in punctured animals (6), whereas the considerable contribution of cuticle mechanics to overall stiffness predicts that mutations which increase cuticle stiffness should decrease force sensitivity.

Materials and Methods

Fabrication of PR Microcantilevers.

We designed and fabricated single-crystal silicon PR cantilevers suitable for mechanical stimulation and characterization of nematodes as described previously (15, 34). Cantilevers were oriented in the 〈110〉 direction such that the applicable modulus of silicon is E_Si〈110〉 = 168 GPa and included designs of varying dimensions (length, L: 2–6 mm; width, w: 80–400 μm; thickness, t: 15 μm) and spring constants (k_c = E_Si〈110〉wt³/4L³: 0.137–2.2 N/m). We defined U-shaped piezoresistors (200 μm long, 20 μm wide, with a 20-μm gap) using photolithography (all patterns used standard Shipley 3612 or 220 photoresist processing) and boron ion implantation [energy, 50 keV (1 eV = 1.602 × 10⁻¹⁹ J); dose, 5E15 cm⁻²]. We implanted the piezoresistors at the cantilever root, near the top surface or the location of maximum mechanical bending stress (σ = 6FL/wt²). We simulated the distribution of carriers after piezoresistor processing using TSuprem4 (resulting junction depth, 1.84 μm and peak concentration were used in sensitivity and noise predictions). Sensor noise was determined by collecting signals conditioned with a passive high-pass filter (0.007 Hz) with a signal analyzer (HP3562).

The microfabrication process [see supporting information (SI) Movie 1 for fabrication details] started with silicon-on-insulator (SOI) wafers (15 μm device layer and 500 nm buried oxide) with the following key process parameters. Piezoresistors were doped with a boron ion implant of 5E15 cm⁻² and unstrained conductive traces doped to 1E16 cm⁻². A thermally grown isolation oxide of 2,100 Å wet at 1,000°C (20 min) plus 1,000°C inert N₂ anneal (5 min) annealed implant damage and diffused the dopant atoms. Finally, an H₂ forming gas anneal at 400°C (2 h) set contacts. Devices were released from the wafer by scribing and breaking snap tabs at the base of the die. We glued cantilevers to custom-printed circuit boards with epoxy (Devcon) and attached 10-μm glass beads (Duke Scientific) to their tapered tips with UV adhesive (Loctite 352; Henkel Technologies). The bead provided a controlled contact geometry (Fig. 3 A and B). We deposited 1 μm of Parylene C on the bead and cantilever to minimize attraction forces.

Force and Displacement Control.

Cantilever velocity and position were controlled and monitored by a piezoelectric actuator with built-in capacitive sensor (PIHera P-622.Z; Physik Instrumente). Resistance changes proportional to applied force were conditioned with a Wheatstone bridge and Analog Devices AD620 instrumentation amplifier (Fig. 3C). Voltages proportional to force and actuator travel were recorded by using a real-time controller (CompactRIO Field Programmable Gate Array (FPGA) System; National Instruments).

We implemented a displacement and force feedback loop with a bandwidth of 100 kHz; the capacitive sensor on the actuator and the PR cantilever can be used for displacement and force feedback, respectively (Fig. 3C). Only displacement feedback was used in this study. Rise time, overshoot, and steady-state error of the response were adjusted by changing the proportional integral derivative (PID) parameters.

Sample Preparation.

PDMS.

We prepared test samples (6.4 ± 0.5 mm thick; n = 5) by pouring 50 ml of mixed, degassed PDMS (1:10 wt/wt ratio of curing agent to PDMS; SYLGARD 184; Dow Corning) into a 10-cm petri dish. Samples were cured at room temperature (24 h).

Agarose.

We prepared 2–8% wt/vol agarose samples (type I; Sigma–Aldrich) in standard C. elegans saline (see ref. 35) and stored samples wrapped in Parafilm at 4°C before testing. Agarose samples were 4.9 ± 0.6 mm (n = 10) thick.

