Translational motion

Cells have an internal biochemical polarity that sets the direction of their motion. This polarity arises due to asymmetry between the back of the cell where myosin contractions pull the rear, and front where filopodia and/or lamellipodia extend the cell frontier [23]. The polarity of cell i is the velocity that the cell would have in the absence of interactions with other cells. The polarity vector has a magnitude of (equivalent to ), roughly consistent with typical single-cell speeds [24, 25]. We evolve cell positions according to over-damped Newtonian equations of motion:

(1)

where is an inter-cellular force (e.g. adhesion and volume exclusion) by cell j on cell i, and the sum is over the neighbors that are within the interaction cutoff distance r_c from the cell i (the interacting neighbors j of cell i are denoted as ). If the distance between two interacting cells exceeds the interaction cutoff distance r_c they cease to interact – modeling a disconnection of cell-cell adhesion. The force arises due to neighboring cells interacting with a harmonic spring:

(2)

where is the distance between cells i and j, and r_eq is a parameter that sets an equilibrium distance between cells. The force exerted by cell j on cell i is given by . Here, the spring constant k, which we can think of as setting the cell-cell adhesion strength, is expressed in units of 1/time – implicitly this means that we have absorbed a friction coefficient into the value of k instead of writing it in Eq 1. We note that the spring interaction does not depend on the particle orientation – the effect of the orientation is solely on the the cell’s ability to sense the field direction and on cell-cell orientational alignment.

Cell polarity and field sensing

We assume that cells estimate the field direction and then polarize in the direction of their estimate. (This neglects any dynamics of reaching this estimated direction [26, 27].) We also assume that precision of a cell’s estimate depends on its orientation such that it has highest precision when oriented orthogonally to electric field, and lowest precision if its long axis is parallel to the field. We model this estimation process as the cell drawing its estimated direction from a von Mises distribution centered around the true electric field angle with respect to the x-axis (which is 0), with a width controlled by the precision which depends on cell i’s orientation, i.e. . The von Mises distribution is a generalization of the normal distribution to orientations [28], and has a form – i.e. in the limit of large it corresponds to a normal with variance – and we will often refer to errors in terms of . The cell polarizes in the direction of its estimate :

(3)

(see Fig 1b).

The precision of the estimate is orientation-dependent. We hypothesize that cells exhibit the highest precision when perpendicular to the electric field (vertical orientation) and lowest precision in parallel (horizontal) orientation:

(4)

where the cell orientation is the smallest angle between the cell’s long axis and the x-axis (Fig 1b). Hence, a cell i in vertical orientation () would have – or an angular error of – while in horizontal alignment () would have a precision of – or an angular error of . The is the orientation-independent intrinsic variance – the baseline accuracy in estimation – and characterizes the additional variance stemming from the anisotropy of cell shape. The orientation dependence of precision is most pronounced when the difference between values of and is substantial (see Fig 1c).

Cells can make an estimation of the electric field multiple times over our simulation. How often should we expect this estimation to happen? Galvanotaxis – in our best current understanding – requires redistribution of proteins or other sensor molecules on the cell membrane [15, 16]. Membrane proteins on a cell of radius R redistribute via diffusion in a characteristic time , where D is the protein diffusion constant. Averaging time-correlated estimates of the electric field direction for a duration less than does not improve accuracy [16]. Therefore, we assume that cells estimate field direction every minutes and keep the polarity to be constant until a new estimate is made. We estimate to be on the order of 10 minutes [15, 16], though this depends on the cell type and properties of the sensor protein.

Cell reorientation

Elongated cells in confluence exhibit a tendency to align with neighboring cells, resulting in the formation of a local nematic order [20–22, 29] (see Fig 1d). Additionally, when exposed to an electric field, many cell types, even in isolation, will follow the field but orient themselves orthogonally to it [3, 6, 10–12]. In other words, on average, cells align their long axis perpendicularly to their velocity vector (depicted in Fig 1e). We incorporate these tendencies into the equations of motion for , the orientation of the long axis of cell i:

(5)

The first term in Eq 5 represents the cell’s orientation tending to rotate to be away from the direction of the time-averaged velocity , with a rate of alignment . Here, we define as the angle between and horizontal axis, where is the velocity averaged over past period T, i.e. if is the instantaneous velocity of the cell i then .

