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[EQUATION] The ring detaches if the PFs break before an unzipping occurs, i.e. with probability that [MATH] [EQUATION] where [MATH] is the probability density function for [MATH] . Evaluating the integral with respect to [MATH] |
[EQUATION] Binomially expanding the integrand, integrating term-by-term and substituting back into Eq. we obtain [EQUATION] where [MATH] |
Time varying applied forces We now consider an oscillating applied force of the form [EQUATION] Provided the period is sufficiently long [MATH] our quasi-static approximation for the ring position should give an accurate estimate for its probability density [MATH] |
The depolymerization velocity will be retarded according to Eq. , relating [MATH] to [MATH] Protofilament model under oscillating force |
The probability that the MT does not unzip in a time [MATH] after the time at which the last unzipping occurred [MATH] is [EQUATION] |
Since Eq. 14 depends explicitly on [MATH] , we perform an average over [MATH] , appropriately weighted, to give the complementary distribution of times between unzipping events |
[EQUATION] involving a normalization constant [MATH] To calculate the runtime as in Eq. , we first determine the probability the unzip time exceeds the curled PF breaking time |
[EQUATION] The probability density of [MATH] is [EQUATION] Finally the runtime is [EQUATION] where [MATH] is the mean number of steps before detachment. |
Binding model under oscillating force The generalization of Eq. to the case of time-varying force (Eq. 13 ) is straightforward, [EQUATION] |
where [MATH] and [MATH] are now time-dependent potential gradients, according to Eq. 13 with Eq. Results and Discussion We identified the following parameters using data reported in the experimental literature; [MATH] nm/s 29 |
[MATH] [MATH] [MATH] 10 ; for the protofilament model we assume [MATH] nm and [MATH] to be typical. Table 1 in Franck et al. ( 12 (see Table S1 in the Supporting Material) lists velocities at [MATH] pN and [MATH] pN. Using the velocity data we fit to obtain [MATH] nm and [MATH] nm/s. As has already been mentioned the r... |
Combining the available data for velocity and detachment frequency we find, on average, [MATH] s and [MATH] s, for [MATH] pN and [MATH] pN respectively. To fit the binding model for [MATH] we choose [MATH] nm, as a reasonable distance over which an attraction might act, and find |
[MATH] [MATH] . Independently, we fit [MATH] for the protofilament model and find [MATH] [MATH] . Fitting these parameters to just two data points does not provide strong evidence for these particular values. However, uncertainty in the exact parameter values should not detract from the main value of this work; to prov... |
Variation of intrinsic depolymerization velocity The protofilament model exhibits the most sensitivity to the intrinsic (bare) MT depolymerization velocity [MATH] , as is shown in Fig. |
. For the protofilament model, [MATH] is strongly dependent on [MATH] and consequently [MATH] . The binding model, on the other hand, is only weakly dependent on [MATH] (see |
Supporting Material ), and on this range of [MATH] we can assume that depolymerization is quasistatically slow with respect to ring diffusion. The result can be understood physically by realising that as [MATH] increases, the rate of PF unzipping [MATH] also increases, while [MATH] remains constant making it less likel... |
An experimental test that might be able to distinguish which model operates could be achieved, e.g., by addition of a depolymerization inducing agent, such as [MATH] or XMCAK1. |
Changing of diffusion coefficient The diffusion constant [MATH] of the ring is determined by the ring’s dimensions and the roughness of the binding energy landscape along the MT, rather than the magnitude of the binding energy itself. A more rough landscape reduces the mobility of the ring. Fig. |
shows the effect of the diffusion constant on the runtime for both models. Only the binding model is sensitive to change in [MATH] , having reduced runtime with faster diffusion. This is because the increased mobility of the ring increases the chance it is able to scale the potential barrier constraining it to the MT. |
Although it may be possible to alter [MATH] biochemically, for example by phosphorylation 30 , it is difficult to do so independently of [MATH] . Decreasing [MATH] may be better accomplished by attaching a long inert polymer to the complex to increase viscous drag. |
Effect of time-varying loading force The runtime in the binding model is sensitive only to the instantaneous force provided [MATH] , see the low frequency portion of Fig. |
. If [MATH] pN then [MATH] is on the order of 1 kHz. The runtime in the protofilament model is sensitive to the time over which changes in [MATH] persist. If the force is oscillating with a long period then the rate of detachment will be greater in the high force part of the cycle than if the period is short. This is b... |
