{'fraction_non_alphanumeric': 0.03327224957498951, 'fraction_numerical': 0.011392488905570397, 'mean_word_length': 4.396640057190516, 'pattern_counts': {'":': 0, '<': 10, '<?xml version=': 0, '>': 2, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 11, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
On the Properties of a Bundle of Flexible Actin Filaments in an Optical Trap

We establish the Statistical Mechanics framework for a bundle of Nf living and uncrosslinked actin filaments in a supercritical solution of free monomers pressing against a mobile wall. The filaments are anchored normally to a fixed planar surface at one of their ends and, because of their limited flexibility, they grow almost parallel to each other. Their growing ends hit a moving obstacle, depicted as a second planar wall, parallel to the previous one and subjected to a harmonic compressive force. The force constant is denoted as trap strength while the distance between the two walls as trap length to make contact with the experimental optical trap apparatus. For an ideal solution of reactive filaments and free monomers at fixed free monomers chemical potential, we obtain the general expression for the grand potential from which we derive averages and distributions of relevant physical quantities, namely the obstacle position, the bundle polymerization force and the number of filaments in direct contact with the wall. The grafted living filaments are modeled as discrete Wormlike chains, with Factin persistence length, subject to discrete contour length variations to model single monomer (de)polymerization steps. Rigid filaments, either isolated or in bundles, all provide average values of the stalling force in agreement with Hill's predictions, independent of the average trap length. Flexible filaments instead, for values of the trap strength suitable to prevent their lateral escape, provide an average bundle force and an average trap length slightly larger than the corresponding rigid cases (few percents). Still the stalling force remains nearly independent on the average trap length, but results from the product of two strongly L dependent contributions: the fraction of touching filaments and the single filament buckling force.

intrinsic large stiffness of these filaments, most of the existing models discard their flexibility and treat them as infinitely stiff. In this paper, following the lines set by [2][3][4], we investigate the role of flexibility in the process of reversible work production by F-actin filaments. Over the last decades, the underlying mechanism which enables cells to produce forces has been extensively studied, both theoretically and experimentally.
Originally, in the early 80's, a purely thermodynamic approach was followed by T.L. Hill [5,6]. The system under study is an almost incompressible (1D) polymer, actually a linear structure of length L = L(N, F, T ) consisting of N self-assembled monomers of size L/N ≈ l 0 at temperature T which is confined by a compressive force F . By considering a supercritical chemical equilibrium between the confined polymer's monomers and a free monomer solution with monomer chemical potential µ 1 at density ρ 1 , Hill showed that F = k B T lnρ 1 /l 0 , wherê and the critical state, with free monomer solution density ρ 1c and chemical potential µ 1c , corresponds to the thermodynamic state in which the same polymer has a propensity neither to grow nor to shrink in the absence of any external force. If the formula is adapted to a bundle of N f rigid and parallel actin filaments, one gets a stalling force given by For F-actin d = 2.7 nm is the contour elongation due to the addition of a subunit. 1 Later Brownian Ratchet models (BRM) [8][9][10][11][12][13][14][15] have been formulated in order to provide a more mechanistic interpretation of the action of a bundle of rigid filaments against a loaded obstacle. The ratcheting mechanism is played by the intercalation of a monomer between the filament tip and the pushing barrier whenever thermal fluctuations of the obstacle open a gap between them wide enough to allow for a polymerization event to occur. Attempts to include filament tip flexibility by adding a supplementary ratcheting mechanism have been published some years ago [10,11]. Under the hypothesis that thermal fluctuations of the obstacle are fast compared to the frequency at which monomers attach/detach, the Brownian Ratchet models for rigid filaments provide a value of the stalling force in agreement with Hill's prediction, irrespectively of the disposition of the filaments seeds [8,[13][14][15].
