Cytoskeletal Mechanics Regulating Amoeboid Cell Locomotion.

Bego~na Alvarez-Gonz alez Research Assistant Mechanical and Aerospace Engineering Department, University of California, San Diego, La Jolla, CA 92093-0411 e-mail: [email protected]

Effie Bastounis Postdoctoral Fellow Division of Cell and Developmental Biology, University of California, San Diego, La Jolla, CA 92093-0411

Ruedi Meili Research Scientist Mechanical and Aerospace Engineering Department, Division of Cell and Developmental Biology, University of California, San Diego, La Jolla, CA 92093-0411

Juan C. del Alamo Associate Professor Mechanical and Aerospace Engineering Department, Institute for Engineering in Medicine, University of California, San Diego, La Jolla, CA 92093-0411

Richard Firtel Distinguished Professor Division of Cell and Developmental Biology, University of California, San Diego, La Jolla, CA 92093-0411

Juan C. Lasheras1 Distinguished Professor Mechanical and Aerospace Engineering Department, Institute for Engineering in Medicine, Bioengineering Department, University of California, San Diego, La Jolla, CA 92093-0411 e-mail: [email protected]

1

Cytoskeletal Mechanics Regulating Amoeboid Cell Locomotion Migrating cells exert traction forces when moving. Amoeboid cell migration is a common type of cell migration that appears in many physiological and pathological processes and is performed by a wide variety of cell types. Understanding the coupling of the biochemistry and mechanics underlying the process of migration has the potential to guide the development of pharmacological treatment or genetic manipulations to treat a wide range of diseases. The measurement of the spatiotemporal evolution of the traction forces that produce the movement is an important aspect for the characterization of the locomotion mechanics. There are several methods to calculate the traction forces exerted by the cells. Currently the most commonly used ones are traction force microscopy methods based on the measurement of the deformation induced by the cells on elastic substrate on which they are moving. Amoeboid cells migrate by implementing a motility cycle based on the sequential repetition of four phases. In this paper, we review the role that specific cytoskeletal components play in the regulation of the cell migration mechanics. We investigate the role of specific cytoskeletal components regarding the ability of the cells to perform the motility cycle effectively and the generation of traction forces. The actin nucleation in the leading edge of the cell, carried by the ARP2/3 complex activated through the SCAR/WAVE complex, has shown to be fundamental to the execution of the cyclic movement and to the generation of the traction forces. The protein PIR121, a member of the SCAR/WAVE complex, is essential to the proper regulation of the periodic movement and the protein SCAR, also included in the SCAR/WAVE complex, is necessary for the generation of the traction forces during migration. The protein Myosin II, an important F-actin cross-linker and motor protein, is essential to cytoskeletal contractility and to the generation and proper organization of the traction forces during migration. [DOI: 10.1115/1.4026249]

Introduction

Migrating cells exert traction forces. These traction forces are necessary in order to perform the locomotion process and are involved in the generation of the signaling events. Cell motion is involved in multiple processes such as the response to infection and inflammation, wound healing, embryogenesis, angiogenesis, and metastasis [1,2]. The cytoskeleton of a cell serves as its structural framework, which determines its shape and consists of a network of protein filaments [3]. Cell traction forces are generated by actin polymerization, by cross-linking proteins, regulatory and motor proteins, and by adhesion molecules. They vary in magnitude and organization depending on the type of cell and environment. Cells move either individually or collectively. In the case of single cell migration, there are two distinct types of locomotion: amoeboid and mesenchymal. Mesenchymal migration is 1 Corresponding author. Manuscript received July 7, 2013; final manuscript received October 9, 2013; published online June 5, 2014. Assoc. Editor: Ellen Kuhl.

Applied Mechanics Reviews

characterized by high adhesion to the substrate. Usually the adhesions formed by mesenchymal migrating cells are integrin mediated and focal adhesions are clearly defined. This stronger adhesion leads to higher contractile traction forces [4]. The characteristic shape of the mesenchymal migrating cells is elongated [5]. In 3D matrices, this migration is proteases-dependent2 and proteolysis and degradation of the extracellular matrix occur. Mesenchymal migration is a slow migration mode. Amoeboid migration is characterized by low adhesion to the substrate and lack of mature focal adhesions. Consequently, the traction forces exerted by these cells are also low [4]. The adhesions in amoeboid migration are non-integrin or weak-integrin mediated [6]. The characteristic shape of the amoeboid migrating cells is rounded or ellipsoidal [5,7]. Amoeboid migrating cells are highly motile and protease-independent in 3D matrices. Typically cells performing amoeboid migration have the ability to change cell shape (blebbing, elongation, or bending). Amoeboid migration can be subclassified in two types depending on the mechanism of forward 2

Cells secrete metalo-proteases at the front to degrade the extracellular matrix.

C 2014 by ASME Copyright V

SEPTEMBER 2014, Vol. 66 / 050804-1

Downloaded From: http://appliedmechanicsreviews.asmedigitalcollection.asme.org/ on 01/28/2016 Terms of Use: http://www.asme.org/about-asme/terms-of-use

extension of the plasma membrane: blebbing (cells move by extending membrane blebs) and protrusion of actin-rich pseudopods (3D fingerlike protrusions) [5]. In both mesenchymal as well as amoeboid single cell migration, the cells move in a cyclic manner. In the case of mesenchymal migrating cells, the cycle is defined by protrusion of the leading edge (lamellipodium), adhesion of the lamellipodium to the substratum at its ventral part, focal adhesion formation, contraction of the cell body by interaction of F-actin and myosin, and retraction of the cell body and nucleus when the adhesions to the substratum are weakened or degraded [7]. The characteristics of the cycle stages in amoeboid migration are somewhat similar. It also comprises protrusion consisting of pseudopods extension, contraction of the cell body, retraction of the rear part, and relaxation [8]. Examples of mesenchymal migrating cells are fibroblasts, smooth muscle cells, epithelial cells, endothelial cells, and stem cells. The group of amoeboid migrating cells is very large and heterogeneous including unicellular eukaryotic cells such as Dictyostelium discoideum as well as individually migrating metazoan cells such as neutrophils and dendritic T cells. Amoeboid cells are highly motile, and the contractile traction forces are exerted at the sides, front and back of the cell body [8–10]. In contrast, mesenchymal migration is slow and characterized by nascent adhesions at the leading edge, which mature into focal adhesions and eventually disassemble at the trailing edge [11], and generate traction forces under the lamellipodium. This review is focused on amoeboid locomotion, for cells performing this type of migration focal adhesions and stress fibers are not detected, as in the case of migratory neutrophils, T cells [10] and Dictyostelium discoideum cells [12]. The mechanism of generation of traction force in neutrophils is still not well characterized and there are few quantitative measurements. Eun Shin et al. [10] studied neutrophils migration over 3.5 kPa gels and found that traction forces are located at both the leading and the trailing edges of the cells where they oscillate with a defined periodicity. However, Smith et al. [9] found that neutrophils migrating over 9 kPa gels exert traction stresses mostly in the uropod3 of the cells oriented in the direction of the chemoattractant gradient, where the traction forces are higher in magnitude and more strongly concentrated. They observed that the motility seems to be organized and initiated in the rear part of the cell [9], in contrast to strongly adherent, slow moving cells such as fibroblasts. Net force and migration speed in neutrophils seem not to be strongly correlated and their motility is largely integrin-independent [13]. The generation of traction forces in neutrophils requires myosin II [10]. Dendritic T cells are amoeboid migrating cells which migrate from sites of inflammation to secondary lymphoid organs where they initiate the adaptive immune response. There are few measurements of traction forces exerted by migrating dendritic cells, and they seem to be weaker in magnitude and different in distribution compared to the traction forces exerted by neutrophils. In these cells, the traction forces are more concentrated at the leading edge than around the nucleus or at the rear part of the cell, and are dependent on acto-myosin contraction [14]. Dendritic cells can migrate without the use of integrins in vitro and in vivo [15] or use short-lived integrin-based adhesions in the leading filopodia [16]. Cell migration is controlled by the dynamic regulation of the F-actin cytoskeleton [17]. The mechanics of cell migration involves the polymerization of F-actin filaments towards the front and pushing of the cell membrane forward. The Rho family of small GTPases [18] regulates the protrusion formation through the activation of the ARP2/3 complex through the SCAR/WAVE complex [19]. The next needed step for the migration mechanism is the adhesion of the cell to the substrate. Amoeboid cells generally do not form well characterized adhesive structures, for Dictyostelium the origin of the adhesions is not clear [20,21] and 3

Opposite side to the leading edge in polarized leukocytes.

