Cost–benefit analysis of the mechanisms that enable migrating cells to sustain motility upon changes in matrix environments

Cells can move through extracellular environments with varying geometries and adhesive properties. Adaptation to these differences is achieved by switching between different modes of motility, including lamellipod-driven and blebbing motility. Further, cells can modulate their level of adhesion to the extracellular matrix (ECM) depending on both the level of force applied to the adhesions and cell intrinsic biochemical properties. We have constructed a computational model of cell motility to investigate how motile cells transition between extracellular environments with varying surface continuity, confinement and adhesion. Changes in migration strategy are an emergent property of cells as the ECM geometry and adhesion changes. The transition into confined environments with discontinuous ECM fibres is sufficient to induce shifts from lamellipod-based to blebbing motility, while changes in confinement alone within a continuous geometry are not. The geometry of the ECM facilitates plasticity, by inducing shifts where the cell has high marginal gain from a mode change, and conserving persistency where the cell can continue movement regardless of the motility mode. This regulation of cell motility is independent of global changes in cytoskeletal properties, but requires locally higher linkage between the actin network and the plasma membrane at the cell rear, and changes in internal cell pressure. In addition to matrix geometry, we consider how cells might transition between ECM of different adhesiveness. We find that this requires positive feedback between the forces cells apply on the adhesion points, and the strength of the cell–ECM adhesions on those sites. This positive feedback leads to the emergence of a small number of highly adhesive cores, similar to focal adhesions. While the range of ECM adhesion levels the cell can invade is expanded with this feedback mechanism; the velocities are lowered for conditions where the positive feedback is not vital. Thus, plasticity of cell motility sacrifices the benefits of specialization, for robustness.


A detailed description of the cell motility model
There have been numerous studies investigating the factors determining the efficacy of cell motility, and probing the dynamics of underlying processes. The mechanisms behind different protrusion types are investigated with models of single bleb formation [28--32], and actin polymerisation dynamics [33--35]. Cell-matrix interactions are modelled with defining the ECM fibres explicitly or as a continuum density [36--40]. While excluding cell shape changes, these models provide insights on the influence of global cytoskeletal dynamics, adhesion requirements, and ECM fibre structures on cell motility. However, these models and others that include cell shape changes, do not explore the possibility of multiple modes of cell motility [41,42] and the description of multiple motility modes with pre--defined assumptions on cell--ECM interactions do not allow for investigation of emergent cell behaviour [43].
We share the goal of previous studies that aimed to identify the efficiencies of different motility modes [43], and the discretisation of the cell surface places our model closer to previous pseudopod based motility models [41]. Our model differs by the inclusion of multiple protrusion types, and emergent nature of motility modes. In our computational model of single cell motility, the cell surface is discretized as a set of nodes with viscoelastic linkers in between. Each of the viscoelastic linkers is composed of a spring for myosin contractility, a spring for membrane tension, and a spring--dashpot combination (Kelvin--Voigt body) for the viscoelastic response of the actin network, all connected in parallel. The physical properties of the actin and myosin linkers are determined by the local protein concentrations at the linked nodes, while the membrane tension is a global property of the cell. Internal cell body responds to the deviations from cell's ideal size with changes in internal cell pressure, and applies a viscous response to the movement of the cell surface. The nucleus of the cell is composed of a nuclear lamina, similar in construction to the actin cortex of the cell surface, and a 2 viscoelastic interior, similar to the rest of the internal cell body. The nucleus is stiffer, and more viscous than the rest of the cell interior.
The protrusion formation is closely linked with both the current physical state of the cell body, and the protein concentration of the cell surface. Lamellipodia formation rate depends on local myosin concentration, mimicking the inhibitory cross talk between Rho and Rac. It should be noted the model is in 2D, and does not support the distinction between lamellipod, filopod, or a pseudopod in terms of their respective 3D structures. The behaviour of the actin polymerisation based protrusions is modelled to resemble lamellipodia behaviour, and as such, the terminology "lamellipodia" is used. The blebs are formed with the interplay between the intracellular pressure, and local ERM concentration linking the plasma membrane to the actin cortex: when local fluctuations of the ERM levels bring the cortex--membrane adhesion strength below the current intracellular pressure, the membrane detaches from the cortex, forming a new bleb [15].
The ECM filaments are defined explicitly, allowing for investigation of varying ECM topologies. We have detailed modelling for the adhesion to the ECM, and physical interactions with the fibres in terms volume exclusion. We do not focus on direct friction based mechanisms, which have been suggested as effective motility modes under confinement [38], and this could provide tolerance to zero adhesion levels under confined continuous environments. The polarity is defined as a shift of myosin content ( Figure S1G). In--line with experimental observations ( Figure S1E--F), in all the simulations, the cell "rear" is defined as the region with high myosin concentration.
We also have directly tested scenarios where the cells are polarised in lamellipodia formation without myosin polarity ( Figure S1H). Under the tested conditions, polarity in lamellipodia formation without polarisation of myosin is not sufficient to give consistent directionality to the cell. Typically, the cells form lamellipodia in both directions, albeit with lower initiation rates on one side. In the absence of polarised contractile forces (myosin polarity), the cell can neither terminate the protrusions towards the "wrong" direction, nor detach its "rear" consistently. Thus, although the cells can still generate considerable movement, it is in arbitrary directions (Figure S1Hii/v). In an average of multiple simulations, on unconfined surfaces, the cells cannot generate movement, due to detachment at low adhesions (Adhesion level 5), and 'noise' at high adhesion levels ( Figure S1Hii). Under confinement with high adhesion, the mean velocities are at the same levels as to the environments where lamellipodia are of little effect, such as discontinuous environments, or low adhesion ( Figure S1H, compare panel v with iii&iv). Our results do not eliminate possibility of alternative contractility--independent mechanisms that terminate the 'un-desired' lamellipodia and disassemble cell--ECM adhesions at the cell rear. Such mechanisms would permit directional movement of the cell.
In this work, the density of agents defining the cell is reduced for faster simulations. Parameter fitting procedure is the same as detailed in our previous study ( Figure S7). The resulting parameters are the same as given in Table S1 of Tozluoğlu et. al., 2013; except for internal cytoskeleton viscosity, , set to 3x10 --4 µN s µm --2 , nucleus viscosity, , set to 6x10 --4 µN s µm --2 , adhesion force per unit cortex--membrane linker protein set to 120 pN, and lamellipodia formation rate at 1.0 myosin level, , set to 0.01 sec --1 .