Why do we choose the cell to orient perpendicular to the time-averaged velocity – or equivalently, to the displacement of the cell over a time T? First, experiments observe that galvanotaxing cells often are oriented with long axis perpendicular to the field [3]. As we have emphasized above, and discussed systematically by [17], for cells to benefit from anisotropic sensing, the cell’s orientation must depend on its past measurement of the field orientation. The displacement of the cell over some time is a natural first representation of the cell’s past measurements, and we think a cell could compute it simply. For instance, integrins are anchored to the substrate, so focal adhesions have an anchor point to the substrate while the cell moves. This means that the cell’s displacement directly affects the relative motion of focal adhesions and the cell’s interior since actin filaments connect to these focal adhesions and reorganize to exert strong contractile forces, guiding the cell’s direction of motion, internal organization, and overall orientation. This interplay also creates a feedback loop: forces generated by the cytoskeleton feed back into focal adhesions, further modulating the migration direction. The information from pulling and anchoring dynamics of focal adhesions could be used by cells to estimate the migration direction and align orthogonally. For this reason, other groups have assumed that cells develop polarity in a direction that reflects displacement over a timescale T [30, 31] – our approach here is somewhat similar, but separates out cell orientation instead of polarity. The choice of modeling cells expanding perpendicular to their velocity has been used earlier both in our work on single-cell galvanotaxis [26] and earlier models of cell motility [32] and self-propelled deformable particles [33, 34]. These models were largely proposed on the grounds of symmetry and simplicity but these effective models may also arise from more detailed biochemical modeling of coupling between cell polarity and cell shape or more general reaction-diffusion models [34–37].

Within our model, the choice of the timescale T will be important, because the instantaneous velocity of a cell is highly variable, influenced by the current estimate of the electric field direction and interactions with neighboring cells. As the timescale T is increased, cells utilize larger displacements to estimate migration direction and will be an increasingly reliable indicator of the net migration direction. If we assume that cells employ the reorientation mechanism proposed in our model, we could expect a cell-type dependent variation in averaging times T since different cells have different timescales for reaching steady-state orientations during electrotaxis [3, 38]. In our study we conduct simulations for various T to establish value that results in meaningfully reliable estimations for our parameter settings.

The second term in Eq 5 nematically aligns a cell’s long axis with that of its neighbors j with a rate of alignment – i.e. it ensures that cells’ long axes are either parallel or antiparallel. Finally, D_r is the rotational diffusion coefficient which describes spontaneous reorientation of the cell, and is a Langevin noise term with Gaussian probability distribution that has a zero mean and time correlation .

In this paper, we always simulate motion in the presence of an electric field. However, we note that in absence of electric field, corresponding to complete uncertainty about field orientation , cells would sample polarization directions uniformly. In this case, a single cell would essentially be undergoing a random walk, with a new orientation chosen every .

Sufficient anisotropy in sensing accuracy and alignment to average velocity is necessary to see benefits of favorable alignment

We hypothesized that cells are more accurate sensors when perpendicular to the field (“vertical”). To what extent does this assumption enhance cells’ directional motion in a cluster? There are two key ingredients for sensing anisotropy to improve cluster accuracy: 1) the difference between sensing in the vertical orientation and the horizontal orientation must be large, and 2) cells must manage to reach the vertical orientation. Within our model, the term that drives cells toward the vertical orientation is the alignment perpendicular to velocity, controlled by . We modulate the rate of alignment to the average velocity () across varying cell cluster sizes (N) for varying and in Fig 2. We initially set the alignment rate to nearest neighbors () to zero to isolate the effect of , and will study nematic alignment effects later. From Fig 2, we see initially that an increase in group size (N) generally correlates with an improvement in directionality, regardless of the combinations of variances and alignment rates.

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Fig 2. The average directionality for various combinations of isotropic and anisotropic variances and alignment rates to velocity.

