Statistical and mechanical analysis of multi-pseudopodial locomotion in a testate amoeba, Arcella sp.

Abstract

Adherent eukaryotic cells typically exhibit amoeboid locomotion through actin polymerization and bleb-driven mechanisms. However, testate amoebae, which enclose their bodies within a shell, exhibit variation in these locomotion types. This study focused on Arcella, a representative testate amoeba that pulls its shell using multiple pseudopods extending from a single aperture on the ventral side. Arcella may be found in peatlands and freshwater, where it adapts its movement to various substrates. We characterized its movement on glass as well as hard, and soft gel substrates through detailed observation. The results indicated a higher randomness in motion on the soft gel, which was influenced by the pseudopodial elongation direction. Additionally, we evaluated the relationship between movement direction and traction stress. The dipole moment of the traction stress field determined the axis of motion, whereas quadrupole moments were correlated with forward and lateral movements. Although some relationships between multipole moments and velocity were shared with other cells, Arcella exhibited unique characteristics in its movement mechanism, which likely occurred due to its use of multiple pseudopods alongside its shell.

A kind of testate amoeba, Arcella sp., employs multiple pseudopods for locomotion. We investigated the characteristics of pseudopod extension and the traction stress in the direction of movement.

1. Introduction

Amoeboid locomotion is ubiquitous in adherent eukaryotic cells from unicellular to multicellular organisms. Their locomotion relates to a variety of biological phenomena, including wound healing,^1)–3) development,^4)–7) cancer metastasis,^8),9) and immune responses.^10),11) Also, various protists, single-celled eukaryotes, exhibit amoeboid locomotion.^12)–25) Interestingly, amoeboid movement is not exclusive to eukaryotes, as bacteria also exhibit amoeboid-like motility.^26),27) The mechanisms of amoeboid locomotion are classified in two types: bleb-driven and actin polymerization-driven modes.^28)–32) The contraction of actomyosin on the cell cortex generates a driving force in bleb-driven amoeboid locomotion. The driving force in actin polymerization-driven locomotion relies on actin polymerization at the tip of the pseudopods.^{29),33)–35)} Cell adhesion and actomyosin contractility affect amoeboid modes.^34),36) Despite extensive studies on these two locomotion mechanisms, some protists exhibit modes that defy the conventional modes of amoeboid locomotion. A notable example of this phenomenon is Arcella, a member of the Arcellinida in the supergroup Amoebozoa.^37),38)

Arcella has a unique extracellular hemispherical shell consisting of chitin, which offers protection from predators, parasites, and environmental stresses, such as drying.^39)–43) Arcella extends several pseudopods through a singular ventral aperture to move, while pulling its shell. Consequently, its locomotion significantly diverges from that of the shell-less amoeboid cells. In general, the nature of the attached substrate affects amoeboid movement.^44)–47) Arcella is frequently encountered in peatlands and freshwater environments,^48),49) where it adheres to diverse substrates, which suggests the potential for variations in locomotory characteristics based on various attachment substrates, which may play an integral role for survival. In this study, we characterized the locomotion of Arcella on glass and gel substrates through detailed observation and by analyzing movement direction and pseudopodia formation. Furthermore, we measured the stress fields that Arcella applied to substrates and analyzed the forces involved in forward and lateral movements to elucidate the underlying mechanisms governing Arcella locomotion.

2. Results

2.1. Direction of pseudopod extension and cell movement.

Arcella has a shell with an aperture at the center of the bottom surface (Fig. 1A). It moves by extending pseudopods through the aperture. The pseudopods extend toward the movement of the cell on the glass substrate (Fig. 1A). Similarly, on the gel substrate, the cells primarily moved with their pseudopodia protruding forward; however, sometimes their pseudopodia protruded in other directions (Fig. 1B). When Arcella moved on the glass substrate from the horizontal direction, the pseudopods protruded near the substrate (Fig. 1C) or they extended into the surrounding water before attaching to the substrate (Fig. 1C). After the cells had adhered, they moved in the direction of the pseudopods.

Fig. 1.

Observation of Arcella on glass and gel substrates from vertical and horizontal perspectives. (A) Arcella features a shell with an aperture (marked in blue) located at the center of its bottom surface, through which it extends pseudopods. The pseudopods (indicated by arrows) extended toward the direction of locomotion on the glass substrate. (B) A cell on a gel substrate extending its pseudopods in the direction of movement (arrows); however, at times, it also protruded pseudopods in other directions (arrowheads). (C) Arcella was observed moving on the glass substrate from a horizontal view. The pseudopodia protruded close to the substrate (arrowheads) or in the surrounding water before attaching to the substrate (arrow). Following adhesion, the cell moved in the direction of the pseudopods. The scale bars in A, B, and C are 50 µm.

To evaluate the relationship between the direction of movement and extension of the pseudopods, the cell’s center of mass was fixed at the origin and aligned with the direction of movement with the positive X-axis. The shapes of the pseudopods were then superimposed as they developed over time (Fig. 2A and G). The elongating pseudopods exhibited a bulging tip in the forward direction, whereas the pseudopods were more pointed and thinner during the contraction process (Fig. 2F and L). For some specimens, the pseudopods extended backward on the gel substrate (Fig. 2K). Further, the temporal changes in pseudopod length and angle were examined. The direction of movement was set to 0 rad as shown in Figs. 2B and H. The cells did not change speed significantly on the glass substrate (Figs. 2C and 3E blue). Most pseudopods repeatedly protruded and retracted around 0 rad (Fig. 2B). This trend was conserved among the samples (Fig. 2D, n = 4 cells). However, the distribution of pseudopods was dispersed in other directions on the gel substrate along with forward and protruding (Fig. 2J, n = 7 cells). The probabilities of anterior pseudopod appearance were 98.1% ± 11.3% and 60.7% ± 30.2% (median ± SD) on the glass and gel substrates, respectively. Cells migrating on the gel substrate showed decreased migration. We refer to the slow intervals as the arrest phase (Fig. 2I; 0–50 s and 150–250 s) and to the active intervals as the migration phase (Fig. 2I; 50–100 s and 250–300 s, Fig. 3E red). Newly forming and disappearing pseudopods were more likely to occur during these behavioral mode transition periods (Fig. 2H and I).

Fig. 2.

Positional analyses of pseudopodia formation. The superimposed images of the pseudopods were analyzed with the center of mass fixed at the origin of the cell, and the direction of movement aligned with the positive X-axis (A and G). (A) The direction of cell movement is fixed toward the right direction, with α representing the angle formed with the direction of motion. Arcella on a glass substrate extending its pseudopods in the direction of movement (v in A). (B) The region containing extracted pseudopods was converted to polar coordinates, in which the radial coordinate was R and the angular coordinate was α in A. The horizontal axis represents time, whereas the vertical axis represents α. The color indicates the normalized length of the pseudopods. (C) The temporal development of cell speed in A and B. (D) The probability of pseudopod presence indicated that pseudopods extend towards the direction of movement on the glass substrate (n = 4 cells). The bin size is π/36 rad. (E) A typical example of pseudopod elongation development in the posterior region of the cell on the glass substrate is shown, as well as in the anterior region (F). In panels E, F, K, and L, the white dashed lines represent the surface of the shells, whereas the dashed blue lines denote the tips of the pseudopods. The arrows in panels E, F, K, and L are 30 s. (G) Arcella on a gel substrate primarily extended its pseudopods in the direction of locomotion, but occasionally extended them in other directions. (H) The temporal development of the extracted pseudopod angles in polar coordinates is displayed. (I) This panel illustrates the cell development speed in G and H. (J) The probability of pseudopod presence relative to the angle showed that the pseudopods were extended in various directions on the gel substrate (n = 7 cells). The bin size is π/36 rad. (K) A typical example of pseudopod elongation development in the posterior region of the cell on the gel substrate is shown, as well as in the anterior region (L). The scale bars in panels A and G are 50 µm, whereas the scale bars in E, F, K, and L are 20 µm.

Fig. 3.

Statistical characterization of Arcella locomotion. (A) The trajectory of cells was tracked for 5 min on a glass substrate (n = 5 cells). (B) Data for the trajectories were collected on a hard gel substrate (n = 10 cells) and (C) on a soft gel substrate (n = 7 cells). (D) Mean squared displacements (MSDs), which represent the temporal and ensemble average of all trajectories of the cells as a function of lag time. (E) The velocity of typical cells on the substrates. (F) The probabilities of speeds on the substrates are shown. The bin sizes are 0.08 µm/s. (G) The distribution of turn angles over a 10-s interval on the substrates. The bin sizes are π/36 rad. (H) The velocity autocorrelation function (vACF) for the movements of the cells on the substrates. The solid lines represent the mean values, whereas the dashed lines indicate the fitted curve. The transparent regions indicate the standard deviation.

