Dynamics and length distribution of microtubules under force and confinement
aa r X i v : . [ q - b i o . S C ] D ec Dynamics and length distribution of microtubules under force and confinement
Bj¨orn Zelinski ∗ , Nina M¨uller † and Jan Kierfeld ‡ Physics Department, TU Dortmund University, 44221 Dortmund, Germany (Dated: August 8, 2018)We investigate the microtubule polymerization dynamics with catastrophe and rescue events forthree different confinement scenarios, which mimic typical cellular environments: (i) The micro-tubule is confined by rigid and fixed walls, (ii) it grows under constant force, and (iii) it growsagainst an elastic obstacle with a linearly increasing force. We use realistic catastrophe modelsand analyze the microtubule dynamics, the resulting microtubule length distributions, and forcegeneration by stochastic and mean field calculations; in addition, we perform stochastic simulations.Freely growing microtubules exhibit a phase of bounded growth with finite microtubule length anda phase of unbounded growth. The main results for the three confinement scenarios are as follows:(i) In confinement by fixed rigid walls, we find exponentially decreasing or increasing stationary mi-crotubule length distributions instead of bounded or unbounded phases, respectively. We introducea realistic model for wall-induced catastrophes and investigate the behavior of the average lengthas a function of microtubule growth parameters. (ii) Under a constant force the boundary betweenbounded and unbounded growth is shifted to higher tubulin concentrations and rescue rates. Thecritical force f c for the transition from unbounded to bounded growth increases logarithmically withtubulin concentration and the rescue rate, and it is smaller than the stall force. (iii) For microtubulegrowth against an elastic obstacle, the microtubule length and polymerization force can be regulatedby microtubule growth parameters. For zero rescue rate, we find that the average polymerizationforce depends logarithmically on the tubulin concentration and is always smaller than the stall forcein the absence of catastrophes and rescues. For a non-zero rescue rate, we find a sharply peakedsteady-state length distribution, which is tightly controlled by microtubule growth parameters. Thecorresponding average microtubule length self-organizes such that the average polymerization forceequals the critical force f c for the transition from unbounded to bounded growth. We also investigatethe force dynamics if growth parameters are perturbed in dilution experiments. Finally, we showthe robustness of our results against changes of catastrophe models and load distribution factors. PACS numbers: 87.16.Ka, 87.16.-b
I. INTRODUCTION
Microtubules (MTs) are one of the main componentsof the cytoskeleton in eukaryotic cells. Their static anddynamic properties are essential for many cellular pro-cesses. MTs serve as pathways for molecular motor pro-teins [1] and contribute to cell stiffness [2]. Dynamic MTsplay a crucial role in the constant reorganization of thecytoskeleton, and single MTs can generate polymeriza-tion forces up to several pN [3]. These forces are usedin various intracellular positioning processes [4], such aspositioning of the cell nucleus [5] or chromosomes duringmitosis , establishing cell polarity [6], or regulation of cellshapes [7, 8]. In many cellular processes, MTs establishand maintain a characteristic length in response to forcesexerted, for example, from the confining cell cortex [7].The fast spatial reorganization of MTs is based on thedynamic instability: Polymerization phases are stochasti-cally interrupted by catastrophes which initiate phases offast depolymerization; fast depolymerization terminatesstochastically in a rescue event followed again by a poly- ∗ [email protected] † [email protected] ‡ [email protected] merization phase [9]. This complex dynamic behaviorwith catastrophes and rescue events is central to a rapidremodelling of MTs in the cytoskeleton, but it also affectstheir ability to generate polymerization forces. We willshow that, in general, the dynamic instability decreases the average polymerization force of a single MT.In this article we theoretically investigate the polymer-ization dynamics of a single MT under force or confine-ment and in the presence of the MT dynamic instability.We use a coarse-grained polymerization model with dy-namic instability and characterize spatial and temporalbehavior in three different scenarios, which mimic typ-ical cellular environments that can also be reproduced in vitro : (i) Confinement:
The MT is confined betweenfixed rigid walls, which cannot be deformed by the micro-tubule. Such confinement is realized in fixed solid cham-bers [10]. (ii)
Constant force:
A constant force is actingon the MT. Constant forces can be realized by opticaltweezers with a force clamp control [11]. (iii)
Elastic ob-stacle:
The microtubule grows against an elastic obstacle,which resists further growth by a force growing linearlywith displacement. Elastic forces can be realized by op-tical tweezers without force clamp [12, 13]. For all threeconfinement scenarios (i)–(iii) we focus on the resultinglength distributions of MTs and for scenarios (ii) and(iii), we calculate the polymerization force that a singleMT can generate.Dynamic MTs also initiate regulation processes or aresubject to regulation. Dynamic MTs can activate or de-activate proteins upon contacting the cell membrane [14],or they can activate actin polymerization within the cellcortex [15, 16]. At the same time, polymerizing MTs arealso targets of cellular regulation mechanisms [17], whichaffect their dynamic properties.The dynamic instability of MTs enables various regu-lation mechanisms of MT dynamics. Catastrophes andrescues result from the hydrolysis of GTP-tubulin withinMTs. When GTP-tubulin is incorporated into the tipof a growing MT, it forms a stabilizing GTP-cap. Theloss of this GTP-cap due to hydrolysis of GTP-tubulinto GDP-tubulin, causes a catastrophe [9]. In livingcells, there are various microtubule associated proteins(MAPs) that either stabilize or destabilize microtubulesand regulate microtubule dynamics both spatially andtemporally [18]. Recently, the importance of MAPsin connection with the plus end of growing MTs hasbeen recognized [19]. Stabilizing MAPs bind to assem-bled MTs, thereby reducing the catastrophe rate or in-creasing the rescue rate. Destabilizing MAPs such asOP18/stathmin bind to GTP-tubulin dimers, thus de-creasing the available GTP-tubulin concentration, whichin turn decreases the growth velocity of the GTP-cap andmakes catastrophes more likely. Therefore, such mecha-nisms can regulate basic parameters in our model, suchas the available GTP-tubulin concentration or the rescuerate, and we will systematically study their influence onthe generated polymerization force for the three confine-ment scenarios (i)–(iii).The paper is structured as follows: In Sec. II, the MTmodel and the basic notation are introduced. We alsodiscuss the catastrophe model and the underlying hy-drolysis mechanism in the absence and in the presenceof a resisting force. Section III deals with the simulationmodel. In Secs. IV, V, and VI, results for the three sce-narios (i)–(iii) are presented and discussed. In Sec. VII,the elastic obstacle is reconsidered using an alternativecatastrophe model based on experimental measurementsto show that our results are robust with respect to thischange in the catastrophe model. In Sec. VIII, we showthat our results are also robust with respect to possiblegeneralization of the force-velocity relation by introduc-ing load-distribution factors. Section IX contains a finaldiscussion and outlook.