050804-2 / Vol. 66, SEPTEMBER 2014

other amoeboid migrating cells that form weak-integrin mediated adhesions have integrins distributed in the plasma membrane [22]. The last step involved in the migration mechanism is the detachment of the rear of the cell, carried by the contractile forces produced by the acto-myosin network [23]. There is a growing need to measure the traction forces in order to unravel the mechanics controlling cell motion. Several methods have been developed for this purpose over the years. Traction forces in individual cells can be detected using microscopy-based techniques that rely on the measurement of the deformation of thin membranes, the measurement of the deflection of microfabricated structures, or the measurement of the deformation of elastic hydrogels [24]. The use of microfabricated structures has been extensively used for the calculation of the traction forces [25]. Cells adhere to an array of flexible microfabricated posts by a discrete number of adhesions, and the force exerted can be calculated from the deflection of the pillar using simple mechanical considerations [26]. Microposts have been used to elucidate the influence of substrate stiffness and spread area in the traction forces [27], and the effect of substrate rigidity on stem cell morphology, traction force generation, focal adhesion organization, and differentiation [28]. The use of magnetic micropost systems allows the study of mechanotransduction and the relation between traction forces exerted by the cells and external forces [29,30]. The measurements of traction forces using micropost have been mostly applied to mesenchymal migrating cells which exert high forces, such as fibroblasts [25,26,31], endothelial cells [27] and smooth muscle cells [30]. One shortcoming of the micropost technique is that it predetermines the shape of the cell and enforces the localization of the adhesions at specific discreet points where the pillars are located. A further advancement is the cell traction force microscopy which can measure the traction forces exerted in the entire cell spreading area. Cell traction force microscopy is, at present, a reliable method for determining the traction forces exerted by a cell in a two-dimensional (2D) substrate surface [32–34]. The study of adhesion formation and traction force generation triggers the construction of computational models that allows their prediction [31,35]. Models that predict adhesion geometry and traction force organization have the power to quantify how different mechanical and biochemical properties affect traction force generation and cell migration and cast light on the elucidation of the relationship between the biochemical aspects of cell functioning and the mechanics of cell migration [36–38]. In this paper, we focus on the mechanics of the chemotactic migration of the amoeba Dictyostelium discoideum over flat surfaces. The material presented here is heavily based on the work done by the authors on the bio-mechanics of amoeboid cell locomotion and it is not intended to be a comprehensive review of cell motility. Dictyostelium is used as a model system since the signaling pathways involved in chemotaxis are highly conserved in some mammalian cells such as leukocytes [39,40]. In order to unravel the role of the various proteins composing the cytoskeleton in the generation of traction forces, we measure the traction forces that mutant cell lines lacking specific proteins exert and compare them to the forces exerted by wild-type cells. In this way, we can determine the role of each molecule in the organization and magnitude of the traction forces exerted over the substrate during migration. Specifically, this review will concentrate on the role that the proteins SCAR and PIR121 and Myosin II (MyoII) play in generating traction forces. We chose these proteins because SCAR and PIR121 are members of the SCAR/WAVE complex that activates the actin branch nucleation on the leading edge of the cells, and MyoII is essential in the actomyosin contractility mechanisms of the cellular cytoskeleton. The pseudopod formation is driven by actin polymerization at the leading edge of migrating cells [41]. Mathematical models to study the force generation at the leading edge of the cell conclude that the polymerizing actin in the front needs to be cross-linked in some way; otherwise, the force of polymerization would drive the filaments rearward instead of driving the protrusion forward [42]. There are various proteins that regulate the actin assembly and Transactions of the ASME


disassembly in cells. The SCAR/WAVE complex activates the Arp2/3 complex activity. The Arp2/3 is a seven protein complex that promotes the dendritic nucleation of actin filaments that elongate only from their barbed ends [43]. The Arp2/3 complex binds to the sides of pre-existing filaments and promotes the growth of a new actin filament with a characteristic angle of 70 deg70 deg [42]. The growth of these filaments produces the force to protrude the membrane and transmit force though the cytoskeleton to the substrate, thus generating the movement [44]. In chemotactic neutrophils, the Arp2/3 complex redistributes to the region that receives maximal chemotactic stimulation, and it colocalizes with sites of actin polymerization [45]. Myosin is an actin-based molecular motor that is an important component of many forms of cellular locomotion [46]. It is expected that the generation of force for cellular translocations requires actin-myosin interaction [47]. MyosinII is concentrated in the posterior cortex of migrating cells [48,49], and it is also located transiently in the tips of retracting pseudopods [50]. This suggests that it plays an essential role in the dynamics of pseudopods as well as filopodia, lamellipodia, and other cellular protrusions [50]. To examine the role of MyoII in the mechanics of migration, we study Myosin II null cells, mhcA–, which lack the MyoII cross-linking and motor function, and Myosin II essential light chain null cells, mlcE–, which have impaired the MyoII motor function. By examining these two mutant cell lines, we can clarify the functioning of the motor and cross-linking activities of MyoII in the generation of forces in Dictyostelium cells.

2

Traction Force Microscopy Method

Several methods have been developed to characterize the dynamics of cells when migrating over flat substrates [32]. Most of these methods are based on measurements of the deformation of a flat elastic substrate on which the cells crawl, and the subsequent calculation of the traction forces from the measured deformation. Dembo et al. used the classical solution of the elastostatic equation for a homogeneous, semi-infinite medium found by Boussinesq for the elastic solid in the half-space [51,52]. They calculated the traction forces applied in a discretized mesh where the cell is located and they used a linear relation between the deformation and traction forces in order to calculate them (perfectly elastic medium). They used markers embedded in the substrate and used the relation between the traction forces and the displacement of the markers given by the following integral transform [52]: ðð gba ðmp rÞTb ðrÞdr1 dr2 (1) da ¼ where the nine functions gba ðmp rÞ are the coefficients of the Green’s tensor that gives the displacement of the substratum in the a-direction generated by a force in the b-direction. Butler et al. [53] traction force cytometry method is based on Fourier analysis, and consists of an analytical calculation of the Fourier transform of the Boussinesq solution. Butler et al. calculate the traction forces by performing the Fourier transform of the displacement field and applying the solution to the Boussinesq equation in Fourier space to obtain the forces through the inverse Fourier transform [53]. Butler et al.’s method has two approaches: the unconstrained method which gives the traction forces calculated with no restriction, and the constraint method which confines the locations of the traction forces to the location of the cell. Butler et al.’s and Dembo et al.’s [51] methods do not take into account the effect of the finite thickness of the substrate and consider it semi-infinite. There is a third method developed by Yang et al. [54], which consists of using a 3D finite element method. The effect of a finite substrate is incorporated in this approach, and the displacements calculated in the outside of the cell are not taken into consideration. Our review focuses on the traction force cytometry method developed by del Alamo et al. [16], which consists of an exact, Applied Mechanics Reviews

computationally efficient solution of the elastostatic equation based on Fourier expansions that obtains the traction forces explicitly as functions of the substrate deformations. Our method takes into account the substrate thickness, which increases the accuracy of the Boussinesq solution and allows for nonzero net forces. It also considers the distance between the measurement plane and the surface of the substrate. 2.1 Substrate Fabrication. We have used two types of elastic substrates. In the experiments using the MyoII mutant cell lines, the substrate used is a gelatin gel with yellow latex beads of 0.1 lm diameter (FluoSpheres; Molecular Probes, Eugene, OR) embedded in it. The thickness of the substrate is around 100 lm. In the experiments using the SCAR/WAVE mutants, the substrate used is a polyacrylamide gel of 5% acrylamide and 0.06% bis-acrylamide coated with 0.25 mg/ml collagen [55,56]. The gels consist of two layers: the bottom layer contains no beads, and the upper one contains 4 ll of 2% carboxylate yellow latex beads of 0.1 lm diameter (FluoSpheres; Molecular Probes, Eugene, OR). The thickness of the gel is approximately 40 lm. To functionalize the surface of the gel and transform it to be physiologically compatible to live cells, we coat the surface of the gel with Type II collagen through coupling to the polyacrylamide gel with 1 mM sulfo-SANPAH activated by using UV light. Figure 1(a) shows a sketch of the experimental configuration. Figure 1(e) shows an image of the substrate embedded with beads acquired with a microscope. Similar images to Fig. 1(e) are used to calculate the deformation exerted by the migrating cells over the substrate. 2.2 Measurement of the Young’s Modulus of the Substrate. The Young’s modulus of the substrate that we use is approximately 1000 Pa, as determined from measurements of the indentation of a tungsten carbide sphere [58]. The Young’s modulus of the substrate is determined by measuring the static indentation depth Dz of a tungsten carbide sphere (R ¼ 150 lm, W ¼ 1.898 lN, Hoover Precision, East Gramby, CT) slowly deposited on the substrate. Figure 2 shows a sketch of the experimental procedure used to measure the indentation created by the ball. We apply the equation found by Keer [58] that calculates the Young’s Modulus, E, as a function of the thickness, h, the indentation depth z, the radius R, the apparent weight of the sphere W and the Poisson ratio r: E¼

3ð1 r2 ÞWR 4a30

where a0 is solved from a 3 a 5 a2 a0 0 0 0:098 z ¼ 0 1 0:504 0:225 R h h h

(2)

(3)

We determine z as the depth where the beads displaced by the carbide sphere come into focus in a z-stack of images with a distance between planes of 0.4 lm. The in-focus beads are detected using the SOBEL function in MATLAB. The range of Poisson ratios, r, that has been reported for polyacrylamide gels is between 0.3 and 0.49 [59–61]. The Young’s modulus obtained in our experiments, assuming the Poisson ratio to be 0.3, has an average of E ¼ 910 Pa and a standard deviation of 360 Pa. 2.3 Identification of the Cell Contour. Cell contours are determined from differential interference contrast (DIC) images. The image processing is performed with MATLAB (Mathworks Inc, Natick, MA). First we remove the static imperfections and bright nonuniformity by using the average image of the time-lapse series. We apply a threshold to the resulting image in order to obtain the most intense structures of the image and subsequently apply two dilations and erosions to obtain the contour of the cell. Figures 1(b)–1(d) shows the process for the identification of the cell contour. Figure 1(b) shows the DIC image acquired with the SEPTEMBER 2014, Vol. 66 / 050804-3


Fig. 1 (a) sketch of the configuration of the experiment. Substrate with an upper layer embedded with beads where the cells are moving. (b) DIC image taken with the microscope to identify the cell contours. (c) dilation and erosion application to determine the cell contour from the DIC image. (d) cell contour determination after a second dilation and erosion application. (e) image of the substrate embedded with beads used to calculate the deformation induced by the migrating cells. (f) displacement field for a wild-type cell at an instant of time. The arrows (color red online) represent the direction of the displacements and the contours underneath (color blue online) represent the magnitude of the tractions. This figure is taken from [16].