2.2. Statistical characteristics of cell movement.

Arcella moved more actively on the glass and hard gel substrates compared with the soft gel substrate (Fig. 3A–C). An analysis of lag time versus the mean squared displacement (MSD) revealed that MSD value increased as a power of time (Fig. 3D). Specifically, when the lag time was set at 60 s, the exponents for the cells moving on the glass, hard gel, and soft gel substrates were 1.71 ± 0.11, 1.93 ± 0.12, and 1.39 ± 0.24 [mean ± standard deviation (SD)], respectively. The square roots of the MSD were 32.69 ± 7.05, 50.29 ± 11.29, and 15.41 ± 7.30 µm (mean ± SD) for the glass, hard gel, and soft gel substrates, respectively, at a lag time of 60 s. This indicated that the cells traveled approximately two and three times as far in 60 s on the glass and hard gel substrates, respectively, compared with the soft gel substrates. The velocities of cells on the glass and hard gel substrates were 0.55 ± 0.05 and 0.75 ± 0.05 µm/s (mean ± SD), respectively. However, during the migration phase on the soft gel substrate, the velocity was 0.65 ± 0.03 µm/s (mean ± SD), whereas it dropped to 0.14 ± 0.01 µm/s (mean ± SD) during the arrest phase (Fig. 3E and F). The turn angle distribution over 10 s was more dispersed on the soft gel substrate, with variances of 0.55, 0.48, and 0.93 rad for the glass, hard gel and soft gel substrates, respectively (Fig. 3G). Next, the relationship between lag time and velocity auto-correlation function (vACF) revealed that there was no significant difference in relaxation time between the glass and soft gel substrates, with values of τ = 27.05 ± 25.31 s [mean ± SE (standard error), n = 5 cells] for the glass substrate and τ = 28.85 ± 10.95 s (mean ± SE, n = 7 cells) for the soft gel substrate. However, the value for the hard gel substrate was higher at τ = 97.11 ± 82.08 s (mean ± SE, n = 10 cells). The zero-order term was 0.64 ± 0.07 (mean ± SE, n = 5 cells) on the glass substrate, 0.28 ± 0.31 (mean ± SE, n = 10 cells) on the hard gel substrate, and 0.22 ± 0.06 (mean ± SE, n = 7 cells) on the soft gel substrate. These results indicated that Arcella moved more randomly on the gel substrate compared with the glass and the hard gel substrates.

2.3. Stress field and direction of cell movement.

Amoeboid locomotion occurs when force is applied to the adhesive substrate through pseudopods.^50)–56) To elucidate the mechanisms underlying Arcella motility, we measured the traction stress exerted by Arcella. Figure 4A shows the typical traction stress observed during locomotion. During the arrest phase, traction stress was applied at various locations. During the migration phase (Fig. 4B, 20–40 s), traction stress was observed at the anterior and posterior positions of the cell (Fig. 4A, 20 s and 35 s). This traction stress was directed toward the center of the cell, regardless of whether it was in the arrest or migration phase. Averaging the traction stress field during the migration phase revealed a distinct anteroposterior and inward traction stress (Fig. 4C).

Fig. 4.

Traction stress field and dipole analysis. (A) This panel illustrates a typical traction stress field exerted by Arcella from the end of the arrest phase to the migration phase. The red arrows indicate the direction of cell movement, whereas the blue arrows represent the major dipole. The angle φ denotes the angle between the velocity vector and major dipole. The background color indicates the magnitude of the traction stress. Time is displayed in the upper right corner of each panel. (B) This panel shows the temporal changes of speed relative to panel A. (C) Time and sample averaged traction stress fields indicate distinct anteroposterior and inward traction stresses. The migration direction of the cells was fixed to the right (positive direction of the X-axis). (D) The relationship between lag time from the maximum velocity and |cos φ| suggests that the angle between velocity and major dipole is near during the migration phase (n = 15, 10 cells). The black line represents the mean value, whereas the gray area indicates standard deviation. The red region indicates a positive correlation, whereas the blue region signifies a negative correlation. (E) The correlation between the eigenvalue of the major dipole and migration speed demonstrates the strongest correlation at a lag time of −46 ± 10 s, with a high correlation value maintained when the lag time was zero (n = 15, 10 cells). The black line represents the mean value, and the gray area shows the standard deviation. The scale bars in A and C are 50 µm.

A multipole analysis^53)–56) developed by Tanimoto and Sano,⁵³⁾ was performed to determine the relationship between traction stress distribution and migration direction, which evaluated the relationship between motion and traction by describing the complex traction force distribution using simpler moments (details in Materials and Methods). The dominant direction of contraction, the major dipole eigenvector (blue arrow in Fig. 4A), was determined by diagonalization of the force dipole matrix. The angle between the major dipole eigenvector and the cell velocity v (red arrow in Fig. 4A) was defined as φ. The relationship was plotted between lag time from the time at maximum velocity and |cos φ| (Fig. 4D). The results showed that |cos φ| reached its maximum value of 0.92 ± 0.14 s (mean ± SD) before 16.75 ± 13.73 s (mean ± SD) from the time of maximum velocity, which indicated that the force distribution was oriented closer to the direction of the cells, approximately 17 s before the magnitude of velocity reached its peak. The correlation, denoted as C(Δt), was calculated between the major dipole eigenvalue e_major and magnitude of the velocity vector. The most significant correlation was observed at a delay time of −46 ± 10 s (mean ± SE, n = 15, 10 cells), with a correlation value is 0.79 ± 0.02 (mean ± SE, n = 15, 10 cells); Fig. 4E). This delay represents the difference in time when the cell exerts maximum traction stress and when the speed reaches its peak. Although there was a time delay, direction of the major dipole eigenvector and cell velocity were almost similar during the migration phase (Fig. 4D). For the following analysis, we defined the direction of the major dipole eigenvector as the direction of motion.