II. MICROTUBULE MODELA. Single MT dynamics
The MT dynamics in the presence of its dynamic in-stability is described in terms of probability densities andswitching rates [20, 21]. In the growing state, a MT poly-merizes with average velocity v + . The MT stochasticallyswitches from a state of growth (+) to a state of shrink-age ( − ) with the catastrophe rate ω c . In the shrinking state, it rapidly depolymerizes with an average velocity v − ≃ · − m / s (Table I). With the rescue rate ω r the MT stochastically switches from a state of shrinkageback to a state of growth. We model catastrophes andrescues as Poisson processes such that h τ + i = 1 /ω c and h τ − i = 1 /ω r are the average times spent in the growingand shrinking states, respectively. The stochastic timeevolution of an ensemble of independent MTs, growingalong the x-axis, can be described by two coupled mas-ter equations for the probabilities p + ( x, t ) and p − ( x, t )of finding a MT with length x at time t in a growing orshrinking state, ∂ t p + ( x, t ) = − ω c p + ( x, t ) + ω r p − ( x, t ) − v + ∂ x p + ( x, t )(1) ∂ t p − ( x, t ) = ω c p + ( x, t ) − ω r p − ( x, t ) + v − ∂ x p − ( x, t ) . (2)In the following, we will always use a reflecting boundaryat x = 0: A MT shrinking back to zero length undergoesa forced rescue instantaneously. This corresponds to v + p + (0 , t ) = v − p − (0 , t ) . (3)A more refined model including a nucleating state hasbeen considered in Ref. [22]. For a constant and fixedcatastrophe rate ω c , eqs. (1) and (2) together with theboundary condition (3) can be solved analytically on thehalf-space x >
0, and we can determine the overall prob-ability density function (OPDF) of finding a MT withlength x at time t , P ( x, t ) ≡ p + ( x, t ) + p − ( x, t ) [20, 21].The solution exhibits two different growth phases: aphase of bounded growth and a phase of unboundedgrowth.In the phase of bounded growth the average lengthloss during a period of shrinkage, v − h τ − i = v − /ω r , ex-ceeds the average length gain during a period of growth, v + h τ + i = v + /ω c . The steady-state solution of P ( x, t )assumes a simple exponential form P ( x ) = | λ | − e − x/ | λ | with an average length h x i = | λ | and a characteristiclength parameter λ ≡ v + v − v + ω r − v − ω c , (4)with λ − < λ − =0, where the average length gain during growth equalsexactly the average length loss during shrinkage, v + ω r = v − ω c , (5)such that h x i diverges.In the regime of unbounded growth ( λ > P ( x, t ) asymptotically approaches a Gaussian distribu-tion [20] P ( x, t ) ≈ √ πD J t exp (cid:18) − ( x − Jt ) D J t (cid:19) (6)centered on an average length which approaches lineargrowth h x i ≈ Jt with a mean velocity J and with diffu-sively growing width h x i − h x i ≈ D J t with a diffusionconstant D J . The average growth velocity is given by J = v + ω r − v − ω c ω r + ω c (7)because the asymptotic probabilities to be in a growing orshrinking state are π + = ω r / ( ω c + ω r ) and π − = ω c / ( ω c + ω r ), respectively. The diffusion constant D J is D J = ω c ω r ( v + + v − ) ( ω c + ω r ) . (8)The transition between the two growth phases can beachieved by changing one of the four parameters of MTgrowth, ω c , ω r , v + , or v − . In the following, we willuse catastrophe models, where the catastrophe rate ω c is a function of the growth velocity v + , which in turnis determined by the GTP-tubulin concentration via theGTP-tubulin on-rate ω on (assuming a fixed off-rate ω off ).Moreover, experimental data suggest that v − is fixed tovalues close to ∼ − m/s (Table I). As a consequence,there are two tunable control parameters left, the GTP-tubulin concentration or, equivalently, the tubulin on-rate ω on and the rescue rate ω r . These are the controlparameters we will explore for MTs in confinements andunder force. These parameters are also targets for regu-lation by MAPs, such as OP18/stathmin, which reduces ω on by binding to GTP-tubulin dimers or MAP4, whichincreases the rescue rate ω r . B. Force-dependent catastrophe rate
In a growing state, GTP-tubulin dimers are attachedto any of the 13 protofilaments with the rate ω on , whichis directly related to the GTP concentration. We explorea regime ω on = 30 , ...,
100 s − , see Table I. GTP-tubulindimers are detached with the rate ω off = 6 s − [23] suchthat we can typically assume ω on ≫ ω off . In the absenceof force or restricting boundaries, the velocity of growthis given by v + (0) = d ( ω on − ω off ) . (9)Here d denotes the effective dimer size d ≈ / ≈ . ω c is determined by the hydroly-sis dynamics of GTP-tubulin [9]. When GTP-tubulinis incorporated into the tip of a growing MT, it formsa stabilizing GTP-cap. In a chemical model, the loss ofthis GTP-cap due to hydrolysis of GTP-tubulin to GDP-tubulin directly causes a catastrophe. However, recentresearch indicates that the “structural plasticity” of theMT lattice can play a role for the kinetics of catastrophes[24]. This structural plasticity mechanism is based on the assumption that GDP-tubulin prefers a curved configu-ration, which generates additional mechanical stresses inthe MT by hydrolysis. Also in the presence of structuralplasticity, the loss of the GTP-cap has a destabilizingeffect, but the kinetics leading to a catastrophe can bemore complicated because the initiation of a catastro-phe event is similar to the nucleation of a crack in thestressed MT lattice within this model. In this article, wefocus on purely chemical catastrophe models and neglectmechanical effects on the catastrophe kinetics.Within a chemical catastrophe model, the loss of theGTP-cap due to hydrolysis of GTP-tubulin to GDP-tubulin triggers a catastrophe immediately. Therefore,the catastrophe rate ω c is given by the first-passagerate to a state with vanishing GTP-cap and has beendiscussed within a model with cooperative hydrolysis[25, 26], where GTP-tubulin is hydrolyzed by a com-bination of random and vectorial mechanisms; similarmodels have also been discussed for hydrolysis in F-actin [27, 28]. In random hydrolysis, GTP-tubulin ishydrolyzed at a random site within the GTP-cap witha rate per length r ⋍ . · m − s − , while in vecto-rial hydrolysis, only GTP-tubulin with adjacent GDP-tubulin is hydrolyzed. This results in hydrolysis frontspropagating through the microtubule with average ve-locity v h ⋍ . · − m / s. The inverse catastrophe ratecan then be calculated as the mean first-passage time to astate with zero cap length, as a function of hydrolysis pa-rameters and v + . With v = v + − v h , D = 0 . d ( v + + v h )and γ = 0 . vD / r − / the exact analytical result forthe dimensionless catastrophe rate α = ω c D − / r − / isgiven by the smallest solution of Ai ′ ( γ − α ) = − γAi ( γ − α ) . (10)Here Ai denotes the first Airy function and Ai ′ its deriva-tive [29]. We solved eq. (10) numerically and obtained ahigh order polynomial for the function α = α ( γ ). Thispolynomial is used in simulations and analytical calcula-tions to compute the catastrophe rate ω c = α ( γ ) D / r / as a function of the growth velocity v + , while the hydrol-ysis parameters v h and r are fixed.Under a force F , the tubulin on-rate ω on is modifiedby an additional Boltzmann factor [30] and the force-dependent growth velocity becomes v + ( F ) = d [ ω on exp ( − F d/k B T ) − ω off ] . (11)Here F d is the work that has to be done against theforce F to incorporate a single dimer of size d ; k B is theBoltzmann constant and T = 300 K the temperature. Inthe following we use the dimensionless force f ≡ F/F with F = k B T /d, (12)in terms of which the force-dependent growth velocity isgiven by v + ( f ) = d (cid:2) ω on e − f − ω off (cid:3) . (13)The characteristic force F has a value F = k B T /d ≈ f stall = ln ( ω on /ω off ) (14)is defined by the condition of vanishing growth velocity v + ( f stall ) = 0. We typically have f stall ≃ . , ..., F stall ≃ , ...,
20 pN for ω on = 30 , ...,
100 s − . The stallforce is the maximal force that the MT can generate inthe absence of catastrophes. We will investigate how theforces that can be generated in the presence of catastro-phes compare to this stall force.The velocity-dependence of the catastrophe rate as cal-culated from eq. (10) gives rise to a force-dependence ω c = ω c ( v + ( f )). We assume that this is the only effectof force on the catastrophe rate [39]. As a result, thecatastrophe rate increases exponentially, when v + ( f ) isdecreased by applying a force f , but a finite value is main-tained at v + ( f ) = 0, which is ω c ( v + = 0) ≈ . − . Weassume that v − is independent of force. For qualitativeapproximations, the force-dependence of the catastropherate can be described by an exponential increase abovethe characteristic force F , ω c ( f ) ∼ ω c ( f = 0) e f , (15)In Sec. VII we introduce an alternative catastrophe modelwhich is based on experimental measurements. The ex-ponential approximation (15) applies to the catastrophemodel described above as well as to the alternative catas-trophe model, see Fig. 11. Our results are robust for allcatastrophe models with an exponential increase abovethe characteristic force F . Our results do not directlyapply to more elaborate multi-step catastrophe modelswith more than two MT states [31]. III. SIMULATION MODEL
In the simulations we solve the stochastic Langevin-like equations of motion for the length x ( t ) of a singleMT using numerical integration with fixed time steps ∆ t and including stochastic switching between growth andshrinkage. In a growing state x ( t ) is increased by v + ∆ t ,while in a state of shrinkage, it is decreased by v − ∆ t .In the growing state, v + is calculated from eq. (9) forzero force and from eq. (11) under force. In each timestep a uniformly distributed random number ξ ∈ [0 , ω r,c ∆ t . If ξ < ω r,c ∆ t the MT changesits state of growth. The catastrophe rate ω c is calculatedfrom the high order polynomial obtained from eq. (10) asmentioned above. To assure ω r,c ∆ t ≤ t = 0 . d = 0 . r = 3 . · m − s − , v h = 4 . · − m / s, k B = 1 . · − J / K, T = 300 K, and ω off =6 s − , are fixed, see Table II, except for ω on , which isvaried in the range ω on = 30 −
100 s − , and ω r , whichis varied in a range ω r = 0 . − . − , see Table I.Averages are taken over many realizations of stochastictrajectories. FIG. 1: (a): Schematic representation of the confinementand possible MT configurations. From top to bottom: A MTgrowing with v + ; MT shrinks with v − . MT in a state ofgrowth and stuck to the boundary wall with v + = 0 and ω c,L .(b): Schematic representation of a single MT growing againstthe elastic obstacle. From top to bottom: MT shrinks with v − . MT under force F ( x ) = k ( x − x ) with f ( x ) ≡ F ( x ) /F , v + [ f ( x )], and force-dependent catastrophe rate ω c [ f ( x )]. IV. CONFINEMENT BETWEEN FIXED RIGIDWALLS
A single MT is confined to a one-dimensional box offixed length L with rigid boundary walls at x = 0 and x = L as shown schematically in Fig. 1(a) [32, 33]. Thereis no force acting on the MT but within the box catastro-phes are induced upon hitting the rigid walls. We proposethe following mechanism for these wall-induced catastro-phes: When the MT hits the boundary at x = L , itsgrowth velocity v + has to reduce to zero, which leads toan increase of the catastrophe rate to ω c,L ≡ ω c ( v + = 0).Since ω c,L is finite, wall-induced catastrophes are notinstantaneous but the MT sticks for an average time1 /ω c,L to the boundary before the catastrophe, whichis in contrast to previous studies [34]. For the averagetime spent at the boundary before a catastrophe, we find ω − c,L ≈ .
