Effects of microtubule mechanics on hydrolysis and catastrophes
EEffects of microtubule mechanics on hydrolysis andcatastrophes
N M¨uller and J Kierfeld
Department of Physics, TU Dortmund University, 44221 Dortmund, GermanyE-mail: [email protected], [email protected]
Abstract.
We introduce a model for microtubule mechanics containing lateral bondsbetween dimers in neighboring protofilaments, bending rigidity of dimers, and repulsiveinteractions between protofilaments modeling steric constraints to investigate theinfluence of mechanical forces on hydrolysis and catastrophes. We use the allostericdimer model, where tubulin dimers are characterized by an equilibrium bending angle,which changes from 0 ◦ to 22 ◦ by hydrolysis of a dimer. This also affects the lateralinteraction and bending energies and, thus, the mechanical equilibrium state of themicrotubule. As hydrolysis gives rise to conformational changes in dimers, mechanicalforces also influence the hydrolysis rates by mechanical energy changes modulatingthe hydrolysis rate. The interaction via the microtubule mechanics then gives rise tocorrelation effects in the hydrolysis dynamics, which have not been taken into accountbefore. Assuming a dominant influence of mechanical energies on hydrolysis rates,we investigate the most probable hydrolysis pathways both for vectorial and randomhydrolysis. Investigating the stability with respect to lateral bond rupture, we identifyinitiation configurations for catastrophes along the hydrolysis pathways and valuesfor a lateral bond rupture force. If we allow for rupturing of lateral bonds betweendimers in neighboring protofilaments above this threshold force, our model exhibitsavalanche-like catastrophe events.PACS numbers: 87.16Ka, 87.16.A-, 87.16.Ln Submitted to:
Phys. Biol.
Keywords: microtubules, dynamic instability, hydrolysis, catastrophes
1. Introduction
Microtubule (MT) dynamics is essential for many cellular processes, such as celldivision [1], intracellular positioning processes [2], e.g. positioning of the cell nucleus[3] or chromosomes during mitosis, establishing cell polarity [4], or regulation of cellshapes [5, 6]. An important feature of MT dynamics is their dynamic instability,which is the stochastic switching between phases of growth and rapid shrinkage [7].Polymerization phases terminate in catastrophes, where the MT switches to a state ofrapid depolymerization. Depolymerization phases are terminated by rescue events, in a r X i v : . [ q - b i o . S C ] J un ffects of microtubule mechanics on hydrolysis and catastrophes ◦ for straight GTP-dimers to 22 ◦ for curved GDP-dimers [13].In the lattice model, both states of the tubulin dimers are slightly bent and hydrolysisweakens the lateral interaction strength between dimers in neighboring protofilaments[9, 10, 11]. At present, the experimental evidence is not sufficient to rule out any of thetwo models. Also combinations of both models are possible, where hydrolysis affectsboth the intra-dimer angle (allosteric) and the interaction strength between laterallyneighboring dimers (lattice).The influence of tubulin dimer hydrolysis onto the mechanics of the MT latticesuggests that, vice versa, mechanical forces and torques acting on tubulin dimers or theMT structure also affect hydrolysis rates. This effect has not been considered in theliterature before, and we will use the allosteric model with intra-dimer bending to studythis coupling of hydrolysis and MT mechanics. Similar investigations should be done inthe future for the lattice model.Moreover, recent experiments also suggest that catastrophes are initiated in a multi-step process, which involves not a single rate-limiting event but a chain of at least twoevents, which are probably related to hydrolysis events [14]. Mechanical forces mightbe one possible way to orchestrate such a chain of hydrolysis events. Our results on theinfluence of MT mechanics on the hydrolysis pathway will give hints on the mechanismof catastrophe initiation.There are numerous models for the growth dynamics of MTs, which either ignorecatastrophe events and focus on the growing phase of MT dynamics [15, 16, 17, 18, 19],or which include catastrophes as explicit stochastic switching events on a macroscopiclevel following [20]. These approaches include details of the hydrolysis mechanism intothe model for the catastrophe rate [21, 22, 23, 24]. One focus of these approaches wasthe explanation of the behavior of MT growth dynamics under force. ffects of microtubule mechanics on hydrolysis and catastrophes ffects of microtubule mechanics on hydrolysis and catastrophes
2. Methods
Each protofilament consists of αβ -tubulin heterodimers of length d = 8nm. MTs consistof 13 protofilaments forming a hollow tube of (outer) radius R o = 12 . β -tubulin and is hydrolyzed in the polymerized state.In the following, we will employ the so-called allosteric model and assume that GTP-tubulin dimers are straight and assemble into straight protofilaments, whereas GDP-tubulin tends to form curved protofilaments with curvature radius 21nm in the typicalram’s horn configurations [13]. Protofilament curvature can be caused both by inter-and intra-dimer bending [9, 12]. If we assume that there is only intra-dimer bendingduring hydrolysis, a GDP-dimer acquires a bent configuration with an equilibrium angleof 22 ◦ . The tube formed by the 13 protofilaments is stabilized by lateral bonds betweentubulin dimers in neighboring protofilaments. These bonds are assumed to be identicalfor GTP- and GDP-dimer within the allosteric model.GTP-tubulin assembles into protofilaments (i.e., has a high polymerization rateand a low depolymerization rate), whereas GDP-tubulin tends to disassemble (i.e., hasa depolymerization rate much higher than the polymerization rate) [39]. Within the MTprotofilaments, GTP-tubulin is hydrolyzed into GDP-tubulin after a certain waiting timedepending on the exact hydrolysis mechanism and hydrolysis rates. This gives rise toMTs consisting of an unhydrolyzed GTP-cap at the growing end and hydrolyzed GDP-tubulin behind this cap. Comparison of recent experimental measurements of MT lengthfluctuations on the nanometer scale and a random hydrolysis model suggests a cap sizeof ∼
40 GTP-dimers corresponding to 3 layers of GDP-dimers [40, 43]. The structureof the cap will depend on the hydrolysis mechanism. For random hydrolysis, wheredimers hydrolyze completely independent from each other [7, 43], the cap boundary isnot sharp (in the sense that there are many GTP-/GDP-boundaries in the cap region)with an exponentially decaying GTP-tubulin concentration. For a vectorial mechanism,where hydrolysis propagates as a “wave” because hydrolysis can only happen if theneighboring dimer is already in its hydrolyzed GDP-state [44], the cap boundary issharp by definition. For cooperative models incorporating both mechanisms [21, 22, 36], ffects of microtubule mechanics on hydrolysis and catastrophes θ ( i )1 θ ( i )3 θ ( i )4 θ ( i )1 - θ ( i )2 - θ ( i )3 - θ ( i )2 (b)(a) (c) z x xy l Rφ (1) v (1) j v (13) j v (12) j w (12) j w (13) j w (1) j αβββαββα αββαββαβα α plus endminus end plus endminus end Figure 1.