microscope, while Figs. 1(c) and 1(d) show the contour detection after the application of one and two dilations and erosions, respectively. The coordinates of the centroid and the principal axes of each cell are calculated using standard MATLAB functions. The front and rear of the cell are determined as the two parts in which the cell is divided by its minor axis, with the front pointing towards the direction of motion. 2.4 Displacement Field Measurement. The substrate deformation is measured from the displacements of the fluorescent beads embedded into it. In order to do that, we use a technique similar to Particle Image Velocimetry [62,63], which consists of performing the cross correlation between two images, one in which the substrate is not deformed and another in which the cell is deforming the substrate. In order to minimize systematic errors, the images are taken in the plane where the fluorescence intensity is maximum. In order to increase the accuracy of the displacement measurement, we perform the cross correlation in small windows of 16 16 pixels with a 50% overlap. To increase the accuracy of the displacement estimation, we use interpolation to obtain the maximum peak in the cross correlation function with sub-pixel accuracy. The Gaussian interpolation works for correlation peaks and is standard in PIV processing. Thus, we use a Gaussian regression by minimizing the norm of the parabolic fit to the logarithmic values of the correlation function. This gives a resolution of 2 lm approximately. Figure 1(f) shows the displacement field distribution for a wild-type cell at a particular instant of time. 2.5 Solution of the Elastostatic Equation by Using Fourier Series. The traction forces are calculated by solving analytically the elasticity equation of equilibrium for a linear, homogeneous, isotropic element of thickness h in a Cartesian coordinate system

with the x and y axes parallel to the base of the substrate, which is located at z ¼ 0. When the cell migrates over the flat substrate, it generates a displacement field, ~ u ¼ ðu; v; wÞ. The measurement plane is located at a distance from the surface h0 . The equations governing the displacement field are rðr ~ uÞ þ D~ u¼0 ð1 2rÞ

The boundary conditions are no slip at the bottom of the substrate, ~ u ¼ 0 at z ¼ 0, since the substrate is attached to a glass coverslip, and the measured displacements at the surface, ðuh0 ; vh0 Þ at z ¼ h0 . The other boundary condition is the assumption that the traction forces exerted by the cells in the perpendicular direction to the substrate surface are negligible, this is szz ¼ 0 at z ¼ h. We assume periodicity in the horizontal directions and solve the elastostatic equation analytically by using Fourier series [16] ðu;v;wÞ¼

1 1 X X

â;b ðzÞexpðiaxÞexpðibyÞ (5) ½^ ua;b ðzÞ;^ va;b ðzÞ; w

a¼1 b¼1

where a and b are the wave numbers in the x and y directions, and â;b ðzÞ are the complex Fourier coefficients of u, uâ;b ðzÞ, vâ;b ðzÞ, w v, w. We obtain a first-order ordinary differential equation for the zfunctional form of these Fourier coefficients d^ ua;b ¼ Aa;b uâ;b dz where

050804-4 / Vol. 66, SEPTEMBER 2014

3 uâ;b 6 vâ;b 7 7 6 7 6 w 6 â;b 7 6 du ^ a;b 7 7 6 7 ¼6 6 dz 7 7 6 dv ^ 6 a;b 7 7 6 6 dz 7 5 4^ dwa;b

(6)

2

uâ;b

Fig. 2 Measurement of the Young’s modulus with the calculation of the indentation produced by a tungsten carbide ball. This figure is taken from Ref. [57].

(4)

(7)

dz and Aa;b is a matrix, whose coefficients only depend on a, b, and r. Transactions of the ASME


Fig. 3 Pole force calculation. The pole force at the front Ff is calculated by integrating the tractions stresses in the front half of the cell, n>0. The pole force at the back Fb is calculated by integrating the traction stresses in the back half of the cell, n0

Fb ¼

ð

~ sðx; yÞdS

(12)

n 0 indicates the front of the cell and n < 0 indicates the back of the cell. 2.8 Extension to Calculation of the Traction Forces in the Three Dimensions. Until recently, traction force microscopy methods were focused on measuring the traction forces exerted by Applied Mechanics Reviews

the cells only in the direction parallel to the plane where the cells were attached, while the perpendicular component was neglected. The role of the z-component was supposed to be negligible assuming that the predominant orientation of the contractile fibers of the cytoskeleton is mainly horizontal. However, since cells and their environment are three-dimensional, it is important to measure the traction forces in the perpendicular direction to the substrate as well. There are some methods that have been developed recently in order to measure the three-dimensional traction forces; they reveal that the perpendicular forces to the substrate are not negligible. Hur et al. [64] developed a method to measure this perpendicular component as well as the in-plane components by using a finite element method and applying the constitutive equations (elastic stress-strain law). Franck et al. [65] calculated the traction forces exerted by 3T3 fibroblasts from the deformation applied on the substrate directly from the constitutive equations for a linearly elastic and incompressible medium. Delano€e-Ayari et al. [66] measured the perpendicular component of the traction forces exerted by Dictyostelium cells, through the calculation of the deformation applied by using a traditional PIV algorithm. They solved the force field exerted by the cell on the substrate by applying the Boussinesq equation in a semi-infinite medium and assuming a Poisson ratio of 0.5, so as the vertical deformation in the surface is simply related to the vertical force as [52] ð (13) uz ðrÞ ¼ dr0 Gzz ðr r 0 ÞFz ðr 0 Þ The method that we use to calculate the traction forces in the perpendicular direction to the substrate surface in addition to the in plane forces was developed by del Alamo et al. [67]. The deformation of the substrate needs to be measured in the three dimensions. In order to do this, we use a confocal microscope to acquire z-stacks of images, close to the surface of the substrate which have fluorescent beads embedded in it. We calculate the deformation produced by the migrating cells over the substrate by crosscorrelating each instantaneous z-stack with a reference z-stack in which the substrate is not deformed. We compute the threedimensional deformation field by solving the elastostatic equation, Eq. (4). We solve this equation in a similar way to the 2D method, explained previously, using discrete Fourier series. However, in this case, the boundary conditions used for the solution of Eq. (4) are zero displacements at the base of the substratum, ~ u ¼ 0 at z ¼ 0, and the measured displacements, ðuh0 ; vh0 ; wh0 Þ, at z ¼ h0 , as shown in Fig. 4. The procedure that we use is similar to the one for the two-dimensional solution but extended to three dimensions [67]. This refined method allows for the calculation of the three dimensional traction forces. Moreover, it makes it possible to characterize the validity of the results obtained with the previously described 2D methods that calculate the traction forces in the substrate plane, but neglect the out-of-plane component of the traction stresses. Analysis of the error made in the calculation of the tangential stresses to the substrate by using 2D methods reveals that depending on the values of several parameters involved in the experiment, such as cell size, substrate thickness and Poisson ratio, the error in the calculated horizontal traction forces can vary considerably. In most of the cases, the error of the horizontal traction forces calculated with 2D methods is large. Only by using some specific combinations of these parameters did we find that the error of the horizontal traction forces measured with the two-dimensional method is small, and the measurements are not highly affected by neglecting the out-of-plane component of the traction forces [67]. The stresses exerted by cells that are fully encapsulated in 3D matrices have been measured by Legant et al. [68] by tracking the displacements applied to fluorescent beads embedded into an elastic hydrogel. They used finite element analysis and linear elasticity theory and calculated the Green’s function by applying unit tractions to each side on the surface of the cell mesh and solved SEPTEMBER 2014, Vol. 66 / 050804-5


Fig. 4 Sketch of the experimental configuration for the measurement of the three-dimensional deformation, where a z-stack of images, Dz, is acquired with the confocal microscope, and boundary conditions applied for the calculation of the traction forces in the three dimensions. This figure is taken from Ref. [67].

the finite element equations to calculate the induced bead displacements.

3

Motility Cycle Definition and Phase Identification

Generally, cell migration is a cyclic process [69]. When migration is triggered by a chemoattractant, cells elongate toward the direction of the movement and form a new protrusion [70]. The extended protrusions can be broad lamellipodia, spike-like filopodia [69], or pseudopods [71]. Amoeboid cells migrate by following a series of well-defined steps that result from periodic oscillations of the cell length and the strain energy exerted on the substrate on which they migrate. There are a series of wellestablished steps that are repeated in cell migration. The cyclic movement in amoeboid migration has been described in Dictyostelium cells [16,72], in leukocytes [73], and in neutrophils [74]. This movement consists of periodic repetitions of pseudopod protrusions and retractions of the cell’s rear [75]. The four phases in which the cycle can be divided are: protrusion, contraction, retraction, and relaxation [8]. Figure 5 shows the cyclic evolution of the cell length for a wild-type Dictyostelium cell, and the four stages in which the cycle is divided. For each of the stages, it shows the average traction forces exerted by this cell. To demonstrate the existence of a well-defined motility cycle, the degree of periodicity of the individual records for the length and the strain energy can be determined by fitting the data in a nonlinear, least squares sense with a sine wave and then calculating their cross correlation [76]. We calculate the role that the MyoII contractility and the dendritic F-actin polymerization play on the ability of the cells to perform the motility cycle periodically. In order to calculate the effect of MyoII, we study mlcE– cells, which are mutants lacking the MyoII motor activity and mhcA– cells, which are cells that lack the MyoII heavy chain. Note that mlcE– cells have the motor activity of MyoII mostly silenced [77]; however, MyoII can still cross-link F-actin [78] and mhcA– cells have both the motor and cross-linking functions of MyoII impaired. To study the importance of the dendritic F-actin polymerization, we examine mutant cells lacking the proteins from the SCAR/WAVE complex SCAR (scrA–) and PIR121 (pirA–). In order to compute the role of MyoII contractility and the dendritic F-actin polymerization on the periodicity of the motility cycle, we compared the temporal evolution of both the cell length 050804-6 / Vol. 66, SEPTEMBER 2014

Fig. 5 The central image shows the periodic evolution of the cell length over time, the black color indicates the protrusion phase, the red color indicates the contraction phase, the green colors indicate the retraction phase, and the blue color the relaxation phase. The surrounding images show a sketch of each of the cycle phases and the average stress map for this cell at each of the four phases of the motility cycle. This figure is taken from Ref. [8].