From an experimental perspective, the determination of the stalling force could be realized, in principle, by interpolating/extrapolating data of the bundle growing or shrinking velocity versus load [15][16][17][18][19][20] and determining the zero velocity conditions. This route is in practice very difficult to follow given the noise level and interferences with hydrolysis of ATP-actin. Hence, Footer et al. [21] used an optical trap set-up to measure the forces generated by the elongation of a few parallel-growing actin filaments in contact with a rigid microfabricated barrier, equilibrium being established between the bundle polymerization force and the trap restoring force directly proportional to the trap length. We observe that this set up represents in principle a true stable equilibrium state, as long as temperature and free monomer chemical potential are kept fixed and the implied chemical reactions are reversible with no filament escaping laterally along the obstacle wall [4,21,23]. Footer et al. monitored the growth of approximately eight actin filaments and found a stationary force significantly smaller than the value predicted by Hill's theory. The force measured in this experiment was of the order of the F H s from Eq.(2) expected for a single filament: the interpretation of this unique (as far as today) and important experiment probing stalling conditions is still missing even if some possibles causes have been evoked [21]. In ref. [26] Carlsson investigated the effects of hydrolysis and irreversible conversion of ATP-complexed in ADP-complexed actin monomers on stalling conditions, within the framework of the BRM with a L dependent load. Still considering fully rigid filaments, he found that the hydrolysis could account for the experimental observation in Footer's experiments. However due to the lack of experimental informations about the ADP off rate, the theoretical predictions remain inconclusive. Flexibility effects could also give rise to a decrease of the bundle force at stalling as observed in experiments because the bundle can "buckle". This was invoked by the authors to justify the results [21], although the arguments remained rather qualitative. Along these lines, in a simulation approach of filament growth against a constant load [23], flexibility was found to prevent the establishing of a true stationary non-equilibrium state since beyond some length, semi-flexible filaments can loose contact with the wall and grow parallel to it, hence reducing the force they are able to provide. This phenomenon has been called "pushing catastrophe" in the context of constant-load experiments [23] and "escaping filaments" in a study restricted to equilibrium conditions [4].
Recently, the force exerted by Brownian fluctuations of a grafted semi-flexible polymer, modelled as a Wormlike chain (WLC) with fixed contour length, upon a rigid wall has been calculated both analytically and by Monte Carlo method, finding a force, entropic in origin, which exhibits a universal behavior in the stiff limit [2]. The discrete version of this model (d-WLC) has then been extended to the case of a bundle of independent "living" filaments growing in contact with a rigid fixed wall in a reactive canonical ensemble [3]; within this statistical mechanical description several general features have been derived, namely the equilibrium filament size distribution and the associated average equilibrium force exerted on the opposite wall. Along these lines, a recent study [4] has extended this analysis to the reactive grand canonical ensemble, specified by temperature T , volume V and free monomers chemical potential µ 1 , for a single grafted living semi-flexible filament modeling F-actin, hitting a fixed wall.
The natural extension towards the properties of a bundle of parallel semi-flexible actin filaments pressing against a mobile loaded wall is the subject of the present work. As already mentioned we limit here to equilibrium statistical mechanics and we consider an external load increasing with the position of the obstacle in order to focus on a true equilibrium state. We will consider the case of a load increasing linearly with L, to mimic the experimental relevant case of a bundle in a harmonic optical trap [21]. Furthermore, we disregard the ATP-ADP conversion through hydrolysis which introduces an inherent irreversible process hence a non-equilibrium situation which remains to be studied. Within our approach, we establish the physical conditions to avoid the escaping filaments regime for flexible filaments, a task that in vivo is performed by specific proteins (capping, fascin). We characterize the effects of flexibility at equilibrium by comparing relevant properties, such as the average trap width, bundle force and number of active filaments, for a F-actin bundle using either flexible or rigid filaments. Moreover, the statistical mechanics foundations of Hill's expression, Eq.(2), of the stalling force are analyzed in depth.
The paper is organized as follows. In section II we define the physical system and, in particular, the homogeneous and in-registry bundles of living filaments based on the disposition of their seeds; we set-up the statistical mechanics framework for a bundle of filaments in the fixed-wall reactive grand canonical ensemble, and derive the expression of the average relevant properties of the system. In section III the moving-wall "ensemble" is introduced for a restoring hookean force acting on the wall to make contact with the experiment realized by Footer et al. [21]. We close the section defining the meaning of force measurement in terms of ensemble averaging. In section IV we briefly recall our model of F-actin filaments [2,4] and extend the non-escaping filament regime criteria of ref. [4] to the present case of the optical trap. Section V presents our results. We first characterize the flexibility effects by comparing rigid and flexible models for single filaments and then for a bundle of filaments. We also characterize the behavior of flexible filaments bundles in a wide range of parameters and physical conditions, going from the quasi-rigid filament behavior for short filaments to near the threshold of the escaping regime, revealing an intermediate regime characterized by a growing cooperativeness between filaments to produce the equilibrium force. In section VI we discuss and suggest an explanation for the experimental results [21] and draw few conclusive remarks.

