text stringlengths 128 2.05k |
|---|
Corollary 3.3 Let [MATH] be as in Assumption 3.1 [MATH] as in 15 ), [MATH] the loss of significance rate, [MATH] as in ( 18 ) and [MATH] the Lyapunov exponent. Then, |
[EQUATION] holds for all [MATH] if [MATH] exists. Before the proof of the theorem can be presented, the following lemma is needed. |
Lemma 3.1 Let [MATH] and [MATH] Then for all [MATH] [EQUATION] holds. Proof. There is nothing to prove in the case [MATH] . So let [MATH] . Two cases are considered. |
1st case: [MATH] Then the inequality reads [MATH] which is equivalent to [MATH] Since [MATH] the assertion follows. 2nd case: [MATH] Then the inequality reads |
[MATH] which is equivalent to [MATH] A sufficient condition to prove the assertion is [MATH] which is equivalent to [MATH] This was already proven in the first case. |
Now everything is prepared to prove Theorem 3.2 Proof of Theorem 3.2 Let [MATH] [MATH] and [MATH] a constant with [MATH] for all [MATH] Starting with Equation ( 14 ) and iterating gives |
[EQUATION] Define [EQUATION] and [EQUATION] Then, [MATH] follows for all [MATH] A sufficient condition for the algorithm to terminate is given by |
[MATH] . Hence, [EQUATION] follows with [MATH] . Using the Assumption 3.1 leads to [EQUATION] By definition, [MATH] follows and hence |
[EQUATION] Next let [EQUATION] for [MATH] and [MATH] , and furthermore [EQUATION] then [MATH] follows for all [MATH] . This gives |
[EQUATION] Let [MATH] be the smallest number such that [MATH] Then consider [MATH] Let [MATH] be a Lipschitz constant of [MATH] , then [MATH] holds for all [MATH] . Consequently, there exists some [MATH] such that |
[MATH] holds for all [MATH] This inequality leads to [MATH] Inserting gives [EQUATION] Now let [MATH] and [MATH] be given. Then choose |
[MATH] such that [MATH] holds. Then for all [MATH] , the above lemma gives [EQUATION] Consider the sequence [MATH] . Observe that, first the sequence [MATH] is increasing and second if [MATH] |
for some [MATH] , then [MATH] or [MATH] . There are two cases. 1st case: [MATH] is bounded. Then, there exists some constant [MATH] such that [MATH] holds for all [MATH] Choose now [MATH] small enough such that [MATH] holds. Then, compute the upper limit to [MATH] By taking the upper limit of ( 20 ), |
[MATH] follows. 2nd case: [MATH] is not bounded. Then, for any [MATH] and any [MATH] there is some [MATH] with [MATH] . Since, by definition, |
[MATH] holds as well as [MATH] , the inequality [MATH] follows. This shows [MATH] Next it is stated that for all [MATH] and [MATH] |
[MATH] holds. This shows [MATH] Assume otherwise. Then, for some [MATH] and [MATH] first [MATH] holds and second [MATH] holds for all |
[MATH] . Using ( 20 ), the first expression gets [MATH] . Choose [MATH] small enough such that [MATH] holds for all [MATH] . Then, for sufficiently high [MATH] |
[MATH] follows. In the second statement, the sum can be split the following way: [MATH] . The first addend on the left side is bounded form below by |
[MATH] , the second addend is bounded from below by [MATH] . Hence, [MATH] follows, but this is a contradiction. In the end, it is shown that, if [MATH] |
holds, the algorithm presented here is optimal with respect to the loss of significance rate. This means that no algorithm with the specification presented at the beginning of this section has a lower loss of significance rate than the algorithm presented in this section. |
Proposition 3.8 Let [MATH] be an orbit of the dynamical system [MATH] and [MATH] . Then, for any [MATH] there exists an [MATH] such that for any [MATH] there is some |
[MATH] such that the following holds. Let an initial value [MATH] be given and consider the corresponding orbit [MATH] . Then, [EQUATION] |
holds. Proof. Let [MATH] be given. Then there exists some [MATH] such that [MATH] holds for all [MATH] . For given [MATH] there is some [MATH] with |
[MATH] , otherwise [MATH] would not exist. Consider now an orbit [MATH] with [MATH] for [MATH] where [MATH] is a Lipschitz constant of [MATH] |
and [MATH] arbitrary. Then, for [MATH] , the following estimation holds. [EQUATION] where [MATH] Iterating finally gives [MATH] Now determine some constant [MATH] such that |
[MATH] holds for all [MATH] the following way. A short calculation shows that this is equivalent to [MATH] Using [MATH] finally gives [MATH] as a sufficient condition. Set [MATH] . Let [MATH] be given. Set |
[MATH] , then [EQUATION] and hence [EQUATION] follows. Proposition 3.9 Let [MATH] be as in Assumption 3.1 and [MATH] given such that [MATH] exists and |
