integrated analysis of cell shape and movement in moving frame · 04/11/2020 · yusri dwi...

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© 2019. MANUSCRIPT SUBMITTED TO JOURNAL OF CELL SCIENCE (2019) 00, jcsxxxxxx. doi:10.1242/jcs.xxxxxx

RESEARCH ARTICLE

Integrated Analysis of Cell Shape and Movement in MovingFrameYusri Dwi Heryanto1, Chin-Yi Cheng1, Yutaka Uchida2, Kazushi Mimura3, Masaru Ishii2, RyoYamada1

ABSTRACT

The cell movement and morphological change are two interre-lated cellular processes. The integrated analysis is needed toexplore the relationship between them. However, it has beenchallenging to investigate them as a whole. The cell trajectorycan be described by its speed, curvature, and torsion. On theother hand, the 3-Dimensional (3D) cell shape can be stud-ied by using a shape descriptor such as Spherical harmonic(SH) descriptor which is an extension of Fourier transform in 3Dspace.

To integrate these shape-movement data, we propose amethod to use a parallel-transport (PT) moving frame as the3D-shape coordinate system. This moving frame is purely deter-mined by the velocity vector. On this moving frame, the move-ment change will influence the coordinate system for shapeanalysis. By analyzing the change of the SH coefficients overtime in the moving frame, we can observe the relationshipbetween shape and movement.

We illustrate the application of our approach using simula-tional and real datasets in this paper.

KEYWORDS: cell shape, cell movement, moving frame, spheri-cal harmonics, integrated analysis

INTRODUCTIONThe cell movement and morphological change are highly inte-grated. They share many biological mechanisms controlled bythe cytoskeleton, cell membrane, membrane proteins, and extra-cellular matrix(Friedl and Wolf, 2009; Ridley, 2003). Almost allforms of the cell active movement need forces that are gener-ated from dynamic shape change(Lämmermann and Sixt, 2009).Moreover, the difference in the shapes and sizes of motile cellsreflects their movement pattern(Keren et al., 2008; Lämmermannand Sixt, 2009). To study the shape-movement of the cell as awhole connected process, the integrated analysis is essential.

The first step in the shape analysis is shape normalization. Cellsare 3D objects that usually have arbitrary spatial positions, direc-tions, and scales in the 3D-space. However, these shapes may

1Unit of Statistical Genetics, Center for Genomic Medicine, Graduate School ofMedicine Kyoto University, Kyoto, Japan 606-8507, Japan.2Department of Immunology and Cell Biology,Graduate School of Medicine andFrontier Biosciences, Osaka University, Osaka, 565-0871, Japan.3Graduate School of Information Sciences, Hiroshima City University, Hiroshima,731-3194 Japan.

Authors for correspondence: ([email protected];[email protected])

5/11/2020

be variations of the same shape and should be recognized as thesame one. Shape normalization will put the object in the com-mon frame of reference and make shape analysis more robust tothese isometric transformations (i.e. translation, scale, rotation,and reflection).

From all isometric transformations, orientation is usually thehardest one to be normalized. Some shapes with a distinguishableaxis of orientation such as an ellipsoid are easy to orient usingtheir longest axis as the axis of orientation. Yet, many shapes donot have a clear axis of orientation. Many studies investigated thistopic with different approaches. The most well-known approach isthe principal component analysis based approach(Shilane et al.,2004). However, these methods do not provide robust normal-ization and can produce poor alignments(Kazhdan et al., 2003;Kazhdan, 2007). Another approach is to transform each shape intoa function and then calculate the rotation that minimizes the dis-tance between the two functions(Makadia and Daniilidis; Makadiaet al., 2004). The weakness of this approach is the alignmentresults are inconsistent and depend on the initial pose of the model.