29 s. The catastrophe rate at the wall, ω c,L , ismuch higher than the bulk catastrophe rate ω c ( v + ). For ω on = 50 s − we find ω c,L /ω c ≃ Q + and Q − of finding the MT stuck tothe boundary in a growing state and in a shrinking state,respectively. The stochastic time evolution of Q + ( t ) and Q − ( t ) is given by: ∂ t Q + ( t ) = − ω c,L Q + ( t ) + ω r Q − ( t ) + v + p + ( L ) (16) ∂ t Q − ( t ) = + ω c,L Q + ( t ) − ω r Q − ( t ) − v − ∆ Q − ( t ) . (17)The quantity v + p + ( L ) is the flow of probability from theinterior of the confining box onto its boundary and isgiven by the solution of eq. (1) and (2) for x = L , while( v − / ∆) Q − is the probability current from the boundaryback into the interior, where ∆ denotes a small interval inwhich the flow v − Q − can be measured. This implies thatthere is a boundary condition v − p − ( L, t ) = ( v − / ∆) Q − for the backward current density at x = L , in additionto the reflecting boundary condition (3) at x = 0. Anidentical model for wall-induced catastrophes has beenintroduced in Ref. [22] recently.In the steady state and in the limit ∆ ≈ Q + ≈ v + ω c,L p + ( L ) (18) Q − ≈ , (19)and v − p − ( L, t ) = ( v − / ∆) Q − = v + p + ( L ). Eq. (18)shows that there is a non-zero probability Q + of find-ing a MT in a state of growth and stuck to the bound-ary, which is given by the flow of probability from theinterior of the confining box onto its boundary dividedby the average time being stuck to the boundary. Incontrast, eq. (19) states that there is no MT in a shrink-ing state and stuck to the wall. This is intuitively clearsince a MT undergoing a catastrophe begins to shrinkinstantaneously. In the steady state, we solve eqs. (1),(2) and (18) simultaneously with the additional nor-malization R L ( p + ( x ) + p − ( x )) dx + Q + = 1. We find v + p + ( x ) = v − p − ( x ) and P ( x ) = N e x/λ (cid:18) v + v − (cid:19) (20) Q + = N v + ω c,L e L/λ (21)with λ from eq. (4) and a normalization N − = λ (cid:18) v + v − (cid:19) (cid:16) e L/λ − (cid:17) + v + ω c,L e L/λ . (22)Equation (20) shows that we find an exponential OPDF P ( x ) in confinement with the same characteristic length | λ | . If the growth is unbounded in the absence of con-finement, which corresponds to λ − >
0, the OPDF isexponentially increasing; if the growth is bounded in theabsence of confinement, which corresponds to λ − < in vivo experiments, both exponentially increasing [35] and ex-ponentially decreasing OPDFs [20] have been found.In the following we focus on the case λ − > h x i = Z L xP ( x ) dx + Q + L = N (cid:26)(cid:18) v + v − (cid:19) λ (cid:20) e L/λ (cid:18) Lλ − (cid:19)(cid:21) + L v + ω c,L e L/λ (cid:27) . (23)In the limit of instantaneous wall-induced catastrophes, Q + ≈
0, we obtain h x i L ≈ − e − L/λ − λL , (24) i.e., the average MT length h x i /L depends on the twocontrol parameters ω r and ω on only via the ratio L/λ .This scaling property is lost if wall-induced catastrophesare not instantaneous because eq. (23) then exhibits addi-tional v + - and thus ω on -dependencies. Within our modelthe increased catastrophe rate at the boundary gives riseto an increased overall average catastrophe rate ω c,eff = ω c ( v + ) + Q + ( ω c,L − ω c ( v + )) , (25)for which we find ω c,eff ≃ .
03 s − for L = 1 µ m and ω c,eff ≃ .
006 s − for L = 10 µ m as compared to ω c ≃ . − for these conditions.We set the length of the confining box to L = 1 µ mand L = 10 µ m, which are typical length scales in ex-periments [10, 11] and cellular environments [5], and wecalculate h x i and Q + as functions of ω on and ω r . Theparameter regimes displayed in Figs. 2 and 3 correspondto regimes L/λ ≫ L = 10 µ m and L/λ ≪ L = 1 µ m. Results obtained from stochastic simulationsagree with analytical findings (Figs. 2 and 3). It is clearlyvisible that the size L of the confinement has a significantinfluence on h x i , mainly via the ratio L/λ .The probability Q + to find the MT at the wall in-creases with increasing rates in the range of Q + ≈ , ..., .
03 and exhibits only a weak dependency on L ,see Figs. 3. Even for maximum rates, the probability offinding a MT in a growing state and stuck to the wallis limited to several percent, due to the large catastro-phe rate ω c,L at x = L . Therefore, in most cases wall-induced catastrophes can be viewed as instantaneous,and the approximation (24) works well. For increasingon-rate ω on or rescue rate ω r , the ratio L/λ approaches
L/λ ≈ Lω r /v − from below. According to the approx-imation (24), the mean length h x i then increases andapproaches h x i /L ≈ / (1 − e − Lω r /v − ) − v − /Lω r frombelow. For L = 10 µ m, we have L/λ ≫ P ( x ) ∼ e x/λ . Theratio h x i /L saturates at a high value h x i /L ≈ . , ..., . L/λ ≫ L .In contrast, for L = 1 µ m, we have L/λ ≪
1, and L istoo small to establish the characteristic exponential de-cay of the length distribution. The length distribution P ( x ) is almost uniform, and the ratio h x i /L ≈ . , ..., . h x i /L = 1 / V. CONSTANT FORCE
In the second scenario a constant force F is appliedto the MT and the right boundary is removed, so thatthe MT is allowed to grow on x ∈ [0 , ∞ [. Accordingto eq. (13) the growth velocity under force is smaller,but it remains constant for fixed f . With eq. (10) thisresults in a higher, but also constant, catastrophe rate ω c [ v + ( f )] > ω c [ v + (0)]. Since v − and ω r are independentof force, the stochastic dynamics of the MT is described FIG. 2: The average length h x i as a function of ω on and ω r forconfinement by fixed rigid walls. Data points are results fromstochastic simulations, lines are analytical results (23). Toprow: The average length h x i as a function of ω on for differentvalues of ω r = 0 .
03 s − ( ⊡ ) , .
05 s − ( (cid:4) ) , . − ( ⊙ ) , . − ( • )and 0 . − ( △ ). (a) L = 10 µ m. (b) L = 1 µ m. Lower row:The average length h x i as a function of ω r for different valuesof ω on = 25 s − ( (cid:4) ) ,
50 s − ( ⊡ ) ,
75 s − ( N ) ,
100 s − ( ⊡ ). (c) L =10 µ m. (d) L = 1 µ m. FIG. 3: The probability Q + to find the MT at thewall as a function of ω on and ω r for confinement byfixed rigid walls. Data points are results from stochas-tic simulations, lines are analytical results (18). Toprow: Q + as a function of ω on for different valuesof ω r = 0 .
03 s − ( ⊡ ) , .
