Lymphocytes have been described to perform different motility patterns such as Brownian random walks, persistent random walks, and Lévy walks. Depending on the conditions, such as confinement or the distribution of target cells, either Brownian or Lévy walks lead to more efficient interaction with the targets. The diversity of these motility patterns may be explained by an adaptive response to the surrounding extracellular matrix (ECM). Indeed, depending on the ECM composition, lymphocytes either display a floating motility without attaching to the ECM, or sliding and stepping motility with respectively continuous or discontinuous attachment to the ECM, or pivoting behaviour with sustained attachment to the ECM. Moreover, on the long term, lymphocytes either perform a persistent random walk or a Brownian-like movement depending on the ECM composition. How the ECM affects cell motility is still incompletely understood. Here, we integrate essential mechanistic details of the lymphocyte-matrix adhesions and lymphocyte intrinsic cytoskeletal induced cell propulsion into a Cellular Potts model (CPM). We show that the combination ofde novo cell-matrix adhesion formation, adhesion growth and shrinkage, adhesion rupture, and feedback of adhesions onto cell propulsion recapitulates multiple lymphocyte behaviours, for different lymphocyte subsets and various substrates. With an increasing attachment area and increased adhesion strength, the cells’ speed and persistence decreases. Additionally, the model predicts random walks with short-term persistent but long-term subdiffusive properties resulting in a pivoting type of motility. For small adhesion areas, the spatial distribution of adhesions emerges as a key factor influencing cell motility. Small adhesions at the front allow for more persistent motility than larger clusters at the back, despite a similar total adhesion area. In conclusion, we present an integrated framework to simulate the effects of ECM proteins on cell-matrix adhesion dynamics. The model reveals a sufficient set of principles explaining the plasticity of lymphocyte motility.

During immunosurveillance, lymphocytes patrol through tissues to interact with cancer cells, other immune cells, and pathogens. The efficiency of this process depends on the kinds of trajectories taken, ranging from simple Brownian walks to Lévy walks. The composition of the extracellular matrix (ECM), a network of macromolecules, affects the formation of cell-matrix adhesions, thus strongly influencing the way lymphocytes move. Here, we present a model of lymphocyte motility driven by adhesions that grow, shrink and rupture in response to the ECM and cellular forces. Compared to other models, our model is computationally light making it suitable for generating long term cell track data, while still capturing actin dynamics and adhesion turnover. Our model suggests that cell motility is affected by the force required to break adhesions and the rate at which new adhesions form. Adhesions can promote cell protrusion by inhibiting retrograde actin flow. After introducing this effect into the model, we found that it reduces the cellular diffusivity and that it promotes stick-slip behaviour. Furthermore, location and size of adhesion clusters determined cell persistence. Overall, our model explains the plasticity of lymphocyte behaviour in response to the ECM.

Modelling cell-matrix adhesions

The computational model is based on the Act model [23,24], which is an extension of the Cellular Potts model (CPM, [25,26]) that efficiently simulates persistent, amoeboid cell motility emerging from feedback mechanism inspired by detailed insights into actin-polymerisation driven cell motility. Depending on its parameter settings, the model produces a variety of realistic cell shapes, and reproduces realistic cell polarisation and cell trajectories. In short, the model is two-dimensional, and can be interpreted as a projection of the three-dimensional cell and the underlying substrate from the top. The Act extension keeps track of recent “actin polymerisation” inside the cell, which is represented throughAct values at each lattice site. Novel protrusions get a highAct value and cell protrusions at sites with locally highAct values are favoured. Two important parameters for this are λ_Act, the weight of the Act model that can be interpreted as the maximum protrusive force of the actin network, and Max_Act, the maximum Act value, interpretable as the lifetime of an actin subunit within the actin network (seeTable 1). Here we extend the model with dynamical cell-matrix adhesions (Fig 1). Full detail is given in the Methods section.

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Table 1. List of parameters involved in adhesion dynamics and values used for simulations.

https://doi.org/10.1371/journal.pcbi.1009156.t001

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Fig 1. Overview of the adhesion processes within the model.

A) Top: overview of a simulated cell. Red to yellow shading indicates the Act-level of each lattice site according to the colour bar. Darker coloured lattice sites contain an adhesion. Bottom: the same cell with in blue the region where new adhesions can form, as the local Act-levels exceeds the 0.75 Max_Act threshold. Both: Arrows point to area with one lattice site with high Act-level due to a recent extension of the cell (top, red), but the geometric mean of Act-levels does not exceed the threshold and hence new adhesions cannot form there (bottom, grey). B) Visual summary of adhesion processes. Dark coloured circles indicate lattice sites containing an adhesion. 1) New adhesions can form spontaneously with probabilityp_s at cell lattice sites where the geometric means of Act values exceeds the threshold of 0.75 Max_Act (blue region). 2) An adhesion patch can grow by Eden growth. A random neighbour of an adhesion site is selected. When it does not contain an adhesion yet, the patch extends into that lattice site with probabilityp_e. 3) To break an adhesion, retractions must overcome an energy cost ΔH = λ_adh. 4) Adhesions can also unbind spontaneously, depending on the number of neighbouring lattice sites without adhesions and probabilityp_d.

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In this two-dimensional model we consider integrin bonds between the cell and the underlying substrate, by which the cell adheres to the substrate. They are represented by a binary number in each lattice site, which indicates if at this site there is an active adhesion or not. In biological cells, the formation of new adhesions occurs at the cell’s leading edge during actin polymerisation and pseudopod protrusion [27–29], as well as through formation and expansion of adhesions in the middle region of the cell. In absence of forces pulling the whole cell off of the substrate, due to the two-dimensional representation of the cell, only the adhesions at the edges of the cell will affect cell motility. Nevertheless, we will need to consider formation and degradation of adhesion in the middle of the cell as well, because–as will become apparent when we consider the dynamics of the model–a large patch of adhesions at the cell edge will still resist a series of unbinding events better than a few isolated adhesions. In addition, in an extended version of the model (see Section Including the effects of cell-substrate adhesion on cell protrusion) we consider a feedback of the adhesions on actin polymerization.

