Mesoscopic model for filament orientation in growing actin networks: the role of obstacle geometry
MMesoscopic model for filament orientation in growing actinnetworks: the role of obstacle geometry
Julian Weichsel
1, 2, ∗ and Ulrich S. Schwarz † Bioquant and Institute for Theoretical Physics, University of Heidelberg, Germany Department of Chemistry, University of California at Berkeley, United States (Dated: October 31, 2018) a r X i v : . [ phy s i c s . b i o - ph ] A p r bstract Propulsion by growing actin networks is a universal mechanism used in many different biologicalsystems, ranging from the sheet-like lamellipodium of crawling animal cells to the actin comet tailsinduced by certain bacteria and viruses in order to move within their host cells. Although the coremolecular machinery for actin network growth is well preserved in all of these cases, the geometryof the propelled obstacle varies considerably. During recent years, filament orientation distributionhas emerged as an important observable characterizing the structure and dynamical state of thegrowing network. Here we derive several continuum equations for the orientation distribution offilaments growing behind stiff obstacles of various shapes and validate the predicted steady stateorientation patterns by stochastic computer simulations based on discrete filaments. We use anordinary differential equation approach to demonstrate that for flat obstacles of finite size, twofundamentally different orientation patterns peaked at either ±
35 or +70 / / −
70 degrees exhibitmutually exclusive stability, in agreement with earlier results for flat obstacles of very large lateralextension. We calculate and validate phase diagrams as a function of model parameters and showhow this approach can be extended to obstacles with piecewise straight contours. For curvedobstacles, we arrive at a partial differential equation in the continuum limit, which again is in goodagreement with the computer simulations. In all cases, we can identify the same two fundamentallydifferent orientation patterns, but only within an appropriate reference frame, which is adjusted tothe local orientation of the obstacle contour. Our results suggest that two fundamentally differentnetwork architectures compete with each other in growing actin networks, irrespective of obstaclegeometry, and clarify how simulated and electron tomography data have to be analyzed for non-flatobstacle geometries. ∗ Electronic address: [email protected] † Electronic address: [email protected] . INTRODUCTION The growth of actin networks is a generic propulsion mechanism occurring in a largevariety of biological systems, ranging from the protruding lamellipodia of animal cells to theactin comet tails recruited by pathogens like the bacterium
Listeria monocytogenes or thevirus
Vaccinia within their host cells. Due to its central importance, the molecular basis ofthis process is well preserved over a wide range of different species [1]. Although a large num-ber of accessory proteins is known to be involved in actin dynamics on the cellular scale, inregions close to the leading edge the dynamics of network growth are determined by a smallnumber of key reactions. Here the interplay between three fundamental processes deter-mines the structure of the growing network: polymerization of filamentous actin, branchingmediated by the protein complex Arp2/3 and binding of capping proteins to the filamenttip preventing further growth [2, 3]. The fact that these processes are highly conservedis impressively demonstrated by the observation that many intracellular pathogens rely onthem for efficient infection and spread in the cytoplasm of their hosts [4–6]. Even function-alized plastic beads, rods and discs as well as lipid vesicles and oil droplet in purified proteinsolutions containing this minimal set of molecules have been shown to be propelled in vitro by polymerizing actin networks [7–12].While the underlying molecular basis is very similar in all of these cases, the geometryof the different propelled objects is very different. For instance, the highly curved shape ofrelatively small, almost ellipsoidal pathogens like
Listeria monocytogenes or Vaccinia virusis very different from the relatively flat shape of the leading edge of a crawling cells. Theexact shape of the membrane in migrating cells is very dynamic, but certainly corrugatedon different scales, thus in this case a flat obstacle shape can only be a first approximationand corrugated obstacle contours are also of large interest. In a growing actin network, newfilaments nucleate by branching off from existing mother filaments at a characteristic anglearound 70 ◦ set by the molecular geometry of the Arp2/3-complex. Importantly, branchingcan occur only close to the surface, as it depends on the presence of surface-bound nucleationpromoting factors (NPF) like WASP or ActA [13]. Whether the daughter filament becomesa productive member of the growing actin gel depends on the way its direction is orientedrelative to the direction of growth and how the object is shaped. Therefore obstacle geometryis a crucial determinant of the resulting structural organization of the network.3ne essential function of growing actin networks is to generate force against externalload. Force-velocity relations for growing actin networks have been measured in differentexperimental setups and for various obstacle shapes [14–20]. Unexpected discrepancies inthe results indicate that obstacle shape and the resulting difference in network organizationalso has an effect on force generation. Modern electron tomography leads to visualiza-tion and analysis of filamentous actin networks in ever greater detail [21, 22]. While earlyelectron microscopy data for the lamellipodia of fish keratocytes suggested a dendritic net-work of relatively short actin filaments [23], recent electron tomography data has revealed amore diverse structural organization, with relatively long filaments connected by few branchpoints into different and spatially extended filament subsets [22, 24]. In addition, it hasbeen demonstrated that the structural organization of the network strongly depends on theprotrusion speed, with fast growing networks dominated by two symmetric diagonal filamentorientations and slowly growing networks featuring more filaments in parallel and orthogonalto the leading edge [25, 26]. It is to be expected, that in the future the effect of obstaclegeometry on network structure can be quantified by such methods as well.In order to achieve a complete understanding of the structure of growing actin networks,different modeling approaches have been developed during the last decade [27], rangingfrom microscopic ratchet models [28, 29], rate equations for filament growth [30–32] andlarge-scale computer simulations of filament ensembles [33–36] to continuum theories ofhow elastic stress propels the obstacle [37] and multi-scale models combining several ofthese model classes [20, 38]. From some of these studies it has emerged that one centralquantity characterizing the structural organization of growing actin networks is the filamentorientation distribution, which can be directly compared with experimental results [26, 31,39, 40].In the following, we study a model which allows us to predict the filament orientation dis-tribution as a function of obstacle geometry. Our reference point will be stochastic computersimulations of growing actin networks incorporating the molecular processes of branching,capping and filament polymerization [32]. For computational simplicity and deeper insight,these will be compared to different versions of a rate equation model [30–32]. For actingrowth behind flat and laterally widely spread obstacles the steady state organization hasbeen predicted earlier to be either a ±
35 or +70 / / −
70 degree filament orientation pattern,with mutually exclusive stability determined by the model parameters [31, 32]. In this pa-4er, we will extend this analysis to also predict the effect of finite size and geometry of thepropelled object. In particular, we will derive and validate a partial differential equation,which is valid also for curved obstacle shapes.The article is organized as follows: In Sec. II we introduce the model and explain howwe analyze it. To validate our results, we compare different versions of a deterministiccontinuum model to stochastic computer simulations of filamentous networks. Subsequently,the impact of different piecewise straight ( linear ) obstacle shapes on the resulting networkorientation patterns is analyzed in Sec. III, while specific examples for curved ( nonlinear )obstacle shapes are the subject of Sec. IV. Our main result is that the competition betweentwo fundamentally different network architectures persists for finite-sized, piecewise linearand curved obstacles.