. Sigmoidal increase of [MATH] would be a signature of a system that depends on a second time (1/ [MATH] ), like the protofilament model; insensitivity of [MATH] to frequency would imply a binding-style coupling. |
Conclusion Our results indicate a power stroke does contribute to the effective force generated during depolymerization but only becomes dominant at over 2 pN load. We show how a faster depolymerization mechanism must operate at lower loads and argue that the Dam1 ring suppresses depolymerization when it is close to th... |
We have shown that either of two rather different Dam1-MT coupling mechanisms might be operating under piconewton loads. Both models have comparable performance under load; their differences only become apparent under novel experimental conditions. Structural studies cannot resolve the question of which model operates ... |
It is important to note that the present work has neglected in vivo factors such as microtubule-associated proteins (MAPs) or kinases. However, some of these factors operate to increase or reduce the depolymerization rate of the microtubule, a parameter included in the model. We therefore expect the general results to ... |
# Source: arxiv 1005.1159 # Title: Polymorphic Dynamics of Microtubules # Sections: all # Downloaded: 2026-03-03T05:15:06.994661+00:00 |
Polymorphic Dynamics of Microtubules Abstract Starting from the hypothesis that the tubulin dimer is a conformationally bistable molecule - fluctuating between a curved and a straight configuration at room temperature - we develop a model for polymorphic dynamics of the microtubule lattice. We show that tubulin bistabi... |
pacs: 87.16.Ka, 82.35.Pq, 87.15.-v Microtubules are the stiffest cytoskeletal component and play versatile and indispensable roles in living cells. They act as cellular bones, transport roads Genref MTs and cytoplasmic stirring rods MTStirringRod Microtubules consist of elementary building blocks - the tubulin dimers -... |
ActtiveMTBending or thermal fluctuation experiments Pampaloni Taute Keller MTs display length dependent, even non-monotonic apparent stiffness Taute . (iii) They exhibit unusually slow thermal dynamics in comparison with standard semiflexible filaments Taute Janson |
The most bizarre and controversial feature (ii) has been the subject of much debate and some theoretical explanation attempts based on low shear stiffness modulus have been put forward |
Frey . However a careful reanalysis of clamped MT experiments, Figs [MATH] reveals two features not captured by these initial models: the lateral end-fluctuations scale as [MATH] |
while the relaxation times scale as [MATH] . This exotic behavior naively suggests the presence of a limited angular hinge at the MT clamping point. On the other hand artifacts that could trivially lead to a ”hinged behavior” (like loose MT attachment and punctual MT damage) were specifically excluded in experiments Pa... |
Amos . Unfortunately their clear, seminal observations were subsequently forgotten for decades leading to much of the confusion about MTs we are witnessing today. |
Polymorphic MT Model . Starting from assumptions [MATH] we model the tubulin dimer state by a two state variable [MATH] (the tubulin dimer in the ”straight”/”curved” state, cf. Fig. 1a) at each lattice site with circumferential PF index [MATH] [MATH] number of PFs) at longitudinal arclength centerline position [MATH] .... |
[EQUATION] where the integration in [MATH] goes over the annular MT cross-section with [MATH] [MATH] the inner and outer MT radii, with |
[MATH] the energy difference between the [MATH] and [MATH] state and [MATH] the monomer length, [MATH] the ”Ising” cooperative coupling term along the PF contour and with the polymorphism induced prestrain |
[MATH] EpsilonPol where [MATH] is the strain generated in the curved state. The latter can be estimated from the switched PF curvature |
[MATH] Multistable Tub EM to be [MATH] . For an isotropic Euler-Kirchhoff beam, the actual material deformations are related to the centerline curvature via [MATH] with [MATH] the radial vector in the cross-section |
Upon inspection it becomes clear that the phase behavior (straight or curved state stability) is contained in the interplay of the first two terms |
[MATH] and [MATH] while the thermal dynamics is governed by the [MATH] rd [MATH] which rules over defect behavior (cf. Fig. 1d). To understand the basic behavior we first consider a short MT section along which the PFs are in a uniform state [MATH] [MATH] const. can be dropped). Furthermore we resort to the single bloc... |
[MATH] . This ansatz was successfully used by Calladine in modelling bacterial flagellin polymorphic states Calladine . In this approximation the energy density becomes |
[EQUATION] with the bending modulus [MATH] and the polymorphic curvature [MATH] with [MATH] . The MT phase behavior depends on the polymorphic-elastic competition parameter |