We mimic formation of adhesions at the leading edge as follows: in the Act model, the leading edge is marked by lattice sites with high Act values. We thus assume that the probability of adhesion formation is proportional to the Act-value. More precisely, defining as the Moore neighbourhood of lattice site () restricted to the same cell as site, a lattice site receives an adhesion with probabilityp_s (Fig 1A and 1B, process 1) if the geometric mean of Act-value, with exceeds a threshold. Note that in our model,p_s lumps together the effect of multiple biological processes, including the rate at which integrin molecules bind to their ligands in the extracellular matrix, and the concentration of integrins at the cell front. A detailed biophysical model suggests that local integrin-substrate bonds favour the formation of adjacent bonds, because they stabilise the membrane, reducing membrane fluctuations [30]. We simplify patch expansion phenomenologically using the Eden model [31] of radial colony growth. Empty lattice sites adjacent to an existing adhesion site join the adhesion patch with probabilityp_e (Fig 1B, process 2). Unbinding of cell-matrix adhesions occurs spontaneously or as a result of cellular forces. More specifically, adhesion patches rupture from the edges of the patches [17]. We model this process only at the edge of the cell, where cellular contraction forces are sufficient to break bonds. Integrin bonds are known to show catch-slip bond behaviour, meaning that initially the bond strengthens with increase of force, but will still break if enough force is applied. Here, we neglect this specific behaviour and associate a single required energy cost of λ_adh with the rupture of adhesions at the retracting edge (Fig 1B, process 3). λ_adh is given by the binding affinity between integrins and their ligands and the concentration of integrins bound to the ECM. In addition, we assume that association with adjacent adhesions reduces the spontaneous unbinding rate. The probability that an adhesion site unbinds thus becomes, where indicates the Moore neighbourhood of lattice site (Fig 1B, process 4). Our model thus represents the dynamic behaviour of cell-matrix adhesions.

All in all, our proposed model extension for adhesions is relatively simple and computationally light. All adhesion dynamics are governed by the four parametersp_s,p_e,p_d and λ_adh. An overview of all the relevant parameters is shown inTable 1. Estimates of length scale, time scale, and the parameter Max_Act, as well as the relative magnitudes of λ_Act and λ_adh can be found inS1 Text. The simulations were performed using Tissue Simulation Toolkit ([32]; source code inS1 Data). An interactive version in Artistoo [33] of the model running in a web browser is inS2 Data.

Model reproduces walking motility and sustained attachment leading to pivoting motility

We first investigated the influence of the cell adhesion formation probabilityp_s, and adhesion strength λ_adh on the motility patterns.Fig 2A–2D summarize the variety of patterns observed in the model. We observe roughly two types of cell motility patterns. If λ_adh is low to moderate (Fig 2A and 2C,S1 Video) the substrate adhesions do not affect retractions of the cell, and the model behaves, therefore, practically like the standard Act-model [23] which mostly reproduces sliding motility [24] For larger values of λ_adh, the simulated cells show less persistent motility (Fig 2B and 2D,S1 Video). For high values of λ_adh andp_s cells form sustained adhesions. For these parameter values, the adhesions form easily and require much energy to break, such that cells will remain stuck in place on the matrix. However, they can still make protrusions around them, leading to pivoting motility (Fig 2D,S1 Video).

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Fig 2. Simulations of the model showing different motility types.

A-D) First column, a display of a single cell at 5000 MCS interval snapshots combined with the cell centre’s trajectory. Each trajectory starts in the centre of the field and periodic boundaries are used. Second column, a close-up of the cell with the adhesions displayed in a darker colour. E-H) A plot of the cell’s speed and percentage of the cell’s area containing adhesions corresponding to the track on the left. Vertical dashed lines indicate the times of the snapshots on the left. Parameters are: A,E) λ_adh = 20,p_s = 0.004, B,F) λ_adh = 100,p_s = 0.004, C,G) λ_adh = 20,p_s = 0.02, D,H) λ_adh = 100,p_s = 0.02. Furthermore,p_d = 0.0008 for all cases. These simulations are also available asS1 Video.

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To further characterize the cell motility and the effect of the dynamic adhesion patches, we measured the cell speed alongside with the summed area of the adhesions divided by the total area of the cell (Fig 2E–2H). Note that this percentage adhesion area should be interpreted as the percentage of the bottom area of the cell that is adhered to the underlying substrate. First we looked at the effect ofp_s and λ_adh independently, expecting thatp_s regulates the adhesion area. Indeed, at relatively low values ofp_s = 0.004 (Fig 2E and 2F), the cells formed low adhesion area, and at higher values ofp_s = 0.02, we observed a larger adhesion area (Fig 2G and 2H).p_s thus positively correlated with the percentage of adhesion area. We expect that λ_adh regulates cell speed. Indeed, at relatively low values of λ_adh = 20 (Fig 2E and 2G) the average speed was higher and also fluctuated less than at higher values of λ_adh = 100 (Fig 2F and 2H), as indicated by the mean squared logarithmic difference (E:0.00313 and G:0.00442 versus F:0.0072 and H:0.0167; see distributions at the left). λ_adh thus negatively correlated with cell speed. Interestingly, inFig 2B and 2F we observed sliding-stepping-like behaviour: the cells frequently slowed down followed by burst of acceleration. However, the simulations did not display the cyclic expansive and contractile shape changes observed in experiments [16] and reproduced in more complex models [19].