II. MODEL DEFINITION
Motivated by established biological observations, in our model we assume Arp2/3 tonucleate daughter filaments from preexisting mother filaments at a characteristic relativebranching angle around 70 ◦ [2, 23, 41]. The exact value varies with biological system andanalysis methods (for example, a recent value from electron tomography data for the lamel-lipodium of fibroblasts is 73 ± ◦ [22]), but is not essential for our theory. Although theexact mechanism for the activation of Arp2/3-mediated branching is not yet well estab-lished and subject of current research, it is generally accepted that Arp2/3 is active onlynear the obstacle, where it is activated by nucleation promoting factors (NPFs) such as theWiskottAldrich syndrome protein (WASP) or the bacterial actin assembly-inducing protein(ActA) [13, 42, 43]. As we are interested in actin network architecture in close proximityto the surface of a propelled obstacle, we will focus on actin dynamics within the first fewten nanometers from the obstacle surface in which filament branching, capping and barbedend polymerization are expected to dominate actin dynamics, while filament depolymeriza-tion and decapping can be neglected [44]. Fig. 1(a) sketches the geometrical arrangementstudied here. The yellow branching region within a vertical distance d ⊥ br from the obstaclesurface indicates the domain in which filament bound Arp2/3 is able to interact with NPFs,which themselves are active only close to the surface. Thus filament branching occurs in thisbranching region. Once filament barbed ends have left this domain due to their retrograde5ow they are not able to nucleate new daughter filaments anymore and will eventually beoutgrown by the bulk network. As those filaments subsequently do not impact the orien-tation distribution at the leading edge, we do not explicitly account for their further fateanymore. If the lateral extension d =br of the obstacle is large compared to the width d ⊥ br ofthe branching region, it is possible to neglect the process of filament barbed ends growingout of the branching region horizontally. However, for relatively small obstacles, for instanceviral pathogens, the lateral dimension of the propelled particle d =br can become comparableto its vertical extension d ⊥ br and the finite lateral size of the branching region has to be takeninto account. Our main quantity of interest is the filament orientation angle θ relative tothe surface normal.In the following, we will validate our theoretical approach by comparing results fromtwo complementary implementations, namely stochastic computer simulations and a rateequation approach, which have been introduced before for flat obstacles of large lateralsize [32]. Motivated by the flat nature of the lamellipodium and also for computationalsimplicity, our modeling is restricted to two dimensions. Representative snapshots of suchsimulations for flat and curved obstacle geometries are given in Fig. 1(b) and (c), respectively.Briefly, we simulate a network of infinitely stiff rods. This simplification is justified as thereaction kinetics that eventually determine the stable orientation distribution in steadystate are active within a narrow branching region along the obstacle only. Because filamentsegments spanning over this region are very short compared to their persistence length(nanometers versus micrometers), local bending undulations are of minor importance here.The remainder of the filaments is considered to be entangled in the bulk actin network,which effectively acts as a base for the protruding filaments. However, filaments embeddedin semidilute actin solutions have been found to exhibit substantial bending fluctuationsin the past [45, 46] and therefore we effectively account for a finite uncertainty in filamentorientation by incorporating a distribution of relative branching angles between mother anddaughter filaments in the model as is discussed below. Each uncapped filament is growingdeterministically at its barbed end with a fixed velocity v fil . Thus we implicitly assumea constant density of actin monomers at the leading edge. Polymerization is quantizedsuch that filaments extent by one building block of length δ fil per unit time. Apart frombranching, individual filaments do not interact and hence the local filament density does notalter polymerization. Filament barbed ends within the branching region close to the leading6dge are possible candidates for stochastic branching and capping events. While capping isassumed to be a first order reaction in the number of actin barbed ends in the reaction zone,for branching we assume zeroth order. This is motivated by the expectation that the supplyof activated Arp2/3 is strongly limited by the availability of NPFs at the leading edge andthus an effective zeroth order branching rate emerges for sufficiently high filament densityor low capping rate [47, 48]. In the opposite limit of low filament density, the branchingreaction is expected to become first order in the number of filaments; this effectively yields anautocatalytic description [30], that is stationary only at one unique network growth velocity.Therefore in this limit, transitions in the filament orientation distribution are not accessible.However, it has been shown that both, first and zeroth order branching models, can beincorporated within a unified theoretical framework and that the crossover from one regimeto the other does not have a direct impact on filament orientation in steady state [48]. Asactin networks in most experimental setups are growing against a finite load of the obstacleand non-constant force-velocity curves are observed, the zeroth order branching regime isexpected to dominate the branching kinetics eventually.The orientation of new filament branches is chosen from a normalized linear combina-tion of two Gaussian probability distributions for the angle between mother and daughterfilament with means at ± ◦ and standard deviation 5 ◦ . Red filaments in the illustrationare actively growing, while blue filaments indicate capped barbed ends that neither branchnor polymerize anymore and will eventually be outgrown by the bulk network and leave thesimulation box at the bottom. In Fig. 1(b) the top boundary of the yellow branching regiondefines a rigid obstacle, which excludes volume and thus prevents polymerization above itsboundary. As a consequence the fastest filaments which are growing in close to verticaldirection are stalled by the obstacle. The vertical velocity of the obstacle is a parameter ofthe model and set to a constant value v nw directly. This growth velocity should be regardedas the effective outcome in steady state, determined by the details of filament-obstacle inter-actions (including regulation by biochemical factors). Steady state growth is possible withinthe velocity range, 0 < v nw < v fil . Under these conditions a well defined filament numberand orientation distribution evolves.As a powerful analytical alternative to the stochastic framework introduced above, wealso develop a deterministic rate equation for dendritic network growth based on earlierapproaches of this kind [30–32]. The evolution of the orientation distribution N ( θ, t ) of7ncapped filament barbed ends integrated over the whole branching region evolves in timeby branching and capping events and by filaments growing out of the branching region. Thistranslates into the following ordinary differential equation: ∂N ( θ, t ) ∂t = ˆ k b (cid:90) W ( θ, θ (cid:48) ) N ( θ (cid:48) , t ) d θ (cid:48) (cid:124) (cid:123)(cid:122) (cid:125) branching − k c N ( θ, t ) (cid:124) (cid:123)(cid:122) (cid:125) capping − k gr ( θ, v nw ) N ( θ, t ) (cid:124) (cid:123)(cid:122) (cid:125) outgrowth . (1)Here again, capping is assumed to be a first order reaction proportional to the number ofexisting filaments with the proportionality constant, a reaction rate k c . The probabilityof branching at a given angle is determined by the weighting factor distribution, W ( θ, θ (cid:48) ),which is modeled as a linear combination of Gaussians with maxima at branching angles ± ◦ between filaments and standard deviation σ = 5 ◦ . θ is the filament orientation anglemeasured relative to the vertical direction. The branching reaction is independent of theabsolute number of existing filament barbed ends in this model (i.e. a zeroth order reaction).Hence, the branching rate ˆ k b is normalized by the total number of new filament ends,ˆ k b = k b W tot , W tot = (cid:90) (cid:90) W ( θ, θ (cid:48) ) N ( θ (cid:48) , t ) d θ (cid:48) d θ . (2)The reaction rate k b indicates the number of new branches per unit time. Most importantlyfor the specific scope of this work, obstacle velocity and shape enter Eq. (1) via the outgrowthrate k gr . As filaments are polymerizing with a constant velocity v fil in their individualdirection, some filaments will not be able to keep up with the moving obstacle and thusleave the branching region. The precise expression for the rate of outgrowth k gr stronglydepends on obstacle geometry and alters the resulting steady state orientation distributionsas will be analyzed in detail in the following. III. LINEAR OBSTACLE SHAPE