as well as the strain energy of wild-type cells and the mutant cell lines with defects in these specific processes. 3.1 Importance of Myosin II Contractility and F-Actin Polymerization in the Ability to Perform a Periodic Motility Cycle. MyoII does not seem to influence the cells ability to perform the motility cycle in a periodic manner since both mutant cell lines mlcE– and mhcA– cells can still perform the motility cycle periodically as wild-type cells [8]. On the other hand, cells lacking the protein PIR121 lose the periodicity in the execution of the motility cycle, while cells lacking the protein SCAR can perform the motility cycle periodically. For scrA– cells, the degree of periodicity (DOP) of the cell length evolution, defined as the correlation between the cell length and the best-fitting sine wave, it is similar to that of wild-type cells; whereas that for pirA– cells, is significantly lower [76]. The time evolution of the cell length for pirA– cells is not periodic and exhibits more random and uncoordinated oscillations [76]. Therefore, we concluded that the protein PIR121 of the SCAR/WAVE complex is necessary for the periodic execution of the motility cycle. 3.2 The Motility Cycle Period Determines the Migration Speed. The relationship between the average velocity of a cell is proportional to the frequency of the oscillations in its length and strain energy. Therefore, the translational velocity of a cell is determined by the frequency at which it is able to perform the motility cycle. This relation between the motility cycle period and the average velocity suggests the following: v¼

k ¼ kf T

(14)

in which k (step length) is a constant equal to the average length advanced by a cell in a cycle. This relationship holds for all the mutant cell lines that we have studied and which perform a motility cycle. However, the step length can vary depending on the cell line. The velocity-frequency slope in the mutant cell lines is smaller than in wild-type cells, which indicates that the mutant cells perform an average shorter step length per cycle as shown in Fig. 6. The average distance moved per cycle for wild-type cells is higher than the one moved by the mutant cell lines. For WT cells the constant k is equal to 18.5, whereas for mlcE– and mhcA– cells this constant k is equal to 16.5 while for scrA– mutants it is equal to 16.1 [8,76]. Figure 6 shows the relationship between the average velocity and the motility cycle frequency for multiple WT, mhcA–, mlcE–, and scrA– cells and the constant k for each cell line. 3.3 Phase Average Maps of Traction Forces. For each cell line we obtain statistically relevant information by calculating the Transactions of the ASME


Mj N P P

ðn; gÞ ¼

j¼1 k¼1

Wij ðtk Þ~ sj ðn; g; tk Þ

Mj N P P j¼1 k¼1

Fig. 6 Average velocity versus motility cycle frequency (determined from the variations of the cell length) shows a linear relationship in WT, mlcE , mhcA , and scrA cells. The blue, red, green, and black circles denote WT, scrA , mlcE , and mhcA cells, respectively. The velocity-frequency slope, k, for each of the cell lines is represented by the dotted lines, in blue and red for WT and scrA cells, respectively, and magenta for both mhcA and mlcE cells. This figure is a combination of figures from Refs. [8,76].

average traction force maps and average cell shapes in each of the phases of the motility cycle. The phase averages are calculated from the instantaneous maps of traction stresses after dividing the experimental time-lapse data and arranging each instant of time in the corresponding phase. The sorting procedure has three steps. First, we calculate the quasi-periodic time evolution of the length of the cell L(t). Second, a human user selects the peaks and valleys of the L(t) time history. Third, a computer algorithm automatically divides each cycle of L(t) into the following phases: • • • • •

phase 1: protrusion, during which the cell length is increasing. phase 2: contraction, during which the cell length is near a local maximum. phase 3: retraction, during which the cell length is decreasing. phase 4: relaxation, during which the cell length is near a local minimum. phase 1 if aðLmax Lmin Þ < LðtÞ Lmin < ð1 aÞðLmax Lmin Þ (15)

•

•

and tmin < t < tmax

(16)

jLðtÞ Lmax Þj < aðLmax Lmin Þ

(17)

where n and g are the spatial coordinates and Pðn; g; tk Þ is the instantaneous traction stress field generated by the cell jth at the time tk . The weight function Wi ðtk Þ is set equal to 1 when the jth cell is in the ith phase of the cycle and set equal to 0 otherwise. Before computing the phase average maps, we converted the instantaneous traction stress field into a cell-based, dimensionless coordinate system ðn; gÞ that takes into account that the shape and orientation of the cell changes with time. The cell-based representation involves aligning the longitudinal axis of the cell with the horizontal axis ðnÞ and rescaling the coordinates with the halflength of the cell as shown in Fig. 7(a). This coordinate system allows us to compile statistics with data that comes from different cells at different instants of time and calculate the average traction forces in a common reference system, as shown in Fig. 7(b). The origin of the cell-based coordinate system is located at the instantaneous centroid of the cell, whose instantaneous coordinates in the laboratory frame are ðxc ðtÞ; yc ðtÞÞ. The ðn; gÞ coordinates are n¼

½x xc ðtÞ cos uðtÞ þ ½y yc ðtÞ cos uðtÞ LðtÞ=2

(22)

g¼

½y yc ðtÞ cos uðtÞ þ ½x xc ðtÞ cos uðtÞ LðtÞ=2

(23)

where x and y are the coordinates in the laboratory reference frame and u is the angle between the longitudinal axis of the cell and the horizontal axis of the laboratory reference frame as shown in Fig. 7(a). Since the distance is scaled with the instantaneous half-length of the cell, LðtÞ=2, in the cell-based reference system, the length of the cell is always between n ¼ 1 and n ¼ 1. The traction stresses are scaled with ðLðtÞ=2Þ2 and have dimensions of force. To calculate the phase average contour of the cell, we applied the same procedure to a scalar function Pj ðn; g; tÞ so that at each instant of time P ¼ 1 inside the two-dimensional projection of the cell and P ¼ 0 outside of it.

phase 2 if

phase 3 if

4 Role of the F-Actin Polymerization and Actomyosin Contractility in the Generation of Traction Forces

and tmax < t < tmin

(19)

jLðtÞ Lmin Þj < aðLmax Lmin Þ

(20)

Using the method previously explained for the calculation of the phase average stress maps, where the average of the stresses and the average contour of the cell are calculated at each stage of the motility cycle, it is possible to perform a quantitative comparison of the contour and traction stress maps of wild-type and mutant cell lines. With these results, we can determine the contribution of each specific protein to the generation of the traction forces during the motility cycle.

where tmin and tmax are the instants of time associated with the nearest local minimum and maximum of L(t). To avoid overlaps between phases, it is necessary that a ranges between 0 and 0.5. We use a ¼ 0:2, but we have checked that the threshold used has a negligible effect in the results of the phase average traction maps. Once a phase has been assigned to each time point of our time-lapse experiments, we calculate the average maps of traction forces for each of the phases. Mathematically, we define the average map of traction forces corresponding to the ith phase of the motility cycle of a set of N cells, j ¼ 1,…, N, using Mj temporal observations for the jth cell, as

4.1 Myosin II Effect on the Generation of Traction Forces. The MyoII complex contains two heavy chains (MhcA) [79], two regulatory light chains (MlcR) [80], and two essential light chains (MlcE) [77]. Myosin II is a motor protein that plays an important role as a cytoskeletal crosslinker [81]. The motor activity is activated by the phosphorilation of the regulatory light chain [82] and requires the essential light chain [77]. The substrate used for the calculation of the traction forces exerted by mhcA– and mlcE– cells was an elastic gelatin gel [16]. In all the cell lines analyzed during all the phases of the cycle, the cells contract from the periphery inward toward the cell center.

aðLmax Lmin Þ < LðtÞ Lmin < ð1 aÞðLmax Lmin Þ (18)

•

(21) Wij ðtk Þ

phase 4 if


SEPTEMBER 2014, Vol. 66 / 050804-7


Fig. 7 (a) conversion of the instantaneous stress map of a cell into a cell-based reference system. x and y are the coordinates in the laboratory reference frame, n and g are the coordinates in the cell based reference frame. / is the angle between the longitudinal axis of the cell and the horizontal axis of the laboratory reference frame, L is the length of the cell, and xc and yc are the coordinates of the center of the cell in the laboratory reference frame. The arrow indicates the direction of the velocity, V , of the cell at this instant of time. (b) the first column indicates the calculation of the average traction forces in the cell-based reference frame for this cell at this instant of time. The origin is located at the center of the cell and the length of the cell is always between n ¼ 1 and n ¼ 1. The second and third columns indicate the components of the average traction forces parallel (x-axis component) and perpendicular (y-axis component) to the cell major axis, respectively.

Fig. 8 The upper row shows the traction forces exerted in each of the phases of the motility cycle by WT cells, the second row shows the traction forces exerted in each of the phases of the motility cycle by mlcE cells, and the third row shows the traction forces exerted in each of the phases of the motility cycle by mhcA cells. This figure is taken from Ref. [8].

The traction stresses generated by wild-type cells are concentrated in two well-defined areas in the front and the back of the cell [83]. The cell performs a simultaneous contraction of the substrate toward its center and the two regions where the traction forces are 050804-8 / Vol. 66, SEPTEMBER 2014

produced probably correspond to the regions where the cell adheres to the substrate observed by Weber et al. [84]. The overall spatial distribution of stresses in mlcE– cells is similar to that of wild-type cells, although their magnitude is lower by Transactions of the ASME


Table 1 Pole forces and elastic energy values for WT, mlcE , and mhcA cells migrating over gelatin substrates, average values, and standard deviation. The pole forces are calculated in pN, and the strain energy in nN lm.

Table 2 Pole forces and elastic energy values calculated for WT, pirA , and scrA cell lines migrating over polyacrylamide substrates, average values, and standard deviation. The pole forces are calculated in pN and the strain energy in nN lm.

