Microtubule-based actin transport and localization in a spherical cell
MMicrotubule-based actin transport and localization in a spherical cell
Marco Saltini a and Bela M. Mulder Institute AMOLF, Science Park 104, 1098 XG Amsterdam, The Netherlands
The interaction between actin filaments and microtubules is crucial for many eukaryotic cellularprocesses, such as, among others, cell polarization, cell motility and cellular wound healing. Theimportance of this interaction has long been recognised, yet very little is understood about both theunderlying mechanisms and the consequences for the spatial (re)organization of the cellular cytoskele-ton. At the same time, understanding the causes and the consequences of the interaction betweendifferent biomolecular components are key questions for in vitro research involving reconstitutedbiomolecular systems, especially in the light of current interest in creating minimal synthetic cells. Inthis light, recent in vitro experiments have shown that the actin-microtubule interaction mediated bythe cytolinker TipAct, which binds to actin lattice and microtubule tip, causes the directed transportof actin filaments. We develop an analytical theory of dynamically unstable microtubules, nucleatedfrom the center of a spherical cell, in interaction with actin filaments. We show that, depending onthe balance between the diffusion of unbound actin filaments and propensity to bind microtubules,actin is either concentrated in the center of the cell, where the density of microtubules is highest, orbecomes localized to the cell cortex. a Current affiliation:Uppsala University, Department of Ecology and Genetics, Animal Ecology, Norbyv¨agen 18D, 752 36 Uppsala, Sweden a r X i v : . [ q - b i o . S C ] S e p I. INTRODUCTION
Microtubules and actin filaments are dynamic polymers and components of the eukaryotic cytoskeleton. Both areinvolved in spatial processes at the scale of the cell. Microtubules are best known for their role in motor protein-mediated directed intracellular transport and forming the mitotic spindle – the machinery for segregating the duplicatedchromosomes prior to cell division, while actin is strongly associated with cell locomotion and deformation, e.g. duringmany developmental processes. Although historically actin and microtubules have been studied independently fromeach other, it has more recently been acknowledged that the interaction between these two species is crucial in processessuch as, among others, cell division, cell growth and migration, cellular wound healing and cell polarization [1–5].Actin and microtubules can interact in many different ways: examples are steric repulsion, passive crosslinking,and microtubule growth guidance by actin bundles [6, 7]. The interaction between actin filaments and microtubuleshas been shown to influence the spatial organization of each others [8]. Here we focus specifically on the interactionsmediated by a class of proteins collectively called cytolinkers. Some cytolinkers, like, e.g., MACF1, have been observedto bind both to actin filaments and microtubules [9–12] suggesting that the spatial organization of the former can beinfluenced by the dynamics of the latter.This possibility was explored in recent in vitro experiments [10, 13, 14] employing the engineered protein constructTipAct [6] specifically designed to bind to both actin and microtubules. These experiments revealed a dual actionof TipAct. It was shown to bind to the plus-end of growing microtubules mediated by the presence of end bindingproteins EB3. This results in a decrease of the microtubule growth speed, and an increase in their catastrophe rate,both effects effectively shortening the microtubule. On the other hand, TipAct also tracks the growing microtubuletip and, hence, transports any bound actin in the direction of microtubule growth. Experiments in planar quasi-2Dgeometries as well as spherical droplets revealed a distinct TipAct-dependent impact of microtubules on the spatialdistribution of the actin.While experimental and theoretical studies aimed at understanding the underlying microscopic mechanism behindthe transport of actin through TipAct-mediated coupling to dynamic microtubules are currently underway [15], thequestion of the macroscopic effects of the resulting actin transport have to date not been addressed. Here we considerthe latter question in a minimal model consisting of dynamic microtubules nucleated from a point-like MicrotubuleOrganizing Center (MTOC) [16] located at the center of a spherical cell. This cell contains a finite amount of actinwhich either diffuses freely in the cytosol or directly binds to microtubule tips, i.e., the presence of the cytolinkermediating this coupling is implicit. Our primary aim is to elucidate the resulting spatial distribution of the actin andto determine which of the model parameters are the main determinants of this distribution.This paper is structured as follows. We first introduce a stochastic model of microtubules undergoing dynamicinstability in a three-dimensional cell. Within this cell actin filaments can either diffuse free of any driving forces or bebound to microtubule plus ends and subsequently transported towards the boundary of the cell. We show that, underthe assumption that the dynamics of microtubules is not influenced by the binding of the actin filaments, the dynamicequations of the model decouple. Under this assumption, we obtain numerical solutions for the spatial distribution ofboth the microtubules and the actin. We show, supported by a dimensional analysis, that while the spherical geometryby default would promote the actin to be concentrated in the cell center where the density of microtubules is highest,a combination of low actin diffusivity coupled to a high propensity to bind microtubules causes the actin distributionto be cortical. Finally, we discuss why, for model parameters in the range of biological relevance, the model cannotnaively be approximated by an advective-diffusive model in which the microtubule dynamics acts as an overall forcefield that pushes the actin filaments towards the surface of the cell.
