Modeling cell crawling strategies with a bistable model: From amoeboid to fan-shaped cell motion
E. Moreno, S. Flemming, F. Font, M. Holschneider, C. Beta, S. Alonso
MModeling cell crawling strategies with a bistable model: Fromamoeboid to fan-shaped cell motion
Eduardo Moreno ∗ , Sven Flemming † , Francesc Font ‡ , Matthias Holschneider §3 , CarstenBeta ¶ and Sergio Alonso (cid:107) Department of Physics, Universitat Politècnica de Catalunya, 08028 Barcelona, Spain. Institute of Physics and Astronomy, University of Potsdam, 14476 Potsdam, Germany. Institute of Mathematics, University of Potsdam, 14476 Potsdam, Germany. Centre de Recerca Matemàtica, Campus de Bellaterra, Edifici C, Bellaterra 08193,Barcelona, Spain. Max Planck Institute for Dynamics and Self-Organization, 37077 Göttingen, Germany.May 29, 2020
Abstract
Eukaryotic cell motility involves a complex network of interactions between biochemical componentsand mechanical processes. The cell employs this network to polarize and induce shape changes that giverise to membrane protrusions and retractions, ultimately leading to locomotion of the entire cell body.The combination of a nonlinear reaction-diffusion model of cell polarization, noisy bistable kinetics, anda dynamic phase field for the cell shape permits us to capture the key features of this complex systemto investigate several motility scenarios, including amoeboid and fan-shaped forms as well as intermediatestates with distinct displacement mechanisms. We compare the numerical simulations of our model to livecell imaging experiments of motile
Dictyostelium discoideum cells under different developmental conditions.The dominant parameters of the mathematical model that determine the different motility regimes areidentified and discussed.
Keywords : Pattern formation,
Dictyostelium discoideum , cell motility, amoeboid crawling, keratocyte motion
The biochemical and biophysical mechanisms involved in cell motility have been extensively studied during thepast years. They are among the most intriguing problems in cell biology, ranging from single cells to multicellularorganisms. Before the cell begins to move, it has to define a front and a back to specify an axis of propagation.This process is known as cell polarization [41]. It sets the direction in which protrusions are formed that drivethe cell forward.Cell locomotion has been extensively studied using keratocytes, which move in a highly persistent fashionand adopt a characteristic fan-like shape [42]. Also neutrophils have been intensely investigated. They dis-play a less persistent movement with more frequent random changes in direction that is known as amoeboidmotility [27]. A well-established model system to study actin-driven motility in eukaryotic cells is the socialamoeba
Dictyostelium discoideum ( D. discoideium ) [4]. The cells of this highly motile single-celled microor-ganism typically display pseudopod-based amoeboid motility but also other forms, such as blebbing motility orkeratocyte-like behavior have been observed.Many aspects of cell motility such as cytoskeletal mechanics [25], intracellular signaling dynamics [9, 52], ormembrane deformation [1] have been modeled using mathematical and computational methods. Cell polarityformation, which is a key features of motility mechanisms to determine the front and back of the cell, often showsbistable dynamics. A reaction-diffusion system with bistable kinetics is thus a common choice to model theintracellular polarity dynamics [30]. Bistable conditions of an intracellular dynamical process can be obtained ∗ [email protected] † svenfl[email protected] ‡ ff[email protected] § [email protected] ¶ [email protected] (cid:107) [email protected] a r X i v : . [ q - b i o . CB ] M a y y a mass-controlling mechanism between the cytosolic and membrane attached concentrations of biochemicalcomponents [43, 49]. This may be relevant at different levels of the cytoskeleton, for example, when differentforms of actin are involved [8, 53, 7] or at the level of the related signaling pathways, involving phospholipids andenzymes at the cell membrane [39, 3]. Cell polarity may be also induced by an external chemical gradient [29].There are several mathematical tools to simultaneously model the pattern formation process inside the celland the dynamics of the cell border, which is required to obtain a full description of a crawling cell. One of themost commonly employed methods to model such a free-boundary problem is to introduce an additional phasefield, which is one inside and zero outside the cell and keeps the correct boundary conditions while the bordersare moving [31], even in the limit of the sharp interface between the interior and the exterior of the cell [12].The first attempts to employ a phase field modeling to study cell locomotion where applied to keratocytemotility [55, 61, 54] because the persistence of motion of these cells facilitates the implementation of the model.These models have also been extended to discuss, for example, the rotary motion of keratocytes [13] and theinteractions among adjacent cells [34].Later, the use of the phase field has been extended to model other generic properties of the moving cells[47, 32] and, in particular, has been also employed to describe the random motion of amoeboid cells, such asneutrophils or D. discoideum . The phase field approach has been employed to model the viscoelasticity of thecell [44, 45], the effect of biochemical waves in the interior of the cell [58], as well as wave-induced cytofissionof cells [19]. The random motion of the cell requires a stochastic bistable process in combination with a phasefield [2] to successfully recover the fluctuating displacements and shape deformations. Such a stochastic bistablemodel is able to capture the cell-to-cell variability observed in the motion patterns of amoeboid
D. discoideum cells by tuning a single model parameter [2]. However,
D. discoideum cells are known to show a more diversespectrum of motility modes, for example when certain genes are knocked out, when phosphoinositide levels areartificially altered, or under specific developmental conditions. This includes a phenotype which is reminiscent ofkeratocyte motility, where the cell adopts a fan-like shape and moves persistently, perpendicular to the elongatedaxis of the cell body – the so-called fan-shaped phenotype – and a form where cells adopt a pancake-like shape,moving erratically without a clear direction of polarization [5, 40, 15].Here, we perform a systematic analysis of a previously introduced model that is based on a stochasticbistable reaction-diffusion system in combination with a dynamic phase field [2]. Such phenomenological modelmay provide a better understanding of how to relate the experimental parameters to specific cellular behaviors,because cell-to-cell variability often masks such relation. Along with the model, we analyze experimental dataof a non-axenic
D. discoideum wildtype cell line (DdB) that carries a knockout of the RasGAP homologueNF1 (DdB NF1 null cells). In this cell line, amoeboid and fan-shaped cells are observed, depending on thedevelopmental conditions. A detailed comparison of the experimental data to simulations of the stochasticbistable phase field model is presented. By tuning the intensity of the noise and the area covered by the bistablefield, the model simulations recover similar motility phenotypes as observed in experiments, ranging from highlypersistent fan-shaped cells to standard amoeboid motion. Furthermore, the simulations predict intermediateunstable states and also a transition from straight to rotary motion of the fan-shaped cells. These forms ofmotility that have so far been neglected in
D. discoideum , were also observed in the experimental data and aresystematically studied in the framework of our mathematical model.
