Why A Large Scale Mode Can Be Essential For Understanding Intracellular Actin Waves
WWhy A Large Scale Mode Can Be Essential For Understanding Intracellular Actin Waves
Carsten Beta, Nir S. Gov, and Arik Yochelis
3, 4, ∗ Institute of Physics and Astronomy, University of Potsdam, 14476 Potsdam, Germany Department of Chemical and Biological Physics, Weizmann Institute of Science, Rehovot 76100, Israel Department of Solar Energy and Environmental Physics,Blaustein Institutes for Desert Research (BIDR), Ben-Gurion University of the Negev,Sede Boqer Campus, Midreshet ben-gurion 8499000, Israel Department of Physics, Ben-Gurion University of the Negev, Be’er Sheva 8410501, Israel (Received June 17, 2020)During the last decade, intracellular actin waves have attracted much attention due to their es-sential role in various cellular functions, ranging from motility to cytokinesis. Experimental methodshave advanced significantly and can capture the dynamics of actin waves over a large range of spatio-temporal scales. However, the corresponding coarse-grained theory mostly avoids the full complex-ity of this multi-scale phenomenon. In this perspective, we focus on a minimal continuum modelof activator-inhibitor type and highlight the qualitative role of mass-conservation, which is typicallyoverlooked. Specifically, our interest is to connect between the mathematical mechanisms of patternformation in the presence of a large-scale mode, due to mass-conservation, and distinct behaviors ofactin waves. ∗ [email protected] a r X i v : . [ q - b i o . CB ] J un I. INTRODUCTION
Biological pattern formation refers to the emergence of complex spatiotemporal variations in living systems thatare typically far from thermodynamic equilibrium [1–3]. Even though these systems can differ in composition andscales, they share many similarities and generic phenomena that are observed in a wide variety of natural settings,such as stationary periodic patterns of pigments on animal skins, spiral waves in biological cells and cardiac arrhyth-mia, or swarming phenomena in bacterial colonies and in flocks of birds or fish. The theoretical study of biologicalpattern formation can be roughly divided into two time periods, namely: ( i ) The second half of the twentieth century,following the seminal works by Turing on morphogenesis [4] and by Hodgkin and Huxley (HH) on action potentialsin the giant squid axon [5], and ( ii ) the beginning of the twenty-first century, where an ever increasing amount ofquantitative biological data provided the basis for more detailed mechanistic models of biological systems.During the first period, theoretical studies were largely limited to a few prototypical reaction–diffusion (RD) oractivator–inhibitor (AI) models [6], such as the FitzHugh–Nagumo (FHN) [7, 8], Gierer–Meinhardt [9], and Keller–Segel equations [10]. Based on the relative simplicity of these models, e.g. the FHN model as compared to the HHequations, and their relation with models of inanimate matter, e.g. the Swift-Hohenberg model of thermal fluidconvection [11] and the Gray–Scott model of chemical reactions [12–14], several pattern formation methodologieshave been advanced [1], such as weakly nonlinear and singular-perturbation methods. These models provided deepinsights into universal aspects of pattern formation phenomena and generic relations to applications were substan-tiated, such as frequency locking and spiral waves in the cardiac system. The second time period has manifesteda gradual shift of research interests towards specific detailed biological and medical systems [15, 16], including,for example, micron-scale intracellular waves, the development of tissues and organs, sound discrimination in theauditory system, and pathologies such as cancer metastasis. In particular, systems in these contexts are generally de-scribed by elaborate, system-specific models that are less amenable to mathematical analysis than earlier toy modelsof pattern-formation.Consequently, despite the common pattern-formation thread that connects these different biological and medicalapplications, it became difficult to navigate through the vast number of distinct models and approaches, particularlyin cases where technical jargon makes it difficult to adopt cross-disciplinary integration between different commu-nities including biophysics, computational biology, mathematical biology, biological chemistry, dynamical systems,and numerical analysis. Even though the formulation of complete models for biological systems is currently unre-alistic, uncovering partial mechanisms that drive pattern-formation phenomena remains of utmost importance forunderstanding functional aspects of living systems and for developing technological and medical applications, suchas drugs or implants. Moreover, mechanistic studies of pattern forming systems are also fertile sources of new math-ematical questions that advance the development of analytical and numerical methods [2, 17–25], which, in turn,contribute new insights into the original applications.In this perspective, we will focus on intracellular actin waves (IAW), a topic that recently gained much interest notonly in the biological context but also as an inspiring showcase of active matter. More specifically, we are interested inIAW that are affected by a large scale mode — a situation that arises due to conservation of actin monomers (over thetime-scale of the IAW phenomenon). We note that phenomena such as Ca + waves are, in general, beyond the scopeof this perspective as they involve the transport of ions between the cell interior and the extracellular space (whichacts as an infinite reservoir) [26–28], unless conservation can be accounted for [29]. Moreover, we emphasize that weaim to provide a perspective and not a comprehensive review, as such reviews are already available, e.g., [30–36].The perspective is organized as following. In Section II, we introduce the rich phenomenology of IAW and themodeling aspects that are associated with mass conservation. Then, we present in Section III the theoretical aspectsof a large scale mode in the context of physicochemical settings and also indicate its significance to IAW as a blueprintfor conservation of actin monomers on the times scales at which many actin dynamics processes operate. Finally,we discuss in Section IV why incorporation of mass-conservation is a plausible qualitative step in unfolding therobustness of IAW mechanisms, and in Section V we conclude by emphasizing the theoretical strategies for modelingand control of wave persistence as a potential roadmap toward applications in synthetic biology. II. INRACELLULAR ACTIN WAVES