In this paper, we propose a novel method to solve this alignmentproblem and integrate the analysis of shape and movement at thesame time by aligning the 3D objects using the parallel-transport(PT) moving frame(Bishop, 1975; Hanson and Ma, 1995). Thismoving frame is constructed primarily using the velocity vector.Therefore, the PT moving frame is more natural for moving bio-logical objects as the frame of reference. For illustration, we canorient a car or a plane using an axis that is parallel with their move-ment direction. In other words, the movement direction become theaxis of the basis. If there is some movement change, it will alsochange the basis of the shape. Hence, this approach can bridge theanalysis of shape and movement. To illustrate our approach, weuse both of simulated and real datasets.

Related work

The study of dynamic morphological change of motile cells wasstarted from the study of 2-dimensional (2D) shapes. The 2Dmigration is characterized by the unilateral adhesion to a sub-strate and a spread-out cell shape with a leading lamellipod(Ridley,2003; Keren et al., 2008). Lee et al. proposed the graded radialextension (GRE) model which described the locomotion and shapechange of keratocytes that consistent with the half-moon shapeand its gliding motion (Lee et al., 1993). This model assumed thatthe local cell extension (protrusion or retraction) are perpendicularto the cell edge and the cell maintains its shape and size duringmoving by a graded distribution (e.g. a maximum near the cellmidline to a minimum towards the sides). Further study found thatkeratocytes inhabit a low dimensional shape space(Keren et al.,2008). The Cellular Potts Model (CPM) is another popular model

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The copyright holder for this preprintthis version posted November 5, 2020. ; https://doi.org/10.1101/2020.11.04.369033doi: bioRxiv preprint

https://doi.org/10.1101/2020.11.04.369033

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RESEARCH ARTICLE Journal of Cell Science (2019) 00, jcsxxxxxx. doi:10.1242/jcs.xxxxxx

for dynamic, irregular, and highly fluctuating cell shapes(Granerand Glazier, 1992; Niculescu et al., 2015; Rens and Edelstein-Keshet, 2019). A cell in CPM is defined over a multiple latticesites region with the constraint on the cell area. In this model, thecell is exposed to an energy function from adhesion between cellsand resistance of cells to volume changes. Then, the cell motionand shape change comes from the minimization of this energyfunction(Marée et al.).

Even though the study on 2D space is important, the cells liveand move in the 3D environments. A 3D shape is more challengingto study due to higher degrees of freedom. A common approach isto represent the 3D object surface as a shape descriptor. By exam-ining the change of the shape descriptors over time, we can studythe dynamic of the shape change.

Spherical harmonic (SH) descriptor is a widely used descrip-tor to study the 3D cell shape(Shen et al., 2009; Ducroz et al.,2012; Du et al., 2013; Medyukhina et al., 2020). It is a spher-ical analogue of the 1D Fourier series. It considers a surfaceas a function on the unit sphere which can be represented asa set of unique coefficients. In the contexts of shape-movementanalysis, SH was used to analyzed amoeboid cell motion(Ducrozet al., 2012; Du et al., 2013) and to perform shape classifi-cation of motile cells(Medyukhina et al., 2020). As mentionedbefore, any shape analysis including the SH descriptor needs tobe robust to rotational transformation. A rotation invariant versionof the SH descriptor is commonly used to address this prob-lem(Kazhdan et al., 2003). However, this method suffers infor-mation lost because they described a shape in a transformationinvariant manner (e.g. disregard the effect of rotation)(Kazhdanet al., 2003).

In this paper, we propose a shape-movement analysis using theSH descriptor in the parallel-transport moving frame. In contrast tothe previous approach, we keep the effect of rotation by aligned allthe shapes in the common frame of reference based on the parallel-transport moving frame. Another contribution from our approach,we can observe and quantify the relationship between the move-ment and the shape of the cell. It is possible because the movingframe is constructed from the velocity vector of the object.

METHODSOverview

The schematic diagram of our approach is shown in FIG. 1. Thetrajectory of the cell was smoothed using Gaussian Process regres-sion. From this smooth curve, we constructed moving frames ateach sample point and extracted the trajectory properties such asthe speed, curvature, and torsion at each sample point. For each 3Dobject from each timepoint, we change the coordinate basis fromthe standard basis to the moving frame basis. In this moving frame,the cell direction become the new x-axis. We performed the SHdecomposition on this new basis to obtain SH coefficients. By ana-lyzing the change of these coefficients over time, we can observethe dynamic of shape change when it was moving. We performedthe statistical analysis of the shape and movement features to studytheir relationship.