We begin our analysis with linear obstacle shapes as illustrated in Fig. 1(a) and (b). Theobstacle (and therefore the leading edge of the network) moves with a constant velocity v nw towards the top. Our region of interest, the branching region, extends vertically to adistance d ⊥ br from the leading edge of the network. The lateral obstacle width is indicatedby d =br . 8 . Flat large obstacle For obstacles that are laterally widely extended, for instance the leading edge at the frontof the lamellipodium in a migrating cell, we have d =br (cid:29) d ⊥ br and thus horizontal filamentoutgrowth can be neglected locally. Mathematically, this means that we can use periodicboundary conditions in the horizontal direction. This case has been analyzed before in asimilar manner as done below for finite-sized and curved obstacles [32], and therefore werecapitulate the most important results as a reference case. In this simple case, outgrowthof filaments from the branching region can only occur in negative normal direction. Whilethe network grows with velocity v nw in positive normal direction, single filaments grow withvelocity v fil in their individual direction. The projected polymerization velocity depends onthe filament orientation, as v ⊥ fil ( θ ) = v fil cos θ . (3)Filaments having a larger absolute orientation than the critical angle θ c = arccos (cid:18) v nw v fil (cid:19) , (4)are thus not able to keep up with the speed of the obstacle and are subject to outgrowthwith a rate k ⊥ gr ( θ, v nw ) = | θ | ≤ θ c v nw − v fil cos θ ( d ⊥ br / if | θ | > θ c . (5)The factor 2 results from the assumption that new filaments branch-off from existing fila-ments on average in the center of the branching region of vertical width d ⊥ br . Outgrowthvanishes for filaments with an orientation smaller than a critical angle θ c and from thereincreases to its maximum at | θ | = 180 ◦ .We solve Eq. (1) numerically by introducing 360 angle bins and iterating the equationsuntil a steady state is achieved, which then can be compared to the results of the stochasticcomputer simulations. In order to achieve a deeper understanding, we also introduced acoarse-grained version of Eq. (1) that can be treated analytically. In Eq. (1), the number offilament ends in the branching region with angles between θ and θ + dθ is given by N ( θ, t ) dθ .By extending this integration to sufficiently large (∆ θ = 35 ◦ ) angle bins N ¯ θ = ¯ θ +∆ θ/ (cid:90) ¯ θ − ∆ θ/ N ( θ (cid:48) , t ) d θ (cid:48) , (6)9nd further assuming that branching is restricted to pairs of angle bins with a relative angledifference of 70 ◦ , a system of five coupled ordinary differential equations results: ∂N − ◦ ∂t = 12 ˆ k b N ◦ − (cid:0) k c + k ⊥ gr (70 ◦ ) (cid:1) N − ◦ (7) ∂N − ◦ ∂t = 12 ˆ k b N +35 ◦ − (cid:0) k c + k ⊥ gr (35 ◦ ) (cid:1) N − ◦ (8) ∂N ◦ ∂t = 12 ˆ k b ( N − ◦ + N +70 ◦ ) − k c N ◦ (9) ∂N +35 ◦ ∂t = 12 ˆ k b N − ◦ − (cid:0) k c + k ⊥ gr (35 ◦ ) (cid:1) N +35 ◦ (10) ∂N +70 ◦ ∂t = 12 ˆ k b N ◦ − (cid:0) k c + k ⊥ gr (70 ◦ ) (cid:1) N +70 ◦ , (11)with ˆ k b = k b W tot = k b N − ◦ + N − ◦ + N ◦ + N +35 ◦ + N +70 ◦ . Here we have also assumed that branching of filaments with orientations | θ | > . ◦ canbe neglected. The five equations are symmetric around 0 ◦ , and thus only three of themare independent. Nonlinearity and the coupling of all five equations is introduced by thebranching term due to the zeroth order branching reaction.By algebraically solving Eq. (7))–(Eq. (11) for the stationary state, we obtain two phys-ically meaningful solutions. The first solution, N ss35 − ◦ = 0 N ss35 − ◦ = k b 14( k c + k ⊥ gr (35 ◦ )) N ss350 ◦ = 0 N ss35+35 ◦ = k b 14( k c + k ⊥ gr (35 ◦ )) N ss35+70 ◦ = 0 , (12)represents a dominant ±
35 degrees orientation distribution in the steady state while thesecond solution, N ss70 − ◦ = k b k c + k ⊥ gr (70 ◦ ) − (cid:113) k c ( k c + k ⊥ gr (70 ◦ ) ) ( k ⊥ gr2 (70 ◦ ) − k ) N ss70 − ◦ = 0 N ss700 ◦ = k b 1 − (cid:114) k c+ k ⊥ gr(70 ◦ )2 k c k c − k ⊥ gr (70 ◦ ) N ss70+35 ◦ = 0 N ss70+70 ◦ = k b k c + k ⊥ gr (70 ◦ ) − (cid:113) k c ( k c + k ⊥ gr (70 ◦ ) ) ( k ⊥ gr2 (70 ◦ ) − k ) , (13)10orresponds to a competing +70 / / −
70 pattern.The stability of these two fixed points can be analyzed with respect to the parameters k b , k c , k ⊥ gr (35 ◦ ), k ⊥ gr (70 ◦ ) and d ⊥ br using linear stability analysis. The result of this analysis isindependent of the branching rate k b and therefore this parameter has no influence on thestability of the system. According to Eq. (12) and Eq. (13), however, the total number offilaments in steady state is proportional to k b and therefore this parameter can be used toadjust the model to experimentally measured densities. The two parameters k ⊥ gr (35 ◦ ) and k ⊥ gr (70 ◦ ) are not independent, but rather both of them are determined by the obstacle velocity v nw as given in Eq. (5). In the following, we will omit the ill-defined cases k c = k ⊥ gr (35 ◦ ) = 0and k c = k ⊥ gr (70 ◦ ) = 0, for which the filament number diverges. We find that the stabilityof both fixed points changes when k c k ⊥ gr (70 ◦ ) = k + 4 k c k ⊥ gr (35 ◦ ) + 2 k ⊥ gr2 (35 ◦ ) (14)and that either one is asymptotically stable, while the other is a saddle. Thus the simplemodel suggests that exactly two possible network architectures exist with mutually exclusivestability.Eq. (14) can now be used to obtain the respective network velocity v nw of the transition.For a critical angle θ c ≥ ◦ (i.e. k ⊥ gr (70 ◦ ) = k ⊥ gr (35 ◦ ) = 0), Eq. (14) is never satisfied (notethat k c > ◦ ≤ θ c < ◦ (i.e. k ⊥ gr (70 ◦ ) > ∧ k ⊥ gr (35 ◦ ) = 0), v nw = 12 k c d ⊥ br + v fil cos(70 ◦ ) , for 35 ◦ ≤ θ c < ◦ . (15)satisfies Eq. (14). For network velocities with a critical angle θ c < ◦ (i.e. k ⊥ gr (70 ◦ ) >k ⊥ gr (35 ◦ ) > v nw 1 , = (cid:0) − k c d ⊥ br + 8 v fil cos(35 ◦ ) (cid:1) ± (cid:113) k c d ⊥ br (cid:0) k c d ⊥ br + 16 v fil cos(35 ◦ ) − v fil cos(70 ◦ ) (cid:1) , for θ c < ◦ . (16)It turns out that solution Eq. (15) is valid for network bulk velocities v fil cos(35 ◦ ) ≥ v nw >v fil cos(70 ◦ ), while for solution Eq. (16) to be valid v nw > v fil cos(35 ◦ ) has to be satisfied.This is never the case for the negative square-root in Eq. (16) and so we can neglect thissolution in the following. In order to further simplify our equations, we define the referencevelocity u c ≡ k c d ⊥ br resulting from the capping rate. Eq. (15) and Eq. (16) then become v nw v fil = 12 u c v fil + cos(70 ◦ ) , (17)11nd, v nw v fil = (cid:16) − u c v fil + 8 cos(35 ◦ ) (cid:17) + (cid:114) u c v fil (cid:16) u c v fil + 16 cos(35 ◦ ) −
16 cos(70 ◦ ) (cid:17) , (18)respectively. Thus the capping rate emerges as an essential parameter through the effectivevelocity u c .In Fig. 2(a) the stability diagram from the analytical analysis is illustrated, showingthe regions in which either the ±
35 degree distribution is asymptotically stable and the+70 / / −
70 pattern is a saddle or vice versa. In order to compare these results to the fullnumerical solution of Eq. (1), we define the relative difference of filaments in the angle binaround 0 ◦ to the one around 35 ◦ as an appropriate order parameter in steady state, O = N ◦ − N ◦ N ◦ + N ◦ = [ − , +1] . (19)For a perfect +70 / / −
70 distribution this parameter will approach +1, while for the com-peting ±
35 pattern it will approach −
1. The transition between the two patterns is definedwhen the order parameter changes its sign. According to this definition of the transitionpoint, it is now possible to numerically solve the continuum model Eq. (1) and to classifyeach observed stationary state as one of the two phases. These results are presented as aphase diagram in Fig. 2(d) and were obtained for different branching rates k b , which indeedhas no significant influence as expected from the analytical considerations. In the solutionof the full continuum model the capping rate only has a very limited influence on the stripe-like pattern of the phase diagram. Contrarily, in the phase diagram resulting from linearstability analysis (Fig. 2(a)), the ±
35 pattern vanishes at large capping rate. This artifactof the reduced rate equation model is introduced by the assumption that filaments with anorientation larger than 87 . ◦ do not branch. As the full rate equation does not share thisassumption, it does not predict the elimination of the ±
35 pattern for large capping rate.
