The motion of cilia could be further understood by creating and analysing a large scale model with similar stiffness and property values. Varying amounts of vibrating HEXBUG robots, representing motion and active particles, were placed in series with different segment lengths connecting them. The oscillating motion of a ciliium was able to be replicated and analyzed using autotracing and autotracking programs. The amplitude of the model increased with increasing length, while increasing length had an inconclusive effect on period. Furthermore, the visual path of a marked end segment repeated with each oscillation recorded

Utterback, Toby

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Washington University St. Louis: Open Scholarship

Washington University in St. Louis Washington University Open Scholarship Mechanical Engineering and Materials Science Independent Study Mechanical Engineering & Materials Science 12-21-2022 Physical Model of Cilia Motion Toby Utterback Washington University in St. Louis Follow this and additional works at: https://openscholarship.wustl.edu/mems500 Recommended Citation Utterback, Toby, "Physical Model of Cilia Motion" (2022). Mechanical Engineering and Materials Science Independent Study. 210. https://openscholarship.wustl.edu/mems500/210 This Final Report is brought to you for free and open access by the Mechanical Engineering & Materials Science at Washington University Open Scholarship. It has been accepted for inclusion in Mechanical Engineering and Materials Science Independent Study by an authorized administrator of Washington University Open Scholarship. For more information, please contact digital@wumail.wustl.edu. Physical Model of Cilia MotionMEMS 400: Independent StudyToby UtterbackLab Instructor: Dr. Philip BaylyDecember 21st, 2022Mechanical Engineering and Materials Science - McKelvey School of EngineeringWashington University in St. Louis1 Brookings Drive, MSC 1185-208-125 St. Louis, Missouri 63130ABSTRACT:The motion of cilia could be further understood by creating and analysing a large scale modelwith similar stiffness and property values. Varying amounts of vibrating HEXBUG robots, representingmotion and active particles, were placed in series with different segment lengths connecting them.The oscillating motion of a ciliium was able to be replicated and analyzed using autotracing andautotracking programs. The amplitude of the model increased with increasing length, while increasinglength had an inconclusive effect on period. Furthermore, the visual path of a marked end segmentrepeated with each oscillation recorded.INTRODUCTIONMotile cilia are hairlike structures found on the surface of cells. The beating nature of ciliamoves fluid and aids in general development within the body. The motion and structure of ciliaincluding different types of imaging have highlighted a ”9+2 axoneme comprising nine outer doubletmicrotubules and a central apparatus” [1]. The physical model created will be used to learn moreabout how the axoneme moves and the importance of its movement. Scaling up from realisticallysized cilia allows properties and theories about cilia movement to be tested.Hexbug Properties. To introduce vibrations and motion into the model, HEXBUG nano®robots were used to simulate active particles. The HEXBUG can move due to a combination of arotating motor and twelve bent rubber legs. The motor ”rotates a counterweight that causes forces upand down” [2]. The rotation of this counterweight is believed to only be in one direction. When theforce is downwards the legs are pressed down. On the reverse, when the force from the weight is up,the legs are uncompressed. This leg motion produces the movement and bouncing of the HEXBUG.The direction of the bugs is random and the power is dependent on the battery power [3].Stiffness Calculations. A nylon rod was used to calculate the force produced by the HEXBUG.The properties of the nylon rod were found by using length, added weight, and difference in height ofa tube. Young’s modulus was calculated using the following beam displacement equation.