05 s − ( (cid:4) ) , . − ( ⊙ ) , . − ( • ) and0 . − ( △ ). (a) L = 10 µ m. (b) L = 1 µ m.Lower row: Q + as a function of ω r for different values of ω on = 25 s − ( (cid:4) ) ,
50 s − ( ⊡ ) ,
75 s − ( N ) ,
100 s − ( ⊡ ). (c) L =10 µ m. (d) L = 1 µ m. by eq. (1) and (2) with the same solutions P ( x, t ) as inthe absence of force, but with a decreased velocity ofgrowth v + ( f ) and an increased catastrophe rate ω c ( f )[20, 21]. In particular, we still find two regimes, a regimeof bounded growth and a regime of unbounded growth.In the regime of bounded growth P ( x, t ) is again ex-ponentially decreasing, and the force-dependent averagelength is h x ( f ) i = | λ ( f ) | with the corresponding force-dependent length parameter λ ( f ) ≡ v + ( f ) v − v + ( f ) ω r − v − ω c ( f ) (26)as compared to eq. (4) in the absence of force. Inthe regime of unbounded growth h x ( f ) i increases lin-early in time with the force-dependent mean velocity J ( f ) = [ v + ( f ) ω r − v − ω c ( f )] / [ ω r + ω c ( f )], cf. eq. (7). TheMT length distribution P ( x, t ) assumes again a Gaussianform (6) where also the diffusion constant D J ( f ) followsthe same eq. (8) with force-dependent growth velocity v + ( f ) and catastrophe rate ω c ( f ).In the presence of a constant force f , the transitionbetween bounded and unbounded growth is governed bythe force-dependent parameter λ ( f ). The regimes ofbounded and unbounded growth are now separated bythe condition λ − ( f ) = 0, which is shifted compared tothe case f = 0, see Fig. 4(a). The inverse length parame-ter λ − ( f ) is a monotonously decreasing function of force f and changes sign from positive to negative values forincreasing force f . Therefore λ − ( f c ) = 0 or v + ( f c ) ω r = v − ω c ( f c ) , (27)defines a critical force for the transition from unboundedto bounded growth. A single MT exhibiting unboundedgrowth ( λ − (0) >
0) in the absence of force undergoesa transition to bounded growth with λ − ( f ) < f > f c . On the other hand,starting with a combination of on-rate ω on and rescuerate ω r and a force f , which results in bounded growthwith λ − ( f ) <
0, the MT can still enter the regime ofunbounded growth by increasing ω on or ω r so that theforce f becomes subcritical, λ − ( f ) > f < f c .Rewriting condition (27) as v + ( f c ) = v − ω c ( f c ) /ω r > v + ( f ) decreases with f , it follows that thecritical force is always smaller than the stall force, f c 12 ln (cid:18) v + (0) ω r v − ω c (0) (cid:19) ∼ 12 ln (cid:18) ω on dω r v − ω c (0) (cid:19) (28)which shows that the critical force grows approximatelylogarithmically with on-rate ω on (note that the catastro-phe rate in the absence of force decreases with ω on as FIG. 4: (a): Phase boundary between bounded(B) and unbounded growth (UB) as a function of ω on and ω r for MT growth under constant force. Datapoints for f = 0( ⊡ ) , . (cid:4) ) , . ⊙ ) , . • ) , . △ ) , . N )represent results from simulations, lines represent solu-tions of v + ( f ) ω r = v − ω c ( f ) for a constant force f .(b): Critical force f c as a function of ω on for ω r =0 . 03 s − ( ⊡ ) , . 05 s − ( (cid:4) ) , . − ( ⊙ ) , . − ( • ). Data pointsrepresent results from simulations, lines represent the solu-tion of eq. (27) for a fixed combination of ω on and ω r . ω c (0) ∝ /ω on [26]) and rescue rate ω r . A negative f c for small on-rates and rescue rates signals that the MT isfor all forces f > f c as a functionof the on-rate ω on and for different rescue rates ω r fromsolving condition (28) numerically and from stochasticsimulations. Agreement between both methods is good.The condition λ − ( f ) = 0 specifies the boundary be-tween bounded and unbounded growth at a given force f .In Fig. 4(a), the resulting phase boundary is shown as afunction of ω on and ω r . There is good agreement betweennumerical solutions of λ − ( f ) = 0 and stochastic simula-tions. With increasing force, the boundary between thetwo regimes of growth shifts to higher values of ω on and ω r , and forces up to F ∼ . · F can be overcome by asingle MT in the parameter regimes of ω on and ω r con-sidered. VI. ELASTIC FORCE In the third scenario, an elastically coupled barrier isplaced in front of the MT as shown in Fig. 1(b), whichmodels the optical traps used in Refs. [11, 13] or the elas-tic cell cortex in vivo . If the barrier is displaced from itsequilibrium position x by the growing MT with length x > x , it causes a force F ( x ) = k ( x − x ) resisting furthergrowth. For x < x there is no force. We use x = 0 µ min the case of vanishing rescue rate and x = 10 µ m inthe case of finite rescue rate and a spring constant k inthe range 10 − N / m (soft) to 10 − N / m (stiff as in theoptical trap experiments in [13]).An elastic force F ( x ) = k ( x − x ) represents the sim-plest and most generic x -dependent force. Whereas for aconfinement of fixed length or a constant force, the MTlength x was the only stochastic variable, the force F ( x )itself is now coupled to x and becomes stochastic as well. Therefore, not only are the MT length distributions of in-terest but also the maximal and average polymerizationforces which are generated during MT growth. A. Vanishing rescue rate We first discuss growth in the absence of rescue events, ω r = 0. This situation corresponds to optical trap exper-iments [11, 13], which are performed on short time scalesand no rescue events are observed. In a state of growththe MT grows against the elastic obstacle with velocity v + [ f ( x )] and f ( x ) increases. For simplicity we suppressthe x -dependency in the notation in the following. At amaximal polymerization force f max , the MT undergoesa catastrophe and starts to shrink back to zero and thedynamics stop due to missing rescue events. No steadystate is reached. Since switching to the state of shrinkageis a stochastic process, the maximal polymerization force f max is a stochastic quantity which fluctuates around itsaverage value. We calculate the average maximal poly-merization force h f max i within a mean field approach.Here h ... i denotes an ensemble average over many real-izations of the growth experiment.Because no steady state is reached in the absence ofrescue events, we have to use a dynamical mean fieldapproach, which is based on the fact that the MT growthvelocity dx/dt = v + ( f ) is related to the time evolution ofthe force by df /dt = ( k/F ) dx/dt . In mean field theory,this results in the following equation of motion for h f i , ddt h f i = kF v + ( h f i ) , (29)where we used the mean field approximation h v + ( f ) i ≈ v + ( h f i ). With the initial condition h f i (0) = 0 we find atime evolution h f i ( t ) = ln h (1 − ω on /ω off ) e − t/τ + ω on /ω off i (30) ≈ f stall + ln [1 − exp( − t/τ )] (31)with a characteristic time scale τ = F /dkω off ≈ ... s for k ≈ − ... − N / m. For long times t ≫ τ , eq. (30) approaches the dimensionless stall force h f i = f stall , see eq. (14), which is the maximal polymer-ization force in the absence of catastrophes. The approx-imation (31) holds for ω on /ω off ≫ t =1 /ω c ( h f max i ). Together with eq. (30), this gives a self-consistent mean field equation for the maximal polymer-ization force h f max i , h f max i = ln h (1 − ω on /ω off ) e − /ω c ( h f max i ) τ + ω on /ω off i . (32)The maximal polymerization force h f max i is alwayssmaller than the stall force f stall as can be seen from FIG. 5: Average maximal polymerization force h f max i foran elastic obstacle and in the absence of rescues as a functionof ω on for different values of k = 10 − N / m( N ), 10 − N / m( (cid:4) )and 10 − N / m( • ). Data points represent results from simula-tions, solid lines are solutions of eq. (32). Error bars repre-sent the standard deviation of the stochastic quantity h f max i .Dashed line: dimensionless stall force f stall = ln ( ω on /ω off ). eqs. (30,31). Since ω on /ω off ≫ ω c τ ≫ h f max i ≈ ln (cid:18) ω on ω off τ ω c ( h f max i ) (cid:19) = f stall − ln [ τ ω c ( h f max i )] . (33)For a catastrophe rate increasing exponentially above thecharacteristic force F , eq. (15), we find h f max i ∼ 12 ln (cid:18) ω on dkF ω c (0) (cid:19) , (34)i.e., the maximal polymerization force grows logarithmi-cally in ω on (note that the catastrophe rate in the absenceof force decreases as ω c (0) ∝ /ω on [26]), see Fig. 5 for k = 10 − N / m. Within a slightly different catastrophemodel obtained from experimental data and discussed insection VII, this logarithmic dependence can be shownexactly.Fig. 5 shows h f max i as a function of ω on . Analyti-cal results from eq. (32) agree with numerical findingsfrom stochastic simulations. The maximal polymeriza-tion force h f max i increases with increasing k , see eq. (34),but it remains smaller than the stall force f stall . Stochas-tic simulations show considerable fluctuations of f max ,which are caused by broad and exponentially decayingprobability distributions for f max and which we quantifyby measuring the standard deviation h f i − h f max i .For increasing k , probability distributions become morenarrow and mean field results approach the simulationresults for h f max i . B. Non-zero rescue rate For a non-zero rescue rate ω r , phases of growth, inwhich f ( x ) increases and which last 1 /ω c ( f ) on average,are ended by catastrophes which are followed by phasesof shrinkage. Shrinking