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Physica A 365 (2006) 481–490 www.elsevier.com/locate/physa
Cell adhesion protein decreases cell motion: Statistical characterization of locomotion activity L. Diambraa,, L.C. Cintraa, Q. Chenb, D. Schubertb, L. da F. Costaa a
Instituto de Fı´sica de Sa˜o Carlos, Universidade de Sa˜o Paulo, Caixa Postal: 369, cep: 13560-970 Sa˜o Carlos SP, Brazil b The Salk Institute, 10010 N. Torrey Pines Road, La Jolla, CA 92037, USA Received 2 March 2005; received in revised form 30 July 2005 Available online 15 November 2005
Abstract This manuscript uses a statistical-mechanical approach to study the effect of the adhesion, caused by the modifier of cell adhesion (MOCA) protein on cell locomotion. The MOCA protein regulates cell–cell adhesion, and we explore its potential role in cell movement. We present a series of statistical descriptions to characterize cell movement, and find that MOCA affects the statistical scenario of cell locomotion. In particular, MOCA enhances the tendency of joint motion and decreases overall cell motion. Furthermore, we observe that non-interacting cells that express the MOCA protein have smaller mean velocities than interacting cells, and seem to exhibit normal diffusion. In contrast, control cells exhibit anomalous diffusion independent of interactions with other cells. Furthermore, we observe that in many cases the velocity distribution tails are longer than those predicted by the Maxwell–Boltzmann distribution, indicating that cell movement is more complex than molecules. r 2005 Elsevier B.V. All rights reserved. Keywords: Anomalous diffusion; Cell motion; Cell adhesion
1. Introduction Cell motility has an important role in many biological processes. On the basis of Abercrombie’s work (1970) [1] and subsequent studies, the cell motility cycle can be defined mainly by five steps: (i) the cells polarize towards a chemo-attractant signal, eliciting localized actin polymerization [2,3]; (ii) the cell surface rearranges to form a protrusion [4]; (iii) the protrusion contacts the extracellular matrix or a neighboring cell to form an adhesion site [5]; (iv) myosin-based contraction develops tension between the adhesion sites [6]. Additional signals determine the physiological consequences of the last step. In cell movement the cell’s trailing edge detaches to reinitiate the cycle. In the absence of detachment, the cells remain spread and stationary. How intracellular pathways control each step remains unclear and is the focus of intense study.
Corresponding author. Present address: IP&D, Universidade do Vale do Paraı´ ba. Av. Shishima Hifumi 2911, cep: 12244-000, Sa˜o Jose dos Campos SP, Brazil. Fax: +55 16 3373 9879. E-mail address:
[email protected] (L. Diambra).
0378-4371/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physa.2005.10.006
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The locomotion of cells is involved in many physiological processes, such as the immune response [7], tumor spreading [8], cell sorting [9–11] and nervous system development [12]. It also plays a key role in the pattern formation during most stages of development [13]. Cell movement is highly dependent both upon their surface properties and the environment in which they are embedded. Cells moving as part of a compact cell aggregate interact strongly with each other, as well as interactions with the surrounding non-cellular environment. The locomotory activity of individual cells in media involving low cell densities is almost free of cell–cell interactions, but highly dependent on environmental effects. In both cases the balance of adhesion between cells and the extracellular environment plays a key role in defining the motion of individual cells. By offering powerful models capable of dealing with the movement of particles subject to intrinsic and external effects while accounting for several types of random behavior, statistical physics represents a unique perspective from which to approach cell movement [14–18]. By using concepts derived from statistical physics, it is possible to show that cells moving at random (random walk) are characterized by simple diffusive processes. This simple type of movement provides the null hypothesis for the movement of cells in the absence of intrinsic or extrinsic influences. Previous work on cell locomotion has focused upon characterizing the dynamics of both single cells or groups of cells. These studies have observed normal diffusive motion and Maxwellian velocity distributions [9,16,18]. Recently, however, Upadhyaya et al. [14] reported anomalous diffusion associated to collective cell motion in Hydra cells. This paper describes the motion of genetically modified human kidney 293T cells and of a control cell line to clarify the locomotive roles of the protein MOCA. MOCA is an abundant protein found in the hippocampus and other cortical areas of the brain [19]. Like DOCK-180, a protein that is involved in cell shape and movement [19,20], MOCA has an SH3 domain and two Crk-binding motifs that mediate protein kinase signaling. Since over-expression of DOCK-180 increases cell migration [21], MOCA might affect cell motion. Recent observations suggest that MOCA may induce cytoskeletal reorganization and change cell adhesion by regulating the activity of Rac1 and N-cadherin [22,23]. 