The functions of many cells are tied to their ability to dynamically change their shape, mostly via the spatiotempo-ral organization of their actin cytoskeleton. Examples of this include diverse cell types, such as human neutrophils,fish keratocytes, or the social amoeba
Dictyostelium discoideum . Among the most prominent dynamical patterns inthe actin cytoskeleton are IAW that have attracted much of attention over the past decade [31]. These waves areassumed to play a role in several essential cellular functions, among them cell locomotion, cytokinesis, and phago-cytic uptake of extracellular matter. Many competing models at different levels of complexity have been developed migration after formin inhibition were also similar to those of thecontrol DCs (Fig. 2 E ). The ROCK inhibitor Y27632 did not sig-nificantly affect the frequency of actin wave formation (Fig. 7 E andMovie S20). Finally, the actin wave propagation speed was signif-icantly reduced after both formin and ROCK inhibition (Fig. 7 F and Movies S20 and S21).From the Arp2/3 inhibition data, we can conclude that Arp2/3 is required for the formation of the actin polymerization waves.This is also supported by the higher wave nucleation rate withformin inhibition, which would lead to more actin monomersavailable for Arp2/3-driven actin wave polymerization. We canfurther conclude that ROCK activity has little influence on actinwave generation but alters the wave propagation. These data areconsistent with the effects of these inhibitors on the DC trajec-tories reported above (Fig. 2). Conclusions
In this study, we confined immature DCs by 2 parallel slides to2 dimensions, and we tracked them over several hours. The DCsperformed random walks that were divided into 2 differentstates. In the persistent state, the DCs were polarized and movedcontinually along curved trajectories, with a mean radius of61 μ m. In the diffusive state, the DCs were not polarized, andthey showed short irregular displacements. Biphasic migrationpatterns have been observed in several cell types before. Whenconfined to 1-dimensional channels, DCs were shown to switchbetween moving and not moving states (45). The bacterium E. coli switches between “ runs ” and “ tumbles, ” and fish keratocytesswitch between continuous random walks and continuouslyturning states in which they move for an extended period of timein circles with a radius comparable with the cell size (46, 47).However, both the form of the trajectories and the mechanismsbehind these biphasic behaviors are different (6, 7). Maiuri et al.(16) attributed the biphasic migration pattern of DCs to fluctu-ating “ polarity cues ” that determined the direction of the actinflow. This flow, in turn, was responsible for propelling the cells.The relative strength of the stochastic polarity cues and the actinflow then determined whether the cells moved continuously, dif-fusively, or in a biphasic manner. The origin of the polarity cues,however, remained unspecified. Together with our theoreticalanalysis, our experimental data show that polarity cues can begenerated by the cytoskeleton in a process of self-organization.Remarkably, self-organization leads to the emergence of in-termittent waves, which suggests a possible deterministic originof the random cell migration. In our theoretical description ofthe actin dynamics, many of the molecular details of the regu-lation of actin polymerization were not included to concentrateon its essential features. The main feature is negative feedbackbetween actin filaments and actin nucleators, which is in agree-ment with previous studies (17, 48, 49). Indeed, interfering withArp2/3 or formins had large effects on the actin polymerizationwaves. Inhibition of formins led to short-lived waves and a higherwave nucleation rate. In contrast, inhibition of Arp2/3 completelysuppressed the formation of actin polymerization waves. However, t41:06min0:08min 0:28min 0:40min0t0 t1 t2 t3t0+t1 t1+t2 t2+t3 t3+t4 t0+t4 ABC T i m e LengthLength W ave s p ee d - ( m m i n ) T i m e D E
Fig. 6.
Actin wave nucleation and propagation in migrating immature DCs. ( A ) Representative epifluorescence image sequences of actin wave formationand progression in an immature DC. Red arrows indicate origin of the wave. TIRF images of Lifeact-GFP were acquired every 2 s. (Scale bar, 20 μ m.) ( B ) Overlayof cellular outline of 2 representative consecutive images displayed in A , illustrating the actin wave progression. Red indicates initial time t ; green indicates t + t t C , Left ) Representative wave analysis of wave from fluorescence TIRF images of Lifeact-GFP in the cell shown in A at t
4. Red and green lines show the location along which the kymograph was generated (top left to bottom right). ( C , Center and
Right ) Representative kymographs of wave propagation. Arrows indicate wave front. (Scale bars, 1 μ m [length]; 10 s [time].) ( D ) Mean actin wave speedsobtained for 4 representative individual cells. Number of waves analyzed per cell: 11 (cell 1), 13 (cell 2), 4 (cell 3), and 8 (cell 4). Box plot displays mean speed(middle line) and SEM (box). ( E ) Contours of representative actin wave shown in A during propagation. Stankevicins et al. PNAS | January 14, 2020 | vol. 117 | no. 2 | B I O P H Y S I C S A ND C O M P U T A T I O N A L B I O L O G Y C E LL B I O L O G Y D o w n l oaded a t UN I VE R S I T AE T S - B I B L P O T S D A M on M a r c h , Time (min)
20 40 60 80 100 C e ll a r e a ( μ m ² ) A=A min A ≥ A crit A ≥ A crit A ≥ A min membrane synthesis two new waves A ≥ A crit wave becomes unstable wave-mediatedcytofission D BAF E
0s 10s 85s
0s 218s 356s 495s lifeact/H2B C
113 500 - - - < area ( μ m²) b=0.05 p=0.25 SimulationExperiments P r o b a b ili t y o f d i v i s i o n ( % ) P r o b a b ili t y o f d i v i s i o n ( % ) b=0.05 p=0.25 N u m b e r o f c e ll s Nuclei per cytofr ission fragment
Experiment G b=0.025 p=0.25 Simulation
Mother cellarea ( μ m²)Mother cell Fig. 4.