II. METHODS
In this section, we set up a stochastic model of dynamic microtubules undergoing dynamic instability in a sphericalcell, where actin filaments diffuse and can interact with the plus end of microtubules. Since preliminary experimentshave revealed that the interaction between actin and microtubules via TipAct can change the dynamic properties ofthe latter [13], we first define a general model that accounts for this change in the dynamics. Nevertheless, our maininterest lies in the actin transport mechanism observed in the experiments performed in the droplet, where no changein the dynamics of microtubules has been recorded [14]. Therefore, we will provide a solution of the model under theassumption that the dynamic properties of microtubules do not change as a consequence of the interaction with actinfilaments. Although we present a three-dimensional model, some of the results presented in this work can easily begeneralized to fewer dimensions.
The model: dynamic microtubules and diffusing actin
The model is based on the Dogterom and Leibler model for microtubules undergoing dynamic instability [17].It consists of M microtubules undergoing dynamic instability in an homogeneous 3-dimensional sphere of radius R , interacting with A actin filaments diffusing with diffusion constant D , and with reflecting boundary conditionsat the boundary r = | r | = R . As we are interested in studying the interaction between actin and microtubules,and in particular in the transport of actin by the plus end of microtubules, in our model we ignore both thepolymerization/depolymerization and the nucleation of the actin filaments, and we model them as dimensionlessparticles, see Figure 1A.All microtubules are isotropically nucleated at position r = 0 in the growing state, with growing velocity v + = v + ˆr ,where ˆr is the unit vector in the radial direction. Microtubules can switch from the growing to the shrinking statewith constant catastrophe rate k c . Then, microtubules shrink with velocity v − = − v − ˆr . A shrinking microtubuleeither switches from shrinking to growing state with constant rescue rate k r , or it completely depolymerizes. Whencomplete depolymerization occurs, reflective boundary conditions at r = 0 for microtubules implement their immediatere-nucleation in the growing state. Finally, we impose reflective boundary conditions at r = R as well, with bothgrowing and bound microtubules switching to the shrinking state directly upon reaching the cortex of the cell, seeFigure 1BC.When the plus-end of a microtubule is within a range of interaction s of an actin filament, the microtubule tip andthe filament can interact leading the actin filament to bind with binding rate k b , and the microtubule entering thebound state. The bound filament is subsequently transported by the microtubule plus end towards the cortex of thecell, with transport velocity v b = v b ˆr , where v b < v + , in accordance with experimental measurements [13]. Theseexperiments also revealed that the interaction between actin and microtubules has a second effect on microtubuledynamics, i.e., it increases the catastrophe rate. Here, however, for simplicity’s sake we decide to ignore this effect.In fact, in the next section we will show how this model always reaches the steady-state and, as a consequence, theproperties of microtubules are solely defined by the ratio between growing speed and catastrophe rate. We thereforedecided to keep the catastrophe rate constant, and change the growing speed in order to take into account the changeddynamics observed in the experiments. A bound actin filament can unbind in three distinct ways: i) it simply detacheswith a constant unbinding rate k u , ii) the microtubule to which it is bound undergoes a catastrophe with rate k c and releases the filament, iii) or when the microtubule hits the surface of the cell, see Figure 1BC. The values of thedynamic parameters used in the model were chosen in agreement with the experimental measurements, see Table I. A B C
Figure 1. Schematic of the model. (A) Microtubules (green lines) undergoing dynamic instability and actin filaments (blue dots)diffusing in a 3-dimensional sphere. (B) Schematic of the dynamics of microtubules and their interaction with actin filaments.(C) Boundary conditions at the cortex of the cell and at the centre.
Dynamic equations
We assume that the interaction process that leads to actin binding is fast compared to both diffusion of the actinand the growth of a microtubule. Moreover, we assume that the interaction range s is small enough that the densitydistribution of growing microtubule plus ends is approximately constant within it. Therefore, if m + ( t, r ) is thedistribution of the position of free growing microtubule tips, and a ( t, r ) the distribution of the position of free actin Parameter Description Numerical value Units v + Free-growth speed 0 . µ m s − v b Transport speed 0 . µ m s − k c Catastrophe rate 0 .
005 s − k b Binding rate 0 . − s Actin-microtubule interaction volume 0 . µ m k u Unbinding rate 0 .