[MATH] . Consider an algorithm computing an initial segment [MATH] of the orbit of [MATH] with relative error [MATH] for some [MATH] [MATH] Then the algorithm has a loss of significance rate [MATH] |
Proof. Let [MATH] be given and [MATH] big enough such that the previous proposition holds for some [MATH] Choose some [MATH] [MATH] . Let [MATH] be big enough such that |
[EQUATION] holds, where [MATH] such that [MATH] for all [MATH] Consider [MATH] as the initial value of another orbit [MATH] Then, with the above proposition, |
[EQUATION] follows. Condition ( 21 ) can also be written as [MATH] Consider now some precision [MATH] , the algorithm actually is working with. Assume for simplicity further that for the initial value [MATH] holds and assume without loss of generality [MATH] . Then, first, |
[MATH] holds. Second, the above condition gives [MATH] since [MATH] holds. In other words, the above condition gives an upper bound |
[MATH] on the needed precision [MATH] . Assume furthermore that [MATH] is big enough such that [MATH] and [MATH] is computed with the demanded precision, that is |
[MATH] and [MATH] holds. Using 22 ) gives [MATH] But this is a contradiction since [MATH] So the upper bound on [MATH] calculated above is still too small. Hence, [MATH] |
must hold. Since Condition ( 21 ) also holds for any [MATH] and the same [MATH] as well as for any [MATH] [MATH] follows for all [MATH] . Computing [MATH] finally gives the assertion. |
Furthermore, if [MATH] holds, then Corollary 3.3 gives [MATH] for the algorithm presented at the beginning of this section. Using the above proposition then leads to the following theorem. |
Theorem 3.3 Let [MATH] be as in Assumption 3.1 and [MATH] given such that [MATH] exists and [MATH] holds. Consider an algorithm computing an initial segment [MATH] |
of the orbit of [MATH] with relative error [MATH] . Then this algorithm has a loss of significance rate greater or equal to that of the algorithm specified by the recursion ( 13 ) and 14 ). |
Conclusions In this paper, two main issues are addressed. First it is shown that a mathematically rigorous treatment of the computability aspects of the iteration of a real function in terms of arbitrary-precision floating-point arithmetic including automated error analysis is straightforward. Also, this treatment is i... |
Second, the results show that the Lyapunov exponent, a central quantity in dynamical systems theory, also finds its way into complexity theory, a branch in theoretical computer science. In dynamical systems theory, the Lyapunov exponent describes the rate of divergence in the course of time of initially infinitesimal n... |
. The reason is that due to the boundedness of the phase space, any two different orbits cannot separate arbitrarily far away. However, the loss of significance rate shows that the Lyapunov exponent has on long time scales not only an asymptotic significance but also a concrete practical one. Acknowledgments The author... |
# Source: arxiv 1004.1368 # Title: Force transduction by the microtubule-bound Dam1 ring # Sections: all # Downloaded: 2026-03-03T05:15:13.759629+00:00 |
Force transduction by the microtubule-bound Dam1 ring Abstract The coupling between the depolymerization of microtubules (MTs) and the motion of the Dam1 ring complex is now thought to play an important role in the generation of forces during mitosis. Our current understanding of this motion is based on a number of det... |
Key words: Brownian ratchet; burnt bridges; DASH; protofilaments; mitosis; motors Introduction Mitosis is the mechanism of cell division in eukaryotic cells. In mitosis, chromosomes condense and are arranged at the center of the cell by the mitotic spindle. Microtubules (MTs) are protein fibers, composed of [MATH] para... |
. To achieve segregation, depolymerizing kinetochore-attached microtubules (KMTs) must generate forces, e.g., to overcome chromosomal drag in the cytosol . There is evidence that mitotic MT force generation occurs in the absence of MT minus-end directed motor proteins |
and when minus-end depolymerization is inhibited . Previously, a hypothetical sleeve had been proposed to couple MT depolymerization to kinetochores . A 10-protein complex, purified from budding yeast , called Dam1 (or DASH) has been observed to form rings around MTs . Dam1 rings have been observed tracking depolymeriz... |
10 and an optical trap has been used to measure force-distance traces for Dam1-coated polystyrene beads attached to depolymerizing MTs |
11 12 13 . Intriguingly, Dam1 has been shown to be essential for chromosome segregation in budding yeast 14 15 and important for avoiding mis-segregation problems in fission yeast 16 |
