Excitable actin dynamics and amoeboid cell migration
EExcitable actin dynamics and amoeboid cell migration
Nicolas Ecker ∗ Department of Biochemistry, University of Geneva, 1211 Geneva, Switzerland andDepartment of Theoretical Physics, University of Geneva, 1211 Geneva, Switzerland
Karsten Kruse † Department of Biochemistry, University of Geneva, 1211 Geneva, SwitzerlandDepartment of Theoretical Physics, University of Geneva, 1211 Geneva, Switzerland andNCCR Chemical Biology, University of Geneva, 1211 Geneva, Switzerland (Dated: December 14, 2020)
Abstract
Amoeboid cell migration is characterized by frequent changes of the direction of motion andresembles a persistent random walk on long time scales. Although it is well known that cell migrationis typically driven by the actin cytoskeleton, the cause of this migratory behavior remains poorlyunderstood. We analyze the spontaneous dynamics of actin assembly due to nucleation promotingfactors, where actin filaments lead to an inactivation of the nucleators. We show that this systemexhibits excitable dynamics and can spontaneously generate waves, which we analyse in detail.By using a phase-field approach, we show that these waves can generate cellular random walks.We explore how the characteristics of these persistent random walks depend on the parametersgoverning the actin-nucleator dynamics. In particular, we find that the effective diffusion constantand the persistence time depend strongly on the speed of filament assembly and the rate of nucleatorinactivation. Our findings point to a deterministic origin of the random walk behavior and suggestthat cells could adapt their migration pattern by modifying the pool of available actin. ∗ [email protected] † [email protected] a r X i v : . [ phy s i c s . b i o - ph ] D ec . INTRODUCTION The ability of cells to migrate is one of their most fascinating characteristics. Duringmesenchymal migration, cells persistently polarize and adhere to the substrate, which leadsto persistent directional motion [1, 2]. In contrast, during amoeboid migration, cells fre-quently change their polarization and hence their direction of motion. They also adhereless strongly to the substrate than cells during mesenchymal migration. Amoeboid migra-tion can be observed for the soil amoeba
Dictyostelium discoideum and for immune cells,for example, dendritic cells. The random walk performed during amoeboid migration is animportant aspect of immune cells’ task to scan the organism for pathogens. The origin ofthe random polarization changes during amoeboid migration is largely unknown [3] and itis not clear to what extent cells can control the characteristics of their random walk.Molecular noise is an obvious candidate for generating random migration [4, 5]. Theprocesses involved in generating migration are indeed subject to noise due to the stochasticnature of molecular reactions. However, these stochastic events take place on length andtime scales that are small compared to those characteristic of cellular random walks. It isnot obvious how cells could influence the strength of this noise and hence their migrationbehavior. Fluctuating external cues could also generate random walks. Indeed, cells respondto a multitude of external signals, notably, chemical or mechanical gradients, and adapt theirmigration accordingly. Here, the cells have a certain degree of control as they can tune thestrength of their responses. However, cellular random walks have been observed in theabsence of external cues [6–8]. Finally, there is the possibility that cells generate internalpolarization cues, which would give them the maximal possible control over their behavior.In this context, spontaneous actin polymerization waves have been proposed to provide suchinternal cues [9].Actin is an important constituent of the cytoskeleton, which drives cell migration. Itassembles into linear filaments called F-actin, with two structurally different ends. Thisstructural polarity of actin filaments is exploited by molecular motors that transform thechemical energy released during hydrolysis of adenosine-triphosphate (ATP) into mechanicalwork. The assembly and disassembly of F-actin is regulated by various cofactors. Forexample, formins and the Arp2/3 complex nucleate new filaments. Actin depolymerizingfactor (ADF)/cofilin, on the other hand, can promote their disassembly. Interestingly, there2s evidence for feedback between the actin cytoskeleton and the activity of these regulatorycofactors. For example, nucleation promoting proteins have been reported to be less activein regions of high F-actin density [10, 11]. Such a feedback can lead to spontaneous actinpolymerization waves [12–16]. Such waves are present during migration [12, 14, 17, 18], andtheoretical analysis has shown that they can be sufficient to cause cell motility [9, 17, 19, 20].From a physical point view, spontaneous actin polymerization waves are akin to wavesin excitable media. Early indications of this connection were given in [12, 13, 16]. Furthersupport came from the observation that actin polymerization waves exhibit a refractoryperiod [14, 21]. More recently, the actin network/cytoskeleton of
D. discoideum was shownto be poised close to an oscillatory instability [18]. The dynamics of excitable systems isexemplified by the FitzHugh-Nagumo system, which is a very much simplified version of theHodgkin-Huxley equations describing action potentials traveling along the axons of nervecells.In this work, we analyze the description of actin polymerization waves proposed inRef. [15]. We clarify its connection to the FitzHugh-Nagumo system and characterize thewaves it generates. Furthermore, we use a phase-field approach [22, 23] to study the impactof actin polymerization waves on cell migration. Here, the phase field is an auxiliary fieldthat distinguishes between the inside and outside of a cell. We analyze in detail a recentlyintroduced current for confining proteins to the cell interior [9]. Finally, we explore the rela-tion between the system parameters and the characteristics of the random walks generatedby chaotic polymerization waves.