Overall, the behaviour of our model resembles observations by Rey-Barroso et al. [16] that B cells on fibronectin with fluctuating adhesion areas showed walking behaviour, and cells with large and sustained adhesion surfaces displaced very little and the adhesion patch did not displace. The behaviour of the model also agrees with observations of Jacobelli et al. [17] that T cells displaying a gliding motility with higher adhesion area have lower speed than cells with a walking motility with lower adhesion area. An important difference is that the gliding cells in experiments show a large adhesion patch at the front where they assist in cell protrusion, whereas in our model the bulk of the adhesions appears more to the back, where they mostly reduce cell retraction probability (Fig 2C and 2G).

Adhesions slow down cell motility and reduce diffusivity

The examples shown inFig 2 andS1 Video suggest that higher adhesion area correlates with reduced cell speed and reduced cell diffusivity. To characterize this potential correlation, we analysed 1000 independent runs for different combinations ofp_s and λ_adh, and measured the average cell speed and adhesion area.Fig 3A shows the mean speed of the cells as a function ofp_s and λ_adh, andFig 3B shows the diffusivity. In both panels, symbols with the same value of λ_adh are connected and symbols of the same colour have the same value ofp_s.Fig 3A shows that increasing the value of λ_adh decreases cell speed and that the average adhesion area increases with the value ofp_s. Furthermore, for high values of λ_adh, the effects ofp_s on cell adhesion are larger, and for high values ofp_s, the effects of λ_adh on cell speed are also larger.Fig 3B plots the diffusivity of the cells and adhesion area as a function ofp_s and λ_adh. As a crude measure for diffusivity we took the slope of the mean squared displacement curves from 11500 to 24000 MCS, assuming the MSD has entered a linear regime in that time period as discussed in more detail below. Similar to the cell speed, increasing λ_adh results in lower diffusivity and the effect is larger for higher values ofp_s. Moreover, the effect of λ_adh is larger on cell diffusivity than on cell speed. The drop in instantaneous cell speed (from highest to lowest inFig 3A a reduction to about one half) is modest compared to the drop in diffusion coefficient (Fig 3B, from highest to lowest a reduction to about one fiftieth). Cell diffusivity is determined by instantaneous cell speed and persistence of movement direction. Thus, this observation suggests that the drop in diffusivity is for a large part caused by a loss of cell persistence.

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Fig 3. Mean speed, diffusion coefficient, and mean adhesive area change by increasingp_s and λ_adh.

Mean speed (A) and diffusion coefficient (B) plotted against mean percentual adhesion area for different values of parametersp_s and λ_adh. Each dot represents the mean of 1000 independent simulations. Different colours indicate differentp_s, where shades from light to dark and marker symbol indicate λ_adh ∈ {20, 40, 60, 80, 100}. For reference, the mean speed and diffusion coefficient of the Act model without any adhesions are indicated by a black circle. Error bars indicate 95% confidence intervals.

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Reduced diffusivity of adhesive cells is due to reduced persistence times

To further characterize the cause of the reduced diffusivity at high values ofp_s and λ_adh, we fitted the mean squared displacement curves with a persistent random walker model, the Fürth equation [34,35]:(1)whereν_th is the walker’s speed andγ₁ is its persistence time (S1 Fig). The persistent random walker model fits well with the MSD curves at long time scale, but it fails at the short time scale where the cell trajectories are mainly determined by the random fluctuations in the CPM. We therefore extendedEq 1 with a term for translational diffusion [36],(2)

The extended Fürth equation (Eq 2,S1 Fig) gives good fits for most cases (Fig 4A–4C). Extending the Fürth equation with anomalous diffusion instead of translational diffusion, i.e., adding the termD_Tt^β, does not result in better fits, despite the additional parameter (S1 Fig). The persistence times, as obtained from the extended Fürth equation (Eq 2), indicate that indeed the reduced diffusivity can be attributed for a large part to a reduced persistence time with higher adhesion energies and large adhesive areas (S2 Fig). However, for the highest values of λ_adh = 80 to λ_adh = 100Eq 2 still does not fit well with the data (Fig 4D). We attempted to improve the fit by increasing the initialization period left out of the MSD computation, in order to compute the MSD of cells closer to their dynamic equilibrium in both Act model dynamics as well as adhesion-extension dynamics. This barely improved the fit and suggests that cell motility in this regime cannot be correctly described by a persistent random walker with translational diffusion.

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Fig 4. MSD fits either a persistent random walker or a subdiffusive persistent random walker.

Log-log plot of MSD for the four scenarios inFig 2, with fits of Eqs2 and4. Parameters are: A) λ_adh = 20,p_s = 0.004, B) λ_adh = 100,p_s = 0.004, C) λ_adh = 20,p_s = 0.02, D) λ_adh = 100,p_s = 0.02.

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Strongly adhesive cells show subdiffusive behaviour

To characterise the diffusive behaviour of strongly adherent cells at high λ_adh values we fitted the data with the MSD of a fractional Klein-Kramers (FKK) process. Previously the fractional Klein-Kramers (FKK) process was proposed to describe the motility of transformed Madin-Darby canine kidney cells [37]. The MSD of the FKK process was given by,(3)whereE_α,3 is the generalized Mittag-Leffler function andη is a noise term. The standard Fürth model (Eq 1) is a special case of the FKK-process forα = 1 andη = 0, whereα parameterizes the long-term diffusive behaviour and is restricted to 0 <α ≤ 1 [38]. The same MSD expression, except for the noise term, has also been derived from a fractional Langevin equation of motion, which only holds for 1 <α < 2 [39]. Since we already concluded that translational diffusion plays a significant role in the short-time scale of the CPM, we assume that the noise term is due to translational diffusion, thus obtaining,(4)which reduces toEq 2 forα = 1. Fort → ∞, Eqs3 and4 can be approximated by [37]. So forα > 1, the long-term behaviour is subdiffusive, whereas forα < 1, the long-term behaviour is superdiffusive.