009 s − D Free actin diffusion coefficient 0 . µ m s − R Radius of the cell 10 µ m M Total number of microtubules 10 - A Total number of actin filaments 5 · -Table I. Model parameters. The choice for the numerical values is in agreement with the experimental measurements of thesame quantities (Alkemade et al., in preparation). filaments, the overall interaction strength can be given by K int ( t, r ) = k b s m + ( t, r ) a ( t, r ) . (1)Then, the dynamic equations for actin filaments and microtubules - including shrinking m − ( t, r ) and bound b ( t, r )microtubules as well, are4 πr ∂m + ( t, r ) ∂t = − v + · ∇ πr m + ( t, r ) − k c πr m + ( t, r ) − k b s πr m + ( t, r ) a ( t, r )+ k u πr b ( t, r ) + k r πr m − ( t, r ) , (2)4 πr ∂m − ( t, r ) ∂t = − v − · ∇ πr m − ( t, r ) + k c πr (cid:2) m + ( t, r ) + b ( t, r ) (cid:3) − k r πr m − ( t, r ) , (3)4 πr ∂b ( t, r ) ∂t = − v b · ∇ πr b ( t, r ) − ( k c + k u ) 4 πr b ( t, r ) + k b s πr m + ( t, r ) a ( t, r ) , (4) ∂a ( t, r ) ∂t = D ∇ a ( t, r ) − k b s m + ( t, r ) a ( t, r ) + ( k u + k c ) b ( t, r ) . (5)The first three equations are the evolution transport equations for the radial distributions of the plus end of growingmicrotubules, shrinking microtubules, and bound microtubules respectively. The last equation is a diffusion equationfor the free actin particles with a loss term due to the capture of actin filaments by microtubule tips, and a sourceterm due to the release of actin from the microtubule tip to the pool.Since the system possesses spherical symmetry, we make use of spherical coordinates, i.e., r = ( r, θ, φ ). Note that,given our assumptions of homogeneity and isotropy, all quantities of the model only depend on the radial coordinate r . Furthermore, it has been shown that the length distribution of microtubules undergoing dynamic instability in aconfined volume always reaches a steady-state, regardless of the choice of the dynamic parameters [18]. More generally,the length distribution of microtubules undergoing dynamic instability in presence of any limiting factor such as, e.g.,finiteness of free tubulin, always reaches a steady-state [19]. Hence, we reasonably assume that our system alwaysreaches the steady-state, and we restrict the study of Eqs. (2-5) to that situation. Finally, since in the experimentsno rescues have been observed [13], we set k r = 0. In this way, once a microtubule undergoes a catastrophe, its fateis determined as it cannot be rescued. Therefore, we make the final assumption that v − (cid:29) v + , i.e., as soon as amicrotubule undergoes a catastrophe, it suddenly completely depolymerizes, and, then, it is immediately re-nucleatedagain. In this way, the number of microtubules in the shrinking state - and hence the related distribution, vanishesand Eq. (3) is redundant. This choice is also motivated by the fact that backward transport of actin filaments byshrinking has been very rarely observed. Therefore, including shrinking microtubules would only raise the complexityof the model without providing further insight.Under these assumptions, we can rewrite Eqs. (2-5) in the steady-state as0 = − v + dd r (cid:2) r m ( r ) (cid:3) − k c r m ( r ) − k b s r m ( r ) a ( r ) + k u r b ( r ) , (6)0 = − v b dd r (cid:2) r b ( r ) (cid:3) − ( k c + k u ) r b ( r ) + k b s r m ( r ) a ( r ) , (7)0 = D r dd r (cid:20) r dd r a ( r ) (cid:21) − k b s m ( r ) a ( r ) + ( k u + k c ) b ( r ) , (8)where m ( r ) ≡ m + ( r ). This set of ordinary differential equations is supplemented by the boundary conditions definedby the properties of the model. Indeed, the sudden renucleation of every microtubule that undergoes a catastropheimplies v + m (0) = v + m ( R ) + v b b ( R ) + k c M, (9)and b (0) = 0 . (10)The release of actin filaments from bound microtubules when they reach the cell surface, together with the reflectiveboundary condition for the free diffusing actin, imply D ∇ a ( r ) · (cid:98) r | r = R = v b b ( R ) . (11)Finally, conservation of probability implies a normalization condition for both actin filaments and microtubules4 π (cid:90) R d r r [ m ( r ) + b ( r )] = M, (12)4 π (cid:90) R d r r [ a ( r ) + b ( r )] = A. (13) Solution