Schematic representation of the MT model. (a) Only the αβ − junctionof tubulin dimers can bend. Straight βα -segments are grouped to stiff rods of length d and diameter d/ θ ( i ) j . Hydrolysis causes a shift of the preferred bending angle θ ( i )0 ,j from 0 ◦ to 22 ◦ at the αβ − junction. (c) Lateral bonds are modeled by springinteractions. The interaction points lie on the surface of the tubulin dimers. we get a cap structure consisting of many GTP-islands with a characteristic dependenceof the island size distribution and cap size on the cooperativity parameter [36, 37].Because GTP-tubulin dimers are straight, they stabilize the MT as they donot stress lateral bonds between protofilaments. Intrinsically curved GDP-dimers,on the other hand, stress the lateral bonds and destabilize the MT lattice. In thefollowing, we consider a MT with a fixed length, a stabilizing GTP-cap of fixed sizeand with a sharp cap boundary for simplicity (typically 3 layers of GTP-dimers). Weformulate a mechanical model, which (i) incorporates dimer bending and lateral bondsbetween neighboring protofilaments, (ii) gives mechanically stable tubular structures inagreement with experimentally observe MT structures, and (iii) is as simple as possible.MTs are hollow cylinders consisting of 13 protofilaments with outer radius R o =12 . R i = 8 . d/ M + 1 αβ -tubulin dimers, which are longitudinally bound by intra-dimer andinter-dimer bonds. The total length of the MT is L = ( M + 1) d with the dimer length d . We only allow for intra-dimer bending at the αβ -junction within each dimer, seefigure 1a. The MT conformation is then described by bending angles of the αβ -dimerin the i th protofilament ( i = 1 , ...,
13) in the j th layer, where j = M corresponds to ffects of microtubule mechanics on hydrolysis and catastrophes Table 1.
MT geometry parameters.dimer size d d/ R o R i R l r z the GTP-capped MT end and j = 1 to the GDP-end, see figure 1b. We only allow forintra-dimer bending and regard the inter-dimer bonds as straight and fixed.For the mathematical description, we can then group the β -tubulin in layer j − α -tubulin in layer j into a straight “rod” j oriented with angle θ ( i ) j in protofilament i .In the following we will describe the MT configuration in terms of these rods, as shownin figure 1a. We fix θ ( i )0 = 0 and parametrize the MT configuration by M angles θ ( i ) j ( j = 1 , ..., M ) for each protofilament i . Vice versa, the bendable junction between rods j − j in protofilament i corresponds to the bendable αβ -junction of the dimer j with bending angle ∆ θ ( i ) j = θ ( i ) j − θ ( i ) j − . We note that the last β -tubulin monomer of thelast tubulin dimer at the plus end in layer j = M is not contained in any rod. Therefore,the hydrolysis state of the last tubulin dimers at the plus end of each protofilament willhave no effect in our model, as further discussed below.The minus end of every protofilament is fixed in the xy -plane. For simplicity, wedo not take into account the helical pitch. We choose the MT axis as z -axis and define v ( i )0 = R cos φ ( i ) R sin φ ( i ) (1)as starting points of the first layer of rods. The polar angle of the i th protofilament is φ ( i ) = i · π/
13. Thus, the ending position v ( i ) j of the i th rod in the j th layer is given by v ( i ) j = v ( i ) j − + d cos φ ( i ) sin θ ( i ) j sin φ ( i ) sin θ ( i ) j cos θ ( i ) j . (2)We assume that dimers or rods can only be displaced in radial direction, i.e., all polarangles φ ( i ) j = φ ( i ) = i · π/
13 are fixed and independent of the layer number j . Theazimuth angles θ ( i ) j of rods describe the radial displacements of dimers and are theconfigurational variables that are determined by mechanical energy minimization.We consider MT configurations that contain m complete layers of hydrolyzed GDP-tubulin and m + 1 complete layers of GTP-tubulin ( M + 1 = m + m + 1); we focus on4 layer GTP-caps with m = 3. In accordance to the allosteric model we assume GTP-dimers to be straight with an equilibrium angle θ = 0 ◦ and GDP-tubulin bent withan equilibrium angle θ = 22 ◦ . We consider deviations from these preferred equilibrium ffects of microtubule mechanics on hydrolysis and catastrophes κ . Therefore, we define the longitudinal bending energy stored in the MT lattice as E long = E bend = M (cid:88) j =1 13 (cid:88) i =1 κ (cid:16) θ ( i ) j − θ ( i ) j − − θ ( i )0 ,j (cid:17) , (3)with θ ( i )0 = 0 fixed and the bending rigidity κ of individual tubulin dimers.To obtain a tubular structure, we have to introduce lateral bonding. A reasonableand simple assumption is that only dimers in neighboring protofilaments interactlaterally (reminiscent of some form of molecular bonds forming). Mechanical stabilitywill require to introduce two contributions to these lateral interactions: one attractivebinding contribution in form of a harmonic bond with a finite rest length andan additional hard-core-like repulsion, which basically prevents configurations withinterpenetrating dimers.For the attractive binding contribution, we assume that neighboring tubulin dimersinteract via bonds between specific interaction points on the surface of the dimers. Weuse only one lateral bond per dimer, i.e., each dimer or each rod has two interactionpoints, which we locate on the surface at the top of each rod, see figure 1a. We modeltubulin dimers and rods with a spherical cross section of radius d/ l , see figure 1c. We considerlateral stretching from the rest length l to be harmonic with a characteristic springconstant k . The rest length l can be determined by geometry; with the mean radius R = ( R o + R i ) / . d/
4, weobtain l = 2(sin π/ R − ( d/ π/ (cid:39) . . (4)If the equilibrium angle θ ( i )0 ,j of a tubulin dimer changes from 0 ◦ to 22 ◦ by hydrolysis,this will strain the lateral bonds in the MT lattice. Because the interaction points forlateral bonds are located at the top of each rod (or at mid-height of each dimer, seefigure 1a), hydrolysis of the last β -tubulin monomer of the last tubulin dimer at theplus end ( j = M ) does not strain the last bond. Therefore, the hydrolysis state of thetubulin dimers at the plus end of each protofilament at j = M will have no effect on MTmechanics in our model. If we consider MT-configurations with GTP-caps consistingof m + 1 complete layers of GTP-tubulin, only m GTP-layers have an effect on MTmechanics. We therefore ignore the last layer of β -tubulin monomers at the plus end inthe following, in particular in the MT hydrolysis patterns shown in figures 5 – 8 below.We point out that the last tubulin layer has a stabilizing effect also in our model asthe stabilizing lateral bonds are contained in our model (at the top of the last rod, seefigure 