In the cases whereEq 2 fits well, we obtainα ≈ 1 (Table 2). However, for the cases wereEq 2 fits badly,Eq 4 has a better fit andα > 1 (Fig 4,Table 2) indicating subdiffusive behaviour. This behaviour corresponds to cases where the cells are strongly attached to the matrix, pivoting around their adhesion patch. Thus, these cells move persistently on a local scale as they have a single protrusion front, but they move subdiffusively on a longer timescale as they stay within a confined area.

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Table 2. Fitted values ofα fromEq 4 for different values of λ_adh andp_s.

https://doi.org/10.1371/journal.pcbi.1009156.t002

Including the effects of cell-substrate adhesion on cell protrusion

So far, the model can explain (i) sliding and stepping, and (ii) pivoting cell behaviour as observed in Ref. [16]. However, key differences between the behaviour of the extended Act-model with real cell motility remain. In particular, in the extended model the cell-substrate adhesions only reduce the probability of cell retraction, whereas in biological cells, the adhesions near the cell front are instrumental for pulling the cell forward. For this reason, the current model still fails to explain (iii) the floating phase that was also observed in Ref. [16], i.e., the observation that B cells with a low adhesive area or no adhesive area on a fibronectin substrate show low displacement compared to cells with dynamic attachment [16].

To also recapitulate such floating behaviour, we extended the model as follows. In presence of a stable adhesion, the forces generated by actin polymerization are transferred onto the matrix leading to protrusion. With reduced adhesions, actin polymerisation more often leads to treadmilling. Thus, in presence of cell-matrix adhesion, actin polymerisation more efficiently translates to cell protrusion [40–42]. We model this effect using a prefactorf to λ_Act, such that it dynamically alters the strength by which the Act-values affect the cell protrusions, and hence the protrusion force. For simplicity, the protrusion efficiency increases linearly with the cell’s total adhesion area. Furthermore, we assume that there is a threshold adhesion area at which the adhesive force suffices to withstand the forces of actin polymerization. Hence, we define,(5)withb the baseline protrusion efficiency ands the threshold adhesion area. Thus the effect of the adhesion area on the propulsion strength only affects cell motility if the adhesion area is below or near the thresholds. A schematic overview is shown inFig 5.

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Fig 5. Schematic representation of the effect of adhesion on cell propulsion strength.

Colour schemes are similar toFig 1A. Arrow width corresponds to the effective protrusion strengthfλ_Act. A) In the absence of adhesions, the propulsion prefactorf is equal to the base levelb. B) Below the saturation points,f increases linearly with adhesion area. C) Above the adhesion area saturation points, prefactorf, and thus effective protrusion strengthfλ_Act, are maximal.

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Extended model reproduces all three phases of cell motility

To test if the new model indeed reproduces floating motility alongside the other two phases of motility observed in Ref. [16], we focus on parameter combinations that result in adhesion areas below or around the thresholds. For adhesion areas above the thresholds, the model behaviour does not change. We chooses = 0.1 and a baseline protrusion efficiencyb = 0.5. From the previous section, we know thatp_s is the main parameter controlling adhesion area, so we chosep_s ≤ 0.004.

The model shows a variety of behaviours depending on the value ofp_s (Fig 6,S2 Video). For very low values ofp_s = 0.001 (Fig 6A), cells make only a small number of tiny adhesion patches and thus have a small adhesive area. Furthermore, they explore relatively small areas (Fig 6A and 6B). Nevertheless, their trajectories can still be described well withEq 2 or withEq 4 withα = 0.974, so the type of motility can still be classified as a persistent random walk, albeit with lower persistence time (S3 Fig). For increased adhesion formation probabilityp_s = 0.0025, stick-slip behaviour is observed (Fig 6C). Interestingly, in contrast to the initial model, the cells accelerate as the adhesion areas grow and they decelerate when they have lost their adhesions (Fig 6D). Forp_s = 0.004 (Fig 6E), the mean adhesive area approaches the threshold. In this case, the diffusivity and persistence are lower compared to the model without the adhesion-protrusion feedback, but the cell speed is comparable (Figs6F and7A).

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Fig 6. Simulations of the model with adhesion-propulsion feedback.

A-C) First column, a display of a single cell at 5000 MCS intervals combined with trajectory of the cell centre. Each trajectory starts in the middle and periodic boundaries are used. Second column, a close-up of the cell with the adhesion displayed in a darker colour. D-F) A plot of the cell’s speed and percentage of the cell’s area containing adhesions corresponding to the track on the left. Parameters are: A,D) λ_adh = 100,p_s = 0.001; C,F) λ_adh = 100,p_s = 0.0025; B,E) λ_adh = 100,p_s = 0.004. Furthermorep_d = 0.008 for all cases. These simulations are also available asS2 Video.

https://doi.org/10.1371/journal.pcbi.1009156.g006

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Fig 7. Mean speed, diffusion coefficient and adhesion area differ between the model with and without adhesion-propulsion feedback.