phases last 1 /ω r on average, andduring shrinkage the elastic obstacle is relaxed and f ( x )decreases. After rescue, the MT switches back to a stateof growth. In contrast to the case without rescue events,the system can attain a steady state. In this steady state,the average length loss during shrinkage, v − /ω r , equalsthe average length gain during growth, v + ( f ) /ω c ( f ), andthe MT oscillates around a time-averaged stall length h x i ,which is directly related to the time-averaged polymer-ization force by h f i = ( k/F )( h x i − x ). In the following,the steady state dynamics and the average polymeriza-tion force are characterized. We start with an analysisof the full master equations focusing on the stationarystate followed by a dynamical mean field theory, whichcan also be applied to dilution experiments.In the presence of a x -dependent force f ( x ), the masterequations for the time evolution of p + , − ( x, t ) become ∂ t p + ( x, t ) = − ω c ( x ) p + ( x, t ) + ω r p − ( x, t ) − ∂ x ( v + ( x ) p + ( x, t )) (35) ∂ t p − ( x, t ) = ω c ( x ) p + ( x, t ) − ω r p − ( x, t ) + v − ∂ x p − ( x, t ) , (36)which differ from eqs. (1) and (2) by the x -dependence ofgrowth velocity and catastrophe rate. Both growth ve-locity v + ( x ) = v + [ f ( x )] and catastrophe rate ω c ( x ) = ω c { v + [ f ( x )] } become x -dependent via their force-dependence. Therefore, also the force-dependent lengthparameter λ ( f ) from eq. (26) becomes x -dependent viaits force-dependence, λ ( x ) = λ [ f ( x )]. Eqs. (1) and(2) are supplemented by reflecting boundary conditions v + (0) p + (0 , t ) = v − p − (0 , t ) at x = 0, similar to eq. (3).For the steady state, eqs. (35) and (36) are solved onthe half-space x > x = 0, and we can calculate the overall MT lengthdistribution P ( x ) = p + ( x ) + p − ( x ) explicitly, P ( x ) = N (cid:18) v − v + ( x ) (cid:19) e x /λ (0) exp (cid:20)Z xx dx ′ /λ ( x ′ ) (cid:21) (37)with a normalization N − = Z ∞ dx (cid:18) v − v + ( x ) (cid:19) e x /λ (0) e R xx dx ′ /λ ( x ′ ) , (38)where λ ( x ) = λ ( f = 0) in the force-free region x < x and λ ( x ) = λ [ f ( x )] for x > x and, likewise, v + ( x ) = v + ( f =0) for x < x and v + ( x ) = v + [ f ( x )] for x > x . Thisimplies e x /λ (0) e R xx dx ′ /λ ( x ′ ) = e x/λ (0) and, thus, a simpleexponential dependence of P ( x ) for x < x . A similarOPDF has been found for dynamic MTs in the presenceof MT end-tracking molecular motors [36].With increasing length x , also the force f ( x ) increasesand, thus, v + [ f ( x )] decreases and ω c [ f ( x )] grows expo-nentially. If x becomes sufficiently large that the con-dition λ − [ f ( x )] < P ( x ) startsto decrease exponentially. In this length regime the MTundergoes a catastrophe with high probability. Becausethe distribution always decreases exponentially for suffi-ciently large x , a single MT growing against an elasticobstacle is always in the regime of bounded growth re-gardless of how large the values of ω on and ω r are chosen.This behavior is a result of the linearly increasing force,which gives rise to arbitrarily large forces for increasing x in contrast to growth under constant or zero force, wherea MT can either be in a phase of bounded or unboundedgrowth as mentioned above.The behavior is also in contrast to length distribu-tions in confinement between fixed rigid walls, where wefound a transition between exponentially decreasing andincreasing length distributions: The elastic obstacle typ-ically leads to a non-monotonic length distribution witha maximum in the region x > x . (as long as the on-rate ω on and rescue rate ω r are sufficiently large andthe obstacle stiffness k sufficiently small). While res-cue events (and an exponential decrease in the growthvelocity v + [ f ( x )]) cause P ( x ) to increase exponentiallyfor small MT length, catastrophes are responsible foran exponential decrease for large x . The interplay be-tween rescues and catastrophes gives rise to strongly lo-calized probability distributions with a maximum. Figs.6 (a-d) show the steady state distribution P ( x ) obtainedfrom eq. (37) for different values of ω on and ω r . Wechose k = 10 − N / m and x = 10 µ m. In the steadystate, a stable length distribution with a well definedaverage length h x i = R ∞ P ( x ) xdx is maintained al-though the MT is still subject to dynamic instability.The length distributions drop to zero for large x , where λ − ( x ) ∼ − ω c ( x ) /v + ( x ) and ω c ( x ) /v + ( x ) increases expo-nentially with increasing force.The most probable MT length x mp maximizes the sta-tionary length distribution (37). Because v − ≫ v + ( x )and using the approximation of an exponentially decreas-ing growth velocity, v + [ f ( x )] ≈ v + (0) e − f ( x ) , which isvalid for ω on ≫ ω off (see eq. (13)), we obtain a condition λ − ( x mp ) = − ∂ x f ( x mp ) = − k/F or v + ( f mp ) ω r − v − ω c ( f mp ) = − ( k/F ) v − v + ( f mp ) (39)for the corresponding most probable force f mp =( k/F )( x mp − x ).For an exponentially increasing catastrophe rate abovethe characteristic force F , eq. (15), we find f mp ∼ 12 ln (cid:20) v + (0) ω r v − ω c (0) (cid:18) kv − F ω r (cid:19)(cid:21) (40)We can distinguish two limits: (i) For a soft obstacle with kv − /F ω r ≪ f mp is identicalto the critical force f c for MT dynamics under constantforce, see (28), because the right hand side in the condi-tion (39) for f mp can be neglected and we exactly recover condition (27) for f c . The most probable MT length thus“self-organizes” into a “critical” state with f mp ≈ f c ,and a MT pushing against a soft elastic obstacle gen-erates the same force as if growing against a constantforce. This force grows logarithmically in the on-rate ω on and the rescue rate ω r . (ii) For a stiff obstacle with kv − /F ω r ≫ 1, on the other hand, the most probableforce is larger than the critical force, f mp ≫ f c , and theMT growing against a stiff obstacle generates a higher force. This limit can also be realized for vanishing rescuerate ω r , and for kv − /F ω r ≫ f mp ≈ h f max i from eq. (34) with v + (0) ≈ ω on d . Thisforce grows logarithmically in the on-rate ω on . Further-more, if f mp becomes negative for small on-rates and res-cue rates (leading to λ − (0) < − k/F , see eq. (40)) thestationary length distribution has no maximum, see forexample Figs. 6(a,b) at the lowest on-rates.With respect to the MT’s ability to generate force thetwo limits can be interpreted also in the following way: F is the characteristic force above which the catastro-phe rate increases exponentially. For kv − /F ω r ≪ 1, theaverage length loss during a period of shrinkage, v − /ω r ,is much smaller than the length F /k , which is the dis-placement x − x of the elastic obstacle under the charac-teristic force F . Therefore, the MT tip always remainsin the region x > x under the influence of the force fora soft obstacle with kv − /F ω r ≪ 1, whereas it typicallyshrinks back into the force-free region x < x before thenext rescue event for a stiff obstacle kv − /F ω r ≫ 1. Theforce generation by the MT can only be enhanced by res-cue events if rescue takes place under force in the regime x > x . Therefore, we find an increased polymerizationforce f mp ≈ f c ≫ h f max i as compared to the force f max without rescue events discussed in the previous sectiononly in the limit kv − /F ω r ≪ 1, i.e., for a soft obstacle orsufficiently large rescue rate. In the limit kv − /F ω r ≫ f mp ≈ h f max i .By comparing the condition (27) or v + ( f c ) = v − ω c ( f c ) /ω r for the critical force f c , the condition (39) or v + ( f mp ) = v − ω c ( f mp ) /ω r (1 + kv − /F ) < v − ω c ( f mp ) /ω r for the most probable force f mp , and the condition v + ( f stall ) = 0 for the stall force, see eq. (14), it followsthat f c ≤ f mp ≪ f stall (41)i.e., force generated against an elastic obstacle is be-tween critical and stall force but typically well below thestall force, which is the maximal polymerization force inthe absence of catastrophes. Therefore, the stall length x stall = ( F /k ) ln ( ω on /ω off ) + x is always much largerthan the most probable MT length x mp at the maximumof the stationary length distribution, see Fig. 6(a). Thisshows that the dynamic instability reduces the typicalMT length significantly compared to simple polymeriza-tion kinetics.In order to quantify the width of the stationary dis-0tribution P ( x ) we expand the exponential in (37) up tosecond order about the maximum at x mp . To do so wefirst expand λ − ( x ) up to first order: λ − ( x ) ≈ − kF (cid:20) v + ( x mp ) ω r + v − ω c ( x mp ) v + ( x mp ) v − (cid:21) ( x − x mp )(42)where we used v + [ f ( x )] ≈ v + (0) e − f ( x ) , which is validfor ω on ≫ ω off (see eq. (13)), and where we approxi-mated the catastrophe rate by an exponential ω c [ f ( x )] ≈ ω c (0) e f ( x ) according to eq. (15) resulting in ω ′ c [ f ( x )] ≈ kω c [ f ( x )] /F . The prime denotes a derivative with re-spect to the length x . Using the expansion (42) in eq.