𝐸𝐼 =𝐹𝑚 ∗ 𝐿33 ∗ 𝛿 ∗ 10003(1)1where E is the Young’s modulus [Pa], 𝐹𝑚 is the weight [N], L is the length of rod [m], and𝛿 is thedifference in height [m].To find the Young’s modulus, the area of inertia for a solid cylinder is calculated with eqn. 2𝐼 =𝜋 ∗ 𝑑464(2)where I is the calculated area moment of inertia for solid cylinder [𝑚4], and d is the diameter of therod [m].Force Calculations. To measure the force produced by a HEXBUG, a nylon rod was connectedto the HEXBUG and the displacement was measured. To find the force eqn. 3 was used.𝐹ℎ𝑒𝑥𝑏𝑢𝑔 =𝛿 ∗ 3 ∗ 𝐸𝑟𝑜𝑑 ∗ 10002𝐿3𝑟𝑜𝑑(3)where 𝐹ℎ𝑒𝑥𝑏𝑢𝑔 is the force produced by a HEXBUG [𝑁], 𝛿 is the displacement [m], 𝐸𝑟𝑜𝑑 is the Young’smodulus of the rod [GPa], 𝐿𝑟𝑜𝑑 is the length of the rod [m].Scaling of Physical Model. The scale of the physical model was based on previousexperiments done by Zheng and coauthors. The experiment conducted by Zheng et al. used sevenHEXBUGs in series to represent active particles. The HEXBUGSs were connected by a laser-cutrubber chain with slots for each HEXBUG. For the physical model discussed in this paper, rubbertubing was used to connect 3d printed HEXBUG casings. For the laser cut model, the width andheight of the segments were constant with a varied length between segments. The different segmentlengths varied the motion of the HEXBUG chain [4]. Using the dimensions of rubber segments thatproduce an oscillating motion and rubber tubing with a set Young’s modulus and cross-sectional area,estimates the length of the rubber tubing needed to produce oscillations. This relation is done bycalculating the stiffness of the rubber segments (eqn. 4) and setting it equal to the stiffness of therubber tubing as shown in eqn 5.𝐾𝑠 =𝑊𝑠∗𝐻3𝑠12 ∗ 𝐸𝑠𝐿𝑠(4)where 𝐾𝑠 is the stiffness [N*m], 𝑊𝑠 is the width [m], 𝐻𝑠 is the height [m], 𝐸𝑠 is the Young’s modulus[Pa], 𝐿𝑠 is the length [m].2𝐾𝑠 = 𝐾𝑡 (5)The 𝐾𝑡 is the stiffness of the rubber tubing [N*m]. The s is used to indicate rubber segments and thet indicates the rubber tubing. With the stiffness known, the estimated length needed for oscillationsusing rubber tubing is found using eqn. 6.𝑙𝑡 =𝜋∗(𝑑4𝑡 −𝑑4𝑖 )∗100064 ∗ 𝐸𝑡𝐾𝑡(6)where 𝑙𝑡 is the length [mm], 𝑑𝑡 is the outer diameter [m], 𝑑𝑖 is the inner diameter [m], 𝐸𝑡 is the Young’smodulus [Pa]. This length value is the starting point for creating a range of length values that couldproduce oscillations.METHODSStiffness Properties. Initially, a nylon rod was going to be used in between the HEXBUGsto add stiffness to the model. The properties of the nylon rod available were unknown. To find theproperties, a set length of the rod was placed over the side of a table with the other end taped down.The pre-weight free end height was measured from the ground to the middle of the rod cross-section.A 10 g weight was taped with a string on the free end of the rod. The post-weight height was measuredto find the height difference. This was repeated with a 20 g and 50 g weight. The length of the rod waschanged two more times for a total of nine trials. The length and weight values used were set to preventtoo much bending. This rod material was not considered for the segments between the HEXBUGsbecause the average Young’s modulus was too large and would have required large segments lengths.Instead, this tubing was used to find the force produced by a HEXBUG because its properties wereknown.Force Properties. Three different types of robots were considered to act as active particlesdue to their force values. Each robot had a rotating motor and rubber legs. The rod previously usedwas measured at a set length and one end was tabled down on the table. The free end was placed sothat the rod was linear and the distance from the tip to the side of the table was measured. A bug was3taped to the end of the free end perpendicular to the rod. After the bug turned on and stabilized inone bend rod position, the distance from the middle of the bug to the side of the table was measured.The distance was the only unknown property needed to find the