II. BUNDLE OF LIVING, SUPERCRITICAL GRAFTED FILAMENTS IN A BOX
We consider a bundle of N f independent (mutually non-interacting) stiff filaments enclosed in a box of constant transverse area A and constant height L with two parallel and opposite walls located at x = 0 and x = L; the filaments, according to the discrete WLC model (material points and bonds) with bond length d and persistence length l p , are anchored normally to the first wall and can grow towards the second wall. We consider the obstacle at L as a hard wall, i.e. no filament articulation point (in particular the filament tip) can overlap the wall region beyond L. The filaments are immersed in an ideal solution of free monomers (material points not interacting with each other) at chemical potential µ 1 . Single monomer polymerization and depolymerization events give to the filaments their living character with probabilities satisfying chemical equilibrium. We showed [4] that, as a result of the chemical equilibrium, the free energy total differential of our confined system can be expressed as where S is the system entropy, p N and p T the total normal and tangential pressures exerted by the wall on the system, µ 2 is the chemical potential of grafted dimers, N t = N 1 + N f n=1 j n is the total number of particles (free plus bonded monomers) and µ 1 is the chemical potential of the free monomers, which results conjugated to N t as a consequence of chemical equilibrium. The last two terms on the right hand side arise from iteratively applying to the free energy differential of a mixture of all chemical species the equilibrium condition, µ i+1 = µ i + µ 1 , where µ i is the chemical potential of the grafted filament of size i. We are interested in supercritical conditions, to be defined more precisely below, when the filaments tend to grow in bulk (polymerization rate greater than depolymerizing rate) but reach an equilibrium as a result of the obstacle wall capacity to reduce the polymerization rate of hitting filaments.
The longitudinal disposition of the filament seeds (first two monomers) at the grafting surface represents a significant characteristic for a bundle. Its influence on the structural properties of the systems, often discussed within the context of multi-filament brownian ratchet models [8,9,11], will be discussed in the next sections. In absence of an experimental information we limit our analysis to the two usually adopted models: we call a bundle homogeneous when the seeds are regularly distributed over a distance d centered at x = 0 (the position of the grafting wall) while we call a bundle in-registry when all seeds are aligned at x = 0. Labelling h n the longitudinal position of the seed of the n-th filament, we set The distance between the first monomer of filament n and the wall at x = L is given by L n = L − h n . Following notations of ref. [4], the contour length of a filament of j monomers is L c,j = (j − 1)d where d is the bond length. The minimum number of monomers in a filament is taken to be two, at least two monomers are needed to specify the growth direction kept perpendicular to the transverse surface A. The maximum number of monomers in a filament with its first monomer at x = h n , before it feels the presence of the obstacle at L is The second critical filament size index introduced in [4] z * n = int πL n 2d (6) corresponds to a contour length equal to a quarter of a circle of radius L n ; a filament with a number of monomers larger than z * is considered an escaping filament since in supercritical conditions, for planar conformations, it may grow unhindered in the direction parallel to the obstacle [4]. Let q jn (L n ), and q 0 jn be the partition functions of a single grafted filament n (with seed at x = h n ) having size j n , respectively in presence and in absence of the wall. We define the wall factors α(j n , L n ) of each specific filament as: α(j n , L n ) = q jn (L n ) q 0 jn = 1 2 j n z n < 1 z n < j n < z * n (7) The size-independent chemical equilibrium constant for the (de)polymerization reaction in the bulk system is, considering two grafted filaments (i.e. their q 0 i have not to be divided by the volume, see Eq.(10.6) of ref. [7]) of arbitrary sizes i − 1 and i, where q 1 = V /Λ 3 is the free monomer partition function and Λ(T ) = βh 2 /2πm is the free monomer thermal de Broglie wavelength. Combining Eqs. (7,8), the filament partition can be expressed as The explicit expression of the grand-canonical partition function for a single non escaping living filament in the reactive-grand-canonical ensemble has been derived in ref. [4] (see Eqs.(6-31) of that paper). The extension to a bundle of independent living filaments is straightforward: we need to sum over all possible N t the canonical partition function, Q R (A, L, T, N t , N f ), involving N t monomers and N f filaments, properly weighted by the corresponding absolute activities, The grand-canonical partition function Eq.(11) can be further expressed as the product of single filaments and free monomers partition function. Indeed using Eqs. (7,11) one gets: where ρ 1 = e βµ1 /Λ 3 since the monomers are a perfect gas, ρ 1c = K −1 0 andρ 1 = ρ 1 /ρ 1c and Ξ f ree (A, L, T, µ 1 ) is the free monomer ideal gas partition function in the accessible volume at same temperature and chemical potential. The free energy βΩ R = − ln Ξ R takes the form which is the natural generalization of the single filament case in ref. [4]. In the r.h.s. of Eqs (15,16), the µ 1 dependence follows from the link betweenρ 1 and the free monomer chemical potential given in Eq.(1). The use ofρ 1 instead of µ 1 is very common in the biophysics literature as it has a direct interpretation as the ratio of polymerization and depolymerization rates in the bulk [4] and as it allows more compact expressions. This applies in the last term in the bundle free energy Eq.(16) where we have introduced the partition function D(L n ,ρ 1 ) of the single living filament of index n This partition function is directly linked to the probability P(j n |L n ,ρ 1 ) = α(j n , L n )ρ jn for filament of index n to have a size j n ∈ [2, z * n ] given the seed-wall distance L n and the reduced densityρ 1 [3,4]. The knowledge of the set of D(L n ,ρ 1 ) for the filaments in the bundle allows to compute all equilibrium properties of the bundle. Moreover our notations allow to treat both flexible and rigid filaments: for the rigid case α(j n , L n ) = 1 j n z n 0 j n > z n D(L n ,ρ 1 ) = zn jn=2ρ j 1 =ρ where | · · · | means the integer part of the argument.