II. ACTIN DYNAMICS
In this section, we present the description of the actin cytoskeleton developed in Refs. [9,15, 20]. After establishing the dynamic equations, we discuss their relation to the FitzHugh-Nagumo model (FHN) and show that oscillations and waves emerge spontaneously in oursystem. Finally, we characterize the waves shape, length and propagation velocity.3 . The dynamic equations
Amoeboid cell migration is driven by the actin cytoskeleton, which is mostly concentratedin the actin cortex, a layer beneath the plasma membrane. The cortex thickness is a fewhundred nanometers [24–26] and thus much smaller than the lateral extension of a cell( > µ m). In this work, we aim at describing the actin cytoskeleton adjacent to the substrateand thus use a two-dimensional geometry.We use the continuum description of Refs. [9, 15, 20] for the actin dynamics, where theactin density is captured by the field c . The alignement of actin filaments can lead to(local) orientational order in the system. This effect is captured by the orientational orderparameter p , which is similar to the nematic order parameter of liquid crystals. In thedynamic equations, all terms allowed by symmetry up to linear order and up to first orderin the derivatives are considered, such that ∂ t c = − v a ∇ p − k d c + αn a (1) ∂ t p = − v a ∇ c − k d p . (2)Here, v a is the average polymerization speed and k d an effective degradation rate, see Fig. 1.Note, that this description neglects flows of the actin network [27] that could, for example,be generated by molecular motors. We also neglect a possible diffusion term that wouldaccount for fluctuations in the actin dynamics. We have checked that our results are notaffected qualitatively for sufficiently small diffusion constants. Equations (1) and (2) canalso be obtained by coarse-graining a kinetic description [20].The last term of Eq. (1) is a source term that describes nucleation of new actin filaments.For the conditions present in cells, new actin filaments hardly form spontaneously. Instead,specialized proteins assist in this process. Examples are members of the formin family or theArp2/3 complex. These proteins can be in an active or an inactive state and their spatialdistribution in a cell can change with time. In this way they can contribute essentially toorchestrating the organization of the actin cytoskeleton. We introduce the densities n i and n a to describe these actin nucleation promoting factors - ’nucleators’ for short -, where theindices refer to the inactive and active forms, respectively. Active nucleators generate newactin at a rate α , hence the form of the last term in Eq. (1).The dynamic equations for the fields n a and n i capture their transport by diffusion and4heir activation and inactivation dynamics. On the time scales that are relevant for thedynamics we study in the remainder of this work, nucleator synthesis and degradation canbe neglected. Consequently, the dynamic equations should conserve the number of nucleatingproteins, (cid:82) A ( n a + n i ) dA = An tot = const , where A is the cell area adjacent to the substrate.We write ∂ t n a = D a ∆ n a + ω (cid:0) ωn a (cid:1) n i − ω d cn a (3) ∂ t n i = D i ∆ n i − ω (cid:0) ωn a (cid:1) n i + ω d cn a . (4)The diffusion constants for active and inactive nucleators are D a and D i , respectively. Spon-taneous activation of nucleators occurs at rate ω . There is some experimental evidence for apositive feedback of nucleator activation [28], such that active nucleators promote the activa-tion of further nucleators. We capture this effect by the parameter ω . Nucleator deactivationcan occur spontaneously. Furthermore it has been proposed that nucleator deactivation canbe induced by factors that are recruited by actin filaments [10, 11, 28, 29]. We assume thatthe latter dominates [28] and neglect spontaneous deactivation. Actin induced deactivationis controlled by the parameter ω d .To fully determine the dynamics of the fields c , p , n a , and n i , Eqs. (1)-(4) have to becomplemented by boundary conditions. In this section, we use periodic boundary conditionsto study the intrinsic actin dynamics. Later we will add the presence of the cell membrane Membrane αω kd ω ω d va Inactive NucleatorActive NucleatorActin Monomer
FIG. 1. Schematic representation of the actin dynamics captured by Eqs. (1)-(4). Blue circlesrepresent inactive nucleators. They are spontaneously activated at rate ω , a process that is oftenassociated with membrane binding. The activation rate is enhanced by already active nucleators,represented by green circles, which is captured by the parameter ω . Active nucleators generatenew actin filaments (red) at rate α . The latter grow at velocity v a and spontaneously disassembleat rate k d . Furthermore, actin filament attract factors that inactivate nucleators. This complexprocess, which can involve several proteins, is captured by the rate ω d . ω − and space by (cid:112) D i /ω . We use the same notation for the rescaled parametersas in Eqs. (1)-(4), such that the non-dimensionalization corresponds to setting ω = 1 and D i = 1 . Unless noted otherwise, we use in the following the parameter values given inTable I. TABLE I. Nondimensional parameter values used in this work unless indicated otherwise. Thelength and time scales are chosen such that the ensuing dynamics is comparable to that of immaturedendritic cells [9].Parameter Meaning Value D a Diffusion constant of active nucleators · − v a Effective actin polymerization speed . - . k d Effective filament degradation rate ω Cooperative binding strength of nucleators · − ω d Detachment rate of active nucleators . − . α Actin polymerization rate n tot Average total nucleator density 700 L System length 1.3 N g Number of grid points per dimension 256 ω − Time scale . s (cid:112) D i /ω Length scale 63.5 µ m D Ψ Phasefield relaxation / surface tension coefficient · − κ Phasefield timescale modifier (cid:15)