Multiplying Eq. (8) by r and adding it to Eq. (7) yields D dd r (cid:20) r dd r a ( r ) (cid:21) = v b dd r (cid:2) r b ( r ) (cid:3) , (14)which implies r (cid:20) dd r a ( r ) − v b D b ( r ) (cid:21) = const., (15)and, using the boundary condition (11), dd r a ( r ) = v b D b ( r ) ∀ r ∈ [0 , R ] . (16)Since b ( r ) is a distribution and, therefore, always positive, from Eq. (16) it follows that a ( r ) is monotonically increasingin the radial direction. Similarly, if we add Eq. (6) to Eq. (7), we finddd r r (cid:2) v + m ( r ) + v b b ( r ) (cid:3) = − k c r [ m ( r ) + b ( r )] . (17)Given that we are interested in the transport mechanism rather than in the change of microtubule dynamics, we willhereafter work under the assumption that the dynamic properties of the microtubules do not change upon bindingactin filaments. This means that we assume v + = v b ≡ v. (18)In this regime, Eq. (17) is solvable for m ( r ) + b ( r ), and the solution is m ( r ) + b ( r ) = m πr e − kcv r , (19)where we used the normalization condition (12), i.e., M = 4 π (cid:82) R d r r [ m ( r ) + b ( r )] , to find the integration constant m = k c v (cid:16) − e − kcv R (cid:17) − M. (20)As Eq. (19) is the steady-state length distribution for non-interacting microtubules [17], we observe that if the interactionbetween actin filaments and microtubules does not affect the dynamic properties of the latter, the microtubule steady-state length distribution remains the same as in the non-interacting system. Interestingly, this result is not dependenton the kind of process behind the switching from the growing to the bound state and vice-versa , as it only depends onthe growing speed and the catastrophe rate of microtubules.Therefore, Eqs. (16) and (19) allow us to disentangle the set of differential equations (6-8), and to write a closed-formexpression for a ( r ) a (cid:48)(cid:48) ( r ) + (cid:20) r + k u + k c v + k b sv a ( r ) (cid:21) a (cid:48) ( r ) − k b sD m πr e − kcv r a ( r ) = 0 . (21)Even though Eq. (21) is still not analytically solvable, its non-dimensionalization highlights relevant features. Indeed,we first set the unit of length for microtubules l = vk c . (22)Then, we define the dimensionless parameters: transport length τ = k c / ( k u + k c ), binding rate β = k c k b s/v , anddiffusion coefficient δ = k c D/v . We also introduce the dimensionless length x = r/l , and its distribution f ( x ) = l a ( r ).The equation for f becomes f (cid:48)(cid:48) ( x ) + (cid:20) x + 1 τ + βf ( x ) (cid:21) f (cid:48) ( x ) − βδ µ πx e − x f ( x ) = 0 , (23)where µ = lm . Eq. (23) shows that once the typical length for microtubules and the radius of the cell are set, thebehaviour of the distribution of the actin filaments is uniquely determined by three parameters, namely the transportlength τ , the binding rate β , and the diffusion coefficient δ .We can make Eq. (23) suitable for numerical integration by working with the cumulative function F ( x ) = (cid:90) x d y πy f ( y ) , (24)from which it follows f ( x ) = 14 πx F (cid:48) ( x ) . (25)Indeed, Eq. (23) becomes F (cid:48)(cid:48)(cid:48) ( x ) + (cid:20) − x + 1 τ + β F (cid:48) ( x )4 πx (cid:21) F (cid:48)(cid:48) ( x ) + (cid:20) x − τ x − βx F (cid:48) ( x )4 πx − βδ µ πx e − x (cid:21) F (cid:48) ( x ) = 0 , (26)with boundary conditions (cid:20) F (cid:48)(cid:48) ( x ) − x F (cid:48) ( x ) (cid:21) x =0 = 0 , (27)and F (cid:0) R/l (cid:1) = q, (28)coming from Eqs. (10) and (13), respectively, and where q is the number of free actin filaments. Notice that, although q is an unknown quantity, we can numerically find its value by observing that q = A − (cid:90) R/l d x πx δf (cid:48) ( x ) . (29)As Eq. (26) is a third-order differential equation, a third boundary condition is in principle required for its solution.However, any integration constant provided by the third condition would be cancelled out by subsequently taking thederivative to find f ( x ). III. RESULTS
Here, we show the results of our model under the assumption that microtubules do not change their dynamicproperties when they are bound to actin filaments, i.e., with the parameter choice (18). The were obtained throughnumerical integration of (26), with dynamic parameters listed in Table I. In particular, our choice for the growingspeed is v + = v b = v = 0 . µ m s − .