Mechanisms of force transduction Several models have been proposed to explain how the Dam1 ring can couple the kinetochore to a depolymerizing MT so as to produce a force. Over two decades ago Hill proposed the first quantitative model describing how a depolymerizing microtubule could be harnessed for the production of... |
17 18 19 and, following the discovery of the Dam1 ring, was extended to reflect current structural knowledge and incorporate the hypothesis that Dam1 forms rigid transient links to the MT |
20 . Another independent model postulated an electrostatic attraction maintaining the rings position at the tip of the MT 21 combined with a powerstroke. All recent models include a combination of the following features: 1) the intrinsic diffusion of the Dam1 ring; 2) an effective powerstroke due to curling PFs; 3) an ... |
In what follows we describe two distinct “minimal” models, both of which describe a functional Dam1-mediated force transduction system. In the protofilament |
model the splaying PFs at the depolymerizing end physically prevent the ring from sliding off. In binding models an attraction between the ring and MT provides an energy barrier preventing detachment. The two models are not mutually exclusive – a hybrid model, incorporating both contributions, may also apply although o... |
22 23 , that certainly play important additional roles in vivo Some previous studies have incorporated a powerstroke, arising from the motion of PFs, in driving the motion of the Dam1 ring |
17 20 21 . Indeed, it has been demonstrated that PFs can push a bead attached to the side of a MT 24 , with a force of about 5 pN per 1–2 PFs. However, it is also known that models in the “burnt bridges” 25 class require no powerstroke per se |
to generate motion 26 . Rather, purely diffusive Brownian motion can be rectified if “bridges” (here segments of MT) are lost (depolymerize) after they have been crossed. No instantaneous physical force is required, although the resultant rectified Brownian motion does give rise to a force in the thermodynamic sense. S... |
Generalized model We seek to analyze a general model that includes both a diffusive burnt bridge mechanism and a powerstroke, in order to determine their relative contribution. Here the powerstroke involves a depolymerization event which “unzips” PFs and moves the position of the last unbroken section of MT; a new sect... |
[MATH] , whenever the ring has diffused a distance [MATH] from the end. This can be thought of as the depolymerization velocity of a bare MT because, in this case, there is no Dam1 ring anywhere on the MT. We make no prior assumptions as to which contribution dominates, rather we determine this by fitting the parameter... |
Our model involves a clear distinction between two mechanisms (only) and represents the simplest possible model capable of explaining this data. It can be biophysically motivated on the grounds that the Dam1 ring interacts with neighboring tubulin and so the rate of PF unzippering at the MT end should depend on how clo... |
24 . This is because, with a powerstroke-only model, we would not expect the velocity of Dam1 ring to be significantly slowed under a force as low as |
[MATH] pN; the data shows a significant slowing. We find that the length scale [MATH] controlling burnt-bridge reactions, a free fit parameter, is close to the axial length of a tubulin dimer. We speculate that this may provide indication of “crack”-like splitting of the MT, as discussed below. |
Model The Dam1 ring complex is reported to be capable of axial movement with respect to the MT 10 . Therefore, we treat the Dam1 ring as a particle undergoing one-dimensional Brownian motion in a potential [MATH] (shown for two different models in Fig. ). The fully intact MT extends away from the depolymerizing end for... |
). The following Fokker-Plank equation 28 determines the probability density [MATH] for the ring’s position relative to the (moving) end, |
[EQUATION] where [MATH] is the diffusion constant of the ring. This approach is appropriate providing the depolymerization velocity [MATH] of the MT is not too fast (bounds given later in this section), otherwise we must instead treat this as a full moving boundary problem. Since the microtubule depolymerization is her... |
In the following we assume the Dam1 ring is sufficiently stable that it can only dissociate by slipping off the tip. For simplicity we restrict our analysis to continuous depolymerization processes only and discount the possibility of rescue and polymerization. Although it would be straightforward to include such proce... |