B. Flat finite-sized obstacle
For flat, but finite-sized obstacle shape, filaments might also leave the branching regionhorizontally. Therefore a second outgrowth rate needs to be incorporated into the rate12quation (Eq. (1)): k =gr ( θ, v nw ) = v nw tan θ ( d =br / if | θ | ≤ θ c v fil sin θ ( d =br / if | θ | > θ c . (20)We find that for this scenario, it is still possible to analyze the stability of the fixed pointsof Eq. (7)–Eq. (11). The only difference that accounts for the change in obstacle geometryat this point is that a combination of outgrowth rates k gr ( θ ) = k ⊥ gr ( θ ) + | k =gr ( θ ) | replacesthe term for exclusively orthogonal outgrowth k ⊥ gr ( θ ) considered in the previous section. Forfilaments growing in the N ◦ angle bin, both outgrowth rates vanish, because these filamentsare growing at a high enough velocity parallel to the lateral boundaries of the obstacle.The same arguments as used before also apply for the stability analysis of the steadystates here. Hence, we can begin with Eq. (14) and evaluate the stability of the two steadystates according to the adjusted outgrowth rate. To determine the transitions, we will followa similar strategy as before. Starting from small network velocities v nw , we will treat thedifferent possibilities for the outgrowth rates given in Eq. (5) and Eq. (20) in a case-by-caseanalysis.For θ c ≥ ◦ , the orthogonal outgrowth rates vanish, as all relevant filament orientationsgrow faster towards the top than the obstacle and are slowed down to v nw in this direction.The outgrowth rates for the different orientation bins therefore read, k ⊥ gr ( θ ) = 0 and k =gr ( θ ) = v nw tan θ ( d =br /
2) for θ = [35 ◦ , ◦ ] . (21)Inserting k gr = k ⊥ gr + k =gr in Eq. (14) and omitting again the ill-defined cases of k b = 0, k c = 0and d =br = 0 here (and additionally d ⊥ br = 0 in the following), it can be shown that Eq. (14)is never satisfied.Increasing the network velocity to the point where 35 ◦ ≤ θ c < ◦ leads to outgrowthrates, k ⊥ gr (35 ◦ ) = 0 and k ⊥ gr (70 ◦ ) = v nw − v fil cos(70 ◦ )( d ⊥ br / , (22)in orthogonal direction according to Eq. (5) and, k =gr (35 ◦ ) = v nw tan(35 ◦ )( d =br /
2) and k =gr (70 ◦ ) = v fil sin(70 ◦ )( d =br / , (23)in the lateral direction according to Eq. (20). These rates yield a quadratic equation for the13etwork velocities v nw that satisfy Eq. (14),0 = v (cid:104) (35 ◦ ) u c2 r br2 (cid:105) + v nw (cid:104) ◦ ) u c r br − u c (cid:105) + (cid:104) − v fil sin(70 ◦ ) u c r br + 2 v fil cos(70 ◦ ) u c (cid:105) , for 35 ◦ ≤ θ c < ◦ . (24)Here, the reference velocity, u c = k c d ⊥ br , together with a new parameter for the length scaleratio in lateral and orthogonal direction, r br ≡ d =br /d ⊥ br , are identified to determine stability.For a critical angle θ c < ◦ , the adjusted outgrowth rates read, k ⊥ gr ( θ ) = v nw − v fil cos θ ( d ⊥ br /
2) and k =gr ( θ ) = v fil sin θ ( d =br /
2) for θ = [35 ◦ , ◦ ] . (25)These rates together with Eq. (14) and the effective parameters u c and r br can be simplifiedto a second quadratic equation defining the network velocities at the transitions,0 = v (cid:104) u (cid:105) + v nw (cid:104) u c − v fil cos(35 ◦ ) u + 16 v fil sin(35 ◦ ) u r br (cid:105) + (cid:104) v fil cos(70 ◦ ) u c − v fil sin(70 ◦ ) u c r br − v fil cos(35 ◦ ) u c + 8 v fil sin(35 ◦ ) u c r br + 8 v cos (35 ◦ ) u − v cos(35 ◦ ) sin(35 ◦ ) u r br + 8 v sin (35 ◦ ) u r (cid:105) , for θ c < ◦ . (26)According to Eq. (24) and Eq. (26), the regions of stability for the two different stationaryorientation patterns in parameter space are illustrated in Fig. 2(b). As the finite obstaclewidth introduces an additional independent parameter r br , the full parameter space is nowthree dimensional. Network growth velocities v nw that fulfill Eq. (24) together with thecondition v fil cos(35 ◦ ) ≥ v nw > v fil cos(70 ◦ ) or Eq. (26) at v nw > v fil cos(35 ◦ ) are identifiedas transition points between the different orientation distributions in the parameter spacespanned by k c , r br and v nw . In the limit of large length scale ratio, r br → ∞ (i.e. d =br (cid:29) d ⊥ br ),lateral outgrowth can be neglected and the phase diagram approaches the results for periodicboundary conditions as given in Fig. 2(a). For relatively small r br (cid:46)
7, the lateral outgrowthof intermediate filament orientations at around ± ◦ is sufficiently large to prevent thisorientation pattern from being stable in this model, independent of the growth velocity (cf.Fig. 2(c)).Using the order parameter Eq. (19), we numerically sampled the parameter space accord-ing to Eq. (1) with adjusted outgrowth. The isosurface O ( v nw , k c , r br ) = 0 is extracted fromthe three dimensional data and shown in Fig. 2(e). In the limit of large length scale ratio, the14esults coincide well with the case of periodic boundary conditions (cf. Fig. 2(d)). It is alsoconfirmed that for small values of r br the ±
35 pattern is not stable anymore at intermediatevelocities (cf. Fig. 2(f)). Thus the overall agreement between the simple analytical modeland the full numerical solution of the rate equation approach is surprisingly good.Fig. 3 compares filament orientation distributions obtained in steady state from stochasticnetwork simulations, the full continuum model and the simplified continuum model. Theparameters of the two cases shown in (a) and (b) do not differ in their network velocity v nw ,but rather only in the obstacle geometry, in this case given by length scale ratio r br = d =br /d ⊥ br .For small r br = 3 at intermediate velocities, the ±
35 pattern of the network is not stableanymore and rather the network organizes in the alternative +70 / / −
70 distribution (cf.Fig. 3(a)). For larger ratio r br = 20, the network organizes in the ±
35 degree distribution (cf.Fig. 3(b)). In the stochastic simulations, outgrowth rates are not explicitly incorporated,but rather emerge as a direct consequence of the obstacle geometry. Thus the computersimulations nicely validate both, the full and reduced continuum approaches, and all threeseem to capture the essential physical mechanisms determining network structure.