2. Materials and methods 2.1. Cells and time-lapse We used clones of human kidney 293T cells (denoted 293T), stably transfected to express MOCA (293MOCA) [24]. Cells were cultured on a laminin surface in a chamber on a microscope, which captured images of cell movement. The laminin surface was made by exposing tissue culture plates to a solution of 10 mg per ml of mouse laminin in phosphate buffered saline overnight at room temperature, followed by washing the dishes twice with culture medium. Since the phenotypic characteristics of the cells reflect their culture state, we used exponentially growing cultures at identical sparse cell densities of 105 cells per 35 mm culture dish. The cells stop dividing when they reach confluence and form a monolayer. Since culture medium and substratum influence cell shape and movement, we used the same experimental conditions for all experiments. The maintenance of temperature and medium pH were accomplished by a heated stage and an enclosed culture chamber in which a humidified mixture of CO2 and air was passed. The chamber was a modified cell culture chamber (Physitemp, Boston, MA), and the experiment was done at 37 1C. Experiments were recorded and translated into digital form by a time-lapse system at an interval of 35 min per frame of 1022 1280 pixels. The time-lapse system consisted of a digital camera (Hamamatsu) attached to an inverted microscope (Leitz DMIRB, using a 16 phase contrast lens), and software for image capture (OPENLAB by Improvision). We took a total of three movies, two corresponding to 293-MOCA (denoted MOCA-A and MOCA-B), and one corresponding to the control cells 293T. The data from MOCA-A and MOCA-B will be pooled and hereafter will be denoted as MOCA, while data from control cells will be denoted by CONTROL. Before being statistically characterized, the movies were preprocessed to remove noise, artifacts and enhance the contrast. Later, each individual frame is extracted from the video sequence in order to facilitate the cell segmentation and the reconstruction of cell trajectories. The task of separating cell from background (segmentation) was done using a semi-automatic procedure, where preliminary segmentation is performed by a software application and then improved through human intervention. This software application was developed in Delphi in order to assist the operator to mark the soma center of mass for each moving cell to reconstruct the
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cell trajectory. Fig. 1 shows some trajectories superposed on the initial acquired frame. Table 1 shows the number of cells followed and the corresponding number of frames in each experiment. 2.2. Statistical methods With the aim of applying statistical approaches to the study of cellular locomotory activity, long-term cell migration patterns were recorded in monolayer cultures. The two-dimensional cell position was extracted from frames taken each 35 min ðDt ¼ 35Þ from time-lapse movies by using a computer program. The trajectory of cell i is denoted by ~ ri ðtÞ ¼ fxi ðtÞ; yi ðtÞg. We estimated the velocities as the mean velocity ~ vi ðtÞ ¼ ð~ ri ðt þ DtÞ ~ ri ðtÞÞ=Dt. For each experiment we studied: (i) the mean square displacement, (ii) the average cell displacements, (iii) the temporal and spatial correlation functions of velocities, and (iv) the distribution of velocities and cell displacements. The mean square displacement hr2 ðtÞi in a given experimental is hr2 ðtÞi ¼ hððxi ðt0 þ tÞ xi ðt0 ÞÞ2 þ ðyi ðt0 þ tÞ yi ðt0 ÞÞ2 Þii;t0 ,
(1)
where the average considers all M cells and all possible t0 using overlapping intervals [25]. The mean square displacement scale is hr2 ðtÞi ¼ Dta , where D is effective diffusivity, a is an exponent that indicates normal diffusion, like a random walk (a ¼ 1), or anomalous diffusion (aa1). We measured the exponent a by determining the linear parameters of the plot loghr2 i vs log t: the slope is a and the y-axis intercept is log D. The total displacement of cells was estimated through the average total path length hLi traveled by the cells in each case. L is the sum of the arc length covered by the cell, and can be estimated as the sum of the distance between the successive positions along the trajectory. In this way we avoid characterizing cell motion only by D, which could lead to wrong conclusions when one compares normal and anomalous diffusion processes. Low values of D associated to anomalous diffusion cannot be related to slow locomotion.
Fig. 1. Some cell trajectories superposed onto the initial acquired frame corresponding to experiment A. Inset: a sample trajectory enlarge containing 13 successive positions of cell i.