According to Eq.(3), the partial derivative of the grand potential with respect to L gives the total normal pressure exerted on the wall. Using Eqs. (15,16) we thus get where Introducing f ⊥ (L n ,ρ 1 ), the equilibrium force exerted by the n th living filament of the bundle on the wall, we can write: ] the corresponding potential of mean force related to the presence of the wall, is the mean force exerted by a filament of fixed size j on the wall distant L n from its seed. Note that for rigid filaments the concept of force becomes ill-defined since the potential of mean force of a filament of contour length L c,j is either zero for L n ≥ L c,j or infinite for L n < L c,j . Correspondingly P(j n |L n ,ρ 1 ) goes to zero for L n < L c,j . The partial derivative of Ω R with respect to µ 1 , gives the average number of monomers in the system where we usedρ 1 = K 0 e βµ1 Λ −3 and is the average bundle size.
To define the number of filaments hitting a wall, it is appropriate to define a new filament relative-size probability, Q(m n |L,ρ 1 ), relative to the distance from the wall position, where the index m n ≡ j n − z n runs in the interval m n ∈ [2 − z n , z * n − z n ]: Q(m n |L,ρ 1 ) = Q(j n − z n |L n ,ρ 1 ) = P (j n |L n ,ρ 1 ) In terms of the absolute-and relative-size distributions, the expected total number of filaments N 0 touching the 8 FIG. 1: In this schematic representation of the optical trap set-up, the bead colloid used in the experiment, monitored during the filaments growth, is pictured as a horizontal moving wall plane connected to a spring of elastic constant T . The filaments are grafted to a horizontal fixed wall plane and grow against the moving one, exerting an upwards force. The compressed spring, with physical rest length equal to LR, exerts a downwards force T L, where L is the distance between the moving and the grafting planes. The moving wall divides the total volume Vtot = LRA into two sub-volumes, which are in global equilibrium at a fixed temperature T and are surrounded by a large reservoir containing a solution of G-actin (represented by little blue spheres) at fixed chemical potential µ1 and fixed density⇢1. At equilibrium, the bundle size fluctuates because of polymerization and depolymerization events.
Chamber I with length L encloses some free monomers and the bundle with filaments pressing on the moving wall which is further subjected to a restoring force  T L (represented by the spring in chamber II) modeling the trapping mechanism a↵ecting the colloidal particle. Chamber II of complementary length L R L contains only free monomers which exert some pressure on the separating wall. The moving wall is here representing the colloidal particle used in the experiment [15] and, given its mesoscopic size, its thermal fluctuations can be analyzed within a thermostatistical approach [24] by analyzing the grand potential of the system at fixed L with given  T . The total free energy will be given by: where ⌦ 1 (A, L, T, N f , µ 1 ) is the grand potential of the first sub-volume containing the bundle of N f filaments, ⌦ 2 (A, L R L, T, µ 1 ) is that of the second sub-volume containing only the solution of free monomers and 1 2  T L 2 is the contribution due to the compressional energy stored in the spring. Note that the arguments of ⌦ O.T. are only the thermodynamic variables (A, L, L R , T, N f , µ 1 ): we have added after the semi-column symbol the trap force constant  T because of the huge importance of this external parameter in the grand-canonical optical trap ensemble average properties.
Eq.(29) is strictly valid at thermodynamic equilibrium. It will be applicable to experimental situations only if this condition is satisfied. The grand potential for the ideal free monomer solution is simply while the grand-potential of the first chamber is given by Eq. (17). Summing these terms to the elastic free energy and discarding irrelevant constant terms, one easily obtains the expression for the optical trap grand potential: FIG. 1: In this representation of the optical trap set-up, the colloid used in the experiment, monitored during the filaments growth, is pictured as a horizontal moving wall connected to a spring of elastic constant κT . The filaments are grafted to a horizontal fixed wall and grow against the moving one, exerting an upwards force. The compressed spring, with physical rest length equal to LR, exerts a downwards force κT L, where L is the distance between the moving and the grafting planes. The moving wall divides the total volume Vtot = LRA into two sub-volumes, which are in global equilibrium at a fixed temperature T and are surrounded by a large reservoir containing a solution of G-actin (represented by little blue spheres) at fixed chemical potential µ1 and fixed densityρ1.
fixed obstacle at given position L, is the sum of the probability for each filament of the bundle to have a size larger than z n , hence where Θ(j n − z n − 1) is an indicator which is unity if the argument of Θ is non negative and zero if it's negative. In the case of rigid filaments, N 0 vanishes.