1a). However, this stabilizing effect is independent of the hydrolysis state of thelast layer and only depends on the hydrolysis states of the layers below. This point ofview can be further justified because GTP-dimers at the plus end cannot hydrolyze asthe last β -monomer has no inter-dimer contact [45].In addition to the attractive lateral binding forces, we apply a strongly repulsivehard core interaction to avoid overlapping of tubulin dimers, which we model by a ffects of microtubule mechanics on hydrolysis and catastrophes r − -potential. Accordingly, we define the total lateral mechanical energy by E lat = E spring + E hc = M (cid:88) j =1 13 (cid:88) i =1 k (cid:104)(cid:12)(cid:12)(cid:12) v ( i ) j − w ( i ) − ( v ( i − j + w ( i − ) (cid:12)(cid:12)(cid:12) − l (cid:105) + M (cid:88) j =1 13 (cid:88) i =1 k (cid:48) · (cid:32)(cid:12)(cid:12)(cid:12) v ( i ) j − v ( i − j (cid:12)(cid:12)(cid:12) − d (cid:33) − , (5)where ± w ( i ) are the vectors from the center of the spherical dimer cross section to thetwo interaction points on the dimer surface, k is the spring constant for the bindingforce and k (cid:48) the strength of the repulsion.As the stable configuration of a physical system is characterized by minimum freeenergy, we numerically calculate the minimum of the total energy functional E = E long + E lat = E bend + E spring + E hc (6)with respect to the 13 · ( m + m ) variables θ ( i ) j .In the following, we often measure energies in units of the dimer bending rigidity κ .This shows that the dimensionless energy E/κ and, thus, our MT model is characterizedby three remaining control parameters: the ratios k/κ and k (cid:48) /κ control the MTmechanics; the ratio κ/k B T of dimer bending rigidity and thermal energy controls therelevance of stochastic thermal fluctuations in comparison to mechanical forces. Allrelevant model parameters for MT geometry are summarized in table 1. No direct experimental measurement of the microscopic mechanical parameters k and κ is available so far. However, there are a number of experiments determining themacroscopic elastic moduli of the MT lattice which could be related to the microscopicmechanical parameters. The lateral bond elasticity k can be deduced from measurementsof the macroscopic shear modulus [25]. Moreover, molecular dynamics (MD) simulationsof short MT protofilaments [25, 26, 27, 28] can eventually give more direct information onthe molecular scale parameters κ and k . The requirement to obtain a stable equilibriumMT structure, which is tubular and agrees with the experimentally observed geometrywill put further constraints on the three parameters κ , k , and k (cid:48) of our model.The lateral bond elasticity k can be related to the macroscopic shear modulus G of the MT, which has been experimentally measured as G (cid:39) / nm = 2MPa (at T = 37 ◦ C) [46]. The two-dimensional shear modulus of the MT lattice is G D = Gh ,where h (cid:39) e sh = hdrGα / r = 2 πR/ (cid:39) . R = ( R o + R i ) / . z (cid:39) . ffects of microtubule mechanics on hydrolysis and catastrophes r l , hel l z z (a) (b) Figure 2.
Deformation of lateral bond springs by shearing the MT lattice. between neighboring protofilaments. The rest length of lateral bond springs becomes l , hel = ( l + z ) / (cid:39) . α gives rise toelongation of the lateral springs from their rest length l , hel by ∆ l = ∆ zz /l , hel , where∆ z = αr is the relative displacement of protofilaments induced by shearing, see figure2. Therefore, the shear energy per dimer can also be written as e sh = 12 k ∆ l = 12 k z r l , hel α . Comparison with e sh = hdrGα / k = hdl , hel z r G (cid:39) . k B T nm − . (7)In order to estimate the bending rigidity κ of single tubulin dimers, we canuse existing MD simulation results on the distribution of protofilament curvatures.In [27], thermal fluctuations of the curvature radius of a protofilament consisting ofthree GTP- or GDP-tubulin dimers have been investigated by MD simulations and adistribution of protofilament curvature radii has been measured. A single protofilamentwith N p = 3 coupled dimers will behave as a short semiflexible polymer of length N p d .Each of the N p − αβ -junctions with bending angle ∆ θ contributes a bending energy e bend , j = κ ∆ θ = κd c j , where c = ∆ θ/d is the local curvature. The mean curvature c m = N p − (cid:80) N p − j =1 c j of the protofilament exhibits Gaussian fluctuations with (cid:104) c m (cid:105) = (cid:104) c j (cid:105) = κd /k B T and, thus, is distributed according to p c ( c m ) ∝ exp( − κd c m / k B T ).Then we can calculate the corresponding distribution of mean curvature radii R = 1 /c m , p ( R ) = p c ( c m ( R )) 1 R ∝ R exp (cid:32) − d κ k B T R (cid:33) , (8) ffects of microtubule mechanics on hydrolysis and catastrophes Log k /κ l -
15 10 5 00.010.020.030.00 - - -
15 10 5 00.010.020.030.00 - - E spring /EE hc /E Log k /κl Figure 3.
The fraction of hard-core energy E hc /E of the total energy in comparisonto the fraction of spring energy E hc /E as a function of the dimensionless hard coreparameter ¯ k (cid:48) = k (cid:48) /κl for two different values k/κ = 0 . − and k/κ = 0 . − (inset) corresponding to strong and weak lateral bond springs. and find a maximum at R = d κ/ k B T . In the MD simulations in [27] a mostprobable radius R max (cid:39) κ = 2 R k B T /d (cid:39) . k B T (9)for the dimer bending rigidity κ .In [25, 27], the dimer bending rigidity κ was determined by applying equipartitionto the mean square thermal fluctuations of bending angles as measured in the MDsimulations of protofilaments containing three dimers. This approach gives much highervalues κ = 1 . × pN nm ∼ k B T [25] and κ ∼ − k B T [27], which differfrom each other and our above estimate (9). Currently we have no explanation for thesediscrepancies.We also have to fix the value of the hard core parameter k (cid:48) . Because the hard corerepulsion only serves as auxiliary interaction in order to avoid an unphysical overlappingof neighboring dimers, we want to use values for k (cid:48) which do not influence the MTequilibrium configuration appreciably. Therefore, we want to choose k (cid:48) such that thehard core interaction energy E hc remains much smaller than the the bending energy E bend and the spring energy E spring . In figure 3, we show the relative contributionsof both energies to the total energy. For values k (cid:48) /κl < − the hard coreinteraction has negligible influence on the MT equilibrium configuration. We thereforeuse k (cid:48) /κ = 10 − nm in the following.Finally, the value of k/κ is constrained by the requirement that the equilibrium MT ffects of microtubule mechanics on hydrolysis and catastrophes (a) (b)(c) (d)(e) (f) Figure 4.