Cell type

Pole force (pN)

Strain energy (nN lm)

Type

Pole force (pN)

Strain energy (nN lm)

WT mlcE– mhcA–

387.5 6 305.6 348.5 6 132.9 284.68 6 159.9

0.0681 6 0.0946 0.0382 6 0.0191 0.0488 6 0.05

WT pirA– scrA–

1080 6 440 1240 6 680 600 6 280

0.64 6 0.35 0.860 6 0.41 0.230 6 0.11

a factor of approximately 2 [8], as shown in Fig. 8. The stresses produced by mhcA– cells are similar in magnitude to those of mlcE– cells, but in mhcA– cells the stress pattern is not focused in two separate areas. In addition, it is located closer to the cell boundaries than in wild-type or mlcE– cells. The difference between the mhcA– and mlcE– cells suggests that the lack of organization in the stress patterns of the mhcA– cells may result from the loss of the cross-linking function of MyoII. Therefore, the MyoII cross-linking function is crucial in regulating the spatial organization of the traction forces. Table 1 shows the values of the pole forces and strain energy for WT, mlcE–, and mhcA– cells moving over gelatin substrates. The pole forces and strain energy exerted by the mutants cell lines are lower than the ones exerted by WT cells. Figure 9 shows the decomposition of the traction forces exerted by WT, mlcE–, and mhcA– cells into their components parallel and perpendicular to the main cell axis (direction of motion). These results show that in WT and mlcE– cells, the contribution of the parallel stress component to the total stress field is much higher than the contribution of the perpendicular component. However, in mhcA– cells, the magnitude of the lateral component is similar to the magnitude of the axial component [8]. Phase averaged forces exerted during each stage of the motility cycle show that the overall time evolutions of the stress patterns during the phases of the motility cycle are similar: they are minimal during the relaxation phase, increase during protrusion, reach their maximum during contraction, and decrease during retraction. These results support the conclusion that WT, mlcE–, and mhcA– cells move by implementing similar motility cycles. There is a marked difference in the durations of each of the four phases. In wild-type cells, all the phases are considerably shorter than in mhcA– and mlcE– cells, which suggests that the contractile function of MyoII is an important factor in determining the overall speed [8]. During protrusion, mhcA– cells produce comparatively low, spread-out traction stresses at their rears compared to WT cells. During retraction, the shape of wild-type and mlcE– cells

becomes much wider at the front than at the back, whereas in mhcA– cells, the width is similar at the front and back. Traction forces exerted by randomly migrating WT, mhcA–, and mlcE– calculated by Lombardi et al. [85] using the method developed by Dembo et al. [51] present an asymmetry that may be important to the development of a rapid directional movement. They measure high tractions in the rear of the cells and low tractions in the front. Their study shows that the rate at which the traction force asymmetry is developed is determinant to the speed of the cell. The fact that the force distribution of randomly moving cells is asymmetric with largest forces in the uropod and little in the anterior pseudopod could be related to the roles suggested for myosin-II consisting of the cell’s rear part squeezing and detachment of their posterior adhesions from the substrate [86]. These results are different from what we observed, since we find equilibrium of forces and the pole forces that the cell exerts in the front are equal in magnitude to the ones exerted in the rear part. The difference could be due to the fact that we are studying chemotactic cells that are moving directionally toward a chemoattractant and the mechanics of force generation could be different in the preaggregating and aggregating stages of Dictyostelium. 4.2 Effect on the Traction Forces and F-Actin Levels of the SCAR and PIR121 Proteins, Members of the SCAR/ WAVE Complex. The SCAR/WAVE pentameric complex is highly conserved in Dictyostelium and includes the proteins PIR121 (Sra-1/CYFIP/GEX-2), SCAR (WAVE), HSPC300, ABI1, and NAP1 (Hem2/KETTE/GEX-3). The SCAR/WAVE proteins activate the Arp2/3 actin polymerization [87–89] and catalyze the formation of new actin filaments from actin monomers under tight temporal and spatial control from multiple intracellular signals [90]. In the absence of the SCAR/WAVE proteins, the formation of pseudopods is greatly diminished [87]. We have investigated the role of the proteins SCAR and PIR121 of the SCAR/WAVE complex in the generation of the traction forces exerted by Dictyostelium cells during chemotactic migration.

Fig. 9 The upper row shows the component of the traction forces exerted in the direction of the major axis of the cell by WT, mlcE , and mhcA cells. The second row shows the component of the traction forces exerted in the direction perpendicular to the major axis of the cell by WT, mlcE , and mhcA cells. This figure is taken from Ref. [8].


SEPTEMBER 2014, Vol. 66 / 050804-9


Fig. 10 The first row shows the traction forces exerted in each of the phases of the cycle by WT cells. The second row shows the traction forces exerted in each of the phases of the cycle by scrA mutant cells. This figure is taken from Ref. [76].

Fig. 11 The upper row shows the component of the traction forces exerted in the direction of the major axis of the cell by WT, pirA , and scrA cells moving over polyacrilamide substrates. The second row shows the component of the traction forces exerted in the perpendicular direction to the major axis of the cell by WT, pirA , and scrA cells. This figure is taken from Ref. [76].

The substrate on which the experiments with scrA– and pirA– cells were performed was an elastic polyacrylamide gel [76]. The organization pattern of the traction forces in the WT, scrA– and pirA– cells is polar. The three cell lines attach at the front and the back and contract inward. The traction forces measured in the polyacrylamide gels are bigger than the ones measured in the gelatin gels. This may be due to an increased attachment on these substrates with respect to the adhesion on the gelatin ones. We measured the traction forces exerted by scrA– and pirA– mutants in order to assess the role of F-actin polymerization. Table 2 shows the values of the pole forces and strain energy exerted by WT, pirA–, and scrA– cells moving over polyacrylamide substrates. The pole forces and strain energy generated by pirA– cells are slightly higher than in WT cells, whereas in scrA– cells these values are remarkably diminished compared to WT cells. We calculated the traction forces in each of the phases of the motility cycle for the scrA– mutant cells and found that the traction patterns are similar in all the phases to the wild-type cells as 050804-10 / Vol. 66, SEPTEMBER 2014

shown in Fig. 10, suggesting that the mechanics of migration of scrA– cells is similar to that of wild-type cells [76]. The pirA– cells do not move periodically; therefore the decomposition of the cycle in phases is not relevant. We also found that in WT cells, as well as in pirA– and scrA– cells, the average stress pattern consists of two pole forces localized at the front and back of the cell. The magnitude of the traction stresses is approximately 50% weaker in scrA– cells whereas in pirA– cells it is slightly larger than in wild-type cells [76]. Figure 11 shows that although the stress patterns for all cell lines are similar, the leading edge of both SCAR/WAVE mutant cell lines exerts negligible traction stresses, and the location of the maximum in the frontal traction stresses in the mutant cells is shifted toward the center of the cell away from the leading edge compared to WT cells. The spatiotemporal distribution of F-actin measured by using the F-actin reporter Lifeact shows that the strength of the traction stresses for all cell lines correlates very well with the specific F-actin level of each cell line [76]. Transactions of the ASME


5 Validation of the 2D Measurements of the Traction Forces with the Improved 3D Method Measurement of the perpendicular traction forces to the substrate reveals that this out-of-plane component of the traction forces is as large as the in-plane component or even larger, and therefore it cannot be neglected. The 2D method is not accurate when calculating the tangential traction forces due to the assumption that szz ¼ 0. The error made when applying the 2D method to the measurement of the tangential traction forces is generally high. We found that the error made in the calculation of the tangential traction forces with the 2D method depends on the value of several parameters. The main parameters that determine the differences between the tangential forces measured with the 2D and 3D methods are the ratio between the cell length and substrate thickness, D=h, the Poisson ratio of the substrate, r, and the ratio between the magnitudes of the in-plane to out-of-plane displacements generated by the cells in the substrate, W0 =U0 . However, there are two experimental configurations in which the error made with the 2D method is small, depending on the ratio between the cell length and substrate thickness and the value of the Poisson ratio. This is the case when the Poisson ratio, r, is close to 0.5, and the ratio between cell length and substrate thickness, D=h, is less than 1. Another configuration that gives small differences is when D=h is large, 100 or larger, and r is lower than 0.3. Regarding the ratio between the magnitude of out-of-plane displacements and in-plane displacements magnitude, there are specific cases in which the accuracy of the measurements with the 2D method is good. This condition occurs when the ratio between the magnitude

of the out-of-plane and in-plane displacement fields, W0 =U0 , is very low. One case is when W0 =U0 < 0:2 and D=h < 10 and another case is when r is close to 0.5 [67]. In our experiments, r is close to 0.5 and D=h is lower than 1. In this manner, these conditions are included into one of the cases in which the error is not large. To obtain an estimation of the error made with our 2D method for the tangential traction forces exerted by Dictyostelium cells moving over the polyacrylamide substrates of approximately 40 lm thickness, we calculated the tangential forces with our 3D method and compare them to the 2D method. With our 3D method, we calculate all the components of the stress tensor and the stresses exerted on the surface of the substratum are obtained through the Cauchy law s ¼ T ~ n

(24)

where T is the three-dimensional stress tensor and ~ n is the unit vector normal to the surface of the substrate. The surface of the gel when it is deformed by a cell exerting forces over it is given by z ¼ h þ wðx; y; hÞ (25) And the vector normal to the deformed surface is ð@x w; @y w; 1Þ ~ n ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð@x wÞ2 þ ð@y wÞ2 þ 1

(26)

Fig. 12 (a) horizontal traction forces obtained by using the 3D method, (b) horizontal traction forces obtained by using the 2D method, and (c) difference between the horizontal traction forces calculated by using the 3D and 2D methods. The red color indicates that the traction forces calculated with the 3D method are bigger than the ones calculated with the 2D method, and the blue color indicates the opposite, that the traction forces calculated with the 3D method are lower than the ones calculated with the 2D method.