force values.Scaling Model. The MATLAB file lengthcalculator.m uses the known property values fromthe Zheng et al. paper and user inputs [4]. When run, a popup tab appears asking for the outerdiameter of the rod (mm), inner diameter of the rod (mm), and Young’s modulus of material (DURO).This allows the user to use different materials in the tube shape. Once input, the script uses the knownproperties to find stiffness, as shown in eqn. 4. That is set as the stiffness of the new material, eqn.5. The final output is 𝐿𝑒𝑛𝑔𝑡ℎ𝑛𝑒𝑒𝑑𝑒𝑑𝑚𝑚, which is the length needed to keep the same stiffness as thematerial known to oscillate.Creating Hexbug Casing. The HEXBUG required a casing that was able to hold it in motionand connect to other casings. A SolidWorks model of a HEXBUG was found online. This modelwas used to cut out the shape on an extruded rectangle. This rectangle was cut down to replicatethe shape of the HEXBUG. The size of the connection part of the casing had to be adjusted due tothe material for the in-between segments. A rubber tubing was chosen because it had a low enoughYoung’s modulus to allow smaller connecting segments between casings. The connection part wasa hollow cylinder. Midway down the cylinder, a hole was made. This hole allowed for a pin to gothrough the rubber tubing and secure the segments together. The pin was also important because itallowed chains with varying amounts of HEXBUGS to be experimented with.Experiment. Once all of the parts of the chain’s properties had been made and analyzed, theexperiment could be run. The experiment was run on a smooth, white table to allow for contrast andoptimal movement. The first rubber tubing segment was weighed down on one side and pinned tothe first HEXBUG casing. The HEXBUG casings were all pinned in series with the rubber tubingin between. There were a total of eight casings. A camera was set up overhead of the experimentto record the motion of the series. For the first set of trials, 10/16” segments were used. Once alleight of the HEXBUGS were pinned and turned on, time was given to let the oscillation start. Oncethe oscillations started, three to four full oscillations were recorded with the camera. Then the endHEXBUG was unpinned, shortening the series to seven. Time was given for the oscillations to adjust4and then the model was recorded. This was repeated until there were three HEXBUGS left, for a totalof six trials. This process was also repeated for 11/16” and 12/16” segments.Video modificaiton. Before the videos were traced, different video modifications had to beperformed. The video camera used recorded at 29.87 fps. The number of frames collected per videowas too much to efficiently trace. videoDOWNsample.m is a MATLAB script created by Dr. LouisWoodhams of Washington University in St. Louis. This program reduces the number of frames in avideo by a set number. For this experiment, the videos were all reduced so only one of every ten framesare saved in a new video. shape.m is a Matlab script that replaces a set area with a white rectangle ineach frame. The tracing program relies on contrast, so a dark object not part of the waveform coulddistract the program. The output of this program is a new video with the white rectangle added.Tracing. The tracing MATLAB script, 𝑎𝑢𝑡𝑜𝑡𝑟𝑎𝑐𝑒 𝑓 𝑢𝑛ℎ𝑒𝑥𝑏𝑢𝑔𝑠.𝑚, was created by Dr.Woodhams. The input of this program is the video being traced, the spatial resolution, and theframes per second. When the program is run, the user is given the first frame of the video and roughlytraces the cilium. Once traced, the program will use the contrast to trace the rest of the frames. To getthe program to trace correctly for each video, different tracing factors such as the number of segmentsalong the length and number of theta values had to be adjusted. The relevant output of this programwas the length, period, and amplitude. plotThetaArray.m, another MATLAB script created by Dr.Woodhams, takes the output from the autotrace program and makes a GIF of the movement of thecilium.Autotrack. autotrack.m, made by