1. Introduction

When the extracellular molecules are taken up by a live cell as forming vesicles, they are gradually delivered from the membrane into the cytoplasmic area, following the biochemical pathway. This process is called endocytosis [1]. The vesicle is a basic unit of intracellular transport, and the movement of vesicle is closely related to the understanding of the intracellular transport mechanism [2]. Therefore, for various biomedical applications related to the intracellular transport mechanism, such as drug delivery, many researches have been conducted to elucidate the precise movement of vesicle [3–5].

Earlier approaches revealed that the vesicles are actively transported via the interaction with cytoskeletons, such as microtubule and actin filament, mainly based on the purified system where the vesicle is coupled with a single transporting filament recruited by motor protein [6,7]. Those studies explained that a vesicle shows linear movement while walking along the cytoskeletal filament. Recent researches are more focused on the visualization and analysis of vesicle trajectories in a live cell with improved imaging techniques [8–10], since the vesicles in a live cell inevitably interact with multiple cytoskeletons rather than a single filament.

Since the size of vesicles is smaller than one hundred nanometers in diameter in case of common receptor-mediated endocytosis, the trajectory of vesicle is acquired based on fluorescence imaging microscopy. FIONA, Fluorescent Imaging with One-Nanometer Accuracy, theoretically enabled the localization of a single fluorophore with one nanometer precision in two dimensions [11]. Also, recent advances in imaging technique such as dual focus optics now allow us to estimate three-dimensional trajectory of a vesicle in a live cell with high spatiotemporal resolution in practical imaging condition [12–14]. Figure 1 shows an example of single vesicle trajectory in three dimensions acquired via the dual focus optics.

 

Fig. 1 An example of complex trajectory of vesicle movement acquired by three-dimensional microscopy. (A) Cell image taken in phase contrast microscopy. (B) Vesicle labeled with quantum dot imaged via dual focus optics [13]. The x, y coordinates and intensity information are exploited for acquiring three-dimensional position data. (C) Three-dimensional trajectory of the target vesicle tracked for 10 s at 100 Hz frame rate. The complexity in trajectory reflects the fluctuating vesicle dynamics in cytoplasm, transferring among multiple intracellular structures.

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However, the movement trajectories of vesicle acquired via advanced imaging techniques still apparently consist of complex cluster of position data, in that the data not only include measurement error but also the vesicle actually interact with complicated structure of interwound cytoskeletons while being transported. Therefore, for detecting the detailed feature of the vesicle movement in a complex cytoskeleton network, one of the most important tasks is to establish a criterion which distinguishes the active transport of vesicle from random diffusion. For this purpose, several researches suggested mathematical methods, based on mean squared displacement (MSD) [15], angle correlation [16], and further with Hidden Markov Model (HMM) [17], in order to determine the active transport of a vesicle where it shows linear and persistent movement. Those studies have provided insight into building statistical model for recognizing the active transport, yet they only concern the vesicle movement itself, and thus the information between the vesicle and cytoskeleton is not included.

In this research, we present a novel numerical analysis method for vesicle movement in terms of the interaction between vesicle and cytoskeleton. Our method features intuitiveness and robustness in detecting active transport section from three-dimensional vesicle trajectory by estimating the orientation and position of the cytoskeleton, which is defined as a linear movement section. The process of the presented method is as shown in Fig. 2. First, noise filter is applied in order to remove the noise from the acquired position data. Then, the movement trajectory is divided into active transport section and random diffusion section, based on the linearity and persistency in the traveling direction. Next, the orientation and position of cytoskeleton are estimated by finding the linear axis during the active transport, via linear regression using principal component analysis. Lastly, the data points are projected onto the estimated location of cytoskeleton, which can be exploited to reveal the precise movement of the vesicles on the cytoskeleton. With the simulation, the performance of the proposed analysis method is proved. The proposed numerical analysis method is expected to help understand the complex trajectory data acquired from various microscopy imaging, unveiling the detailed shape of the vesicle movement in a complicated cytoskeleton network.

 

Fig. 2 Process overview of the presented numerical analysis method. (A) Raw position data distribution ${\text{P}}_i^{*}$ in three dimensions. (B) Noise reduction using Weierstrass transform with the filter size of 16 data points to acquire the filtered position data P_i. (C) Recognition of the linear section (colored in pink) by determining the local curvature. Nonlinear sections are colored in mint green. The orientation and position of cytoskeleton in linear section (red line) are estimated via linear regression based on PCA. (D) Analysis of vesicle movement on the estimated cytoskeleton is conducted by projecting the data points using vector analysis.

