Correction for Bolmin Et Al., Nonlinear Elasticity and Damping Govern

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Correction

SYSTEMS BIOLOGY, APPLIED PHYSICAL SCIENCES Correction for “Nonlinear elasticity and damping govern ultrafast pnas.2014569118 (Proc. Natl. Acad. Sci. U.S.A. 118, e2014569118). dynamics in click beetles,” by Ophelia Bolmin, John J. Socha, The authors note that, due to a printer’s error, Fig. 1 appeared Marianne Alleyne, Alison C. Dunn, Kamel Fezzaa, and Aimy A. incorrectly. The corrected figure and its legend appear below. Wissa, which was first published January 19, 2021; 10.1073/ The online version has been corrected.

A CORRECTION B C D

E F

Published under the PNAS license. Published April 19, 2021.

www.pnas.org/cgi/doi/10.1073/pnas.2102790118

PNAS 2021 Vol. 118 No. 17 e2102790118 https://doi.org/10.1073/pnas.2102790118 | 1of1 Downloaded by guest on September 28, 2021 niern n ehnc,Vrii eh lcsug A24061; VA Blacksburg, Tech, Virginia Mechanics, and Engineering a Bolmin Ophelia beetles click in ultrafast dynamics govern damping and elasticity Nonlinear oy sarsl,vr e pigmcaim aebe iden- been PNAS have mechanisms spring the few throughout very distributed result, and a internal As are body. springs organisms because most challenging is in output power mechanical in amplify lobes shell jaw-like two for closing flytrap, by ∼ Venus The insects (10–12). captures themselves pollen, instance, defend body and to seeds of or disperse recoil to feed, plants elastic to some Fast by 9). used also 8, is (1, elements shut posi- ( snapping exoskeletal faster mandibles open distributed times the the 600 the than of more recoil to occurs elastic springs mandibles the contrast, their In tion. loading store mechanical when ( example, in exoskeleton for slowly results ants, the energy Trap-jaw recoil between timescales elastic amplification. difference fast release output The very be and power motion. latch, to the storage the energy powers of energy spring elastic release the allow upon of systems and, with stored, Such muscles slowly the latches. of limitation and output actuation power springs this mechanical the overcome com- amplifying to complex by most evolved materials have animals’ and animals muscles, some structures Instead, by (7). directly actuators mon generated be acceler- cannot achieving to of capable up are shrimps, ations beetles mantis click froghoppers, and Fleas, ants, trap-jaw predators. escape or mote, S dynamics release beetles click repeatedly. our accelerations enhance extreme small achieve to strategies to release organisms use and animals other storage energy to to demon- the applied of observation tools understanding be The motion can movement. here from the strated causing starting forces motion, the this identifying extreme in anal- used of the methods for ysis guidelines The analytical and phase. experimental release provide study energy posi- the angular during the dominant driving tion the respectively, be forces, to elastic as identified and damping beetle are buckling the snap-through model and damping and Quadratic oscillator. motion single-degree-of-freedom of nonlinear a equation the rapid identification, through determine system mechanism nonlinear we and spring analysis the spectral Using to recoil. contributes hinge four of the structures in internal hinge’s abruptus the of Elater energy motion observe the the to analyze of imaging X-ray and (forces) synchrotron high-speed dynamics use We governing release. kinematics the motion the investigate detail We and latch- release. motion: energy clicking and the loading, of ing, phases the we quantify Here, the and not. depth, identify have in first unbending studied fast been enabling has mechanisms motion physical jumping in the results clicking While motion thoracic jump. a clicking the a in quick at this resulting unconstrained, quickly, bodies When extremely their motion. the unbend bend then to overcome bee- and latches to Click hinge and muscles. latches springs their use 2020) and of 13, tles limitations July springs review output for use power (received 2020 mechanical animals 10, December approved small and OH, Many Columbus, University, State Ohio The Denlinger, David by Edited and 61801; eateto ehnclSineadEgneig nvriyo liosa raaCapin raa L61801; IL Urbana, Urbana–Champaign, at Illinois of University Engineering, and Science Mechanical of Department 0 s(10). ms 100 dniyn h nryrlaemcaim nml s to use animals mechanisms release energy the Identifying m ftefsetadms xrodnr oin fthe of motions loco- hunt, extraordinary to animals most small by employed and are kingdom fastest animal the of ome 01Vl 1 o e2014569118 5 No. 118 Vol. 2021 d -a cec iiin ron ainlLbrtr,Lmn,I 60561 IL Lemont, Laboratory, National Argonne Division, Science X-ray | oe amplification power 10 | 6 ycrto -a imaging X-ray synchrotron pcmn.W hweiec htsf cuticle soft that evidence show We specimens. a,1 m/ s 2 onJ Socha J. John , eetdy(–) uheteemotions extreme Such (1–6). repeatedly ∼ 0 s ydfrigprso their of parts deforming by ms) 400 | itiue springs distributed b aineAlleyne Marianne , ∼ . s,rsligin resulting ms), 0.6 | c eateto noooy nvriyo liosa raaCapin raa IL Urbana, Urbana–Champaign, at Illinois of University Entomology, of Department c lsnC Dunn C. Alison , ulse aur 8 2021. 