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A Stochastic Model for Actin Waves in Eukaryotic Cells

A stochastic model of spontaneous actin wave formation in eukaryotic cells that includes positive feedback between the actin network and filament nucleating factors on the membrane is developed and analyzed. Simulation results show that the model can produce a variety of actin network behavior depending on the conditions. Actin spots of diameter about 0.5 µm can be formed and persist for tens of seconds at low actin concentrations, and these spots may either shrink and die or grow and develop into fully-developed propagating waves. The model correctly captures the vertical profile of actin waves along line scans through wave fronts, as well as the separation between the region enclosed by circular actin waves and the external area. Our results show how the complicated actin behavior depends on the amounts and state of various membrane molecules. Author Summary Locomotion of eukaryotic cells is a complex process that involves the spatio-temporal control and integration of a number of sub-processes, including the transduction of chemical or mechanical signals from the environment, local and global modification of the cytoskeleton, and translation of the intra- and extracellular signals into a mechanical response. In view of the complexity of the processes, understanding how force generation and mechanical interactions with the surroundings are controlled in space and time to produce cell-level movement is a major challenge. Recent experimental work has shown that a variety of actin waves propagate within cells, both under normal conditions and during re-building of the cytoskeleton following its disruption. Controlled disruption and re-building of the actin network has led to new insights into the key components involved in actin waves, and here we develop a stochastic model that can qualitatively and quantitatively describe the dynamical behavior of such waves.

propagation [34]. The cytoplasmic part of the F-actin wave is assembled based on the known processes 1 in actin network formation, which include filament branching, severing, capping, and end-wise polymer-2 ization and depolymerization. The author concluded that wave propagation depends on three major 3 processes: auto-catalytic polymerization of F-actin, random filament orientation, and slow recruitment of 4 NPF from the cytoplasm to the membrane. However there are restrictions imposed by the assumptions, 5 for instance, any actin cluster whose nearest distance from the membrane is larger than 60 nm is assumed 6 to disappear and is removed from the simulation. Thus the model can give some insight into the dynamics 7 near the surface, but cannot reproduce the full height of the network. In addition, the propagation of 8 the waves is driven in part by diffusion of the network along the membrane. 9 To date, none of the existing models includes coupling between F-actin polymerization and membrane-  NPF is activated by free filament barbed ends in close proximity to the membrane, the activated state 17 NPF* is converted to the recovery state NPF** after participation in filament branch nucleation, and 18 NPF** slowly reverts to the native inactive state. It is known that NPFs can diffuse on the membrane, 19 and a major conclusion from our analysis is that the spread of de novo filament nucleation above the 20 basal level is driven by lateral diffusion of NPF* that is activated locally by F-actin, and this drives 21 expansion of actin waves. This differs from a recent model [34] wherein network expansion depends on 22 filament elongation in the direction parallel to the plane of the membrane. Our simulation results agree 23 qualitatively with the characteristics and various behaviors of actin waves in vivo, and in particular, show 24 that localized, highly-branched actin spots emerge at sites where NPF is activated. At low levels of NPF 25 activation an actin spot either grows for a short time and then dies out, or it forms an expanding wave. 26 The fate of an actin spot is determined by two competing processes: NPF activation by NPF-stimulated 27 F-actin nucleation and conversion of NPF* into NPF** after F-actin nucleation. Mobile actin spots can 28 form randomly, and adjacent actin spots may develop into circular actin waves within the region over 29 which a wave has passed. This continual generation of actin spots and their maturation into waves agrees with the observations in substrate-attached cells [2]. Our results also show that the dynamics are those of 1 an excitable medium, in that when multiple wave fronts collide they annihilate each other and eventually 2 die out, as observed in vivo [9]. An intriguing result of our model is that when a wave propagates to 3 the border of a region in which indigenous nucleation sites are present, the wave stalls and eventually 4 decays, which replicates the in vivo behavior when it reaches to the border of the SA portion of the 5 cell membrane [9]. The simple positive feedback loop between F-actin and NPF in the current model 6 can easily be extended by incorporating details of signaling networks such as the Ras/PI3K/Factin and 7 integrin/PI3K/Rac pathways as they become known, so as to more realistically describe the coupling 8 between membrane adhesion and F-actin waves. 14 Initially G-actin, Arp2/3, coronin and capping protein (CP) are distributed uniformly in the cytosol, 15 where they diffuse freely, whereas NPF's diffuse on the membrane. Indigenous nucleation sites at which 16 filaments can nucleate are fixed on the membrane, but this is energetically unfavorable and hence only 17 occurs at a low basal rate. These nucleation sites could be viewed as membrane-anchored filament 18 nucleators such as formin or ponticulin. We call the filaments nucleated at these sites backbone filaments, 19 to contrast them with the branched filaments initiated on backbone filaments by Arp2/3 complex. The 20 barbed end of a backbone filament is attached at the nucleation site, perhaps by Myo-IB (cf. Fig. 3) and 21 is able to elongate until CPs cap the barbed end and free nucleation sites for a new round of backbone 22 filament nucleation. We assume that backbone filament nucleation requires free nucleation sites, but not 23 NPF activation, and thus a basal level of F-actin nucleation and polymerization is always present on the 24 membrane. In the experimental context this only occurs in the SA portion of the membrane.