Wave-mediated binary cytofission. ( A ) Probability of a cytofission event to happen within the first 16 min of a simulation, depending on the cellsize. The areas of simulated cells are multiples of a reference cell size of 113 µ m , given by a disc of 12 µ m in diameter. For each cell size 40 independentsimulations with ↵ = 1, b = 0.05, p = 0.25, and u = 1 were analyzed. Error bars represent the SD and are calculated assuming a binomial distribution.( B ) Simulation of a cell with an area of 339 µ m and the same parameter values used in A . The wave splits into two parts and leads to cytofission.( C ) Size evolution of a growing cell over time in a numerical simulation (corresponding to Movie S17). Once a critical size of about four to five times thesize of a single cell is reached, the cell divides via wave-mediated fission into 2 cells of at least the size of a single cell. The graph shows only the size of thelarger daughter cells over eight generations. ( D ) Analysis of the probability of wave-mediated cytofission within the first 16 min of observation for fusedDdB NF1 KO cells of different sizes. Cells were categorized according to their area into four groups: < µ m (10 cells), 250 to 500 µ m (21 cells), 500 to750 µ m (5 cells), and 750 to 1,000 µ m (5 cells). ( E ) The actin wave in a fused DdB NF1 knockout cell with two nuclei expressing Lifeact-GFP and histoneH2B-RFP becomes unstable and splits into two independent waves that move in opposite directions and induce cytofission. ( F ) Histogram of the numberof nuclei in 55 cytofission fragments obtained from live cell imaging experiments with DdB NF1 knockout cells expressing Lifeact-GFP and histone H2B-RFP.( G ) Schematic of wave-mediated binary cytofission in a growing cell. A min is the minimal cell area and A crit the critical cell size where wave-mediatedcytofission starts to occur. (Scale bars, 10 µ m.) more detailed descriptions, our model does not aim at eluci-dating specific molecular mechanisms. Instead, we designed areduced model, based on a generic nonlinear wave generator,that highlights the minimal degree of complexity required todescribe how cortical waves drive the fission of adherent cells.Our model captures all our observations very well, including thefan-shaped phenotype of the daughter cells, their characteristicrange of sizes, the lateral instability of waves that collide with thecell border, and unsuccessful fissions for wave segments belowa critical size. Moreover, our analysis demonstrates that wavedynamics need to be appropriately balanced between bistableand excitable regimes (reflected in the choice of model param-eter b ) to reproduce the pinch-off behavior observed in ourexperiments. Note that bistability was also identified as a keyelement in describing the dynamics of circular dorsal ruffles,actin-based ring-shaped precursors of macropinocytic cups (46).We believe that a phenomenological modeling approach thatidentifies the minimal dynamical features needed to recover theexperimental observations will be particularly beneficial for guid-ing future efforts to reconstitute primitive cytofission scenarios insynthetic systems. The daughter cells that emerged from wave-mediated fissionresembled fan-shaped cells that were first observed in knock-out cells deficient in the aggregation-related amiB gene (21).Recently, it was shown that increased RasC or Rap1 activity, aswell as development at very low cell densities, can also induce aswitch to the fan-shaped phenotype (22, 23). After wave-mediatedcytofission, the ventral membrane of the emerging fan-shaped cellis entirely filled with a wave segment that is known to be richin active Ras (47). This confirms the key role of increased Rasactivity for fan-shaped motility. We thus conclude that the fan-shaped phenotype is generally associated with a stable drivingwave segment that covers the ventral cell membrane. † This is inagreement with earlier conjectures (12) and has also been sug-gested by recent modeling of transitions between amoeboid and † Due to their elongated shape and their highly persistent motion, these cells havealso been described as “keratocyte-like.” However, to avoid confusion with actualkeratocyte fragments (27) that show a very different cytoskeletal organization,we use the recently introduced term “fan shaped” to denote this wave-drivenmotility phenotype (22). | D o w n l oaded a t UN I VE R S I T AE T S - B I B L P O T S D A M on A p r il , migration after formin inhibition were also similar to those of thecontrol DCs (Fig. 2 E ). The ROCK inhibitor Y27632 did not sig-nificantly affect the frequency of actin wave formation (Fig. 7 E andMovie S20). Finally, the actin wave propagation speed was signif-icantly reduced after both formin and ROCK inhibition (Fig. 7 F and Movies S20 and S21).From the Arp2/3 inhibition data, we can conclude that Arp2/3 is required for the formation of the actin polymerization waves.This is also supported by the higher wave nucleation rate withformin inhibition, which would lead to more actin monomersavailable for Arp2/3-driven actin wave polymerization. We canfurther conclude that ROCK activity has little influence on actinwave generation but alters the wave propagation. These data areconsistent with the effects of these inhibitors on the DC trajec-tories reported above (Fig. 2). Conclusions
In this study, we confined immature DCs by 2 parallel slides to2 dimensions, and we tracked them over several hours. The DCsperformed random walks that were divided into 2 differentstates. In the persistent state, the DCs were polarized and movedcontinually along curved trajectories, with a mean radius of61 μ m. In the diffusive state, the DCs were not polarized, andthey showed short irregular displacements. Biphasic migrationpatterns have been observed in several cell types before. Whenconfined to 1-dimensional channels, DCs were shown to switchbetween moving and not moving states (45). The bacterium E. coli switches between “ runs ” and “ tumbles, ” and fish keratocytesswitch between continuous random walks and continuouslyturning states in which they move for an extended period of timein circles with a radius comparable with the cell size (46, 47).However, both the form of the trajectories and the mechanismsbehind these biphasic behaviors are different (6, 7). Maiuri et al.