Area conservation strength β Actin-membrane interaction coefficient . · − A Mean cell area . . Spatially homogeneous solutions Consider the case of homogeneous protein distributions. The constraint on the nucleatordensity thus is n a + n i = n tot = const , where n tot is the average total nucleator density.According to Eq. (2), the polarization field is decoupled from the other fields and will tendto zero, p → , for t → ∞ . The remaining dynamic equations become ∂ t c = − k d c + αn a (5) ∂ t n a = (cid:0) ωn a (cid:1) ( n tot − n a ) − ω d cn a , (6)where we have used n i = n tot − n a .Equations (5) and (6) are reminiscent of the FitzHugh-Nagumo (FHN) system [30, 31].In its general form, the latter is given by [32]: (cid:15) ∂ t w = v − aw (7) ∂ t v = − w + I + f ( v ) . (8)Equation (7) describes generation of the ’carrier’ w by the ’driver’ v and degradation of w with rate a . Here, (cid:15) (cid:28) is a small parameter, such that the dynamics of w occurs on longertime scales than the one of v . The second equation captures inhibition of v by w and I is anexternal stimulus. Finally, f ( v ) describes a feedback of v on its own production: in general,it promotes generation of v for small values of v , whereas it inhibits its production for largervalues of v .A typical specific choice of f is f ( v ) = v − v . In that case, the system essentially dependsonly on the parameter a and the external stimulus I , because variations in (cid:15) do not affectthe dynamics qualitatively as long as (cid:15) is small. Although the stimulus can depend on time,for the time being, we consider the case of constant I . Information about the asymptoticbehavior can be obtained by analyzing the nullclines in phase space, that is, the curvesdefined by the respective conditions ˙ v = 0 and ˙ w = 0 in the ( v, w ) -plane. Intersections ofthe two nullclines correspond to fixpoints of which there are either one or three. In the lattercase, the system is bistable as two fixpoints are stable against small perturbations, whereasthe third is unstable, see Fig. 2A.In the case that there is one fixpoint, it can be stable or unstable against small pertur-bations. If it is unstable, the system exhibits a limit cycle and asymptotically oscillates, see7ig. 2B. In the opposite case, the FHN system can present excitable dynamics, that is, eventhough the fixpoint is stable against small perturbations, sufficiently large perturbationsinduce an ’excursion’ in phase space, before returning to the fixpoint, see Fig. 2C. Thisbehavior can be observed, when the intersection of the two nullclines is left to the minimumor right to the maximum of the v -nullcline. If the intersection is between the two extrema,the system spontaneously oscillates, see Fig. 2B.The similarity between the actin-nucleator dynamics, Eqs. (5) and (6), and the FHNsystem becomes evident when choosing c = w , n a = v , (cid:15) = α , a = k d /α , I = n tot , and f ( v ) = − v + ωIv − ωv . The two dynamical systems differ in that the term − w of Eq. (8)corresponds to − ω d vw in Eq. (6). Lastly, in contrast to v and w in the FHN system, whichcan take any real value, we now have w ≥ and ≤ v ≤ n tot . Note that, in the FHNsystem, I is an external signal and can depend on time, while the corresponding term n tot in the actin-nucleator system is a constant. - 2 - 1 0 1-3-2-1012 vw - 2 - 1 0 1-101234 vw - 2 - 1 0 1-101234 vw n a c n a n a A B CD E F c c FIG. 2. Phase space diagrams for spatially homogenous dynamics. A-C) Phase space for theFitzHugh-Nagumo equations (7) and (8) with a = 2 , I = 0 (A), a = 0 . , I = 2 (B), and a = 0 . , I = 2 (C). D-F) Phase space for the dynamic equations (5) and (6) with k d = 5 , α = 50 (D), k d = 50 , α = 400 (E), and k d = 80 , α = 400 (F). Other parameters as in Table I. In each case, thenullclines are shown in red, the vector fields as blue arrowheads and an example trajectory in black.For the FHN equations, the diagrams show a bistable case (A), a limit cycle (B) and an excitablecase (C). For Eqs. (5) and (6) we present limit cycles (D, E) and an excitable case (F). For theseequations, there is no bistable case. α is not a small parameter.Let us now take a closer look at the nullclines. Analogously to ˙ w = 0 for the FHN system, ˙ c = 0 yields a linear relation between c and n a and the n a -nullcline exhibits the characteristicS-shape of ˙ v = 0 . The nullclines of our system intersect exactly once in the region c ≥ and n a ≥ , such that there is only one fixpoint ( c , n a, ) , independently of the parametervalues. To see this, note first that the c -nullcline is a straight line through the origin. Nowconsider the function c ( n a ) defined by the nullcline ˙ n a . If there were parameter values forwhich three intersection points existed, then there would be some tangent to c ( n a ) with anegative y-intercept c y . However, for any value n a ≥ the value c y is given by c y = ωn a + 2 n tot − n a , (9)which is always positive as the number of active nucleators is bounded from above by thetotal number of nucleators, n tot ≥ n a , proving the above statement.If the fixpoint is unstable against small perturbations, the system exhibits oscillationsas mentioned above, see Fig. 2D, E. In case, ( c , n a, ) is stable, the system can amplify afinite perturbation, but will eventually