Table 1 Some important features of the data including hLi and the exponent a considering all cells in each experiment Experiment
# of frames
M (# of cells)
hvi [mm= min]
hLi [mm]
a
MOCA CONTROL
31 þ 41 37
53 þ 43 33
0:245 0:042 0:304 0:056
246 302
1:14 0:01 1:13 0:01
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When a ¼ 1, the cells undergo normal diffusion motion like a random walk. Anomalous diffusion can also be identified by the following two situations: (i) ao1, corresponding to sub-diffusion, and (ii) a41, which corresponds to super-diffusion. Anomalous diffusion can be induced by temporal or spatial correlations [26]. For this reason, we computed the temporal and spatial correlation functions of the velocities following [18]: ZðtÞ Zð0Þ ZðrÞ CðrÞ ¼ Zð0Þ
CðtÞ ¼
with ZðtÞ ¼ h~ vi ðt0 þ tÞ ~ vi ðt0 Þi,
ð2Þ
with ZðrÞ ¼ h~ vi ð~ ri Þ ~ vj ð~ rj Þi; r ¼ jri rj j.
ð3Þ
The spatial correlation can be used to address the following question: do two neighboring cells travel together more frequently than cells that are distant from one another? We also studied the underlying thermodynamics of the motion by computing the experimental histogram PðX t Þ of cell displacement X t ¼ xðt0 þ tÞ xðt0 Þ during time interval t. We fit the experimental distributions with a Gaussian function and with the form corresponding to anomalous diffusion proposed by Tsallis and Bukman [27]: PTB ðrÞ ¼ Z1 ð1 bð1 qÞr2 Þ1=ð1qÞ .
(4)
The one-component velocity distributions Pðjvk jÞ (where jvk j ¼ jrk ðt0 þ tÞ rk ðt0 Þj=t, k ¼ x; y) were also computed and fitted them with the Gaussian function and with the same functional form used in Ref. [14] where anomalous diffusion was reported f t ðvÞ ¼ að1 þ bv2 Þ1=1q .
(5)
3. Results The following paragraphs describe our analysis of the statistical properties of the cell motion. Table 1 shows the number of cells that were tracked in each experiment, the mean velocities hvi, the average total path length hLi, and a exponents. The top panels of Fig. 2 show the mean square displacement hr2 i as a function of time corresponding to MOCA cells (triangles) and control cells (circles). In order to avoid cell–cell adhesion effects on the characterization of the cell motion, we have distinguished between two situations: considering only non-interacting (NI) cells (open symbols) and considering all (AC) cells (filled symbols) in each particular experiment. For all cells, both control and MOCA cells display an anomalous diffusion behavior as indicated by the exponent a1:1341:0. In contrast, the exponent a associated to NI MOCA cells is very close to 1.0, indicating that cells execute a random walk suggesting normal diffusion. However, better statistic will be necessary before confirmation if this normal diffusion behavior is due to the low number of NI MOCA cells tracked in the experiment. The mean square displacements of MOCA cells are smaller than for control cells, and NI MOCA cells is smaller than for cells that are in cell–cell contact. Additionally, the ANOVA test reveals that NI MOCA cells mean velocity ðhvi ¼ 0:23 0:03 mm= minÞ is significatively smaller than NI control cells mean velocity ðhvi ¼ 0:28 0:05 mm= minÞ, with a significance level po104 . These results show that MOCA expression slows cell locomotion on laminin surfaces relative to control cells independently of cell–cell contact. It should be noted that the mean velocities observed for non-interacting cells are smaller than the mean velocities that include cells touching each other (see Table 1). It is an interesting result as it would be expected that the NI cells move quicker and more randomly than cells in contact with others. We have also computed the average total length hLi in each case. Table 1 shows the average path length for each experiment. In the control case, hLi was 302 mm, thus substantially higher than for the MOCA case, where the hLi value was 246 mm, indicating that the cell displacement in the control case is greater than in 293-MOCA cells. We have also calculated the temporal and spatial correlation functions of the velocities. Fig. 3 shows the temporal auto-correlation of velocities, as expected, at large times the cell velocities are uncorrelated. However, we have not been able to determine the decay rate due to the large sampling period used in our experiment. Fig. 4 depicts the spatial correlation function of velocities for MOCA cells (filled circles) and for
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10000 AC
NI
< ρ2 > (µm2)
MOCA CONTROL
1000
100
100
1000 Time (min)
Fig. 2. MSD logðhz2 iÞ versus logðtÞ plots for MOCA cells (triangles), and control cells (circles). Both fitted lines have slope around 1.13, which would represent anomalous diffusion. Solid symbols were obtained from the average over all cells (AC), while open symbols were obtained from average over non-interacting (NI) cells. Notice the effect of MOCA protein on NI cells whose associated slope is compatible with normal diffusion.