Isotropic component () changes across rows (top to bottom) and anisotropic component () across rows (left to right) with specific values shown at right side and bottom of the figure (i.e. the panel (f) show directionalities for ). The averages are over 40 simulations at the interaction strength of k = 0.2 . For each simulation the reported directionality is the steady state average over the final 5 hours of simulation as shown in Fig 1f. Results for different values of alignment rates to average velocity are color coded rad/min (black), rad/min (purple), and rad/min (orange). The averaging time T for velocity is set to 1h, and the number of cells N was set to 4, 16, 32, 48, 64, and 96. The shaded areas represent standard errors, although they may not be easily discernible due to their small size.

https://doi.org/10.1371/journal.pone.0325800.g002

In Fig 2, we observe that the alignment rate to velocity – which controls to what extent the cell has a vertical orientation – only has a relevant effect in the upper-right four graphs, which are the cases where isotropic error is relatively low and anisotropic error is relatively high. This is what we would expect. If the anisotropic variance is large while the isotropic error small, cells experience reduced precision if they significantly deviate from the favorable vertical orientation. Consequently, the advantages of faster alignment to average velocity become pronounced (see Fig 2b and c). By contrast, if cells have near-perfect estimates of the field direction independent of orientation (e.g. panel a, and ), then the cells within groups have near-perfect directionality independent of rates of alignment. Similarly, if cells’ error is dominated by the isotropic component, e.g. , cells experience a reduction in precision for all orientations, leading to an overall low level of directionality and no strong dependence on (Fig 2d–f).

For the remainder of this paper, we adopt and as our default parameters – panel f of Fig 2, S1 Movie. We selected these values because they yield a directionality trend, with respect to the number of cells, that is roughly consistent with the findings of Li et al. [7] for MDCK cells. The ratio of variance between the horizontal () and vertical () orientations, which is are roughly consistent with plausible numbers motivated by studies of chemotaxis [13], where the ratio of variance between the horizontal direction and vertical direction is for a cell with aspect ratio 2. Comparable variations in sensing accuracy are also observed in the results on galvanotactic sensing [14], though these depend on both the field’s magnitude and the cell’s interpretation of it.

Why does directionality improve with increasing N? Initially, this may not be surprising – previous models of collective gradient sensing often found that the directionality of the cluster center of mass increases with cell number [39]. The cluster center of mass will almost always have a decreased noise as the number of cells increases simply from the law of large numbers because the center of mass motion reflects the average of many noisy motions of individual cells (see, e.g. [40]; we repeat the basic argument in our S1 Appendix later). However, the directionalities we plot in Fig 2 are for single cells – not the center of mass. Why should directionality increase here? We argue this arises from cell-cell adhesion in our model. Even though the motion of the center of mass is independent of adhesion strength k, the spring strength k controls the degree to which an individual cell follows the center of mass within a given time. In the limit of very large k, the cell cluster is essentially a rigid body – each cell perfectly follows the center of mass. On the other hand, for very small values of k, each cell behaves almost as an independent unit, only weakly tracking the group’s center of mass. Hence, we expect minimal benefits from increasing cluster size in the case of weak adhesion but much stronger N dependence in the case of strong adhesion. We repeat the simulations of Fig 2 with very weak (k = 0.05 ) and strong (k = 1 ) adhesion strengths. Consistent with our expectations, directedness improves as cell groups get larger for strong adhesive forces (S1 Fig, S2 Movie) while there are only small benefits of increasing cell numbers for weakly interacting cells (S2 Fig, S3 Movie). The essential role of cell-cell adhesion to gain a benefit in group galvanotaxis is consistent with experimental measurements showing E-cadherin is necessary for MDCK group galvanotaxis [7]. We will explore the role of adhesion strength in more detail later.

Increasing velocity averaging time improves vertical alignment of cells and directionality

Because cells align perpendicular to their average velocity, and cells are better sensors when perpendicular to the field, there is a natural feedback between orientation and velocity. As cells follow the applied field more accurately, their average velocity becomes more aligned to the electric field. This, in turn, means the cells become increasingly orthogonal to the field, improving their ability to sense the field. Because the instantaneous velocity of a cell will reflect both its self-propulsion and forces from neighboring cells, it will fluctuate around the true direction of the field. We expect that the velocity averaged over some time T will thus have less variability and more reliably point toward the field – so we expect that increasing the averaging time will increase the cell directionality as well as making the cells increasingly vertically-oriented.