Mechanical equilibrium configuration of MTs for k/κ = (a) 10 − , (b)1 . × − , (c) 2 × − , (d) 10 − , (e) 10 − , and (e) 10 − nm − ( k (cid:48) /κ = 10 − nm ).MTs are stabilized by a 3 layer GTP-cap ( m = 10, m = 3). forms a tubular structure. We find that for weak lateral springs, i.e., too small values of k/κ the stabilizing effect of a GTP-cap is lost and MTs spontaneously acquire a stronglybent shape similar to the ram’s horn configuration of depolymerizing catastrophic MTs.We studied this effect systematically by calculating MT configurations of minimalmechanical energy for k/κ in the range k/κ = 10 − nm − ... − , see figure 4. Becausesuch forms have not been observed experimentally, we conclude that a reasonable lowerbound for k/κ is k/κ ≥ . − . (10)Note that with our above estimates (7) for k and (9) for κ we find k/κ (cid:39) . − ,which is far above this bound. The much larger values of κ that have been obtained in[25, 27] as discussed above would lead to a ratio k/κ close to the lower bound (10).Because of the considerable uncertainty in the estimates for k and κ , we willinvestigate values k/κ = 0 . − close to the lower bound as an example for weaklateral springs and k/κ = 0 . − as an example for strong lateral springs in thefollowing. In this section, we want to discuss alternative models. In a conceptually simpler modelwe could put single interaction points for the lateral bond springs at the centers ofthe tubulin dimers. We investigated such simpler models and found that equilibrium ffects of microtubule mechanics on hydrolysis and catastrophes et al. [41, 47] also contains a bending energyand lateral bonds, which are modeled as harmonic springs around their equilibriumlength. In [41, 47], four interaction points and two lateral springs are introducedper dimer and the helical structure is taken into account in contrast to our model.Moreover, only inter-dimer bending is considered. Molodtsov et al. use much largervalues k ∼ k B T nm − than our estimate k (cid:39) . k B T nm − , see (7). Their valueis obtained from an estimate of the activation energy for bond rupture, assuming acertain one-parameter form of the lateral bond potential, which tightly couples theparameter for bond elasticity to the bond rupture forces. We rather used values forthe bond elasticity confirming with the macroscopic shear modulus of a protofilamentsheet following [25] and consider the bond rupture force as an independent parameter(see section 3.2 below). Molodtsov et al. then also estimate much higher values of κ byapplying a similar constraint for k/κ as we obtained above (10), violation of which givesrise to strongly outward bending cap configurations even in the presence of a GTP-cap.In contrast, we tried to obtain an estimate for κ from microscopic MD simulation data.The mechanical model proposed by Van Buren et al. [30, 43] also contains abending energy and lateral bonds, which are modeled as harmonic springs of zero restlength. In addition, the model also includes longitudinal harmonic bond springs to modelprotofilament stretching and torsion elasticity, i.e., an elastic energy for the angles φ ( i ) j ,which are fixed to φ ( i ) j = φ ( i ) = 2 πi/
13 in our model.Finally, we want to point out that the MT bending rigidity κ MT and, thus, itspersistence length, is not directly related to the dimer bending rigidity κ in mechanicalmodels. It is mainly related to the stiffness of longitudinal bonds between neighboringdimers on the same protofilament, which are stretched and compressed in bendingdeformations of the tube. Longitudinal bond stiffnesses are absent in our model as wedo not consider shape fluctuations of the whole tube but are included in the models usedin Refs. [41, 43, 47] and have also been discussed in Ref. [48]. Longitudinal bonds arefound to be much stiffer than lateral bonds. In Ref. [48] it has been shown that the MTpersistence length becomes length dependent as it contains both bending contributionsrelated to longitudinal bonds and shear contributions related to the lateral bonds. Wedo not expect the longitudinal bond stiffness, the lateral bond stiffness k or the dimerbending rigidity κ themselves to be length dependent. In the allosteric model, hydrolysis of GTP-tubulin dimers to GDP-tubulin dimers withinthe MT leads to bending of the dimer, i.e., a change in the equilibrium dimer angle θ . ffects of microtubule mechanics on hydrolysis and catastrophes θ as the reaction coordinate of hydrolysis such that, in the absenceof mechanical forces, hydrolysis of single dimers is characterized by a free energy profile F h ( θ ) with a free energy minimum corresponding to a GTP-dimer at θ = 0 ◦ and asecond minimum corresponding to the GDP-dimer state at θ = 22 ◦ . Because of thefree energy ∆ G GTP (cid:39) k B T released by hydrolysis of GTP within the MT lattice [39],the second minimum is lower by at most F h (0 ◦ ) − F h (22 ◦ ) < ∆ G GTP (cid:39) k B T .Because GTP hydrolysis can generate mechanical forces within the MT, mechanicalforces also influence the hydrolysis rate of GTP-dimers. This influence can be quantifiedusing Bell theory [49]. For each equilibrium angle θ ≡ θ ( i )0 ,j of a given dimer we canalso calculate the mechanical total equilibrium energy E ( θ ) of the MT lattice. Therelevant free energy profile for hydrolysis, that takes into account the mechanics of thesurrounding MT lattice, is F h ( θ ) + E ( θ ). Using θ as a reaction coordinate therewill be a rate limiting free energy barrier ∆ F h for an intermediate 0 ◦ < θ , max < ◦ .The height and the exact angle θ , max of this energy barrier will also be modified by thecorresponding mechanical contribution to ∆ F h +∆ E . If we neglect a possible shift of thebarrier angle, we have ∆ E = E ( θ , max ) − E (0 ◦ ). We have currently no information on thedetailed free energy profile F h ( θ ) of the hydrolysis reaction; we only have informationon the hydrolysis rates themselves, which should be related to the barrier height ∆ F h via the Arrhenius law. To proceed, we simply assume θ , max = 11 ◦ in the following.One important point regarding the influence of the mechanics of the MT lattice ontothe hydrolysis rates is the following: differences between hydrolysis rates at differentsites in the MT lattice are not governed by ∆ F h , which is identical for all MT latticessites, but rather by the mechanical contribution ∆ E to the barrier, which we want toestimate now.We know that the total mechanical energy is a quadratic function of θ via thebending energy contribution in (3). In order to determine ∆ E for hydrolysis we haveto increase θ from 0 ◦ to θ , max to obtain the saddle point value ∆ E . Because weassume that a single hydrolysis reaction is faster than mechanical relaxation of the MTlattice, we increase θ for fixed values of the configuration angles θ ( i ) j . We assume anapproximately linear dependence E ( θ ) − E (0 ◦ ) ≈ ( E ( δ ) − E (0 ◦ )) θ /δ for a small angle δ , and estimate ∆ E numerically using δ = 1 ◦ and assuming θ , max = 11 ◦ for the positionof the barrier. This results in the estimate∆ E = E (11 ◦ ) − E (0 ◦ ) (cid:39) E (1 ◦ ) − E (0 ◦ )) (11)(calculated for fixed values of all configuration angles θ ( i ) j ). According to its definition,∆ E will be small (eventually even negative) for hydrolysis of a given dimer if there aremechanical forces in the MT lattice that pull the dimer into its bent GDP-configuration.This mechanical shift of the free energy barrier for the hydrolysis reaction will ffects of microtubule mechanics on hydrolysis and catastrophes r h of a dimer according to r h (∆ E ) = r h (0) exp( − ∆ E/k B T ) (12)following Bell [49]. In particular, this leads to site-dependent hydrolysis rates, whichdepend on the position of the dimer in the MT lattice via the mechanical forces actingon it at that position. We conclude that the interaction via the mechanics of the MTlattice can give rise to possible correlation effects in the hydrolysis dynamics, which havenot been taken into account before.Our simulation algorithm for the most probable hydrolysis pathway (see section 3.1below) will be based on two assumptions regarding the time scales for hydrolysis