SEPTEMBER 2014, Vol. 66 / 050804-11


Ust ¼

ðL ðW 0

Fig. 13 Time evolution of the tangential strain energy obtained with the 3D method in blue and time evolution of the total strain energy obtained with the 2D method in red

nT T ~ nÞ~ n ~ sn ¼ ð~

(27)

~ st ¼ ðT ~ nÞ ~ sn

(28)

In our experiments, the values of @x w and @y w are small, ranging between 0.05 and 0.1. Thus, ~ n ’ ð0; 0; 1Þ, and the normal and tangential stresses can be approximated by sn ¼ ð0; 0; szz Þ and st ¼ ðsxz ; syz ; 0Þ. With the results obtained for the traction forces in the tangential direction to the substrate, st , using our 3D method, we can calculate the accuracy of the measurements that we obtained with our 2D method. Figures 12(a) and 12(b) show the average traction forces in the horizontal plane for a wild-type cell that are obtained with our 3D and 2D methods. Figure 12(c) shows the error that is made by using the 2D method for this cell at each point. The 2D method underestimates the axial forces, parallel to the cell’s major axis, and overestimates the lateral forces, perpendicular to the cell’s major axis. The tangential q stresses ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiare ffi calculated in both the 2D and 3D s2xz þ s2yz . The error in the tangential stresses

method as: st ¼

calculated with the 2D method can be defined as follows: 2ð ð 6 errort ¼ 6 4

31=2 2 3D ½s2D t ðx; y; hÞ st ðx; y; hÞ dxdy7 7 100 ðð 5 2 ðx; y; hÞ dxdy ½s3D t

(29)

This definition of the error is normalized in a way that errort would be 100% if s2D is 0 and s3D is different than 0. The error t t that is made for this cell in the tangential traction stresses by using the 2D method is 21.2%. The strain energy exerted by the cell, when calculating the stresses in the three directions, is defined as Us total ¼

ðL ðW 0

½sxz u þ syz v þ szz wdxdy

(30)

0

where L is the length of the cell and W is the width of the cell. This total strain energy can be decomposed in its normal, Usn , and tangential components, Ust , in order to compare the results for the tangential strain energy with the total strain energy calculated with the 2D method. The tangential component of the strain energy is defined as 050804-12 / Vol. 66, SEPTEMBER 2014

(31)

This definition is equivalent to the definition of the total strain energy exerted by the cell when using our 2D method. Figure 13 shows the evolution of the total elastic energy calculated by using the 2D method and the tangential strain energy calculated with the 3D method over time for the specific wild-type cell. In the specific case of our experimental conditions, the measurement for the total strain energy calculated with our 2D method is not very different from the tangential strain energy calculated with our 3D method, mainly because the cell length is smaller than the substrate thickness. This may not be the case for a different type of cell and a different substrate. However, the total magnitude of the strain energy, that also takes into account its normal component, is much larger than the total strain energy computed with the 2D method, and the error in the magnitude is around 80%.

6 The stresses in the perpendicular, sn , and tangential, st , directions to the deformed substrate are obtained following Cauchy’s law:

½sxz u þ syz vdxdy

0

Conclusions

Several traction force cytometry methods have been developed to measure the traction forces exerted by migrating cells. Most of them are based on the measurement of a substrate deformation induced by the cells. This review is focused on the traction forces exerted by migrating Dictyostelium cells measured using the Fou rier Traction Cytometry method developed by del Alamo et al. [16]. The oscillation of the cell’s length and the strain energy in Dictyostelium cells is cyclic, and the cycle is composed of four distinguished stages. The proper polymerization of the actin filaments in the leading edge of migrating cells regulated by the proteins SCAR and PIR121, members of the SCAR/WAVE complex, and the Myosin II contractility are critical regulators in the mechanics of cell migration. They are implicated in the periodic performance of the migration cycle and in the generation and organization of the traction forces. The protein PIR121 included in the SCAR/WAVE complex that activates the Arp2/3 complex, responsible for the branch nucleation of new actin filaments in the leading edge of the cell, is essential for the periodic repetition of the cycle phases. pirA– mutant cells cannot execute the motility cycle periodically. However, mutant cells lacking the protein SCAR included in the SCAR/ WAVE complex too, are able to execute the motility cycle in a periodic manner. The motor protein MyoII is essential to the generation of the traction forces. The cross-linking function of MyoII is responsible for the front to back organization of the traction forces. mhcA– mutants do not exhibit a polar organization of the traction forces whereas mlcE– mutant cells conserved this polar organization characteristic of WT cells. These findings show that the crosslinking function of this protein is essential for the proper organization of the traction forces. The SCAR protein is essential to the generation of traction forces. scrA– mutant cells exert much lower traction forces than WT cells, whereas the protein PIR121 does not affect the generation of traction forces in a significant manner.

Acknowledgment This work was supported by NIH Institute of General Medical Sciences{FundingSource} 5R01GM084227. BAG acknowledges the support of Ibercaja foundation and the Rita L. Atkinson Graduate Fellowship. EB acknowledges the support of the NHLBI Training Grant on Integrative Bioengineering of Heart, Vessels and Blood.

Nomenclature MyoII ¼ myosin II mhcA– ¼ myosin II heavy chain null cells Transactions of the ASME


mlcE– ¼ myosin II essential light chain null cells pirA– ¼ PIR121 null cells scrA– ¼ SCAR null cells

References [1] Ausprunk, D. H., and Folkman, J., 1977, “Migration and Proliferation of Endothelial Cells in Preformed and Newly Formed Blood Vessels During Tumor Angiogenesis,” Microvasc. Res., 14(1), pp. 53–65. [2] Bagorda, A., Mihaylov, V., and Parent, C. A., 2006, “Chemotaxis: Moving Forward and Holding on to the Past,” Thromb. Haemos., 95(1), pp. 12–21. [3] Cooper, G. M., and Hausman, R. E., 1997, The Cell: A Molecular Approach, Sinauer Associates, Sunderland, MA. [4] Lammermann, T., and Six, M., 2009, “Mechanical Modes of ‘Amoeboid’ Cell Migration,” Curr. Opin. Cell Biol., 21, pp. 636–644. [5] Friedl, P., and Wolf, K., 2009, “Plasticity of Cell Migration: A Multiscale Tunning Model,” J. Cell Biol., 188(1), pp. 11–19. [6] Huttenlocher, A., and Horwitz, A. R., 2011, “Integrins in Cell Migration,” Cold Spring Harbor Perspectives in Biology, 3(9), p. a005074. [7] Mannherz, H. G., Mach, M., Nowak, D., Malicka-Blaszkiewicz, M., and Mazur, A., 2007, “Lamellipodial and Amoeboid Cell Locomotion: The Role of ActinCycling and Bleb Formation,” Biophys. Rev. Lett., 2(1), pp. 5–22. [8] Meili, R., Alonso-Latorre, B., del Alamo, J. C., Firtel, R. A., and Lasheras, J. C., 2010, “Myosin II is Essential for the Spatiotemporal Organization of Traction Forces During Cell Motility,” Mol. Biol. Cell, 21, pp. 405–417. [9] Smith, L. A., Aranda-Espinoza, H., Haun, J. B., Dembo, M., and Hammer, D. A., 2007, “Neutrophil Traction Stresses are Concentrated in the Uropod During Migration,” Biophys. J., 92(7), pp. 58–60. [10] Shin, M. E., He, Y., Li, D., Na, S., Chowdhury, F., Poh, Y.-C., Collin, O., Su, P., de Lanerolle, P., Schwartz, M. A., Wang, N., and Wang, F., 2010, “Spatiotemporal Organization, Regulspatiotemporal Organization, Regulation, and Functions of Tractions During Neutrophil Chemotaxis,” Blood, 116, pp. 3297–3310. [11] Beningo, K. A., Dembo, M., Kaverina, I., Small, J. V., and Wang, Y.-L., 2001, “Nascent Focal Adhesions are Responsible for the Generation of Strong Propulsive Forces in Migrating Fibroblasts,” J. Cell Biol., 153, pp. 881–888. [12] Friedl, P., Borgmann, S., and Brocker, E.-B., 2001, “Amoeboid Leukocyte Crawling Through Extracellular Matrix: Lessons from the Dictyostelium Paradigm of Cell Movement,” J. Leukocyte Biol., 70(4), pp. 491–509. [13] Jannat, R. A., Dembo, M., and Hammer, D. A., 2011, “Traction Forces of Neutrophils Migrating on Compliant Substrates,” Biophys. J., 101, pp. 575–584. [14] Ricart, B. G., Yang, M. T., Hunter, C. A., Chen, C. S., and Hammer, D. A., 2011, “Measuring Traction Forces of Motile Dendritic Cells on Micropost Arrays,” Biophys. J., 101, pp. 2620–2628. [15] Lammermann, T., Bader, B. L., Monkley, S. J., Worbs, T., Wedlich-Soldner, R., Hirsch, K., Keller, M., Forster, R., Critchley, D. R., Fassler, R., and Six, M., 2008, “Rapid Leukocyte Migration by Integrin-Independent Flowing and Squeezing,” Nature, 453, pp. 51–55. [16] del Alamo, J. C., Meili, R., Alonso-Latorre, B., Rodriguez-Rodriguez, J., Aliseda, A., Firtel, R. A., and Lasheras, J. C., 2007, “Spatiotemporal Analysis of Eukaryotic Cell Motility by Improved Force Cytometry,” Proc. Natl. Acad. Sci., 104(33), pp. 13343–13348. [17] Stricker, J., Falzone, T., and Gardel, M. L., 2010, “Mechanics of the F-Actin Cytoskeleton,” J. Biomech., 43, pp. 9–14. [18] Spiering, D., and Hodgson, L., 2011, “Dynamics of the Rho-Family Small Gtpases in Actin Regulation and Motility,” Cell Adhes. Migrat., 5(2), pp. 170–180. [19] Sasaki, A. T., and Firtel, R. A., 2006, “Regulation of Chemotaxis by the Orchestrated Activation of RAS, PI3K, and TOR,” Eur. J. Cell Biol., 85, pp. 873–895. [20] Fey, P., Stephens, S., Titus, M., and Chisholm, R., 2002, “Sada, A Novel Adhesion Receptor in Dictyostelium,” J. Cell Biol., 159, pp. 1109–1119. [21] Uchida, K., and Yumura, S., 2004, “Dynamics of Novel Feet of Dictyostelium Cells During Migration,” J. Cell Sci., 117, pp. 1443–1455. [22] Friedl, P., Entschladen, F., Conrad, C., Niggemann, B., and Z€anker, K., 1998, “Cd4þ t Lymphocytes Migrating in Three-Dimensional Collagen Lattices Lack Focal Adhesions and Utilize Beta1 Integrin-Independent Strategies for Polarization, Interaction With Collagen Fibers and Locomotion,” Eur. J. Immunol., 28(8), pp. 2331–2343. [23] Ananthakrishnan, R., and Ehrlicher, A., 2007, “The Forces Behind Cell Movement,” Int. J. Biol. Sci., 3(5), pp. 303–317. [24] Li, B., and Wang, J. H.-C., 2010, “Application of Sensing Techniques to Cellular Force Measurement,” Sensors, 10, pp. 9948–9962. [25] Li, B., Xie, L., Starr, Z. C., Yang, Z., Lin, J.-S., and Wang, J. H.-C., 2007, “Development of Micropost Force Sensor Array With Culture Experiments for Determination of Cell Traction Forces,” Cell Motil. Cytoskeleton, 64, pp. 509–518. [26] Mathur, A., Roca-Cusachs, P., Rossier, O. M., Wind, S. J., Sheetz, M. P., and Hone, J., 2011, “New Approach for Measuring Protrusive Forces in Cells,” J. Vacuum Sci. Technol. B, 29(6), 06FA02. [27] Han, S. J., Bielawski, K. S., Ting, L. H., Rodriguez, M. L., and Sniadecki, N. J., 2012, “Decoupling Substrate Stiffness, Spread Area, and Micropost Density: A Close Spatial Relationship Between Traction Forces and Focal Adhesions,” Biophys. J., 103, pp. 640–648. [28] Yang, M. T., Fu, J., Wang, Y.-K., Desai, R. A., and Chen, C. S., 2011, “Assaying Stem Cell Mechanobiology on Microfabricated Elastomeric