Running simulations while varying the averaging time T, we do see that cells become increasingly vertical and directional as T increases (Fig 3). This dependence on averaging time, naturally, only happens if , i.e. that cells orient perpendicular to their averaged velocity. The simulations of Fig 3 are with our default adhesion level of k = 0.2 . Further increases in T would dampen fluctuations even more, and in the limit of large T, the would align with the electric field. Consequently, cells would have information about the true direction of the electric field and would orient orthogonally to it. This would cause cells to assume almost vertical orientations at higher alignment rates , and in this regime, directionalities would plateau. However, we believe that biologically averaging over T>1 hour is unlikely, as cells have been shown to respond to electric fields on a timescale of approximately 15 minutes [3], depending on the cell type.

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Fig 3. Cell orientations and directionalities as a function of velocity averaging time.

In the left column, the absolute value of the cell orientation angle is presented, while the right column displays the directionality. The reported values represent averages across 40 simulations of 64 cells conducted with an interaction strength of k = 0.2 and . The shaded areas represent standard errors of the mean.

https://doi.org/10.1371/journal.pone.0325800.g003

Cells connected with stiffer springs (k = 1 ) will have faster dynamics of cell-cell interactions, and we see pronounced improvement in alignment and directionality even for smaller averaging times (see S3b Fig). Weakly interacting cells move more like individual units and while increasing averaging times improves alignment a little bit it does not relevantly improve the directedness of cells (S3a Fig). This suggests that the relevance of the averaging time is to integrate over fluctuations of cell position due to relative motion from one cell to another, so that the averaged velocity better reflects the cluster center of mass motion.

Nematic alignment to neighbors improves vertical alignment and directedness

Following experimental motivation [18, 21, 41], we have assumed that our cells’ long axes have a nematic alignment controlled by . Can cells use this alignment to work together to align themselves perpendicular to the field, and increase directionality? We simulate cluster migration over a broad spectrum of alignment rates to nearest neighbors and average velocity in Fig 4. In addition to directionality and extent of vertical alignment of cells , we quantify the overall alignment of group of cells using the nematic order parameter Q [18, 41]

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Fig 4. Effect of rates of alignment to velocity () and neighbors () on directionality (a), absolute value of orientation angle (b), and order parameter (c).

Each grid value represents an average result over 40 simulations conducted with 64 cells at the interaction strength of k = 0.2 with an averaging time T = 1 h with colorbars indicating corresponding numeric values. Example simulation snapshots for alignment rate tuples of (i), rad/min, rad/min; (ii), rad/min, rad/min; (iii), rad/min, rad/min; (iv), rad/min, rad/min; (v), rad/min, rad/min; (vi), rad/min, rad/min, also shown in panel c. Cells are colored according to their orientation shown on the colorbar. For all simulations and .

https://doi.org/10.1371/journal.pone.0325800.g004

(6)

where represents the angle of cell orientation, and the averaging is performed across the cell population. Q = 1 means the long axes of the cells are perfectly aligned – but not necessarily vertically aligned – while Q = 0 if is uniformly distributed.

The cell directionality depends on both alignment to velocity and alignment to neighbors (Fig 4a). Directionality is maximal when alignment to velocity is in our intermediate range ( rad/min) but alignment to neighbors is high ( rad/min). The directionality largely reflects the degree to which cells successfully reach a vertical orientation (Fig 4b). We know that in the absence of any alignment to velocity , long axes of cells are essentially randomly oriented relative to the field. Consistent with this idea, we see our lowest directionality and lowest at low rad/min. Increasing alignment to neighbors while keeping low does make cells line up nematically (Fig 4c plots Q, and the difference is dramatic in comparing points i and ii). However, increasing with low fails to induce the vertical alignment necessary to significantly improve directionality (Fig 4b).