Trajectory and Moving Frame of the Mass Center

Gaussian Process smoothingWe obtained the trajectory by calculating the mass center of the3D object at every timepoint. The mass center of the object isdefined as the arithmetic mean of their vertices. Thus, we had

(x(t), y(t), z(t)) as the position of the mass center at the time t.Then, we defined sx = (x(1), .., x(n)), sy = (y(1), .., y(n)), andsz = (z(1), .., z(n)). For each sequence si; i = x, y, z, we stan-dardized the sequence so that it has mean zero and standarddeviation one. From hereafter, the subscript i refers to one of theelement of x, y, z, unless otherwise stated.

We modeled each of the sequence as ssti (t) = fi(t) + ε wheressti is a standardized sequence. The function fi(t) is an underlyingsmooth function and ε is a Gaussian noise with mean zero andvariance σ2.

The function fi(t) can be approximated by Gaussian Pro-cess (GP) Regression(Rasmussen and Williams, 2006). In the GPregression, we have mean function and kernel function as hyper-parameters. As our sequence had zero mean, we chose the zeromean function. And because we need a smooth function, we usedSquared-Exponential (SE) kernel function.

The SE kernel itself has two parameters: (1) the lengthscaleparameter that determines how far the influence of one sam-ple point to its neighbor points, (2) the variance parameter thatdetermines the mean distance from the function’s mean.

We used the function fi(t) to interpolate new data pointsbetween two consecutive elements of sequence ssti (t) and ssti (t+1). The number of new data points determines the smoothness ofthe new sequence.

Next, we performed the inverse transform of standardizationby multiply ssti by the standard deviation of si,then add it withthe mean of si to produce a near-smooth sequence ri. Then, wedefined r(t) = (rx(t), ry(t), rz(t)) as the smooth trajectory.

Parallel transport moving frameOne of the main ideas in our approach is a frame of referencewhich is moving along with the curve and telling us the maindirections of the movement (Fig S1). We made this frame ofreference using parallel-transport (PT) moving frame algorithmfrom Hanson and Ma (1995). In brief, we calculated a tangentvector Ti for each point i on the curve. Then, set an initial nor-mal vector N0 which is perpendicular to the first tangent vectorT0. For each sampled point, we calculate the cross product ofU = Ti ×Ti+1. If the length ‖U‖ = 0 (i.e. both vectors are par-allel) then Ni+1 = Ni. Otherwise, to obtain Ni+1, we rotate Ni

around the vector U by the angle θ = Ti ·Ti+1 using the rotationmatrixRot(U, θ) defined on Hanson and Ma (1995). The completealgorithm is shown in Algorithm 1.

Orientation normalizationWe change the coordinate basis from the standard basis to the PTmoving frame basis using simple linear algebra transformation:

v∗ =

Tx Nx Bx

Ty Ny By

Tz Nz Bz

−1 v (1)

where v∗ and v are the coordinates of the vertex in moving frameand standard basis respectively. The subscript x, y, and z here arethe first, second, and third component of the vectors T,N, and B.

Speed, curvature, and torsionFrom the smooth trajectory, we can extract the trajectory featuressuch as speed, curvature, and torsion as defined on Patrikalakis andMaekawa (2009). In summary, speed is the distance traveled perunit of time. Curvature measures the failure of a trajectory to be a

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Fig. 1. The schematic diagram of integrated analysis of shape and movement in moving frame.