18, January Published doi:10.1073/pnas.2014569118/-/DCSupplemental.y https://www.pnas.org/lookup/suppl/ at online information supporting contains article This 1 license .y PNAS the under Published Submission.y Direct PNAS a is article This interest.y performed competing no A.A.W. declare and authors The resources; new provided A.A.W. and and specimens administration.y M.A. and K.F., project A.C.D., paper; K.F. insect M.A., the J.J.S., and contributed O.B., wrote J.J.S., data; A.A.W. A.A.W. analyzed O.B., A.A.W. and and research; O.B. M.A., tools; designed reagents/analytic O.B., A.A.W. and research; J.J.S., performed O.B., contributions: Author hard a on and unconstrained is body beetle’s click 1 the When (Fig. (3). unbending by hinge the mass along body their 1 fascinated bend to (Fig. them have enables tho- that the beetles hinge, adaptation, racic morphological Click unique a and unknown. of because springs scientists is use 1 mechanism (Fig. that output spring animals power mechanical of amplify to group latches under-studied an dissipation dae), and storage energy the compo- on material strategies. of architecture, as effects such and the properties, sition evaluating mesoscale motions. and sys- and stimuli ultrafast a microscale external simulating to to to lead response essential forces, tem’s also that governing are contributions analyses the can Dynamics-based their identify we motion, dynamics, detail the of and considering equations kine- is, the combining explaining that derive By to kinetics, generated. critical with are is matics accelerations which extreme that (kinetics), forces how the movement kinematics- of However, the identification the another. cause enable not one do analyses with based interaction movements part their body different and the Variations describe series. help analysis time metrics the these acceleration in is, and that velocity, kinematics, position, on the focus of systems as such Most on to (13). studies referred organisms (LaMSA) also spring-actuated systems, latch-mediated biological power-amplified in tified owo orsodnemyb drse.Eal bli2ilni.d rawissa@ or [email protected] Email: addressed. illinois.edu.y be may correspondence whom To hs,wihepan h h ete a ail perform rapidly can damage. beetles significant without the click why this explains release the which govern phase, that forces release elastic and char- quickly damping we the and spring methods, acterize store analytical the specialized to of using use Finally, portion energy. beetles a the discover synchrotron that we high-speed mechanism Second, using release imaging. motion and clicking X-ray loading, the latching, of the phases identify we fast clicking First, Here, extremely the (i.e., motion). the movements. beetles click of by the dynamics performed maneuver and of bending kinematics kinematics the study lim- the we been observing have mechanical to ultrafast animals perform ited these amplify to involving that latches Studies and movements. animals springs using among output power are beetles Click Significance nti ae,w ou ncikbels(oepea Elateri- (Coleoptera: beetles click on focus we paper, this In A lc ro) n hnt ceeaetercne of center their accelerate to then and arrow), black , a ae Fezzaa Kamel , https://doi.org/10.1073/pnas.2014569118 A ahdarw xrml quickly extremely arrow) dashed , b eateto Biomedical of Department d n iyA Wissa A. 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Fig. 1. Phases of the clicking motion and characterization of the peg tip and the soft cuticle displacement during the loading and energy release phases. (A) Adult E. abruptus click beetle general appearance (with legs removed) and lateral view projections of the hinge extracted from the X-ray recordings. Lateral view projections show the interaction between the peg, the mesosternal lip, and the cavity during the clicking motion. The landmark points tracked at the peg tip (green star), along the soft cuticle situated behind the peg (blue markers), and along the upper part of the cavity (yellow markers) are highlighted. The clicking motion can be divided into three phases: latching (i and ii), loading (iii and iv), and energy release (v–viii). During the latching phase, the peg slides along the cavity (i) and latches on the mesosternal lip (ii). Then, the loading phase begins as the soft cuticle contracts (iii). While the soft cuticle is contracting (i.e., displacing dorsally), the peg slowly rotates about the contact with the mesosternal lip (iii and iv, purple arrow). At the end of the contraction (iv), the latch is released (unlatching, v), and the peg slips and slides along the mesosternal cavity as the soft cuticle recoils (vi). During the energy release phase, the peg oscillates along the mesosternal cavity (vi and vii), until coming to a rest inside the cavity (viii). When the peg reaches the interior of the cavity, the soft cuticle and the mesosternal lip come in contact (vii and viii). Colored arrows indicate the movement direction of the peg (i and v–vii) and the soft cuticle (iii and vi). (B) Position of the peg tip of specimen 1 in x and y coordinates during the clicking motion. At t = 0.208 s, the peg is unlatched, and fast oscillations occur. (C) Fast damped oscillations of the peg tip in x and y coordinates during the energy release phase. (D) Duration of the energy release phase of the clicks of the four E. abruptus specimens. (E) Displacement of the soft cuticle and duration of the loading phase for the four specimens. The blue colors represent the start of the loading phase (0 s), and the red colors represent positions at the end of the loading phase (0.15 s). (F) Recoil of the soft cuticle at the beginning of the energy release and duration of the unlatching and elastic recoil for specimens 1 and 2. The blue colors (t = 0.1545 s) represent the start of the energy release phase (unlatching), and the red colors represent the first contact between the soft cuticle and the mesosternal lip, that is, the end of the elastic recoil at t = 0.1553 s. Error bars represent ± 1 SD. surface, this fast unfolding motion results in a jump (3, 14, 15). describes, in detail, the loading phase of the clicking motion The body bending and unbending motion is referred to as the and analyzes the energy release kinematics and dynamics in click click or clicking motion because it generates an audible “click- beetles. This work expands our knowledge of power-amplified ing” sound, similar to snapping one’s fingers. The thoracic hinge systems in nature, and provides a path for the analysis of the is composed of two conformal parts: the peg and the mesosternal dynamics of ultrafast motions by applying physics-based ana- lip (Fig. 1A), which together form a mechanical latch (4). When lytical tools such as spectral analysis and nonlinear system the latch is engaged, the beetle stores elastic energy in a still identification. unknown distributed spring; the recoil of the spring leads to the clicking motion (3, 4, 16). Although the kinematics of the jump Results have been studied (3, 14–18), as have the latch geometry and Clicking Motion Phases. The high-speed X-ray recordings of the mechanics (4), the unique clicking motion that enables the jump hinge allow for the visualization and identification of the phases has not been described, let alone analyzed in detail. Specifically, of the clicking motion. The clicking motion can be divided into the phases of the click have never been determined, nor have three phases: latching, loading, and energy release (Fig. 1). Prior the elastic energy storage and release mechanisms. As the click- to the click, the peg is at rest in the cavity (Fig. 1 A, i). The latching motion happens at a much faster timescale than the jump, it ing phase starts as the beetle rotates the anterior body (head and requires specialized analysis tools to characterize the movement prothorax) dorsally (Fig. 1A, black arrow). The rotation produces (kinematics) and the forces (dynamics). a bend in the body such that the head and thorax are angled In this study, we ask the following questions: 1) What are upward relative to the posterior body (Fig. 1A). The peg slides the phases of the clicking motion? 2) What are the elas- anteriorly out of the cavity (Fig. 1 A, i), and sets in place on the tic energy storage and release mechanisms in click beetles? mesosternal lip, forming a “latch” (Fig. 1 A, ii) (4). 3) Can the energy release mechanism be inferred from the During the loading phase, the bend in the body is maintained latch dynamics during the energy release phase? This study as the latch is locked, and energy is stored in the system. The

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APPLIED PHYSICAL SYSTEMS BIOLOGY SCIENCES are coupled by the cavity’s geometry. As such, the motion of For the remainder of the paper, the angular position of the peg the peg can be fully described using one independent param- tip, θr , is shifted to set the final equilibrium position to zero and eter, the radius r or the angular position θr , and the cavity’s facilitate the dynamics analysis calculations. The shifted angular geometry, which represents a nonlinear mapping that couples position of the tip is referred to as θ (Fig. 3A). r and θr . Therefore, we modeled the system as an oscillator Spectral analysis of the θ response. The angular position of the with one degree of freedom, using the angular position θr as peg tip θ during the energy release phase of specimen 1 is shown the independent parameter because it represents the anterior in Fig. 3A. For all clicks recorded, the angular position, θ, is body’s oscillations. The nonlinear spatial mapping of the cavity’s asymmetrical with respect to the final equilibrium, and two linear geometry was obtained by tracing the upper part of the cavity decay envelopes are fitted. The Fourier and the wavelet trans- in the X-ray images, following the yellow landmarks shown in forms are used to uncover the spectral behavior of the angular Fig. 1A. Fig. 2D compares the experimental response of r to its position response, θ. The Fourier transform shows that the dom- response derived from the experimental response of the angular inant frequencies of the angular position response signal are at 0, position of the peg tip θr and of the cavity’s geometry mapping. 352, and 703 Hz (Fig. 3B), which are also revealed in the wavelet The period of oscillation and the amplitude of the r response can transform plot (Fig. 3C). The zero frequency observed in the be recovered from the cavity’s geometry mapping and the exper- angular position response θ reflects the asymmetry of the sys- imental angular position θr (Fig. 2D), supporting the assumption tem, previously shown in Fig. 3A. The frequency of 352 Hz is the that the peg oscillations can be modeled as a one-degree-of- fundamental frequency (or first harmonic) and has the highest freedom oscillator. We attribute the deviations in amplitude in amplitude in the Fourier transform plot and the darkest shade Fig. 2D to resolution limits in measuring the cavity’s geometry. in the wavelet transform plot (Fig. 3 B and C). The frequency

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Fig. 3. The angular position of the peg tip kinematics, spectral analysis, and nonlinear system identification during the energy release phase. (A) Repre- sentative angular position of the peg tip θ as a function of time during the energy release (specimen 1) with the final equilibrium position set to zero. Two linear decay rates with different slopes are observed for all specimens. The peg tip oscillations are asymmetric with respect to the final (resting) equilibrium. (B) Discrete Fourier transform of the angular θ response of specimen 1 and of a linear time-invariant oscillator. The fast Fourier transform of the angular position response, θ, shows the fundamental frequency at 352 Hz, a second harmonic at 703 Hz, and a zero frequency. In comparison, the Fourier transform of the linear time-invariant oscillator (the dashed line) shows a fundamental frequency only. (C) The wavelet transform plot of the θ response of specimen 1 shows the decay in amplitude of the fundamental frequency and the second harmonic as time increases. (D) The normalized damping force for all responses (specimen 1 and 2) is nonlinear and follows the same quadratic curve, with a coefficient a = 1.8∗10−5.(E) The normalized elastic force is also nonlinear and shows two regions, for negative and positive positions of the peg tip θ. A very stiff region is shown for negative θ and corresponds to the angles where the soft cuticle and the mesosternal lip are in contact. A region of negative stiffness is observed between 0◦ and 25◦ (note that the x axis is decreasing). The elastic force curve is characteristic of a snap-through buckling system, where P, E, and F are three equilibrium points. P is commonly referred to as the critical point, and F is the final equilibrium point (resting position at the end of the motion). (F) Representative potential energy curve (specimen 2, release A). The two equilibria F and E are stable, and P is unstable. The motion sketches represent the positions of the peg at the different θ angles during the energy release phase.