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To initiate the dynamics, a small number of NPF's are activated locally to mimic the signaling 26 transduced from the SA portion of the membrane. Activated NPF (NPF*) then recruits Arp2/3 and 27 G-actin sequentially onto the membrane and forms a complex, which generates a filament branch upon 28 binding to either a backbone filament or a previously-formed branch filament, either of which is called a 1 mother filament. After nucleation of a new branch, Arp2/3 remains attached to the pointed end of the 2 new filament until coronin binds and subsequently releases Arp2/3 from it, whereas NPF* released from 3 the daughter filament is converted into NPF**, which slowly recovers to the inactive NPF state. NPF is 4 activated into NPF* by a free barbed end of a branched filament when the latter is within a threshold 5 distance (L nucl zone ) from the membrane. The simplified network involving feedback between barbed 6 ends and NPF's is summarized in Fig. 6. 7 We assume that all filaments are vertically aligned, with the barbed ends facing the membrane. 8 Filaments do not diffuse or move in directions parallel to the membrane plane, but do undergo vertical 9 shifts due to elongation at the barbed end abutting the membrane. All filaments associated to the same 10 nucleation site via Arp2/3-mediated branching constitute a local actin cluster (LAC). Filaments in a LAC 11 are assumed to interconnected, which may be due to the tightness of inter-filament space or cross-linkage 12 by other proteins which are not explicitly modeled here. It is assumed that elongation of any constituent 13 barbed end against the membrane moves the entire LAC upwards. We have to clarify if we assume 14 all filaments are tethered.

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Actin wave generation and propagation 17 We show in what follows that the stochastic model described above, which comprises the actin assembly 18 machinery and coupled membrane activity, predicts the emergence and propagation of actin waves that 19 are qualitatively and quantitatively in agreement with experimental observations. An example of one 20 realization of the waves is shown in Fig. 7, in which the wave is initiated by activating NPF on a 21 0.1 µm × 0.1 µm membrane patch (one computational cell on the membrane) at the lower left corner.

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The density of F-actin in each compartment on the x-y plane is the total F-actin measured in monomer 23 equivalents within 100 nm of the SA surface and projected onto the membrane plane. This is to be 24 compared with the TIRF imaging of F-actin waves on the SA membrane in vivo shown in Fig. 1. This 25 external perturbation of NPF activity is meant to mimic localized spontaneous membrane activity, which 26 could arise as the downstream effect of the integrin-mediated adhesion signaling cascade or PIP 3 signaling 27 via Ras/PI3K pathway. Local NPF activation leads to actin polymerization and triggers formation of 28 an expanding wave, as seen in the snapshot of wave dynamics at 5 seconds in Fig. 7. If we were to 1 initiate a wave in the center of a larger domain, the result would be an approximately circular wave that 2 propagates radially outward (See online video S3).