The second-order moment of traction stress, represented by the force quadrupole M_ijk, was calculated to determine the intricate relationship between traction stress and motion. This tensor consists of six independent components (Fig. 5A). The correlation P_ijkl between each component and both the direction of motion and the direction orthogonal to motion was determined (Fig. 5B). The correlation value P is constrained, such that 1 ≦ P ≦ ∞, with a value of P = 1 indicating no correlation. An increase in P indicates a stronger correlation. Correlation analysis between the motion vector v₁ and each secondary moment (left in Fig. 5B) revealed a strong correlation at P(1,2,2;1). This finding suggested a significant positive correlation between M₁₂₂ and the direction of motion v₁, indicating that the traction stress direction toward the center of the cell, which was exerted perpendicularly to the direction of cell motion, was greater on the posterior side than anterior side. Focusing on the negatively correlated component (blue bar in Fig. 5B), we observed a substantial value at P(2,2,1;1). This suggests that the traction force opposing the direction of cell motion spreads more perpendicularly to the direction of motion in the anterior portion of the cell compared with that in the posterior region.

Fig. 5.

Quadrupole analysis of the traction stress field of Arcella during the migration phase. (A) The horizontal axis of the figure represents the direction of the major dipole, while the vertical axis indicates the direction perpendicular to it. The subscripts i, j, and k in M_ijk refer to the indices x_i, x_j, and T_k in Eq. [3]. The values of i, j, and k can be either 1 or 2, each representing the relevant value of x and y components, respectively. The force quadrupole consisted of six independent elements. (B) The correlation, denoted as P_ijkl, between each quadrupole component and both the direction of motion and direction orthogonal to the motion, was limited to the range 1 ≦ P ≦ ∞. A value of P = 1 indicates no correlation, whereas an increase in P signifies a stronger correlation. The correlations between the motion vector and each secondary moment are shown on the left. The correlations between the direction orthogonal to the motion and each secondary moment are shown on the right. The value of P is represented in red when P_equal was greater than P_opposite and in blue when P_equal was less than P_opposite.

Subsequently, the vector v₂, orthogonal to the direction of motion, was examined with the stress distribution (Fig. 5B). P(1,2,1;2) exhibited the strongest positive correlation, which indicated that while moving perpendicularly to the predominant direction of motion, the traction force parallel to the dominant direction toward the center of the cell was amplified on the opposite side of the perpendicular direction to the predominant motion.

Taken together, these experiments provide insight into the statistical characteristics of the motility of the testate amoeba, Arcella, on glass and gel substrates, as well as the formation of pseudopods. Furthermore, the relationship between the distribution of traction stresses transmitted to the substrate during this locomotion and the direction of motion was elucidated.

3. Discussion

We conducted a detailed examination of the amoeboid locomotion of the testate amoeba Arcella, which utilizes multiple pseudopods that extend from a central aperture located at the ventral side of its shell⁴³⁾ (Fig. 1). Their locomotion differs from the extensively studied amoeboid locomotion.^{5),22),24),52),57)–60)} In nature, Arcella traverses diverse substrates, and the characteristics of its locomotion may vary significantly depending on the substrate encountered. In this study, we examined the movement dynamics of Arcella on glass and gel (soft and hard) substrates.)

Our results indicated that Arcella exhibited a relatively constant migratory velocity on glass and hard gel substrates, whereas it underwent distinct phases of migration and arrest when situated on a soft gel substrate (Fig. 3E). Over the course of a 60-second observation period, Arcella traversed approximately twice the distance on the glass and thrice the distance on a hard gel substrate compared with that on the soft gel substrate (Fig. 3D). Furthermore, cells migrating on a glass substrate demonstrated a propensity to extend their pseudopods in the direction of movement (Fig. 2A and B); however, cells on a soft gel substrate occasionally directed their pseudopods in various orientations (Fig. 2G and H). This behavioral divergence may be linked to the presence of migration and arrest phases, along with the observed differences in turn angle characteristics.