[MATH] Physically, [MATH] measures the ratio between polymorphic energy of tubulin switching and the elastic cost of the transition. For [MATH] all the PFs are in the (highly prestrained) state [MATH] while for [MATH] all them are in the state [MATH] - both corresponding to a straight MT. For |
[MATH] we have coexistence of [MATH] locally (meta) stable states : straight ( [MATH] or [MATH] ) and curved state with [MATH] . For [MATH] with [MATH] the curved state is the absolute energy minimum and the straight state is only metastable. Therefore in this regime, the ground state of a microtubule bearing natural l... |
giving us an estimate for the radius of curvature [MATH] . This compares favorably with an estimate of observed helices [MATH] from Venier . The helical stability and the magnitude of the protofilament curvature [MATH] |
Multistable Tub EM with a typical protein Young modulus [MATH] allows us also a simple estimate of the transition energy per monomer [MATH] to [MATH] In general, the energy in Eqs. gives rise to a complex behavior and we focus on basic phenomena. It turns out that a most remarkable deviation from standard wormlike chai... |
Polymorphic Phase Dynamics. To better understand the central phenomenon, we define at each MT cross-section the complex polymorphic order parameter |
[MATH] where [MATH] denotes the ”polymorphic modulus” and [MATH] the ”polymorphic phase” (cf. Fig. 1b). The polymorphic state can then be described by the local (complex) centerline curvature |
[MATH] with [MATH] and [MATH] the natural lattice twist that varies with PF number Kinesin Rotation . This gives rise to a helical MT shape described by the curvature [MATH] and torsion |
[MATH] . For large acting forces both the polymorphic phase [MATH] and amplitude [MATH] will vary along the contour, however for small (thermal) perturbations the phase fluctuations will be dominant Phase Footnote Based on this and on the observation of stable helical states |
Venier we will now assume [MATH] and write the total energy of the MT whose centerline deflection is described by a complex angle [MATH] |
(deflection angles in x/y direction) as follows: [EQUATION] The first energy term is the ”wormlike-chain” bending contribution [MATH] . The second term is the polymorphic phase energy [MATH] with the polymorphic phase stiffness [MATH] which can be related to the density of double defects with energy [MATH] (cf. Fig. 1d... |
The most unusual property of a polymorphic chain is reflected in the rotational invariance of [MATH] . The broken cylindrical to helical symmetry of the straight state is restored by the presence of a ”Goldstone mode” |
[MATH] CommentContiuity consisting of a rotation of [MATH] by an arbitrary angle [MATH] in the material frame (cf. Fig. 1c). This mode that we will call the ”wobbling mode” is a fundamental property of a helically polymorphic filament. The wobbling mode leads to dramatic effects on chain’s fluctuations and is the clue ... |
Persistence Length Anomalies. Among several definitions of the persistence length Persistence Footnote we consider for direct comparison with experiments Pampaloni Taute , the lateral fluctuation persistence length [MATH] |
with [MATH] the transverse displacement at position [MATH] of a MT clamped at [MATH] and [MATH] the statistical average. It is easy to see from Eq. that for small deflections, [MATH] decouples into independent elastic and polymorphic displacements [MATH] , such that |
[MATH] with [MATH] where [MATH] . The coherent helix nature of the MT observed in Venier and the absence of a plateau in [MATH] imply |
Plateau Footnote that [MATH] (the helix wave length). In that limit we obtain [MATH] Whereas for an ideal WLC [MATH] is position and definition independent, the polymorphic fluctuations induce a strong position / distance dependence - a behavior that could be interpreted as ”length dependent persistence length”. Indeed... |
[MATH] the persistence length displays a non-monotonic oscillatory behavior around a nearly linearly growing average value [MATH] . This oscillation is related to the helical ground state while the linear growth [MATH] is associated to the conical rotation of the clamped chain (wobbling mode), cf. Fig. 1c, with an angl... |
[MATH] . For [MATH] the saturation regime with a renormalized [MATH] with [MATH] is reached. The theory can now be compared with the experimental data |
Pampaloni Taute (cf. Fig. 2) that reveal several interesting characteristics in agreement with predictions. In particular the mean linear growth of [MATH] (single parameter fit [MATH] gives |
[MATH] ) and the non-monotonic [MATH] dependence Taute are well captured by the theory. The linearly growing experimental spread of [MATH] with [MATH] is likely linked to the spread of [MATH] in the MT lattice populations Kinesin Rotation . The large length plateau |
[MATH] is not reached even for longest MTs ( [MATH] in agreement with coherent helices Venier . Our best comparison between theory and experiments (cf. Fig. 2) gives |