Algorithm 1: Parallel transport moving frame algorithmsInput: 1. Tangent vectors Ti, i = 0, 1, ..., n;

2. An initial normal vector N0,N0 ⊥ T0

Output: Vectors Ni and Bi, i = 0, 1, ..., n where Ti,Ni,Bi are orthogonalfor i = 0 : (n− 1) do

U = Ti ×Ti+1;if ‖U‖ = 0 then

Ni+1 = Ni;else

U = U \ ‖U‖;θ = arccos(Ti ·Ti+1) ; // 0 ≤ θ ≤ πNi+1 = Rot(U, θ)×Vi;

endendBi = Ti ×Ni, i = 0, 1, ..., n;

straight line, while torsion measures the failure of a trajectory tobe planar. Taken together, the curvature and the torsion of a spacecurve are analogous to the curvature of a plane curve.

Spherical Harmonic Decomposition

After orientation normalization, shapes were decomposed Spheri-cal Harmonic (SH) transform. To perform SH decomposition, weneed to map the object surface to the unit sphere. Before it, wenormalize the volume of all objects to one.

Spherical parameterizationWe used the mean-curvature flow spherical parameterizationmethod from Kazhdan(Kazhdan et al., 2012) to maps the object

surface to a unit sphere. The result of spherical parameterization isa continuous and uniform mapping between a point on the objectsurface and a pair of the latitudinal-longitudinal coordinate (θ, φ)on a unit sphere:

v(θ, φ) = (x(θ, φ), y(θ, φ), z(θ, φ)) (2)

where (x(θ, φ), y(θ, φ), z(θ, φ)) is the Cartesian vertex coordi-nates.

Spherical Harmonic expansionOn the unit sphere, each of the (x(θ, φ), y(θ, φ), z(θ, φ) can beapproximated using the real form spherical harmonic(SH) series:

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x(θ, φ)y(θ, φ)z(θ, φ)

≈∑lmax

l=0

∑lm=−l Cx(l,m)Ym

l (θ, φ)∑lmax

l=0

∑lm=−l Cy(l,m)Ym

l (θ, φ)∑lmax

l=0

∑lm=−l Cz(l,m)Ym

l (θ, φ)

(3)

Yml (θ, φ) =

√2K−ml P−ml (cos(θ)) sin(−mφ) if m < 0

K0l P

0l (cos(θ)) if m = 0

√2Km

l P|m|l (cos(θ))cos(mφ) if m > 0

(4)

Kml =

√(2l + 1)

4π

(l −m)!

(l +m)!(5)

where Pml (θ, φ) is the associated Legendre polynomial.

The coefficient of degree l, order m, C(l,m) can be obtainedusing standard least-square estimation. Using x(θ, φ) as an exam-ple, assume that the number of vertices is n and xi = x(θi, φi).We need to find the coefficients Cx = (c1, c2, · · · , ck)T wherecj = Cx(l,m). The index j for l ∈ (0, · · · , lmax),m ∈ (−l, · · · , l)is obtained from the equation j = l2 + l +m+ 1. We can obtainthe coefficients by solving equation 6.

y1,1 y1,2 · · · y1,ky2,1 y2,2 · · · y2,k

......

. . ....

yn,1 yn,2 · · · yn,k

c1c2...ck

=

x1x2...xn

(6)

where yi,j = Yml (θ, φ), j = l2 + l +m+ 1, and k = lmax +

1. After obtaining the coefficients for each of Cx, Cy , and Cz ,we can bundle it into one feature vector of the shape, C =(Cx, Cy, Cz).

Shape characteristics measuresEach coefficient of the expansion retains a shape informationcorresponding to a particular spatial frequency. The increasingdegree of l describes the finer scales of shape information. Thedirection of shape changes can be detected in each of three setsof coefficients, especially Cx(l = 1,m = 1) for deformation inthe x-direction, Cy(1, 0) for y-direction, and Cz(1,−1) for z-direction. If Cx(1, 1) = Cy(1,m = 0) = Cz(1,−1) and the othercoefficients are zero then the object is a perfect sphere. Based onthese three coefficients, we defined three eccentricity index:

Exy =Cx(1, 1)

Cy(1, 0)(7)

Exz =Cx(1, 1)

Cz(1,−1)(8)

Eyz =Cy(1, 0)

Cz(1,−1)(9)

Exy , Exz , and Eyz are the measurements of how much theobject deviates from being sphere in xy, xz, and yz plane, respec-tively (Fig. S2). We can exclude theEyz because the calculation ofExyand Exz already contains all of the eccentricity information.