1.0 MOCA CONTROL
Temporal Correlation C(t)
0.8
0.6
0.4
0.2
0.0
-0.2 0
50
100
150
200 250 Time (min)
300
350
400
Fig. 3. Temporal auto-correlation function of the cell velocities, CðtÞ. The low sampling rate does not allow us to estimate the correlation decay.
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1.0
MOCA
0.8
0.6
0.4
Spatial Correlation C(r)
0.2
0.0 0
50
100
150
1.0
200
250
CONTROL
0.8
0.6
0.4
0.2
0.0 0
50
100
150
200
250
r (µm) Fig. 4. Spatial correlation of the cell velocity for MOCA cells (solid circles in top panel) and for control cells (open circles in bottom panel). The solid curves are exponential functions fitting the data. The correlation length MOCA cells is 23:3 mm, while in control cells 13:8 mm.
control cells (open circles), and the solid lines correspond to the exponential function fitted to the experimental data. In both cases, at large distances the cell velocities are uncorrelated. At short distances, we found that velocities of MOCA cells are more correlated than control cells. In particular, the correlation length in the MOCA case was 23:3 mm, while in the control cells only 13:8 mm, which is similar to the approximate cell size (between 10 and 20 mm). This means that MOCA cells travel together more frequently than control cells. A further suggestion of the correlated type of anomalous diffusion arises from examining the distribution of cell displacements PðX t Þ during interval of time t. If the distribution of cell displacements scales as PðX t Þ ¼ X b for large X t with b43, then the origin of the anomalous diffusion is a consequence of correlations [26]. t On the other hand, broad distributions are associated with Levy-type anomalous diffusion. Fig. 5 displays the distributions corresponding to MOCA cells (filled circles) and control cells (open circles) at two different times. The solid curves correspond to the form proposed by Tsallis–Bukman, while the dashed line corresponds to the Gaussian form. At t ¼ 35 min both experimental distributions are in agreement with the Tsallis–Bukman distribution, with similar q-values (1.39 for MOCA cells and 1.34 for control cells). The Tsallis–Bukman distribution has an asymptotic power law behavior for large X, i.e., PðX t ÞX b t . The exponent b associated with MOCA cells is 3.5, while for control cells we found b ¼ 3:75. The exponent b corresponding to NI MOCA cells is 2:75o3, while in the case of NI control cells we found b ¼ 3:8, which is
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1000 MOCA t=245 min
MOCA t=35 min
Histogram P (Xt)
100
10
1
1000 CONTROL t=35 min
CONTROL t=245 min
Histogram P (Xt)
100
10
1 -60
-40
-20
0 Xt (µm)
20
40
60
-150
-100
-50
0 Xt (µm)
50
100
150
Fig. 5. Histogram distribution PðX t Þ of the cell displacements X t ¼ xðt0 þ tÞ xðt0 Þ along the axis x for MOCA (solid circles) and control cells (open circles) during time intervals t ¼ 35 min (left panels) and t ¼ 245 min (right panels). The solid curves correspond to the distribution form (4), while dashed curves correspond to Gaussian distribution. At t ¼ 35 min (4) fits the experimental data better than the Gaussian both in MOCA and in control cells. For t ¼ 245 min the distribution corresponding MOCA cells seems to converge to a Gaussian. In all cases, collective drift was not observed.
very similar to the case that includes cell–cell contact. For larger time intervals ðt ¼ 245 minÞ, the experimental distribution associated with control cell also follows the Tsallis–Bukman form with q ¼ 1:6 and the associated tail has the exponent b ¼ 4:1. However, the experimental distribution associated to MOCA cells seems to converge to a Gaussian form for larger times. The tail associated is poorly fitted by both Gaussian and Tsallis–Bukman (with q ¼ 1:56) functions. Fig. 6 shows the probability distributions of one-component cell velocity Pðjvk jÞ for MOCA cells (triangles) and control cells (circles). Solid lines correspond to the respective fitting by the function f t ðvÞ while the dashed
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0.1
Distribution of velocities
0.01
1E-3 (a)
(b)
(c)
(d)
0.1
0.01
1E-3 0.1
1 |vk| (µm/min)
0.1
1 |vk| (µm/min)
Fig. 6. Probability distribution of one-component cell velocity (jvi j ¼ jri ðt0 þ tÞ ri ðt0 Þj=t where i ¼ x; y), for MOCA cells (triangles in panels a and c) and for control cells (circles in panels b and d). Solid symbols were obtained considering the velocities of all cells (a,b), while open symbols were obtained considering only non-interacting cells (c,d). The solid curves are fits with function (5), while dashed curves is a fit with a Gaussian distribution.