[MATH] corresponding to [MATH] (proteins with [MATH] up to [MATH] exist 15 GPa Tubes ) and a helix wave length [MATH] This is close to the expected [MATH] |
corresponding to the twist Kinesin Rotation of the predominant [MATH] PF MTs fraction in the in-vitro MTs preparation of Pampaloni Taute . It turns out that [MATH] is larger than in previous studies [MATH] where however polymorphic fluctuations were neglected. The absence of the plateau also allows a lower estimate of ... |
[MATH] and the coupling constant [MATH] Polymorphic Phase Dynamics. To describe the MT fluctuation dynamics we consider the total dissipation functional [MATH] which is composed of an internal dissipation contribution [MATH] and an external hydrodynamic dissipation |
[MATH] with [MATH] the lateral friction constant, [MATH] the solvent viscosity, [MATH] and [MATH] the MT radius and length. The time evolution equation of the phase variable |
[MATH] and elastic displacement [MATH] is given by the coupled Langevin equations [MATH] and [MATH] with [MATH] the thermal noise term. In general this dynamics is highly non-linear however in the experimentally relevant regime where the behavior is dominated by the wobbling mode the equations simplify greatly and we e... |
[MATH] with the relaxation time [MATH] . For small lengths, [MATH] is dominated by internal dissipation while for large lengths [MATH] A careful analysis of the experimental data Taute reveals in fact the latter scaling. An independent single exponent fit gives [MATH] |
with [MATH] . Using the value [MATH] from Fig. 2 and [MATH] with [MATH] Time Footnote we find the theoretical value [MATH] that can be compared with the fit of experimental data (Fig. 3) [MATH] . The excellent agreement of both the exponent and the prefactor leads us again to the strong conclusion that in these experim... |
[MATH] from the limit value of [MATH] for [MATH] . For the available data [MATH] [MATH] Taute this plateau-regime is not yet fully developed and we can only provide an upper estimate from the data |
[MATH] Conclusion . The MT fluctuations are well described - both dynamically and statically - by the bistable tubulin model and the reason for appearance of MT helices becomes obvious. The otherwise mysterious lateral fluctuations reflected in [MATH] and [MATH] scaling are mere consequences of the ”wobbling motion” of... |
# Source: arxiv 1005.3301 # Title: Microtubule Dynamics and Oscillating State for Mitotic Spindle # Sections: all # Downloaded: 2026-03-03T05:15:18.543613+00:00 |
Microtubule Dynamics and Oscillating State for Mitotic Spindle Abstract We present a physical mechanism that can cause the mitotic spindle to oscillate. The driving force for this mechanism emerges from the polymerization of astral microtubules interacting with the cell cortex. We show that Brownian ratchet model for g... |
pacs: 87.17.Ee, 87.16.Ka, 05.40.-a Introduction Establishment of geometrical polarity during mitosis which is a result of asymmetric cell division is a fundamental fact in many live systems. The understanding of underlying molecular mechanisms for this phenomena is an important and challenging issue in biophysics celli... |
Microtubules growing from two organizing centers inside the cell, form a reliable scaffold for the mitotic spindle and make the mechanics of the cell. In the animal cells during mitosis, microtubules are nucleated from centrosomes, organizing centers near nucleus cell . Some microtubules grow from one centrosome toward... |
It is understood that the spindle motion during the cell division is responsible for asymmetric cell division. To gain insight into the cellular mechanisms by which the spindle oscillates and repositiones during the mitosis, extensive studies are carried out on the C. Elegans embryo grillexper . All experiments have cl... |
Considering a solid spindle body and astral microtubules, interacting with cell cortex, two different deriving mechanisms for spindle motion are possible. The first is the pushing and pulling forces related to the motor proteins connecting the microtubule tips to the cell cortex and the second is the pushing mechanism ... |
julicher , where, they have shown that the resulting forces make the spindle oscillation possible. Regarding the second mechanism, the existence of pushing forces is experimentally investigated and theoretically studied MTpushing MTkozlow PNAS_dogterom . In such systems the effects of microtubule bending and buckling a... |
In this article we concentrate on spindle body motion with deriving forces originated from the microtubule and cell cortex interaction. We present a simplified theoretical model with minimum requirement to generate an stable oscillating state for spindle body. In section II, we briefly review the governing equations fo... |