01 0.5 R R2010 0.5 R R2000.5 0.5 R R1 10
01 0.05 R 0.1 R2010 0.05 R 0.1 R2000.5 0.05 R 0.1 R1 10
010 0.5 R R20010 0.05 R 0.1 R20
AE BF CG DH D i s t r i bu ti on [ μ m - ] D i s t r i bu ti on [ μ m - ] Figure 2. Distribution of the positions of (A) growing microtubule plus ends, (B) bound microtubule plus ends, (C) free actinfilaments, and (D) all actin filaments (free and bound). Panels (E), (F), (G), (H) show the above distributions in the centre ofthe cell, when r ∈ (0 , . R ). Figure 2 shows the distribution of the positions of the tips of the growing and bound microtubules, the free actinfilaments, and all actin filaments (both free and bound), respectively. The figure shows that in the biologically relevantrange of parameters, the interaction between actin and microtubules plays a minimal role in changing the steadystate distribution of the free actin filaments, as the resulting distribution is roughly uniform (Figure 2CG). However,if we consider the total distribution of actin filaments - i.e. including both free and bound filaments, the resultingdistribution exhibits a marked peak at the centre of the cell (Figure 2DH). The peak is caused by the very high densityof microtubule tips at r → v/k c = R . The observed diluted microtubule density at the cortex is solely aconsequence of the three-dimensional geometry.To test whether the binding rate and the diffusion coefficient play an important role in the distribution of the actinfilaments, we numerically solve Eq. (23) for different τ , β , and δ . As expected, figure 3AB show that both a highbinding rate and a low diffusion constant have the effect of localizing free acting filaments closer to the cell surface,while changing the transport length τ does not seem to have a significant influence on f ( x ), see Figure 3C. Figure3AB highlights another interesting fact: except for very high values of β , scaling β with a certain factor z , has roughlythe same effect on the actin distribution as scaling δ with the factor 1 /z . This means that there exists a range ofvalues for parameter β such that βf ( x ) (cid:28) x + 1 τ , (30)see Figure 4. As a consequence, for small β Eq. (23) can be approximated with f (cid:48)(cid:48) ( x ) + (cid:18) x + 1 τ (cid:19) f (cid:48) ( x ) − βδ µ πx e − x f ( x ) = 0 , (31) A B C
012 012 00.951.10.8
Figure 3. Distribution of the positions of free actin filaments when (A) δ is tuned, (B) β is tuned, and (C) τ is tuned ( τ = 0 . τ = 0 . τ = 0 . τ = 0 . δ = 1, β = 3 . · − , τ = 0 .
36, from Table I.
A B C
Figure 4. Comparison between (straight line) x + τ , and (dashed line) βf ( x ) for three different choices of β . (A) For modelparameters in the biologically range of values, the latter is neglectable compared to the former. Reference values are δ = 1, β = 3 . · − , τ = 0 .
36, from Table I. without significant loss of accuracy in the solution. Intriguingly, the biologically relevant limit for β is the limit (30).Therefore, from Eq. (31) and Figure 3C we can observe that the system is fully characterized by only one parameter:the binding/diffusion ratio γ ≡ β/δ . Figure 5 shows the distribution of all actin filaments - both free and bound tomicrotubules, for different choices of γ . In particular, we can observe that a high binding/diffusion ratio increases theactin density at the surface of the cell, while decreasing the effect of the spherical geometry of trapping of actin at thecentre of the cell. In fact, for γ = 0 . r →
0, there are no free actin filaments close to the origin to attach to. Thus, a high bindingrate, promoting the transport of the filaments towards the cortex as it enhances the interaction of the latter with themicrotubule tips, coupled to a low diffusion coefficient, preventing the actin redistribution over the whole volume ofthe cell including the central volume, can create an actin density profile with a maximum at the cortex.