return to the fixpoint,see Fig. 2F. before performinga linear stability analysis of the fixpoint, we first obtain a physical picture of the necessaryconditions for an instability based on the nullclines.The fixpoint can only be unstable, when the n a -nullcline c ( n a ) exhibits two extrema for n a > . Explicitly, the nullcline is given by c ( n a ) = n tot − ωn a + ωn tot n a − n a ω d n a . (10)Consequently, lim n a →∞ c ( n a ) = −∞ and lim n a → + c ( n a ) = + ∞ . To determine whether the n a -nullcline is monotonously decreasing, we consider the positive roots of the derivative c (cid:48) = ∂c/∂n a . They are determined by − n tot − ωn a + ωn tot n a . (11)9his equation always has a negative real solution. Two positive roots can only exist if thediscriminant of the polynomial is negative. This leads to ωn > . In that case, the tworeal roots take the form n ± a = n tot ± π − sin − (cid:16) − ωn (cid:17) . (12)The value of n + a is always positive and n − a is always negative, because the argument of thesine function takes values between π/ and π/ . The second positive root is n a = n tot sin − (cid:16) − ωn (cid:17) . (13)In conclusion, the fixpoint ( c , n a, ) is unstable and the system oscillates for ωn > andif n a < n a, < n + a .We now turn to a linear stability analysis of the fixpoint. For the dominating growthexponent s of the perturbation, we find s = a − k d + (cid:112) ( a − k d ) − αω d n a, , where a = − − ωn a, +2 ωn tot n a, − ω d c only depends on k d /α . By increasing the nucleationrate α while keeping k d /α = const the nullcline remains unaffected. For k d > a the real partof the eigenvalue becomes negative, leading to a stationary state. Thus, k d < a is the lastcondition for the presence of oscillations in our system. The oscillation frequency ω F closeto the instability can be estimated from the imaginary part of the growth exponent s of asmall perturbation through ω F = (cid:61) ( s ) = √ αω d n a, . C. Wave solutions
After having analyzed the dynamic equations (1)-(4) for spatially homogenous fields, wenow turn to the general case and study the system in a domain of size L with periodicboundary conditions in the x - and y -direction. Then, the system can generate a variety ofspatially heterogeneous solutions, including planar traveling waves and stationary patterns,see Fig. 3 and Supplementary Movies 1,2. In the following we will determine the parameterregion, in which these patterns exist and characterize the shape of planar waves.10 . Linear stability analysis We start our analysis by investigating the stability of the homogenous steady state againstsmall spatially heterogeneous perturbations. The homogenous state is characterized by c ( x ) = c = αn a /k d , p ( x ) = p = 0 , and n i, = n tot − n a, with (1 + ωn a, ) n i, − ω d c n a, = 0 . (14)As shown above there is only one positive solution n a, ≤ n tot to this equation, such thatthere is a unique homogenous stationary state.Consider c ( x, y, t ) = c + δc ( x, y, t ) and similarly for the fields p , n a , and n i . Linearizingthe dynamic equations with respect to the steady state and expressing the perturbations interms of a Fourier series, δc = (cid:80) ∞ n,m = −∞ ˆ c nm e − i ( q x,n x + q y,m y ) and similarly for δ p , δn a , and A BC D y yyy x xxx
FIG. 3. Snapshots of solutions for the actin concentration c to Eqs. (1)-(4) in two dimensions withperiodic boundary conditions. A, B) Travelling planar waves for D a = 0 . , ω d = 0 . , v a = 0 . (A) and D a = 0 . , ω d = 0 . , v a = 0 . (B). Green arrows indicate the direction of motion.The disclinations in (B) might heal after very long times. C, D) Stationary Turing patterns for D a = 0 . , ω d = 0 . , v a = 6 . (C) and D a = 0 . , ω d = 0 . , v a = 9 . (D). For different initialconditions a pure hexagonal pattern of blobs can appear. All other parameters as in Table I. n i with q x,n = 2 πn/L and q y,m = 2 πm/L , leads to ddt ˆ c nm = − iv a ( q x,n ˆ p x,nm + q y,m ˆ p y,nm ) − k d ˆ c nm + α ˆ n a,nm (15) ddt ˆ p x,nm = − iv a q x,n ˆ c nm − k d ˆ p x,nm (16) ddt ˆ p y,nm = − iv a q y,m ˆ c nm − k d ˆ p y,nm (17) ddt ˆ n a,nm = − D a (cid:0) q x,n + q y,m (cid:1) ˆ n a,nm + (cid:0) ωn a, (cid:1) ˆ n i,nm + 2 ωn i, n a, ˆ n a,nm − ω d ( c ˆ n a,nm + ˆ c nm n a, ) (18) ddt ˆ n i,nm = − (cid:0) q x,n + q y,m (cid:1) ˆ n i,nm − (cid:0) ωn a, (cid:1) ˆ n i,nm − ωn i, n a, ˆ n a,nm + ω d ( c ˆ n a,nm + ˆ c nm n a, ) . (19)The solutions to these equations are of the form ˆ c ∝ e s nm t etc, where s nm are the growthexponents of the modes ( n, m ) . If s nm > , then a heterogeneous steady state emerges. Ifinstead, (cid:60) ( s nm ) > and (cid:61) ( s nm ) (cid:54) = 0 , then an oscillatory state, that is, either a standing ora traveling wave, can be expected.Our numerical solutions indicate that all instabilities in our system are super-critical suchthat there is no coexistence of different states that are not linked by a symmetry transfor-mation. Close to the instability, the wavelength λ of the unstable determines the wavelength of the emerging pattern. This remains true in a large region beyond the instability,see Fig. 4. The wave length depends only weakly on the actin assembly velocity v a , Fig. 4A,B, and not on the nucleator inactivation rate ω d , Fig. 4D, E. It increases with the diffusionconstant D a , Fig. 4C, and decreases with the cooperativity parameter ω , Fig. 4F.In contrast to the wave length, we only get a poor estimate of the wave’s propagationvelocity from the linear stability analysis. In the following we use a variational ansatz todetermine the wave form and propagation velocity of plane waves.