3
The wave initiated at the corner travels at a speed between 0.1 and 0.2 µm per second, and the F-actin 4 density behind the wave gradually decays, thereby producing a relatively stable profile about 1-2 µm in 5 width. A circular wave separates the underlying membrane into three regions: the membrane beneath the 6 wave, the area enclosed by the wave, and the outside area ahead of the wave. Though it is difficult to see 7 in the figure, both the interior and exterior regions are populated with small amounts of actin filaments 8 that arise via spontaneous filament nucleation activity at the nucleation sites. However, these two regions 9 differ in their molecular composition in that a large proportion of NPF on the interior membrane is the 10 recovery form NPF**, whereas in the exterior region the NPF is in the native inactive state, as will be 11 shown later. When the wave reaches the boundary it decays slowly because the interior does not recover 12 fast enough to facilitate reversal of the wave. Following passage of a wave the dynamics take one of two 13 forms. In one form, the entire membrane is covered with a low level of F-actin generated by spontaneous 14 nucleation at distributed nucleation sites. Alternatively, some free barbed ends of branched filaments 15 remain following passage of the wave and serve as sites for the initiation of new waves, as will be shown 16 later.

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Further details of wave propagation can be understood quantitatively by examining a cross-section 18 of a propagating wave. An example of the time-evolution in Fig. 8 shows the outward propagation 19 of a roughly unimodal shape of F-actin along the diagonal line from the initial NPF activation site at 20 the left lower corner. By tracking the wavefront location we estimate an average speed slightly above 21 0.13 µm/s. One sees that the wave front is steeper than the back, reflecting the rapid increase in 22 filament density due to the positive feedback loop. The basal level of F-actin ahead of the wave is 23 small compared to that within the wave, and the area well behind the wave also relaxes to a basal level 24 of F-actin corresponding to spontaneous filament polymerization at nucleation sites. Since continual the wave, beyond 4 µm shown in both figures, is devoid of Arp2/3-mediated branched filament nucleation due to the lack of activated NPF* there, but has the potential for filament nucleation due to the presence 1 of a large amount of inactive NPF (blue line). Within the steep leading edge of the wave, which is around 2 3.5-4 µm, the membrane is populated with a low level of activated NPF*, while behind the wave, between 3 2.5 and 3.5 µm, most of the NPF is in the recovery state (green line). There is little NPF * in this region, 4 hence little branched filament nucleation by NPF*/Arp2/3/actin complex, and this leads to the decay 5 of F-actin behind the peak. The region well behind the peak has some NPF that has recovered, but the 6 majority of the NPF is in the recovery state. Where there is some NPF in the inactive state, any residual 7 free barbed ends of branched filament could activate it and thus generate catalytic filament branching 8 locally. This is why one sometimes sees a new round of wave initiation and propagation long after the 9 wave has passed -an example of this will be shown in later section on colliding waves.

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Since TIRF imaging only captures the density of labeled actin within ∼100-200 nm of the surface it 11 does not reflect the entire structure and evolution of the waves. All previous mathematical models of 12 actin waves only reproduce the F-actin within the range of TIRF imaging, but the optimized simulation 13 algorithm we use enables us to explore the entire wave structure. A stochastic realization of the entire 14 wave structure as it evolves, from which the previous images were extracted, is shown in Fig. 10.

15
This example shows clearly that an actin filament cluster grows rapidly where NPF's are activated.

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The filaments grow vertically into the cytoplasm due to insertion of actin monomers at the SA portion of 17 the membrane Simultaneously, the network expands over the adherent membrane, a consequence of NPF 18 diffusion from the network covered area into the neighboring region. As the wave propagates it attains a 19 stationary shape with a rapidly-increasing network density at the front and a slowly-decaying density at 20 the rear, with an overall width in the range of 1-2 µm. Parametric studies of the current model indicate that increased NPF diffusion, elevated spontaneous 1 filament nucleation, and amplified monomer concentration all lead to increases in the wave speed.