III. OPTICAL TRAP ENSEMBLE
Footer et al. [21] measured the force exerted by a bundle of approximately eight F-actin filaments by opposing to the growing filaments a colloidal particle subjected to a restoring force. The force, linear in the colloid displacement from its position in absence of the bundle, was generated by trapping the colloid within an optical trap apparatus and was indirectly measured by monitoring the displacement of the colloid during the growth of the bundle. For large enough time the colloid position reached a stationary state because of the harmonic restoring force. Figure 1 schematically shows an equivalent set-up. In a large volume filled with free monomers at fixed chemical potential µ 1 and hence at fixed grand-canonical average densityρ 1 , and in contact with a heat bath at temperature T , consider a central volume defined by a transverse area A and a fixed length L R . The central volume of size V = AL R is bounded on one side by a fixed wall of area A into which the filaments are grafted. Additionally, this volume is partitioned into two chambers of common transverse area but of variable heights, by a mobile hard wall parallel to the grafting wall which can only move vertically. Chamber I with length L encloses some free monomers and the bundle with filaments pressing on the moving wall which is further subjected to a restoring force κ T L (represented by the spring in chamber II) modeling the trapping mechanism affecting the colloidal particle. Chamber II of complementary length L R − L contains only free monomers which exert some pressure on the separating wall. The moving wall here represents the colloidal particle used in the experiment [21]. The total free energy of this system is given by: where Ω 1 (A, L, T, N f , µ 1 ) is the grand potential of the first sub-volume containing the bundle of N f filaments, Ω 2 (A, L R − L, T, µ 1 ) is that of the second sub-volume containing only the solution of free monomers and 1 2 κ T L 2 is the contribution due to the compressional energy stored in the spring.
Eq.(29) is valid at thermodynamic equilibrium and it will be applicable to experimental situations when this condition is satisfied. The grand potential for the ideal free monomer solution is while the grand-potential of the first chamber is given by Eq. (14). Summing these terms to the elastic free energy the expression for the optical trap grand potential is It is now convenient to define an equilibrium distribution for the variable L through One can further define the joint distribution function p(L, The measurement of the force in the experiment [21] has been obtained indirectly through that of the colloid position. In these conditions the measured force must be compared to the average over the L distribution P O.T. (L), namely For the average colloid position one gets where the first term vanishes noting that P O.T. At the relevant length scales (a few microns at most) actin filaments are "semiflexible polymers" with large bending rigidity. The wall distance range of interest, L ∼ 25d ÷ 90d, comparable to the filaments contour length L c , is rather smaller than actin persistence length at room temperature p = 5370d (d = 2.7 nm is half of the size of the G-actin monomer) so that actin filaments behave as stiff chains. As in ref. [4], we adopt the living version of the discrete Wormlike Chain (d-WLC) model for F-actin. This model is particularly suited for our investigation since we can adopt the universal expression of Gholami et al. [2] for the entropic force produced by "dead filaments" as far as the filament contour length remains within the non-escaping regime (see ref. [4] and the following subsection). In the weak-bending (L ∼ < L c ), stiff-chain (L c p ) regime the relevant adimensional variable is the reduced compression and the universal expression for the force-compression law of a continuous WLC of contour length L c is found to be and with λ k = (2k − 1) π 2 .f (η) starts from zero atη = 0 (L = L c ) and rapidly increases withη up to a unity plateau reached aroundη = 0.25. Higher compressions do not increase the response force of this model. Note that the above behavior strictly concerns a continuos WLC. The extension of the WLC model to living filaments requires the use of a discrete WLC model whose contour length changes in a quantized fashion upon chemical events. As shown in ref. [4], the use of the above theory for d-WLC model introduces negligible errors as far as the compressionη does not reach the breakdown of the weak-bending regime and the occurrence of the escaping regime for the living extension of the model.