There are two key features in Fig 4a that we want to highlight. First, at sufficiently large , increasing alignment to neighbors () enhances directionality. This suggests that cells effectively share information through their nematic alignment. Secondly, the dependence of directionality on alignment to velocity () is not monotonic. At low , cells fail to orient relative to the field and gain no significant directionality benefit. However, at excessively high values of , directionality decreases again. There are two complementary factors driving this phenomenon. First, as increases, cells rapidly align their long axis to be exactly perpendicular to their own averaged velocity (see S6b Fig), neglecting information from their neighbors. Second, even though cells align well to the perpendicular direction of their own averaged velocity, the are themselves quite noisy estimates of the angle perpendicular to the electric field (S6c Fig). Our picture of the non-monotonicity of directionality on essentially depends on the size of the terms in Eq 5. At small , the cell’s orientation is dominated by rotational noise and cell-cell alignment, and can have no correlation to the field direction – leading to low directionality. At intermediate , though, there is an effective balance between the alignment to velocity and alignment to neighbor orientations that allows a cell to incorporate information from its neighbors about the “correct” orientation – allowing the cell to benefit from averaging out the noisiness of multiple cells’ . At very large , the velocity alignment term in Eq 5 dominates all other terms, and cell i’s orientation precisely follows its own – but because is noisy, this leads to a lower directionality than the case with intermediate .

We can probe the origin of this nonmonotonicity further by changing and in two different ways. If we increase the degree of anisotropy by raising to 6 while keeping , this does not alter the patterns in directionality, alignment, and order (S7a-c Fig). With this change of parameters, we have only made the cells’ errors in sensing larger – so we expect that the estimates of the ideal cell orientation will also become noisier, so we expect to still see the non-monotonic dependence on . Conversely, if we maintain and decrease to 0.5, the accuracy of cells in any orientation significantly improves. This makes a more reliable estimate of the favorable vertical orientation – so cells with large , who are only using their own and not sharing information, may still reach near-optimal orientations. In this case, the non-monotonic dependence on is mostly suppressed (S7d-f Fig). We argue the key requirement for non-monotonic behavior to occur is that a single cell’s is noisy enough that the cell deviates significantly from the ideal perpendicular orientation. This will occur at sufficiently large and .

Our key trends are somewhat robust to varying degrees of cell adhesiveness. Whether considering cells with weak adhesion (k = 0.05 , S4 Fig) or those bound by stronger forces modeled as taut springs (k = 1 , S5 Fig), the influence of on vertical alignment is clear. In both scenarios, low values fail to produce significant vertical alignment across a broad range of alignment rates to neighbors . However, for cells interconnected by taut springs, the non-monotonic trend is weaker – rapid alignment at large doesn’t greatly impede directionality. This suppression of the nonmonotonic trend in at high cell-cell adhesion is consistent with our picture above for the origin of nonmonotonicity – large cell-cell adhesion strength decreases the noise of the time-averaged velocity, because each cell is more aligned with the cluster’s center of mass motion.

Cell-cell adhesion is crucial for robust directedness of cells

We have noted earlier in this paper that cell-cell adhesion may play a role in controlling directionality of collective galvanotaxis. What do experiments suggest? Research from Min Zhao’s group demonstrated that disrupting E-cadherin junctions in MDCK I cells with the monoclonal antibody DECMA-1 leads to increased cell speeds but diminished galvanotactic directionality [7]. Similarly, keratinocyte speeds rise upon the disruption of cell adhesions by DECMA-1 [8]. However, the impact on E-cadherin blockade by DECMA-1 on directionality varies significantly with the initial strength of cell-cell adhesion: disruption of E-cadherin junctions enhances the directionality of cells with strong adhesions (high calcium in media), has negligible effects on cells with medium adhesive interactions, and reduces directionality in cells with weaker adhesions (low calcium media). Can we understand why groups of cells might, depending on context, either have an increase or decrease in directionality as adhesion strengths are varied?