All experiments were performed with non-axenic
D. discoideum
DdB NF1 KO cells [10], which were cultivatedin 10 cm dishes with Sorensen’s buffer (8 g KH PO , 1.16 g Na HPO , pH 6.0) supplemented with 50 µ MMgCl , 50 µ M CaCl and Klebsiella aerogenes at an OD of 2. The cells expressed Lifeact-GFP via theepisomal plasmid SF99, which is based on a new set of vectors for gene expression in non-axenic
D. discoideum strains [50]. Plasmids were transformed as described before [50] with an ECM2001 electroporator using threesquare wave pulses of 500 V for 30 ms in electroporation cuvettes with a gap of 1 mm. G418 (5 µ g/ml) andHygromycin (33 µ g/ml) were used as selection markers.The phenotype of DdB NF1 KO cells differs between individual cells of a population and especially betweendifferent developmental stages. When cultivated in buffer supplemented with bacteria, cells are in the vegetativestate and the predominant phenotype is amoeboid with very little movement due to the abundance of bacteria.After several hours of starvation, cell enter the developed state and the probability to observe a fan-shapedphenotype is increased. Preparation of the cells forexperiments therefore differed between experiments. Cellswere washed to removethe bacteria and (i) suspended in Sörensen’s Buffer immediately after washing to obtainmainly amoeboid cells with high motility or (ii) starved for 3-6 hours to obtain a high percentage of fan-shaped cells. After starvation, cells were seeded in microscopy dishes at low density for imaging. Usually inthe beginning of an experiment, many cells showed the amoeboid or the intermediate phenotype with regular2witches from amoeboid to fan-shaped motility and vice versa. The percentage of fan-shaped cells increasedover time and the fan-shaped phenotype became more stable. An increase in the number of fan-shaped cellsduring development has also been described for the D. discoideum Ax2 AmiB knockout strain (Asano et al.,2004). Note however, that the effects of cell development on the phenotype of DdB NF1 KO cells showed ahigh day-to-day variability and best results were accomplished with fresh K. aerogenes cells. The cells weretransferred to a 35 mm glass bottom microscopy dish (FluoroDish, World Precision Instrumnets) and dilutedto a concentration enabling imaging of single cells. For imaging an LSM 780 (Zeiss, Jena) with a 488 nm argonlaser and a 63x or a 40x oil objective lens were used. We investigate different types of cell motility based on a minimal model that couples a concentration fieldaccounting for the complex biochemical reactions occurring in the interior of the cell to an auxiliary phasefield describing the evolution of the cell shape. The use of a phase field is a well-established approach to dealwith problems of evolving domains/geometries without the need of explicitly tracking the domain boundaries,which has been exploited to tackle moving boundary problems of different nature such as crack propagation[11], solidification [51], or fluid interface motion [20]. In particular, the versatility of this approach has beenused for modelling cell-shape evolution and locomotion [54, 58, 13]. The model we use here has been previouslyintroduced in [2] and will be summarized below.In what follows we will consider the dynamics of a generic activatory biochemical component at to thesubstrate-attached cell-membrane and, thus, restrict ourselves to an idealized 2D geometry. The phase field φ ( x , t ) smoothly varies between the values of φ = 1 inside and φ = 0 outside the cell, respectively. The phasefield allows to implicitly impose no-flux boundary conditions at the cell border, which we assume to be where thephase field takes the value of φ = 0 . . Following the work by Shao et al. [54], the phase field evolves accordingto the equation τ ∂φ∂t = γ (cid:18) ∇ φ − G (cid:48) ( φ ) (cid:15) (cid:19) − β (cid:18)(cid:90) φ dA − A (cid:19) |∇ φ | + α φ c |∇ φ | , (1)where G ( φ ) = 18 φ (1 − φ ) is a double well potential. The phase field equation is the result of a force balanceinvolving forces of different nature acting on the cell body. The terms with |∇ φ | affect the border of the cell,while the others affect the volume. The first term on the right hand side of eq. (1) corresponds to the surfaceenergy of the cell membrane, where γ is the surface tension (note that value is obtained assuming a cell heightof 0.15 µ m [54]) and (cid:15) the mathematical definition of the width of the cell boundary. The second term ensuresthat the cell area is kept close to A . The last term represents the active force of the biochemical field c ( x , t ) onthe cell membrane. The parameters that control the impact of the area conservation constraint and the activeforce, β and α , respectively, are kept constant in our simulations. The term on the left hand side of eq. (1)accounts for cell-substrate friction.A complete derivation of eq. (1) can be found in [55].Cell polarization is implemented in our model by assuming noisy bistable dynamics resulting from a non-linear reaction-diffusion equation for the concentration c ( x , t ) . The biochemical field c ( x , t ) represents a di-mensionless generic concentration variable that accounts for different subcellular components that promote thegrowth of filamentous actin (F-actin) such as active Ras, PI3K or PIP [56]. It is related to the intensity of theLifeact-GFP marker for F-actin in our experiments.Imaging experiments with D. discoideum typically show rich dynamical patterns in the cell cortex and atthe cell membrane. Deriving a detailed model that captures the full complexity of the underlying biochemicalreactions is unfeasible. To overcome this difficulty deriving a detailed model of the reactions inside the cell andaiming for mathematical simplicity, we take a similar approach as in previous studies [43, 13, 2] and formulatea simple reaction-diffusion equation, where the non-linear reaction kinetics leading to bistability is modelled bya cubic polynomial in the variable c ( x , t ) . In addition, we introduce a term accounting for degradation of thebiochemical component c . The equation reads ∂ ( φc ) ∂t = ∇ ( φD ∇ c ) + φ [ k a c (1 − c )( c − δ ( c )) − ρ c ] + φ (1 − φ ) ξ ( x, t ) , (2)where k a is the reaction rate, ρ the degradation rate, and D the diffusivity of the