The difference between shape i and j, d(i, j), can be calculatedusing any distance metrics intended for real-valued vector spaces.The most common is the L2 norm distance:

d(i, j) = ‖Ci − Cj‖2

where Ci and Cj are SH coefficients for shape i and j,respectively. Here, we calculated the the rate of shape changewhich is defined as d(t, t+ 1) where t, t+ 1 are two consecutivetimepoints.

Statistical analysisFor each cell in the real dataset, we calculated the median, medianabsolute deviation, 25th percentile, and 75th percentile of thespeed, curvature, torsion, shape change rate, Exy , and Exz fromall timepoints. We also counted the number of peaks from thecurvature and torsion graph. These values are used as features(26 features in total). Then, we standardized all these features tohave mean zero and standard deviation one. To show the corre-lation from each features, we calculated Kendall rank correlationcoefficient.

We use UMAP (McInnes et al., 2018) for visualization in a 2Dplot. The UMAP parameters that we used: number of neighbors =20, number of components = 2, minimum distance = 0.1, and theEuclidean distance as metric.

K-nearest neighbor (KNN) classifier was trained on the shape-movement features on the PT moving frame and on the standardbasis to show the performance of our approach. We varied thehyper-parameter k = 3, 5, 10, 15, 20, 25. A stratified 10-fold cross-validation was used to give a set of 10 accuracy scores for eachhyper-parameter k. From each iteration of cross-validation, wecalculated the difference of accuracy scores between these twoapproaches. The One sample T-test was performed to test whetherthe accuracy difference was significantly different from zero ornot.

Datasets

Simulational datasetWe manually created a 3D cell object that moves along a path usingopen-source software Blender (Community, 2018). The datasetconsists of 250 time points and a 3D object at each time point.The 3D objects were saved as triangular mesh objects. In oursimulation, the cell moves in the accelerate-decelerate-accelerate-decelerate pattern. The cell starts from a near-spherical shape, pro-truding a pseudopod when accelerating, and back to near-sphericalshape when decelerating (Movie 1,Movie 2a). The volume of thecell is fixed to be one. We chose the shape object at timepointt = {1, 10, 20, . . . , 250} as the observation data. The rest of thedata were used as the holdout data. The true trajectory of the cellis defined as the center of the mass of the cell at each of 250timepoints.

Real datasetThe real dataset consists of 3D objects from microscope imagesof neutrophils. We isolated the neutrophils from the bone marrowof LysM-EGFP mice. Erythrocytes were excluded from harvestedbone marrow cells using ACK lysing buffer and density gra-dient centrifugation(800xG for 20 minutes) using 62.5 percentpercoll. The isolated neutrophils were mixed with collagen with3× 104 cells/µl. Next, we dropped collagen-cell mixture(10 ul) onthe glass bottom dish. After the collagen-cell mixture turned into

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(a) Trajectory smoothing (b) PT Moving Frame

Fig. 2. Trajectory of simulated cell.(a)The Gaussian Process reconstructed the original trajectory from the observation.(b) The vectors T together with thevectors N and B construct moving frames for each point on the curve.

a gel, we add culture medium to the dish. The dish was incubatedat 37 °C for 2 hours. Before the imaging, the culture medium wasreplaced with the imaging medium. After the cells were stimulatedwith Granulocyte- Macrophage Colony-Stimulating-Factor(GM-CSF) 25 ngml−1, Lipopolysaccharide(LPS) 10 µgml−1, or (Phor-bol 12-myristate-13-acetate(PMA) 1 µgml−1, we immediatelyperformed imaging for 90 minutes at 1-minute intervals usingtwo-photon excitation microscope (Nikon A1R MP). The imag-ing was performed at 45 µm (15 stacks) in 3 µm steps in the Z-axisdirection.