II Microtubule’s dynamics and Brownian ratchet mechanism Let us consider a bundle of stiff and noninteracting microtubules growing from a centrosome (nucleating center). The life of these microtubules can be seen as two different phases: the growth and the shrinkage phases. To describe the stochastic transition of micr... |
[EQUATION] where [MATH] and [MATH] stands for the monomer length. Despite the above equations for microtubule dynamics in the bulk, the microtubules have different evolving equations on the centrosome (nucleation cite), and also on the cell cortex. To express the dynamical equations for the microtubules on the centroso... |
[EQUATION] Correspondingly the boundary equations at the position of the cell cortex that is located at a fixed position [MATH] , can be written as dogterom2 |
[EQUATION] where [MATH] represents the catastrophe frequency on the obstacle. For the next step, we present the steady state solutions to the above system of equations. Using matrix techniques for difference equations and for a wall that is fixed at position [MATH] , the solutions read dogterom2 |
[EQUATION] and the boundary values are: [EQUATION] with [EQUATION] where [MATH] The growth and shrinkage velocities that we have used in above descriptions, are assumed to be the velocities for free microtubules in the bulk. For a microtubule tip reaching an obstacle, polymerization process can persist by pushing which... |
[EQUATION] where [MATH] is the polymerization velocity in the presence of the load and [MATH] stands for the polymerization velocity in the absence of the load. The stall force [MATH] , is the threshold force that stops the polymerization process. The stall force is given by [MATH] , where [MATH] and |
[MATH] are polymerization and depolymerization rates. One should note that the bulk polymerization velocity is related to the microscopic rates by: |
[EQUATION] Defining friction coefficient by [MATH] , we can rewrite the force velocity as: [MATH] In the next section we will use above description about microtubule dynamics and Brownian ratchet mechanism to analyze the dynamics of spindle body. |
III Dynamics of spindle body Astral microtubules with the mitotic spindle constitute our one dimensional model shown in Fig. . In our model, spindle structure, behaves like a rigid body. Dynamical instability of astral microtubules makes stochastic transition between growth and shrinkage phases. During the polymerizati... |
[MATH] and the corresponding number at right side by [MATH] , and using the linearized Brownian ratchet response function, the net polymerization force acting on the spindle body can be written as: |
[EQUATION] which [MATH] is the spindle’s velocity. Beside the above force, the effects due to the complex properties of cellular fluid should be added. The viscoelastic properties of the cytoplasmic fluid can be modeled by defining an effective viscous drag coefficient ( i.e. |
[MATH] ) and an effective elastic modulus for the spindle body ( i.e. [MATH] ). The effects of other astral microtubules in the real three dimensional configuration are also included in this elastic constant, [MATH] . As a result, the governing dynamical equation for the spindle body reads: |
[EQUATION] The elastic term acts as a centering force, it enforces the spindle body to choose an equilibrium position [MATH] Due to the dynamic instability of microtubules structure, a growing microtubule may change to a shrinking one even if it has reached the cortex surface. However, this happens with a new catastrop... |
[EQUATION] which [MATH] is the total number of microtubules. In the limiting case, where the spindle body doesn’t move, these equations are the same as Eq. multiplied by [MATH] as a total number of microtubules . |
To solve above dynamical equations and consequently obtain the dynamical behavior of spindle body, we assume that the system eventually reaches an oscillating steady state that is a steady state solution which oscillates around an equilibrium state. We denote the equilibrium value of spindle position by [MATH] and the ... |
[EQUATION] where [MATH] is a small deviation from the equilibrium position. In this case the number of attached microtubules can be expanded as below: |
[EQUATION] We note that the time scale for the polymerization process is smaller enough than the time scale for the spindle body motion. This fact allows us to assume that for a definite value of [MATH] , the number of attached microtubules is related to the length [MATH] through the steady state relations presented in... |
[EQUATION] then we will have: [EQUATION] and [EQUATION] If we put the above information in Eq.( ) and Eq.( III ), we may derive [MATH] which can be used in Eq.( ). In the first order of perturbation expansion, our governing dynamical equation will be: |
[EQUATION] As one can see the overall motion of the spindle body is determined by a viscose term [MATH] and an inertial term [MATH] . These effective parameters are given by: |
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