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In light of the effects of latrunculin on the network, it is particularly important to understand how 3 the G-actin concentration influences the wave dynamics. It is observed in vivo that small and relatively 4 stationary actin spots first appear on the membrane in the early phase of F-actin reassembly when the 5 monomer-sequestering drug latrunculin is washed away. The free polymerizable G-actin is presumed 6 to be low in the cytosol at that stage since latrunculin gradually releases the bound G-actin following 7 washout. The underlying mechanism leading to spots is not established, but one possibility is that the 8 stochastic membrane-ECM interaction excites local signaling on the membrane, leading to downstream 9 actin polymerization [35]. In the following numerical experiment, we randomly choose nine membrane 10 patches for NPF activation with a G-actin level as low as 0.1 µM. Only four of nine membrane spots give 11 rise to detectable actin networks, as can be seen in Fig. 11. The TIRF density of F-actin is low, and 12 more interestingly, it takes up to 50 seconds to develop actin spots ∼ 0.5 µm in diameter, which suggests 13 that they are essentially stationary. 14 JFH modify later We also did a parametric study of the dependence of wave speed on monomer 15 concentrations. For each actin concentration, twenty realizations of the stochastic simulation were gen-16 erated for quantification of the mean wave speed. As shown in Fig. 12, at low concentrations the mean 17 speed is approximately linearly related to the actin concentration, whereas at high concentrations the 18 speed approaches a saturation value. At low G-actin concentration the generation of barbed ends is low, 19 and accordingly NPF activation is low, which weakens the wave propagation. At the other extreme of 20 high G-actin concentration the barbed-end generation by NPF*-Arp2/3-actin complex may saturate as 21 NPF* is consumed and depleted locally, i.e., the positive feedback between F-actin and NPF saturates.

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In that regime a higher G-actin level will not elevate NPF activation further, and thus the speed will 23 reach a plateau value.

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Determinants of the wave shape 25 Two additional experimentally-measurable quantities are the width and density of the wave, which con-26 sists of a rising front and decaying back, as shown schematically in Fig. 1. A comparison of the distribution 27 of various NPF states and downstream F-actin in Fig. 8 and 9 shows that the rapid increase in network 28 density at the wave front occurs in a region of NPF*-rich membrane. The conversion of NPF to NPF* and its subsequent consumption by filament branch generation determines the width of the wave from the 1 leading edge to the peak, whereas the maximal density and height of the wave is determined by both of 2 elongation rate at the barbed ends and duration of local filament nucleation. In contrast to the wave front, 3 the wave back lies over the NPF*-depleted membrane, where the supply of Arp2/3-mediated filament 4 generation is diminished. In addition, barbed end capping and fast depolymerization of filament pointed 5 ends exposed by coronin-mediated Arp2/3 removal causes the network to shrink from the cytoplasmic 6 side. Therefore, the rapid action of coronin in removing Arp2/3 from filament branches, together with 7 fast depolymerization at pointed ends, will reduce the width of the wave back. An increase in the CP 8 concentration also accelerates filament turnover and leads to a decrease in the width of the wave back. At 9 very high CP concentrations propagation can be blocked, since CP's also cap barbed ends at the leading 10 edge, thereby weakening the positive feedback loop between F-actin and NPF. In summary, the wave 11 density and height are determined by the elongation rate at the barbed end and the NPF* consumption 12 rate, whereas the width of the wave back is primarily controlled by the pointed-end depolymerization 13 rate.
14 Wave annihilation and reformation at multiple sites 15 One characteristic of actin waves on SA membranes is that they annihilate each other when they collide. 16 Two processes in the current model produce such wave behavior: positive feedback of F-actin polymer-17 ization from NPF activation at the wave front, and reduction of filament polymerization at the wave 18 back, due primarily to the slow recovery of NPF**. Simulations shown in Fig. 13 confirm this -there 19 four wave fronts generated at the four corners of the domain expand toward each other, and when they 20 collide they annihilate each other.