andmechanical relaxation: (i) a single hydrolysis reaction is fast compared to mechanicalrelaxation, which was the basis for calculating the modulated hydrolysis rate (12), and(ii) mechanical relaxation is faster than the time between successive hydrolysis events(set by the hydrolysis rate r − h ) such that we relax the MT lattice mechanically betweensuccessive hydrolysis events. Within the MT GTP-dimers are hydrolyzed into GDP-dimers. The backward reactionis not observed. This puts a constraint on the mechanical model parameters because itimplies that the total mechanical energy difference during hydrolysis of a certain dimer,∆ E h = E (22 ◦ ) − E (0 ◦ ), has to be much smaller than the chemical energy ∆ G GTP (cid:39) k B T released by GTP hydrolysis in the absence of any mechanical forces [39], ∆ E h (cid:28) ∆ G GTP .We consider the situation where the MT is still stabilized by the GTP-cap to atubular configuration such that all angles θ ( i ) j are small. Then, hydrolysis of a singledimer (in layer j and protofilament i ) increases its rest angle and strains the surroundingMT lattice. If the lateral interaction springs are sufficiently strong to stabilize a tubularMT configuration, the change ∆ E spring in the lateral spring energies by this hydrolysiscan be neglected as compared to the change ∆ E bend in bending energy such that thetotal mechanical energy increase during hydrolysis is approximated by∆ E h ≈ ∆ E bend = κ (cid:16) θ ( i ) j − θ ( i ) j − − ◦ (cid:17) − κ (cid:16) θ ( i ) j − θ ( i ) j − (cid:17) ≈ κ ◦ ) . (13)Then, the condition ∆ E h (cid:28) ∆ G GTP (cid:39) k B T implies an upper bound on the dimerbending rigidity κ , κ (cid:28) G GTP ◦ ) (cid:39) k B T rad − . (14)Our above estimate (9), κ (cid:39) . k B T , from analyzing MD simulation results forprotofilament curvature radii distributions obeys the constraint (14). The otherestimates κ ∼ k B T from [25] and κ ∼ k B T from [27], however, violate thisconstraint. If κ -values violating the upper bound (14) are confirmed experimentally in ffects of microtubule mechanics on hydrolysis and catastrophes E h ≤ ∆ G GTP , we can distinguish two limits forthe hydrolysis rates: (i) if ∆ E (cid:28) k B T , the influence of the mechanical shift ∆ E of thefree energy barrier for the hydrolysis reaction is small, and we expect purely “chemical”models as defined in the introduction to be essentially correct. (ii) If ∆ E (cid:29) k B T , onthe other hand, the mechanical shift ∆ E dominates the hydrolysis rates according to(12) and we expect a strong interplay between MT mechanics and hydrolysis.In limit (ii), the dimer to be hydrolyzed next is mainly determined by the mechanicalshifts ∆ E of the free energy barrier for the hydrolysis reaction; the GTP-dimer with thesmallest ∆ E will be hydrolyzed next, eventually under additional restrictions dependingon whether we consider a vectorial or random hydrolysis mechanism. Only if severalGTP-dimers have a similar ∆ E , the next hydrolysis event will be stochastic amongthese dimers. Therefore, the order of hydrolysis of GTP-dimers within the MT lattice(the hydrolysis pathway) is mainly determined by the hydrolysis mechanism (vectorialor random) and the mechanical energies ∆ E and exhibits much less stochasticity ascompared to limit (i), as discussed below in the results section.We will now explore under which condition we can expect such mechanicallydominated hydrolysis order, i.e., under which conditions ∆ E ≥ k B T holds. Themechanical shift ∆ E of the energy barrier for the hydrolysis reaction should be smallerthan total mechanical energy change ∆ E h during hydrolysis, ∆ E ≤ ∆ E h resulting in acondition ∆ E h ≥ ∆ E ≥ k B T or (using (13)) κ ≥ k B T ◦ ) (cid:39) k B T rad − . (15)For smaller values of κ , we expect to find hydrolysis rates, which are essentiallyindependent of mechanical forces developing in the MT lattice.Our above estimate (9), κ (cid:39) . k B T , from analyzing MD simulation results forprotofilament curvature radii distributions violates condition (15) only weakly. Theother estimates for κ from [25, 27], however, give much higher values of κ supportingmechanically dominated hydrolysis rates.
3. Results
We use the mechanical model introduced above to investigate how mechanical forcesinfluence and direct hydrolysis pathways and how catastrophe events emerge if lateralbond rupture is included.
We explore results for a mechanically dominated hydrolysis rate where the mechanicalshift of the free energy barrier for the hydrolysis reaction is much larger than the thermalenergy 1 k B T . Therefore, our results will apply if both bounds (14) and (15) are met, i.e., ffects of microtubule mechanics on hydrolysis and catastrophes k B T ≤ κ (cid:28) k B T of dimer bending rigidities. This regime is conceptuallyinteresting because correlation effects in the hydrolysis dynamics introduced via themechanics of the MT lattice become maximal. Moreover, this parameter range forthe tubulin dimer bending rigidity κ cannot be ruled out at present because differentestimates for κ are deviating and not reliable. Our above estimate κ (cid:39) . k B T , see(9), is close to the lower boundary of the considered κ -range.If hydrolysis is mechanically dominated (∆ E (cid:29) k B T ), the hydrolysis pathways,i.e., the order of hydrolysis of dimers within the MT, is mainly determined by themechanical shift ∆ E of the energy barrier for the hydrolysis reaction: the dimer tobe hydrolyzed next with highest probability is the GTP-dimer with the smallest ∆ E among all GTP-dimers accessible by the hydrolysis mechanism. For random hydrolysisall GTP-dimers are accessible, for vectorial hydrolysis only the GTP-dimers at the capboundary. Because the dimensionless energies ∆ E/κ only depend on the parameter k/κ ,the hydrolysis pathway is entirely determined by this mechanical parameter for bothmechanisms. As a result of mechanically dominated hydrolysis, the choice of a certainhydrolysis pathway becomes much less stochastic: from the large number of chemicallypossible pathways only relatively few have considerable statistical weight if the influenceof mechanics is dominant for ∆ E (cid:29) k B T . Therefore, the concept of a most probablehydrolysis pathway , which is the hydrolysis pathway with the highest statistical weight,is reasonable in this limit.In the simulation, we determine the most probable hydrolysis pathway by thefollowing algorithm: We relax the mechanical forces in the MT lattice by energyminimization for a given hydrolysis state. Next, we calculate the mechanical energyshifts ∆ E for each GTP-dimer which can be hydrolyzed according to the assumedhydrolysis mechanism (random or vectorial). We calculate ∆ E according to (11). Then,out of these GTP-dimers, we choose the one with the minimal ∆ E and, thus, the highesthydrolysis rate according to (12) to be hydrolyzed next. Then, we relax the MT latticeagain mechanically, and so on.The condition ∆ E (cid:29) k B T is necessary but not sufficient to select a hydrolysispathway uniquely, i.e., render the hydrolysis deterministic: if several GTP-dimers havea similar ∆ E , the next hydrolysis event will be essentially stochastic among these dimers.The hydrolysis pathway becomes deterministic only if also differences ∆∆ E between the∆ E for different GTP-dimers within the MT are much larger than the thermal energy,i.e., ∆∆ E (cid:29) k B T .As discussed in the introduction, the correct hydrolysis mechanism is not exactlyknown. Therefore we consider both vectorial and random hydrolysis separately in thefollowing. We address the question how the mechanical forces direct the hydrolysispathway for both mechanisms and as a function of the parameter k/κ . We will considertwo exemplary values: k/κ = 0 . − close to the lower bound (10) as an examplefor weak lateral springs and k/κ = 0 . − as an example for strong lateral springs inthe following.In order to isolate effects of MT mechanics onto the hydrolysis pathway, we ignore ffects of microtubule mechanics on hydrolysis and catastrophes n hyd = 0 1 2 3 4 5 6 78 9 10 11 12 n hyd = 13 14 15 16 17 18 19 20 21 22 23 24 25 n hyd = 26 27 28 29 30 31 32 33 34 35 36 37 - ∆ E [ κ ] Figure 5.