biochemical component.The last term on the right hand side introduces noise at the cell membrane, which allows us to account for thestochastic nature of the reaction-diffusion processes occurring within the cell. The noise intensity along with thereaction rate are key parameters in our model that allow the transition between different forms of cell motility.The stochastic variable ξ ( x, t ) follows an Ornstein-Uhlembeck dynamics dξdt = − k η ξ + η , (3)3able 1: Parameter values for the numerical model.Parameter Value Units Meaning Value Reference D µm /s Diffusion coefficient [2] k a s − Reaction rate [2] ρ s − Degradation rate [2] σ s − Noise strength [2] τ pN sµm − Membrane dynamics time-scale 1[55], 2.62[61] γ pN Surface tension 1[55] (cid:15) µm Membrane thickness 1[55] β pN µm − Parameter for total area constraint [2] A µm Area of the cell [2] δ M µm − Strength of the global feedback input [2] k η s − Ornstein-Uhlembeck rate [2] α pN µm − Active tension 0.5-3[61] Co
28, 56, 84 µm Maximum area coverage by c [2]where η is a Gaussian white noise with zero mean average (cid:104) η (cid:105) = 0 and a variance of (cid:104) η ( x, t ) η ( x (cid:48) , t (cid:48) ) (cid:105) = 2 σ δ ( x − x (cid:48) ) δ ( t − t (cid:48) ) .The reaction diffusion equation aims at reproducing the pattern activity on the substrate-attached cellmembrane observed in our experiments. A control of the size of the patterns is important since the patternedarea rarely covers the entire cell membrane. Previous experiments with giant D. discoideum cells revealed that,after a critical size is reached, wave patterns tend to modify their shape rather than actually growing into largerareas [23]. Therefore, a dynamic control in the form of a global feedback on the parameter δ ( c ) is implemented.It affects the pattern dynamics and prevents the system to be covered completely by c depending of the valueof C . The control term reads δ ( c ) = δ + M (cid:18)(cid:90) φ c dA − C (cid:19) , (4)where the parameter C represents the average area covered by component c . The control mechanism shown ineq. (4) dynamically changes the value of the unstable fixed point of the system. This enforces that the amountof component c inside the cell is constant on average. In our simulations, we will specifically show the effect ofchanging C on the cell trajectories.We integrated Eqs. (1)-(4) on a square domain of 300 ×
300 pixels with periodic boundary conditions usingstandard finite differences. The pixel size is given by ∆ x = ∆ y = 0 . µ m and the integration time stepis ∆ t = 0 . s. The values and definitions of the parameters of the model can be found in Table 1. Celltrajectories and velocities were obtained by finding and tracking the center of mass of cells from the numericalsimulations.In the following sections, we will study and analyze different cell motility modes. The transitions betweenthese modes are obtained by varying three parameters: the noise intensity, the average membrane coverage withthe activatory component c , and the activity rate of the biochemical field. In general, the bistable kinetics of c will drive the formation of patches of high concentration of c on a background of low c concentration. Thecoherent effects of these patches on directed cell locomotion will be disturbed and interrupted by the impact ofnoise, which will favor nucleation events and the formation of new patches in other regions of the membrane.Therefore, the dynamics of our model can be qualitatively understood as a competition between the coordinatedeffects of pattern formation and the randomizing impact of noise on cell locomotion. We performed a systematic study of the model described in the previous section. By modifying the valuesof the biochemical reaction rate k a , the intracellular area covered by the concentration c , defined as C , andthe noise strength σ , we obtained a diverse set of cell shapes, trajectories, and speeds. An overview of thestudied cases is presented in Figure 1, where different cell shapes and average speeds are shown in the planespanned by the parameters σ and C . We identified four different types of motility with distinct shapes andtrajectories: Amoeboid cells, characterized by a motion parallel to the elongation axis, fan-shaped cells thatmove perpendicular to the elongation axis, intermediate states that combine features of both amoeboid andfan-shaped types, and oscillatory cells, where the concentration c is almost homogeneously distributed insidethe cell with only small fluctuations at the border.The transitions between the four cell types, summarized in Figure 1, are presented in more detail in Figure 2,where cell shapes are shown for k a = 2 s − and k a = 5 s − separately as a function of the parameter C , which4igure 1: Phase diagram of the cell shapes obtained from the model and their respective average speedsfor different values of the spatio-temporal noise, the reaction rate, and the maximum area coverage of theconcentration c . The intracellular concentration c is proportional to the intensity of sky blue color. Cell speedwas computed as the sum of the velocities at each time step divided by the number of time steps. Coloredboxes classify cells into four types: Amoeboid cells inside the red box, fan-shaped and similar cases inside thegreen box, intermittent cases at the transition from the amoeboid to the fan-shaped regime inside the yellowbox, and oscillatory cells inside the purple box. 5igure 2: Phase diagrams of the cell shape obtained from the model simulations when varying area coverageand noise intensity. The computational cells are obtained for k a = 2 s − (A) and k a = 5 s − (B). The intracellularconcentration c is proportional to the intensity of sky blue color.is changed in smaller increments here. In both diagrams, the simulations in the left top panels show cell typesthat mimic the vegetative and starving states of D. discoideum , analyzed in more detail in [2]. The rightbottom panels in Figure 2 A and B correspond to stable fan-shaped and rotating fan-shaped cells, respectively,which are shapes related to particular mutations of
D. discoideum [5, 40]. In between these two limits, differentdynamical regimes have been obtained during the computational study of our model, some of which have beenalso observed in our experiments with
