Transformation from spots to waves in a model of actin pattern formation
TTransformation from spots to waves in a model of actin pattern formation
Stephen Whitelam , , Till Bretschneider and Nigel J. Burroughs Molecular Foundry, Lawrence Berkeley National Laboratory, 1 Cyclotron Road, Berkeley, CA 94720, USA Systems Biology Centre, University of Warwick, Coventry, CV4 7AL, UK
Actin networks in certain single-celled organisms exhibit a complex pattern-forming dynamicsthat starts with the appearance of static spots of actin on the cell cortex. Spots soon becomemobile, executing persistent random walks, and eventually give rise to traveling waves of actin.Here we describe a possible physical mechanism for this distinctive set of dynamic transformations,by equipping an excitable reaction-diffusion model with a field describing the spatial orientationof its chief constituent (which we consider to be actin). The interplay of anisotropic actin growthand spatial inhibition drives a transformation at fixed parameter values from static spots to movingspots to waves.
Introduction. Dictyostelium discoideum ( Dicty ) is anamoeba known to generate spectacular patterns throughorganized multicellular aggregation [1]. This organiza-tion is made possible by
Dicty ’s ability to move, which inturn is regulated by polymerization of the protein actininto oriented networks within individual amoebae [2]. In-terestingly, these networks exhibit their own distinctivepatterns. When treated with the drug latrunculin, actinnetworks within
Dicty degrade, rendering the amoeba im-mobile. Upon removal of latrunculin, actin networks re-polymerize through a complex pattern-formation processthat appears to consist of three stages [3, 4], summarizedin Fig. 1. First, immobile circular spots of actin form onthe cell membrane. Second, spots acquire a persistentdiffusive motion. Third, wave-like actin structures rem-iniscent of cell ‘leading edges’ appear and coexist withspots. As waves strike the cell periphery the amoebarecovers its ability to move.This paper describes a mechanism that might under-pin the remarkable dynamic transformation from staticspots to moving spots to waves. Our starting pointis the recognition that spots (both static and moving)and waves have been seen in chemical systems [5], andthat such patterns can be described by equations mod-eling reacting and diffusing chemicals [6]. Here we con-sider an excitable ‘activator-inhibitor’ reaction-diffusionmodel that describes, in different regions of its parame-ter space, stationary spots and moving waves. In orderto interpret the chief constituent of this model (the ‘ac-tivator’) as substrate-bound actin in
Dictyostelium , weequip it with an additional variable that describes localactin fiber orientation. We find that the resulting modelexhibits a series of nonequilibrium transformations, atfixed parameter values, from stationary spots to movingspots to traveling waves. This transformation is drivenby the interplay of inhibition (which permits localizedspots) and the directional bias imparted to actin poly-merization by local fiber orientation, the latter emergingfrom a spontaneous breaking of fiber symmetry. Our re-sults support the idea that emergent actin patterns invivo are fostered by an excitable medium [7, 8], and il-lustrate the dynamical richness accessible to an intrinsi-cally anisotropic chemical species that suffers inhibition. Models of inhibited but spatially isotropic chemicals candescribe striking transitions between static and movingspots, but only as their parameters are varied [9]; tradi-tional models of anisotropic actin polymerization (tread-milling) do not possess as solutions static, size-limited,isolated spots.
FIG. 1: Dynamic actin structures in the substrate-attachedcortex of a
Dictyostelium cell, as visualized by TIRF mi-croscopy. LimE∆coil-GFP was used to selectively label F-actin. The cell was incubated for 15 min under 5 µ M latrun-culin A to depolymerize F-actin. The first frame (0 min) istaken 16 minutes after reducing latrunculin A concentrationto 1 µ M; the first stationary actin spots are visible (scale bar10 µ m). Spots become more numerous and mobile (13 minframe; selected traces over a period of 30 sec shown in red).At 16 min a prominent spiral wave appears (see dashed out-line). Bottom right: 4 frames of the counterclockwise-rotatingspiral wave superimposed in different colors (red, green, blue,white; first and final images separated by 45 sec). We present our model below. We discuss the originof spot stability, and demonstrate numerically that lo-cal fiber orientation can both induce a spot to move anddistort a spot into a wave. The resulting dynamical evo-lution of the model captures several features of the spotsand waves seen in vivo . Model.