We examined the vACF associated with turning behavior (Fig. 3H). The relaxation times in the direction of motion were similar between the glass and soft gel substrates, whereas it was higher for hard gel substrate. These results indicated that Arcella exhibited greater straight movement and higher speed on the hard gel surface than that exhibited on the glass. The reason is unclear; however, one possibility is that, in addition to its elastic modulus, the chemical structure of the substrate surface itself may affect cell adhesion and alter movement characteristics. The zeroth-order term exhibited a significant disparity, approximately three times greater on the glass substrate compared with that on the soft gel substrate. Previous studies have shown that the vACF of amoeboid locomotion in protists and mammalian cells may be represented by the sum of two exponential functions.^{59),61)–65)} Shorter relaxation time is associated with the characteristic duration of pseudopodia formation, whereas longer relaxation time corresponds to the duration of directional movement. In this study, the observed relaxation time fits with the shorter relaxation time, which is considered the characteristic duration for pseudopod formation. The time required for pseudopod formation at the front of the cell was less than 60 s on the glass and soft gel substrates (Fig. 2F and L), which is consistent with the relaxation time value. In contrast, relaxation time on the hard gel was three times more compared with that on the glass and soft gel. These results suggest that the duration of pseudopod formation on the hard gel is longer compared with that on the glass and soft gel. The zero-order term indicated the extended relaxation time associated with the duration of directional motion. Although the precise measurement for the longer relaxation time was not feasible because of the sustained directional movement of Arcella, our results indicated that the zero-order term exhibited a higher value on the glass substrate compared with that on soft gel substrates. This difference suggested that the directional motion on the glass substrate had a longer relaxation time compared with that on the soft gel substrate.

The traction stress field exerted by cells on their substrates during amoeboid locomotion plays an important role in dictating the directionality of their movement. Studies into the traction stress of amoeboid motion have involved individual cells and multicellular systems, with some analyzing cellular dynamics in two- and three-dimensional environments.^{52),55),56),66)–70)} In amoeboid locomotion, the effect of inertial forces and viscous resistance from the ambient fluid is typically negligible. This results in a net sum of the traction stress vectors exerted by the cell upon the substrate, which equates to zero.^53),56),71) Consequently, the distribution of traction stress shows an inherent asymmetry critical for the regulation of cell motility and directional movement.^53),54),56) Our studies into the traction stress of Arcella revealed that cells generate a centripetal force, with pronounced magnitudes observed at the anterior and posterior regions of the cell, in contrast to the lateral orientation (Fig. 4C). Except for amoeboid movement of non-adherent cells in a confined space⁶⁹⁾ and keratocytes involved in the repair of fish epidermis,^72),73) most amoeboid cells display such a stress field,^50)–54) and Arcella appears to use a similar traction field. Traction force microscopy enables the measurement of the deformation of the gel substrate, resulting in cellular movement on the gel. In these conditions, cellular behavior was characterized by distinct migration and arrest phases (Fig. 3E). For our analysis, we focused on cellular dynamics during the migration phase in which the absolute value of traction stress varied significantly spatially and temporally (Fig. 4A). Arcella transmits traction stress to the substrate at a maximum level of several hundred Pascals, which is lower compared with that observed in mammalian cultured cells, such as fibroblasts,^54),71) but is comparable to that in Dictyostelium discoideum.⁵³⁾ When the movement direction of the cell during the migration phase was fixed, and averaged the traction stress of the samples over time (Fig. 4C), the distribution of this stress primarily reflected centerward stress, with absolute traction stress values increasing in the anterior and posterior regions of the cell. This distribution of stress may affect the cell’s directional axis of movement. To examine this in detail, we performed a dipole moment analysis. The angle between the dominant stress vector and velocity vector was measured, and the results indicated that they were most closely aligned at approximately 17 s before the fastest motion of the cell (Fig. 4D). During the migration phase, which lasted for several seconds, the direction of the dominant traction force aligned with the velocity vector at the beginning of the migration phase, which was also observed in D. discoideum.⁵³⁾ The correlation between the magnitude of the predominant traction stress and speed of motion revealed that the traction stress reached its maximum approximately 1 min before the peak speed of the cell. This high level of traction stress was maintained until just before the cell speed reached maximum (Fig. 4E). These results indicated that traction stress in the direction of motion increased before the movement onset and remained elevated during locomotion. Thus, the dipole moment likely determined the direction of motion.

What type of force distribution determines the front-back and left-right motion of a cell? To answer this, we examined the relationship between the quadrupole moments of the traction stress and velocity vector (Fig. 5B). The P(1,2,2;1) exhibited the highest positive correlation, which indicated that the element of centripetal traction stress perpendicular to the direction of cell motion was higher at the posterior region of the cell compared with that at the anterior region (Fig. 6: left panel). This strong positive correlation of P(1,2,2;1) was also observed in D. discoideum.⁵³⁾ Moreover, P(2,2,1;1) exhibited a negative correlation (Fig. 5B), which). indicated that the element of centripetal traction stress in the direction of the cell locomotion axis was more widely perpendicular to the direction of motion in the anterior part of the cell compared with that in the posterior portion. Although the M₁₁₁ exhibited the highest positive correlation with the velocity of D. discoideum,⁵³⁾ in Arcella, the M₁₁₁ had a low negative correlation with its velocity [P(1,1,1;1) in Fig. 5B] because Arcella extends its pseudopodia forward on a soft gel substrate; however, the direction of pseudopod elongation varied considerably (Fig. 2J). The results were also supported by the significant negative value of P(2,2,1;1) (Fig. 6).