Figure 5. Distribution of the cumulative positions of actin filaments, bound and free, for four different choices of γ = β/δ . Thereference value is γ = β /δ = 3 . · − , from Table I. IV. COMPARISON TO AN ADVECTION-DIFFUSION MODEL
At first sight, one could argue that our proposed transport mechanism could well be approximated by a diffusive-advective mechanism with a velocity field directed towards the cortex of the cell with a speed proportional to thedensity of microtubules. In this case, microtubules act as a position-dependent background that push actin filamentstowards the external volume of the cell.Thus, the steady state dynamics of the system is described by the equation0 = D ∇ a ( r ) − ∇ · [ V ( r ) a ( r )] , (32)with V ( r ) = u m ( r ) ˆr , and where u is an appropriate constant. The density of microtubules, in absence of bindingand unbinding to actin filaments, is the steady state density m ( r ) = m πr e − kcv r , already discussed in Eq. (19). Conservation of the number of actin filaments imposes the following boundary andnormalization conditions for a ( r ) D ∇ a ( r ) · (cid:98) r | r = R = um ( R ) a ( R ) , (33) A = (cid:90) R d r πr a ( r ) . (34)By combining Eqs. (32), (33), and (34) we find the expression for the density of actin filaments, i.e., a ( r ) = a exp (cid:18) uD (cid:90) r d t m πt e − kcv t (cid:19) , (35)where a = A (cid:34)(cid:90) R d r exp (cid:18) uD (cid:90) r d t m πt e − kcv t (cid:19)(cid:35) − . (36)The integral in Eq. (35) diverges at r = 0. Therefore, the density of actin filaments approaches zero as r → u , as we show in Appendix VII B. F r ee ac ti n f il a m e n t s d i s t r i bu ti on Position
Figure 6. Distribution of the position of free actin filaments, for the transport-diffusion model (dashed line), and for theadvection-diffusion model (solid line), for model parameters from Table I, and u = 5 · − µ m /s. V. DISCUSSION
We introduced a minimal model for the interplay between actin filaments and dynamical microtubules, based on theexperimentally observed TipAct-mediated interaction between actin filaments and microtubule plus ends, and the0subsequent transport of the former by the latter during microtubule polymerization. Focusing on the question to whatextent such a transport mechanism could spatially reorganize the cytoskeleton, we identified the ratio of the bindingpropensity of actin filaments to the microtubules over their free diffusion coefficient as the key parameter determiningthe spatial organization of actin filaments. Our analysis showed that a high binding/diffusion ratio can overcome thedefault trapping of actin filaments at the microtubule-dense centre of the cell, causing actin re-localization to the cellcortex.At first sight, one might naively argue that this transport mechanism could well be approximated by a diffusionmechanism with a velocity field directed towards the cortex of the cell with an advective speed proportional to thedensity of microtubules. However, it is readily seen that this system would exhibit a divergent pushing force at r → m ( r ) diverges at the centre of the cell because of the three-dimensional geometry. That would result in a completeabsence of free actin filaments at the centre of the cell, and in a monotonic increase of the total actin distribution fromthe centre to the boundary. This is in contrast with our full transport model, where - for low binding/diffusion ratio,we observe trapping of actin at the center of the cell due to the high density of microtubules.However, it is important to underline that the very high peak in the total actin distribution close to r = 0 is purelythe consequence of the divergence of the distribution of free microtubules. This divergence, mathematically inherent tothe design of our model, could be in principle be removed by assuming a finite size for the centrosome located at r = 0,and treating microtubules as objects with a finite diameter of about 25 nanometers [20] rather than one-dimensionalentities. Further analytical investigations should therefore aim at testing to what extent the details of the geometryinfluences the result.Although our model gives us insights about how to tune the binding/diffusion ratio in order to obtain localizationof actin at the cortex of the cell, it as yet fails in giving a complete description of the spatial organization of thecytoskeleton that includes the interaction among its components. Indeed, we limited ourselves to microtubules inthe bounded-growth regime and in particular in the limit of fast depolymerization. Including shrinking microtubulesand rescues from the shrinking to the growing state in the system, would add a further degree of complexity to theset of Eqs. (6-8), making them no longer analytically tractable. Furthermore, in our model we have ignored bothactin-actin interactions, driving the organization in networks at the cortex, as well as the dynamic instability andthe nucleation of actin filaments. Therefore, a full description of the system, including shrinking microtubules andinteraction between different actin filaments, most likely requires a more brute-force simulation approach.Nevertheless, our theoretical predictions on the response of the system to the change of the binding/diffusion ratio,could in principle be validated by experiments, as transport in three-dimensional domain has recently been observed indroplets [14]. While directly manipulating the transport and binding rates may be harder to achieve experimentally,tuning the diffusion coefficient by changing the viscosity of the medium seems feasible. Another possibility could be tochange the typical length of actin filaments. Indeed, it has been observed that the diffusion coefficient of an actinfilament is inversely proportional to its length [21]. Engineering longer or shorter actin filaments could then test ourhypothesis that a high diffusion coefficient is correlated to the localization of actin close to the cortex.While our model is neither aimed at nor able to describe currently known mechanism of microtubule-actin organization in vivo , we hope that our work could serve as an inspiration for future more integrated analytical, computational, andexperimental research on how the interactions between different cytoskeletal components determine spatial structure ofthe cytoskeleton. At the very least, it may provide useful insights towards the design of reconstituted minimal systemsaimed at reproducing some traits of living cells, which are dependent on a properly organized actin cortex. In thatlight it would also be interesting to in future consider, e.g., the role of actin nucleators such as formins, which havebeen observed to interact with microtubules, and specifically with growing microtubule plus ends [22]. VI. ACKNOWLEDGEMENTS
The work of MS was supported by the ERC 2013 Synergy Grant MODELCELL. The work of BMM is part ofthe research program of the Dutch Research Council (NWO). We acknowledge many helpful discussions with CelineAlkemade, Gijsje Koenderink and Marileen Dogterom (Delft Technical University).