(16) attributed the biphasic migration pattern of DCs to fluctu-ating “ polarity cues ” that determined the direction of the actinflow. This flow, in turn, was responsible for propelling the cells.The relative strength of the stochastic polarity cues and the actinflow then determined whether the cells moved continuously, dif-fusively, or in a biphasic manner. The origin of the polarity cues,however, remained unspecified. Together with our theoreticalanalysis, our experimental data show that polarity cues can begenerated by the cytoskeleton in a process of self-organization.Remarkably, self-organization leads to the emergence of in-termittent waves, which suggests a possible deterministic originof the random cell migration. In our theoretical description ofthe actin dynamics, many of the molecular details of the regu-lation of actin polymerization were not included to concentrateon its essential features. The main feature is negative feedbackbetween actin filaments and actin nucleators, which is in agree-ment with previous studies (17, 48, 49). Indeed, interfering withArp2/3 or formins had large effects on the actin polymerizationwaves. Inhibition of formins led to short-lived waves and a higherwave nucleation rate. In contrast, inhibition of Arp2/3 completelysuppressed the formation of actin polymerization waves. However, t41:06min0:08min 0:28min 0:40min0t0 t1 t2 t3t0+t1 t1+t2 t2+t3 t3+t4 t0+t4 ABC T i m e LengthLength W ave s p ee d - ( m m i n ) T i m e D E
Fig. 6.
Actin wave nucleation and propagation in migrating immature DCs. ( A ) Representative epifluorescence image sequences of actin wave formationand progression in an immature DC. Red arrows indicate origin of the wave. TIRF images of Lifeact-GFP were acquired every 2 s. (Scale bar, 20 μ m.) ( B ) Overlayof cellular outline of 2 representative consecutive images displayed in A , illustrating the actin wave progression. Red indicates initial time t ; green indicates t + t t C , Left ) Representative wave analysis of wave from fluorescence TIRF images of Lifeact-GFP in the cell shown in A at t
4. Red and green lines show the location along which the kymograph was generated (top left to bottom right). ( C , Center and
Right ) Representative kymographs of wave propagation. Arrows indicate wave front. (Scale bars, 1 μ m [length]; 10 s [time].) ( D ) Mean actin wave speedsobtained for 4 representative individual cells. Number of waves analyzed per cell: 11 (cell 1), 13 (cell 2), 4 (cell 3), and 8 (cell 4). Box plot displays mean speed(middle line) and SEM (box). ( E ) Contours of representative actin wave shown in A during propagation. Stankevicins et al. PNAS | January 14, 2020 | vol. 117 | no. 2 | B I O P H Y S I C S A ND C O M P U T A T I O N A L B I O L O G Y C E LL B I O L O G Y D o w n l oaded a t UN I VE R S I T AE T S - B I B L P O T S D A M on M a r c h , A C B
FIG. 1. Examples of intracellular actin waves. (A) Actin wave nucleation and propagation in a migrating immature dendriticcell. Red arrows indicate origin of the wave, scale bar 20 µ m. (B) Overlay of contours of representative actin waves shown in (A)during propagation. (C) Wave-mediated binary cytofission in a Dictyostelium discoideum cell, scale bar 10 µ m. An actin wave ina cell with two nuclei becomes unstable and splits into two independent segments that move in opposite directions and inducea cytofission event. (A) and (B) are reproduced from [43], (C) is reproduced from [44]. Copyright 2020 National Academy ofSciences. to describe cortical actin waves, mostly relying on coupled nonlinear AI equations. Even though intracellular actinwaves involve a large number of interacting molecular species as well as multiple local and global interactions, pro-totypical AI models have been shown to capture many features of the overall dynamics. However, important effectsdue to mass conservation constraints have been hitherto largely neglected. A. Phenomenology from experiments
Actin waves are characterized by propagating of cytoskeletal regions that are enriched in filamentous actin andactin-related proteins. Depending on the cell type, IAW may differ in their biochemical composition and dynamics,including different wave morphologies and propagation speeds. One of the earliest examples of IAW was reportedfrom cultured neurons that show propagation of fin-like actin-filled membrane protrusions along their axon [37].They were found to depend on actin polymerization and have been associated with neural polarization [38, 39].Similar fin-like actin wave also emerge in non-neural cell types when cultured on thin fibers [40]. Also adherentcells that are attached to flat substrates may display traveling wave-like protrusions of their cell shape. They areparticularly prominent when moving laterally along the cell border, such as in mouse embryonic fibroblasts [41] orat the leading edge of fish keratocytes [42].Traveling actin waves have also been observed at the dorsal and ventral sides of adherent cells. In neutrophils,small dynamic wave fragments emerge that organize cell polarity and leading edge formation [45]. Larger ring-shaped waves were found to travel across the substrate-attached bottom membrane of
D. discoideum cells [46]. Theyenclose a region that is structurally distinct from the cortical area outside the actin ring [47, 48] and their dynamicsoften shows rotating spiral cores and mutual annihilation upon collision [49, 50] but they could not be initiated byexternal receptor stimuli [51]. While understanding the rich dynamics of IAW is challenging on its own right, thereare prominent applications and functional properties that stimulate further studies of IAW in different contexts:
Motility.:
Recently, clear evidence was reported that actin waves directly impact the motility of immune cells, seeFig. 1A,B. In particular, dendritic cells that move in an amoeboid fashion and search the human body forpathogens, display a random walk pattern that can switch between diffusive and persistent states of motion, adirect consequence of the intracellular actin wave dynamics [43];
Cell division.:
In oocytes and embryonic cells of frog and echinoderms, excitable waves of Rho activity in conjunc-tion with actin polymerization waves were observed shortly after anaphase onset, providing an explanationfor the sensitivity of the cell cortex to signals generated by the mitotic spindle [52]. Similarly, in metaphasemast cells, concentric target and spiral waves of Cdc42 and of the F-BAR protein FBP17 were found to set thesite of cell division in a size-dependent manner [53]. IAW can also act as the force-generating element thatdirectly drives the division process in a contractile ring-independent form of cytofission. This was observedin
D. discoideum cells beyond a critical size, where waves that collide with the cell border not only inducestrong deformations of the cell shape but also trigger the division into smaller daughter cells — a cell cycle-independent form of wave-mediated cytofission, see Fig. 1C [44];
Macropinocytosis.:
While functional roles in phagocytosis and motility have been proposed [54, 55], recent geneticstudies suggest a relation to macropinocytosis [56]. This is supported by similarities between the basal actinwaves and circular dorsal ruffles (CDR) [57, 58]. The latter also adopt a ring-shaped structure but mean-der across the apical membrane, where they induce membrane ruffles that were related to the formation ofmacropinocytic cups [59];
Cancer.:
Macropinocytosis has been also identified as an important mechanism of nutrient uptake in tumor cells [60].Specifically, inability of cells to undergo efficient macropinocytosis, e.g., thorugh disordered IAW behavior orsuppressed activity via pinning of IAW to cell boundaries [58], has been associated with cancerous pheno-types [61, 62].Despite intense studies over the past years, the molecular details of IAW mechanisms remain largely unclear andmost likely vary between different cell types.
B. Modeling approaches of actin waves
Following the numerous experimental observations of IAW in different cell types and during different cellularfunctions, many model equations have been proposed to describe this phenomenon. Here, we will briefly describethe main types and features of theoretical models that have been employed while referring the reader to [31, 34, 35,63, 64] for more details.The growth of the cortical actin network within IAW is a complex dynamical process that involves many compo-nents that perform a coordinated set of functions, giving rise to the formation of a three-dimensional network of actinfilaments, that propagates along the cell membrane. This process involves the activation of actin associated proteinssome of them membrane bound, that initiate the nucleation of actin polymerization, branching of actin filaments,cross-linking and bundling, as well as severing and depolymerization. There are very few theoretical models thatattempt to give a molecular-scale description of the IAW phenomenon where all of these processes are described.One example for such a model that describes the waves at the scale of the individual actin filaments is given in[65]. While providing detailed pictures of the actin network, it is difficult and time-consuming to use such modelingto extract understanding regarding the large-scale dynamics of the IAW. Such modeling efforts could in the futureinclude more molecular components [66, 67], on larger length and time scales, and provide a platform for theoreticaladvances in this field, that works in conjunction with filament-scale experimental data [48].Since the IAW have widths in the range of hundreds of nanometers, propagate over tens of microns and persistover hours, it is natural to describe them using coarse-grained models that avoid prescribing the molecular-scaledetails of the actin network. As will be shown, many of these models agree with some qualitative or even quantitativefeatures of the observed IAW in cells. It is therefore difficult at present to reach a clear consensus regarding thevalidity of these models. Comparisons in between such models is complicated since they often include differentcomponents and it is not clear if and which of those components play a fundamental role in the emergence of IAWor can be neglected otherwise.Among the coarse-grained models we can find a small class of models that contain biophysical elements, suchas forces and/or the membrane shape, which play a key role in the mechanism that drives the propagation of theIAW. One example is well demonstrated by Gholami et al. [68], who show that the dynamics of the actin polymer-ization/depolymerization drive the oscillatory propagation of waves. When actin filaments polymerize against thecell membrane, they exert a protrusive pressure on the membrane, which pushes the membrane forward and theactin network backwards. The interplay between the rate of actin polymerization and the rate at which the actinfilaments are cross-linked into a stable gel-like network, determine if the cortical actin is stable or whether it exhibitsan unstable oscillatory regime.Another group of biophysics-based models contain curved membrane proteins that nucleate the cortical actinpolymerization [69–73]. In these models, the curved proteins flow/adsorb to the membrane regions that have acurvature similar to their intrinsic shape, and their concentration is therefore affected by the membrane deformationsthat are induced by the forces exerted by the actin cytoskeleton. These forces include the protrusive force of actinpolymerization, as well as contractile forces due to myosin-II mediated contractility. Recently, also models combiningan RD kinetics coupled to mechanical properties through the impact of curved actin nucleators and/or membraneshape and tension were introduced [42, 74]. Other models combine the RD dynamics with a physical effect, such thatthe directed or random lateral actin polymerization can physically drive the treadmilling of the IAW componentsalong the membrane [75]. The advantage of the biophysical class of models is that they can naturally account for theobserved effects of physical parameters on the IAW, such as membrane tension [68, 74] or the contractile forces ofmyosin-II motors [76].In many cases however, RD equations that include both positive and negative feedback loops, are sufficient todemonstrate the formation of propagating waves, fronts, or localized pulses. These models exhibit different levelsof complexity and different numbers of components. In the simplest cases, generic activator-inhibitor models ofFHN-type were proposed. In particular, they were used together with a local-excitation, global-inhibition (LEGI)mechanism to account for the response of the receptor-mediated signaling pathway and the downstream actin cy-toskeleton to external cues [77–79]. Other basic RD-models describe the actin dynamics, including the monomericand filamentous species, and one form of an actin activator, using the filamentous actin itself as a source of negative[65] or positive [80] feedback. More complex models include different numbers of activators of actin polymerization,inhibitors, and their complex network of interactions [55, 81, 82]. Yet, in general, RD equations are not subjected toconservation of mass although often some of the components are conserved, for example when they represent twodifferent forms of the same protein [83]. In other cases, the actin is conserved as it is converted from monomeric tofilamentous forms and back, see for example [58, 84]. In what follows, we address the qualitative role of conserva-tion, which is reflected by the existence of a large scale mode, on the dynamics of IAW, using as much as possiblegeneric principles, i.e., extracting conclusions that are qualitatively independent of the specific molecular details thatare included in the model.