C. Obstacle with tilted straight contour
It is not trivial to find an explicit expression for the outgrowth rate out of the branchingregion behind a skewed linear obstacle, which is rotated according to a constant skew angle ϕ , with − ◦ < ϕ < +90 ◦ . However, for reasonably small skew angle ϕ and width ofthe orthogonal branching region d ⊥ br an approximation can be obtained within a rotatedcoordinate frame, that is adjusted to the constant obstacle skew. Fig. 4(a) shows a sketchof the obstacle geometry we are interested in at this point. The branching region is givenby a rhomboid, with rotated (or skewed) upper and lower side. The orthogonal width ofthe branching region d ⊥ br is again defined parallel to the lateral sides of the obstacle, whileits horizontal width is given by d =br . Network velocity v nw is defined as before in the verticaldirection. To find an explicit expression for the rates of filament outgrowth, a rotation ofthe coordinate frame to the point where the skew angle ϕ of the branching region vanishessimplifies the situation (Fig. 4(b)). In this frame the obstacle appears flat horizontally, verysimilar to the setup analyzed before in Sec. III B. The finite skew angle ϕ , however, manifestsin a non-vanishing lateral network growth velocity ˜ v =nw , while the vertical propulsion speed15f the obstacle is the orthogonal part of the network velocity in this frame, ˜ v ⊥ nw . (Whereapplicable, we will denote variables defined in the rotated coordinate frame by ˜.) In thefollowing we will refer to the original coordinate system Fig. 4(a) as the lab frame and tothe rotated picture Fig. 4(b) as the obstacle frame .Using the transformation to the obstacle frame for a given skew angle ϕ , we can nowdeduce the relevant parameters. The filament orientation angles ˜ θ are given relative to theobstacle, i.e. for ˜ θ = 0 ◦ filaments are growing vertically in the rotated frame. The twocomponents of obstacle growth velocity are given by,˜ v ⊥ nw ( ϕ ) = v nw cos ϕ and ˜ v =nw ( ϕ ) = v nw sin ϕ . (27)The two obstacle diameters, which scale the vertical and horizontal outgrowth rates are,˜ d ⊥ br ( ϕ ) = d ⊥ br cos ϕ and ˜ d =br ( ϕ ) = d =br cos ϕ . (28)According to these definitions, we can now reformulate the continuum description in theobstacle frame. The rate equation model Eq. (1) for the temporal evolution of orientationdependent filament density remains unchanged with respect to the transformed orientationangles ˜ θ , however with adjusted outgrowth rate k gr ( θ, v nw ) given by the sum of orthogonaloutgrowth, ˜ k ⊥ gr (˜ θ, ϕ, v nw ) = | ˜ θ | ≤ ˜ θ c˜ v ⊥ nw ( ϕ ) − v fil cos ˜ θ ( ˜ d ⊥ br ( ϕ ) / if | ˜ θ | > ˜ θ c , (29)and the absolute value of lateral outgrowth,˜ k =gr (˜ θ, ϕ, v nw ) = ˜ v =nw ( ϕ )+˜ v ⊥ nw ( ϕ ) tan ˜ θ ( ˜ d =br ( ϕ ) / if | ˜ θ | ≤ ˜ θ c˜ v =nw ( ϕ )+ v fil sin ˜ θ ( ˜ d =br ( ϕ ) / if | ˜ θ | > ˜ θ c , (30)with, ˜ θ c = arccos (cid:18) ˜ v ⊥ nw ( ϕ ) v fil (cid:19) . (31)The lateral outgrowth rate compared to Eq. (20) is biased by the finite horizontal velocity ofthe network in the obstacle frame ˜ v =nw . This additional feature also breaks the symmetry inthe resulting steady state filament orientation distributions. The sign of the outgrowth ratedetermines, whether filaments are growing out to the left (˜ k =gr <
0) or the right (˜ k =gr > ϕ = 20 ◦ , fast and slow network velocityin the lab frame v nw , and small and large horizontal obstacle diameter as indicated by thelength scale ratio r br (Fig. 5 also shows results obtained with a PDE-model which we willintroduce below). If the results are interpreted directly in the adjusted obstacle frame, verysimilar orientation patterns as the familiar ±
35 and +70 / / −
70 peaked distributions arerealized in steady state. In the limit of very large length scale ratio r br , outgrowth in thelateral direction can be neglected again and the resulting patterns resemble orientation pat-terns for a network growing behind a large obstacle, only now within the rotated obstacleframe (cf. Fig. 5(b) & (d)). For small r br in combination with a relatively high obstaclevelocity, horizontal outgrowth cannot be neglected and the orientation distribution has tobe interpreted in the lab frame (Fig. 5(a)). In between a transition occurs where the solu-tions from stochastic simulations already have to be interpreted in the obstacle frame, whileresults from the rate equation model are still to be interpreted directly in the lab frame(Fig. 5(c)). The reason for this is the spatial resolution in the lateral filament position,which is introduced in filament based stochastic simulations and neglected in the continuummodel. Laterally nonuniform spatial filament distributions lead to differences in the effectiveoutgrowth of filaments to the sides of the branching region. In order to include a similarspatial resolution in the continuum model, in the following we will incorporate an additionaladvection term in the equation. In this way it will also become possible to treat nonlinearobstacle shapes within the continuum description. IV. NONLINEAR OBSTACLE SHAPE
In this section, we focus on actin network dynamics in the tail of curved (i.e. nonlinear)obstacle geometries. The definitions of the relevant parameters stay the same in this geome-try as introduced in Fig. 1(a). Fig. 1(c) features a representative steady state network fromstochastic simulation. In the following, we will stepwise extend the continuum model Eq. (1)17ith the goal to also incorporate nonlinear obstacle shapes in the description. We will dothis first within an adjusted ordinary differential equation (ODE) model in a piecewise-linearapproximation of the obstacle surface and then within a partial differential equation (PDE)model, where an advection term will explicitly govern lateral filament growth in the rateequation, additional to the reaction terms.
A. Piecewise-linear obstacle approximation
Nonlinear obstacle shapes can be treated within a piecewise-linear approximation for theobstacle surface in which the ODE-model from above is still applicable. The construction ofthis approximation is sketched in Fig. 4(d)–(f) for a parabola-like curved obstacle surface.The branching region is divided laterally in n sections of equal size d =br ,i = d =br /n ( i =1 , . . . , n ). The orthogonal width of the branching region in each section remains unchangedat d ⊥ br ,i = d ⊥ br . The resulting obstacle approximation is a combination of lateral sectionswith skewed linear shape according to skew angles ϕ i , very similar to the obstacles discussedbefore. The individual sections (i.e. the ODE-model equation of each section) are coupledto their direct neighbors via the lateral outgrowth rates k =gr ,i to the left and right boundariesas indicated in the sketch. In case of periodic lateral boundary conditions of the obstacle,sections i = 1 and i = n are also coupled via outgrowth. Individual outgrowth rates withinthe local obstacle frame of each section can then be formulated according to Eq. (29) andEq. (30) as before.Assuming a single linear obstacle is the simplest approximation of a nonlinear geometry(Fig. 4(d)). To increase accuracy, the obstacle can be subdivided laterally in ever smallersubsections. Due to the left-right symmetry of many biologically relevant obstacle shapes(e.g. the parabola-like shape in the sketch) it is sufficient to obtain results in one half-space of the obstacle geometry. This translates into a problem of n/ θ → − θ . Forinstance in case of a triangular approximation of the nonlinear obstacle (Fig. 4(e)), onlythe outgrowth behavior at one boundary needs adjustment compared to the problem of a18kewed linear obstacle shape that was already discussed before. Fig. 6(a) and Fig. 7(a),show steady state filament orientation distributions for a laterally nonperiodic parabola andperiodic-cosine shape obtained by the ODE-model in triangular approximation respectively.The nonlinear obstacle shapes together with their respective triangular approximations areillustrated in Fig. 4(c). To parametrize nonlinear parabola and cosine geometries in thiscontext, we are using the left hand side skew angle of the triangular obstacle approximation¯ ϕ in combination with the horizontal width of the obstacle indicated by r br . B. Reaction-advection equation