2. Wave form
We start by rewriting the dynamic equations (1)-(4). First of all, we combine the equa-tions for the actin density c and the polarization p to obtain one equation for the density.Furthermore, we exchange n i for N = n a + n i . Finally, we consider solutions in a reference12rame moving with the wave velocity v . We will use periodic boundary conditions withperiod Λ . We thus arrive at v − v a Λ ∂ x c + (cid:16) v Λ ∂ x + k d (cid:17) ( k d c − αn a ) (20) − vN = 1Λ ∂ x N − − D a Λ ∂ x n a − vn tot (21) − v Λ ∂ x n a = D a Λ ∂ x n a + (cid:0) ωn a (cid:1) ( N − n a ) − ω d cn a , (22)where we have scaled space by Λ , such that the period is equal to , see App. A.Equations (20) and (21) are linear and can be solved as soon as n a is known, see App. A.To solve the nonlinear Eq. (22) we make the following ansatz for a right-moving wave in theinterval [ − / , / n a ( a , a , a , a , x ) = a e − a x (1 + tanh[ a x ])(1 − x ) a , (23)where a to a are variational parameters. We constrain a and a to vary in the intervals [5 , and [30 , , respectively, whereas a can take on the values , , ; we do not imposeany constraints on a . Note that the test function (23) does not fulfill the periodic boundarycondition. However, since a , a (cid:29) , n a ( a , a , a , a , ± / ≈ .In our ansatz , the active-nucleator density n a increases according to the exponentialpolynomial x a e a x at the front of the wave. In this region actin is nucleated and increasescorrespondingly. The trailing region of the wave is defined by a decrease of the active ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ v a λ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ λ ■■■■■■■■■■■■■■■■■■■■■■■■■■ n tot λ v a ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ω λ ■ ■■ ■ ■ ■ ■ ■ ■ ω d λ ■ ■■ ■ ■ ■ ■ ■ ■ ω d λ A B CD E F FIG. 4. Wavelength as a function of system parameters. Orange dots represent values obtained fromnumerical solutions in two spatial dimensions with periodic boundary conditions ( L = 1 . ), bluelines are the results of a linear stability analysis, see Sect. II C 1. Parameter values are ω d = 0 . (A), ω d = 0 . (B), v a = 0 . and ω d = 0 . (C), v a = 0 . (D), v a = 0 . (E), and v a = 0 . and ω d = 0 . (F). All other parameters as in Table I. a x ) . This decrease results from a threshold actinconcentration beyond which nucleator inactivation occurs at a higher rate than nucleatoractivation. Due to the large value of a , the nucleator density drops sharply to zero andalso the actin density decays exponentially in the trailing region. The corresponding decaylength is v/k d , see App. A.After solving the linear equations (20) and (21), we calculate an error by integrating thedifference between the left and the right hand sides of (22) over the whole period: Err ( a , a , a , a , v ) = (cid:90) − | F ( n a , c, N ) | dxF ( n a , c, N ) = v Λ ∂ x n a + D a Λ ∂ x n a − ω d cn a + (cid:0) ωn a (cid:1) ( N − n a ) . (24)Minimizing the error yields values for the variational parameters a to a , v , and Λ .In Figure 5A, we compare a solution obtained by the variational ansatz and by numer-ically solving the dynamic equations (1)-(4). The agreement is very good with the largestdeviations being present at the front of the wave. Similarly, the parameter dependence ofthe wave speed is reproduced well by our variational ansatz, Fig. 5B, C. The wave speedis essentially independent of the actin polymerization speed v a as long as v a (cid:46) , which isconsistent with our earlier remark that the wave dynamics is driven by the nucleator activ-ity rather than actin assembly. Furthermore, the wave speed increases with the parameter ω d describing nucleator inactivation by actin. Indeed, as ω d increases, nucleators are morerapidly inactivated, such that they become available for activation at the wave front.
3. Stationary patterns
In addition to planar traveling waves, the dynamic equations (1)-(4) can also producestationary patterns, see Fig. 3C, D. These Turing patterns appear if v a (cid:38) and consisteither of ’blobs’ of high or low active nucleator densities or of labyrinthine stripes of highactive nucleator density. These structures can coexist in the same system. Since our focusin this work is on actin waves, we refrain from discussing these states further.14 xc(x) v a v SimulationVariational ansatz 0..25 0..3 0..35 0..40.00.51.01.52.02.5 ω d v SimulationVariational ansatz
AB C
FIG. 5. Shape and velocity of traveling waves in one dimension. A) Actin and active nucleatorconcentrations c (green) and n a (orange) for v a = 0 . and ω d = 0 . . Dots are from a numericalsolution, solid lines are obtained from the variational ansatz Eq. (23) and the solution Eq. (A6).B, C) Wave speed as a function of the actin polymerization velocity v a (B) and the nucleatorinactivation parameter ω d (C). All other parameters as in Table I. III. CELL MOTILITY FROM ACTIN POLYMERIZATION WAVES
Having analyzed the intrinsic actin dynamics, we now turn to a characterization of cellmigration patterns emerging form spontaneous actin waves. We start by introducing aphase-field approach for describing the cellular domain. It contains a novel current forconfining the nucleators to the cell interior. We then describe migration patterns and studythe dependence of their characteristics on the system parameters.
A. Phase-field dynamics
Similar to previous work on cell motility, we use a phase-field approach to define thedynamic cell shape [22, 23]. A phase field is an auxiliary scalar field with values rangingbetween 0 and 1, which are called the pure phases of the system. We treat values of 0 asbeing outside of the cell and values of 1 as being inside. The phase-field dynamics is given15y [22, 23] ∂ t Ψ = D Ψ ∆Ψ + κ Ψ(1 − Ψ)(Ψ − δ ) − β p · ∇ Ψ , (25)where δ = 12 + (cid:15) (cid:18)(cid:90) A Ψ dA (cid:48) − A (cid:19) . (26)The term proportional to κ derives from a free energy with minima at the pure phases. Theyare separated by an energy barrier at Ψ = δ . Conservation of the cell area/volume can beachieved by adjusting the value of δ as described in Eq. (26): The actual cell area is givenby (cid:82) A Ψ dA (cid:48) , it’s target area by A . If the cell is bigger than A , then δ > . such thatthe overall cell area shrinks and vice versa . For sufficiently large values of κ , the transitionbetween the two pure phases is sharp.The transition region between the two pure