Fig. 6.

Schematic diagram of Arcella movement and traction stress distribution. Black and magenta arrows indicate the direction of movement and traction stress, respectively. During forward motion (left panel), the elements of the traction stress perpendicular to the direction of cell movement are higher at the posterior region of the cell compared with the anterior region. Moreover, in the anterior part of the cell, the elements of traction stress along the direction of locomotion are more widely distributed perpendicularly to the direction of motion compared with the posterior part. In lateral migration (right panel), the centripetal traction stress perpendicular to the direction of lateral motion in the posterior position of lateral motion was greater compared with that in the anterior position of lateral motion.

Then, we compared the elements of the velocity vector perpendicular to the direction of the predominant motion and quadrupole moment (Fig. 5B). During lateral migration, P(1,2,1;2) was the most dominant factor, which indicated that the centripetal traction stress perpendicular to the direction of lateral motion in the posterior position of lateral motion was greater than that in the anterior position of lateral motion). (Fig. 6: right panel). For D. discoideum, no component of the quadrupole moment showed a significant correlation with lateral migration.⁵³⁾ Therefore, Arcella has a unique stress distribution relative to lateral migration.

This difference was noted because in D. discoideum, traction was generated at the posterior end of the cell away from the center of mass, whereas Arcella generated traction from the pseudopods that extend forward from its center of mass. The density of the Arcella shell is higher than water, and the shell is in direct or indirect contact with the adhesive substrate. The anterior pseudopods pulled the shell forward, creating friction between the shell and the substrate. This friction was expected to generate traction stress at the posterior region of the cell, which suggested that the traction stress was produced just below the shell near its center of mass. Additionally, in some mammalian cells that used actin polymerization for amoeboid locomotion, M₁₁₁ was negatively correlated with the direction of movement, similar to Arcella.^54),56) Further studies on more cell types are required for understanding locomotion modes and traction stress distribution.

In this study, we observed and quantified the elongation of the Arcella pseudopod during amoeboid motion and elucidated the statistical properties of this movement. Moreover, we measured and analyzed the traction stress exerted on the substrate during this motion and determined the relationship between the direction of amoeboid movement and traction stress field. The results of this study indicated that the traction stress applied by Arcella to the substrate during locomotion shares some similarities with other naked amoebae and cultured mammalian cells, while exhibiting some distinct differences. Further detailed studies of various cell types are necessary to elucidate the mechanisms underlying amoeboid locomotion. These studies will provide a cohesive understanding of the diverse mechanisms of amoeboid motility relevant to various biological phenomena.

4. Materials and Methods

4.1. Cell culture.

Arcella sp. was collected from a pond in the Tomakomai experimental forest of Hokkaido University (42°41′11.2′′N, 141°37′35.1′′E). The isolated Arcella sp. was cultured in Prescott & James solution⁷⁴⁾ containing lettuce extract (2.92 µM CaCl₂·2H₂O, 2.14 µM KCl, 2.92 µM K₂HPO₄, 1.13 µM MgSO₄·7H₂O, and 1 g/L cereal grass media from Ward’s Science) in 60-mm plastic petri dishes at 25 °C. The cells were transferred twice a week to fresh medium. For the experiments, the cells were in culture for at least one day following transfer.

4.2. Observation of cells.

We prepared a chamber to observe cells on a glass substrate from the vertical direction. The chamber consisted of a spacer (a silicon sheet with a 300-µm thickness and a 20-mm diameter hole) sandwiched between two cover glasses. The inner space was filled with Prescott & James solution, and the cells were introduced in this space for observation. To observe cells on a gel substrate, a soft acrylamide gel (soft gel (2.42 kPa) was prepared consisting of): 4% acrylamide (w/v), 0.075% N,N′-methylenebisacrylamide (w/v), 1.5% ammonium persulfate (w/v), and 0.35% N,N,N′,N′-tetramethylethylenediamine (v/v) in water. The hard gel (484.27 kPa) contained): 30% acrylamide (w/v), 0.8% N,N′-methylenebisacrylamide (w/v), 1.5% ammonium persulfate (w/v), 0.35% N,N,N′,N′-tetramethylethylenediamine (v/v) in water with 470-µm thickness. The cells were placed into the observation chamber, left for 30 min, and observed under an inverted microscope (Eclipse Ti2, Nikon) with an objective lens (20 × Plan Apo VC, Nikon) and an sCMOS camera (ORCA-Fusion, Hamamatsu). A second chamber was prepared to observe the cells from the side. A smaller cover glass (24 mm × 36 mm) was sandwiched between two larger cover glasses (24 mm × 50 mm). The three sides were sealed without the cover glass using Vaseline. The cells were left for at least 30 min after putting them into the inner space of the chamber and observed the cells from the side using an objective lens (10 × Plan Apo, Nikon), a tube lens (TTL180-A, Thorlabs), an sCMOS camera (ORCA-spark, Hamamatsu), and a halogen light (LA-100USW, Hayashi Repic) as light source.