VII. APPENDIX
In this section we show that, in one-dimension, our model can well be approximated by an advective-diffusive modelwhere microtubule density acts as an overall velocity field that pushes actin filaments towards the cortex of the cell.1
A. One-dimensional transport of actin filaments
In the one-dimensional case, the steady-state dynamic equations (6-8), under the assumption that no changes occurto the dynamics of microtubules when they bind actin filaments, can be rewritten as0 = − v dd x m ( x ) − k c m ( x ) − k b s m ( x ) a ( x ) + k u b ( x ) , (37)0 = − v dd x b ( x ) − ( k c + k u ) b ( x ) + k b s m ( x ) a ( x ) , (38)0 = D d d x a ( x ) − k b s m ( x ) a ( x ) + ( k u + k c ) b ( x ) , (39)with boundary and normalization conditions vm (0) = vm ( R ) + vb ( R ) + k c M, (40) b (0) = 0 , (41) D dd x a ( x ) (cid:12)(cid:12)(cid:12)(cid:12) x = R = vb ( R ) , (42) (cid:90) R d x [ m ( x ) + b ( x )] = M, (43) (cid:90) R d x [ a ( x ) + b ( x )] = A. (44)Similarly to the three-dimensional case, Eqs. (6-8) can be disentangled to find a closed-form expression for the densityof free actin filaments, i.e., a (cid:48)(cid:48) ( x ) + (cid:20) k u + k c v + k b sv a ( x ) (cid:21) a (cid:48) ( x ) − k b sD k c v M e − kcv x − e − kcv R a ( x ) = 0 . (45)The latter equation can be numerically solved to obtain the density of actin filaments, see Figure 7. F r ee ac ti n f il a m e n t s d i s t r i bu ti on Position
Figure 7. Distribution of the position of free actin filaments, for the transport-diffusion model (black dashed line), and for theadvection-diffusion model (red solid line), for model parameters from Table I, and u = 10 − µ m /s. B. One-dimensional advection-diffusion model
We now show that this result can be well approximated by an advective-diffusive mechanism in which microtubulesact as a velocity field that pushes actin towards the cortex of the cell. Hence, in this case, we can consider actinundergoing a diffusive process with a drift force proportional to the microtubule density m ( x ), and with reflectiveboundary conditions at both boundaries x = 0 , R .Therefore, steady-state dynamic equations for microtubules and actin are0 = − v dd x m ( x ) − k c m ( x ) , (46)and 0 = D d d x a ( x ) − dd x [ um ( x ) a ( x )] , (47)where u is an appropriate constant. Normalization and boundary conditions are M = (cid:90) R d x m ( x ) , (48) A = (cid:90) R d x a ( x ) , (49)and dd x a ( x ) (cid:12)(cid:12)(cid:12)(cid:12) x =0 = 0 = dd x a ( x ) (cid:12)(cid:12)(cid:12)(cid:12) x = R . (50)From Eqs. (46) and (48) it follows that m ( x ) = m e − kcv x , (51)where m = k c v (cid:16) − e − kcv R (cid:17) − M. From Eq. (47), instead, it follows that D dd x a ( x ) − um ( x ) a ( x ) = const., (52)i.e., a first order linear differential equation, the solution of which is a ( x ) = exp (cid:20) − uD vk c m e − kcv x (cid:21) (cid:20) c − c uD Ei (cid:18) uD vk c m e − kcv x (cid:19)(cid:21) (53)where c and c are integration constants and Ei ( z ) = (cid:90) z −∞ d y e y y , (54)is the exponential integral. Reflecting boundary conditions at x = 0 and x = R imply c = 0 , (55)whilst, from the normalization condition, A = (cid:90) R d x a ( x ) = c vk c (cid:20) Ei (cid:18) − uD vk c m (cid:19) − Ei (cid:18) − uD vk c m e − kcv R (cid:19)(cid:21) , (56)3we obtain c = A vk c (cid:104) Ei (cid:16) − uD vk c m (cid:17) − Ei (cid:16) − uD vk c m e − kcv R (cid:17)(cid:105) . (57)The distribution of the actin filaments is then a ( x ) = A vk c (cid:104) Ei (cid:16) − uD vk c m (cid:17) − Ei (cid:16) − uD vk c m e − kcv R (cid:17)(cid:105) exp (cid:20) − uD vk c m e − kcv x (cid:21) . (58)Figure 7 shows that in 1D the approximation of considering the transport of actin as the result of a velocity fieldthat drives the filaments toward the surface of the cell is reasonable, suggesting that the discrepancy observed in thethree-dimensional case emerges from the geometry of the system. Unfortunately, at present we lack a mechanisticdescription of the system that would enable us to derive a suitable value of the velocity u from the other modelparameters, which would be useful avenue of further research. [1] S E Siegrist and C Q Doe. Microtubule-induced cortical cell polarity. Genes & Development , 21(5):483–496, 3 2007.[2] F Huber, A Boire, M Preciado L´opez, and G H Koenderink. Cytoskeletal crosstalk: when three different personalities teamup.