(37), we obtain an approximately Gaussian length distri-bution P ( x ) ≈ N (cid:18) v − v + ( x ) (cid:19) e x /λ (0) × exp (cid:20) ( x mp − x ) σ (cid:21) exp (cid:20) − ( x − x mp ) σ (cid:21) (43)with a width σ = F k (cid:20) v + ( x mp ) v − v + ( x mp ) ω r + v − ω c ( x mp ) (cid:21) ≈ (cid:18) F k (cid:19) (cid:18) F ω r kv − (cid:19) − (44)where we used the saddle point condition (39) in thelast approximation and the exponential approximations v + [ f ( x )] ≈ v + (0) e − f ( x ) and ω c [ f ( x )] ≈ ω c (0) e f ( x ) .Again we have to distinguish the two limits of soft andstiff obstacles: (i) For a soft obstacle with kv − /F ω r ≪ σ ≈ F v − / kω r . This shows that the width ofthe length distribution decreases with increasing ω r butis roughly independent of the on-rate ω on , as can alsobe seen in the series of simulation results shown in Figs.6. Closer inspection of the simulation results shows thatthe width of the stationary length distribution P ( x ) isslightly decreasing with the on-rate ω on . (ii) For a stiffobstacle with kv − /F ω r ≫ 1, on the other hand, we find σ ≈ ( F /k ) , which only depends on obstacle stiffness.All in all, σ is monotonously decreasing for increasingstiffness k .For a soft obstacle kv − /F ω r ≪ 1, high rescue ratesthus lead to a sharply peaked length distribution P ( x )and suppress fluctuations of the MT length around x = x mp and we expect h x i ≈ x mp to a very good approxi-mation. This property of a sharp maximum in P ( x ) willmake the mean field approximation that is discussed inthe next section very accurate.If the obstacle stiffness k is increased the most probableMT length x mp = x + f mp F /k approaches x , and aconsiderable probability weight is shifted to MT lengths x below x (see Fig. 6 (e)). The average length approachesand finally drops below x . This signals that the forcegenerated by the MT is no longer sufficient to push theobstacle out of its equilibrium position x . The obstacle now serves as a fixed rigid boundary and P ( x ) approachesthe results eq. (21) and (22). The dynamics of a singleMT within confinement can therefore be seen as a specialcase of the dynamics in the presence of an elastic obstacle,i.e., for small ω on and ω r or for large spring constants k .So far we have quantified the generated force by themost probable force f mp . The generated force can alsobe quantified by the average steady-state force h f i = R ∞ f ( x ) P ( x ) dx . Using the stationary distribution (37)with normalization (38) we can calculate h f i ; results areshown in Fig. 7 in comparison with the most probableforce f mp , which is determined numerically from the max-imum of P ( x ), and the stall force f stall in the absence ofdynamic instability from eq. (14). For h f i , there is ex-cellent agreement with stochastic simulations over thecomplete range of parameter values. The results clearlyshow that the dynamic instability reduces the ability togenerate polymerization forces since, even for large val-ues of ω on and ω r , the average force h f i is always smallerthan the stall force. Nevertheless forces up to F ∼ . F can be obtained in the steady state for realistic param-eter values. Comparing h f i and f mp we find h f i ≤ f mp ,and both forces become identical, h f i ≈ f mp , in the limitof large rescue rates or a soft obstacle kv − /F ω r ≪ P ( x ) become sharplypeaked, see Fig. 6. Comparing different combinations of ω on and ω r and the corresponding forces, one finds thatthe influence of the on-rate ω on on force generation ismore significant than the influence of the rescue rate ω r .For ω on = 100 s − , a four fold increase of the rescue rate ω r gives rise to an increase of h f i by a factor of ∼ . ω r = 0 . − , a four fold increase of the on-rate ω on results in an amplification of the force h f i by a fac-tor of ∼ 9. These results can be explained within a meanfield theory presented in the next section. C. Mean field approach (non-zero rescue rate) In the following, we show that we can reproduce manyof the results for the average polymerization force h f i for non-zero rescue rate using a simplified mean fieldapproach. Using the mean field approach, we can alsoaddress the time evolution of the average force h f i , forexample, in dilution experiments. Since the switchingbetween the two states of growth is a stochastic pro-cess, the length x and the force f ( x ) are stochastic vari-ables. Therefore, the velocity of growth v + [ f ( x )] and thecatastrophe rate ω + [ f ( x )] also become stochastic vari-ables which, in the steady state, fluctuate around theiraverage values. Within the mean field approach we ne-glect these fluctuations and use h v + [ f ( x )] i = v + ( h f i )and h ω + [ f ( x )] i = ω + ( h f i ). In the mean field approxi-mation, the average time in the growing state is givenby 1 /ω c ( h f i ) and the average growth velocity is v + ( h f i ).The average time in a shrinking state is 1 /ω r . There-fore, the mean field probabilities to find the MT grow-ing or shrinking are p + = ω r / [ ω r + ω c ( h f i )] and p − =1 FIG. 6: Stationary MT length distribution P ( x ) in the steady state for growth against an elastic obstacle with ω on =25 s − , 50 s − , 75 s − , 100 s − and different values of ω r . We set k = 10 − N / m and x = 10 − m. (a) ω r = 0 . 03 s − . (b) ω r = 0 . 05 s − . (c) ω r = 0 . − . (d) ω r = 0 . − . Dashed line represents x . In picture (a) the stall length x s for ω on = 25 s − ,obtained from simple polymerization kinetics, is indicated by an arrow. (e): P ( x ) for ω on = 50 s − , ω r = 0 . 05 s − and differentvalues of the spring constant k . FIG. 7: Average steady state force h f i as a func-tion of ω on for growth against an elastic obstacle with ω r = 0 . 03 s − ( ⊡ ) , . 05 s − ( (cid:4) ) , . − ( ⊙ ) , . − ( • ) and k =10 − N / m. Solid lines: h f i = R ∞ f ( x ) P ( x ) dx with P ( x ) givenby eqs. (37) and (38). Dashed lines: h f i calculated from meanfield equation (46). Dotted lines: most probable force f mp ,measured in simulations, for ω r = 0 . 03 s − and ω r = 0 . 05 s − .Also shown is the dimensionless stall force f stall obtained fromsimple polymerization kinetics (14). ω c ( h f i ) / [ ω r + ω c ( h f i )], respectively. This results in thefollowing mean field average velocity v of a single MTunder force: v ( h f i ) = v + ( h f i ) ω r − v − ω c ( h f i ) ω r + ω c ( h f i ) . (45)In the steady state the barrier is pushed so far that h f i stalls the MT. We require v ( h f i ) = 0 and obtain thecondition v + ( h f i ) ω r = v − ω c ( h f i ) (46)for the stationary state. This condition corresponds toa force, where the average length gain during growth, v + ( h f i ) /ω c ( h f i ), equals the average length loss duringshrinking, v − /ω r . From the mean field equation (46),the average steady state force, h f i can be calculated asa function of ω r and ω on . The average length h x i canbe obtained from the relation h f i = ( k/F )( h x i − x ). FIG. 8: Average steady state force h f i as a func-tion of ω on for growth against an elastic obstacle with ω r = 0 . 03 s − ( ⊡ ) , . 05 s − ( (cid:4) ) , . − ( ⊙ ) , . − ( • ) and k =10 − N / m. Solid lines: h f i = R ∞ f ( x ) P ( x ) dx with P ( x )given by eqs. (37) and (38). Dashed lines from bottomto top: h f i calculated from mean field equation (46) for ω r = 0 . 03 s − , . 05 s − , . − and 0 . − . Results obtained from the mean field equation (46) matchnumerical results from stochastic simulations very well asshown in Fig. 7.The mean field condition (46) is identical to the condi-tion (27) for the critical force f c for MT dynamics underconstant force such that h f i = f c , (47)which can be interpreted as “self-organization” of theaverage MT length or the average force to the “critical”state. Therefore, the curves presented in Fig. 7 for h f i are identical to the curves shown in Fig. 4 (b) for f c .This also allows us to take over the results we derivedfor the critical constant force f c . Using the approxi-mation of an exponentially decreasing growth velocity, v + [ f ( x )] ≈ v + (0) e − f ( x ) , which is valid for ω on ≫ ω off (see eq. (13)), and an exponentially increasing catastro-phe rate above the characteristic force F , eq. (15), we2find h f i ∼ 12 ln (cid:18) v + (0) ω r v − ω c (0) (cid:19) . (48)which is identical to the result (28) for f c .Comparing with the stall force and the most probableforce, we use relation (41) and find h f i = f c ≤ f mp ≪ f stall . (49)In the limit of a soft obstacle, kv − /F ω r ≪ 1, the averageforce h f i approaches the most probable force h f i ≈ f mp ,whereas the mean field average force h f i is always smallerthan the stall force f stall in the absence of dynamic in-stability from eq. (14).Finally, we discuss the limits of validity of the meanfield approximation. The mean field approximation isbased on the existence of a pronounced maximum inthe stationary MT length distribution P ( x ), which con-tains most of the weight of the probability density P ( x ).It breaks down if this maximum broadens or vanishes,such that a considerable amount of probability densityis shifted below x into the regime of force-free growth.Then the MT typically shrinks into the force-free region x < x during phases of shrinkage such that the grow-ing phase explores the whole range of forces starting from f = 0 up to f > h f i , and the approximation of a constantaverage force f ≈ h f i during growth is no longer fulfilled.For small spring constants k or large values of ω r , thelength distribution P ( x ) assumes a Gaussian shape withwidth σ , see eqs. (43) and (44). When k is increased fora fixed combination of ω on and ω r , the average length h x i approaches x as h x i − x ∝ /k , whereas the width σ ofthe length distribution only decreases as σ ∝ / √ k in theregime of a soft obstacle kv − /F ω r ≪ 1, as can be seenfrom eq. (44). Therefore, an increasing amount of proba-bility density is shifted below x , where no force is actingon the MT ensemble (see Figs. 6)(a) and (e)). The meanfield approximation is only valid for spring constants k which fulfill h x i− x ≫ σ/ ω on and ω r . With h f i = ( k/F )( h x i − x ) this is equivalent to acondition h f i ≫ kσ F ≈ (cid:18) F ω r kv − (cid:19) − / (50)according to eq. (44). This condition can only be fulfilledin the limit of a soft obstacle with kv − /F ω r ≪ 1. Forthe validity of the mean field approximation we thereforerecover the condition that the average length loss duringa period of shrinkage, v − /ω r , is much smaller than thetypical displacement F /k of the elastic obstacle underthe characteristic force F . Then the MT tip always re-mains in the region x > x under the influence of theforce. FIG. 9: (a): Average force h f i ( t ) as a function of time for k = 10 − N / m, ω r = 0 . 