B. Non-escaping filaments criteria
Living filaments in supercritical conditions tend to grow indefinitely unless some external agent stops the preferential polymerization process. For continuously growing filaments we cannot define statistical equilibrium but at most a stationary non-equilibrium state. The growth of completely rigid filaments ( p → ∞) can always be arrested by a rigid obstacle provided the external force applied to the obstacle is strong enough to balance the bundle action. For semi-flexible filaments the situation is more complex because a filament that can bend, can also laterally escape and grow indefinitely because of the supercritical conditions. If this situation occurs we cannot use Equilibrium Statistical Mechanics to describe our system. In order to avoid the escaping state we have imposed that each filament in the bundle cannot have more than a maximum number of monomers z * n = int (πL n /2d) (see Eq. (6)). Imposing a  (2)). The divergence from the dashed lines, for L greater than L l = pd/ lnρ1, corresponds to the breakdown of condition (45) and then to the appearance of lateral escapes. Inset: magnification of the same data forρ1 = 2.5 only for both the homogenous (red, dot-dashed line) and the in-registry (green continuous line) bundle. In limiting case of in-registry bundle of fully rigid filaments, this contribution to the free energy has a discontinuous pattern, as shown in the inset as black dots. See Eq. (19). maximum number of monomers per filament, however, will bias the properties of the system unless the probability for j n = z * n to occur be negligibly small for all filaments in the bundle since α(z n , L n ) = 1. Following ref. [4], the non-escaping regime condition on the reduced density at fixed L is found to be:ρ At fixedρ 1 this relation establishes a maximum amplitude of the box to avoid the z * bias. In figure 2 we display the bundle contribution to the free energy βΩ bun (L, t,ρ 1 , N f ) + N f ln q20Λ 6 n=1 ln D(ρ 1 , L n ) for a bundle of N f = 16 filaments. We report results for both homogeneous and in-registry bundles for actin at the given room temperature persistence length, and for two values of the free monomers reduced densityρ 1 . We observe that for the specific d-WLC model of living filaments with force law given by Eq.(40) the bundle contribution to the free energy is roughly linear with L up to aρ 1 -dependent value of the box size L above which the presence of escaping filaments drives the system towards a different, unjustified, linear behavior. In the main figure results for both type of bundles, homogeneous and in-registry, appear to be superposed. However at the magnified scale of the inset an almost discontinuous behavior with period d is seen for the in-registry case. At an even finer scale (not shown), the same behavior can be detected for the homogeneous bundle, although with a period of d/N f . The nature and the origin of this behavior directly relies on the equilibrium force expression Eq.(21) [4]. An infinitesimal change of L, by a fraction of d, is accompanied by a strong variations of the α(j n , L n ) factors for filament lengths touching the wall, and hence by a large modification of the equilibrium size distribution P(j n |L n ,ρ 1 ) and of the strength of the mean forcef j (L) exerted on the wall by a filament with size j. The dashed lines in the main panel of Fig.2 correspond to the linear behavior −F H s L based on Hill's mean field prediction of Eq.
(2). This shows a close similarity of Hill's result with our free energy Ω bun (L), up to theρ 1 -dependent crossover to the escaping regime. To avoid escaping, for given values of the parameters p andρ 1 , we have to impose a maximum size of the box; basing on the criterium given by Eq.(45), we have to choose a length L < L max (ρ 1 , p ) < p d/ lnρ 1 . A good choice for L max follows imposing in Eq.(44) a ratio of probabilities at most equal to 0.001, giving L max = 89.8d and 70.2d atρ 1 = 1.7 and 2.5 respectively.
In the optical trap apparatus the box size is a random variable which, by using Eqs.(31, 32) and the linearity shown in figure 2, results to be Gaussian with a variance give by σ L = k B T /κ T . Therefore a safe choice for the average box size of the optical trap has to be Knowing L OT , Eq.(47) provides a condition for the minimum value of the κ T that can be used for given N f andρ 1 . A weaker trap would let the filaments become too long and eventually escape. This is not the full story. To produce useful work, actin filaments in the usual conditions should not become shorter than L m in = 70 nm which corresponds to a minimum number of monomers ∼ 25, as discussed by Mogilner [12]. Thus another constraint for the average optical trap size is: This additional constraint implies κ T < κ T,max .

V. RESULTS
In this section we first compare rigid and flexible models for a single chain and a bundle of 8 filaments, the typical number in the experiments of ref. [21]. Later we will investigate more in details flexible bundles for various number of filaments and for various average trap amplitudes. We stick on a single value of the reduced density,ρ 1 = 2.5, a typical value for in-vitro experiments [15,21]. A more complete characterization of our F-actin model in different conditions is provided in the Supplementary Material.
where |x| indicates the integer part of a real variable x, σ = d √ βκ T −1 and for flexible and rigid case respectively, the latter again in perfect agreement with Hill's prediction. As for the flexible case, we note that the trap length for the in-registry bundle is slightly smaller than for the homogenous bundle which reflects an effective larger stiffness of the in-registry disposition with respect to the homogenous disposition of seeds. However, as for the homogeneous case, the distribution gets more localized for increasing N f . The relative deviation of L O.T. from L H at given N f is smaller than in the corresponding homogenous case.