Dr. Woodhams, is another tracing program that tracks thebody motion of a focused shape in a video. In this case, the user is able to circle one of the HEXBUGS.The output is a visual trace of where the HEXBUG was in all of the frames on one image. This wasdone with each of the HEXBUG videos.5RESULTS AND DISCUSSIONStiffness and Force Results. The average calculated Young’s modulus value for the rod was2.19 GPa. This is similar to the known Young’s modulus value for nylon of 2.7 GPa. Table 1 showsthe force values of three different types of robotic bugs.Table 1 Force Values of Different Robot BugsAverage Force Value (N)Nanoflash Large 0.0169Nanoflash Small 0.0171Nano Nitro 0.0229The Nano Nitro had the largest force, but it also bounced back and forth at the maxdisplacement. For the final model, the Nanoflash small was chosen.Scaling Model. The dimensions and properties of the rubber segments and the rubber tubingare shown in Table 2 and Table 3. The rubber segments are from the Zheng el al. paper [4]. Therubber tubing is the material used as segments for this model.Table 2 Properties of Rubber SegmentsWidth (m) Height (m) Length (m) Youngs Modulus (Pa)4.25E-03 4.25E-03 8.50E-03 2.39E+05Table 3 Properties of Rubber TubingOuter Diameter(m) Inner Diameter(m) Young’s Modulus (Pa)3.20E-03 1.60E-03 1.00E+06Using lengthcalculator.m the length needed was found to be 6.31𝒎𝒎. This length does notinclude the 6.7 mm of the HEXBUG casing. The total length of each segment needed to be around19.71 mm (∼ 12/16 in). The length of segments used in this experiment were 10/16 in, 11/16 in, 12/16in.6Hexbug Casing. Figure 1 and 2 show the Solidworks model for the HEXBUG casing.Figure 1 Top View of HEXBUG CasingThe shape of the HEXBUG was cut out so that the HEXBUG has a snug fit. The dotted linesare hollowed cylinders.Figure 2 Rotated View of HEXBUG CasingThe holes in the connection parts are for inserting a pin to hold the rubber segments.Autotrace Results. Figure 3 shows eight frames from the trial with 12/16 in segments andfour HEXBUGS.7Figure 3 Frames from Trial with Four HEXBUGSFigure 4 shows eight frames from the trial with 12/16 in segments and eight HEXBUGS.Figure 4 Frames from Trial with Eight HEXBUGSFigure 5 shows four frames from a .Gif of the traced motion. This trace is from the trial with11/16 in segments and six HEXBUGS.8Figure 5 Frames from Autotrace TracingThere will be a link to all of the Gifs of the traces in the Appendix. The most important valuesfrom Autotrace were period and amplitude. Figure 6 shows the period vs. calculated length of thecilia model in mm.Figure 6 Period vs. Calculated Length of Cilia ModelThe calculated length of cilia was found by the length of the HEXBUG casings and the lengthof the segments in between the casings. Each point signifies the amount of HEXBUGs includedstarting with three HEXBUGs on the left to eight HEXBUGs on the right. For all of the differentsegment lengths, from three to five HEXBUGS, the period increased with the length of the ciliamodel. This means that as the cilia model increased the time in-between cycles increased. The 11/16”and 12/16” segment experiment’s periods peaked at five HEXBUGs and then slightly decreased. Thelargest period of the 11/16” segment trials and the 12/16” segment trials were respectively, 18.4 s9and 23.7 s. The increasing length of the cilia model did not seem to increase the period of 11/16”and 12/16” segment trials for over five HEXBUGs. The 10/16” segment trial’s period continued toincrease with the length of cilia until peaking at seven HEXBUGs. This means that as the length ofthe cilia model increased, the period also increased until over seven HEXBUGs. The 10/16” segmenttrials largest period of 35.4 s was at a length of 442.72 mm (7 HEXBUGs). Figure 7 shows theamplitude vs. calculated length of the cilia model.Figure 7 Amplitude vs. Calculated Length of Cilia ModelFor each of the experiments, as the calculated length of the cilia model increased, the amplitudeincreased. The amplitude is the deviation from the median, indicating more