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2. Numerical method for vesicle movement analysis

Vesicles show linear movement when they are transported by a single cytoskeleton filament, due to the linearity of the cytoskeleton structure [18]. Also, because the active transport of vesicle occurs when the vesicle interacts with cytoskeleton, one of the intuitive methods to distinguish the active transport of a vesicle is to extract the linear section from complex movement data points. To do this, we suggest a numerical approach to detect the linear section from the acquired position data distribution.

2.1. Linear section detection

Prior to detecting the linearly distributed data points from complex vesicle movement trajectory, it is required to reduce the noise of the raw position data. For this purpose, Gaussian moving average is applied as a noise filter to each raw x, y, and z coordinates. Applying Gaussian filter is also called as Weierstrass transform in mathematics, which is achieved by transforming input data points via convolution with Gaussian function [19]. The convolution scheme is required in order to prevent the latency involved in the smoothing process. Therefore, the data points in three dimensions are individually filtered by Weierstrass transform function as in Eq. (1).

By applying Weierstrass transform to three-dimensional data points according to Eq. (1), it is possible to obtain Gaussian weighted moving average of the raw data points set. Now consider the filtered position as a point P_i(x_i, y_i, z_i) and the corresponding vector r_i ∈ ℝ³. In order to define a linear section with the filtered data point set P = {P₁, P₂, ..., P_N} and corresponding vector set r = {r₁, r₂, ..., r_N}, the interval where the data points show persistent linearity is required to be distinguished. To do this, we define a linear domain based on the local curvature of the trajectory, which can be investigated from the local angles. As shown in Fig. 3, the local angle θ_i is defined between the two vectors r_i+m − r_i and r_i−m − r_i, the three consecutive data points separated by the size of Gaussian filter m, as a criterion for partial coherence in the direction of travel. The angle θ_i can be calculated as in Eq. (3), by computing the inner product.

As shown in Fig. 3, if θ_i = π, the local trajectory is completely linear, while ${\theta }_i=\frac{\pi }{2}$ means the perpendicularly changed local trajectory. Therefore, the local movement of a vesicle between P_i−m and P_i+m can be considered as relatively linear and persistent when θ_i is closed to π, compared to the case where θ_i is far smaller than π. Here we introduce a threshold angle θ_c, in order to determine and extract the local linear domains from the entire trajectory data. As shown in Fig. 3, a series of data points forming θ_i ≥ θ_c is recognized as a persistent movement section, otherwise the data point set is treated as a random movement section.

 

Fig. 3 Determination of the local curvature in order to recognize a linear section. The angle θ_i is defined as an angle between two vectors, consisting of the data points separated by the filter size m. Note that the closer value of θ_i to π implies the more linear and persistent movement of a vesicle.

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In fact, the size parameter m for data separation and the threshold angle θ_c contain a trade-off between noise removal and excessive smoothing. Therefore, it is required to investigate the range of m and θ_c before we apply this noise reduction method to actual data. In order to provide practical reason for determining proper values of both parameters m and θ_c, simulation is conducted as shown in Fig. 4. In the simulation, three different linear segments are prepared in space as shown in Fig. 4(A), mimicking the crossed cytoskeletons located in a cell. Since the size of endocytic vesicle is up to 100 nm in diameter assuming common micropinocytosis [20], vesicle is expected to be detected around the cytoskeleton within the range of its radius. Therefore, the vesicle trajectory along the segments are generated randomly to be located between 0 and 100 nm from the base segment, as shown in Fig. 4(B). In order to determine the appropriate range of the parameter m, the angle θ_i which is defined as Fig. 3 is calculated with varying m, from 1 to 40 data points. The line colored in orange shown in Fig. 4(C) represents the θ_i calculated on the ground truth. As shown in Fig. 4(D), the proper value of m can be determined by estimating the ratio of occupation with respect to the area defined by the ground truth. When the threshold angle θ_c is equal to $\frac{3\pi }{4}$, the optimal value m is around 16 data points for the simulation.