4.3. Statistical analysis of cell movement.

The images were binarized, including pseudopodia and shell, from the vertical direction using the image intensity to obtain the cell shape. In addition, we extracted the center of mass of the shell without pseudopods and defined it as the position of the cell r(t) by binarization according to image intensity. To fix the position and the movement direction to the center and positive X-axis, respectively, the coordinates and angles of the cell image with pseudopods were changed using the time series for the cell position. Thereafter, we removed a circular region with a 50-µm radius (the average radius of the cell shell) from the center images to obtain the images of the pseudopods (Fig. 2A and G). For further analysis, we transformed the extracted pseudopod region into polar coordinates (radial coordinate R and angular coordinate α; Fig. 2A, B, G and H). The length of the pseudopods was measured at each time and angle (Fig. 2B and H). The relationship between the angle and the length of the pseudopods was averaged over time and across all samples (Fig. 2D and J). Additionally, we calculated the probability density for the direction of pseudopod extension by averaging over time without considering the sample average. The absolute value of the angle (α) was classified between the pseudopod direction and the movement direction as anterior if it was less than π/2 radians, and as posterior if it was greater. The proportion of anterior pseudopods relative to the total pseudopod extension directions was defined as the probability of anterior pseudopod appearance. The velocity of cells was calculated as v(t) = (r(t + Δt) − r(t − Δt))/2Δt, where Δt = 5. Histograms for the velocities were fitted by the least squares method as Gamma distribution. The MSD at lag time Δt was calculated as follows: MSD(Δt) = ⟨|r(t + Δt) − r(t)|²⟩_t,i, where ⟨·⟩_t,i denotes the temporal and ensemble average of all trajectories. The turn angle is the angle between v(t) and v(t + Δt), where Δt = 10 s. The velocity autocorrelation function (vACF) was vACF(Δt) = ⟨v(t) · v(t + Δt)⟩_t,i. The results for vACF were fitted to A·exp(−Δt/τ) + C using the least squares method, where τ was the relaxation time and C was the zero-order term.

4.4. Measurement of traction stress.

A disc-shaped acrylamide gel was prepared with 12-mm diameter and 470-µm height. The gel consisted of 4% acrylamide (w/v), 0.075% N,N′-methylenebisacrylamide (w/v), 1.5% ammonium persulfate (w/v), 0.35% N,N,N′,N′-tetramethylethylenediamine (v/v), and 0.01% red fluorescent beads (w/v) (1.0 µm, 580/605 carboxylate modified microspheres, Invitrogen). The gel was placed onto a coverslip, and the suspended cells were seeded into the Prescott & James solution on the gel surface. After 30 min, we observed the motion of fluorescent beads beneath the cells using a confocal laser microscope (Eclipse Ti with A1, Nikon) equipped with a 20 × Plan Apo VC objective lens (Nikon) and an excitation laser at 488 nm at room temperature (25 °C, n = 12). To obtain the deformation vector fields of the substrate caused by cell traction, the normalized cross-correlation was used to analyze the difference in bead positions between the absence and presence of the cells.⁷⁵⁾ The interrogation window, search window, and overlap were set to 19.8 µm × 19.8 µm, 29.7 µm × 29.7 µm, and 19.8 µm, respectively. Based on the displacement vector field, the traction force was estimated using Fourier transform traction cytometry with the Tikhonov regularization method.^{51),71),76),77)} The elastic modulus of the gel (2,054 Pa) was determined using Hertz’s contact equation by placing a 1.5-mm steel ball on the acrylamide gel and measuring the resulting deformation.⁷⁸⁾ Based on the analysis, the acrylamide gel, a type of hydrogel, was considered an incompressible material, with Poisson’s ratio set at 0.5.^76),79)

4.5. Multipole analysis of traction stresses.

Based on previous studies,^53)–56) a multipole analysis was performed to determine the relationship between the traction stress field and the direction of cellular motion. We focused on the first- and second-order moments of the traction stress field. The first-order moment, referred to as the force dipole, is defined as follows:

  

\begin{equation} M_{ij} = \frac{1}{2}\left[\int_{S}x_{i}\,T_{j}dS + \int_{S}x_{j}\,T_{i}dS \right]. \end{equation} [1]

The subscripts i and j denote the direction of the coordinates, and the x and y components were set at 1 and 2, respectively. x_i and x_j represent the position from the center of the cellular mass, and T_i and T_j are the elements of the traction stress. The integration region S is characterized as a circle with a radius of 150 µm centered at the cell mass. The resulting 2 × 2 matrix was diagonalized to identify two pairs of eigenvalues and eigenvectors. The eigenvector corresponding to the larger eigenvalue, e_major, is considered the major dipole, whereas the eigenvector associated with the smaller eigenvalue, e_minor, is identified as the minor dipole. In assessing the correlation between the major dipole and cellular velocity, we calculated the correlation coefficient between the absolute value of cellular velocity and the major eigenvalue. The correlation is defined as follows:

