Both cell-intrinsic and environmental factors constrain speed and persistence in T cell migration

8 9 T cells are key effector cells in the immune system that are well-known for their ability to adapt their shape and 10 behavior to environmental cues. It has been suggested that these highly diverse, context-dependent migration 11 patterns reflect an optimization process – where T cells adjust motility parameters such as speed and persistence 12 to aid their search for antigen. Whereas models investigating such ”search strategies” typically treat speed and 13 persistence as independent variables, one aspect of cell motility was recently found to be conserved across 14 a large variety of cell types: fast-moving cells turn less frequently. This raises the question whether T cells 15 can tune speed and persistence independently of each other. We therefore investigated to what extent this 16 universal coupling between cell speed and persistence (UCSP) shapes the behavior of migrating T cells. We 17 first show that the UCSP emerges spontaneously in an in silico Cellular Potts Model (CPM) of T cell migration. 18 Our model shows a link between the UCSP and cell shape dynamics, which put an upper bound on both the 19 speed and the persistence a cell can reach. We then use the CPM to examine how environmental constraints 20 affect motility patterns of T cells migrating in the crowded environments they also face in vivo, and show that T 21 cells completely lose their speed-persistence coupling when confined in a densely packed environment such 22 as the epidermis. Thus, although our model further highlights the validity of the UCSP in migrating cells, it 23 also demonstrates that environmental factors may overrule this coupling. Our data show that T cell motility 24 parameters are subject to both cell-intrinsic and extrinsic constraints, suggesting that ”optimal” T cell search 25 strategies may not always be attainable in vivo. 26

A CPM models a tissue as a collection of pixels on a grid that each belong to a specific cell (or to surroundings). Pixels randomly try to copy their cell identity into pixels belonging to neighbor cells, with a success probability P copy that depends on the effect the change would have on physical properties of the involved cells (cell-cell adhesion, and deviation from target volume/perimeter, dashed lines). The weighted sum of these energetic effects (∆H) is negative when a copy attempt is energetically favourable. (B) In a CPM with only adhesion, volume, and perimeter constraints, cells only exhibit Brownian motion. Plot shows an example cell track. (C) In the Act model (Niculescu et al., 2015), each pixel has an "activity" that represents the time since its most recent protrusive activity. Copy attempts from active to less active pixels are stimulated (negative ∆H act ), whereas copy attempts from inactive to more active pixels are punished (positive ∆H act ). (D) Act cells alternate between persistent motion and "stops" in which they change direction. Plot shows example tracks of 5 Act cells with overlaying starting point (black dot, t = 0). (E) Displacement plot of CPM cells simulated with and without the Act extension. Brownian motion (without the Act model, gray line) results in a linear curve. Act cell movement appears as brownian motion on large time scales (linear part of red line), but is persistent on smaller time scales (non-linear start of red line). 77 To investigate how external cues affect the cell-intrinsic coupling between speed and persistence in migrating T 78 cells, we need a model that not only reproduces this coupling, but can also simulate realistic cell migration 79 within the complex environments relevant to T cell biology. Such a model must provide a spatial description cells as collections of pixels that move by randomly trying to add or remove pixels at their borders ("copy 83 attempts"). While doing so, cells try to minimize the energetic cost ∆H associated with maintaining their shape 84 and contacts with neighbor cells ( Figure 1A). This allows CPMs to reproduce realistic, dynamic cell shapes and

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-behavior using only a few simple rules and parameters. Their spatially explicit yet simple nature makes CPMs 86 powerful tools for modelling the interactions of individual cells with complex, multicellular environments in a 87 controlled setting. However, the energy that basic CPM cells try to minimize is based solely on adhesion and 88 cell shape. As there is no energetic benefit for consistent movement in any given direction, these cells undergo 89 brownian motion rather than migrating actively ( Figure 1B). 90 We therefore use an extension of the CPM that does allow for active migration (Niculescu et al., 2015). In 91 this "Act-model", pixels newly added to a cell temporarily remember their recent protrusive activity. Copy 92 attempts from active into less active pixels are rewarded via a negative contribution ∆H act to the cost ∆H, Color gradients indicate the active protrusion. Simulated microchannels ("1D" migration) consist of two parallel walls, leaving 10 pixels in between. 2D and 3D simulations were performed in an empty grid of the indicated sizes, with no external constraints on the cell shape. (B) Speed-persistence coupling arises in the Act model on an exponential scale. Plots show data from 1D, 2D, and 3D simulations, respectively (ρ = spearman correlation coefficient). See Materials and Methods for a list of parameters used. (C) Speed-persistence coupling is stronger for cells stratified by max act . Plot shows mean ± SD of persistence time plotted against speed for Act cells migrating in microchannels (1D), grouped by value of max act ; numbers in the plot indicate the corresponding value of λ act . Shaded gray areas in the background indicate regions where the persistence time is lower than the time it takes for the cell to move 10% of its length. until they lose their active protrusion -at which point they wait ("stop") for a new protrusion to form and can switch direction ( Figure S1A in Supplementary Material).