4 of 8 | PNAS Bolmin et al. https://doi.org/10.1073/pnas.2014569118 Nonlinear elasticity and damping govern ultrafast dynamics in click beetles ncnatwt h otctce(i.1 is (Fig. lip cuticle mesosternal soft the region, the this with In contact cavity. the in inside peg the of from 3 defined Fig. 3 in (Fig. shown curve force quadratic a follows force analyzed. clicks all for trends olna lsiiyaddmiggvr lrfs yaisi lc beetles click in dynamics ultrafast govern damping and elasticity Nonlinear al. et Bolmin 3 Fig. of indicated region as shaded region, from the defined second in is region the slope to steep relative the by large is stiffness the D the of derivative as defined acceleration forces. is ferential energy-storing which or elastic force, and to displacement-dependent related forces, often normalized energy-dissipative total or the damping is to related often is where as formulated be can motion the of Thus, equation freedom. the of of degree form single a general with oscillator response, damped tip a peg as the of position angular the of of characteristic is Hz), time system. (703 with damped harmonic a decay second Amplitude increases. the time and as funda- Hz) the (352 of shades frequency lighter indi- mental the 3 by transform Fig. indicated time. wavelet as with decays, The amplitude vary harmonic. frequencies second the how the cates is Hz 703 of iru.Tu,drn h nryrlaepae h e snaps peg equi- the stable phase, a release energy toward the movement during rapid infinitesimal Thus, causes an librium. where load point point, in critical equilibrium A change unstable stable. that are the and F, to unstable, equilibrium, refers is resting point, final the 3 critical and the (Fig. E points as landscape to equilibrium referred energy of commonly approximately potential is positions force The the approximate which zero. at angles are rotation the E by points defined Here, and (posi- concavity). “valleys” P, (negative two “hill” F, by one curve and energy concavity) potential tive the condition in This reflected equilibrium. is unstable neg- an while indicates energy equilibrium, concavity the stable ative a of indicates curvature concavity the 3 positive by plot; 3 (Fig. determined (Fig. plot is force points energy equilibrium the the which in at minima points by or 1) maxima indicated are and a points equilibria Equilibrium A equilibrium. stable distant. unstable non- two one physically exhibits another be system to may buckling system position snap-through that a equilibrium position enables one equilibrium that from adjacent instability rapidly of pass mode to a (i.e., is region buckling through second the 3 of (Fig. rest the 25 between stiffness) in ues positive again (i.e., slope observed positive is A stiffness. negative of negative tive becomes the mesoster- curve that force–displacement the (note elastic and between the cuticle region, of second soft where slope this cavity, the the Within between of lip. entrance contact nal the no at is and outside there peg the of tions (19). terms inertial of presence displace- the Eq. and Per in forces) forces) damping elastic that (e.g., (e.g., forces velocity ment nonlinear on of dependent identification are the enables (19). method method force This restoring the following data kinematic mental force, sys- elastic nonlinear normalized using characterization identification. force tem elastic and Damping 3 and 2 forces (Figs. kinematics angular and o l lcsrcre rmseies1ad2 h damping the 2, and 1 specimens from recorded clicks all For h ieai n pcrlaaye ugs htteresponse the that suggest analyses spectral and kinematic The E G G E 1 scaatrsi fsa-hog ukig(0.Snap- (20). buckling snap-through of characteristic is ) hw htbt ocsaennieradflo h same the follow and nonlinear are forces both that shows at , and stettlnraie eoiydpnetfre which force, velocity-dependent normalized total the is θ F θ 0 = x ◦ = r e ob determined. be to yet are xsi i.3 Fig. in axis and θ , − G coordinate. h omlzddmigforce, damping normalized The 22 a ( 63 E θ θ θ ◦ ˙ = ) + ◦ 0 = etrstorgos h rtrgo is region first The regions. two features to .Teoealsaeo h lsi force elastic the of shape overall The ). G F − 0 a eetmtdfo h experi- the from estimated be can , ◦ ( ◦ a θ to ˙ θ + ) E n orsod otepositions the to corresponds and n,at and, , a 63 a θ sdcesn) hc sindica- is which decreasing), is E , F θ ◦ A szr.Tesaiiyo the of stability The zero. is ) θ n orsod oteposi- the to corresponds and , = ˙ ( and , .Tedmigadelastic and damping The ). θ r 0, = ) D θ ¨ θ A ˙ 2˙ + .Tennierelastic nonlinear The ). 0 = , θ vii r r nw rmthe from known are θ , θ and F ˙ 0 = θ and , F a emodeled be can , C ( a hw htP, that shows ) θ F θ E = ) hw htthe that shows ◦ viii n )with 2) and ) stecircum- the is h second The . and θ ˙ G ,adthus and ), − stetime the is n the and , a 25 θ i.3 Fig. . θ ◦ the , val- [1] F em nteeuto fmto niaeta h ocsatn on acting forces the that indicate motion single (Eq. of term equation a the elastic in an position, namely, terms and angular motion, term the damping of of a terms including equation in derive the equation to differential of us nonlinear enabled form the motion both general the Analyzing of the time. histories and peg time frequency the and both of frequency of position function angular a the as analy- study tip to Spectral used timescale. were exhibiting short techniques dynamic, very sis extremely a is over that oscillations phase multiple release energy an in motion the power to spring a as exoskeleton ani- (22–24). of their category use a to that belong mals beetles click shrimps, mantis and