[29]

[30]

[31]

[32] [33]

[34]

[35]

[36] [37]

[38]

[39]

[40] [41] [42] [43]

[44] [45]

[46] [47]

[48]

[49]

[50] [51]

[52] [53]

[54]

[55]

[56]

[57] [58] [59]

Substrates with Geometrically Modulated Rigidity,” Nat. Protoc., 6(2), pp. 187–213. Sniadecki, N. J., Lamb, C. M., Liu, Y., Chen, C. S., and Reich, D. H., 2008, “Magnetic Microposts for Mechanical Stimulation of Biological Cells: Fabrication, Characterization, and Analysis,” Rev. Sci. Instrum., 79, p. 044302. Lin, Y.-C., Kramer, C. M., Chen, C. S., and Reich, D. H., 2012, “Probing Cellular Traction Forces With Magnetic Nanowires and Microfabricated Force Sensor Arrays,” Nanotechnology, 23, p. 075101. McGarry, J. P., Fu, J., Yang, M. T., Chen, C. S., McMeeking, R. M., Evans, A. G., and Deshpande, V. S., 2009, “Simulation of the Contractile Response of Cells on an Array of Micro-Posts,” Philos. Trans. R. Soc. A, 367(1902), pp. 3477–3497. Wang, J. H.-C., and Lin, J.-S., 2007, “Cell Traction Force and Measurement Methods,” Biomech. Model. Mechanobiol., 6, pp. 361–371. Schwarz, U., Balaban, N., Riveline, D., Addadi, L., Bershadsky, A., Safran, S., and Geiger, B., 2003, “Measurement of Cellular Forces at Focal Adhesions Using Elastic Micro-Patterned Substrates,” Mater. Sci. Eng., 23(3), pp. 387–394. Reinhart-King, C. A., Dembo, M., and Hammer, D. A., 2003, “Endothelial Cell Traction Forces on RGD-Derivatized Polyacrylamide Substrata,” Langmuir, 19(5), pp. 1573–1579. Han, S. J., and Sniadecki, N. J., 2011, “Simulations of the Contractile Cycle in Cell Migration Using a Bio-Chemical-Mechanical Model,” Comput. Methods Biomech. Biomed. Eng., 14(5), pp. 459–468. Banerjee, S., and Marchetti, M. C., 2013, “Controlling Cell–Matrix Traction Forces by Extracellular Geometry,” New J. Phys., 15, p. 035015. Zielinski, R., Mihai, C., Kniss, D., and Ghadiali, S. N., 2013, “Finite Element Analysis of Traction Force Microscopy: Influence of Cell Mechanics, Adhesion, and Morphology,” ASME J. Biomech. Eng., 135(7), p. 071009. Holmes, W. R., and Edelstein-Keshet, L., 2012, “A Comparison of Computational Models for Eukaryotic Cell Shape and Motility,” PLOS Comput. Biol., 8(12), e1002793. Devreotes, P. N., and Zigmond, S. H., 1988, “Chemotaxis in Eukaryotic Cells: A Focus on Leukocytes and Dictyostelium,” Ann. Rev. Cell Biol., 4, pp. 649–686. Charest, P. G., and Firtel, R. A., 2007, “Big Roles for Small Gtpases in the Control of Directed Cell Movement,” Biochem. J., 401, pp. 377–390. Zigmond, S. H., 1993, “Recent Quantitative Studies of Actin Filament Turnover During Cell Locomotion,” Cell Motil. Cytoskeleton, 25, pp. 3309–3016. Borisy, G. G., and Svitkina, T. M., 2000, “Actin Machinery: Pushing the Envelope,” Curr. Opin. Cell Biol., 12, pp. 104–112. Mullins, R. D., Heuser, J. A., and Pollard, T. D., 1998, “The Interaction of arp2y3 Complex with Actin: Nucleation, High Affinity Pointed End Capping, and Formation of Branching Networks of Filaments,” Proc. Natl. Acad. Sci., 95, pp. 6181–6186. Pollard, T. D., 2007, “Regulation of Actin Filament Assembly by arp2/3 Complex and Formins,” Ann. Rev. Biophys. Biomol. Struct., 36, pp. 451–477. Weiner, O. D., Servant, G., Welch, M. D., Mitchison, T. J., Sedat, J. W., and Bourne, H. R., 1999, “Spatial Control of Actin Polymerization During Neutrophil Chemotaxis,” Nat. Cell Biol., 1, pp. 75–81. Spudich, J. A., 1989, “In Pursuit of Myosin Function,” Cell Regul., 1, pp. 1–11. Stites, J., Wessels, D., Uhl, A., Egelhoff, T., Shutt, D., and Soll, D. R., 1998, “Phosphorylation of the Dictyostelium Myosin ii Heavy Chain is Necessary for Maintaining Cellular Polarity and Suppressing Turning During Chemotaxis,” Cell Motil. Cytoskeleton, 39, pp. 31–51. Yumura, S., Mori, H., and Fukui, Y., 1984, “Localization of Actin and Myosin for the Study of Ameboid Movement in Dictyostelium Using Improved Immunofluorescence,” J. Cell Biol., 99(3), pp. 894–899. Fukui, Y., and Yumura, S., 1986, “Actomyosin Dynamics in Chemotactic Amoeboid Movement of Dictyostelium,” Cell Motil. Cytoskeleton, 6, pp. 662–673. Moores, S. L., Sabry, J. H., and Spudich, J. A., 1996, “Myosin Dynamics in Live Dictyostelium Cells,” Proc. Natl. Acad. Sci., 93, pp. 443–446. Dembo, M., Oliver, T., Ishihara, A., and Jacobson, K., 1996, “Imaging the Traction Stresses Exerted by Locomoting Cells With Theelastic Substratum Method,” Biophys. J., 70, pp. 2008–2022. Dembo, M., and Wang, Y.-L., 1999, “Stresses at the Cell-to-Substrate Interface During Locomotion of Fibroblasts,” Biophys. J., 76, pp. 2307–2316. Butler, J. P., Tolic´-Nørrelykke, I. M., Fabry, B., and Fredberg, J. J., 2002, “Traction Fields, Moments, and Strain Energy that Cells Exert on Their Surroundings,” Am. J. Cell Physiol., 282, pp. 595–605. Yang, Z., Lin, J.-S., Chen, J., and Wang, J. H.-C., 2006, “Determining Substrate Displacement and Cell Traction Fields-A New Approach,” J. Theor. Biol., 242, pp. 607–616. Wang, Y. L., and, Pelham, R. J., Jr., 1998, “Preparation of a Flexible, Porous Polyacrylamide Substrate for Mechanical Studies of Cultured Cells,” Methods Enzymol., 298, pp. 489–496. Engler, A., Bacakova, L., Newman, C., Hategan, A., Griffin, M., and Discher, D., 2004, “Substrate Compliance Versus Ligand Density in Cell on Gel Responses,” Biophys. J., 86, pp. 617–628. Alonso-Latorre, B., 2010, “Force and Shape Coordination in Amoeboid Cell Motility,” Ph.D. thesis, University of California, San Diego, CA. Keer, L. M., 1964, “Stress Distribution at the Edge of an Equilibrium Crack,” J. Mech. Phys. Solids, 12(3), pp. 149–163. Chippada, U., Yurke, B., and Langrana, N. A., 2011, “Simultaneous Determination of Young’s Modulus, Shear Modulus, and Poisson’s Ratio of Soft Hydrogels,” J. Mater. Res., 25(3), pp. 545–555.