The continuum limit of an infinitely large number of piecewise-linear subsections, eachof infinitesimal width, yields a PDE-model for nonlinear obstacle shapes. The model equa-tions in this limit are complemented by an additional filament advection term, similar tohydrodynamic models, that incorporates lateral filament growth in Eq. (1), ∂N ( θ, x, t ) ∂t + ∂ ( N ( θ, x, t ) · v =fil ( θ, ϕ, v nw )) ∂x = ˆ k b (cid:90) W ( θ, θ (cid:48) ) N ( θ (cid:48) , x, t ) d θ (cid:48) − k c N ( θ, x, t ) − k ⊥ gr ( θ, ϕ, v nw ) N ( θ, x, t ) , (32)with the horizontal filament growth speed, v =fil ( θ, ϕ, v nw ), and the vertical outgrowth rate inthe lab frame, k ⊥ gr ( θ, ϕ, v nw ). Both, filament growth velocity and outgrowth rate are functionsof filament orientation θ as well as the local obstacle skew angle ϕ ( x ), which is active atlateral position x . Here again, we average outgrowth of filaments in the vertical directionby an effective rate, while horizontal growth is now spatially resolved due to the additionaladvection term.For arbitrary obstacle shape, that can be expressed in terms of an analytic function o ( x ),the local skew angle can be written as, ϕ ( x ) = − arctan (cid:18) ∂o ( x ) ∂x (cid:19) . (33)The two components of the filament growth velocities within the local obstacle frame canthus be written as, ˜ v ⊥ fil (˜ θ, ϕ, v nw ) = | ˜ θ | ≤ ˜ θ c ˜ v ⊥ nw ( ϕ ) − v fil cos ˜ θ if | ˜ θ | > ˜ θ c , (34)19nd, ˜ v =fil (˜ θ, ϕ, v nw ) = ˜ v =nw ( ϕ ) + ˜ v ⊥ nw ( ϕ ) tan ˜ θ if | ˜ θ | ≤ ˜ θ c ˜ v =nw ( ϕ ) + v fil sin ˜ θ if | ˜ θ | > ˜ θ c , (35)with the components of the obstacle velocity, ˜ v ⊥ nw ( ϕ ) and ˜ v =nw ( ϕ ), defined as in Eq. (27) and˜ θ c as in Eq. (31). A transformation (i.e. rotation) from the local obstacle frame into thelocal lab frame at each horizontal position along the obstacle surface subsequently leads tofilament growth velocities, which enter the PDE-model Eq. (32) due to vertical outgrowthand advection, v ⊥ fil ( θ, ϕ, v nw ) = ˜ v ⊥ fil (˜ θ, ϕ, v nw ) cos ϕ + ˜ v =fil (˜ θ, ϕ, v nw ) sin ϕv =fil ( θ, ϕ, v nw ) = − ˜ v ⊥ fil (˜ θ, ϕ, v nw ) sin ϕ + ˜ v =fil (˜ θ, ϕ, v nw ) cos ϕ . (36)Note, that filaments with a positive v ⊥ fil ( θ, ϕ ) are growing vertically in opposite direction tothe obstacle. Using these expressions, we can now rewrite the advection term in Eq. (32)as, ∂ ( N · v =fil ) ∂x = v =fil ∂N∂x + N ∂v =fil ∂x = v =fil ∂N∂x + N ∂v =fil ∂ϕ ∂ϕ∂x , (37)with, ∂v =fil ∂ϕ = − ∂ (cid:0) ˜ v ⊥ fil sin ϕ (cid:1) ∂ϕ + ∂ (˜ v =fil cos ϕ ) ∂ϕ , (38)and substituting Eq. (34), ∂ (cid:0) ˜ v ⊥ fil sin ϕ (cid:1) ∂ϕ = | ˜ θ | ≤ ˜ θ c − v nw sin ϕ + v nw cos ϕ − v fil cos ˜ θ cos ϕ − v fil sin ϕ sin ˜ θ if | ˜ θ | > ˜ θ c , (39)and Eq. (35), ∂ ( ˜ v =fil cos ϕ ) ∂ϕ = v nw cos ϕ − v nw sin ϕ − v nw cos ϕ sin ϕ tan ˜ θ − v nw cos ϕ cos − ˜ θ if | ˜ θ | ≤ ˜ θ c v nw cos ϕ − v nw sin ϕ − v fil sin ˜ θ sin ϕ − v fil cos ϕ cos ˜ θ if | ˜ θ | > ˜ θ c , (40)and using Eq. (33), ∂ϕ∂x = − ∂ o ( x ) ∂x (cid:16) ∂o ( x ) ∂x (cid:17) . (41)As we average outgrowth in the vertical direction by the rate k ⊥ gr ( θ, ϕ, v nw ) in the lab frame,while horizontal growth is now spatially resolved, we have to take into account the finite lat-eral movement of filaments along the obstacle geometry, when considering filaments growing20ut of the branching region vertically. To approximate the filament outgrowth rate accord-ingly, we again assume a locally linear obstacle shape at each horizontal position to find ananalytical expression for the outgrowth rate in Eq. (32), k ⊥ gr ( θ, ϕ, v nw ) = (cid:118)(cid:117)(cid:117)(cid:117)(cid:116) v ⊥ fil2 + v =fil2 (cid:16) d ⊥ br / v ⊥ fil /v =fil − tan ϕ (cid:17) + (cid:16) d ⊥ br / − v =fil /v ⊥ fil tan ϕ (cid:17) . (42)To calculate steady state filament orientation distributions from this PDE-model,Eq. (32)–Eq. (42) can be numerically iterated in discretized space and time using for in-stance a second order upwind scheme for the spatial differential in combination with anEuler method for temporal iteration [49]. For this procedure, we are using a uniform fila-ment distribution in space and orientation as initial condition and, dependent on the specificobstacle shape considered, either periodic boundary conditions or zero filament density atthe respective inflow boundary. C. Linear obstacle shape revisited
The important advantage of using the PDE-model Eq. (32) compared to the initial ODEEq. (1) to solve for steady state filament patterns clearly lies in its additional spatial reso-lution. This benefit not only increases the applicability of the model to a broader range ofobstacle shapes, but also manifests itself when solving for specific parameter combinationsfor simple nonperiodic linear flat obstacles that have been already accessible using the ini-tial ODE equations. Fig. 5(e) illustrates the steady state filament orientation distributionsaveraged over the whole branching region behind an obstacle of small width (i.e. r br = 3).For such laterally small and flat obstacles, the ODE-model predicted the absence of the ± v nw (cf. Fig. 2). Due to their additionalspatial resolution horizontally, the PDE-model as well as results from stochastic networksimulations yield this orientation distribution nevertheless for a small range of parametersat relatively slow obstacle velocity, v nw /v fil = 0 .
2. As shown in Fig. 5(f), the spatial steadystate filament distributions from the PDE-model are nonuniform horizontally and in goodagreement to results from stochastic simulations. This has an impact on the effective lateraloutgrowth of filaments in the model and thus is able to change the final (spatially averaged)filament distributions characteristically as shown in this example.21 . Parabolic and periodic-cosine obstacle shapes
The PDE-model allows us to systematically analyze actin networks in the tail of nonlinearobstacle shapes. As a proof of principle, in the following we have chosen two different obstacleshapes which are motivated by highly relevant biological examples. On the one hand, wemodel a laterally nonperiodic parabolic obstacle shape (blue solid line in Fig. 4(c)) whichis a first approximation for the spherical or ellipsoidal shape of actin-propelled intracellularpathogens. On the other hand, actin growth behind a periodic-cosine geometry (red solidline in Fig. 4(c)) is analyzed, motivated by the laterally widely spread but corrugated leadingedge of a crawling or spreading cell.In Fig. 6 and Fig. 7, the resulting steady state patterns from the three different model-ing approaches (PDE-model, ODE-model in triangular piecewise-linear approximation andstochastic network simulations) are shown for specific parameter combinations. In (a) thefilament orientation distributions spatially integrated over the left side of the symmetricobstacle are shown. Here, the PDE-model corresponds very well to stochastic network simu-lations. Despite the rough piecewise-linear approximation of the nonlinear obstacle shape inthe ODE-model, the results capture the overall characteristic of the resulting network patternvery well. Subfigures (b) illustrate the resulting spatial distributions of certain characteristicfilament angles. For specific parameter cases separate spatial domains emerge over the hori-zontal obstacle dimension, in which the signature of the two alternative steady state networkpatterns, i.e. +70 / / −
70 and ±