 

Fig. 4 Simulation for determining proper noise filter size m in order to extract the linear section. (A) Three consecutive linear segments in space as mimicking the crossed cytoskeletons. (B) Vesicle movement trajectory generated along the segments. (C) Angle θ_i changes when the filter size m varies from 1 to 40 data points. Ground truth represents the angle calculated from the base segments. Note that there exists a trade-off between the noise reduction and excessive smoothing according to the parameter m. (D) The efficiency of the filter size is defined based on the occupation ratio with respect to the ground truth above the critical angle, θ_c. The graph on the right indicates the occupation ratio calculated when ${\theta }_c=\frac{3\pi }{4}$. The optimal value of m is about 16 data points in this simulation.

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The range of angle θ_i to define the persistent movement of a vesicle is suggested as the angle from 0 to $\frac{\pi }{2}$ in the previous research [16], but the local minimum angles shown in Fig. 4(C) imply that $\frac{\pi }{2}$ is too small to recognize two different linear sections. Considering the ideal persistent movement forms π while the returning movement forms 0 between adjacent data points, we conservatively suggest $\frac{3\pi }{4}$ as a threshold angle to determine the linear and persistent movement from complex position data distribution.

Note that the nonlinear movement includes random diffusion and the change in traveling direction. Using the above angle criteria, now we can define the linear sections as the interval where the angle θ_i is persistently larger than $\frac{3\pi }{4}$, while the random diffusion or the corner of direction change is defined when θ_i is smaller than $\frac{3\pi }{4}$. Based on this definition and the m as 16 data points, linear sections in the simulated data point set are detected as shown in Fig. 5(A). The data points colored in pink represent the sections detected as linear and persistent movement while the mint green indicates the groups of data at the corner of the direction change.

 

Fig. 5 Estimation of linear section and corresponding cytoskeleton location. (A) The detected linear section (colored in pink) and the corner of the crossing segments (mint green) by simulation. (B) Estimated orientation of cytoskeleton ê and the scheme for finding perpendicular projection onto the linear axis. Q₁ and Q_n are the starting and ending point of estimated cytoskeleton, respectively. Q_c is determined by the mean coordinates of P_l. Note that the notation Q is used for the points on the linear axis. (C) Comparison between the ground truth (black line) and the estimation result by PCA (red line). The discrepancy in localization is defined using δ, the angle between the estimation and ground truth. The angle ϕ represents the angle between estimated cytoskeletons. (D) Influence of system noise level and the diameter of vesicle upon δ.

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2.2. Estimation on the orientation and position of the cytoskeleton

Now it is possible to estimate the orientation and position of cytoskeleton, in the section where vesicle showed linear and persistent movement. For this purpose, Principal Component Analysis (PCA) is applied to locate the axis of traveling direction, which can be interpreted as the traces of the interaction between vesicle and cytoskeleton. As a statistical technique in regression analysis, PCA produces the principal axis which minimizes the orthogonal errors from data points [21]. In order to apply PCA method, the data point set {P₁, P₂, ..., P_n}, which is distinguished as a linear section, is rearranged as a n × 3 matrix P_l. Then, the covariance matrix C for P_l can be calculated as in Eq. (5).

Additionally, the overall shape of the cytoskeleton can be estimated by perpendicularly projecting the respective initial and final data points in the linear section, P₁ and P_n, onto the evaluated position vector of the cytoskeleton. Since the projection of a point onto a position vector can be computed by the inner product scheme of two vectors, we define the position vector of the estimated cytoskeleton as $\overrightarrow{{\text{Q}}_c{\text{Q}}_c^{\prime }}$, for simplicity. Then, the coordinates of Q_i, the perpendicularly projected point from P_i(x_i, y_i, z_i) ∈ P_l onto $\overrightarrow{{\text{Q}}_c{\text{Q}}_c^{\prime }}$, can be computed as in Eq. (7).

Since the point Q_i can be expressed as Q_i = κ(e_x, e_y, e_z) + (x_c, y_c, z_c) using the size factor κ, the Eq. (7) can be numerically calculated as Eq. (8).

Accordingly, as shown in Fig. 5(B), starting point Q₁ and ending point Q_n, respectively projected from P₁(x₁, y₁, z₁) and P_n(x_n, y_n, z_n) to the cytoskeleton, can be numerically evaluated as in Eq. (10).

Therefore, the line $\overline{{\text{Q}}_1{\text{Q}}_n}$ is the estimated cytoskeleton in terms of the 3D orientation and position for the linear section P_l.