Random hydrolysis: Most probable hydrolysis pathway of a MT with a 3layer GTP-cap with strong lateral bond springs ( k/κ = 0 . − ). Each 13x3 rectanglein the sequence of 38 hydrolysis steps shows the hydrolysis state of the 3 layer GTP-cap (MT plus end on the right side): squares symbolize dimers; grey squares representhydrolyzed GDP-dimers; green squares represent GTP-dimers color-coded for theirrespective ∆ E/κ . The red dot marks the dimer to be hydrolyzed next. The thickblack vertical line marks the lateral bond under maximal force, see figure 9a. the polymerization dynamics and consider hydrolysis in MTs of fixed length with astabilizing GTP-cap consisting of 3 layers ( m = 20 and m = 3). For random hydrolysis the chemical hydrolysis rate r h (0) ofa given GTP-dimer is independent of the hydrolysis state of its neighbors, see (12).Therefore, all GTP-dimers in the cap can be hydrolyzed with equal probability ifmechanics is neglected.For random hydrolysis in the absence of mechanical forces, all GTP-dimers in thecap are accessible for hydrolysis with equal probability, and in a MT with a GTP-capsize of three layers ( m = 3) there are 39! equally probable hydrolysis pathways. Formechanically dominated random hydrolysis, on the other hand, it is always the dimerwith smallest ∆ E among all GTP-dimers, which is hydrolyzed next with the highest ffects of microtubule mechanics on hydrolysis and catastrophes n hyd = 0 1 2 3 4 5 6 78 9 10 11 12 n hyd = 13 14 15 16 17 18 19 20 21 22 23 24 25 n hyd = 26 27 28 29 30 31 32 33 34 35 36 37 - ∆ E [ κ ] Figure 6.
Random hydrolysis: Most probable hydrolysis pathway of a MT with a 3layer GTP-cap with weak lateral bond springs ( k/κ = 0 . − ). Representation asin figure 5. The thick black vertical line marks the lateral bond under maximal force,see figure 9b. probability. As a result there emerges a most probable hydrolysis pathway , which isshown in figures 5 and 6 for a MT with m = 20 and m = 3 and strong springs( k/κ = 0 . − , figure 5) or weak springs ( k/κ = 0 . − , figure 6). These figuresshow the sequence of 38 hydrolysis states of the 3x13 dimer cap (from n hyd = 0 GDP-dimers to n hyd = 37 GDP-dimers). GTP-dimers are represented by green squares,hydrolyzed GTP-dimers by grey squares.If there are several GTP-dimers with similarly small ∆ E , such that ∆∆ E ≤ k B T ,they can be hydrolyzed with comparable probability, and there are other hydrolysispathways which are similarly probable. The values of ∆ E of different GTP-dimers(measured in units of κ ) are indicated in the most probable hydrolysis pathways infigures 5 and 6 by color-coding. The existence of a unique, dark green square in acap indicates the existence of a unique next hydrolysis spot, which is separated by alarge ∆∆ E from other possible hydrolysis spots. Small color differences indicate smallvalues of ∆∆ E , such that the next hydrolysis spot should be chosen essentially stochasticamong the dimers with darkest green colors. Then there exist other pathways competing ffects of microtubule mechanics on hydrolysis and catastrophes n hyd = 1 , , n hyd = 12 , , ,
21 in figure 5). Theseisolated straight GTP-dimers are pulled outward by already hydrolyzed and, thus, bentGDP-neighbors.
For vectorial hydrolysis, only dimers whose longitudinalneighbor (towards the minus end) is already hydrolyzed into GDP-tubulin are accessiblefor hydrolysis. Then there is a sharp interface between GDP- and GTP-dimers, whichpropagates towards the plus end. For mechanically dominated vectorial hydrolysis, theGTP-dimer with the smallest ∆ E among all GTP-dimers at the GTP-GDP interface ishydrolyzed next.Again, for mechanically dominated vectorial hydrolysis, there is a most probablehydrolysis pathway, which is shown in figures 7 and 8 for a MT with m = 20 and m = 3and strong springs ( k/κ = 0 . − , figure 7) or weak springs ( k/κ = 0 . − , figure8) in terms of the sequence of 38 hydrolysis states of the 3x13 dimer cap. For vectorialhydrolysis, there are also GTP-dimers which cannot be hydrolyzed because they are notat the GTP-GDP interface, i.e., there is no hydrolyzed neighboring GDP-dimer towardsthe minus end. These dimers are shown in white color. Only hydrolyzable GTP-dimersat the GTP-GDP interface are represented by green squares, hydrolyzed GTP-dimersby grey squares as before.We recognize marked differences as compared to the random hydrolysis:(i) Because of the vectorial constraint, hydrolysis has to start in the back layerof the GTP-cap towards the minus end, where the GTP-GDP interface is located. Itcannot advance directly to the front layer as for random hydrolysis.(ii) There is a strong tendency to continue hydrolysis on the same protofilament (seestates n hyd = 1 , , n hyd = 9 , ,
11 in figure 7). If the hydrolysis front has advancedto the next layer on one protofilament, the bending forces from the hydrolyzed dimersbehind the front give rise to a strong outward bending moment on the next GTP-dimer,which favors its hydrolysis. This force is transmitted via the lateral bonds to neighboringprotofilaments in the MT lattice. ffects of microtubule mechanics on hydrolysis and catastrophes n hyd = 0 1 2 3 4 5 6 78 9 10 11 12 n hyd = 13 14 15 16 17 18 19 20 21 22 23 24 25 n hyd = 26 27 28 29 30 31 32 33 34 35 36 37 - ∆ E [ κ ] Figure 7.