35, are observed. For a better overview over the completespatially resolved orientation patterns from the PDE-model, Fig. 6(c) and Fig. 7(c) illustratethe two dimensional filament distribution in heat plots, where regions of cooler color indi-cate higher filament density. The apparent coexistence of alternative patterns resolves whenadditional dotted lines are included in the heat map to indicate local filament orientationsthat correspond to the characteristic orientations ˜ θ = 0 ◦ , ± ◦ , ± ◦ within the obstacleframe, locally orthogonal to the obstacle surface along its contour. Although coexistence ofalternative filament orientations is present in the lab frame, an interpretation in the localobstacle frame yields one unique pattern only that is present along the lateral extension ofthe obstacle. 22 . DISCUSSION In this article, we have introduced several theoretical approaches at different levels ofcomplexity with the common goal to predict actin filament orientation patterns at the sur-face of stiff two dimensional obstacles, whose surfaces promote growth of actin networks.This problem is not only central for many important health and disease related biologicalphenomena, such as migration of animal cells or propagation of pathogens in their host, itis also a prominent example for a physical process whose properties emerge on a mesoscopiclength scale in relative independence of the details of the underlying molecular processes.For example, the competition of the two fundamentally different network architectures dis-cussed here does not rely on the exact value of the branching angle, but rather on the factthat any branching angle below 90 degrees can lead to the possibility that fundamentallydifferent network architectures satisfy the simultaneous requirements of forward growth andside branching. Our model focuses on the geometrical aspects of this situation (in particu-lar filament orientation and obstacle shape). In the future, it might be extended by otherimportant aspects of this complex biological system, including the details of the filament-membrane interaction and the role of filament bending [20, 38].In general, all our results were benchmarked against stochastic computer simulationswhich in principle can be used to include many details of the underlying molecular processes.Here we have adopted the established view that polymerization, branching and capping arethe dominating processes in the context of growing actin gels. For relatively simple linearobstacle shapes, a rate equation (ODE) model yielded accurate results and, due to its reducedcomplexity, also allowed for analytical progress. Using this approach, we found that either oftwo competing orientation patterns emerges in steady state and transitions between the twoare triggered mostly by changes in the velocity of the obstacle. A similar result has alreadybeen obtained in earlier studies for a flat obstacle with large lateral extension, where filamentorientation distributions with characteristic peaks at either +70 / / −
70 or ±
35 have beenfound to dominate the steady state of growing networks [31, 32]. These results were nowextended to finite sized obstacles. We find that for very small objects, the +70 / / −
70 patternis dominant, while for larger objects, mutual stability of the two patterns recovers. In caseof obstacles with a tilted straight contour (linear obstacles), again very similar patterns arepredicted. However, for most parameter configurations they have to be interpreted within23 rotated obstacle frame.Curved (nonlinear) obstacle shapes have been analyzed in the ODE-model using apiecewise-linear approximation of the given geometry. As an alternative, a continuous spatialcoordinate in the horizontal direction was introduced to the model equation by incorporat-ing a filament advection term, thus yielding a PDE-model. The additional benefit of thisspatial resolution clearly lies in the gain of accuracy compared to results obtained earlier.For specific parameter combinations, it was shown that the lack of spatial resolution canlead to incorrect predictions regarding the resulting filament patterns in steady state. Anal-ysis of nonlinear obstacle shapes in the PDE-model yielded an apparent coexistence of thetwo competing network patterns at different horizontal positions along the obstacle surfacein the lab frame. A transformation to the obstacle frame, locally at each lateral position,however revealed that one unique orientation pattern is dominating the network structureas before. Again, these results are in good agreement with computer simulations.In order to experimentally test these theoretical predictions, the method of choice wouldbe electron tomography [22, 24, 26], although in the future super-resolution microscopylike dual-objective STORM might become an interesting alternative [50]. In the contextof a rapidly increasing image quality of electron microscopy (EM) data for actin networks,the correct analysis and interpretation of the measured observables becomes increasinglyimportant. Therefore, a detailed understanding of the structural network characteristicsemerging under different situations is indispensable. In this work, we have also shownthat the extracted orientation distribution of actin filament networks from experimentalEM images growing at the surface of nonuniform obstacles can be easily misinterpreted.We have shown that a standard measurement in the lab frame would yield apparently novelpatterns that could not be explained by existing models. For a correct interpretation of suchfindings, we introduced a rotated obstacle frame which lies locally orthogonal to the obstaclesurface. Within this reference frame, the seemingly novel filament orientation patterns arerationalized and can be understood in terms of the two well known filament distributions,peaked at +70 / / −
70 or ±
35. Thus our results also contribute to improving the wayexperimental data can be compared to theoretical predictions.A convenient setup to test our predictions would be the combination of electron to-mography with biomimetic assays with particles of various shapes, which earlier have beenanalyzed mainly in regard to macroscopic variables such as propulsion velocity and shape of24he comet tail [11]. Apart from obstacle shape, the obstacle velocity relative to the filamentpolymerization speed is a central parameter in the model which triggers transitions betweenfilament orientation patterns. This velocity could be adjusted for instance by applying aforce against network growth diminishing steady state growth. Changing the biochemicalreaction rates for capping and branching is expected to be of minor importance and couldbe tested by varying the concentration of purified protein solutions in the assay. As an addi-tional benefit of introducing different obstacle geometries in experiment, this might also leadto a deeper understanding of Arp2/3 activation close to the surface of the obstacle, whichis one of the most important questions still to be clarified in the context of actin-drivenmotility.
Acknowledgments
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Nature Methods , 9(2):185–188, 2012. vnw obsta led=brd?br bran hing regiond=br 70Æ Y Y (a) (b) (c) FIG. 1: Model of a growing actin network behind a rigid obstacle. (a) Sketch of the setup.Branching can occur only in the yellow region. The main quantity of interest is the filamentorientation angle θ relative to the surface normal. (b) Snapshot of a stochastic simulation based onthe reactions of individual filaments for straight ( linear ) obstacle shape. (c) Snapshot of networkgrowth behind a curved ( nonlinear ) obstacle shape. The network is growing in two dimensions.Red filaments are actively growing at their barbed end in direction θ . Blue filaments have beencapped and will eventually be outgrown by the network and leave the box at the bottom. Growthof the fastest filaments is stalled by the obstacle. c r e l . ne t w o r k g r o w t h s peed v n w / v f il ± br = d =br /d ⊥ br r e l . ne t w o r k g r o w t h s peed v n w / v f il ± (a) (b) (c) c r e l . ne t w o r k g r o w t h s peed v n w / v f il k b =5k b =20k b =100 ± br = d =br /d ⊥ br r e l . ne t w o r k g r o w t h s peed v n w / v f il ± (d) (e) (f) FIG. 2: Phase diagram for flat finite-sized obstacles. (a-c) Analytical results from linear stabilityanalysis of the simplified continuum model with 5 angle bins. (d-f) Full numerical solution of therate equation model with 360 angle bins. In (a) and (d), 2D slices through the 3D hypersurface areplotted within the k c - v nw plane in the limit of r br → ∞ , which corresponds to periodic boundaryconditions or very large objects. (b) and (e) show the hyperplane of transition points between thetwo filament patterns in the full 3D parameter space. In (c) and (f), slice plots in the r br - v nw planeare shown along the transparent gray plane in (b) and (e) at k c = 0 . orientation [ ° ] f il a m en t nu m be r pe r ang l e simple modelfull modelstochastic sim v nw /v fil =0.6 −180−150−120 −90 −60 −30 0 30 60 90 120 150 18000.20.40.60.811.21.41.61.82 orientation [ ° ] f il a m en t nu m be r pe r ang l e simple modelfull modelstochastic sim v nw /v fil =0.6 (a) (b) FIG. 3: Comparison of the steady state orientation distributions for actin networks growing in thetail of a flat obstacle of finite size at v nw /v fil = 0 . k b = 20, k c = 0 . d ⊥ br = 2 δ fil and (a) r br = 3, (b) r br = 20. The results were obtained according to three different methods, the simplified continuummodel (bars), the full continuum model (blue solid line) and stochastic simulations (black dashedline). 'd=brd?brvnw ~(cid:18)vnw ~v?nw~v=nw~d?brrotatedframe ~d=br −50 0 5001020 X Y (a) (b) (c) k=gr;1 bran hing regionN1((cid:18);t)d=br;1 k=gr;1d?br;1 bran hing regionk=gr;1 N2((cid:18);t)N1((cid:18);t) k=gr;2 k=gr;2k=gr;1d=br;i '1 (cid:0)'1d?br;i N1((cid:18);t) N2((cid:18);t) N4((cid:18);t)N3((cid:18);t)k=gr;1 k=gr;2 k=gr;3 k=gr;4k=gr;2 k=gr;4k=gr;3'1 '2 (cid:0)'3 (cid:0)'4bran hing regionk=gr;1d=br;id?br;i (d) (e) (f) FIG. 4: (a) and (b) Within a rotated coordinate frame, the continuum model (Eq. (1)) is applicableto treat filament networks growing behind a skewed linear obstacle. (a)