D. discoideum , as we described below.
If we set C = 28 µm , the biochemical component c roughly occupies one quarter of the total cell area, fixedat µm (corresponding to a circular cell with radius 6 µm ). Under these conditions, the concentration c accumulates in the front part of the cell reminiscent of the typical amoeboid shape of D. discoideum cells. In thisscenario, we observe different trajectories depending on the noise intensity. A large noise intensity translatesinto slow and random cell motion, whereas low intensity leads to faster and much more persistent motion.A comparison between the dynamics of living
D. discoideum cells under amoeboid conditions and cell dy-namics generated by our model is shown in Figure 3. The numerical simulations shown in Figure 3A and Ccorrespond to low values of C and a noise intensity of 100 % of the total given in Table 1. Figure 3C, showingtrajectories for k a = 2 s − and k a = 5 s − , demonstrates that the parameter k a controls cell polarization [2]. Avalue of k a = 2 s − induces a diffuse trajectory in a small area of space and a random appearance of protrusionsformed by fluctuating amounts of concentration c along the cell membrane. For k a = 5 s − the cell exploreslarger areas due to a continuous and more stable accumulation of c in one region of the membrane that setsin the direction of motion (see Figure 3A). A representative sequence of snapshots of an experimental obser-vation of amoeboid motion of a D. discoideum cell is shown in Figure 3B, where the accumulation of actin isclearly appearing in the cell front. In addition, we present examples of individual experimentally recorded celltrajectories that cover the behavioral diversity of amoeboid
D. discoideum cells Figure 3D.
By increasing the parameter C to µm , which corresponds to half of the total area of the cell covered bythe biochemical species c , and maintaining the noise intensity between 75 % and 100 % of the total given inTable 1, we find a different motile behaviour in our model simulations. Under theses conditions, the resultsof the numerical simulations resemble the amoeboid shapes described in the previous section, however therepeated appearance of an additional large protrusion strongly modifies the trajectories of the simulated cells,6igure 3: Numerical and experimental results for amoeboid motion. (A) Sequence of several snapshots of anumerical simulation with C = 28 µm and k a = 5 s − . (B) Sequence of several snapshots of a vegetative DdBNF1 KO cell showing the amoeboid phenotype. (C) Example cell trajectories tracked from four simulationsover 600 s (two simulations with k a = 2 s − and two with k a = 5 s − ) It is important to mention that the twosmall trajectories correspond to k a = 2 s − , while the two larger trajectories to k a = 5 s − . Both cases also with C = 28 µm , (D) Examples of cell trajectories tracked from four vegetative DdB NF1 KO cells showing theamoeboid phenotype in experiments over 600 s .see Figure 4A. Initially, the amount of c is concentrated in one region at the cell border, clearly defining a leadingedge. From time to time, a part of the total amount of c changes position at the cell border thus triggeringan instability of the initial leading edge. This drives the formation of a new protrusion, where eventually mostof the total amount of c will accumulate and define a new cell front. In Figure 4B, we show an example ofsimilar intermittent behaviour that was observed in our experiments with D. discoideum cells which frequentlyswitch from amoeboid to fan-shaped motility and vice versa. Figure 4C, where we present a comparison betweenan experimental trajectory and three trajectories obtained from numerical simulations, demonstrates how thisdynamics generates trajectories with abrupt changes in direction.By further increasing the parameter C to µm , corresponding to 75% of the total area of the cell coveredby concentration c , another distinct behaviour is obtained in the numerical simulations, see the purple box inFigure 1. Keeping k a = 2 s − and a noise intensity similar to the previous case, an oscillatory behaviour of thecell border is observed due to saturation of c inside the cell. Noise-driven small displacements and a circular cell shape characterize this regime. It resembles previous experimental observations of a so-called pancakephenotype [18]. For the values of C employed in the previous section but low noise intensity, the shape of the numericallyobtained cells becomes more elongated perpendicular to the direction of motion than parallel to the directionof motion of the cell. Moreover, their elongated shape is stable over time, and they move in a highly persistentfashion. Together these features characterize the so-called fan-shaped motion of D. discoideum cells [40]. Theoverall appearance and motion characteristic of fan-shaped cells share many similarities with keratocytes, eventhough the internal organization of the motility apparatus is clearly different.For C = 56 µm , corresponding to a concentration c covering half of the total cell area, and a noise intensitythat is reduced to 50 % or less of its maximal value, a rounded elongated shape, reminiscent of a keratocytes,is observed in the numerical simulations, see for example the results in Figure 1. The trajectories are straight7igure 4: Numerical and experimental results for the intermediate unstable case (A). Sequence of six snapshotsof a numerical simulation with C = 56 µm , k a = 5 s − . (B) Sequence of snapshots of a DdB NF1 KO cell starvedfor 4 hours prior to imaging, showing the intermediate phenotype. (C) Comparison between the trajectories ofthe center of mass of three numerical simulations (solid lines) and a trajectory of the center of mass of the DdBNF1 KO cell shown in B (dotted line). In all four cases, cells were tracked over s .8igure 5: Numerical and experimental results for fan-shaped motility. (A) Sequence of three snapshots takenfrom a numerical simulation of the computed cell with parameter values C = 84 µm , k a = 2 s − and noiseintensity set to percent. (B) Snapshots of a DdB NF1 KO cell starved for 4 hours prior to imaging, showingthe fan-shaped phenotype. (C) Four examples of trajectories of the center of mass of numerically simulatedfan-shaped cells tracked over s . (D) Four trajectories of the center of mass of DdB NF1 KO cells starvedfor 4 hours prior to imaging, showing the fan-shaped phenotype tracked over 600 s in experiments.and persistent for lower noise intensities and become more erratic for high noise levels.By further increasing the covered area to C = 84 µm , we obtain similar fan-shaped cells. For k a = 2 s − the simulation produces rounded cell shapes that move at a reduced speed in a highly random fashion. Com-paring Figure 5A and B reveals the qualitative similarities between fan-shaped cells obtained from simulationsunder these conditions