In order to evaluate the robustness of this estimation model, especially against the noise level and the diameter of vesicle, the discrepancy between the estimated cytoskeleton position and the ground truth was investigated. As shown in Fig. 5(C), for the estimated cytoskeleton $\overline{{\text{Q}}_1{\text{Q}}_n}$, the corresponding ground truth is $\overline{{\text{G}}_1{\text{G}}_n}$, and the angle between $\overline{{\text{Q}}_1{\text{Q}}_n}$ and $\overline{{\text{G}}_1{\text{G}}_n}$ is defined as δ which represents the estimation error. The distribution of δ according to the change in noise level and vesicle diameter are shown in Fig. 5(D). Noise level includes the system noise which influences the detection stability, and is defined as the standard deviation of the measurement value. From one hundred independent simulations with changing noise level, the median of δ was smaller than 1° up to 10 nm of noise level. Also, the distribution of δ increased according to the size of the vesicle, but the median value of δ was smaller than 1° when the diameter of vesicle is around 100 nm or smaller. Since the recent advances in imaging system such as super-resolution microscopy enabled high spatial resolution imaging and the diameter of vesicle is around 100 nm assuming common endosomes except for phagosomes [24], it is expected that the proposed model can be applied to various vesicle movement trajectory analysis acquired via high spatial resolution microscopy imaging.

Based on the estimated orientation and position of cytoskeleton, the angles between crossing cytoskeletons in an interwound structure also can be calculated. Assuming that a vesicle moves from one cytoskeleton to another as shown in Fig. 5(C), and the estimated start and end coordinates are Q₁ and Q_n for the former and Q′₁ and Q′_n for the latter, the angle ϕ between two cytoskeletons can be computed by using outer and inner product of $\overrightarrow{{\text{Q}}_1{\text{Q}}_n}$ and $\overrightarrow{{\text{Q}}_1^{\prime }{\text{Q}}_n^{\prime }}$, as in Eq. (11).

Note that the angle ϕ is calculated by four-quadrant inverse tangent function, also known as atan2, in order to prevent any confusion in determining the rotation direction [25,26].

2.3. Rotational and translational movement analysis of vesicle on the cytoskeleton

In order to analyze the movement of vesicle in terms of the interaction with cytoskeleton, the data points in the linear section are projected to corresponding estimated location of cytoskeleton, as shown in Fig. 6. The projection is conducted in the same manner described in Eq. (7). Figure 6(A) shows the projected points Q_i and Q_i+1 from P_i and P_i+1, respectively. The instant movement of vesicle around the estimated cytoskeleton at time t_i can be explained based on the relation between two right-handed coordinate systems, Σ_i and Σ_i+1, which are defined as shown in Fig. 6(A). Therefore, the local movement from P_i to P_i+1 can be described by the coordinate system transformation from Σ_i to Σ_i+1, with rotation matrix R_i and translation vector t_i, as shown in Eq. (12).

Note that R_i concerns the vesicle rotation around the axis of cytoskeleton as shown in Fig. 6(B), while t_i indicates the translational displacement along the cytoskeleton orientation. The angle ρ_i denotes the rotation angle of Σ_i+1 with respect to Σ_i around ẑ axis, and ψ_i measures the angle between Σ₁ and Σ_i in a cumulative manner, as described in Fig. 6(B). The cumulative angle ψ_i is a useful term when vesicle shows continuous rotation around cytoskeleton, and it can be calculated as in Eq. (14).

Preparing these angle criteria is important in that vesicles are known to move spirally along cytoskeletons from single-molecule experiments, such as microtubule and actin filament [27,28]. Therefore, it is expected that we can detect any angular motion of the target vesicle on cytoskeleton, using these definitions for ρ_i and ψ_i. Also, since Q_i can be interpreted as the actual position of a vesicle while interacting the cytoskeleton, instant translational velocity v_i along the axis of cytoskeleton between the time t_i and t_i+1 can be calculated as in Eq. (15).

Note that v_i is also an essential factor for identifying the type of motor protein involved in the interaction between vesicle and cytoskeleton.

 

Fig. 6 Vesicle movement analysis on the cytoskeleton. The movement between consecutive two data points can be calculated as a two-dimensional rotation transformation with coherent direction vector ẑ, which is the orientation of the estimated cytoskeleton. (A) Q_i and Q_i+1 are respectively the projected points of the data points P_i and P_i+1 onto the estimated linear axis, which can be interpreted as the positions where a vesicle directly interacts with the cytoskeleton. (B) Relative angular position P_i+1 to P_i around the axis of cytoskeleton can be expressed by the rotation angle ρ_i. The angle ψ_i is defined as the cumulative angle of Σ_i with respect to Σ₁.