The FitzHugh-Nagumo equations [10, 11] stud- a r X i v : . [ q - b i o . S C ] M a y ied by Vasiev [12] describe localized spots and waves indifferent regions of parameter space. We take these equa-tions as our starting point in an attempt to describe sim-ilar patterns seen in Dicty , and augment these equationsto account for the polarity of the fibers comprising poly-merized actin structures. We consider the evolution ofscalar fields u ( x , t ) and v ( x , t ) on a two-dimensional sub-strate (cell membrane) according to the equations ∂ t u ( x , t ) = ∇ · J + h ( u ) − ρv + (cid:112) T u η ( x , t ); (1) ∂ t v ( x , t ) = δ ∇ v + (cid:15) ( u − v ) . (2)Here J ≡ ∇ u + V τ u and h ( u ) ≡ − ku ( u − u )( u − u ).We shall set the parameters (cid:15) , δ , ρ , k , u and u bycomparison with previous work [12], and will explore theeffect of varying V and T u . We consider u ( x , t ) to beproportional to the number of actin fibers per unit area(relative to a reference concentration) at substrate posi-tion x at time t . The field v ( x , t ) acts to degrade actin,and the vector field τ ( x , t ) labels the orientation of theactin network. Equations (1) and (2) (with V = T u = 0)describe classical excitable behavior wherein the auto-catalytic ‘activator’ field u and activator-suppressing ‘in-hibitor’ field v interact to generate localized patterns thatmay propagate spatially [13]. We consider the homoge-neous terms of Equation (1) to describe polymerizationof actin at a rate proportional its local concentration, r pol = k ( u + u ) u , and degredation of actin with rate r deg = − ku u u . We regard the term − ku as a modelof the concentration-limiting effect of steric hinderance.The field η is a Gaussian white noise with zero mean andunit variance; T u ≥ Dicty [4]) that stationary spots are composed of fibers ori-ented principally normal to the substrate (Fig. 2(a)). Weassume that membrane-bound proteins (such as MyoB)recruit Arp2/3 complexes to the growing ends of fibers,and that these complexes in turn initiate the growth ofdaughter fibers at the spot periphery. We argue thatsuch growth is isotropic, and the resulting propagationof material diffusive. By contrast, we expect that wavespossess many fibers aligned in part parallel to the sub-strate and move chiefly by treadmilling [2, 14]. To de-scribe fiber orientation we have introduced a field τ ( x , t ):vanishing τ describes fibers pointing solely in the verti-cal direction, while nonzero τ describes fibers with somecomponent of orientation parallel to the substrate. Weassume that lateral fiber orientation biases the directionof actin growth, which we model using the term V ∇· τ u . V controls the rate of directed polymerization. We fur-ther hypothesize that orientation may be acquired spon-taneously, and therefore require that τ evolve accordingto the equationΓ − τ ∂ t τ ( x , t ) = − δδ τ (cid:90) d x F τ [ τ , u ] + (cid:112) T τ ξ ( x , t ) . (3) Here F τ ≡ h τ − uτ + τ + α ( ∇ · τ ) + α ( ∇ × τ ) is a Landau-esque free energy density describing a field τ that may order in the presence of the field u (pro-vided that the ‘barrier’ to initial ordering, the term in h , is sufficiently small) and that has a tendency to align.Its degree of alignment is determined by a competitionbetween the orientation-inducing terms coupled to α , ,and the orientation-destroying effect of the noise termcoupled to T τ . The field ξ is a Gaussian white noisewith zero mean and unit variance. We set (arbitrarily) α = α = 0 .
2, and regard T τ as our chief measure ofnetwork-alignment propensity. The kinetic prefactor Γ τ controls the rate of network alignment relative to that ofnetwork polymerization. v = u v = h ( u ) . v . u u < u ! u ! u > u ! . . . . f ( u ) . u t t t V = 0 0 .