and the experimentally observed dynamics of fan-shaped D. discoideum cells. The modelsatisfactorily reproduces the experimental features of the cell motion. The four different realizations of trajecto-ries, generated in numerical simulations and presented in Figure 5C, display a similar persistent motion as thestraight cell trajectories observed in experiments, see Figure 5D. These trajectories resemble the trajectories offan-shaped cells with C = 56 µm and intermediate noise intensity as discussed previously. Thus, dependingon the value of the reaction rate and the noise intensity, we see remarkable differences between cell shapes andtrajectories. Numerical simulations with C = 84 µm and k a = 5 s − produce fan-shaped cells with a more elongated andcurved shape. Depending on the noise intensity, different scenarios are obtained ranging from irregular shapesand trajectories at high levels of noise to regular shapes and circular trajectories for lower noise levels (seeFigure 1). Under these conditions the trajectories may also reveal rotational dynamics. In Figure 6A and B,we show examples of rotational dynamics observed in a simulation and in an experiment, respectively, finding aqualitative similarity between them. The corresponding trajectories are displayed in Figure 6C for comparison,along with a third trajectory of another simulation. Despite the differences in radius and frequency of rotationbetween simulations and experiment, the main characteristics of a periodic rotary motion are reproduced. Notethat the concentration patterns inside the simulated cells resemble a half-moon shape, which is typical forfan-shaped cells with both straight and rotational trajectories.The transition from straight to rotational motion for different values of C and k a is shown in a phase diagramin Figure 7. The noise intensity was kept constant at 10 % in all cases. For low values of C the trajectoriesof the simulated cells are straight and only sometimes exhibit a slight curvature, depending on the realization9igure 6: Numerical and experimental results for the rotational fan-shaped case. (A) Sequence of four snapshotsobtained in the numerical simulations for C = 84 µm , k a = 5 s − and noise intensity set to percent. (B)Sequence of snapshots of a DdB NF1 KO cell starved for 5 hours prior to imaging, showing a fan-shaped cell withrotational movement. (C) Comparison of the trajectories of two simulations (solid lines) and one experimentalrealization (dotted line). The three lines correspond to trajectories tracked over more than s .and the parameter values. With increasing parameter C , irregular trajectories are observed combining straightpieces with rapid rotations giving rise to a highly erratic motion. For values of C between 80 µm and 90 µm and k a larger than 3 s − , the simulations produce rotating cells as shown in Figure 6. For even higher values of C , after a small region with curved trajectories, the cell surface is almost saturated with the concentration c ,and due to the low noise intensity, no significant net motion is observed.Thus, we have found that rotational trajectories arise for specific combination of the reaction rate and thearea coverage, and that they are favoured by low values of the noise intensity. Finally, we also investigate ifrotational trajectories can be induced by other factors. In Figure 8, we present a trajectory phase diagramspanned by the diffusion coefficient ( D ) and the surface tension ( γ ). The simulations indicate that rotationalmodes giving rise to circular trajectories are obtained by increasing the diffusion coefficient and by reducing thesurface tension. This analysis agrees with the results obtained with a similar model for keratocyte dynamicsdescribed in [13]. As we described in the previous sections, shapes and trajectories of the cells vary strongly from one caseto another. Amoeboid cells produce fluctuating displacements, fan-shaped cells exhibit also persistent androtational motion. Finally, an intermediate case between amoeboid and fan-shaped phenotypes produces motionwith characteristic features of both cases. The great majority of shapes and dynamics have also been observedin experiments, and a good qualitative agreement between experimental and numerical results was found.10igure 7: Phase diagram representing the entire range of different dynamics obtained from the model atlow noise intensity and as a function of the reaction rate k a versus the maximum area coverage C . Redcircles represent trajectories that combine straight and curvature paths (fan-shaped cells). Blue crosses indicateirregular circling trajectories. Black dots show the region where regular circular trajectories are found. Greenmarks represent curved paths. Pink squares represent trajectories of almost stationary cells (pancakes). Thenoise intensity was set to 10%.Figure 8: Phase diagram in the plane spanned by the diffusion coefficient ( D ) and the surface tension ( γ ) ofthe numerical simulations of fan-shaped cells with straight and circular trajectories corresponding, respectively,to red circles and blue crosses. Parameter values are: reaction rate k a = 5 s − , coverage area C = 84 µm , anda low level of noise intensity (10%). 11n this section we will perform a more quantitative comparison between the experimental and the numericalresults, and we define an index which clearly differentiates amoeboid and fan-shaped cells and permits tocharacterize the transition between both cases. There are several indices that are commonly used in the studiesof cell migration [26], such as the directionality ratio, the mean square displacement (MSD), and the directionalautocorrelation. Here we will compute the directionality ratio to characterize the transition between amoeboidand fan-shape. It is defined as the distance between the starting point and the endpoint of the cell trajectory,divided by the length of the real trajectory. Mathematically is defined as follows: DR = (cid:12)(cid:12)(cid:12) (cid:126)X N − (cid:126)X (cid:12)(cid:12)(cid:12)(cid:80) N − n =0 (cid:12)(cid:12)(cid:12) (cid:126)X n +1 − (cid:126)X n (cid:12)(cid:12)(cid:12) , (5)where X and X N are the initial and final positions, respectively.This ratio is close to 1 for a straight trajectory and close to 0 for a highly curved trajectory.In the following results DR was computed for every ∆ t using the positions of the cell trajectories. First, wecompare the directionality ratio for a set of simulations corresponding to vegetative amoeboid cells ( k a = 2 s − ),which produce low values of the directionality and low velocities because of their random dynamics. Here,the relation between random motion and velocity can be interpreted as follows. A low noise intensity or largeactivity rate will lead to less nucleation events of new patches of c at the membrane, favoring stable movementin one particular direction which