III. ACTIN DYNAMICS AS A CONSTRAINED CONTINUOUS MEDIUM: IMPLICATIONS AND APPLICATIONS
The phenomenology of dissipative waves can be demonstrated through a dynamical systems approach via pro-totypical models, such as FHN. As summarized above, many variants of such activator-inhibitor models have beenused to describe different aspects of cytoskeletal dynamics and in particular the formation of actin waves. Althoughthese are heuristic models, they are analytically tractable and thus allow for fundamental insights into spatiotempo-ral behavior, which cannot be obtained through the analysis of more realistic multi-variable equation sets. Propagat-ing waves are traditionally classified into three universality classes [1, 2, 6, 85]:
Oscillatory dynamics,: which represent traveling waves that develop via a Hopf instability of a uniform steadystate;
Excitability,: corresponding to supra–threshold solitary waves (pulses) that propagate on top of a linearly stableuniform steady state;
Bistability,: which describes traveling domain walls or fronts, i.e., an interface that connects to linearly stable uni-form steady states.While the mathematical mechanisms are distinct, the emerging patterns can show similar characteristics, for exampleall classes may display the formation of spiral waves [2, 85]. Consequently, comparisons to experimental observa-tions can often only be qualitative, making insights uncertain. Moreover, it is not always clear whether the simplifiedmodels comprise the minimal set of qualitative ingredients, e.g., interactions (local vs. non-local), spatial coupling,essential degrees of freedom and feedback loops, finite domain effects, or existence of conserved observable(s). In abroader context, IAW can be classified as AI type media [31, 86], although unlike the typical RD media the numberof actin monomers is conserved over the time scales of wave dynamics. As such, mass conservation is an inherentconstraint of the modeling framework [35, 58, 86, 87], which is generically reflected by coupling to a large scale modein the dispersion relation, as illustrated in Fig. 2.
FIG. 2. Schematic representation of a dispersion relation obtained from infinitesimal periodic perturbations, proportional toexp ( σ t + ikx ) , about a uniform steady state; Re [ σ ] is the perturbation growth rate and k its wavenumber. The right-hand partof the dispersion relation represents the onset of an instability of a finite wavenumber type (often also referred to as Turinginstability), while the left-hand part reflects a conserved quantity and stays always neutral; both parts are model independent.The curves may connect as typically occurs in systems such as (3) or belong to different curves such as for (5). The imaginary partof σ corresponds to stationary nonuniform patterns if zero, and otherwise describes time-dependent solutions. A. Conservation in physicochemical systems
It is convenient to first consider total conservation of an observable, described by the continuity equation ∂ u ∂ t = ∇ · (cid:20) M ( u ) ∇ δ F ( u ) δ u (cid:21) , (1)where u is a scalar observable, M is a mobility function, and F is a free energy. If the free energy contains anintrinsic length scale, like in the phase field crystal model or wetting, stationary periodic and localized patterns mayemerge [88–95]. The mutual aspect is coupling between the large scale mode ( k = x → − x and u → − u . Such behavior arises in systemsthat are being driven out of equilibrium, such as convection [111–114], propagation of flames [115, 116], surfacewaves [117–119] and electro-diffusion in ion channels [120, 121]. In such cases leading order approximations showthat the dynamics can still be enslaved to an oscillatory (Hopf) finite wavenumber mode and a large scale mode [122–126]. While many fundamental advances have been made in understanding the coupling between the complexGinzburg–Landau equation and the large scale mode, e.g. in terms of stability of periodic and solitary waves in onespace dimension and dynamics of spiral waves in two-dimensional systems, several pattern formation issues remainopen [127]. Consequently, since over the time scales on which IAW occur the system is far from equilibrium, it isnatural to assume that a large scale mode due to mass conservation alters the pattern formation mechanism, evenwithout explicit flux conservation. B. Activator–inhibitor patterns with conservation
In general, AI systems are modeled in a similar fashion as chemical reactions [6, 15, 16, 36, 128–131], which are notlimited by supply of new substrates into the reactor ∂ u ∂ t = f ( u , v ) + D u ∇ u , (2a) ∂ v ∂ t = g ( u , v ) + D v ∇ v , (2b)where u is the activator that typically contains an autocatalytic or enzymatic term and a diffusion constant D u , and v is an inhibitor that diffuses with a diffusion constant D v , where typically D v (cid:29) D u . As intracellular processes oftentake place on very different time scales, effective mass conservation may arise, for example, in cases where proteinsynthesis and/or degradation occurs much slower than a particular biochemical reaction of interest. Conservationin AI models is associated with a local conservation of mass ˆ Ω [ u ( x , t ) + v ( x , t )] d x = constant, (3)where Ω is the physical domain, or by writing in (2) g ( u , v ) = − f ( u , v ) . (4)Linear stability analysis about uniform solutions leads to dispersion relations that contain the persistent neutral(large scale) mode, as shown in Fig. 2. As in the case of Eq. 1, also Eq. 4 supports multiplicity of uniform solutionssince u depends on an arbitrarily chosen value of v (or vise versa), and this degenerate degree of freedom appearsas the k = k = w , ∂ u ∂ t = f ( u , v , w ) + D u ∇ u , (5a) ∂ v ∂ t = − f ( u , v , w ) + D v ∇ v , (5b) ∂ w ∂ t = h ( u , v , w ) + D w ∇ w , (5c)where h can be either a linear or a nonlinear functional and essentially does not have to include transport of w via diffusion; these details are naturally determined by the characteristics of the biological system. Equation 5 thusreflects only a partial conservation and has been employed to study the emergence of IAW in the context of CDR [58],where a variety of complex behaviors have been observed experimentally, ranging from distinct types of propagatingfronts to spatiotemporal chaotic spiral waves. IV. DISCUSSION AND EXAMPLE