  

\begin{equation} \text{C}(\Delta t) = \frac{-\displaystyle\sum\nolimits_{t_{\text{start}}}^{t_{\text{end}}}e_{\text{major}}(t)|v(t + \Delta t)|}{\displaystyle \sum\nolimits_{t_{\text{start}}}^{t_{\text{end}}}|e_{\text{major}}(t)||v(t)|}. \end{equation} [2]

Only the data in the region was used, where |v(t)| ≧ 0.5 µm/s to focus on the cells in the migration phase. The t_start and t_end denote the beginning and end times of the migration phase, respectively. Even during the migration phase, we excluded the periods where the peaks of |v(t)| were obscured because they overlapped with other peaks. The time when the velocity reaches its maximum value is defined as 0 s, and Δt represents the time delay from that moment. To determine the directionality of the velocity vector relative to the dominant dipole, the angle between the two vectors was designated as φ. Subsequently, the absolute value of the cosine of φ was calculated to assess their alignment. Next, we focused on the secondary moment, known as the force quadrupole, which is calculated as follows:

  

\begin{equation} M_{ijk} = \int_{s}x_{i}x_{j}T_{k}dS. \end{equation} [3]

The origin of the coordinate system was established at the center of mass of the cell. The major dipole was oriented along the X-axis, with the direction more closely aligned with the velocity vector being classified as the positive orientation. The force quadrupole had six independent elements: x₁²T₁, x₁²T₂, x₁x₂T_1x, x₁x₂T₁, x₁x₂T₂, $x_{2}^{2}T_{1}$, $x_{2}^{2}T_{2}$ $x_{2}^{2}x_{2}^{2}$ (Fig. 3f). To determine the relationship between this force quadrupole and the direction of movement, we introduced a correlation as follows. When the signs of the quadrupole and velocity components (v₁ or v₂) were equal, one was added to P_equal. When they were opposite, one was added to P_opposite. Moreover, using the ratio of these two values, the correlation was defined as follows:

  

\begin{equation} \text{P} = \begin{cases} \text{P}_{\text{equal}}/\text{P}_{\text{opposite}} & \text{$(\text{P}_{\text{equal}} > \text{P}_{\text{opposite}})$}\\ \text{P}_{\text{opposite}}/\text{P}_{\text{equal}} & \text{$(\text{P}_{\text{equal}} < \text{P}_{\text{opposite}})$} \end{cases} . \end{equation} [4]

Conflict of interest

The authors declare no conflicts of interest.

Author contributions

G.M., J-P.R., and Y.N. conducted experiments and analyzed the data. G.M., A.T., and Y.N. sampled Arcella from the environment. G.M., A.T., and M.N. developed the cell culture methods. A.T., M.N., and S.S. contributed to identifying the genus of the cell. G.M., K.S., T.N., and Y.N. led the study design. K.S., T.N., and Y.N. supervised the project. G.M., S.S., K.S., T.N., and Y.N. acquired funding. G.M. and Y.N. wrote the original draft, and all authors reviewed, edited, and approved the manuscript.

Acknowledgements

We thank the Open Facility, Global Facility Center, Institute for Integrated Innovations, Hokkaido University, and the Nikon Imaging Center at Hokkaido University for allowing us to conduct the imaging of Arcella using microscopes. We also thank Hiroshi Orihara for providing valuable insights during the development of this work and Osamu Kishida for his assistance with the sampling.

This work was performed under the Cooperative Research Program of the Network Joint Research Center for Materials and Devices (Y.N.) and the Program for Fostering Researchers for the Next Generation conducted by the Consortium Office for the Fostering of Researchers in Future Generations, Hokkaido University (Y.N.). This work was supported by Japan Society for the Promotion of Science (JSPS) KAKENHI Grant numbers JP24K09388 (Y.N.), JP21H05308 (Y.N.), JP21H05310 (T.N. and K.S.), JP21H05303 (T.N.), JP23H04300 (K.S.), the JSPS Core-to-Core Program, an Advanced Research Networks (Y.N.), the JST Support for Pioneering Research Initiated by the Next Generation Grant number JPMJSP2119 (G.M.) and the Project of Junior Scientist Promotion at Hokkaido University (Y.N.). This work was supported by research funds of the Asahi Glass Foundation (S.S.).

Notes

Edited by Toshio YANAGIDA, M.J.A.

Correspondence should be addressed to: Y. Nishigami, Research Institute for Electronic Science, Hokkaido University, Sapporo, Hokkaido 001-0020, Japan (e-mail: [e-mail]).

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Non-standard abbreviation list

MSD

mean squared displacement

vACF

velocity autocorrelation function