Initiation of blebs on retracting blebs
Layered bleb formations are frequently observed in cancer cells. We improve our model by redefining plasma membrane blebbing to allow the formation of new blebs on top of retracting blebs. This leads to the formation of multi--layered blebs, with multiple decaying bleb necks of different stiffness ( Figure S1B--C), similar to those observed in cancer cells ( Figure S1D). In

Protrusion scores
Protrusion data is collected at 1--second intervals during each simulation.
Spreading lamellipodia score is defined separately for lamellipodia spreading

Cell--ECM adhesion strengthening
The adhesion strengthening is modelled such that a given force applied on an adhesion point defines the equilibrium adhesion protein concentration at that junction. The current adhesion concentration is gradually updated towards the equilibrium adhesion concentration according to the equation Here, !"# is the concentration of cell--ECM adhesion protein at a selected agent at time t; !"# !" is the equilibrium adhesion concentration, induced by the forces applied on the same agent at time t; !"# is the adhesion concentration renewal time of 3 minutes, and is the time step. The equilibrium adhesion concentration is related to the applied forces via equation: 6 Here, !"# !"# is the initial adhesion at a contact site, it defines the minimum strength any given adhesion will be at, regardless of the forces applied; !"# !"# is the maximum adhesion concentration that can be achieved within a single contact zone. !"# is the lower limit of mechanosensing, above which the cell starts reinforcing the adhesion, and !"! is the force magnitude that induces the maximum adhesion cell could reach ( Figure 5A). The subscripts ∥ and ⊥ denote parallel and perpendicular orientations with respect to the surface of adhesion, Red is for overall contractility set to 0.7 and green is for 1.4. Cell polarity is the same as in Figure 4A. A) !"# !"# = 5, !"# !"# = 150, !"# = 0.0 pN, and !"# = 300 pN. B) !"# !"# = 5, !"# !"# = 250, !"# = 0.0 pN, and !"# = 300 pN. C) !"# !"# = 5,  Figure 6, with the high/low adhesion zones defined in inverse order, panels (iii). All cell parameters and adhesion behaviour is the same as Figure 6. Cv) Instantaneous velocity for a cell with mechanosensing, at lower contractility (1.0 compared 1.4 in (iv)), lower membrane--cortex adhesion (0.75 compared to 1.0 in (iv)). All average of at least 10 simulations.

Supplementary Figure 7. Parameterisation of the single cell motility model.
A) i--iv) Heatmaps of parameter fitting score. x--axis shows the increasing inner cell body (cytoskeleton) viscosity. y--axis demonstrates the initial actin fraction 11 that will be left at the detaching cortex, at the onset of bleb initiation. Each heatmap presents fitting scores at different cell cortex--plasma membrane adhesion strength per unit protein, the values are indicated in the titles. Bleb dynamics are fitted to experimental data for each parameter set. The plotted score is the mean of R 2 fit for maximum bleb size, bleb expansion rate, and bleb retraction rates. The selected parameter set in boxed in Aiv. v) Colourbar is valid for (Ai--iv). B) Spreading lamellipodia score for resting cells with overall contractility of 1.0, as a function of lamellipodia initiation rate constant . The experimental measurement is plotted as a red line [2], the selected parameter marked by "*".