05 s − and different values of ω on .Symbols: time dependent average force h f i ( t ) measured insimulations. Solid lines: time dependent average force trajec-tory calculated from eq. (51). (b): Average force h f i ( t ) as afunction of time for k = 10 − N / m, ω on = 50 s − and differentvalues of ω r . Symbols: time dependent average force h f i ( t )measured in simulations. Solid lines: time dependent averageforce trajectory calculated from eq. (51). D. Dynamics and dilution experiments Within the mean field approach we can also derive ananalytical time evolution of the average time-dependentforce h f i ( t ). The time evolution is based on eq. (45),which gives a mean field approximation for the averageMT velocity v ( h f i ) as a function of the average force.On the other hand, the average MT growth velocity isrelated to the time derivative of the average force by ddt h f i = kF ddt h x i = kF v ( h f i ) (51)Using eq. (45) for v ( h f i ), this gives a mean field equationof motion for h f i similar to eq. (29) in the absence ofrescue events. Integrating this equation numerically weobtain mean field trajectories for the average force h f i ( t )as a function of time t . Figs. 9 shows such trajectoriesfor k = 10 − N / m and a initial condition h f i (0) = 0 at t = 0. Also shown in Figs. 9 are results from stochas-tic simulations, which show excellent agreement with themean field trajectories.We now address the question of how fast a single MTresponds to external changes of one of its growth pa-rameters. Here we focus on fast dilution of the tubulinconcentration, which is directly related to the tubulin on-rate ω on . In vivo tubulin concentration can be changedby tubulin binding proteins like stathmin [37], while in invitro experiments, the tubulin concentration can be di-luted within seconds [38]. In the following we give a meanfield estimate of the typical time scale, which governs thereturn dynamics of the MT back to a new steady stateafter the tubulin on-rate is suddenly decreased. In theinitial steady state the average velocity v ( h f i i ) vanishesand the average polymerization force h f i i (and, thus, theaverage length h x i i ) can be calculated from the condition v + ( h f i i ) ω r = v − ω c ( h f i i ), cf. eq. (46), for a given com-bination of ω on and ω r . If ω on is suddenly decreasedthis leads to a sudden decrease in the growth velocity to3 FIG. 10: Average force h f i ( t ) as a function of time t . Symbolsare results obtained from simulations. We set k = 10 − N / m, ω r = 0 . 05 s − and ω on = 75 s − . At t = 20000 s, ω on is diluteddown to ω on = 50 s − . Solid line represents a fit with an ex-ponential decay (53) to the simulated data with fit parameter τ d = 1762 s. Dashed lines indicate the average force in theinitial state f i before dilution and in the new final state f f after dilution. ˜ v + ( f ) < v + ( f ) and an increase of the catastrophe rateto ˜ ω c ( f ) > ω c ( f ), resulting in a negative average veloc-ity v ( h f i ) = [˜ v + ( h f i ) ω r − v − ˜ ω c ( h f i )] / [ ω r + ˜ ω c ( h f i )] < v ( h f i ) < 0. This relaxesthe force from the elastic obstacle, i.e., h f i ( t ) starts todecrease from the initial value f i ≡ h f i i . With decreasingaverage force h f i ( t ), the average growth velocity v ( h f i ( t ))increases again (because ˜ v + increases and ˜ ω c decreases)until the steady state condition ˜ v + ( h f i f ) ω r = v − ˜ ω c ( h f i f )holds again and a new steady state force h f i f < h f i i isreached (s. Fig. 10).The relaxation dynamics to the new steady state af-ter tubulin dilution is therefore governed by the averagevelocity v ( h f i ) given by eq. (45). To extract a charac-teristic relaxation time scale, we expand the average ve-locity v ( h f i ) to first order around the final steady-statepolymerization force f f ≡ h f i f , which is the solution ofeq. (46) with ω r and the decreased tubulin on-rate ω on ,which takes its dilution value. Using v ( f f ) = 0 one findsin first order v ( h f i ) ≈ − (cid:20) v + ( f f ) ω r + v − ω ′ c ( f f ) ω r + ω c ( f f ) (cid:21) ( h f i − f f ) (52)where the prime denotes the derivative with respect tothe force. In the last approximation we used the meanfield condition eq. (46) and v + [ f ( x )] ≈ v + (0) e − f ( x ) ,which is valid for ω on ≫ ω off (see eq. (13)). This ex-pansion is only valid for average forces close to the newaverage polymerization force f f . Using this expansion,the time evolution (51) of the average force after dilutionexhibits an exponential decay h f i ( t ) = f f + ( f i − f f ) e − t/τ d (53)with a characteristic dilution time scale τ d = F k ω r + ω c ( f f ) v + ( f f ) ω r + v − ω ′ c ( f f ) ≈ F k ω r + ω c ( f f )2 v − ω c ( f f ) (54)where we approximated the catastrophe rate by an expo-nential ω c [ f ( x )] ≈ ω c (0) e f ( x ) according to eq. (15), and we used the mean field condition eq. (46). In the limit ω c ( f f ) ≫ ω r , i.e., at forces f f ≫ 1, we obtain the sim-ple result τ d ≈ F / v − k . In general, the relaxation time τ d is proportional to the square σ of the width of thestationary distribution, cf. eq. (44): A narrow length dis-tribution gives rise to fast relaxation to the new averageforce. VII. EXPERIMENTAL CATASTROPHE MODEL So far we have employed the catastrophe rate derivedby Flyvbjerg et al. , to which we will refer as ω c, Flyv inthe following. This expression for the catastrophe ratewas based on theoretical calculations of the inverse pas-sage time to a state with a vanishing GTP-cap, see eq.(10). In order to investigate the robustness of our resultswith respect to changes of the catastrophe model, we nowinvestigate an alternative expression for the catastropherate that has been obtained from experimental results.Throughout this section, we focus on the third confine-ment scenario of an elastic obstacle, and we compare re-sults from the two different catastrophe models for zerorescue rate ω r = 0 and non-zero rescue rate ω r > 0. Inaddition, we restrict the comparison to mean field results,since numerical and stochastic calculations match meanfield results well over the complete range of parameters(see Sec. VI).Experimentally, it has been found that the averagetime h τ + i spent in a growing state is a linear function ofthe growth velocity v + [39]. The force-dependent catas-trophe rate is then given by ω c, Jans ( f ) = 1 av + ( f ) + b (55)with constant coefficients a = 1 . · s m − and b = 20 s. At v + ( f ) = 0, ω c, Jans ( f ) = 0 . 05 s − andfor v + ( f ) = − b/a , the catastrophe rate ω c, Jans ( f ) di-verges. This is in contrast to the theoretical model, where ω c, Flyv ( f ) is finite for all v + ( f ). Also ω c, Jans ( f ) increasesexponentially for forces F > F or f > 1. This com-mon feature is essential and lead to similar results forboth catastrophe models. In Fig. 11, both catastropherates are shown as a function of the dimensionless force f . The catastrophe model (55) is based on experimentaldata and, thus, is phenomenological. It assumes neithera purely chemical model, as in the model by Flyvbjerg etal. , nor a chemo-mechanical model in the sense of “struc-tural plasticity” [24]. A. Vanishing rescue rate We start with the case ω r = 0 without rescue events,and we calculate the average maximal polymerizationforce within the experimental catastrophe model usingthe self-consistent mean field eq. (32), which holds inde-pendently of the choice of catastrophe model (see Sec.4 FIG. 11: Catastrophe rate ω c ( f ) as a function of force f for ω on = 50 s − and ω on = 75 s − . Solid lines: ω c, Flyv from thecatastrophe model by Flyvbjerg et al. Dashed lines: ω c, Jans from the experimental catastrophe model by Janson et al . VI A). As for the catastrophe by Flyvbjerg et al. , wehave ω c, Jans τ ≫ v + ( h f i ) < − b/a , and eq. (32) can be solved explicitlyfor h f max i in this limit. We find an average maximalpolymerization force h f max i ≈ ln (cid:16)h(cid:0) A + B (cid:1) / − A i(cid:17) (56)with A ≡ ( ω on /ω off − adω off − ( ω on /ω off − b − τ τB ≡ ( ω on /ω off − adω on τ . Since ω on /ω off ≫ 1, eq. (56) can be approximated by h f max i ≈ ln ( ω on /ω max ) (57)with ω max ≡ τ ω off h ( adω off − b ) + 4 adω off τ i / − [ adω off − b ](58)For realistic parameter values, we have τ ≫ adω off ≥ b ,and recover the expression (34) derived using the Flyvb-jerg catastrophe model: h f max i ≈ 12 ln (cid:18) ω adω off τ (cid:19) ≈ 12 ln (cid:18) ω on dkF ω c, Jans (0) (cid:19) . (59)In Fig. 12 (a), h f max i as obtained from eq. (32) with theFlyvbjerg catastrophe model and eq. (56) with the ex-perimental catastrophe model are shown as a function of ω on . Results match qualitatively and quantitatively well,although they are obtained from two different catastro-phe models. The maximal polymerization force h f max i always remains smaller than the stall force f stall . 