We analyzed the cells that were captured in a minimal 6 con-secutive timepoints(i.e. 233, 398, 293, and 166 cells in saline,GM-CSF, LPS, and PMA group, respectively).

Method Implementation

The above method was developed in Julia ProgrammingLanguage(Bezanson et al., 2017). We used GaussianPro-cesses.jl(Fairbrother et al., 2018) package for the Gaussian Processsmoothing. For SH coefficient calculation, we used Julia wrapperfor the GNU Scientific Library (GSL)(Galassi, 2009).

For the SE kernel parameter, we set the lengthscale=e1.0 andvariance = 1.0. We note that the choice of the parameters is notnecessarily optimal, but it gives good modeling results in our sim-ulation, as will be shown later. For the SH decomposition, we usedlmax = 6.

RESULTSSimulational dataset

The smooth trajectory from the observation data is shown in FIG.2a. It is difficult to see any difference due to almost perfect recon-struction (The mean squared error = 1.203). The quality of thesmooth trajectory degrades around t=0 and t=250. This is probablydue to the Gaussian Processes had fewer data to process around theboundary (i.e. no data at t < 0 or t > 250 ).

We constructed the PT moving frames on this smooth trajec-tory (FIG. 2b). Orientation normalization was performed using PTframes as the canonical frame of reference (FIG. 3). After reori-entation, we observed that the cell protrude its pseudopod in thedirection of the cells (Movie 2b).

The shape-movement features of the simulational cell is shownon FIG. 4. In these plots, the features of the original data are alsoshown. The pattern that we obtained from observation data aresimilar to the original data, even though the magnitude of the peakis different. These differences due to the data from observationaldata is smaller than the original data.

On the upper part of FIG. 4, the movement behavior ofthe cell is shown on the speed, curvature, and torsion graph.The bimodal graph of the speed graph indicates the accelerate-decelerate-accelerate-decelerate pattern. Meanwhile, the peak onthe curvature and torsion graph indicate the change of the directionbetween t = 100 and t = 130.

On the middle part of FIG. 4, the global deformation patterns ofthe cell are shown as the eccentricity of the shape. Here, Exy andExz changed as time progressed and had values more than one inthe majority of the time. In contrast, the value Eyz did not varya lot from one. These finding indicate that the cell had ellipsoidshape with the longer axis on the direction of the movement. Fur-thermore, the relation between shape and movement can be seenon lower part of the FIG. 4. In this plot, the speed, the Exy , andExz are standardized for comparison purpose. We can easily iden-tify the similarity of the bimodal pattern on the speed, theExy , andExz plot.

Real data

The UMAP was utilized to visualize the shape-movement featuresof the real dataset on 2-dimensional plot. The FIG. 5a and the FIG.5b shows the 2D plot of the features from PT moving frame andthe standard basis. Qualitatively, the features obtained on the PT

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(a) Before reorientation (b) After reorientation

Fig. 3. Reorientation of shape using PT moving frame

Fig. 4. The shape and movement pattern of the cell over time from observation data and its comparison with the original data. (a,b,c) The movement featuresof the cell are characterized by its speed, curvature, and torsion. (d,e,f) The global shape features of the cell is represented by the ratio of spherical harmoniccoefficients. (g) The relation between shape and movement of the cell can be seen in this graph. Here, we standardized the graph of speed, Exy , and Exz

for easy comparison.

moving frame separate each group better than the features fromthe standard basis.

Quantitatively, we performed KNN classifier on the featuresfrom the PT moving frame and the standard basis. The FIG. 6ashow the mean and the standard deviation of the KNN classifieraccuracy. The majority of the accuracy scores on the PT mov-ing frame are superior to the standard basis. Using One SampleT-test, the accuracy difference is significant (i.e. p-value < 0.05)on k = 15, 20, 25 (FIG. 6b).