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In addition, Maiuri et al also observed the UCSP for cells migrating on surfaces ("2D") or within 3D 115 environments (Maiuri et al., 2015). We mimicked these experiments by simulating Act cell migration on large 116 2D and 3D grids without microchannel walls (Figure 2A). In contrast to the uniform, elongated shape observed 117 in channels, Act cells moving in 2D and 3D form protrusions of different shapes (Niculescu et al., 2015) ( Figure   118 2A and see below). 119 We then used these different set-ups to simulate cells with different migratory behavior. Two parameters 120 control migration in the Act model. The first, λ act , tunes the contribution of the positive feedback ∆H act relative 121 to the other energetic constraints (adhesion, volume, perimeter), and can be interpreted as the force exerted on 122 the cell membrane by actin polymerization in the cell's leading edge. When λ act is large, this force can easily 123 push the membrane forward to form a stable protrusion, but when it is small, the actin cytoskeleton has a hard 124 time overcoming other, opposing forces (such as membrane tension). Higher λ act values therefore yield larger 125 protrusions ( Figure S1B in Supplementary Material). The second parameter, max act , determines how long pixels   Figure 2B). Although the correlation was weak in these heterogeneous datasets of Act cells with highly 134 different λ act and max act parameters, it became much stronger when we stratified cells by max act value ( Figure   135 2C). Analysis of speed and persistence in these tracks yielded the same exponential correlation between speed 136 and persistence that was also observed in experimental data (Maiuri et al., 2015). This finding was independent 137 of the choice of max act , as we found similar curves for different values of max act ( Figure 2C). These results 138 demonstrate the existence of a speed-persistence coupling in the Act model.  140 We then harnessed the spatial nature of the Act model to examine the UCSP in more detail. We focused on 141 the 2D and 3D settings (which are less artifical than the microchannel environment and allow cells to take on 142 their preferred shapes), and investigated how the cell's migration mode changed when speed and persistence 143 increased ( Figure 3).

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An important feature of the Act model is that it reproduces different cell shapes and migration modes 145 (Niculescu et al., 2015). Low values of λ act and max act promote the formation of small and narrow protrusions 146 that form and decay dynamically, giving rise to an amoeboid ("stop-and-go") migration mode ( Figure 3A, left).

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By contrast, large values of λ act and/or max act favor the formation of broad, stable protrusions, yielding a more 148 persistent "keratocyte-like" migration mode ( Figure 3A, right).

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Like with the microchannel data ( Figure 2C), we again found a strong exponential correlation between speed  This time, however, the exponential increase in persistence was also accompanied by a transition from amoeboid 152 to keratocyte-like cell shapes (insets in Figure 3B).  (Niculescu et al., 2015)). Amoeboid cells are characterized by small, narrow protrusions that decay quickly and produce stop-and-go motion. Keratocyte-like cells are characterized by more stable, broad protrusions that do not decay easily. (B) Exponential speed-persistence coupling in 2D and 3D spans a transition form amoeboid to keratocyte-like motion. Plot shows mean ± SD of persistence time plotted against speed for different combinations of λ act and max act . Insets show representative cell shapes for the indicated parameters; shaded gray background indicates regions where the persistence time is lower than the time it takes for a cell to move 10% of its length. See also Figure S2A in Supplementary Material. (C) Distributions of the instantaneous speeds of 2D and 3D Act cells reveal a transition through different migration regimes. Cells go from not moving (single peak at speed ∼0 pixels/MCS), via "stop-and-go" motility (bimodal distributions), to near-continuous movement (single peak at high speed). See also Figure S2B in Supplementary Material.
of time a cell spends in "stops" (reflected by a decrease in the size of the first peak). As stops provide an 160 opportunity for the cell to change its direction ( Figure 3A), the reduced "stopping time" at high λ act values 161 explains why Act cells with high λ act values migrate not only faster, but also more persistently.