012 0 . u , v x ( c )( b )( d )( a ) FIG. 2: Stationary and moving model spots. (a) Cartoonof the actin fiber structure we conjecture for stationary andmoving spots in vivo . (b) Homogeneous effective ‘free energy’density f ( u ) from our model (with u = 0 . , k = 4 . u below ( u = 1 [12]; solidline) at ( u = u (cid:63) ≈ .
06; dashed line) and above ( u = 1 . u -field pro-files (1 d cuts through a 2 d box of size 500 ) at three differ-ent times (( t , t , t ) = (5 , , × ) from three differ-ent simulations ( V = 0 , . , . t = 0 so that a spot nucleates, and we impose an ori-entation field τ = ˆ x /
2. The spot is stationary and stablefor V = 0; the spot moves persistently without significantstructural distortion at the two larger values of V . We showat t the v -field profiles also (thin lines). Parameter values: u = 0 . , u = 1 , k = 4 . , (cid:15) = 0 . , δ = 2 . , T u = 0. Equation (2) describes the evolution of an actin-suppressing ‘inhibitor field’. Actin recruits the agent ofits destruction at rate (cid:15)u ; this agent degrades actin at arate − ρv , and is itself degraded at a rate − (cid:15)v . Possiblebiological origins for v include the concentration of actinsevering proteins (e.g. cofilin) and actin capping pro-teins, or the state of hydrolysis of fibers. Here we simplyregard v as a coarse-grained agent of actin degredation.Our model assumes that actin polymerization is not inthermal equilibrium [2]. Intuitive explanation of spot stability.
Other authorshave demonstrated semi-numerically [12] and analyti-cally [15, 16, 17] that equations similar to (1) and (2)with V = 0 admit static spots as steady-state solu-tions in certain parameter regimes (a stationary spotprofile is shown in Fig. 2 (d)). Here we follow ap-proaches detailed in Refs. [9, 16, 17, 18] to put for-ward a simple physical argument for why such spots canexist. At steady state, and with coordinates rescaledsuch that ∇ → ( (cid:15)/δ ) ∇ , Equation (2) has solution v ( x ) = (2 π ) − (cid:82) d x (cid:48) K ( | x (cid:48) − x | ) u ( x (cid:48) ). We expand thissolution to second order in ∇ and insert the resulting ex-pression in the steady-state version of Equation (1). Weregard the equation so obtained as the Euler-Lagrangeequation of the ‘free energy’ functional F u [ u ] = (cid:90) d x (cid:110) f ( u ) + g ρ ( ∇ u ) + c ρ (cid:0) ∇ u (cid:1) (cid:111) , (4)where f ( u ) ≡ a u − a u + a ( ρ ) u ; a ≡ k/ a ≡ k ( u + u ) / a ( ρ ) ≡ ( ku u + ρ ) / g ρ ≡ ( (cid:15)/δ − ρ ) / c ρ ≡ ρ/
2. The homogeneous component f ( u ) de-scribes a first-order phase transition from an empty sub-strate to an actin-covered substrate as the parameter u is increased ( u is roughly proportional to the mean-fieldactin polymerization rate); bulk phase coexistence occurswhen u = u (cid:63) = u + (cid:0) u + 8 ρ/k (cid:1) / . For the param-eters of reference [12] ( ρ = 1, u = 0 .
05 and k = 4 . u (cid:63) = 1 . u was set to unity, im-plying that the system considered there was numericallyclose to coexistence. We show f ( u ) (and its correspond-ing nullclines) in this regime in Fig. 2(b) (Fig. 2(c)). Notethat the activator-inhibitor coupling ρ influences the lo-cation of the mean-field critical point, but not the natureof the transition.The space-dependent terms of Equation (4) admitmodulated patterns when ρ > (cid:15)/δ < ρ ; in thispaper we perform simulations with ρ = 1 , (cid:15) = 0 . δ = 2 .
