Cell Motion Alignment as Polarity Memory Effect
Katsuyoshi Matsushita, Kazuya Horibe, Naoya Kamamoto, Koichi Fujimoto
aa r X i v : . [ q - b i o . CB ] J u l Cell Motion Alignment as Polarity Memory E ff ect Katsuyoshi Matsushita ∗ , Kazuya Horibe, Naoya Kamamoto and Koichi FujimotoDepartment of Biological Sciences, Osaka University, Toyonaka, Osaka, JapanJuly 26, 2019 Abstract
The clarification of the motion alignment mechanism incollective cell migration is an important issue commonlyin physics and biology. In analogy with the self-propelleddisk, the polarity memory e ff ect of eukaryotic cell is afundamental candidate for this alignment mechanism. Inthe present paper, we theoretically examine the polaritymemory e ff ect for the motion alignment of cells on thebasis of the cellular Potts model. We show that the polar-ity memory e ff ect can align motion of cells. We also findthat the polarity memory e ff ect emerges for the persistentlength of cell trajectories longer than average cell-cell dis-tance.Motion alignment plays various roles widely in self-propelled systems including migrating cells[1], mov-ing organisms [2], molecular motors[3], self-propelleddroplets [4] and swarming robots [5]. In particular, thealignment of migrating cells is indispensable for cell orga-nizing in organogenesis, wound healing and immune re-sponse [6, 7, 8]. In these processes, migrating cells exhibitcollective behavior commonly observed in self-propelledsystems [9, 10, 11], including various patterns [12, 13],active turbulence [14], traveling wave excitation [15]. Forthe understanding of these behavior, an important issueis to clarify the alignment mechanism as their underlyingbasis.The alignment mechanisms of other self-propelled sys-tems may give hints for this clarification. In many self-propelled systems including bird flocking [16, 17], thedirect aligning-interaction through visual contact is sup- ∗ [email protected] posed and is expressed as an interaction between vectordegrees of freedom, which are so-called polarity. Unfor-tunately, this direct aligning-interaction is not expected inmigrating cells lacking visual contact. As another can-didate, the peculiar alignment of self-propelled disks ordeformable particles is considerable[18, 19, 20, 21] be-cause it requires only an indirect interaction of polari-ties through excluded volume. In these cases, the di-rect aligning-interaction is not necessary at least for thisalignment. Especially for the disks, the polarity memoryrecording the past motion aligns the subsequent motionthrough collisions even with rotation-symmetric excludedvolume [22, 23, 24, 25].Since the polarity memory recording cell motion iswell known for eukaryotic cells[26, 27], the polaritymemory e ff ect is expected to work in the collective cellmigration of Dictyostelium Discoideum (Dicty)[28, 29],keratocytes[30, 31] and neural crest cell[32] . In the stud-ies of these collective migration [33, 34], since the align-ment was attributed only to the chemataxis[35, 36], thecontribution of polarity memory has been overlooked. Inaddition, even when the chemotaxis is artificially inhib-ited [37, 15], other e ff ects including intercellular adhe-sion [38, 39], shape anisotropy[40, 41, 42] or contactinhibition[43, 44] / activation[45, 46] of locomotion havethe polarity memory e ff ect be experimentally invisible.Therefore, the theoretical examination to evaluate the po-larity memory e ff ect is a powerful method to clarify thealignment mechanism.In this examination, the spontaneous fluctuation in theshape may give considerable di ff erences between eukary-otic cells and self-propelled disks [47, 48], even when theaveraged cell shape is rotation symmetric. In particular,1ince the shape fluctuation due to protrusion [49, 50] orblebbing [51, 52] stochastically propels cells, the fluc-tuation clearly gives the qualitative di ff erence betweencells and self-propelled disks in the propulsion mecha-nism. In fact, the behavior of the cells cannot directlybe deduced from the theory of the self-propelled disks,because cells lose propulsion in unfluctuating shape likethe disks. Therefore, for the theoretical examination, thetheory, only by itself, is insu ffi cient. At least, it should becombined with the cellular model suitably expressing thepropulsion mechanism[53].In the present work, we investigate a model for mi-grating cells with the polarity memory and the interactiononly through rotation-symmetric excluded volume. Weperform Monte Carlo simulations based on the CellularPotts model[54, 55] and thereby confirm that the align-ment mechanism due to the polarity memory is e ff ective.Then, we further examine the shape fluctuation e ff ect onthe alignment and show that the alignment emerges at thecrossover between the persistent length of cell trajectoriesand the average cell-cell distance when the propulsion isstronger than the shape fluctuation.Let us consider the two dimensional Cellular Pottsmodel consisting of migrating cells with the polaritymemory [56]. This model is defined by the Hamiltonian, H = H CC + H CE + H Vol + H Mot (1)for given Potts state { m ( r ) } on the square lattice. Here, r represents a site in the square lattice. m ( r ) takes a numberfrom 0 to the number of cells N . m ( r ) = r is empty. Otherwise, m ( r ) is the index of cellson the site r . For simplicity of examination, we fix N .The first and second terms in rhs of Eq. (1), H CC = Γ X h rr ′ i η m ( r ) m ( r ′ ) η m ( r ′ ) η m ( r )0 , (2) H CE = Γ X h rr ′ i (cid:0) δ m ( r )0 η m ( r ′ ) + η m ( r )0 δ m ( r ′ )0 (cid:1) , (3)represent interface parts of cell-cell and cell-empty space. δ mn is Kroneker δ and η mn is defined by (1 − δ mn ). Γ rep-resents cell-cell interface tension and Γ cell-empty spaceinterface tension. For Γ > Γ [57], cells are suspended.Since the suspended cells interact only with excluded vol-ume, we impose this condition on Γ for our purpose. Thesummations in rhs of these equations are taken over all the neighboring sets which consist of the nearest and nextnearest neighbor sites[54].The third term in the rhs of Eq. (1), H Vol = κ N X m = − P r δ mm ( r ) V ! . (4)represents the balk part. This term expresses the situationwhere the occupation area of cells is maintained to be V ,as empirically observed. Here, κ represents area sti ff ness.The forth term in the rhs of Eq. (1), H Mot = − ε N X m = X r δ mm ( r ) p m · e m ( r ) , (5)represents propulsion [56]. Here, ε is the strength ofpropulsion, p m is the unit vector of the polarity and e m ( r )is the unit vector indicating from the center of the m -thcell R m = P r r δ mm ( r ) / P r δ mm ( r ) to a site r . Notice that thecell takes rotation-symmetric shape even with this term,because this term does not induces tensile stress. p m obeys[30, 58] d p m dt = a τ p ∆ t " d R m dt − p m · d R m dt ! p m . (6)Here, t is time, τ p is the time scale ratio of p m change to R m change and a is the lattice constant. This equation rep-resents the polarity memory during the time of τ p ∆ t in thesense of the solution, p m ( t ) ∼ R t −∞ dt ′ d R m ( t ′ ) / dt ′ exp[ − ( t − t ′ ) /τ p ∆ t ] / a τ p ∆ t [56]. Here, ∆ t represents the time of sin-gle monte carlo steps (mcs) and we set ∆ t = H , the shape fluctuation of cells is re-produced by monte carlo simulation [53]. In this sim-ulation, the state { m ( r ) } is updated to { m ′ ( r ) } as fol-lows: firstly a site r is randomly selected. Then, thestate m ( r ′ ) at a randomly selected site r ′ in its neigh-boring set is copied to the site r . This copy is ac-cepted with the Metropolis probability P ( m ′ ( r ) | m ( r )) = min(1 , exp (cid:8) − β (cid:2) H ( m ( r )) − H ( m ′ ( r )) (cid:3)(cid:9) ), where β is in-verse temperature. Otherwise, this copy is rejected. Wecall this procedure a copy trial. Mcs conventionally con-sist of 16 L copy trials [54], where L is the linear size ofsquare system. The series of mcs expresses the shape fluc-tuation. For each mcs, p m is assumed to be a slow variable2ith R m and is updated once by the Eular method [58].Simultaneously, R m is also updated and is fixed in mcs.In this simulation, we employ the following parame-ters: L is 256 and V is 64. These parameters are chosenso as to realize the suspended state for the tractable N from 64 to 512. These N corresponds to the area fractionof φ from 0.11 to 0.85. Additionally for the same pur-pose, we choose surface tensions Γ = Γ = β of 0.3. Since β ≥ ff ect. Firstly,we decide the simulation strategy on the basis of the the-ory of the self-propelled disk. To the theory, the motionalignment appears only in the low damping parameter γ for the angle of p m , θ m . Therefore, when the polaritymemory e ff ect works, the alignment depends on γ in thesame manner with the disk and a certain threshold γ c forthe alignment is present. By the reproduction of this de-pendence, we evaluate the polarity memory e ff ect.In order to determine γ , let us derive an e ff ective self-propelled disk from this model. Since γ is independent ofintercellular interaction, we consider an isolated cell forsimplicity and ignore the intercellular interaction. In ouroverdamped simulation based on H , we can suppose theLangevin equation of cell motion, d R m dt = ε p m · X r δ mm ( r ) * ∂ e m ( r ) ∂ r + R m + f (7)In the derivation of force in rhs, we use the fact that H Mot is only the term explicitly depending on R m in H . Wealso interpret the R m derivative as the derivative of thecell boundary coordinate r with fixed R m . This is becausethe force is practically exerted on the cell boundary as de-fined in the virtual work due to the deformation of cellshape. We divide this force into the shape-average and-fluctuation parts, where h . . . i R m represents the averageover cell shapes with fixed R m and f is deviation rep-resenting the shape fluctuation of cells. In addition, wereformulate the equation of p m in Eq. (6) as an equationof θ m similar to the self-propelled disk [23]. For this pur-pose, we substitute p m with (cos θ m , sin θ m ) and d R m / dt with | d R m / dt | (cos ψ m , sin ψ m ), where ψ m is the angle of the cell velocity. Then, by multiplying d R m / dt to bothsides of Eq. (6), we read d θ m dt = − a τ p (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) d R m dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) sin( θ m − ψ m ) . (8)The former, Eq. (7) is identical to an overdamped activeBrownian particle [25] rather than the self-propelled disk[22, 23] in the sense of similarity in overdamping fea-ture. In contrast, the latter, Eq. (8) is similar to the self-propelled disk. In the comparison of Eq. (8) with the cor-responding equation in the literature [23], | d R / dt | / a τ p isread as γ .In the self-propelled disk, the damping parameter γ is a control parameter of the motion alignment [22, 23].Naively, the dependence on γ = | d R / dt | / a τ p in our modelis evaluable from the τ p dependence. However, in contrastto self-propelled disk, γ also depends on the cell velocity, | d R / dt | , which is not a model parameter. To consider thise ff ect, we focus on the dependence of the alignment notonly on τ p and but also ε . Here, ε is one of the parameterswhich determine | d R / dt | in Eq. (7) and the dependence of | d R / dt | on ε is | d R / dt | ∼ ε for ε larger than f .Secondarily, we give the method to identify the mo-tion alignment in our examination. Refer to the previouswork[17, 23], we define the motion alignment by finitevalues of the order parameter of p m , P = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) T Z T + T T dt N N X m = p m (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (9)Here, T represents the number of mcs for the relaxationto the steady state. T is the number of mcs for the time av-erage. We empirically employ T = mcs and T = mcs [58]. Here, note that P is not adequate for detection ofheterogeneous alignments of motion, including vortexes.Therefore, to avoid the heterogeneous alignments due tothe boundary e ff ect of the system, we simply employ theperiodic boundary condition.To examine the dependence of the alignment on τ p and ε , we calculate P as a function of τ p for various ε . Foreasily observing the alignment, we choose relatively higharea fraction of cells φ = NV / L ≃ .
43. The data areplotted in Fig. 1(a). P at τ p = ε . Inthis case, the cells randomly move and p m does not alignas shown in Fig. 1(b). With increasing τ p , P is kept to3 < P > t e p (a)(b) (c) t c Figure 1: (a) h P i as a function of time scale ratio τ p inEq. (6) for various strength of motility ε . The di ff erentsymbols indicate di ff erent values of ε . Snapshots for (b) ε = τ p = .
0, and for (c) ε = τ p = .
0. Di ff er-ent color region indicates di ff erent Potts domains, namelycells. Black region represents the empty region. Whitearrows represent the direction of polarity.be 0 for small τ p and then rapidly increases at a thresh-old value of τ p , τ c , excepting ε = P takes high val-ues around unity which indicates the motion alignmentshown in Fig. 1(c). Therefore, even in cells propelled byshape fluctuation, the polarity memory e ff ect works, ex-cepting cases of too small ε ’s.In addition, τ c is almost independent of ε excepting toosmall value. This independence seemingly contradicts theexpectation from the constant threshold γ c = | d R / dt | / a τ c in previous work [23]. In fact, in the linear response | d R m / dt | ∼ ε , the expectation of τ c ∼ ε is not shown inFig. 1(a). Since the constant threshold γ c , which leadsto this contradiction, is based on the theory of rigid self-propelled disk, the rigidity of the disk is a candidate origin of this contradiction. Therefore, to understand this inde-pendence, the shape fluctuation of cells should be consid-ered beyond the theory for the rigid disk.To explore the origin of this independence, we considerthe e ff ect of this shape fluctuation in the motion align-ment. This e ff ect comes from f in Eq. (7) and reducesthe polarity memory e ff ect, through | d R m / dt | and ψ m inEq. (8). Intercollision processes reflect this reducing be-cause it shortens the persistent length of cell trajectories, l p , which is defined for an isolated cell di ff erently fromthe mean free path. Since the intercollision processes alsoreflect the cell density φ , changes of φ interferes the e ff ectof l p . Therefore, by utilizing this interference in the inter-collision processes, we can evaluate the shape fluctuatione ff ect.For this evaluation, we calculate P as a function of τ p for various values of φ with ε = ≫ f ). The resultof P is plotted as a function of τ p for vairous φ in Fig. 2. P commonly takes 0 for small τ p and increases with in-creasing τ p . The alignment threshold, τ c , where P rapidlyincreases, largely decreases with increasing φ in contrastto the independence on ε in Fig. 1(a).Let us consider the mechanism of this strong decreas-ing τ c with increasing φ . From discussions so far, we fo-cus on intercollision processes. In these processes, since p m depends on f