phases determines the position of the cellmembrane. Specifically, we implicitly define the location of the cell membrane by all positions r with Ψ( r ) = 0 . . The term proportional to D Ψ accounts for interfacial tension betweenthe two pure phases and thus the surface tension of the membrane. For cells, surface tensionof the membrane dominates its bending energy [23], which we neglect. Finally, the termproportional to β describes the interaction strength between the phase field and the actinnetwork. The interaction is always directed along the polarization vector, such that themembrane can be pushed outwards or pulled inwards [23]. In our case we do not observepulling to the inside.The dynamics of the actin network and the nucleators is confined to the cell interior bymultiplying the dynamic equations (1)-(4) by Ψ . Conservation of the nucleators is an im-portant aspect of these dynamic equations. Simply multiplying the corresponding transportterm by Ψ violates conservation of the total nucleator amount and also leads to nucleatorsleaking out of the cell interior [20]. Here, we choose a different option and instead modifythe nucleator current at the position of the membrane. For a particle density n , we write ∂ t n = D (Ψ∆ n − n ∆Ψ) (27) = ∇ ( D Ψ ∇ n ) − ∇ ( Dn ∇ Ψ) . (28)This term evidently conserves the total particle number. It can be interpreted as a combina-tion of scaling the diffusion constant with Ψ and introducing an inwards flux proportional to16 at the membrane. This suggests that the expression is efficient for keeping the nucleatorsinside the cell. This is indeed the case as can be seen by solving for the stationary state ofEq. (28), which is given by n ∝ Ψ .In this context, it is also instructive to look at the discretized version of the right handside of Eq. (28). Using the discretized Laplacian (cid:52) n j ≡ ( n j − − n j + n j +1 ) h − , where h isthe discretization length, we get in one dimension: D (Ψ j (cid:52) n j − n j (cid:52) Ψ j ) = D n j +1 Ψ j + n j − Ψ j − n j Ψ j +1 − n j Ψ j − h . (29)From this expression it is evident that nucleators can hop only to a site j inside the cell,i.e., with Ψ j > , see Fig. 6A.In presence of the phase field, the dynamic equations are ∂ t c = Ψ( αn a − v a ∇ · p ) − k d c (30) ∂ t p = − v a Ψ ∇ c − k d p (31) ∂ t n a = D a (Ψ∆ n a − n a ∆Ψ) + Ψ((1 + ωn a ) n i − ω d cn a ) (32) ∂ t n i = Ψ∆ n i − n i ∆Ψ − Ψ((1 + ωn a ) n i − ω d cn a ) . (33)For actin, the diffusion current can be neglected as argued above, and thus its dynamicsis unaffected by the modified diffusion introduced in Eq. (28). However, since the actinconcentration is not a conserved quantity and rapidly degraded in the absence of nucleators,we chose the degradation term to act also outside the cell interior to get rid of any actinthat might have left the cell. B. Actin-wave induced cell trajectories
In Figure 6B we show the phase diagram of the different dynamics patterns of the phasefield’s center r c = (cid:82) r Ψ( r ) d r as a function of the parameters v a and ω d . Five differentdynamic states can be distinguished. Below a critical value of ω d , waves do not emerge inthe system and the center settles into a stationary state. The critical value of ω d dependsonly weakly on v a . There is a second critical value, such that the center r c is again stationaryif ω d is larger than this critical value.Close to the critical values of ω d , the actin-nucleator system forms a spiral wave, seeMovie 3. These spirals are symmetric and do not deform the phase field. They spin around17 fixed point, which coincides with the center r c . Since the dynamic equations are isotropic,solutions with clockwise or counter-clockwise rotations coexist. As the value of ω d is, re-spectively, further increased or decreased, the spiral loses its symmetry. In this case, themotion of the center r c becomes erratic and can be described as a random walk.Three different types of random walks can be identified. First, the center r c can exhibitdiffusive dynamics, see Fig. 6C and Movie 4. Second, it can perform a random walk, wherestraight segments along which the cell moves with constant velocity alternate with segmentsof diffusive motion, see Fig. 6D and Movie 5. Also in the third type of random walk thecell center changes between two states, namely, diffusive or curved motion, see Fig. 6E and a ω d Static DistributionStable SpiralDiffusive MotionPersistent/Erratic MotionCell Splits
ABC D E
FIG. 6. Polymerization waves in presence of a phase field. A) Schematic comparison of the dis-cretized diffusion in absence (left) and presence (fight) of a phase-field, see Eq. (29). B) Phasediagram of migration patterns as a function of the actin growth velocity v a and nucleator inactiva-tion parameter ω d . C-E) Example trajectories with cell outlines drawn at 8 equidistant points intime for v a = 0 . and ω d = 0 . (diffusive migration, C), v a = 0 . and ω d = 0 . (random walkwith straight segments, D), and v a = 0 . and ω d = 0 . (random walk with curved segments, E).Scale bars correspond to a length of 0.3. Other parameters as in Table I. C. Dependence of migration characteristics on parameter values
The random walks discussed above fall into the class of persistent random walks. For apersistent random walk, the velocity of the walker has a finite time autocorrelation, that is,its magnitude and direction persist for a characteristic time τ . Note that there are severalrealizations of a persistent random walk. In a run-and-tumble process, the walker exhibitsperiods during which it moves along straight lines with constant speed. These periodsare interrupted by events during which the walker essentially does not move but changes itsdirection. Another possibility is that the direction of motion and the speed varies constantlyin a smooth way. Inbetween these extremes, the segments of a run-and-tumble motion showscontinuous changes of the velocity. In all cases, the mean square displacement (cid:104) r ( t ) (cid:105) is givenby (cid:104) r ( t ) (cid:105) = 4 Dt + 2( vτ ) ( e − t/τ − . Here, v is the mean velocity of the persistent periodand D is the diffusion constant describing the effective diffusive behavior on very long timescales. In the following we study, how the effective parameters τ , v , and D depend on oursystem parameters.As shown in Figure 7A-C, the persistence time τ , the speed v and the diffusion constant D initially