Lab frame : A generic skewedlinear obstacle with skew angle ϕ . (b) Obstacle frame : The coordinate system is rotated by its skewangle ϕ counter-clockwise. In this frame the initial skew of the system is expressed by an additionalfinite lateral motion at velocity ˜ v =nw of an otherwise flat obstacle moving at ˜ v ⊥ nw . (c) Parabolic (blue)and cosine (red) nonlinear obstacle shapes. The solid lines indicate the nonlinear obstacle shape,while the dashed line corresponds to a piecewise-linear triangular approximation. Horizontallythe parabolic shape has a finite width, while the cosine obstacle shape is analyzed in periodicconditions laterally. (d-f) Piecewise-linear approximations of a parabola-like obstacle geometry. (d)The simplest approximation is a flat linear obstacle. (e) Due to the left-right symmetry in obstacleshape, in higher order approximations of the nonlinear obstacle (e.g. the triangular approximationshown here), only half of the number of sections have to be considered explicitly. The boundarycondition at the center of the box has to be adjusted accordingly. (f) Subdividing the piecewise-linear sections again and again leads to ever higher accuracy in the approximation of the nonlinearobstacle shape and to ever smaller horizontal width of each subsection, d =br ,i = d =br /n . Approachingthe continuum limit, n → ∞ , yields a PDE-model for nonlinear obstacle shapes. orientation [ ° ] f il a m en t nu m be r pe r ang l e pde modelode modelstochastic sim v nw /v fil =0.8 −180−150−120 −90 −60 −30 0 30 60 90 120 150 18000.20.40.60.811.21.41.61.8 orientation [ ° ] f il a m en t nu m be r pe r ang l e pde modelpde obst frameode modelstochastic sim v nw /v fil =0.2 −180−150−120 −90 −60 −30 0 30 60 90 120 150 18000.20.40.60.811.21.41.61.8 orientation [ ° ] f il a m en t nu m be r pe r ang l e pde modelode modelstochastic sim v nw /v fil =0.2 (a) (c) (e) −180−150−120 −90 −60 −30 0 30 60 90 120 150 18000.511.522.533.5 orientation [ ° ] f il a m en t nu m be r pe r ang l e pde modelpde obst frameode modelstochastic sim v nw /v fil =0.8 −180−150−120 −90 −60 −30 0 30 60 90 120 150 18000.511.522.533.5 orientation [ ° ] f il a m en t nu m be r pe r ang l e pde modelpde obst frameode modelstochastic sim v nw /v fil =0.2 −3 −2 −1 0 1 2 300.020.040.060.080.10.120.140.160.18 obstacle x position f il a m en t nu m be r den s i t y pde model stochastic sim 0 ° ± ° ± ° (b) (d) (f) FIG. 5: (a-d) Representative results for the stationary filament orientation distributions of a lineartilted obstacle obtained in the PDE-model (thick red solid line), the ODE-model (Eq. (1)) (bluesolid line) and in stochastic network simulations (black dashed line). Where applicable also thetransformation of the resulting PDE-model distributions to the obstacle frame is shown (thin redline). Here the orientation patterns can be interpreted as ±
35 and +70 / / −
70 patterns, that werealso active in the tail of flat linear obstacles (cf. Fig. 2 & Fig. 3). The active parameters are, ϕ = 20 ◦ , k b = 20, d ⊥ br = 2 δ fil and (a) r br = 3, v nw /v fil = 0 .
8. (b) r br = 100, v nw /v fil = 0 . r br = 3, v nw /v fil = 0 .
2. (d) r br = 100, v nw /v fil = 0 .
2. (e) and (f) Comparison of the steadystate solution for a nonperiodic linear flat obstacle from the ODE-model (Eq. (1)), the PDE-model(Eq. (32)) and stochastic network simulations. (e) For small r br = 3, the ODE-solution (blue line)predicts the absence of a ±
35 orientation distribution (cf. Fig. 2). However, for relatively lownetwork velocity at v nw /v fil = 0 .
2, the spatially resolved PDE-solution (red line) as well as networksimulations (black dashed line) yield such a pattern. (f) The spatial filament distributions are farfrom uniform. Results from PDE-model (solid) and stochastic network simulation (dashed) rathershow an accumulation of filaments at the lateral flanks of the obstacle, and therefore also horizontaloutgrowth rates and as a direct consequence, the resulting orientation distribution averaged overthe whole branching region differs from the prediction of the ODE-model. orientation [ ° ] f il a m en t nu m be r pe r ang l e pde modelode modelstochastic sim v nw /v fil =0.85 −50 0 5000.050.10.150.20.250.30.35 obstacle x position f il a m en t nu m be r den s i t y pde model stochastic sim 0 ° ± ° ± ° −50 −40 −30 −20 −10 0 10 20 30 40 50−180−150−120−90−60−300306090120150180 obstacle x position o r i en t a t i on [ ° ] (1.a) (1.b) (1.c) −180−150−120 −90 −60 −30 0 30 60 90 120 150 18000.20.40.60.81 orientation [ ° ] f il a m en t nu m be r pe r ang l e pde modelode modelstochastic sim v nw /v fil =0.3 −50 0 50−0.0500.050.10.150.20.250.30.35 obstacle x position f il a m en t nu m be r den s i t y pde model stochastic sim 0 ° ± ° ± ° −50 −40 −30 −20 −10 0 10 20 30 40 50−180−150−120−90−60−300306090120150180 obstacle x position o r i en t a t i on [ ° ] (2.a) (2.b) (2.c) FIG. 6: Typical stationary filament distributions in the tail of a nonperiodic parabolic obstacle(cf. blue solid line in Fig. 4(c)) for the parameter combinations, ¯ ϕ = 20 ◦ , r br = 50, d ⊥ br = 2 δ fil , k b = 20, k c = 0 .
05; (1) v nw /v fil = 0 .
85; (2) v nw /v fil = 0 .
3. (a) Filament orientation distributionsspatially averaged laterally over the left hand side half space behind the symmetric obstacle. Forcomparison the results from PDE-model, ODE-model in triangular approximation (as illustratedby the dashed blue line in Fig. 4(c)) and stochastic simulations are shown. (b) Spatial distributionsof characteristic filament angles at θ = 0 ◦ , ± ◦ and ± ◦ . Interpretation in the lab frame indicatescoexistence of the competing ±
35 and +70 / / −
70 patterns at different lateral positions along theobstacle. (c) Heat map of the spatially resolved filament orientation distributions, where coolercolors indicate increasing filament density. Plotting dashed lines at characteristic angles at ˜ θ = 0 ◦ , ± ◦ and ± ◦ in the obstacle frame indicates that in this frame a single familiar pattern dominates.The apparent coexistence of different patterns is artificially introduced due to a misinterpretationof the results in the lab frame. orientation [ ° ] f il a m en t nu m be r pe r ang l e pde modelode modelstochastic sim v nw /v fil =0.5 −50 0 5000.050.10.150.20.250.30.350.40.45 obstacle x position f il a m en t nu m be r den s i t y pde model stochastic sim 0 ° ± ° ± ° −50 −40 −30 −20 −10 0 10 20 30 40 50−180−150−120−90−60−300306090120150180 obstacle x position o r i en t a t i on [ ° ] (1.a) (1.b) (1.c) −180−150−120 −90 −60 −30 0 30 60 90 120 150 18000.20.40.60.811.2 orientation [ ° ] f il a m en t nu m be r pe r ang l e pde modelode modelstochastic sim v nw /v fil =0.1 −50 0 5000.10.20.30.40.50.60.7 obstacle x position f il a m en t nu m be r den s i t y pde model stochastic sim 0 ° ± ° ± ° −50 −40 −30 −20 −10 0 10 20 30 40 50−180−150−120−90−60−300306090120150180 obstacle x position o r i en t a t i on [ ° ] (2.a) (2.b) (2.c) FIG. 7: Same as Fig. 6 for a periodic-cosine shaped obstacle as illustrated by the solid red linein Fig. 4(c). Active parameters are ¯ ϕ = − ◦ , r br = 50, d ⊥ br = 2 δ fil , k b = 20, k c = 0 .
05; (1) v nw /v fil = 0 .
5; (2) v nw /v fil = 0 .
1. Again an apparent coexistence of competing patterns disappearswhen results are interpreted in the obstacle frame, that is locally orthogonal to the obstacle surface.1. Again an apparent coexistence of competing patterns disappearswhen results are interpreted in the obstacle frame, that is locally orthogonal to the obstacle surface.