will result in higher velocities. On the contrary, a high noise or small activityrate will generate more nucleation events at different positions of the membrane, that will compete with eachother, thus pushing the cell membrane in different directions, resulting in lower instantaneous velocities.Second, a set of simulations corresponding to starvation-developed amoeboid cells ( k a = 5 s − ) is considered,which produce intermediate values of the directionality ratio because of their more persistent motion. Finally,we analyze a set of fan-shaped cells, which produce large directionality ratios because of their highly persistentmovement. All these results are presented in Figure 9A, where we see that the three types of cells are locatedin different regions of the parameter space.The numerical results of the directionality ratio are in good agreement with the experimental measurementsof this quantity, see Figure 9B, where the experimental data is plotted for a similar amount of cells. Fan-shapedcells in both cases have large directionality ratios due to their persistent motion. The set of amoeboid cellsnaturally divides into two subsets: one with low directionality ratios and low speeds and the other one withlarger directionality ratios (although lower than in the fan-shaped cases) and speeds spreading over a widerange. In the numerical simulations, we have produced equivalent subsets by changing the parameter k a .Further similarities between the numerical results and the experimental realizations are found when com-paring panels C and D in Figure 9. Here, the average of the directionality ratio over time with their respectiveerrors for the amoeboid and the fan-shaped cells are shown for the numerical simulations in Figure 9C andthe experimental recordings in Figure 9D. It is important to mention that for a better comparison we includedexperiments curves from panel D in C and curves from panel C in D, in both cases as dotted lines and also withtheir respective errors. The good agreement between the two cases is remarkable although a systematic slightdecrease of the directionality ratio appears in the experimental case, which may be related to the more noisydynamics of the cell outline in the experimental measurements.In Figure 9E we display the directionality ratio for several cases obtained in the simulations and for the twoexperimental cases discussed above. We compare the directionality ratios in a box plot representation of thedifferent cases. First, we observe that the circular motion of fan-shaped cells gives rise to the smallest valueof the directionality ratio as expected, because the motion is confined to a small region of space. The secondobservation is that intermediate cases, discussed in the previous sections, give rise also to intermediate valuesof the directionality ratio. The largest directionality ratios are observed for the stable fan-shaped cells. Notethat we did not take into account other experimental cases because of the low number of recordings availablein some of the experiments.Finally, due to the cell shape diversity in experiments and simulations we have also analyzed the shapein a quantitative way. Computing quantities such as aspect ratio, ellipticity, or circularity are commonlycomputed when comparing cell shapes. There exist some earlier works, where cell morphology has been analyzed[17, 36, 33, 21]. In this work, we focus on the circularity measure, which quantifies how closely the shape of amarked region approaches that of a circle. Circularity can be valued between 0 and 1, where 1 represents thevalue of a perfect circle. Mathematically the circularity is defined as follows: CR = πAP , where A is the areaand P is the perimeter of the cell. Along the trajectories of the cells, we calculate in each frame the value ofthe circularity for amoeboid and fan-shaped cells with the images obtained from simulations and experiments.In Figure 10 we show a box plot representing the circularity ratio for the cases mentioned above. Herewe see the tendency of fan-shape cells to oscillate around values close to 1 due to their rounder shape. Onthe other hand, amoeboid cells present a larger variation in the value of their circularity parameter because oftheir irregular fluctuating shape. From the results we also notice a good agreement between simulations and12igure 9: Directionality ratio from simulations and experiments over 600 s . (A) Cell speed versus directionalityratio obtained from numerical simulations of a sample of 20 amoeboid cells (red circles): 10 cells correspondingto k a = 2 , C = 28 µm , and 100% noise intensity (blue shaded area) and 10 corresponding to k a = 5 , C = 28 µm , and 100% noise intensity (red shaded area); and 15 fan-shaped cells (blue crosses). (B) Cell speedversus directionality ratio obtained from experiments with a sample of 16 amoeboid cells (red circles) and 15fan shape cells (blue crosses) . (C) The mean value of the directionality ratio and standard error is plotted overtime for amoeboid (red solid lines) and fan-shaped (blue solid lines) cells obtained from numerical simulations.Experiments curves from panel (D) are shown as dotted lines. (D) The mean value of the directionality ratioand standard error is plotted over time for amoeboid (red solid lines) and fan-shaped (blue solid lines) cellsobtained from experiments. Simulation curves from panel (C) are shown as dotted lines. (E) Box plot of thedirectionality ratio for the data of several simulations and experiments over s .13igure 10: (A) Circularity box plot representation of amoeboid and fan-shaped cells from simulations andexperiments. Values of circularity for the results of 11 simulations and 10 experiments are shown: amoeboidsimulation k a = 2 s − (AS1, n=3), amoeboid simulation k a = 5 s − (AS2, n=3), and fan-shaped simulations(FS, n=5), amoeboid experiments (AE, n=5), and fan-shaped experiments (FE, n=5). (B) Represents the meanvalue of circularity over time for the studied cases. Amoeboid and fan-shape simulations are represented in blueand red crosses, repectively. Red and blue circles corresponds to experiments of amoeboid and fan-shape cell,respectively. The time for all the trajectories is 600 s .experiments for both scenarios. Just a small difference is marked in the fan-shaped case, where we obtained alarger amount of outliers in the analysis of the experimental data. To compare the cell trajectories obtained from model simulations and experimental recordings in more detail,we analyzed the correlation structure of the trajectories of both amoeboid and fan-shaped cells. For this wechose a representation in polar coordinates, so that all the displacement vectors that connect the adjacent datapoints of a trajectory are represented by an amplitude (absolute value of the displacement) and a phase (anglewith respect to the laboratory frame). The time series of the log amplitudes and