C. elegans nematodes.

Wild-type (N2) and mutant animals were cultivated according to ref. 36. Age-synchronized animals were obtained as described in ref. 37 and maintained on standard NGM agar plates for 48–52 h at 20°C until animals reached the L4 or young adult stage. NGM agar contains 21 mM NaCl, 1 mM MgSO₄, 24 mM K₂PO₄, 13 mM cholesterol, 20 g/liter bacto-peptone, and 25 g/liter agar. For stiffness testing, live worms were transferred to an 8% agarose gel by using a sharpened platinum wire, arranged in parallel by a fine dog hair and partially immobilized on the head and tail by using veterinary-grade cyanoacrylate (QuickSeal; WPI) (Fig. 3F). This technique prevents the worm from fully attaching to the substrate and reduces stiffening effects of the glue. The following strains were obtained from the Caenorhabditis Genetics Center: wild type (N2), CB678 lon-2(e678), CB61 dpy-5(e61), and XA901 clh-1(qa901).

Cuticle Puncture and Osmotic Shock.

We evaluated the contribution of internal osmotic pressure to C. elegans stiffness in two ways. First, we measured stiffness before and ≈10 min after puncturing the cuticle with a sharpened glass probe. Second, wild-type and clh-1 worms were subjected to acute hyperosmotic shock before gluing. Worms were cultured for at least 4 weeks on standard or hypertonic plates. (hypertonic plates contained NGM agar supplemented with 179 mM NaCl.) Worms were divided into two groups: controls were tested on standard plates, whereas experimental animals were tested on hypertonic plates. We tested live worms within 1 h of immobilization.

Data Reduction.

To evaluate mechanical properties, we used displacement control to collect F–D data. During loading, actuator increments of 0.2 μm (agarose gel and C. elegans) or 0.5 μm (PDMS) were maintained for 0.5 s. Voltages proportional to cantilever deflection and actuator position were recorded in LabView and converted to displacement from calibrated sensitivities. Indentation depth, δ = x_a − x_c − x₀, was extracted as the difference between actuator moving distance, x_a, deflection of cantilever, x_c, and initial contact point, x₀ (Fig. 7). Deforming force, f_bead, was derived from the output voltage of the cantilever (f_bead = k_cx_c) and referred to zero at x₀.

Fig. 7.

Mechanical model. (A) Schematic of nematode measurement system. (B) Mechanical model used to analyze F–D curves. (C) Adult nematode under a PR cantilever.

Data Analysis: Elastic Modulus of PDMS and Agarose.

F–D curves were fit with a linearized Hertz model describing the contact mechanics of a rigid sphere indenting a flat surface:

where ν is Poisson's ratio, f_bead is the indentation force, R_b is the bead radius, and effective elastic modulus, E, and initial contact point, x₀, were estimated from the deforming F–D relationships by using a least-squares fit method (38). Our parameters lay within the range required to apply Eq. 2; i.e., semi-infinite sample thickness or thick enough to ignore substrate effects (sample thickness, t_s ≥ 13R_b) and indentation depth small enough to maintain material linearity (δ < 0.1t_s) (39). For indentation depths less than the tip radius, the indentation is assumed to have a paraboloid shape (40). In support of these assumptions, we found that effective moduli were constant for indentation depths up to the bead radius of 5 μm (data not shown). Additionally, the Poisson's ratio, ν, of our materials was taken to be 0.5 (39).

Data Analysis: Stiffness of C. elegans Nematodes.

We glued live nematodes to 8% agarose, which provided sufficient hydration for survival. This concentration of agarose was sufficient to allow us to neglect substrate effects because higher agarose concentrations had little if any effect on apparent stiffness (SI Fig. 8). As the nematode body plan (Fig. 1B) suggests, the Hertz model (Eq. 2) is inappropriate for analysis of the linear F–D curves observed during stiffness testing of nematodes (Fig. 5B). Instead, we analyzed nematode F–D curves according to a linear shell model (41): f_bead = k_sδ = k_s(x_a − x_c − x₀). Sample stiffness, k_s, and the initial contact point, x₀, were extracted by fitting the data to the model.

Abbreviations

AFM

atomic force microscopy

F–D

force–displacement

MEMS

microelectromechanical system

OT

optical tweezers

PDMS

poly(dimethyl siloxane)

PR

piezoresistive.