The complex pattern formation exhibited by CDR raises the question about the modeling strategy, specifically,with respect to the minimal set of equations and the necessity of a conserved quantity. As has already been indicatedin Section II B, there are many ways to model IAW but all of them are prone to subjective interpretations.In the absence of a clear physical intuition, since IAW are far from equilibrium phenomena, dynamical systemsoffer an efficient platform for creating an appropriate qualitative framework. More specifically, the study of bifurca-tions may provide the minimal qualitative set of constraints, exactly as phase-transitions allow us to classify manytypes of physical phenomema. On the other hand, bifurcation analysis can also be a tedious task as there may bemany local and global bifurcations that coexist in a given parameter range (as an example we refer the reader to asystematic extension of excitable media by Champneys et al. [143]). Nevertheless, utilizing recent advances in non-linear perturbations [83, 144] and numerical path continuation methods [145–149] it might be possible to navigate t ↑→ x FIG. 3. Space–time plots showing (from left to right) annihilation, reflection/crossover, and “birth” of new pulses followingcollision (a behavior that resembles backfiring), respectively, as obtained from direct numerical integration of the minimal CDRmodel equations [151] that have the same structure as Eqs. 5. No–flux boundary conditions were used. From left to right theamount of actin monomers increases (see details in [151]). The dark shaded color indicates higher values of filamentous actin inthe IAW. Reprinted figure with permission from [151] Copyright 2020 by the American Physical Society. between coexisting bifurcations and a multiplicity of emerging stable and unstable solutions [144, 150]. Next, weturn to conservation and ask whether it may prescribe a fundamental and robust qualitative change, as compared totypical local RD modeling in the absence of conserved quantities. To exemplify this case, we exploit a reduced CDRmodel (of Eqs. 5 type), which has been used to examine solitary wave collisions in the context of IAW [151]. In thereduced CDR model the conserved AI system of Eqs. 5 is replaced by the conservation of the actin monomers, asthey are converted from the monomeric to the filamentous form (and back) which the IAW propagates.Observation of solitary waves dates back to John S. Russell (1834), yet only after the work of Zabusky and Kruskal[152] were solitary waves distinguished by their collision properties [153, 154]: solitons if after collision of two pulses,two pulses emerge (particle-like identity) and dissipative solitons or excitable pulses if they are annihilated. Solitons areoften being discussed in the context of conservative media, which mathematically means exploiting the integrablenature of the governing model equations [107, 155] while, excitable pulses often arise in RD type systems. Althoughcollisions of solitons may involve high spatiotemporal complexity, the outcome of two colliding solitons remainsunchanged (i.e., elastic particle-like dynamics) [156, 157]. On the other hand, the annihilation of excitable pulses afterthe collision is recognized as paramount for electrophysiological function, i.e., it would be impossible to maintaindirectionality, and thus rhythmic behavior, under the reflection of action potentials [158]. Importantly, collision ofpulses implies merging of the pulses in space, i.e, through the formation of a collision zone . This behaviour is distinctfrom interaction between excitable pulses that is due to repulsion and can exhibit dynamics that may resemble asolitonic behavior [159, 160].Also more complex scattering scenarios have been observed in generic RD models suchas, for example, the Gray-Scott model [161, 162]. Note that there exists a vast literature on the latter topic that we donot intend to review in total here. Taken together, the distinction between solitons and excitable pulses is importantfor numerous applications.Yochelis at el. [151] showed that the minimal IAW model, in the class of Eqs. 5, may indeed support rich androbust spatiotemporal dynamics following pulse collisions, in contrast to IAW models which do not contain explicitmass conservation [28, 55, 163–165]: annihilation, reflection, and “birth” of new pulses after reflection, as shownin Fig. 3. In a broader RD context, where similar aspects have been also observed, these dynamics do not requirespecial properties, such as non-locality [166–169], cross–diffusion [170], and heterogeneity [171–173]. Moreover, thephenomenon is robust and occurs over a wide range of parameter values, whereas for a typical RD model withoutmass conservation, such as FHN, somewhat similar dynamics of propagating pulses are observed only in a narrowrange near the onset of an oscillatory Hopf bifurcation about a uniform steady