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Together, these results demonstrate that the exponential speed-persistence coupling holds across different 163 "regimes" of migration. At very low λ act , the cell barely moves at all -as indicated by a single peak at 164 instantaneous speeds of almost zero ( Figure 3C). This corresponds to a cell without protrusions, spending most 165 of its time in "stops". As λ act increases, the cell enters a "stop-and-go", amoeboid migration regime (bimodal Open circles indicate points where the persistence time is lower than the time it takes the cell to move 10% of its length (corresponding to the points in the gray background in Figure 3B). Insets show cell shapes at the indicated parameter values. 169 Interestingly, our data also show the saturation of the persistence at higher cell speeds that was also reported in 170 the experimental data (compare the 2D figures in Figure 3B to the data in (Maiuri et al., 2015)). In fact, this   We therefore focused on this extreme example to examine how environmental structure can affect the UCSP.

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To this end, we simulated T cell migration in the skin as reported previously (Niculescu et al., 2015), placing an 212 Act cell in a grid covered completely with keratinocytes ( Figure 5A). Because of the opposing forces from the 213 surrounding keratinocytes, cells now required higher λ act forces to counter this resistance and start moving 214 ( Figure S4 in Supplementary Material). At sufficiently high λ act values, they once again showed the characteristic Figure S4 in Supplementary Material).

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Unlike Act cells in an unconstrained environment, these Act T cells did not switch from amoeboid to 218 keratocyte-like cell shapes as λ act increased. Cells at high λ act now mostly maintained their amoeboid shape, 219 probing their surroundings with narrow protrusions and migrating in the direction of their longest axis.  In this set-up, increases in λ act were once again associated with a higher speed that gradually saturated at high    Our results have implications for studies investigating the efficiency of T cell search behavior. Motility 286 characteristics such as speed and persistence are not independent parameters that can be tuned to generate 287 "optimal" search behavior, but rather reflect a complex interplay between cell-intrinsic rules such as the UCSP 288 and the topology of the environment. Random walk models of T cell search strategies should therefore consider 289 that speed and persistence are subject to both cell-instrinsic and environmental constraints. As different models 290 can often explain the same data, understanding these constraints will be crucial to select those models that are 291 not only consistent with data, but represent search strategies that T cells might actually adopt in vivo. For our simulations, we used the Act model as described in (Niculescu et al., 2015). Our JavaScript implementation 295 of this model is available at http://github.com/jtextor/cpm. 296 Briefly, the Act model is an extension of the Cellular Potts Model that represents cells as collections of pixels 297 on a 2D or 3D grid. These pixels randomly try to copy their cell's identity into neighbor pixels belonging to other 298 cells. The probability that such a copy attempt will succeed depends on the global energy or Hamiltonian of the 299 system, which is in turn determined by cell-cell adhesion and constraints on the cell's volume and perimeter.

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The success probability of a copy attempt therefore not only depends on the cell's shape, but also on that of  The Act model extends the CPM with positive feedback, such that pixels that were recently added to a cell 304 (= recent "protrusive" activity) remember this activity for a period of max act MCS -during which they become 305 more likely to protrude again. This is accomplished via a negative contribution ∆H act to the Hamiltonian for all 306 copy attempts from a source pixel s in an active region into a target pixel t in a less active region: where GM act (p) of pixel p is the geometric mean of the activity values of all pixels in the (Moore) neighborhood 308 of p. Thus, ∆H act is negative when GM act (s) > GM act (t).

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For a complete description of the CPM, see (Niculescu et al., 2015). For details on parameters used, see   For each experiment, we selected parameters combinations where cells had realistic shapes and migration 318 behavior (see Table 1). Temperature, volume, perimeter, adhesion, and λ V /λ P were chosen such that cells stayed 319 connected even at high max act and λ act values tested ("connectedness" at least 95% for at least 95% of the time 320 for all simulations except skin simulations. See section "Connectedness" below). Except for max act and λ act , 321 parameters were held constant within each experiment. Parameters for 1D and 2D simulations were mostly 322 equal -except for the larger perimeter in 1D to account for the elongated shape of cells in microchannels. For 323 3D simulations, however, we had to select other parameters to account for changes in surface to volume ratio 324 and thus altered relative contributions of the different terms to the total ∆H.