ants the genus ∼ example, the For in significant. ants also trap-jaw is energy duration in release difference and the experimental storage where comparable different systems is power-amplified system a other spring will using to exoskeletal and beetle’s future energy exoskele- click elastic the The the release in setup. and of further store parts studied to Other able be click be motion. also that the to may mechanism likely power ton spring is to distributed hinge use the the in beetles of cuticle component characteristic soft is a the phase Thus, be release recoil. energy elastic cuti- the an soft release of of the energy beginning of the and displacement at loading fast cle the very during the Additionally, cuticle phases. soft the of ments elastic release and limiting store unknown, to capability energy. were their study, beetles of this click understanding to our of prior mechanisms However, spring limited (21). not the speed are they contraction that muscle is by actuators as of advantage mechanisms the spring animals, using For dissipation stored elements. of passive fast release via energy fast enables power elastic specifically, recoil, mechanisms elastic mechanical spring through energy muscle of of amplify use The to output. (13) latch-mediated mechanisms on biologi- rely spring storage beetles other mechanical click like energy systems, amplify Therefore, power-amplified the motion. can cal clicking beetles between the click during difference that power com- time than supports faster 2) release The much (13.8 and and released stored. phase is 1 release is energy energy that specimens it the indicates in ms), of 17.4 duration ms and overall 235 the the of and to duration pared (160 long latching, relatively phase namely, The loading click, release. the energy of and phases loading, distinct three revealed ultrafast ray the characteriza- govern that the forces and damping and motion release. elastic of the the equation of enabled tion phase the release of energy formulation the the during and click (kinematics) (dynamics) in movement mechanism forces the spring of the analysis click- of In-depth the part beetles. of uncovered phases and the motion quantified and ing identified we study, this In Discussion model. the generate to used not movements similar reproduce in Appendix identified forces SI the on 3 based for reproduced Fig. accelerations are circumferential clicks The three 1 study. the which this (Fig. cor- in release, phase modeled response the unlatching not transient of the The to s response. responds 0.001 transient circumferential first the the the of represents for decay the except and acceleration, frequency the well predict aiae yetmtn h icmeeta ceeainfrall for acceleration circumferential ( the releases estimating by 3 Fig. validated in E and F minima local and energy potential the between 0 sadrlaei nls hn1m 2) hs ietrap-jaws like Thus, (22). ms 1 than less in it release and ms 400 ntecikbel,teeatcrci ftesf uil results cuticle soft the of recoil elastic the beetle, click the In displace- large reveal hinge the of images X-ray High-speed X- synchrotron high-speed using motion clicking the Studying h dnicto ehd(etrn oc ehd sdis used method) force (restoring method identification The F . 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APPLIED PHYSICAL SYSTEMS BIOLOGY SCIENCES the system, the elastic and damping forces, do not increase (or nonlinear system identification demonstrate that the damping is decrease) proportionally throughout the motion. Compared to quadratic. Previous work by some of the authors has shown that a linear, time-invariant, spring-mass, damped oscillator, which is the surfaces of the peg and the mesosternal cavity in contact dur- the simplest representation of a single-degree-of-freedom oscil- ing the energy release are very smooth (4). Consequently, dry lator (Fig. 3B, dashed line), the angular position of the peg tip friction is not likely the primary mechanism of energy dissipation. (Fig. 3B, solid line) exhibits two frequencies in addition to the The quadratic damping profile may explain why click beetles can fundamental frequency or first harmonic (i.e., the largest com- perform the clicking motion repeatedly without apparent exter- ponent at 352 Hz): a zero frequency (i.e., a large component at nal damage in the hinge, which would have been expected if dry 0 Hz) and a second harmonic. The zero frequency means that friction was the primary source of energy dissipation. the angular position response is asymmetric, and the second har- Uncovering the elastic and damping forces driving the release monic indicates the existence of a quadratic nonlinearity (25). provides several key insights. First, we show that the clicking The asymmetry of the angular position response, θ, may be due motion can be modeled as a single-degree-of-freedom system to the contact between the mesosternal lip and the soft cuticle during the release phase. This model structure allows us to infer when the peg is inside the cavity, constraining the motion for the energy release dynamics from observations and analysis of negative θ positions (Fig. 1 A, vii and viii; see Energy Release the hinge’s motion, specifically, by analyzing the motion of the Kinematics). While most analyses of extreme motions have been soft cuticle and latch dynamics. Second, the forces identified dur- limited to investigations of kinematics (13, 26), here, we consider ing the release are nonlinear. Knowledge of the dissipation forces the frequency response of the motion to investigate the dynamics profile (quadratic) and