Current Opinion in Cell Biology , 32:39–47, 2 2015.[3] M Dogterom and G H Koenderink. Actin-microtubule crosstalk in cell biology.
Nature Reviews Molecular Cell Biology ,20(1):38–54, 1 2019.[4] M Abercrombie. The bases of the locomotory behaviour of fibroblasts.
Experimental cell research , Suppl 8:188–198, 1961.[5] Alexandra Colin, Pavithra Singaravelu, Manuel Th´ery, Laurent Blanchoin, and Zoher Gueroui. Actin-Network ArchitectureRegulates Microtubule Dynamics.
Current Biology , 28(16):2647–2656, 8 2018.[6] M Preciado L´opez.
In vitro studies of actin-microtubule coordination . PhD thesis, Free University Amsterdam, 2015.[7] Feng Quan Zhou, Clare M. Waterman-Storer, and Christopher S. Cohan. Focal loss of actin bundles causes microtubuleredistribution and growth cone turning.
Journal of Cell Biology , 157(5):839–849, 5 2002.[8] Jessica L. Henty-Ridilla, Aneliya Rankova, Julian A. Eskin, Katelyn Kenny, and Bruce L. Goode. Accelerated actin filamentpolymerization from microtubule plus ends.
Science , 352(6288):1004–1009, 5 2016.[9] H Chen, C Lin, C Lin, R Perez-Olle, C L Leung, and R K H Liem. The role of microtubule actin cross-linking factor 1(MACF1) in the Wnt signaling pathway.
Genes & development , 20(14):1933–1945, 7 2006.[10] M Preciado L´opez, F Huber, I Grigoriev, M O Steinmetz, A Akhmanova, G H Koenderink, and M Dogterom.Actin–microtubule coordination at growing microtubule ends.
Nature Communications , 5(1):4778, 12 2014.[11] X Wu, A Kodama, and E Fuchs. ACF7 Regulates Cytoskeletal-Focal Adhesion Dynamics and Migration and Has ATPaseActivity.
Cell , 135(1):137–148, 10 2008.[12] Kossay Zaoui, Khedidja Benseddik, Pascale Daou, Dani`ele Sala¨un, and Ali Badache. ErbB2 receptor controls microtubulecapture by recruiting ACF7 to the plasma membrane of migrating cells.
Proceedings of the National Academy of Sciencesof the United States of America , 107(43):18517–18522, 10 2010.[13] C Alkemade, M Dogterom, and G Koenderink. In preparation.[14] Kim J A Vendel, Celine Alkemade, Nemo Andrea, Gijsje H Koenderink, and Marileen Dogterom. In Vitro Reconstitution ofDynamic Co-organization of Microtubules and Actin Filaments in Emulsion Droplets. In Helder Maiato, editor,
CytoskeletonDynamics: Methods and Protocols , pages 53–75. Springer US, New York, NY, 2020.[15] C Alkemade, H Wierenga, V Volkov, M Preciado-L`opez, A Akhmanova, M Dogterom, P R ten Wolde, and G Koenderink.Condensation force drives actin transport by growing microtubule ends. - In preparation.[16] B. R. Brinkley. Microtubule Organizing Centers.
Annual Review of Cell Biology , 1(1):145–172, 11 1985.[17] Marileen Dogterom and Stanislas Leibler. Physical aspects of the growth and regulation of microtubule structures.
PhysicalReview Letters , 70(9):1347–1350, 3 1993.[18] Bindu Govindan and William Spillman. Steady states of a microtubule assembly in a confined geometry.
Physical ReviewE , 70(3), 9 2004.[19] Simon H. Tindemans, Eva E. Deinum, Jelmer J. Lindeboom, and Bela M. Mulder. Efficient event-driven simulations shednew light on microtubule organization in the plant cortical array.
Frontiers in Physics , 2:19, 2014.[20] M C Ledbetter and K R Porter. Morphology of Microtubules of Plant Cell.
Science , 144(3620):872–874, 5 1964.[21] P A Janmey, J Peetermans, K S Zaner, T P Stossel, and T Tanaka. Structure and mobility of actin filaments as measuredby quasielastic light scattering, viscometry, and electron microscopy.
The Journal of biological chemistry , 261(18):8357–8362,6 1986.[22] Ying Wen, Christina H. Eng, Jan Schmoranzer, Noemi Cabrera-Poch, Edward J.S. Morris, Michael Chen, Bradley J. Wallar,Arthur S. Alberts, and Gregg G. Gundersen. EB1 and APC bind to mDia to stabilize microtubules downstream of Rho andpromote cell migration.