As for the bundle force and its dependence on N f at given L H , we concentrate here on the homogenous bundles of N f = 8, 16 and 32 filaments. We report in figure 7 the distribution of the bundle force (panel (a)) defined as:  . Along the individual crests, we see that the force decreases for increasing L which is in agreement with the buckling force expression. It is reasonable to assign each crest to a specific number of buckled filaments participating to the average bundle force: in panel (b) the first crest (from lower force value) is due to the a single filament pushing the obstacle, the second crest is due to two filaments in contact with the obstacle, and so on. Therefore the peaks observed in P O.T. (F bun ) arise from the contribution to the bundle force of integer numbers of buckled living filaments. Note that filaments in contact with the obstacle are mostly buckled since the buckling regime of the compression force starts at quite small reduced compression (⌘ ' 0.25). Once a filament is buckled it will oppose a force independent of its compression amplitude the applied compression. However, due to its living character, it can change its contour length and escape from exit the buckling regime by loosing one or few monomers. At the same time, filaments whose tip position is closer to the wall position by less than a monomer size d can increase their contour length by polymerization and, if the filaments sti↵ness is large enough (which at constant`p means for shorter traps filaments), it can enter in the compressed buckled state. We can schematize look at this mechanism by a two-state model which can be thought as a generalization of the rigid model: filaments are either non-active (not in contact with the obstacle and therefore providing zero contribution to the bundle force) or in their buckled state therefore contributing to the force by a finite amount equal to f b (L c )/ L 2 c ⇡ L 2 . The intrinsic flexibility of the filament hence transforms the infinite force of the rigid model into a finite force proportional to the inverse second power of the filament contour length, roughly equal to the wall-seed distance for sti↵ filaments. In this two-states model a living bundle of given length can adjust its the number of active touching filaments to adapt and resist to an external load. . Along the individual crests, we see that the force decreases for increasing L which is in agreement with the buckling force expression. It is reasonable to assign each crest to a specific number of buckled filaments participating to the average bundle force: in panel (b) the first crest (from lower force) is due to the a single filament pushing the obstacle, the second crest is due to two filaments, and so on. Therefore the peaks observed in P O.T. (F bun ) arise from the contribution to the bundle force of integer numbers of buckled filaments. Note that filaments in contact with the obstacle are mostly buckled since the buckling regime of the compression force starts at quite small reduced compression (η 0.25). Once a filament is buckled it will oppose a force independent of the applied compression. However, due to its living character, it can change its contour length and exit the buckling regime by loosing one or few monomers. At the same time, filaments whose tip position is closer to the wall position by less than a monomer size d can increase their contour length by polymerization and, if the filaments stiffness is large enough (which means for shorter filaments), it can enter in the buckled state. We can look at this mechanism by a two-state model: filaments are either non-active (not in contact with the obstacle and therefore providing zero contribution to the bundle force) or in their buckled state therefore contributing to the force by a finite amount equal to In this two-states model a living bundle of given length can adjust the number of touching filaments to resist to an external load.
In the present optical trap system the equilibrium bundle force can be related to the average number of touching filaments as follows where we used Eq. (28) and the approximations L 2 c,j ≈ ( L O.T. ) 2 andf (η jn ) = 1. As the average bundle force roughly equals Hill's stalling force Figure 8 shows the average fraction of touching filaments in the homogeneous bundle, divided by ln(ρ 1 ), as a function of L O.T. 2 for two different densities and various number of filaments. Data points align on a line with slope ∼ 9 × 10 −5 , represented by the straight black line in Figure 8. The red dashed line corresponds to the slope expected according to Eq.(54), slightly smaller (≈ 7.5 × 10 −5 ): the number of touching filaments results slightly larger than predicted by Hill's theory. This is a further effect of filament flexibility. Since the average polymerization force is slightly larger than Hill's stalling force, Eq.(54) underestimates the observed average fraction of touching filaments.