increase with v a and then decrease for larger values of v a . The non-monotonousbehavior of these quantities is a consequence of two competing effects. To see this, let usfirst recall that the wave speed does not increase with increasing v a , Fig. 5B. However,19he polarization of the actin network does increase in this case as can be read of directlyfrom Eq. (2). Consequently, the interaction between the actin field and the membranegets stronger and the membrane deformations are more pronounced. At the same time,the pronounced membrane deformations feed back on the actin waves, which are gettingless regular. Thus, the cell polarization is less efficient, such that the periods of persistentmigration are effectively shortened. At the same time, the migration speed decreases duringthese periods. This is confirmed by the mean instantaneous speed of the cell centers, whichare very similar to the effective speed v , see Fig. 7B.As a function of the parameter ω d , we observe a transition from a persistent to a diffusiverandom walk. Below the transition, the parameters v and D increase with ω d . In contrast,the value of τ depends non-monotonically on ω d ; it first increases and then decreases. Abovea critical value of ω d , we find τ = 0 . For these values, the diffusion constant varies onlyslightly with ω d and is two orders of magnitude smaller than for the persistent random walks.The dependence of v on ω d is linear for the persistent random walks. Note that the valuesof v obtained from fitting the mean square displacement for τ ≈ are not meaningful. Themean instantaneous velocity is again very similar to v for the persistent random walks. Inthe diffusive regime, it still grows linearly with ω d . This is in line with the wave velocity,which increases with ω d , see Fig. 5C. 20 ● ● ● ●● ● ● ● v a D ω d D ● MSD ■ Mean Speed0.2 0.3 0.4 v a v ● MSD ■ Mean Speed0.35 0.4 0.45 ω d v v a τ ω d τ ω = 0.36 d v = 0.38 a A DB EC F ● ● ● ● ● ● ● ● ● -5 -4 -3 -2 ● ● ●● ● ● ● -5 -4 -3 -2 ● ● ● ● ● ● ● ●■ ■ ■ ■ ■ ■ ■ ■ ■ ● ● ●● ● ● ● ■ ■ ■■ ■ ■ ● ● ● ● ● ● ● ● FIG. 7. Effective parameters of random walk trajectories. A-C) Diffusion constant D (A), speed v (B), and persistence time τ (C) as a function of the actin polymerization speed v a . D-F) As(A-C), but as a function of the nucleator inactivation parameter ω d . Values were measured byfitting a persistent random walk model to the mean square displacement (MSD) of the respectivetrajectories. In (B) and (E), also the mean speed measured directly on the trajectories is shown(orange squares). Other parameters as in Table I. IV. DISCUSSION
In this work, we have shown that a deterministic, self-organized system describing theactin assembly dynamics in living cells is capable of generating cellular random walks akinto amoeboid migration [9, 20]. We elucidated its relation to excitable systems by a compar-ison with the FitzHugh-Nagumo system and characterized in detail spontaneously emergingtraveling waves. We recall that the wave propagation speed is independent of the actinpolymerization velocity v a , such that the waves are driven by the nucleator dynamics andnot the actin dynamics.By coupling the actin dynamics to a phase field, we studied the impact of the spontaneousactin dynamics on cell migration. In this context, we introduced a new expression for thenucleator current in presence of a phase field, such that nucleators are confined to the cellinterior. In other phase-field studies of cell migration, conservation of particle numbers is21ypically not an issue and all material leaving the cell interior is simply quickly degraded [23].If nucleators are not conserved, for example, by replacing the concentration of inactivenucetaors n i by a constant, then the density of active nucleators diverges and waves areabsent from the system. In Ref. [20], nucleators that had leaked out of the system werereintroduced into the cell by homogenously distributing them in the cell interior. In contrast,the current − D (Ψ ∇ n − n ∇ Ψ) used in this work acts locally. All phases reported in Ref. [20]are recovered and also the topologies of the phase spaces are the same in both systems withone notable exception: whereas in the present work erratic migration occurred for largervalues of v a and ω d than for persistent migration, it was the opposite in Ref. [20].By analyzing the mean-squared displacement of the simulated cells, we characterizedtheir persistent random walks in terms of a diffusion constant, a persistence time, and thecell speed. We linked these effective parameters to the actin-polymerization speed v a andthe strength ω d of the negative feedback of actin on nucleator activity. It showed that theseparameters had a strong effect on the effective diffusion constant and the persistence time,whereas the cell speed varied only by a factor of two. This suggests that by changing thepool of available actin monomers, cells can control important aspects of their random walks.This might allow notably cells of the immune system patrolling an organism for pathogensto adapt their behavior to the tissue they reside in.A negative feedback of actin filaments on the nucleator activity is essential for the emer-gence of spontaneous actin-polymerization waves. In cells, indirect evidence has been foundfor this negative feedback, but it remains to disentangle the molecular interactions involved.They likely involve the action of small GTPases, which also take part in the signal trans-duction pathways that couple external stimuli to the intracellular actin dynamics. In futurework it will be interesting to couple the actin-nucleator system to such external signals andstudy the ensuing dynamics.Furthermore, it will be interesting to study in future work collective cell migration drivenby spontaneous actin-polymerization waves. Previous phase-field studies revealed how stericinteractions between cells can lead to collective migration [33, 34] and how topographicsurface structures influence this behavior [35]. In the context of our work, one might expectinteresting synchronization phenomena between actin waves in different cells.22 CKNOWLEDGMENTS
We thank Carles Blanch-Mercader for helpful discussions and the Swiss National ScienceFoundation (grant 205321-175996) for financial support.
Appendix A: Wave profile
In this appendix, we determine the actin and nucleator densities for a wave traveling atvelocity v .