state [174, 175]. The distinctionbetween the FHN model and a system of Eqs. 5 type can be elaborated by geometrical intuition, since pulses are oflarge amplitude and thus cannot be unfolded using weakly nonlinear analysis such as in Section III. Argentina etal. [174] showed that in the FHN model a manifold construction about the collision state of two pulses (”collisiondroplet”, Fig. 4(A)) can explain why a Hopf bifurcation may impact the collision zone and thus generate crossoverof pulses (soliton-like behavior). A similar geometric picture shows that mass-conservation in Eqs. 5 changes thenature of the collision zone by addition of a generic two-dimensional neutral manifold (Fig. 4(B)), relating the pulsecrossover behavior to a localized unstable mode and does not require any Hopf bifurcation of the uniform state [174, A BFIG. 4. Excitable solitons, geometric analysis of the dynamics during collision of two pulses. (A) FitzHugh–Nagumo model and(B) an reaction–diffusion model with mass-conservation, of Eqs. 5 type. (A) Reprinted from Publication [174], with permissionfrom Elsevier and (B) from [151], Copyright 2020 by the American Physical Society.
V. CONCLUSIONS
The case of the reduced CDR model discussed above provides a glimpse to the profound impact of mass conser-vation on the dynamics. In conventional FHN-type AI models without mass conservation, colliding pulses typicallyannihilate upon collision. Here, a soliton-like crossover occurs only under special conditions, e.g. near a Hopf point,and thus requires fine-tuning of the parameters. In contrast, if mass conservation is taken into account, propagatingpulses robustly exhibit rich collision scenarios over a wide range of parameters, including crossover and formationof new pulses following collision. Even though this has only been demonstrated for a simple toy model, the uni-versal nature of the underlying bifurcations suggests that a similar behavior will be observed also in more detailed,high-dimensional models of IAW, provided that mass conservation is included, e.g., for mechanochemical wavesunder conservation of calcium [29].The impact of mass conservation on pattern formation in biological systems has recently attracted increasing atten-tion, in particular in the context of well-controlled, confined systems such as the bacterial Min protein oscillator [87].However, many biological systems involve multiple components not all of which are conserved, so that the conse-quences of strict mass conservation as implied by Eqs. 2 and 4 are often relaxed and require a more general view.This is provided, in the simplest case, by adding a third dynamical variable to the system that is coupled to the con-served quantities but does not obey mass conservation itself, see Eqs. 5. It demonstrates that a large scale mode is thekey feature that mass conservation introduces to the system and that triggers specific dynamical properties, such assoliton-like crossover of pulses and the collision-induced birth of new pulses in a wide range of parameters. Eqs. 5,and its resulting dynamics, can serve as motivation for further future studies of the synthesis between classical AImodels and models with complete mass-conservation (such as those used in the context of the Min and Par systems[180]).Similar to neural systems, where annihilation of colliding pulses is essential to maintain directionality of infor-mation transport, we conjecture that also in the case of IAW, the crossover of colliding pulses, which is favored dueto the mass conservation constraint, plays an important functional role. This may be particularly true, when sus-tained wave activity is a key requirement for proper cell functions, as for example in cases where cell locomotion0or nutrient uptake depend on IAW (see Section II A). For traditional excitable pulses that annihilate upon collision,wave activity is likely to get extinguished regularly, thus hampering cellular activities that rely on persistent IAW. Incontrast, soliton-like crossover and collision-induced nucleation of new pulses that are robust properties of a mass-conserved system may ensure prolonged wave activity even in the absence of actively triggered pulse nucleationor local heterogeneities that may serve as pacemakers. Moreover, cells may also actively exploit shifts between pa-rameter regimes of pulse annihilation and soliton-like behavior to control their level of IAW activity, as shown inFig. 3.Finally, the study of simplified models to elucidate generic properties of IAW patterns may also prove useful forthe future design of synthetic cellular systems. A current focus of bottom-up approaches in synthetic biology is tointroduce artificial cytoskeletal structures into membrane vesicles, thus assembling the essential building blocks ofa primitive cell [181, 182]. The logical next step along this line of research will be to endow the artificial cytoskele-tal components with simple pattern forming properties that may ultimately serve as a basis for essential cellularfunctions, such as motility and cytokinesis. This requires a thorough understanding of the key properties that arenecessary to reconstitute the desired wave patterns in a minimal model system. We thus expect that the understand-ing of the essential bifurcations and instabilities that govern the dynamics of IAW to provide a useful guideline forthe future design of artificial cell cortices. [1] M. C. Cross and P. C. Hohenberg, Rev. Mod. Phys. , 851 (1993).[2] L. Pismen, Patterns and interfaces in dissipative dynamics (Springer, 2006).[3] M. Cross and H. Greenside,
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