the elastic force profile (snap-through of the energy release. The spectral analysis informs our choice of buckling, an asymmetric instability) will guide future studies to an appropriate equation of motion structure and enables us to investigate the bending and deformation of the soft cuticle, and connect the discovered phenomena (asymmetry in the position the contact between the soft cuticle and the hinge, and to look graph) with physical observations (dynamic contact between two for shell-like structures in the hinge. Snap-through buckling is body parts). commonly observed in thin shells, and the material properties of During the energy release, the elastic and damping terms such structures are of primary interest for studies on fatigue and (F and G in Eq. 1) of the one-degree-of-freedom oscillator damage mitigation. that describe the angular position of the peg tip were deter- This paper integrates biological knowledge with physics-based mined using nonlinear system identification. The restoring force experimental and analytical tools to examine the click beetle’s system identification method is capable of extracting the total ultrafast clicking motion. The tools applied in this study allow for displacement and velocity-dependent forces, referred to as the the analysis of LaMSA organisms as integrated dynamic systems. elastic and damping forces. However, the exact source and con- This study provides a pathway for the analysis of ultrafast motion straints that contribute to each of these forces would require starting from traditional motion observations (kinematics) and additional experiments. Nonetheless, characterizing the elastic leading to uncovering the dynamics. The use of synchrotron and damping forces reveals the fast energy actuation and energy imaging enables subsequent dynamical analysis without the use dissipation strategies used by click beetles to overcome mus- of invasive measurements, which are difficult to conduct at this cle velocity limitations. For example, understanding the elastic small scale and would likely alter the system and its response. recoil strategy is fundamental to explain the material and muscu- The procedures outlined in this study provide an analysis and loskeletal properties required to generate extremely fast energy modeling scaffold for researchers studying organisms that use release. Additionally, uncovering damping strategies can inform springs and latches to amplify mechanical power output. Such a fatigue and damage considerations in power-amplified systems. scaffold may also lead to the design and fabrication of power- The overall shape of the elastic force, which features a nega- amplified engineered systems. In-depth understanding of the tive stiffness region (Fig. 3E), is characteristic of snap-through forces governing extreme maneuvers will enable the creation buckling systems (20). Our analysis shows that snap-through of a new generation of insect-inspired robots capable of gener- buckling is the dominant elastic recoil mechanism governing the ating and sustaining high-acceleration movements. Such robots angular position of the peg during the energy release phase. In will also serve as research platforms to answer critical questions the click beetle, the use of snap-through buckling as the elastic about their biological counterparts. recoil strategy enables the peg to move between physically distant points extremely quickly, with only a small change in applied Experimental Methods forces, effectively leveraging energy transfer to move between Click Beetle Collection. In July 2018, four click beetles (Elater low-energy states. Snap-through buckling has also been observed abruptus Say, 1825) were collected on permanent research sites in other plants and animals. For example, the Venus flytrap uses owned by the University of Illinois at Urbana–Champaign in snap-through buckling as the governing energy release mech- Champaign County, with permission from the university’s Com- anism to shut its jaw-like shells (10). Unlike the visible snap- mittee of Natural Areas. The beetles were captured alive using through buckling of the Venus flytrap shells, no visible snapping modified black cross-vane panel traps (AlphaScents), coated or deformations were observed in the hinge of the click beetles with the fluoropolymer dispersion Fluon®PTFE (AGC Chem- in this study. Thus, there is a need to examine other parts of the icals Americas, Inc.) to improve trapping efficiency (28). An cuticle in the future to uncover the portions of the body that may experimental lure provided by Jocelyn G. Millar, Department of snap during the energy release phase. Entomology, University of California, Riverside, was used. The The damping force is of quadratic form (Fig. 3D), which con- live insects were kept in plastic containers with bark and soil, and firms the existence of a nonlinear quadratic term in the equation were fed sugar and water ad libitum using a 1.7-mL microcen- of motion as predicted by the spectral analysis. A damping force trifuge tube (Denville Scientific) filled with 10% sucrose solution of quadratic form is usually characteristic of air damping (27). capped with a cotton ball. The animals were identified to species However, quadratic damping could also arise due to the con- following Roache (1960), Johnson (2003), and Evans (2014) tact between the soft cuticle and the mesosternal lip (Fig. 1 A, (29–31). Voucher specimens were deposited in the Illinois Natu- vii) during the energy release oscillations, or due to the contact ral History Survey Insect Collection, Prairie Research Institute, between the peg and very low stiffness compliant elastic anatom- the University of Illinois at Urbana–Champaign. ical elements such as hairs, which have been previously observed on the upper part of the cavity (4). Although the linear decay Motion Capture Setup. The clicking motion of the hinge was of Fig. 3A may indicate that the primary damping mechanism is recorded with both synchrotron X-ray imaging and visible-light dry friction (Coulomb friction), both the frequency analysis and imaging at beamline 32-ID of the Advanced Photon Source