Vectorial hydrolysis: Most probable hydrolysis pathway of a MT with a 3layer GTP-cap with strong lateral bond springs ( k/κ = 0 . − ). Each 13x3 rectanglein the sequence of 38 hydrolysis steps shows the hydrolysis state of the 3 layer GTP-cap (MT plus end on the right side): squares symbolize dimers; grey squares representhydrolyzed GDP-dimers; green squares represent GTP-dimers at the boundary of theGTP-cap, color-coded for their respective ∆ E/κ . White squares represent GTP-dimers, which are not at the cap boundary and, thus, cannot be hydrolyzed in avectorial mechanism. The red dot marks the dimer to be hydrolyzed next. The thickblack vertical line marks the lateral bond under maximal force, see figure 10a. (iii) As for random hydrolysis, there is a preference for GTP-dimers with one or twolateral GDP-neighbors to be hydrolyzed (see, for example, states n hyd = 10 , , , , n hyd = 3 , , ffects of microtubule mechanics on hydrolysis and catastrophes n hyd = 0 1 2 3 4 5 6 78 9 10 11 12 n hyd = 13 14 15 16 17 18 19 20 21 22 23 24 25 n hyd = 26 27 28 29 30 31 32 33 34 35 36 37 - ∆ E [ κ ] Figure 8.
Vectorial hydrolysis: Most probable hydrolysis pathway of a MT with a 3layer GTP-cap with weak lateral bond springs ( k/κ = 0 . − ). Representation asin figure 7. The thick black vertical line marks the lateral bond under maximal force,see figure 10b. GDP-dimers that has to assemble before the hydrolysis front will reach the plus endcould be an explanation of the experimental finding of two rate-limiting steps, whichare involved in the initiation of a catastrophe [14].
During a catastrophe, it is experimentally observed that the protofilaments of a MT fallapart and curl into “ram’s horn” conformations as a result of the 22 ◦ equilibrium angle ofhydrolyzed GDP-dimers. This implies that protofilaments separate during catastrophes,i.e., lateral bonds between dimers in neighboring protofilaments will rupture.Bond rupture is an activated process, and the bond rupture rate under force, r rup ( F ) = r rup (0) exp( F/F rup ), increases exponentially above a characteristic ruptureforce F rup according to Bell theory [49]. In a simplified approach, we can ignorestochastic rupture effects and assume that all lateral bonds under forces F > F rup rupture. One problem is that there is currently no microscopic information on values ffects of microtubule mechanics on hydrolysis and catastrophes (a) (b) n hyd n hyd ¯ F m a x [ / n m ] ¯ F m a x [ / n m ] ∆ ¯ F m a x [ / n m ] ∆ ¯ F m a x [ / n m ] Figure 9.
Random hydrolysis: Maximal lateral bond force ¯ F max = F max /κ andchange ∆ ¯ F max in the maximal lateral bond force after rupture of the maximallystrained bond as a function of the number n hyd of hydrolyzed GTP-dimers. We startwith a MT with a 3 layer GTP-cap. a) Strong lateral bond springs, k/κ = 0 . − ,and b) weak lateral bond springs, k/κ = 0 . − . The location of the lateral bondunder maximal force within the GTP-cap is shown in figures 5 and 6 for strong andweak bonds, respectively, as thick black line. for the rupture force F rup . In [47], an activation energy for the rupture process wasestimated from the depolymerizing velocity of catastrophic MTs as E rup (cid:39) k B T .Together with a characteristic bond length of (cid:39) . F rup ∼ Here we want to take a different approach and ask,which rupture forces can initiate catastrophe-like events within our mechanical model ofthe MT. To answer this question we consider the most probable hydrolysis pathway andcalculate the maximal lateral bond force F max within the MT lattice at every hydrolysisstep. The results are shown in figure 9 for random hydrolysis and figure 10 for vectorialhydrolysis. The location of the lateral bond under maximal force within the GTP-capis shown in the corresponding figures 5–8 as thick black line.For random hydrolysis, mechanical forces favor hydrolysis at the last layer of theGTP-cap towards the plus end. For random hydrolysis and strong lateral bonds, seefigure 9a, we see a strong increase in the maximal bond force F max after hydrolysis of n hyd = 14 dimers. At this point, the first dimer in the second to last layer is hydrolyzedafter complete hydrolysis of the last layer at the plus end. For weak lateral bonds, seefigure 9b, we even see a periodic layer-by-layer pattern in the increase of F max withpronounced jumps at n hyd = 1 , ,
27, where hydrolysis reaches the next layer starting ffects of microtubule mechanics on hydrolysis and catastrophes (a) (b) n hyd n hyd ¯ F m a x [ / n m ] ¯ F m a x [ / n m ] ∆ ¯ F m a x [ / n m ] ∆ ¯ F m a x [ / n m ] Figure 10.
Vectorial hydrolysis: Maximal lateral bond force ¯ F max = F max /κ and change ∆ ¯ F max in the maximal lateral bond force after rupture of the maximallystrained bond as a function of the number n hyd of hydrolyzed GTP-dimers. We startwith a MT with a 3 layer GTP-cap. a) Strong lateral bond springs, k/κ = 0 . − ,and b) weak lateral bond springs, k/κ = 0 . − . The location of the lateral bondunder maximal force within the GTP-cap is shown in figures 7 and 8 for strong andweak bonds, respectively, as thick black line. from the plus end. In the corresponding figures 5 for strong lateral bonds and 6 forweak lateral bonds, the location of the maximally strained lateral bond is shown asthick black line. The maximally strained bond is at the plus end layer of the GTP-capbond next to the same protofilament as the hydrolyzed dimer. Hydrolysis of the firstdimer in a layer increases the spontaneous curvature of that protofilament and, thus,increases the strain on lateral bonds of the same protofilament in the last layer.For vectorial hydrolysis, only dimers at the GTP-GDP interface can be hydrolyzed,and hydrolysis has to advance from the minus end side. The results for vectorialhydrolysis and strong lateral bonds, see figure 10a, show that after hydrolysis of n hyd = 16 dimers, when the third protofilament becomes completely hydrolyzed, thereis a pronounced step-like increase in F max . Similarly, for weak lateral bonds, see figure10b, there is a strong increase after hydrolysis of n hyd = 5 dimers, when the firstprotofilament becomes completely hydrolyzed. A completely hydrolyzed protofilamentprefers to assume its curved equilibrium state and, thus, exerts strong outward forces onneighboring stabilizing protofilaments with GTP-caps. Therefore, with each completelyhydrolyzed protofilament the strain on the lateral bonds increases. For weak bondsa single hydrolyzed protofilament is sufficient to cause a pronounced increase in F max ,whereas for strong bonds several hydrolyzed protofilaments seem to be necessary. Inthe corresponding figures 7 for strong lateral bonds and 8 for weak lateral bonds thelocation of the maximally strained lateral bond is shown as thick black line: the bondunder maximal force occurs at the last layer of the cap next to the completely hydrolyzed ffects of microtubule mechanics on hydrolysis and catastrophes F max depend on the strength of lateral bonds. For strong lateralbonds ( k/κ = 0 . − ) we find F max /κ ∼ . − both for random and vectorialhydrolysis. For κ ∼ k B T well within the considered range 14 k B T ≤ κ (cid:28) k B T ofdimer bending rigidities, this gives maximal forces F max ∼ k/κ = 0 . − ) we find smaller maximal forces F max /κ ∼ . − correspondingto F max ∼ To further characterize thesusceptibility to a mechanical instability, i.e., a rupture avalanche as it happens incatastrophes, we also investigated the change ∆ F max in the maximal lateral bond forceif we rupture the maximally strained bond and calculate the new maximal lateral bondforce on an intact bond, see the lower plots in figure 9 for random hydrolysis and figure10 for vectorial hydrolysis. If ∆ F max is large and positive, there is an increase in themaximal bond force and the possibility of an instability: a rupture force F rup < F max below this maximal force level can give rise to continued bond rupture. Vice versa, anegative value for ∆ F max signals a mechanical stable situation. Remarkably, we find astrong correlation between the pronounced increase in F max and a pronounced increasein ∆ F max to larger positive values, see figure 9 for random hydrolysis and figure 10 forvectorial hydrolysis.This suggests that the pronounced increase in the maximal lateral bond force isthe starting point of a catastrophe-like rupture avalanche, if we choose the value forthe rupture force F rup such that after the increase in F max , the maximal lateral bondforce exceeds the rupture threshold. Using this criterion, a reasonable value for therupture force for random hydrolysis is F rup ∼ . κ nm − for strong lateral bondsand F rup ∼ . κ nm − for weak lateral bonds. For vectorial hydrolysis, this suggestsrupture forces F rup ∼ . κ nm − for strong lateral bonds and F rup ∼ . κ nm − forweak lateral bonds. With values of κ ∼ k B T from the range 14 k B T ≤ κ (cid:28) k B T according to the bounds (14) and (15), the resulting rupture forces are F rup ∼ − . − . n hyd = 5 or 6, i.e., hydrolysis practically stops during such an event. The hydrolysis ffects of microtubule mechanics on hydrolysis and catastrophes (a) (b) (c) (d) (e) Figure 11.