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3. Application to actual vesicle movement data

The proposed numerical analysis method described is now applied to the actual vesicle movement data. As an example case, the movement of an endocytic vesicle in a live cancer cell was tracked using three-dimensional imaging microscopy, and its movement trajectory was analyzed following the processes presented above.

3.1. Three-dimensional position data acquisition

The cell line exploited in the experiment is KPL-4 human breast cancer cell [29], which was kindly provided by Dr. Kurebayashi (Kawasaki Medical school, Kurashiki, Japan). The endocytic vesicles were labeled by adding 4 nM of carboxylate-functionalized quantum dots (Thermo Fisher Scientific, Inc.) to the cells. Quantum dot is a semiconductor nanocrystal, which is widely applied in biological imaging as a fluorescent probe [30]. Since the cell engulfs surrounding molecules including quantum dots, it is possible to detect the position of vesicles by tracking the light emitted from the quantum dot [31,32]. The movement of vesicle in a live cell was imaged through fluorescent microscopy and reconstructed in three dimensions based on the principle of dual focus imaging optics [13]. The x, y coordinates of a vesicle were detected as a position of the Gaussian peak of point source intensity, and the z coordinates were evaluated by comparing the intensities of the point source on dual image planes. In the live-cell imaging, the initial position of the specimen was kept stable by using the axial position locking system [33]. The detection precision in the imaging system is within 5 nm in three dimensions, as shown in Fig. 7(A). Total imaging time was 10 s, and the images were taken at 100 Hz frame rate, which produced one thousand position data points. Note that fluorescent microscopy is not mandatory for applying the proposed analysis method, but the vesicle position data acquired via various imaging system can be exploited as long as the detection noise level and the size of vesicle are within the range of 10 nm and 100 nm, respectively, as explained in Fig. 5(D).

 

Fig. 7 Actual vesicle movement data analyzed by the proposed numerical method. (A) Tracking precision of the imaging system in the experiment. (B) Dual view images of vesicles labeled by quantum dot (red) and corresponding phase contrast image of a live cell (gray scale). One vesicle marked as P_c near the cell edge and the other vesicle P_d near nucleus were selected and tracked. (Scale bar = 10 μm). (C) In the trajectory of P_c, three linear sections (c₁, c₂, and c₃) were detected, and the angles between the linear sections were calculated as 158° and 19°, respectively. Blue arrow indicates the initial point while the yellow arrow refers to the final point of the trajectory. (D) In the trajectory of P_d, three linear sections (d₁, d₂, d₃, and d₄) were detected, and the angles between the linear sections were calculated as 156°, 159°, and 98°, respectively. (E) Cumulative angle ψ_i around the axis of cytoskeleton and corresponding translational distance in each linear section of P_c trajectory. Note that the initial position of vesicle in each linear section is respectively set at 0° and 0 nm, for simplicity. (F) Cumulative angle ψ_i around the axis of cytoskeleton and corresponding translational distance in each linear section of P_d trajectory.

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3.2. Vesicle movement analysis

The result of the presented method applied to a representative trajectory of vesicle movement acquired in a live cell is shown in Fig. 7(B). The red dots represent the location of vesicle labeled by quantum dot, and the shape of the target cell is colored in gray scale, which was taken by phase-contrast microscopy. In order to compare the shape of interaction in a cytoskeleton network, we chose two vesicles in different location in a cell : one vesicle marked as P_c near the cell edge where cytoskeletons form a dense network structure, and the other vesicle P_d found at perinuclear area where a few microtubules run due to the existence of a nucleus. The reconstructed trajectory of P_c and P_d in three dimensions are shown in Fig. 7(C) and Fig. 7(D), respectively. When our linear section finding algorithm is applied to the acquired trajectories, three individual linear sections were recognized for the vesicle P_c and four linear sections were detected for P_d. The linear sections in P_c were labeled as c₁, c₂, and c₃, and the linear sections in P_d were marked as d₁, d₂, d₃, and d₄. Also, the orientation and position of cytoskeleton for each linear section was evaluated as a red segment for both trajectories, as respectively shown in Fig. 7(C) and Fig. 7(D).