1. Actin density
The actin density c and the polarisation field p are given by Eqs. (1)-(4), which in onespatial dimension and after non-dimensionalization read ∂ t c = − v a ∂ x p − k d c + αn a (A1) ∂ t p = − v a ∂ x c − k d p. (A2)Deriving Eq. (A1) with respect to time, we can eliminate the field p and obtain a linearequation for c with an inhomogeneity proportional to n a : ∂ t c + 2 k d ∂ t c + k d c − v a ∂ x c = α ( k d + ∂ t ) n a . (A3)This is the equation for a wave with speed v a , internal friction with k d and a drivingproportional to k d . The source of the wave depends on n a and its time derivative. Wewill assume that the active nucleators move as a solitary wave with velocity v , that is, n a ( x, t ) = n ( x − vt ) .In the reference frame moving with the nucleation wave speed v and normalized by thewavelength L , Eq. (A3) becomes v − v a ( k d L ) ∂ x c − vk d L ∂ x c + c = αk d (1 − vk d L ∂ x ) n a . (A4)The homogeneous solution c h ( x ) to this equation can be written as c h ( x ) = e λx (cid:20)(cid:18) v λ a − c λλ a (cid:19) sinh( λ a x ) + c cosh( λ a x ) (cid:21) , (A5)23here λ = k d Lv/ ( v − v a ) and analogously λ a = k d Lv a / ( v − v a ) . In the above equation, theamplitude of the homogeneous solution is fixed by the conditions c h (0) = c and c (cid:48) h (0) = v .The solution to the in-homogeneous Eq. (A4) with the source term S ( x ) = αk d L v − v a (1 − vk d L ∂ x ) n a is obtained by the method of variation of constants. We write c = AS ( x ) and v = AS (cid:48) ( x ) , where A is the Wronskian of our system and arrive at the full solution c ( x ) = αLvv − v a (cid:90) n a ( x + ξ ) e − λξ (cid:16) cosh( λ a ξ ) − v a v sinh( λ a ξ ) (cid:17) dξ, (A6)where λ = v/ ( v − v a ) and analogously for λ a .The solution corresponds to a fraction of v − v a v of the scaled nucleator density decaying on alengthscale of L − = λ − λ a and a fraction of v + v a v decaying with L + = λ + λ a . The decaying partof the actin wave can be fitted perfectly with the single parameter a (cid:0) v − v a v e − λ − x + v + v a v e − λ + x (cid:1) .Note that the nucleation rate α has no effect on the shape of the wave, but only affects itsamplitude.The solution for the polarization field p is obtained by solving Eq. (A2) for p ( x, t ) ≡ p ( x − vt ) .
2. Total nucleator density
We now rewrite the dynamic equations (1)-(4) for the active and inactive nucleator con-centrations n a and n i in terms of the total nucleator concentration N = n a + n i and n a . Inone spatial dimension and after non-dimensionalization, we have ∂ t n a = D a ∂ x n a + (cid:0) ωn a (cid:1) ( N − n a ) − ω d cn a (A7) ∂ t N = ∂ x N + ( D a − ∂ x n a (A8)In the reference frame of the traveling wave, (A8) becomes − vL ∂ x N = 1 L ∂ x N + D a − L ∂ x n a . (A9)Integrating once and determining the integration constant by integrating once more over theentire system, we arrive at a first order equation for the total amount of nucleators, ∂ x N + vLN = vLn tot + (1 − D a ) ∂ x n a (A10)with n tot being the average total nucleator density.24quation (A10) implies that with a homogeneous total nucleator concentration N = const = n tot , gradients in n a also vanish. Thus, a heterogeneity in the total nucleator con-centrations is necessary to observe waves and wave propagation requires nucleator transport.Furthermore, D a is a measure for how far active nucleators can diffuse around the bulkof the wave while bound before detaching, on a time scale proportional to the wave period τ , thus affecting the wave length. D a needs to be sufficiently smaller than D i to create alength scale difference large enough to enable the formation of the bulk of the wave andmaintain the imbalance in total nucleator concentration, otherwise the constant distributionof proteins is the only solution (as the wave length grows too large, or the imbalance shrinkstoo much to be supported).The solution to Eq. (A10) is given by N ( x ) = n tot + (1 − D a ) (cid:34) n a ( x ) − vL (cid:32) e − vL vL ) (cid:90) − n a ( ξ ) e vL ( ξ − x ) dξ + (cid:90) x − n a ( ξ ) e vL ( ξ − x ) dξ (cid:33)(cid:35) (A11)From this equation we see that there are no waves, when D a = 1(= D i ) .
3. Active nucleator density
Using the solutions for c , Eq. (A6), and N , Eq. (A11), we arrive at a single equation forthe distribution of the active nucleators in the reference frame moving at the wave speed v : D a L ∂ x n a ( x ) + vL ∂ x n a ( x ) = ω d αLvv − v a n a ( x ) (cid:90) n a ( x + ξ ) e − λξ (cid:104) cosh( λ a ξ ) − v a v sinh( λ a ξ ) (cid:105) dξ − (cid:2) ωn a ( x ) (cid:3) n i ( x ) , (A12)where n i ( x ) = n tot − D a n a ( x ) − (1 − D a ) vLe vL − (cid:90) n a ( x + ξ ) e vLξ dξ (A13)is the distribution of inactive nucleators. This non-linear integro-differential equation canbe solved using the variational ansatz of Sect. II C.25 ppendix B: Movie captions Movie 1:
Example of a traveling wave solution to Eqs. (1)-(4) in two dimensions withperiodic boundary conditions for v a = 0 . , ω d = 0 . . Other parameters as in Table I.Disclinations can take very long times to heal. Movie 2:
Example of a Turing pattern generated by Eqs. (1)-(4) in two dimensions withperiodic boundary conditions for v a = 6 . , ω d = 0 . , D a = 0 . . Other parameters as inTable I. Movie 3:
Symmetric spiral wave solution of Eqs. (30)-(33) for v a = 0 . and ω d = 0 . .Other parameters as in Table I. Colors indicate the actin concentration, red line correspondsto Ψ = 0 . . Movie 4:
Asymmetric spiral wave solution of Eqs. (30)-(33) for v a = 0 . and ω d =0 . , leading to diffusive motion. Other parameters as in Table I. Colors indicate the actinconcentration, red line corresponds to Ψ = 0 . . Movie 5:
Wave solution of Eqs. (30)-(33) for v a = 0 . and ω d = 0 . , leading to a dynamicsof the phase field’s center, where straight segments alternate with diffusive segments. Otherparameters as in Table I. Colors indicate the actin concentration, red line corresponds to Ψ = 0 . . Movie 6:
Wave solution of Eqs. (30)-(33) for v a = 0 . and ω d = 0 . leading topersistent random walk of the phase field’s center, where curved segments alternate withdiffusive segments. Other parameters as in Table I. Colors indicate the actin concentration,red line corresponds to Ψ = 0 . . [1] Friedl P, Wolf K : Plasticity of cell migration: a multiscale tuning model.
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