6 of 8 | PNAS Bolmin et al. https://doi.org/10.1073/pnas.2014569118 Nonlinear elasticity and damping govern ultrafast dynamics in click beetles olna lsiiyaddmiggvr lrfs yaisi lc beetles click in dynamics lip ultrafast govern mesosternal damping and and peg elasticity Nonlinear the One of move. motions al. plane head to et recorded sagittal Bolmin free the the that were in validate head and view the animals lateral the and the monitor prothorax, to record used the to view, hinge, visible-light hinge a thoracic the provided the on camera clicks. mesothorax; high-speed focused were second the was A to motion. and click abdomen view the the X-ray during from the constrained ( provided cavity. and camera vise mesosternal high-speed custom the a and in lip, placed mesosternal ( the abdomen. and peg, metathorax, the mesothorax, the and prothorax, and head 4. Fig. antenna further and with leg click the on perform based to filming, unable after were but alive movements, specimen still ( of fps were motions 1,000 mals clicking at eight recorded and were clicking 4 3 Seven specimen fps). release (20,000 second of 2 a motions specimen respectively; for fps, recorded 30,000 was and motion fps 20,000 1 at specimen each 2 for and recorded was release) for and made loading, (latching, were ( recordings specimen Multiple began each cycles. immediately multiple beetles the on, for to turned clicking animal was the beam exposing the by when motion. beam; triggered clicking was the motion during clicking prothorax in the The view of 1/4 lateral about a and providing hinge, up, was 4 body head the (Fig. the of projection side and the beam that the such faced source, positioned were X-ray was the movements individual to Each phase perpendicular body. release the energy constraining 1 by the (Fig. altered able Only motions were jump. unbending beetles a and the before head, bending Thus, the same move. the leaving to perform Corpo- free constrained, Kerr hinge was and Round, abdomen prothorax, were Rods the Wax beetles Only (Utility the ration). wax which dental between using plates fixed 4 adjustable (Fig. parallel vise two custom a in that mounted confirm and insects clicks. the were monitor movements to observed visible- the The used (fps). were second images per frames light 30,000 X- and using 20,000, 1,000, hinge of 4 the (Fig. of light visible motion and the ray record simultaneously were to SA-Z) used FASTCAM (Photron cameras high-speed Two view. screen, scintillator 5 a a keV), cre- and (25 were scale X-rays Images monochromatic centimeter using (32). to ated resolution millimeter spatial the micrometer-scale at with internal insects of motion living the in record and structures view to imag- possibility X-ray the offers Synchrotron ing Laboratory. National Argonne (APS), niiulbelswr nshtzduigntoe and nitrogen using anesthetized were beetles Individual lc etemrhlg n xeietlstp ( setup. experimental and morphology beetle Click × irsoeojcie rvdn 2.4 a providing objective, microscope IAppendix SI C .Xryiae atrdtemstoa,the mesothorax, the captured images X-ray ). hinge B A C hinge .Til eercre tfaerates frame at recorded were Trials ). 1 cm al S1 Table , Dorsal view IAppendix SI Lateral view C .Tevs a opsdof composed was vise The ). .Oecikn motion clicking One ). abdomen and metathorax meso-, prothorax head and al S1 Table , sternal lip × meso- cavity peg . mfil of field mm 3.6 C A xeietlstpue oiaetehnedrn h lcigmto.Tebelswere beetles The motion. clicking the during hinge the image to used setup Experimental ) h oyo lc etecnb iie notosbnt,lne yatoai ig:the hinge: thoracic a by linked subunits, two into divided be can beetle click a of body The ) .Teani- The ). C A B as ) kthhglgtn h ntmclcmoet ftehnei h aitlplane: sagittal the in hinge the of components anatomical the highlighting Sketch ) a edet ihrcneto uiua rtislk resilin like the proteins making cuticle proteins, cuticular and the of chitin the of content between higher cross-linkages softness less a Relative or to (4). due for maintained be critical is be may is which cuticle to cuticle, the stiffer latch cuticle,” of peg’s the “soft the part to the This comparison as phase. in to loading soft referred the peg, during the tracked position to was the dorsal tip Additionally, cuticle motion. peg the of the of phases of all position for the the digitized track Specifically, was to frame. used by were phases. frame release hinge motion, energy thus the and mechanics; around loading latch landmarks of the Anatomical analysis detailed the (4) on of al. focused range et we full Bolmin the by work observe ( ous phases to release used energy record- were and 20,000-fps loading 2) blur the and and motion 30,000- to 1 The due (specimens phase. release, ings release fast energy the resolve the phases), to in loading able and not (latching relatively were click but the the captured during recordings movements slow 1,000-fps clicks multiple The enabled recording. which were level, per radiation images lower 1,000-fps a at The obtained (Xcitex). ProAnalyst using analyzed Analysis. Motion in reported are recorded length body specimens S1 Table and four mass apex the The the to of Absolute). base the (Mitutoyo accuracy from calipers an measured was with using length body (Ohaus) each The scale after mg. 1 Standard measured of Precision was a individual using each trial of mass The stimulus. nCreincodntso pcmn1 eodda 000fps, 30,000 at recorded tip 1, peg specimen and the of of ProAnalyst coordinates the position tracker Cartesian using The each in in (Mathworks). coordinates of cavity Matlab position Cartesian into the The imported in and camera. X-ray extracted lip, the mesosternal was of the view of peg, field the to respect cuticle knowl- soft our beetle’s known. To click not the thinner. is of be structure may chemical exact it the or edge, (33), sclerotized less cuticle 0.5 cm i.1 Fig. 1 mm A . hw h adakpit sdfrtakn with tracking for used points landmark the shows camera (x-ray) speed High- (visible) camera speed High- beetle h ihsedsnhornXryvdo were videos X-ray synchrotron high-speed The X-ray source Top view vise and mirror Scintillator https://doi.org/10.1073/pnas.2014569118 oisS1 Movies and IAppendix SI PNAS S2 .Previ- ). | f8 of 7 ,

APPLIED PHYSICAL SYSTEMS BIOLOGY SCIENCES was filtered using a low-pass filter (cutoff frequency, 800 Hz) imen 2’s release B. SI Appendix, Fig. S2A shows the comparison in Matlab, and the position of specimen 2, recorded at 20,000 between the experimentally driven circumferential acceleration fps, was interpolated in ProAnalyst. The velocity and accelera- (i.e., aθ = r2Aθ¨2A + 2˙r2Aθ˙2A) and the modeled circumferential tion of the peg tip were calculated using the custom derivative acceleration (i.e., aθ = −F2A(θ2A) − G2A(θ˙2A)) for specimen 2 function “deriv” from the System Identification Programs for release A. SI Appendix, Fig. S2 B and C validates the system AirCraft toolbox in Matlab (34). The derivatives were smoothed identification approach by comparing the experimentally driven using a local quadratic least-squares fit to each data point and its circumferential acceleration using θ and r for specimen 1 and four neighboring points. The angular position of the peg tip was specimen 2 release B to the specimen 2 release A modeled cir- shifted to reach zero at the equilibrium position at the end of a = −F (θ ) − G (θ˙ ) the oscillations for the dynamics analysis, to facilitate the calcu- cumferential acceleration (i.e., θ 2A i 2A i , i = 1 lations. The synchrotron X-ray images were also used to acquire or 2B). G morphological data, including the cavity’s geometry (Fig. 1A), was estimated by fitting a quadratic polynomial through the F necessary for the analysis of the energy release dynamics. data for all clicks. was estimated for each click by fitting a piecewise cubit hermit interpolating polynomial. The circumferential G Restoring Force Method and Model Validation. The normalized acceleration was predicted based on the function and on the damping force G and the normalized elastic force F were derived function F estimated from specimen 2’s release A’s data. G and F using the restoring force method (19). We assume that no are the lumped velocity-dependent and displacement-dependent external force is applied to the system during the release (free forces. Various constraints and forces may contribute to either response). The equations of motion in polar coordinates can be or both of these functions. P formulated as aθ = Fθ, where Fθ are the normalized forces ¨ ˙ Data Availability. Video files and raw excel data have been deposited at tangent to the motion, and aθ = rθ + 2˙rθ. As the polar coordi- https://doi.org/10.13012/B2IDB-8033264 V1. nates r and θ are linked through the cavity’s geometry, we define the polynomial functions F and G such that ACKNOWLEDGMENTS. We thank Dr. Alex L. Deriy for his help with the experimental setup at Argonne National Laboratory. We thank Dr. Andrew ¨ ˙ ˙ V. Suarez (Department of Entomology at the University of Illinois at Urbana– rθ + 2˙rθ + F (θ) + G(θ) = 0. [2] Champaign [UIUC]) for his support in obtaining APS beam time. We thank Dr. Alexander F. Vakakis (Department of Mechanical Science and Engineer- F and G were evaluated at the points where θ and θ˙ are zero such ing [MechSE], UIUC) for the discussions about the restoring force method, ˙ ˙ and Alireza Mojahed (MechSE, UIUC) for the discussions about the wavelet that, at θ = 0, G(θ) = −aθ, and, at θ = 0, F (θ) = −aθ after fit- transform plots. We are grateful for Dr. Tommy C. McElrath’s (Illinois Natural ting a piecewise cubit hermit interpolating polynomial (“pchip” History Survey, Prairie Research Institute, UIUC) help for the species identi- function in Matlab) to the experimental data for each click. fication. We would also like to thank Dr. Jocelyn G. Millar (Department of Eq. 2 was used to predict the functions G and F (SI Appendix, Entomology, University of California, Riverside) and Dr. Lawrence M. Hanks (Department of Entomology, UIUC) for providing us with an experimental Fig. S2). G and F outputted by the system identification anal- click beetle lure. The use of the APS was supported by the US Department ysis for specimen 2’s release A were used to reconstruct the of Energy, Office of Science, Office of Basic Energy Sciences, under Contract circumferential acceleration for specimen 1’s release and spec- DE-AC02-06CH11357.

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8 of 8 | PNAS Bolmin et al. https://doi.org/10.1073/pnas.2014569118 Nonlinear elasticity and damping govern ultrafast dynamics in click beetles