the phases are displayedrespectively in Figure 11 and Figure 12. In the case of the log amplitudes, the time series are stationary andfluctuate around a constant mean value. Overall, the magnitudes and time scales of fluctuations are comparablebetween simulations and experiments and also between amoeboid and fan-shaped cases. Only in the case of thesimulated fan-shaped cells the magnitude of fluctuations is smaller. In contrast to the amplitude, the time seriesof the phases is not stationary. They furthermore reflect that the amoeboid cases reorient much more rapidlycompared to the fan-shaped cases, i.e. in the given time interval, the phase drifts over a much larger range forthe amoeboid cells than for the fan-shaped cells. This difference is particularly pronounced for the simulatedfan-shaped cells, where the phase remains almost constant over the entire measurement time, see Figure 12D.From the time series, we computed the corresponding autocovariances. For a real valued scalar time series X i , i = 0 , . . . , n − of length n we take the following estimator of the autocovariance ( | k | < n − ) ˆ γ ( k ) = 1 n − | k | n −| k |− (cid:88) i =0 ( X i + | k | − µ )( X i − µ ) , µ = 1 n n − (cid:88) i =0 X i (6)In Figure 13, the autocovariance of the log amplitude is shown for both experimental and model trajectoriesof amoeboid and fan-shaped cells. For the log amplitude of the model trajectories, we observe average correlationtimes that are slightly larger than for the experimental trajectories; they differ by a factor of approximately two.Nevertheless, in all cases the correlation time is rather short (of the order of seconds). In particular, no significantdifference is observed between amoeboid and fan-shaped cases. Note, however, that the variances differ betweenamoeboid and fan-shaped cases, which is particularly pronounced in the case of the model trajectories.In the case of the phase, we calculated the autocovariance based on the time series of the rate of phasechange between adjacent displacement vectors (the time series of the rate of phase change is stationary, whilethe phase time series is not). In Figure 14, the autocovariance of the rate of phase change is shown for all cases.While in the amoeboid cases and for the simulated fan-shaped cells correlations decay to zero within 4 sec, thefan-shaped cells in the experiment show markedly larger correlation times, see Figure 14B. On the other hand,the variances are comparable in all cases except for the simulated fan-shaped cells, where the variance is by afactor of 100 smaller than in the other cases, see Figure 14D.From our correlation analysis, we thus conclude that the cell trajectories produced by our model simulationscorrectly capture the main correlation structure of the experimental trajectories. Only in the case of the fan-shaped cells, qualitative differences can be observed. In the experimental case, an increased correlation time inthe change rate of the phase leads to smooth and persistent trajectories. The model trajectories, in contrast,do not show increased correlations in the phase change rate. Here, smooth and straight trajectories originatefrom a strongly decreased variance of the phase change rate.14
25 50 75 100 125 150 175543210 l o g d i s p l a c e m e n t (A) (B) time [seconds] l o g d i s p l a c e m e n t (C) time [seconds] (D) Figure 11: Time series of the log amplitudes of displacement vectors. (A) Amoeboid and (B) fan-shaped cellsrecorded in experiments. (C) Amoeboid and (D) fan-shaped cells produced by numerical model simulations.We consider 15 different trajectories for each type of cell.
We have studied a mathematical model that consist of biochemical dynamics in the form of a bistable reaction-diffusion equation including a noise description based on an Ornstein-Uhlenbeck process. The biochemicaldynamics is coupled to a phase field to account for the deformable cell border. The model was previouslyintroduced in [2] to characterize the dynamics of vegetative and starvation-developed amoeboid
D. discoideum cells. In that case, we found good agreement between the cell shape evolution, the intracelullar patterns, andthe center of mass movement.Here, we have now systematically explored the entire relevant range of the parameter space and qualitativelyreproduced different motility regimes observed in
D. discoideum cells. The phase diagram of the model wasexplored by changing parameters such as the noise strength, the coverage area, and the rate responsible forcell polarization, giving rise to a series of different motility scenarios as presented above. The numerical resultsreproduced the dynamics of
D. discoideum cells, which were difficult to catalog for the experimentalists. Ingeneral, the comparison of trajectories, cell shapes, and cell speeds between numerical simulations of the modeland experimental data showed good agreement. We also performed a more quantitative comparison, basedon the correlation structure of the amplitude and phase of the displacement vectors. Also in this case, goodagreement was found and only minor differences in the case of the fan-shaped cells revealed that the modelimposed smoothness of the cell trajectories of fan-shaped cells by reducing the variance of the phase changesrate, while the experimental data shows an increased correlation time instead. In future studies, longer modeltrajectories will be generated to explore anomalous behavior in the diffusive properties that has been recentlyreported also for
D. discoideum cells [37, 16]. We should distinguish between the characteristic times obtainedfor example from the autocorrelation functions and the persistence time, which refer to different concepts. Incontrast with the calculations using autocorrelation functions, the persistence times are typically calculatedfrom mean square displacements (MSD) [57, 28].The behavior of the model critically depends on the choice of the model parameters. Together with noise,realistic dynamics of intracellular patterns and cell shape changes are produced, when correct characteristictemporal scales are used. The parameter C corresponds to the area covered by the biochemical component c . It takes into account membrane deformations due to local accumulation of the biochemical component and,together with the reaction rate, reproduce variations in cell speed and persistence of motion.Our model shows similarities to the bistable reaction-diffusion model coupled to a phase field described in [13],where the motion of keratocytes is investigated. In particular, a systematic study of the transition betweenstraight and circular motion of cells moving in a keratocyte-like fashion, equivalent to our fan-shaped cells, was15