VI. DISCUSSION AND FINAL REMARKS
Before concluding, it is interesting to discuss the experimental situation of ref. [21]: the growth of N f ≈ 8 filaments bundle anchored to a latex bead controlled by a harmonic force from an optical trap apparatus, and pushing against an immobile and impenetrable surface was followed up to the establishing of a stationary state (see figure 4 of reference [21]). Two different sets of experiments at different free monomers density, ρ 1 = 4 and 2 µM were performed, corresponding in our chemically simplified model toρ 1 = 2.5 andρ 1 = 1.7, respectively. The measured latex bead displacement at stationarity was L ≈ 900nm ≈ 300d atρ 1 = 2.5 and to L ≈ 180nm ≈ 70d atρ 1 = 1.7. Knowing the strength of the trap apparatus, 0.0035 k B T /d 2 in the first case and 0.011 k B T /d 2 in the second case, the measured displacements corresponded to an apparent stalling force of F bun exp ≈ 1.6 pN = 1.05 k B T /d at ρ 1 = 2.5 and 1.1 pN = 0.72 k B T /d atρ 1 = 1.7. Those values are in marked disagreement with Hill's formula, Eq.(2), which predicts F H s = 6.31k B T /d and 4.25 k B T /d respectively. The analysis of these experiments was based on the distinction between buckled and unbuckled filaments, using the following criterium: if the detected force was smaller than the single filament buckling force, Eq.(41), at the observed average length of the bundle, then the measured force was considered a proper estimate of the stalling force, otherwise the measured force was regarded as meaningless. The first case (ρ 1 = 2.5) fell within this latter condition, and hence it was not considered. The second case (ρ 1 = 1.7) instead fell into the former condition and therefore was considered a correct measure of the bundle polymerization force. According to our analysis of section IV B which assumes the absence of hydrolysis, we find that the average trap length should be below L max − 3σ L to avoid the occurrence of escaping filaments (see Eq. (47)). Atρ 1 = 2.5 the measured steady bead position in the experiment was much larger than L max , so we can conclude that the bundle was in the escaping regime, in agreement with the interpretation of the authors. In the other case the observed average trap length did not exceed L max , but it still exceeded the boundary given by Eq.(47), as L O.T. ∼ 70d > 90d − 3(0.011) −1/2 d 60d so that we should expect the presence of escaped filaments. The detected force of F bun exp 0.72 k B T /d is about 35% larger than the stalling force of a single filaments k B T lnρ 1 /d. It is possible that the residual force accounts for the elastic force from seven escaped filaments. According to our analysis, what is needed to measure the polymerization force of a 8-bundle atρ 1 = 1.7 is a stronger trap with κ T ≈ 0.1 k B T /d 2 , establishing the equilibrium distance around 40 ÷ 50d well within the non-escaping regime.
In conclusion, in this paper we have developed the Statistical Mechanics formalism to treat a bundle of (de)polymerizing filaments in a box pushing against a mobile wall. Our system is a schematic representation of in-vitro experimental apparatus exploited to measure the force that a bundle of F-actin can exert on an obstacle. Our treatment, limited to equilibrium conditions, requires the external load to increase with the distance between the channel boundaries in order to match the bundle force at stalling, where the bundle growth is stopped and a genuine equilibrium state is established. We have developed the formalism for a simple and flexible model in which we have disregarded direct inter-filaments and filament-solvent interactions which are considered to be irrelevant in the present context. The formalism has been used for two specific filament models under a load increasing linearly with the box size: i) the fully rigid model (1D) which is at the heart of the much celebrated Brownian Ratchet model used in interpreting experimental data for the force-velocity law [15,21] and ii) a model of semiflexible discrete Wormlike chain with persistence length p and monomer size d adapted to F-actin values for which a force-compression law is known from previous studies [2,4]. For the rigid model, we have derived exact expressions for the probability distribution of the mobile obstacle position and its average value, for a single filament and for homogeneous and in-registry bundles. These expressions allow us to discuss the validity of the celebrated Hill's formula for the stalling force. We found that for box sizes beyond ∼ 5d our exact statistical mechanics averages, taken over the optical trap ensemble, do converge asymptotically and exponentially fast to the Hill's prediction based on 1D thermodynamic approach with the relative deviation being already ≤ 1% at L ∼ 10d and ∼ 10 −6 for L ∼ 30d. For narrower boxes ( L < 10d), exact results indicate a markedly different behavior, a boundary effect already noticed in ref. [4]. The consideration of filament flexibility forces us to distinguish the stalling regime, the regime of small optical trap widths in which the external load is able to stop the polymerization of all filaments, from the escaping regime, the regime of larger trap widths where the bent filaments can polymerize freely parallel the obstacle wall and the external load is balanced by the mechanical bending force of the filaments. In the stalling regime we found that, with respect to the rigid filament case, flexibility induces in general a slight increase of the equilibrium optical trap distance and of the stalling force, the amount of which depends on the average optical trap width and on the disposition of the bundle seeds. A marked difference from the rigid model is that the individual filaments are either not in contact with the obstacle, hence not directly active, or in their buckled state providing a finite force roughly proportional to ( L O.T. ) −2 . Therefore the total bundle force results, to a very good level of accuracy, from N f times the average fraction i.e. Eq.(50).

In-registry bundle
For an in-registry bundle of N f filaments the bundle partition function at fixed L is given by the product of N f identical single filament partition functions, Eq.(A1), Therefore the wall position probability distribution in the optical trap is given by Following a procedure similar to that for the single filament, we obtain for the average trap length of this bundle

Homogenous bundle
According to the definition given in the main text, in a homogeneous bundle of N f filaments the filament n starts at The bundle partition function is now and the normalization of P O.T. (L) Now we can split the single integration interval into N f sub-intervals in which each of the N f − 1 terms in the product has constant value In the k-th interval N f − k terms have the value (ρ i 1 − 1) while the remaining k terms have the value (ρ i+1 1 − 1). The generic term of the double sum is therefore changing back variable to x = y + h N f we obtain Hence the normalization of P O.T. (L) for the homogenous rigid bundle is (A18) Correspondingly, for L O.T. we obtain