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To investigate the link between speed and persistence, we analyzed cell tracks with increasing λ act while keeping 327 max act fixed (Table 2). Max act values were chosen to obtain a range of protrusion sizes, from small protrusions to 328 large protrusions occupying most -but not all -of the cell volume (see percentage active pixels analysis below).

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For each max act , a range of λ act values was then chosen such that cells went from completely brownian motion 330 (persistence time ∼ 5 MCS, the time between subsequent measurements of cell location) to maximally persistent 331 motion (persistence time ∼ 10,000 MCS). Persistence times higher than 10,000 MCS were not considered, as such 332   high persistences will likely be underestimated due to the finite total simulation time (50,000 MCS). For skin 333 simulations, keratinocytes were modelled with max act = λ act = 0, and variable λ P , T cells with max act = 20 and 334 variable λ act (Table 3).

Microchannel simulations 336
To simulate migration of cells confined in a 1D microchannel, we created a 2D grid with a height of 10 pixels 337 and a width of 50,000 pixels. Cells were confined by a layer of "barrier" pixels on the top and bottom of the grid, 338 into which copy attempts were forbidden (yielding a total grid height of 12 pixels). Cells were seeded in the 339 middle of the channel for each simulation.    groups of pixels where for every pair of pixels (p 1 ∈ c, p 2 ∈ c), it is possible to walk from p 1 to p 2 via the edges of 361 graph G i . We then define C i as: where V k is the pixel volume of connected component c k in G i , and V i the total volume of cell i. Thus, an 363 unbroken cell -which by definition has only one connected component -has C i = 1, whereas a cell broken into 364 many isolated pixels has C i → 0.    To measure the persistence of a moving cell, consider the vectors v (t) (movement direction at time t) and v (t+∆t)

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(movement direction at time t + ∆t). When the cell moves persistently, we expect that the direction of movement 381 at t + ∆t is similar to that at t, even for relatively large values of ∆t. By contrast, for a cell undergoing random 382 Brownian motion, the direction of v (t+∆t) is probably unrelated to that of v (t) .

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To quantify this, consider the dot product between the movement vectors v t and v t+∆t : Here, cos θ of the angle between vectors v (t) and v (t+∆t) is 1 when the vectors align perfectly (θ = 0) , -1 when 385 they are exactly opposite (θ = 180), and somewhere in between for all other angles. When we take ∆t = 0, 386 equation 3 simplifies to: As v (t) equals the instantaneous speed at time t, the average of this dot product for different values of t 388 with ∆t = 0 is just the squared mean speedv 2 .

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However, when we increase ∆t, the vectors v (t) and v (t+∆t) are no longer perfectly aligned, and their dot 390 product becomes smaller. The rate at which this decay occurs depends on the motility mode of a cell: for a 391 given ∆t, persistent cells will on average have a smaller θ and thus a larger dot product than cells undergoing 392 Brownian motion. Thus, to compute persistence, we first construct the autocovariance curve of the average 393 dot product v (t) · v (t+∆t) as a function of ∆t (using the "overallDot" function of the MotilityLab package). As a 394 measure of persistence, we then compute the half-life τ of this autocovariance curve for which: As the dot product decays more slowly for more persistent cells, high values of τ indicate persistent 396 movement. Note that, as the average v (t) v (t+τ) = v (t) 2 =v 2 , τ is independent of the mean speedv, even 397 though the dot product is not.

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The authors declare that the research was conducted in the absence of any commercial or financial relationships 400 that could be construed as a potential conflict of interest.  Open circles indicate points where the persistence time is lower than the time it takes the cell to move 10% of its length (corresponding to points in the gray region in Figure 2).   Figure S5: Both speed and persistence saturate for T cells moving in skin. Plots show mean ± SD of (A) cell speed, and (B) persistence as a function of λ act , for different values of the tissue rigidity parameter λ P . Open circles indicate points where the persistence time is lower than the time it takes the cell to move 10% of its length (corresponding to points in the gray region in Figure 5).