Catastrophe event for vectorial hydrolysis and weak lateral bond springs k/κ = 0 . − using a rupture force F rup = 0 . κ nm − ( m = 20 and m = 3).The numbers n hyd of hydrolyzed dimers and n cut of cut lateral bonds are (a) n hyd = 5, n cut = 1, (b) n hyd = 6, n cut = 4, (c) n hyd = 6, n cut = 8, (d) n hyd = 6, n cut = 11, (e) n hyd = 6, n cut = 33. (a) (b) (c) (d) (e) Figure 12.
Catastrophe event for random hydrolysis and weak lateral bond springs k/κ = 0 . − using a rupture force F rup = 0 . κ nm − ( m = 20 and m = 3).The numbers n hyd of hydrolyzed dimers and n cut of cut lateral bonds are (a) n hyd = 14, n cut = 1, (b) n hyd = 14, n cut = 6, (c) n hyd = 14, n cut = 15, (d) n hyd = 14, n cut = 24,(e) n hyd = 14, n cut = 45. ffects of microtubule mechanics on hydrolysis and catastrophes n hyd = 5 consists of a single completely hydrolyzed protofilament (see figure8). Therefore, the initial stress distribution in lateral bonds is concentrated on thisprotofilament. The stress due to the preferred curved configuration of this protofilamenttriggers the bond rupture of neighboring lateral bonds, and this protofilament starts topeel off during the catastrophe.For random hydrolysis, see figure 12, typically several protofilaments peel off the MTlattice. Also for this hydrolysis mechanism, the hydrolysis stops during such an event:the entire catastrophe event in figure 12 happens at n hyd = 14. For random hydrolysisthe stress is typically distributed among the entire top layer of lateral bonds becausemechanically dominated hydrolysis proceeds layer by layer; the state n hyd = 14 consistsof a completely hydrolyzed plus end layer of dimers with one additional hydrolysis eventin the next layer, see figure 6. This gives rise to several bond rupture events within thetop layer and several protofilaments peeling off the MT.These different catastrophe characteristics for vectorial and random hydrolysiscould be an interesting issue for future experimental studies.
4. Discussion and Conclusion
We introduced a mechanical model, which gives stable tubular MT structures if astabilizing GTP-cap is present in accordance with experimental observations. Themodel includes intra-dimer bending and one lateral bond per dimer. In order to avoidoverlapping of tubulin dimers we also include a sufficiently strong hard core interaction.We use the allosteric model for the hydrolysis state of tubulin dimers: GTP dimers arestraight; after hydrolysis, a GDP dimer has an equilibrium bending angle of 22 ◦ .We obtained several constraints for the model parameters, the bending rigidity κ and the lateral bond strength k : (i) The ratio k/κ is constrained by a lower bound k/κ ≥ . − , see (10), which ensures that GTP-capped MTs do not spontaneouslyacquire a strongly bent shape similar to the ram’s horn configuration. (ii) The valuefor κ is constrained by an upper bound (14) because hydrolysis is not observed tobe a reversible reaction and the free energy released in hydrolysis should exceed themechanical energy increase in the MT lattice during hydrolysis. (iii) If we additionallyassume that hydrolysis is dominated by mechanical forces in the MT lattice, i.e., typicalmechanical energy changes of the the MT lattice during hydrolysis exceed the thermalenergy k B T , the value for κ is also constrained by a lower bound (15). The bounds (ii)and (iii) define a range 14 k B T ≤ κ (cid:28) k B T of dimer bending rigidities.The mechanical model allows us to investigate the interplay of mechanical forces inthe MT lattice and hydrolysis, which has not been done previously. The interactionvia the mechanics of the MT lattice can give rise to possible correlation effects inthe hydrolysis dynamics, which have not been taken into account before. Underthe assumption of a mechanically dominated hydrolysis reaction the concept of a most probable hydrolysis pathway becomes very useful. We calculated most probablehydrolysis pathways numerically both for random hydrolysis (figures 5 and 6), where all ffects of microtubule mechanics on hydrolysis and catastrophes increases the maximalbond force, which signals an instability with respect to a catastrophe event initiated bylateral bond rupture. If the rupture force value lies within the force range set by thesharp increase of the maximal lateral bond force, we indeed find continued bond ruptureand catastrophe events as observed in experiments, see figures 11 and 12. Using thiscriterion we find rupture force values between 1pN for weak and 5pN for strong lateralbonds. For vectorial hydrolysis, we find catastrophes starting with single protofilamentspeeling off the MT. For random hydrolysis, on the other hand, we typically see severalprotofilaments peeling off the MT. These characteristic differences in our simulationscould motivate further experimental studies of this issue.Our results suggest several routes for future work. Firstly, we only studied MTs offixed length for simplicity. Further investigations will include stochastic polymerizationand depolymerization similar to the models in [30, 43]. Secondly, we assumed so farthat hydrolysis is mechanically dominated and forces on the most probable hydrolysispathway by selecting the next GTP-dimer to be hydrolyzed according to the maximalmechanical energy gain. Future models should be fully stochastic with hydrolysis rates modulated by mechanical energies in order to include all hydrolysis pathways with theirrespective statistical weight into the analysis. Finally, a similar mechanical MT modeland its coupling to hydrolysis should be investigated not only for the allosteric modelbut also for the lattice model of dimer hydrolysis. ffects of microtubule mechanics on hydrolysis and catastrophes Acknowledgments
We thank Bj¨orn Zelinski and Sebastian Knoche for fruitful discussions, and weacknowledge support by the Deutsche Forschungsgemeinschaft (KI 662/4-1).
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