Mean speed (A) and diffusion coefficient (B) plotted against mean percentual adhesion area for different values of parametersp_s and λ_adh. Each dot represents the mean over 1000 independent simulations. Filled symbols are results from the model without adhesion-propulsion feedback, and open symbols show results from the model with adhesion-propulsion feedback. Different colours indicate differentp_s, where shades from light to dark and marker symbol indicate λ_adh ∈ {20, 60, 100}. Error bars indicate 95% confidence intervals.

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Fig 7 shows the average speed (Fig 7A) and diffusivity of cells (Fig 7B) with low adhesive areas for both the initial (filled symbols) and the extended model (open symbols). The cell speed and cell diffusivity of the standard Act model are shown in the figure for reference. The cell speed and cell diffusivity of the initial model follow the same trend as observed inFig 3, both decreasing for increasing values ofp_s and λ_adh. The model converges to the Act model for low adhesion areas as expected (λ_Act = 240 (black cross)). Similarly, the extended model converges to the Act model for low adhesion area (black plus sign), where λ_Act = 120 corresponds to the baseline propulsion strengthb = 1/2. Near the adhesion area threshold ofs = 0.1 both models show the same behaviour. The mean speed and diffusion coefficient increase monotonically between these two extremes: higher adhesion areas lead to higher speed. Whereas the value of λ_adh has little effect on the cell speed in this parameter regime,p_s determines the adhesion area and thus has a large effect on cell speed. Remarkably, there is a slight difference in mean adhesion area between the models. This small effect is likely due to the positive feedback loop between adhesion growth and cell propulsion in the extended model.

In conclusion, by adjusting the parametersp_s and λ_adh, the extended model provides a minimal model explaining the three motility phases of B cells observed on fibronectin: for low cell-matrix attachment the cells have reduced motility (floating motility), for increased cell-matrix attachment the cells form dynamic adhesion areas and display increased motility (sliding-stepping motility), and for the strongest cell-matrix attachment the cells displayed sustained attachment with low displacement (pivoting motility) [16].

Distribution pattern of adhesions influences cell motility by changing the persistence times

So far, we have neglected the effect of adhesion distribution and variation in time on cell motility. However, the dynamics of the distribution and size of the adhesive patches over the cell may have a large effect on cell motility patterns. For example, the distribution of the adhesion clusters differs between walking and sliding T cells [17]. Sliding T cells have a large contact area at the cell front. Walking T cells, in contrast, had multiple distinct adhesion areas that were distributed over the cells, including the rear end of the cell.

To gain insight in the impact of adhesion patch distribution on cell motility, we explored parameter settings resulting in the same adhesion area but with different adhesion cluster size distributions We did this by varying the formation rate for new adhesions (p_s) as well as the adhesion growth rate (p_e).Fig 8 shows the results of two such parameter settings resulting in the same time-averaged adhesion area. The blue parameter set obtains its adhesive area mostly through the formation of new adhesions (p_s >p_e,Fig 8A: blue), whereas the orange parameter expansion of existing adhesion clusters dominates (p_s <p_e,Fig 8A: orange, see alsoS3 Video). The two parameter sets form cells with approximately the same average adhesion area (Fig 8B; t-Test for difference: p = 0.07), but the cluster size distribution differs (Fig 8C). Forp_s >p_e there are many small adhesion clusters (blue line), whereas forp_s <p_e (orange line) small clusters are combined with a few large clusters. Furthermore, the distribution of adhesion along the cell differs (two-sample K–S testD = 0.164,p = 0.0): if the cell forms many small adhesions (blue), adhesions are located preferentially towards the front of the cell, whereas adhesions are located more towards the back of the cell if the cell forms larger adhesions (orange;Fig 8D).

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Fig 8. Adhesion growth dynamics influence adhesion cluster size and localisation, cell speed and MSD.

(A) Example of different adhesion cluster size for different parameter values ofp_s andp_e. Blue:p_s = 0.003,p_e = 0.0015 resulting in multiple small adhesions. Orange:p_s = 0.001,p_e = 0.004, resulting in a small number of larger clusters. Colours in B-F correspond to these parameter settings. (B) Average adhesion area of 1000 cells over time measured after a ‘burn in’ time of 1000 MCS; each data point represents one cell and vertical lines show the average. (C) Distribution of cluster sizes for 1000 independent simulations for each parameter setting on a logarithmic scale. Distribution of blue does not exceed cluster size 20. (D) Distribution of adhesions along the front-rear axis of the cell for 1000 independent simulations for each parameter setting. Axis was determined from the cell’s position at 25 MCS in the past and 25 MCS in the future at 100 MCS intervals. Position 0 indicates the cell’s centre of mass. Dotted lines show the means of the distributions. (E) Distribution of instantaneous speed of 1000 independent simulations for each parameter setting. Mean speed for orange is lower compared to blue. (F) Log-log plot of the MSD. The onset of the second linear regime (log-log slope approximately equal to 1) is marked with an arrowhead in corresponding colours. This regime starts at smallerdt for the orange curve compared to the blue curve, which corresponds to a lowered persistence time compared to blue. Simulations are also available asS3 Video.

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Next, we studied whether the differences in adhesion distribution lead to differences in cell motility. First, the speed distribution has a slightly higher mean for the many-small-cluster (blue) setting, and appears more bimodal than the few-large-cluster (orange) setting (Fig 8E). The MSD curves largely overlap on the short term, but on the long term the few-large-cluster (orange) setting shows an earlier start of the final linear regime. The onset of this regime corresponds to the persistence time, which we obtained by fitting the MSD curves withEq 2 as well asEq 4. Indeed, the few-large-cluster (orange) setting has about 25% lower persistence time than the many-small-cluster (blue) setting, possibly because it is harder to detach a large adhesion patch than a few small ones. Although we cannot fully exclude that these differences are due to the wider distribution of cluster sizes (Fig 8B), these data suggest that not only the total adhesion area, but also the location and distribution of the adhesion clusters influences cell motility. This further shows that the dynamics of cell-matrix adhesion can be a key component of the plasticity of cell motility.