25 50 75 100 125 150 17520100102030 p h a s e [ r a d ] (A) (B) time [seconds] p h a s e [ r a d ] (C) time [seconds] (D) Figure 12: Time series of the phases of displacement vectors. (A) Amoeboid and (B) fan-shaped cells recordedin experiments. (C) Amoeboid and (D) fan-shaped cells produced by numerical model simulations. We consider15 different trajectories for each type of cell.done: a high tension tends to stabilize the cell motion to a straight trajectory, whereas large diffusion coefficientsor small velocities tend to push the cell towards rotation [13]. In our study presented here, we have investigatedthe connection of keratocyte-like behavior and amoeboid motility, apparently associated to different mechanismsand cell types. We studied the transition between the persistent fan-shaped phenotype (keratocyte-like) andthe amoeboid case, also showing intermediate dynamics and directly compared our model to experimentaldata obtained from recordings of
D. discoideum . With respect to the transition between straight and circularlymoving fan-shaped cells our results shown in Figure 8 are consistent with the earlier predictions [13]. Transitionsbetween amoeboid and fan-shaped motility modes have also been described recently in a wave-generating two-component reaction-diffusion model by Cao et al. [15] that specifically emphasizes the role of cell deformationmechanics. While our results are compatible with the findings of Cao et al. , our model clearly shows that therichness of different motility modes does not require intracellular traveling waves but can be already observedfor intracellular kinetics that relies on a single dynamical variable only.In contrast to these simple modeling approaches based on generic reaction-diffusion systems, there arealso more complex descriptions following a different biophysical approach including more detailed biochemicalreactions and mechanical forces. For example keratocyte motion has been extensively studied in [38] combiningbiochemical and mechanical aspects to model how epidermal fish keratocytes form a leading edge, polarize, andmaintain their shape and polarity. There are also more complex models where the transitions between straightand circular trajectories have been studied, see for example [48], where a minimal mechanical model is presentedconsisting of two equations, one for the force balance of the actin network and a second one consisting of areaction diffusion equation that describes the concentration of myosin, demonstrating that transitions occursfor small values of the Peclet number.On the other hand, there are reductionist approaches to keratocyte motion, see for example [13] and [61],which display similar levels of complexity as our model. Both types of descriptions contribute to a betterunderstanding of the experimentally observed dynamics and can be readily extended in different directions.For example, we are currently working on the implementation of more complex biochemical models into thephase field description. In particular, we can extend the model to more closely recover the detailed dynamicsof certain intracellular reactions, such as, for example, the phosphorylation of PIP to PIP or the dynamics ofthe associated kinases and phosphotases that affect cell polarization, membrane deformation, and pseudopodformation [59, 22]. Furthermore, the phase field framework will also allow us to implement cell-cell interac-tions [35], the behavior of cells under confined stimuli [24], in enclosed environments [46, 60], in the presence ofexternal chemical gradients [47], and also in three dimensions [14].Here, we have restricted ourselves to a comparison of our model to D. discoideum cells. However, we found16 c o v a r i a n c e l o g d i s p l a c e m e n t (A) (B) time lag [seconds] c o v a r i a n c e l o g d i s p l a c e m e n t (C) time lag [seconds] (D) Figure 13: Autocovariance calculated from the time series of the log amplitudes of displacement vectors shownin Figure 11. (A) Amoeboid and (B) fan-shaped cells recorded in experiments. (C) Amoeboid and (D) fan-shaped cells produced by numerical model simulations. We consider 15 different trajectories for each type ofcell. Dashed red thick lines correspond to the mean Autocovariances.that our model is also able to describe more diverse situations observed, for example, in keratocytes, whereclose to the transition to circular motion, bipedial motion was observed that relies on local alternation of celldisplacements during persistent motion [6].
In summary, we have studied a model based on a bistable reaction-diffusion equation with Ornstein-Uhlenbecknoise for the intracellular biochemistry, coupled to a dynamical phase field to describe the cell membranedynamics. The results obtained from the numerical integration of the model show that essential features ofamoeboid and fan-shaped motion observed in experiments of motile
D. discoideum cells are reproduced by ourmodel. We found close qualitative agreement between the numerical simulations and the experiments and, insome cases, motility measures such as the directionality ratio even showed quantitative agreement. The study ofthe correlation structure of the cell displacements furthermore allowed us to perform a quantitative comparisonof the cell trajectories from our model simulations with experimental data.Based on our simulations we furthermore conclude that a continuous transition between amoeboid and fan-shaped motions is a realistic scenario, as some of the predicted intermediate states observed in simulationshave been confirmed in experiments with
D. discoideum cells. We speculate that the same model can be alsoemployed to describe the motion of other cell types with different motion strategies such as keratocytes orfibroblasts.
Acknowledgment
E.M. and C.B. acknowledge funding by the Deutsche Forschungsgemeinschaft in the framework of Sonder-forschungsbereich 1294, project B02. S.F. and C.B. gratefully acknowledge funding by the Deutsche Forschungs-gemeinschaft in the framework of Sonderforschungsbereich 937, project A09. S.A. and E.M. thank supportfrom MICINN (Spain), and FEDER (European Union), under project PGC2018-095456-B-I00. E.M. acknowl-edges also financial support from CONACYT. F.F. acknowledges financial support from the
Juan de la Ciervaprogramme (grant IJC2018-038463-I) from the Spanish MICINN, from the Obra Social la Caixa through theprogramme Recerca en Matemàtica Col · laborativa and the CERCA Programme of the Generalitat de Catalunya. c o v a r i a n c e p h a s e c h a n g e r a t e (A) (B) time lag [seconds] c o v a r i a n c e p h a s e c h a n g e r a t e (C) time lag [seconds] (D) Figure 14: Autocovariance calculated from the time series of the phase change rate of displacement vectors.(A) Amoeboid and (B) fan-shaped cells recorded in experiments. (C) Amoeboid and (D) fan-shaped cellsproduced by numerical model simulations. We consider 15 different trajectories for each type of cell. Dashedred thick lines correspond to the mean Autocovariances.
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