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Nonetheless, some actin spots may survive in the wake of waves that collide. These spots are transient, The role of the recovery state 1 One conclusion from our model is that the decay at the back of the wave results from the combination of 2 barbed-end capping by CP's, fast depolymerization at exposed pointed ends, and a lack of new filament 3 generation due to local depletion of NPF* and slow recovery of NPF**. Of these, the lack of NPF* in 4 the negative feedback on F-actin polymerization at the wave back is a dominant factor. This is shown 5 by the fact that waves become transition waves rather than solitary waves, i.e., the waves propagate 6 outward but the network in the back of the wave does not decay, when the recovery rate of NPF** 7 to NPF is increased. The extreme case of this arises when the recovery state is eliminated and NPF* 8 is converted directly to NPF after its participation in filament generation. The dynamics of the two-9 NPF-state model is illustrated in Fig. 17, which demonstrates that a slowly-recovering NPF** state is 10 critical for the observed wave structure. In accordance with this prediction, it is observed experimentally 11 that fluorescence recovery after photo-bleaching of the WAVE protein, one type of NPF, is slow in the  loop, where free barbed ends activate membrane-bound NPF. We suppose that there is a low level of 28 spontaneous filament nucleation at sites on the membrane, and these generate backbone filaments that 1 provide docking sites for Arp2/3-mediated filament branching. The local filament branching is further 2 strengthened by a positive feedback between NPF activation and free barbed ends. The actin assembly 3 machinery in the model comprises a minimal set of proteins -in addition to NPF -that can generate 4 a dynamically evolving actin network: Arp2/3, which nucleates branches and caps pointed ends on the 5 branch, coronin, a factor that contributes to Arp2/3 removal from the network and pointed-end exposure, 6 and CP, a barbed-end capping protein that prevents polymerization. In this minimal model, we predict 7 that the formation and propagation of actin waves can be initiated with a local perturbation that acti-8 vates sites on the membrane, and numerical experiments show that waves annihilate when they collide, 9 and travel at speeds strongly depending on the actin concentration.

10
The simulation results show that the model is able to produce a variety of actin network behavior 11 depending on the conditions. Actin spots of diameter about 0.5 µm can be formed and persist for tens  The model also correctly captures the vertical profile of actin waves along line scans through wave 18 fronts and the separation between the region enclosed by circular actin waves and the external area. 19 The decay of the wave back is caused by the slow recovery of NPF, which becomes NPF** upon branch 20 creation. This slow recovery of NPF, which becomes NPF** upon branch creation, leads to exhaustion 21 of active NPF and the decay of the wave back. In circular actin waves, areas with low and high levels of with reflective boundary conditions on the surface ∂Ω 3d except on the membrane Ω 2d , and there where R's represent various reactions at filament ends, and F bkf ree the concentration of backbone fila- nL,ptag f r (n, n L , 0, p tag ) nL,ptag f r (n, n L , 0, p tag ) The proteins that reside on the membrane are the various states of NPF's and their association 16 with Arp2/3 and G-actin. We allow 2D diffusion for free (non-complexed) states of NPF's, but not for 20 complexes. The dynamics of these state variables satisfy The dynamics of the branched filament is dictated by the filament-end reactions, which include the 6 Arp2/3 removal facilitated by coronin binding and subsequent depolymerization at the pointed end, and 7 polymerization and capping at the barbed end. The detailed evolution follows as For filaments whose lengths n L ≥ 2, one has We use the following benchmark set of parameters for all stochastic simulations of the model, except for 9 cases where selected particular parameters are changed to examine the resulting effect of those parameters.

10
The parameters which are not referenced are chosen to produce experimentally-compatible wave behavior.

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Referenced parameters are chosen either the same as or within the normal range as in the literature.

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There exist three major differences between the parameters used here and those in literature. Firstly,   is, which is legitimate since the on-rate for monomer addition is independent of the filament length. Thus 9 a third random number is needed after the reaction type that occurs is determined in order to decide 10 which reaction within the equivalence class occurs. This treatment reduces the computational cost by 11 2-3 orders of magnitude by making an optimal use of the structure of underlying reaction network [46] .

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In the current discretization of the simulation domain, there are N cmprt = (L x /l x × L y /l y × L z /l z ) 13 compartments in the cytoplasmic domain, which is around 50, 000 in typical computations. Each of 14 the N cmprt computational compartments is considered to be well-mixed, and there are pseudo-reactions 15 corresponding to diffusive hops between compartments. We lump the equivalence class of reactants of the same type in individual compartments into a large equivalence class in the whole domain. Following 1 the MO method, we first determine the waiting time for the next reaction from the prospensity of all 2 the allowable reactions, then decide which equivalent class of reactions over the whole domain occurs, 3 after which we decide which compartment this equivalent class belongs to. A direct search for which 4 compartment the next reaction occurs in consumes a great deal of time to find the target compartment 5 by checking each compartment. We thus developed a search method by subdividing the total N cmprt 6 compartments into N sub subsets. Instead of searching directly for the target compartment for the next 7 reaction, we first search for which subset the target compartment belongs to, then determine the target 8 compartment within that subset. We found that an optimal subdivision of the compartments can reduce 9 the search time by a factor of 5-10.