Linking cell variability and parameter variability

The experimental studies by Rey-Barroso et al. [16] and Jacobelli et al. [17] showed variability in motility among the cells. Other studies have also described variability among genetically identicalDictyostelium discoideum cells in their chemotactic performance [45], and among keratocytes in their shape and speed. Which features of the individual cells underlie such variability in cell motility? By adjusting the adhesion formation rate, adhesion strength and adhesion distribution, we could already capture the different motility modes of B cells on fibronectin and of T cells on casein and ICAM. Still, it would be very interesting to be able to link the changes in these parameters to actual differences between cell populations on different substrates or between individual cells on the same substrate.

An interesting starting point for addressing the variability in lymphocyte motility are measurements of the different subsets of differentiated CD4+ T cells. Th1, Th2, and Th17 subsets have been described to harbour distinct motility properties bothin vitro andin vivo, as well as distinct molecular equipment in terms of adhesion and cytoskeleton dynamics [46,47]. Th1 and Th17 cells show low speed and displacement, have a low expression of integrinα_vβ₃ and show fluctuating, yet high concentrations of Ca²⁺, whereas Th2 cells have high mobility, high levels ofα_vβ₃ and lower and more constant levels of [Ca²⁺]. Interestingly, these differences have been proposed to support distinct search strategies aligning with the fact that these cell subsets target different types of pathogens. The differences in motility and integrin expression correspond well with the observations in our simulations with adhesion-dependent protrusions, where a larger rate ofde novo adhesion resulted in higher motility as well as more adhesive surface. As such, our model predicts a larger adhesive area for Th2 cells than for Th1/Th17 cells, which can be verified experimentally. By further measurements such as measuring integrin expression levels among individual cells by flow cytometry, or monitoring size and distribution of adhesions with super-resolution microscopy approaches, we can specify our model parameters and assess the predictive value of our model. In general, our study provides a mechanistic framework to identify which processes lead to differential motility and then address this experimentally.

Conversely, experimental measurements can elucidate the current estimates of protrusion and adhesion energies. With the current parameter settings, protrusion energy exceeded the adhesion energy, whereas our literature-based estimates suggest the reverse should be true (S1 Text, [48–51]). However, the referenced studies measured adhesion forces in fibroblasts, osteoclasts and CHO cells, which can have adhesive properties distinct from lymphocytes. For instance, a study on adhesion between a T cells and TNF-α stimulated HUVEC cells measured a de-adhesion work of the entire T cell in the same order as our estimates for protrusion-associated work of a single lattice site [52]. This suggests that our estimated adhesion energies based on other cell types might be largely overestimated. However, how this cell-cell adhesion energy translates to adhesion energies on substrates is yet unclear, and new experimental measurements on lymphocyte adhesion energies on substrates can further improve our model.

Motility on multiple time scales

An important feature of our model is the possibility to study cell motility at multiple timescales, especially long-term motility. For long-term behaviour in our model, we mostly observed regular diffusive behaviour or, for more extreme λ_Act andp_s values, subdiffusive behaviour. This corresponds with the pivoting B cells observed by Rey-Barroso et al. [16]. The other extreme, superdiffusive behaviour, has also been observed in mammalian cells. Harris et al. [11] showed that murine T cells display superdiffusive behaviour, but they have only been tracked for a relatively short time (∼10 min), so their diffusive behaviour on longer time scales is unknown. In comparison, our simulated cells display persistent, and hence superdiffusive, behaviour at time scales up to 400 to 1000 MCS, which is in the order of minutes, yet they display regular diffusive behaviour at larger timescales. This highlights the existence of multiple time scales in cell motility. Interestingly, the Madin-Darby canine kidney cell in Dieterich et al. [37] have also been tracked for longer time (∼1000 min) and show three time scales of roughly 0–4 minutes, 4–16 minutes and from 16 minutes onwards, at all of which superdiffusion is observed. Furthermore, cell velocities show long range correlations in time. What causes these long-time correlations and the corresponding long-term superdiffusive behaviour is unclear. Non-trivial behaviour among the factors that determine cell motility might introduce such long-term behaviour, e.g., anomalous rheological properties of the cytoskeleton [53].

An experimental study showed this correlation to be a universal coupling between cell speed and cell persistence (UCSP), mediated by actin flow [43], as actin flow stabilizes cell polarity. In our model, the actin flow is modelled phenomenologically by the Act model [23], which displays this UCSP as well [24]. So the observation that B lymphocytes can display slow persistent motility on fibronectin, and much faster Brownian cell motility seems to disagree with UCSP. Signalling between the cytoskeleton and the matrix may further contribute to the differences in cell motility between the two types of matrices.

Interplay between cytoskeleton and substrate

In our current model, there is only an explicit interaction between the adhesions and the Act-extension. Specifically, only the protrusion efficiency is directly influenced by the presence of adhesion. However, whether other aspects of the actin network, such as life time and turnover rate of actin networks, are influenced by adhesions is not taken into account.

Where the Act-extension mainly models the actin network at the front of the cell, many locomotion-related processes also involve the contractile components of the cytoskeleton, such as myosin-II. Myosin-II contraction pulls the back of the cell towards the front and can increase cell speed [17]. Within the CPM, myosin-II contractility is modelled indirectly through the perimeter constraint and the contact energy between cell and medium. Changing parameter values for both results in altered speeds within our model (S4 Video). Part of this cortical tension is transferred onto the matrix through adhesions [44,54]. An interesting question is whether the cortical tension is also influenced by the presence of adhesions. Furthermore, myosin-II is suggested to be a polarization cue and to be transferred to the back of the cell by retrograde actin flow and could possible also alter persistence of cell polarization [43]. An interesting direction for future research would be to study how the retrograde flow is influenced by cell-matrix adhesions and how this may affect the UCSP that also depends on the actin flow.