On FIG. 7, we plot the importance of each feature on the accu-racy of KNN classifier. The most importance features come fromthe speed features which is the number of the peak on curvature

graph, the number of the peak on torsion graph, the 75th per-centile and median absolute deviation of the speed. Here, the shapefeatures play a minor role to classify the sample. We can see therelationship between each feature from the Fig 8. We found someinteresting observations from it. Torsion and curvature features arenegatively correlated with the speed features. It suggests that thecells that are moving faster are inclined to move straight(i.e. rarelychange their direction and low magnitude of direction change). Wealso found that the fast cells had the rate of shape change higherand protrude more along its movement direction than the slowercells.

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(a) (b)

Fig. 5. The Shape-Movement features embedding on 2D space using UMAP. We utilized UMAP to visualize the high-dimensional shape-movement featureson (a) PT moving frame and on (b) Standard basis.

(a) (b)

Fig. 6. The analysis of shape-movement features on PT moving frame. (a) The comparison of accuracy of KNN classifier across the 10-folds cross-validationtrained on PT moving frame and on standard basis. (b) The accuracy difference of KNN classifier on PT moving frame and on standard basis. The onesample t-test showed that the difference was significantly different from zero for k = 15, 20, 25

CONCLUSIONIn this paper, we have presented a novel application of parallel-transport moving frame to solve the shape alignment problem in3D space and to integrate the analysis of shape and movement.We have shown that our approach can extract shape-movementfeatures of the cell and their dynamic change. We found someinteresting relationships between the movement and the cell shapesuch as the negative correlation between the cell speed and cur-vature/torsion, the fast cell tend to protrude the pseudopod tothe movement direction, and positive correlation between shapechange rate and the cell speed. The shape-movement featureswhich are extracted using our approach also have the potentialto be used for clustering or classification. It suggests that ourapproach can provide new insight into the mechanobiologicalprocess of the cell.

AcknowledgementsWe thank for Hironori Shigeta and Shigeto Seno of the Graduate School of Infor-

mation Science and Technology, Osaka University, for the support in 3D mesh

object creation.

Competing interestsThe authors declare no competing or financial interests.

ContributionConceptualization: Y.D.H, R.Y., K.M.; Methodology: Y.D.H, R.Y.; Validation: Y.D.H,

R.Y.; Formal analysis: Y.D.H; Investigation: Y.D.H, C.Y.C, Y.U.; Software: Y.D.H,

C.Y.C; Data acquisition: Y.U., M.I.; Data curation: Y.U., M.I.; Writing - original draft:

Y.D.H; Writing - review and editing: R.Y.; Visualization: Y.D.H; Supervision: R.Y.;

Project administration: R.Y.; Funding acquisition: R.Y., M.I.

FundingThis study is funded by Core Research for Evolutional Science and Technology

(CREST), Japan with grant numbers JPMJCR1502 and JPMJCR15G1.

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Fig. 7. The permutation importance plot that shows the importance of eachfeature. We found that the movement features was dominantly selected asimportant features.

Fig. 8. The correlation matrix of each features.

Data availabilityThe source code and simulational dataset are available at https://github.com/

yusri-dh/MovingFrame.jl.

SupplementaryMovie 1

Movie 1. The simulational cell manually created using Blender.

Movie 2a

Movie 2a. The simulational cell shape before reorientation. The centers of mass

for each timepoints are normalized to origin.

Movie 2b

Movie 2b. The simulational cell shape after reorientation. The centers of mass for

each timepoints are normalized to origin.

Figure S2

The shape eccentricity calculated using SH coefficients.

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Fig. S1. The construction of PT Moving Frame on each timepoints. The moving frames is moving with the curve and telling us the directions of the objectmovement.

(a) (b) (c)

Fig. S2. The shape eccentricity calculated using SH coefficients. For examples, figure (a) have eccentricity: Exy = 2, Exz = 2, Eyz = 1, figure (b):Exy = 1/2, Exz = 1, Eyz = 2, and figure (c): Exy = 1, Exz = 1/2, Eyz = 1/2.

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integrated analysis of cell shape and movement in moving frame · 04/11/2020 · yusri dwi...

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