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The detailed algorithm is as follows. Suppose that the system has N rct type equivalent reaction classes 11 and that the rate constant of the j-th reaction type is r j . Consider there are N cmprt computational 12 comparments, in the i-th of which there are RA j i possible reactions for reaction type j. Therefore, for the 2. Generate a second random number, and decide which reaction type the next reaction will be from 20 RAtot j and r j according to Gillespie direct method; 21 3. Generate a third random number and decide which reacting compartment the reaction type decided in Step 2 locates in. In this step, instead of checking the N cmprt compartment one by one, we first 23 subdivide the compartments into subsets, determine in which subset the reacting compartment 24 falls, and then within that subset determine the appropriate reaction compartment. In essence this 25 is done as in step one, except that we compute total propensities within subsets and use these to ization is to occur in the reacting compartment, the filaments whose pointed end lies in the 7 comparment may be of different lengths, and thus we must randomly choose one from these 8 filaments.

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The effect of subdividing of the total number of compartments in Step 3 on the computational time 12 is shown in a simulation trial which produce the results as in Fig. 19.      Nucleation sites for the formation of backbone filaments, which are generated by G-actin attached to the site, reside on the membrane. Active NPF* can recruit Arp2/3 complex and G-actin to the membrane, and when the trimeric complex binds to the side of a filament, it generates a new filament branch with its pointed end capped and linked to the mother filament. The cytoplasmic coronin may bind to Arp2/3 complex at the branch junction, and remove it from the branch site and expose the filament pointed end. Free barbed ends in close proximity to the membrane are assumed to activate inactive NPF, and after branch generation NPF* is converted to NPF**, which in turn slowly recovers to NPF. NPF is activated to NPF* by free barbed ends of branched filaments, which in turn promotes barbed end generation and itself becomes inactive NPF**. NPF** slowly recovers to inactive NPF. Free barbed ends can be capped and the filament can depolymerization completely.  Figure 7. A computational TIRF sequence for the formation and propagation of an F-actin wave. The initial G-actin concentration is 10 µM, and half of the NPF is activated at the lower left corner. The color index indicates the total F-actin within 100 nm of the membrane projected to each membrane compartment. The maximal density in the color representation is set to be 70 monomers per compartment, and thus density larger than 70 is colored the same as 70. However, the highest density of F-actin throughout the membrane could be larger than 70.  Figure 9. The distribution of various NPF states at 30 seconds of the wave dynamics. This distribution is along a cross-line through the wave on the x-y plane as in Fig . 8. The y-axis is the density (#/0.01 µm 2 ) of various NPF's.    Figure 15. The distribution of various NPF states at t = 50.0 seconds for the waves shown in Fig. 14. Note that at this time a significant portion of NPF** has recovered to NPF, which is able to generate new waves with free barbed end seeds.   Figure 18. TIRF images of an F-actin wave when NPF is allowed to detach from the membrane and diffuse in the cytosol before re-attaching to the membrane.  Figure 19. The CPU times taken for the simulation with various subdivision sizes for the compartments. The CPU time is the time required for computing the first 5 seconds of the dynamics shown in Fig. 7. Without subdividing the compartments, the computation takes about 750 minutes.

Supporting Information
1 Video S1 2 Time evolution of wave birth and propagation. The wave was initiated with NPF activation at the 3 left bottom corner at t = 0 second. The F-actin density in each simulating compartment is the F-actin 4 within a 100 nm depth of the cytoplasm onto the membrane, which is to be compared with TIRF images 5 of the wave. The condition of this realization of the stochastic model is the same as that in Fig. 7.  The dynamics of wave collision and subsequent new wave formation.