With our default parameters, cell speeds decrease and directionalities increase as adhesion strength is increased (Fig 5a). As we saw above, directionality typically increases with the number of cells in the group – though at the weakest adhesion strength, where cells are often separate from the group, cell number only weakly influences directionality. Our cell speed results are consistent with experiment [7, 8], while the directionality results are consistent with those of [7] on MDCK I cells but do not explain the complicated dependence of directionality on adhesion seen in [8]. Why not? One hypothesis is that changing cell-cell adhesion in the experiment regulates multiple factors at once – so that we should model these experimental changes in cell-cell adhesion as changing multiple parameters in our model. For instance, changing cadherin expression is known to regulate cell shape in complex and context-dependent ways [42]. Similarly, changes in cell-cell adhesion may alter cell-substrate interactions [43]. We explore these two hypotheses in our model, to see if it is possible to create decreases in directionality with increasing adhesion strength, or a non-monotonic trend.

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Fig 5. Dependence of directionality on cell-cell adhesion.

(a) Velocity (top row) and directionality (bottom row) as a function of cell count for different (color coded) spring constants. (b) Directionality for different combinations of isotropic variance and interaction strength k (left). The arrow indicates a linear path of concurrent variations of and k resulting decreasing directionality shown on the right figure. Here . (c) Directionality for different combinations of alignment rate to velocity and interaction strength k (left). The arrow indicates a linear path of concurrent variations of and k resulting decreasing directionality shown on the right figure. Here . The reported values represent averages across 40 simulations with 64 cells . Where applicable the shaded areas represent standard errors of the mean.

https://doi.org/10.1371/journal.pone.0325800.g005

We first study what happens if changing E-cadherin strength simultaneously changes cell size or shape, thus altering the precision of cells’ ability to estimate the field direction [16]. As an example for what could happen if cell size changes as E-cadherin strength changes, we show how directionality varies if we vary both and k in our model in Fig 5b, left. (This is only one possibility; models extending the work of [16] would be necessary to explicitly connect cell size and shape and accuracy.) Increasing the variance of the cells’ estimates naturally leads to reduced directionality across a broad spectrum of adhesion strengths (see Fig 5b, left). As an illustration, suppose that when E-cadherin strength is regulated, this also changes , following the pink arrow in Fig 5b, left. The directionalities that would be measured as E-cadherin is varied are then shown in Fig 5b, right. If we assume that longer keratinocytes have enhanced overall precision of electric field direction estimate (i.e. reduced ), then noting that keratinocytes become more elongated as adhesions weaken [8], this combination of effects could potentially explain the observed increase in directionality upon disrupting cell-cell adhesions. Since adhesion-dependent shape changes are modest [8], cells still interact with approximately the same neighbors. Thus, we kept the interaction range unchanged in our simulations and focused instead on how accuracy varies with adhesion strength.

A second hypothesis is that cell-cell and cell-substrate interactions may compete [43]. Specifically, formation of cell-cell adhesions, which modulate cellular forces and alignment, can locally downregulate cell-substrate adhesions. Conversely, formation of cell-substrate adhesions, which regulate cell movement through adhesion strength and dynamics, can also weaken cell-cell interactions. In our model, this might mean that cells with weaker cell-cell adhesion are more effective at aligning perpendicular to their own velocity – something we attribute to more effective cell-substrate interactions that could promote reorientation by providing stable anchoring point and enabling efficient signal transduction. We plot the effect of varying and k separately in Fig 5c, left. For an alternate view of this effect, we can plot directionality as a function of (S8a-b Fig). Generally, lower adhesion strength corresponds to reduced directionality across different alignment rates to velocity. However, simultaneous variation of k and can reverse this trend. We hypothesize that k and vary simultaneously following the pink arrow in the phase diagram, increasing while decreasing k. This leads to a weakly non-monotonic behavior, with a slight dip in directionality above (Fig 5c, right). This non-monotonic behavior results from the interplay between the spring constant k and . As k decreases, directionality is reduced, while an increase in enhances alignment perpendicular to the field, improving the accuracy of individual cells’ electric field direction estimates and leading to higher overall group directionality. The observed dip in directionality, though not dramatic, aligns with experimental findings of a minor dip under similar conditions.