lines correspond to Gaussian fittings. The function f t ðvÞ with q1:5 fits the experimental probability distributions corresponding to cells which can have cell–cell contacts (filled symbols). Figs. 6a, b and d present a longer tail than that corresponding to Gaussian distributions. This slower decay has also been observed in other experiments (Fig. 3a in [14] and Fig. 3 in [9]). Fig. 6c shows the distribution derived from NI MOCA cells. In contrast to previous cases, it is well fitted for the Gaussian form (or by f t ðvÞ with q ¼ 1:0). We should also note that such a velocity distribution is characterized by smaller velocities than in Figs. 5a and d, in agreement with the mean velocities calculations.
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4. Discussion and conclusions Cell movement requires a series of highly coordinated events that are powered by the actin cytoskeleton and regulated by a complex group of kinases and phosphatases (for review see Ref. [28]). We have recently demonstrated that the expression of MOCA is required for neurite outgrowth in both PC12 cells and central nervous system neurons [23]. The results presented here establish that MOCA expression may be involved with cell–cell adhesion and that it also influences cell movement. This conclusion is also supported by Fig. 7, which depicts the mean distance between cells that were no further apart than 25 mm in the initial frame as a function of time. It is clear from this result that the 293-MOCA cells remain closer together than the control cells. Our results suggest that MOCA protein modifies two important aspects of cell movement. The measurements clearly indicate that MOCA tends to enhance joint motion and to decrease the mean velocities. The last result is consistent with the hypothesis that cells move slower in a more cohesive environment. However, we did not verify the same hypothesis for non-interacting MOCA cells which move in a random walk fashion, and even slower than MOCA cells in contact with each other. Furthermore, the spatial correlations of the cell velocities show that two neighboring 293-MOCA cells travel together more frequently than two control cells, and that their velocities are highly correlated. This is consistent with our data demonstrating that MOCA increases N-cadherin-mediated cell–cell adhesion [23]. In contrast, the over expression of DOCK-180 leads to increased cell migration [21], establishing that the function of the two proteins is distinct. Based upon the exponent a of mean square displacement, the spatial correlation of cell velocities and the narrow distribution of cell displacements (i.e., b is almost always larger than 3), we can conclude that both control cells and genetically modified 293T cells present a correlated-type anomalous diffusive behavior. Such behavior was already found in cellular aggregates [14] although the origin of the anomalous diffusion was probably different. On the other hand, exponents a corresponding to non-interacting MOCA cells suggest a normal diffusive process. This observation shows that the MOCA mediated cell–cell interactions limit the displacement options of the interacting cells relative to single cells. This is likely due to the requirement for 160 MOCA 140
CONTROL
Mean Distance (µm)
120
100
80
60
40
20
0 0
200
400
600 Time (min)
800
1000
1200
Fig. 7. Mean distance between cells whose initial distance was smaller than 25 mm as a function of time for MOCA cells (filled circle) and control cells (open circles).
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coordinating the motion of multiple cells in a single direction, but once a direction is determined, the velocity is faster in cell aggregates. It is important to notice that we were able to observe differences in NI cell behavior due to the sampling period (35 min) used in the study. Cells need about 1 h to respond to the interaction with another cell. Finally, the observation that the experimental distribution tails are longer than those predicted by Maxwell–Boltzmann thermodynamics in all cases studied here, agrees with results in previous studies [9,14]. This fact could be a statistical indication that cell movement is more complex than that of a liquid. Acknowledgements Luciano da F. Costa is grateful to FAPESP (proc. 99/12765-2), CNPq (proc. 308231/03-1) and Human Frontier Science Program for financial support. Luis Diambra thanks the Human Frontier Science Program for his post-doctoral grant. David Schubert and Qi Chen are supported by the National Institute of Health. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28]
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