through | d R m / dt | in Eqs. (7) and (8),cells must maintain their direction of polarity against thedisorder of shape fluctuation for the stable alignment.Therefore, the persistent length, l p , exists as a finite valueand determines the stability of alignments. This situationis contrast to the self-propelled disks having infinite l p due to rigid shape[23]. Hence, for migrating cells, τ c isdetermined not only by γ but also by l p . In small shapefluctuation, the memory time of polarity is typically givenalmost by t R = γ − = a τ p | d R m / dt | − . This time gives thepersistent length, l p ≃ t R (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) d R m dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = a τ p . (10) l p must be larger than the intercellular distance l d for cellsto maintain the direction of motion during intercollisionprocesses. This gives a necessary condition for the align-ment, namely, l p > l d .To intuitively understand this condition, we discuss thetwo serial collision processes shown in Figs. 3(a) and3(b). Firstly, we consider the case of l p > l d . In this case,4 t p < P > f f Figure 2: h P i as a function of time scale ratio τ p in Eq. (6)for various area fraction φ . Di ff erent symbols indicate dif-ferent value of φ . The arrow shows the decreasing of τ c with increasing φ .as shown in Fig. 3(a), the direction of motion after the 1stcollision is maintained until the 2nd collision, because the2nd collision typically occurs in the cell movement lengthof l d . Owing to the polarity memory e ff ect, cells aligntheir motion in the common direction through the 1st and2nd collisions[22, 23]. This aligning repeats in the follow-ing collisions and finally results in the motion alignmentover all cells. Next, we consider the other case of l p < l d .In this case, the polarity relaxes owing to the shape fluctu-ation in the intercollison periods. Therefore, as shown inFig. 3(b), even when the motion of cells temporary alignin the same direction in the 1st collision, then the aligneddirection of motion is lost until the 2nd collision. As a re-sult, the motion alignment does not growth over all cellsand thereby cannot induces the motion alignment over allcells. The marginal case between these cases, l p ≃ l d , de-termines τ c .On the basis of this marginal condition, we can discussthe independence of τ c on ε in Fig. 1(a). For this dis-cussion, notice that since this system corresponds to theactive Brownian particle with high rotational Pe´clet num-ber limit because of the absence of heat bath in Eq. (8)[25], the phase separation does not appear. Therefore, theconfiguration of cells becomes uniform (See Figs. 1(b)and 1(c)) and thereby l d is typically a φ − / d . Here, d = l d = a φ − / d and l p in Eq. (10) are commonly independent of d R m / dt , themarginal condition is also independent of d R m / dt . This is
0 2 4 6 t f f < P > p ( a ) ( b )(c) l d l d l p l p t c f Figure 3: Schematic view of two sequential collision pro-cess of cells (grey circle) for (a) l p > l d and for (b) l p < l d . The circles represent a cell and intracircle arrowsrepresent a polarity direction. Solid arrows indicates thecell motion and their straight regions almost correspondto shorter one in l d and l p . Dashed and dotted lines witharrow heads represent l p and l d . (c) h P i as a function of τ p φ / d . Di ff erent symbols indicate di ff erent value of φ .the origin of the independence of τ c on ε .To confirm this condition, we can use the fact that l p / a = τ c must be equal to l d / a = φ − / d . In this case, τ c de-creases with increasing φ as keeping the value of l p / l d = τ c φ / d . In Fig. 3(c), we replot P in Fig. 2 as a functionof τ p φ / d . In comparison with data in Fig. 2, the datacollapse into a single curve, excepting the cases of higharea fractions φ = φ = ff ects[59, 60]. This result suggests that the crossover of lengthscales between the persistent length, l p , and intercellulardistance, l d , determines τ c .In conclusion, similarly to the self-propelled disks,cells align their motion by a pure polarity memory e ff ect,while the condition for alignment has a slight correction5ue to the shape fluctuation[61]. The polarity e ff ect is apowerful candidate of the alignment mechanism compa-rably with the shape-velocity coupling e ff ect for the softdeformable particles [62, 20, 21]. This implies the biolog-ical function of the polarity memory as the driving forceof the motion alignment. A prominent example is theearly development stage of Dicty [63] and therein Dictyexhibits the extention of memory time [28]. For utilizingthis function in motion alignment, Dicty may strategicallyextends the memory time of polarity in this early stage.We thank fruitful discussions with H. Kuwayama, H.Hashimura, S. Yabunaka, K. Hironaka, T. Hiraoka and R.Ishiwata. We also thank the support on the research re-source by M. Kikuchi and H. Yoshino. This work is sup-poted by supported by JSPS KAKENHI (Grant Number19K03770). References [1]