When the angles between the estimated cytoskeletons, which is notated as ϕ in Fig. 5, were computed via Eq. (11), the estimated cytoskeletons in P_c formed 158° and 19°, while the angles between the estimated cytoskeletons in P_d were 156°, 159°, and 98°. Although we only investigated two example cases, very acute angle (19°) was detected in the case of P_c that was tracked near the cell edge, which implies that the vesicle can transfer between the cytoskeletons crossed with quite acute angles in a complex network structure. It is expected that identification of the cytoskeleton in this type of transfer and the precise relation between the angle ϕ and the intracellular location of the vesicle can be further studied in the future researches.

Also, for each detected linear section, the angular motion of the vesicle on the estimated cytoskeleton was calculated in terms of cumulative angle ψ_i according to Eq. (14). Figure 7(E) represents the change of cumulative angle ψ_i and the corresponding translational distance of the vesicle in the linear section c₁, c₂, and c₃, for the case of P_c. Note that the ψ_i at the initial position of the vesicle on each linear section was set at 0°, and the translational distance of the vesicle on the estimated cytoskeleton was calculated by adding up the $\Vert \overrightarrow{{\text{Q}}_i{\text{Q}}_{i+1}}\Vert$ in Eq. (13). In the same manner, the change of cumulative angle ψ_i and the translational distance for d₁, d₂, d₃, and d₄ in the trajectory P_d were computed as shown in Fig. 7(F).

As a result, as shown in Fig. 7(E) and Fig. 7(F), the cumulative angle ψ_i in the linear sections seemed to be either increasing or decreasing for a certain extent but not until the end of linear section, in both cases P_c and P_d alike. Since the changes in cumulative angle indicates a rotational movement around the cytoskeleton, the detected motion can be interpreted as a composition of both clock-wise and anti clock-wise rotation. This non-uniformity in the direction of rotation implies the possibility that the interaction between vesicle and cytoskeleton in a live cell might not be directly governed by the structure of the cytoskeleton, which induces a regular helical motion around the cytoskeleton as observed in the case of in vitro experiment [27,28]. In addition, while the vesicles were fluctuating in terms of angular motion around the axis of estimated cytoskeleton, they walked along the cytoskeleton showing linearly increasing translational distance as shown in Fig. 7(E) and Fig. 7(F). The mean speeds in the linear sections were distributed from 100 nm/s to 360 nm/s for the case of P_c, while the linear sections in P_d showed values between 250 nm/s and 380 nm/s. Particularly, the translational speed between the adjacent linear sections of P_c changed considerably, in contrast to the small variance of the mean speeds in the case of P_d. Since the speed of vesicle on the cytoskeleton is determined by the interaction between the vesicle and cytoskeleton, which is mediated by motor proteins, the large variance in mean speed found in P_c can imply the difference in the type of involved motor proteins or cytoskeletons. Although there can exist any external influences to the vesicle movement from the cytoskeletons, such as the movement of cytoskeleton itself [34], the movement in the linear section occurs in a sub-second scale, thus the speed here can be interpreted independently from such influences.

Therefore, using our numerical analysis method, complicated movement of vesicle in a cytoskeleton network can be investigated and understood in quantitative terms. If substantial data are collected, it is expected that the unsolved questions regarding the vesicle delivery in a cell, such as the identification of the types of cytoskeleton involved in the vesicle transfer and the relation between the transfer angle and the frequency of transfer, can be answered with a reliable evidence.

4. Conclusion

In this research, we proposed a novel numerical analysis method for detecting precise movement of vesicle in a complex trajectory, focusing on the interaction between the vesicle and cytoskeleton. Since the vesicles are actively transported along cytoskeleton as showing linear movement, our computational method started with locating the linear section in three-dimensional trajectory and estimated the orientation and position of cytoskeleton based on PCA and vector analysis. In the simulation, the proposed method showed high accuracy in detecting the linear sections, proving the effectiveness of the algorithm. Also, we suggested criteria to show the detailed feature of vesicle movement on cytoskeleton, by projecting the position of the vesicle in the linear section onto the estimated cytoskeleton using vector computation. Particularly, we showed that rotational and translational movement of the vesicle on the axis of the cytoskeleton can be explained by the transformation of coordinate system. When the presented method was applied to the actual vesicle trajectory acquired from a live cell, several linear sections were recognized from apparently complex trajectory data. The movement of vesicle on the estimated cytoskeleton was also analyzed in terms of the angular and translational movement. Since the presented numerical analysis method features intuitiveness and simplicity, it is expected to be practically applied to various trajectory data analyses which suffer from noise and complexity, and to open a door for understanding the precise movement of vesicle in terms of interaction with cytoskeleton.