bending. Overall, the11/16” segment trials consistently have higher amplitudes at each different HEXBUG amount, followedby 12/16” and 10/16” segment trials.10Autotrack Results. Figure 8 shows the tracing of the end HEXBUG from three to eightHEXBUGS with 12/16” segments.Figure 8 End Movement of Three to Seven HEXBUGsEach frame shows the end movement with a different amount of HEXBUGs. The first twoframes show the light sway of the chain back and forth, while the longer chains demonstrate a figureeight motion. The larger section on the left motion is consistent throughout all of the experiments.The reason is unknown, but could be connected with the direction of the motor inside the HEXBUGs.11CONCLUSIONThe motion of cilia can be physically modeled on a large scale by using forces from vibrationand calculated stiffness from set segments. Using known material properties, the forces of vibratingHEXBUGs and the stiffness of the rubber tubing were found and connected with 3d printed casings.The motion of the 3D model was able to be recorded and analyzed visually with tracing software; thesoftware allowed the period and amplitude of the motion to be compared between trials with differingsegment lengths and number of HEXBUGs. These findings highlight that is possible to replicate andanalyze the motion of a large-scale cilia model. The future of this work involves using the techniquesand values discovered in this model to create coupled models. Coupled models will contribute tounderstanding the synchronization of a phase and the general motion of the flagellum.12References[1] Rasagnya Viswanadha, W. S. S. and Porter, M. E., 2017, “Ciliary Motility: Regulation of Axonemal Dynein Motors,”Tech. rep., Cold Spring Harb Perspect Biol.[2] Robotics, “HEXBUG Nano How it Works,” accessed November 13, 2011, https://www.youtube.com/watch?v=dhRoqUkhbdU[3] Suphona, T., 2021, “Collective behaviors of automous robots in complex enviroment,” Master’s thesis, ChambersUniversity of Technology, Gothenburg, Sweden.[4] Zheng, E., Brandenbourger, M., Robinet, L., Schall, P., Lerner, E., and Coulais, C., 2021, “Self-oscillation andSynchronisation Transitions in Elasto-Active Structures,” doi: 10.48550/ARXIV.2106.05721, https://arxiv.org/abs/2106.0572113A Additional FiguresAutotrace GIF. Click here to be able to see the all of the GIF from Autotrace.Autotrack Results. :Figure A.1 End Movement of HEXBUGs (10/16”)14Figure A.2 End Movement of HEXBUGs (11/16”)151 MATLAB Code1 %Bug Length Calculator2 %TLU with variables from Zheng et al.3 %Paper Variables4 W p = 4.25e−3; %m (p.1 fig 1)5 L p = W p*2; % m ( based on fig 1c)6 H p = W p; % m (based on fig 1c)7 G p = 0.239e6; %Pa %youngs modulus (p.7)89 % %Variables10 % d t = 3.175; %mm %diameter of rod11 % E r = 2.185 %GPa % Measured Young's Modulus12 % I r = 4.988E−12 %mˆ41314 prompt = {'Enter Outer Diameter of Rod (mm)','Enter Inter Diameter of ...Rod(mm)','Enter Youngs Molulus of Materal(DURO)',};15 dlgtitle = 'Input';16 dims = [1 40];17 definput = {'0','0', '0'};18 answer = inputdlg(prompt,dlgtitle,dims,definput)1920 %d t = str2num(cell2mat(answer(1,1)))/100021 %d i = str2num(cell2mat(answer(2,1)))/100022 %E = str2num(cell2mat(answer(3,1)))2324 d t = .003225 d i = .001626 E = 602728 %E r = exp(E*0.02750−0.6403)*10ˆ6; %Pa29 %E r = exp((E+50)*0.02750−0.6403)*10ˆ630 E r = 1e6 %3132 % if cylinder33 I r = (pi * ((d t)ˆ4 − (d i)ˆ4))/64; %mˆ4341635 %Paper Torsional Constsant −−> torsion is twisting36 w p = W p/2; %m37 l p = L p/2; %m38 h p = H p/2; %m39 J p = w pˆ3*h p*((16/3) − 3.36*(w p/h p)*(1−(w pˆ4/(12*h pˆ4)))); %mˆ4 (p.7)40 C = (G p*J p)/L p; %N*m/rad %torsional stiffness (p.7)4142 %Paper Bending Stiffness43 I p = (W p * H pˆ3)/(12); %mˆ444 C p = (I p * G p)/L p; % N*m45 C b = C p/C;4647 %Model Stiffness48 K t = C p; %N*m49 Length needed mm = (E r * I r *1000) / K t %mm50 Length needed c = Length needed mm/10 %cm17

Physical Model of Cilia Motion

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Physical Model of Cilia Motion

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