5. It is instructive to recognize that while versionsof Equation (4) appear in many different settings [17],the inhibitor-induced spatial modulation of the u fieldclosely resembles the physical mechanism by which sur-factant induces microphase separation of oil in water [18].One might therefore consider the stable spots of theFitzHugh-Nagumo equations studied by Vasiev [12] to beakin to drops of surfactant-coated oil in water. Moreover,extremizing Equation (4) in circular geometry for an as-sumed density profile (adapting the approach of Ref. [18])reveals a characteristic spot size. While a local approxi- mation such as Equation (4) possesses limited predictivepower [17], it does capture the correct trend of variationof spot radius with model parameters (such as (cid:15) ), andprovides an intuitive explanation for why static spots ofwell-defined size exist in this system. ( a ) ( b ) V = 0 . . u , v −
25 0 25 50 x . . . . V = 0 R g t ( c ) !! ! !"" t = 1 × t = 9 × t = 24 . × t = 29 × ( d ) t = 0 . × t = 2 × t = 12 . × t = 17 . × FIG. 3: Model spot-to-wave transformation. (a) An initiallystatic spot moves under the influence of a fixed orientationfield τ = ˆ x / V = 0 .
24) and deforms into a structure re-sembling a traveling wave (profile and image shown 1 . × time units after application of the field). (b) We quantify thisdistortion mechanism by plotting the radius of gyration R g as a function of time for spots exposed to similar fixed τ andvarious values of V . (c) Time-ordered snapshots of u ( x ) froma simulation with fluctuating τ starting from a single spot inthe center of a box. The spot acquires orientation sponta-neously (spot arrow labels spot-averaged τ ), executes a per-sistent random walk (numbers identify periodic box crossings;trajectories and arrows record spot motion), and begins to de-forms into a wave at t ≈ × . (d) A similar simulationwith nine initial spots mimics the transformation from staticspots to moving spots to waves shown in Fig. 1. Effect of fiber orientation upon spots . To determine theeffect of τ upon the behavior of spots, we performed nu-merical simulations of our model for a parameter set thatadmits, at steady state, immobile spot solutions when V = 0 ( u = 0 . , u = 1 , k = 4 . , (cid:15) = 0 . , δ = 2 . τ and V are fixed, a spot moves persistently. Moving spots in ourmodel behave similarly to the moving spots described inRef. [9], colliding both inelastically (at high speeds) andelastically (at low speeds). When V is made sufficientlylarge, a spot deforms into a localized structure (Fig.3(a,b)) resembling the traveling waves seen in the regu-lar FitzHugh-Nagumo model at parameter values distinctfrom those at which spots are found [12]. This exploita-tion of a spot’s apparent underlying instability to wavedeformation, which embodies the notion of nonequilib-rium pattern control discussed in Ref. [21], requires per-sistence of orientation. In the full model, when τ evolvesaccording to Equation (3) (with noise), the transforma-tions from static spot to moving spot and from movingspot to wave can occur on a broad range of timescales.In Fig. 3(c) we show configurations of u ( x ) from a simu-lation (Γ u = 1 , h = 0 . , T u = 0 , T τ = 0 . , V = 0 . , periodically replicated in imitation of bulk sur-roundings. The spot spontaneously acquires orientation(at t ≈ . × ), executes a persistent random walk,and eventually deforms into a wave. Spot trajectoriesare shown in the second and third panels, with periodicboundary crossings labeled in the order they occur. InFig. 3(d) we show images from a similar simulation withnine initial spots. A phase of autonomous mobile spotsis supplanted at later times by an organized collection ofwaves (moving from left to right), mimicking the trans-formation seen in Dicty . Indeed, the model’s dynamicscaptures several qualitative features seen in vivo [1, 4]:spots nucleate spontaneously (when T u (cid:38) . Dictyostelium . Spatially local-ized patterns in reaction-diffusion systems have recentlyreceived considerable attention [6], and the model wepresent demonstrates the unusually rich dynamics acces-sible to an intrinsically anisotropic chemical species. Ourwork suggests that certain cytoskeletal dynamics can in-deed be caricatured by simple reaction-diffusion modelsembodying the idea of excitability [7, 8], supporting aphysical picture (that one might dub the excitoskeleton )in which cytoskeletal pattern formation is driven by theself-organization of excitable solitons of actin and its at-tendant proteins, rather than being orchestrated solelyby biophysical signaling pathways.We thank B.N. Vasiev for discussions. Work atthe Molecular Foundry was supported by the U.S. De-partment of Energy under Contract No. DE-AC02-05CH11231. 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