123 25 50 75 1000123 25 50 75 100 FIG. 12: (a): Average maximal polymerization force h f max i as a function of ω on and ω r = 0 for k = 10 − N / m, 10 − N / mand 10 − N / m (top to bottom) . Dotted line: dimensionlessstall force f stall . Solid lines: h f max i obtained from ω c, Flyv (eq.(32)). Dashed lines: h f max i obtained from ω c, Jans (eq. (56)).(b): Average steady state force h f i as a function of ω on . k =10 − N / m, ω r = 0 . − (top) and ω r = 0 . 03 s − (bottom).Solid lines: h f i obtained from ω c, Flyv (eq. (46)). Dashed lines: h f i obtained from ω c, Jans (eq. (60)). Dotted line: dimension-less stall force f stall . B. Non-zero rescue rate Now we compare both catastrophe models for a non-zero rescue rate, and we calculate the average steadystate force. For the experimental catastrophe rate (55),the mean field equation (46) can be solved explicitly, andthe average steady-state force h f i is given by h f i = ln ( ω on /ω av ) , (60)with ω av ≡ (cid:18) b ad (cid:19) + v − ω r ad ! / − b ad + ω off (61)Again h f i < f stall since ω av > ω off . Fig. 12 (b) show h f i as a function of ω on . For realistic parameter values,we have v − /ω r ≫ b /a and ( v − /ω r ad ) / ≫ ω off , andrecover the expression (48) derived using the Flyvbjergcatastrophe model: h f i ≈ 12 ln (cid:18) ω ω r ad v − (cid:19) ≈ 12 ln (cid:18) v + (0) ω r v − ω c, Jans (0) (cid:19) . (62)In Fig. 12 (b), results for h f i from both catastrophemodels are shown as a function of on-rate ω on . Theaverage steady state force obtained from ω c, Flyv is al-ways slightly larger than h f i obtained from ω c, Jans , since ω c, Jans ( f ) > ω c, Flyv ( f ) for forces smaller than or compa-rable to F . Otherwise, both results agree qualitativelyand quantitatively well. VIII. FORCE-VELOCITY RELATION Finally, we discuss the influence of the force-velocityrelation on the MT dynamics. We restrict our analysis tomean field results obtained for the third scenario, i.e., the5elastic obstacle. A change in the force-velocity relationdirectly modifies the velocity of growth v + ( f ), but it alsoaffects the catastrophe rate ω c ( v + ( f )), which are bothcrucial parts of the MT dynamics. In the following, weemploy a more general form of the force-velocity relation,which is consistent with thermodynamic constraints, andwe show that our results are robust with respect to thisgeneralization.In their investigation of experimental data Kolomeisky et al. used a generalized growth velocity v + ( f, θ ) = d { ω on exp( − θf ) − ω off exp[(1 − θ ) f ] } , (63)which depends on a dimensionless “load distribution fac-tor” θ [40]. The load distribution factor θ ∈ [0 , 1] deter-mines whether the on- or off-rates are affected by externalforce, while keeping the ratio of overall on- and off-rateunaffected. Under force both the tubulin on-rate ω on and the tubulin off-rate ω off now acquire an additionalBoltzmann-like factor. For θ = 1, we obtain again v + ( f )as given by eq. (13). The dimensionless stall force is un-affected by θ and is still given by f stall = ln ( ω on /ω off ). A. Vanishing rescue rate We use the generalized force-velocity relation v + ( f, θ )given by eq. (63) and the catastrophe rate ω c, Flyv ( f ) inorder to calculate the average maximal polymerizationforce h f max i from the self-consistent mean field eq. (32).In Fig. 13 (a), h f max i is shown as a function of the loaddistribution factor θ for k = 10 − N / m and different val-ues of ω on . At θ = 1, the maximal force h f max i equals themaximal polymerization force obtained with v + ( f ) fromeq. (13). With decreasing θ , h f max i increases but remainsbelow the dimensionless stall force. The growth velocity v + ( f, θ ) increases with decreasing θ for a fixed force f and, therefore, the maximal polymerization force h f max i increases. For high tubulin on-rates ω on = 75 − 100 s − and small θ ≈ , . . . , . 2, the maximal polymerizationforce h f max i approaches the dimensionless stall force. B. Non-zero rescue rate For non-zero rescue rate, the average steady state force h f i is calculated from the mean field eq. (46), where weuse the force-velocity relation v + ( f, θ ) (eq. 63) and thecatastrophe rate ω c, Flyv ( f ). In Fig. 13 (b), results for h f i are shown as a function of θ for k = 10 − N / m, ω r = 0 . 05 s − and different values of ω on . At θ = 1, h f i equals the average steady state force obtained with avelocity v + ( f ) taken from eq. (13). The average steadystate force h f i increases with decreasing θ , as explainedabove. For high tubulin on-rates ω on = 75 − 100 s − andsmall θ ≈ , . . . , . h f i again approaches the dimensionless stall force butremains smaller. FIG. 13: (a): Solid lines: Average maximal polymerizationforce h f max i as a function of θ for k = 10 − N / m and differentvalues of ω on . Dashed line: Dimensionless stall force f stall for ω on = 100 s − . (b): Solid lines: Average steady state force h f i as a function of θ for k = 10 − N / m, ω r = 0 . 05 s − anddifferent values of ω on . Dashed line: Dimensionless stall force f stall for ω on = 100 s − . IX. DISCUSSION AND CONCLUSION We studied MT dynamics in three different confiningscenarios: (i) confinement by fixed rigid walls, (ii) anopen system under constant force, and (iii) MT growthagainst an elastic obstacle with a force that depends lin-early on MT length. These three scenarios representgeneric confinement scenarios in living cells or geome-tries, which can be realized experimentally in vitro . Forall three scenarios, we are able to quantify the MT lengthdistributions. In scenario (iii) of an elastic obstacle,stochastic MT growth also gives rise to a stochastic force.For this model, we also quantify the average polymeriza-tion force generated by the MT in the presence of thedynamic instability.The parameter λ , see (4) and (26), governs the MTlength distributions in confinement by fixed rigid walls,and under a constant force. For confinement by rigidwalls we introduced a realistic model for wall-inducedcatastrophes. There is a transition from exponentially in-creasing to exponentially decreasing length distributionsif λ changes sign. The average MT length is increasing forincreasing on-rate and increasing rescue rate, as shownin Figs. 2. Wall-induced catastrophes lead to an overallincrease in the average catastrophe frequency, which wequantify within the model.For MT growth under a constant force, there exists atransition between bounded and unbounded growth as inthe absence of force. This transition takes place wherethe parameter λ ( f ) changes sign. Under force, the tran-sition to unbounded growth is shifted to higher on-ratesor higher rescue rates and determines a critical force f c ,see Figs. 4.MT growth under a MT length-dependent linear elasticforce allows for regulation of the generated polymeriza-tion force by experimentally accessible parameters suchas the on-rate or the rescue rate. The force is no longerfixed but a stochastically fluctuating quantity becausethe MT length is a stochastic quantity. For zero rescuerate, i.e., in the absence of rescue events, we find that6the average maximal polymerization force h f max i before acatastrophe depends logarithmically on the tubulin con-centration and is always smaller than the stall force inthe absence of dynamic instability as shown in Fig. 5.For a non-zero rescue rate, we find a steady state lengthdistribution, which becomes increasingly sharply peakedfor increasing rescue rate and is tightly controlled bymicrotubule growth parameters, see Figs. 6. Interest-ingly, the average microtubule length self-organizes suchthat the average steady state polymerization force h f i equals the critical force for the boundary of bounded andunbounded growth, h f i = f c . Because of the sharplypeaked MT length distribution, the average polymeriza-tion force h f i can be calculated rather accurately withina mean field approach as can be seen in Figs. 7 and 8.The average polymerization force is always smaller thanthe stall force in the absence of dynamic instability.Within this mean field approach, we can also describethe dynamics of the average force, see Figs. 9. This mightbe useful in modeling dilution experiments, where the re-sponse to sudden changes in the on-rate is probed. For this type of experiment, we estimate typical polymeriza-tion force relaxation times.Finally, we show that our findings are robust againstchanges of the catastrophe model (Figs. 12) as long as thecatastrophe rate increases exponentially above a char-acteristic force and that results are also robust againstvariations of the relation between force and polymeriza-tion velocity in the growing phase (Figs. 13), which areobtained by introducing a load distribution factor. X. 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