SEPTEMBER 2014, Vol. 66 / 050804-13


[60] Takigawa, T., Morino, Y., Urayama, K., and Masudab, T., 1996, “Poisson’s Ratio of Polyacrylamide (Paam) Gels,” Polym. Gels Networks, 4(1), pp. 1–5. [61] Li, Y., Hu, Z., and Li, C., 1993, “New Method for Measuring Poisson’s Ratio in Polymer Gels,” Appl. Polym. Sci., 50(6), pp. 1107–1111. [62] Willert, C. E., and Gharib, M., 1991, “Digital Particle Image Velocimetry,” Exp. Fluids, 10(4), pp. 181–193. [63] Gui, L., and Wereley, S. T., 2002, “A Correlation-Based Continuous WindowShift Technique to Reduce the Peak-Locking Effect in Digital PIV Image Evaluation,” Exp. Fluids, 32, pp. 506–517. [64] Hur, S. S., Zhao, Y., Li, Y.-S., Botvinick, E., and Chien, S., 2009, “Live Cells Exert 3-Dimensional Traction Forces on Their Substrata,” Cell. Mol. Bioeng., 2(3), pp. 425–436. [65] Franck, C., Maskarinec, S. A., Tirrell, D. A., and Ravichandran, G., 2011, “Three-Dimensional Traction Force Microscopy: A New Tool for Quantifying Cell-Matrix Interactions,” PLOS ONE, 6(3), 317833. [66] Delano€e-Ayari, H., and Rieu, J. P., 2010, “4D Traction Force Microscopy Reveals Asymmetric Cortical Forces in Migrating Dictyostelium Cells,” Phys. Rev. Lett., 105(24), p. 248103. [67] del Alamo, J. C., Meili, R., Alvarez-Gonz alez, B., Alonso-Latorre, B., Bastounis, E., Firtel, R., and Lasheras, J. C., 2013, “Three-Dimensional Quantification of Cellular Traction Forces and Mechanosensing of Thin Substrata by Fourier Traction Force Microscopy,” PLOS ONE, 8(9), e69850. [68] Legant, W. R., Miller, J. S., Blakely, B. L., Cohen, D. M., Genin, G. M., and Chen, C. S., 2010, “Measurement of Mechanical Tractions Exerted by Cells in Three-Dimensional Matrices,” Nat. Methods, 7(2), pp. 969–971. [69] Ridley, A. J., Schwartz, M. A., Burridge, K., Firtel, R. A., Ginsberg, M. H., Borisy, G., Parsons, J. T., and Horwitz, A. R., 2003, “Cell Migration: Integrating Signals from Front to Back,” Science, 302, pp. 1704–1709. [70] Bailly, M., Condeelis, J. S., and Segall, J. E., 1998, “Chemoattractant-Induced Lamellipod Extension,” Microscop. Res. Tech., 43(5), pp. 433–443. [71] Zhelev, D. V., Alteraifi, A. M., and Chodniewicz, D., 2004, “Controlled Pseudopod Extension of Human Neutrophils Stimulated With Different Chemoattractants,” Biophys. J., 87(1), pp. 688–695. [72] Wessels, D., Vawter-Hugart, H., Murray, J., and Soll, D. R., 1994, “ThreeDimensional Dynamics of Pseudopod Formation and the Regulation of Turning During the Motility Cycle of Dictyostelium,” Cell Motil. Cytoskeleton, 27, pp. 1–12. [73] Murray, J., Vawter-Hugart, H., Voss, E., and Soll, D. R., 1992, “Three-Dimensional Motility Cycle in Leukocytes,” Cell Motil. Cytoskeleton, 22, pp. 211–223. [74] Ehrengruber, M. U., Deranleau, D. A., and Coates, T. D., 1996, “Shape Oscillations of Human Neutrophil Leukocytes: Characterization and Relationship to Cell Motility,” J. Exp. Biol., 199, pp. 741–747. [75] Lauffenburger, D. A., and Horwitz, A. F., 1996, “Cell Migration: A Physically Integrated Molecular Process,” Cell, 84, pp. 359–369. [76] Bastounis, E., Meili, R., Alonso-Latorre, B., del Alamo, J. C., Lasheras, J. C., and Firtel, R. A., 2011, “The Scar/Wave Complex is Necessary for Proper

050804-14 / Vol. 66, SEPTEMBER 2014

[77]

[78]

[79]

[80]

[81]

[82]

[83]

[84]

[85]

[86]

[87]

[88] [89]

[90]

Regulation of Traction Stresses During Amoeboid Motility,” Mol. Biol. Cell, 22, pp. 3995–4003. Chen, T., Kowalczyk, P., Ho, G., and Chisholm, R., 1995, “Targeted Disruption of the Dictyostelium Myosin Essential Light Chain Gene Produces Cells Defective in Cytokinesis and Morphogenesis,” J. Cell Sci., 108, pp. 3207–3218. Xu, X. S., Lee, E., Chen. T., Kuczmarski, E., Chisholm, R. L., and Knecht, D. A., 2001, “During Multicellular Migration, Myosin II Serves a Structural Role Independent of its Motor Function,” Develop. Biol., 232, pp. 255–264. Lozanne, A. D., and Spudich, J. A., 1987, “Disruption of the Dictyostelium Myosin Heavy-Chain Gene by Homologous Recombination,” Science, 237(4805), pp. 1086–1091. Chen, P., Ostrow, B., Tafuri, S., and Chisholm, R., 1994, “Targeted Disruption of the Dictyostelium RMLC Gene Produces Cells Defective in Cytokinesis and Development,” J. Cell Biol., 127, pp. 1933–1944. Laevsky, G., and Knecht, D. A., 2003, “Cross-Linking of Actin Filaments by Myosin II is a Major Contributor to Cortical Integrity and Cell Motility in Restrictive Environments,” J. Cell Sci., 116, pp. 3761–3770. Griffith, L. M., Downs, S. M., and Spudich, J. A., 1987, “Myosin Light Chain Kinase and Myosin Light Chain Phosphatase from Dictyostelium: Effects of Reversible Phosphorylation on Myosin Structure and Function,” J. Cell Biol., 104(5), pp. 1309–1323. Alonso-Latorre, B., del Alamo, J., Meili, R., Firtel, R., and Lasheras., J., 2011, “Strain Energy Modes in Migrating Amoeboid Cells,” J. Cell. Mol. Biol., 4(4), pp. 603–615. Weber, I., Wallraff, E., Albrecht, R., and Gerisch, G., 1995, “Motility and Substratum Adhesion of Dictyostelium Wild-Type and Cytoskeletal Mutant Cells: A Study by Ricm/Bright-Field Double-View Image Analysis,” J. Cell Sci., 108, pp. 1519–1530. Lombardi, M. L., Knecht, D. A., Dembo, M., and Lee, J., 2007. “Traction Force Microscopy in Dictyostelium Reveals Distinct Roles for Myosin Ii Motor and Actin Crosslinking Activity in Polarized Cell Movement,” J. Cell Sci., 120, pp. 1624–1634. Delanoe-Ayari, H., Iwaya, S., Maeda, Y. T., Inose, J., Riviere, C., Sano, M., and Rieu, J.-P., 2008, “Changes in the Magnitude and Distribution of Forces at Different Dictyostelium Developmental Stages,” Cell Motil. Cytoskeleton, 65, pp. 314–331. Steffen, A., Rottner, K., Ehinger, J., Innocenti, M., Scita, G., Wehland, J., and Stradal, T. E., 2004, “Sra-1 and Nap1 Link RAC to Actin Assembly Driving Lamellipodia Formation,” EMBO J., 23, pp. 749–759. Cory, G. O. C., and Ridley, A. J., 2002, “Cell Motility: Braking Waves,” Nature, 418, pp. 732–733. Basu, D., El-Assal, S. E.-D., Le, J., Mallery, E., and Szymanski, D. B., 2004, “Interchangeable Functions of Arabidopsis Pirogi and the Human Wave Complex Subunit Sra1 During Leaf Epidermal Development,” Developmental, 131, pp. 4345–4355. Davidson, A. J., and Insall, R. H., 2011, “Actin-Based Motility: Wave Regulatory Complex Structure Reopens Old Scars,” Curr. Biol., 21(2), pp. R66–R68.

Transactions of the ASME


Cytoskeletal architecture and its evolutionary significance in amoeboid eukaryotes and their mode of locomotion.

Cooperative cell motility during tandem locomotion of amoeboid cells.

Regulation of microtubule dynamics by DIAPH3 influences amoeboid tumor cell mechanics and sensitivity to taxanes.

A continuum theory of the mechanics of amoeboid pseudopodium extension.

Cell locomotion: new research tests old ideas on membrane and cytoskeletal flow.

Cytoskeletal mechanics during Shigella invasion and dissemination in epithelial cells.

Emulating constant acceleration locomotion mechanics on a treadmill.

Stem cell differentiation increases membrane-actin adhesion regulating cell blebability, migration and mechanics.

Discrete Modeling of Amoeboid Locomotion and Chemotaxis in Dictyostelium discoideum by Tracking Pseudopodium Growth Direction.

MIEN1 drives breast tumor cell migration by regulating cytoskeletal-focal adhesion dynamics.

Phospholipids: molecules regulating cytoskeletal organization in plant abiotic stress tolerance.

3 inhibition induces amoeboid-like protrusions in MCF10A epithelial cells by reduced cytoskeletal-membrane coupling and focal adhesion assembly.

Cell mechanics: a dialogue.

Cell migration and division in amoeboid-like fission yeast.

Actomyosin and vimentin cytoskeletal networks regulate nuclear shape, mechanics and chromatin organization.

Common mechanics of mode switching in locomotion of limbless and legged animals.

Mechanics of fibroblast locomotion: quantitative analysis of forces and motions at the leading lamellas of fibroblasts.

HOXA5 inhibits metastasis via regulating cytoskeletal remodelling and associates with prolonged survival in non-small-cell lung carcinoma.

Introduction to the Symposium-Unsteady Aquatic Locomotion with Respect to Eco-Design and Mechanics.

Locomotion- and mechanics-mediated tactile sensing: antenna reconfiguration simplifies control during high-speed navigation in cockroaches.

The mechanics behind cell division.

Substrate mechanics and cell spreading.

Red-cell cytoskeletal abnormalities--implications for malaria.

Cell locomotion. Actin alone in lamellipodia.