Next, the retraction of the rear end is slowed down by adhesions if they do not detach. Our current model uses greatly simplified adhesion patch detachment: a stochastic process of loss of sites, and energy requiring retractions of adhesion sites. It ignores the following two processes: First, myosin-II, besides contracting the rear end, is also involved in the detachment of adhesion patches in T cells [17,55], as it increases the forces exerted on the adhesions. Second, adhesion detachment at the rear of the cell is also regulated by scaffolding proteins talin and moesin. Both compete for binding sites between integrin and the cytoskeleton, but have different properties. While talin connects the cytoskeleton to integrins, moesin inactivates integrins, thereby decoupling the adhesion from the cytoskeleton at the rear end [56]. Considering the myosin-II dependent detachment and rear-end specific detachment in a next model can further enhance the understanding of the role adhesion cluster size and distribution in cell motility.

So far, we have addressed model improvements regarding intracellular processes. However, cell-matrix interactions are also determined by integrin and matrix properties, including mechanistic feedback between the integrins and matrix. Hence, both matrix rigidity and the cell’s ability to generate force influence cell shape and cell motility. When it comes to modelling this feedback, different approaches have been used already in phase-field models of cell motility. In Copos et al. [19], adhesions were modelled as mechanosensitive bonds. In Ziebert et al. [20], adhesions ruptured when they exceeded a maximum length. In Shao et al. [57] the probability of adhesion rupture increased with force. In Löber et al. [21], the matrix deformation was also taken into account, leading to non-trivial motility such as bipedal motility. In Kim et al. [58], a 3D matrix was modelled as individual fibres which a cell can push and pull on. This allows filopodia to sense the matrix stiffness locally, which results in durotaxis.

Modelling matrix deformation or displacement of adhesion sites within the CPM is challenging, but a lot of progress has been made recently. Methods to estimate forces within the CPM cell have been developed, either based on cell shape or on the Hamiltonian [59,60]. The CPM has also been combined with a finite element method to model matrix traction forces with feedback between the CPM and FEM [59,61]. Explicit descriptions of cell-matrix adhesions were recently introduced into this framework [62] to describe tension-dependent growth of focal adhesions. In our future work, this methodology will allow us to study the mechanisms by which substrate compliance affects lymphocyte motility.

In conclusion, we propose a simple mechanism by which the ECM can affect the characteristics of lymphocyte trajectories. To this end, we have introduced a novel CPM model that combines the Act model [23] with dynamic cell-matrix adhesions, generating a large repertoire of cell trajectories (Fig 9). We show that a simple model considering actin-driven protrusion formation in interaction with the dynamic formation and detachment of adhesions to the substrate, suffices to reproduce both persistent random walks as well as short-term persistent but long-term subdiffusive random walks. Thus, our simplified model reproduces the motility patterns observed in individual B cells on a fibronectin substrate, such as reduced motility for non-attached cells, walking motility, and pivoting motility due to sustained attachment, as well as the walking and gliding motility of T cells on ICAM or casein substrate. The computational efficiency of the model allowed us to efficiently study both short-term molecular scales as well as the long-term cell behaviour following from it, providing insight into the molecular parameters that explain the plasticity of cell motility due to interaction with substrates. Our study shows that the interplay between adhesion formation, adhesion expansion and adhesion strength may determine the turn-over of the adhesion area which regulates cell speed and persistence.

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Fig 9. Overview of the motility modes possible in the model and which parameters govern the transitions between them.

For each motility mode, a representative cell and its trajectory are plotted in a persistence versus total adhesion area plane.

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Cellular Potts model

We use the Cellular Potts model (CPM) to represent a cell on a regular, square lattice. Lattice sites are assigned an identity, with if the site belongs to the cells, and if the cell belongs to the medium. The cell can then be defined as the set. The model mimics cell protrusions and retractions through iterative attempts to extend or retract the cell into one of the neighbouring lattice sites, depending on a Hamiltonian function,, representing passive forces acting on the cell, and a number of active, dissipative processes (e.g., actin protrusion). Together, these represent the balance of forces that drive cell motility. More formally, the algorithm selects a pair of adjacent lattice sites, where cells are considered adjacent if they are adjacent orthogonal or diagonal neighbours, i.e., where is defined as the set of the eight first and second-order neighbours of.

The Hamiltonian,, describes the balance of passive forces produced acting on the cell in terms of a system energy,(6)

The first term describes the contact energies between cell and medium. The second term describes an area constraint, withA andA_target being the area and target area of the cell and, similarly, the third term describes a perimeter constraint withP andP_target the perimeter and target perimeter of the cell.

The total change in energy is given by the change in the Hamiltonian, due to the attempted update in addition to the energy, coming from the Act model and giving the energies associated with the detachment of cell-ECM substrate adhesions. The total change in energy determines the acceptance probability of a copy attempt,(7)

Here,T controls the amount of random fluctuations in the system. HigherT will allow more thermodynamically unfavourable copy attempts to be accepted. In the Cellular Potts model, time progresses in Monte Carlo step (MCS), which represents a unit of time that allows each lattice site to be updated once on average.

Cell motility—Act model

Cells move by making protrusions through actin polymerization and form cell extensions like filopodia, pseudopodia, and lamellipodia. Actin polymerization in the CPM has previously been modelled in a phenomenological way in the Act model by Niculescu et al. [23]. This extension adds an extra layer to the CPM, describing the Act values of lattice sites, ranging from 0 to maximum value Max_Act. For lattice site newly added to the cell,. At each MCS, the Act values are decreased by 1 until they reach 0. The term is subtracted from, and can be interpreted as the resulting force from pushing and resistance at the membrane element between and. In, the local geometric mean of Act-values of both the expanding and retracting lattice sites are compared and the lattice site with the highest mean is favoured in the following way:(8)with describing the neighbourhood of lattice site restricted to the same cell, and λ_Act is the weight given to this model component.

Adhesion to the matrix is required to fully translate actin polymerization to cell membrane protrusion [40–42], as it transmits the polymerization force to the matrix. We add feedback between the cell adhesions and actin polymerization, by increasing the force produced by polymerization upon increase in adhesion area. This is only up to a threshold adhesion area, after which the protrusion force remains fully activated. We therefore multiply λ_Act with factorf defined as follows,(9)

Here,A denotes the area of the cell,A_adh the adhesive area of the cell,b the value off when there are no adhesions, ands the fractional adhesive area at whichf saturates.

Cell-substrate adhesions

The adhesions of a cell to the extracellular matrix are modelled as a third layer in the CPM. A lattice site in this layer can either contain no adhesion (), or an adhesion patch (). Adhesion dynamics are governed by four processes:de novo formation of adhesions, adhesion patch expansion, adhesion patch unbinding, and rupture of adhesion through retraction of a cell. We describe each of these processes below.

Implementation

A measure of time in the CPM is the Monte Carlo Step (MCS). Within one MCS, the expectation is that theσ of each lattice site has been updated once. However, many of the proposed neighbouring lattice site pairs share the sameσ and will thus not result in a changed model state. Therefore, we use a rejection-free algorithm that only considers attempts between neighbours of differentσ to speed up simulations [68,69]. Further, the adhesion layer and Act layer of the model are also updated during and after theσ-update. Act-values and adhesion updates regarding the relocation of the cell are executed immediately during theσ-update: e.g., for copy attempts that let a cell retract from a lattice site, we do directly update the Act-values and adhesions of that site. After theσ-update, we update the adhesion layer asynchronously: we iterate, in random order, over the lattice sites within the cell and execute the processes described in the Cell-substrate adhesions subsection. Lastly, we update the Act-layer: every Act-value is diminished by 1 until 0. These three updates together constitute one MCS. The model has been implemented in Tissue Simulation Toolkit and is available inS1 Data.

Simulation parameters

During our different simulations, many parameter values were kept constant (Table 3). All simulations were done on a 300 × 300 lattice with periodic boundaries with a single cell. Parameter values that were not constant are shown inTable 1. For the simulations in Figs2,3 and4,p_d = 0.0008, andp_s and λ_adh varied according to the figure legends. For simulations shown in Figs6 and7,p_d = 0.001 and again λ_adh varied according to the figure legends. The Act-only simulations in Figs3 and7 were run with all adhesion dynamics parameters equal to zero: i.e., λ_adh,p_s,p_e, andp_d were all zero. For all simulations, λ_Act = 240, except for the specific Act-only simulations inFig 7 with λ_Act = 120. For the simulations inFig 8,p_e andp_s were varied, see figure legend. The parameters not mentioned in the figure legend arep_d = 0.0004, λ_adh = 60,b = 0.5,s = 0.12.

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Table 3. List of parameter values kept constant during all simulations.

Values are arbitrary units, unless specified otherwise.

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S1 Fig.Fürth with translational diffusion fits MSD of model better than Fürth without translational diffusion.

Log-log plot of MSD for the four scenarios inFig 2, similar toFig 4, with fits of Eqs1 and2. Parameters are: A) λ_adh = 20,p_s = 0.004, B) λ_adh = 100,p_s = 0.004, C) λ_adh = 20,p_s = 0.02, D) λ_adh = 100,p_s = 0.02.

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S2 Fig.Persistence times obtained from fitting MSD with Fürth with translational diffusion (Eq 2) against adhesion area for different values ofp_s and λ_adh.

Parameters are the same as inFig 3, except that λ_adh has been limited to 20, 40, 60 because of bad fitting withEq 2. For reference, the persistence time of the Act model without the adhesion extension is plotted as the black dot.

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S3 Fig.MSD for initial and extended model with fits of Eqs2 and4.

Parameters are the same as inFig 6, withp_s being varied: (A)p_s = 0.001, (B)p_s = 0.004, and (C)p_s = 0.0025.

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S1 Video.Simulations of the model without adhesion-propulsion feedback.

Videos corresponding toFig 2.

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S2 Video.Simulations of the model with adhesion-propulsion feedback.

Videos corresponding toFig 6.

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S3 Video.Adhesion growth dynamics influence adhesion cluster size and localisation, cell speed and MSD.

Videos corresponding toFig 8.

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S4 Video.Effect of perimeter constraint, λ_P and cell-medium interfacial energyJ_cell,medium on cell behaviour.

Parameter values are identical to the blue parameter settings inFig 8, except where indicated in the video.

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S1 Text.Estimates of parameter units.

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S1 Data.Model implementation in Tissue Simulation Toolkit.

Also deposited athttps://doi.org/10.5281/zenodo.5917626.

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S2 Data.Interactive Model Implementation using Artistoo.

To use the model, openhttps://ingewortel.github.io/2021-motility-from-adhesion/. Alternatively, download the Supporting Data File (also deposited athttps://dx.doi.org/10.5281/zenodo.5914705), unzip the folder “Artistoo” and openArtistoo/active-adhesion/index.html within a web browser.

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