A force [MATH] appears in Eq. . This is the magnitude of the applied force [MATH] on the Dam1 ring while on the MT ( [MATH] ) since the ring must do work to move against this force. Hence, from Eq. it can be shown that, for constant (or slowly varying) [MATH] the probability distribution |
[MATH] is of Boltzmann form [EQUATION] where the ring typically explores a characteristic diffusion length [MATH] from the MT end and positive values of [MATH] here indicate loads pulling in the negative [MATH] direction (towards the MT end). We assume that the depolymerization is quasistatically slow . This is appropr... |
[MATH] . In this case the distribution of the ring position is always close to the equilibrium probability distribution that it would have on an MT that was not depolymerizing. This sets an upper bound on the depolymerization velocity, or equivalently a lower bound on the load force, beyond which our theory is at best ... |
[MATH] with Eq. , we estimate these values to be 500 nm/s and 0.04 pN respectively. Under these conditions the average ring velocity is equivalent to the depolymerization velocity of the MT. |
Force dependent depolymerization velocity The powerstroke and burnt-bridge reactions can be thought of as arising from transitions over an energy barrier of the form shown in Fig. , where the free energy [MATH] of PF curling is shown as varying with protofilament angle [MATH] |
and the distance of the Dam1 ring from the MT tip [MATH] . The figure shows only a putative schematic of the free energy of PF curling reaction, and should not be confused with the potential [MATH] in which the Dam1 ring diffuses. |
PFs may produce a power stroke that pushes the ring with force [MATH] estimated from experimental evidence to be [MATH] pN 24 . This is the slope down the descending valley, diagonally right to left, in Fig. . Provided that the load force [MATH] the powerstroke will give rise to a depolymerization velocity [MATH] that ... |
[MATH] is so much larger than any force considered here, it is reasonable to make the limited assumption that [MATH] is constant for all experimentally measurable load forces of a few pN or less. |
In addition the MT can also depolymerize when the Dam1 ring is further than a critical distance [MATH] from the end of the MT. In this case the burnt-bridge reaction gives rise to a depolymerization velocity [MATH] that is the rate at which the last intact dimer on the MT crosses the lower part of the ridge-like energy... |
The resultant velocity due to both mechanisms is the sum of the probability that the ring is close to the MT end [MATH] , multiplied by the powerstroke velocity, and the probability that it is far [MATH] , multiplied by the burnt-bridge velocity, |
[EQUATION] The velocity follows from Eqs. and [EQUATION] The variation of this velocity with load is shown in Fig. for [MATH] nm/s 29 , and the values [MATH] nm/s and |
[MATH] nm that correspond to the best fit to data 12 . Since a “burnt-bridges”-only model fails to fit the data sufficiently (i.e. [MATH] ) it suggests that a powerstroke plays a role in forced Dam1 motion. It should be noted that, although in this model |
[MATH] as [MATH] , we do not suggest this is a physical feature of the system. Rather it is the consequence of the assumption that protofilaments are perfectly rigid and the powerstroke reaction is asymptotically strong. Our model would need modification for forces approaching |
[MATH] . As discussed later, PFs are estimated to require tens of pN to bend. Two models for Dam1 ring retention We now proceed to calculate the mean time the Dam1 ring will remain on a MT and transduce force . This time is controlled by different physics in the protofilament and the binding models, see Figs. and . How... |
Runtime: Binding model The binding model involves a ring diffusing on a MT according to Eq. , leading to a depolymerization velocity as given in Eq. . However, in order to detach from the MT end the ring must overcome a linear potential imposed by the Dam1-MT binding energy [MATH] as it slides off the end of the ring. ... |
. Previous models invoked a [MATH] that also determined the roughness of the energy landscape through “linkers” 20 whose existence is supported by binding studies 33 . Here we don’t make this assumption, rather [MATH] could be due to less specific interactions without significant energy barriers between neighboring sit... |
34 but, importantly, can vary independently of the diffusion constant [MATH] . This, in turn, is fixed by the smoothness of the underlying energy landscape experienced by the ring as it diffuses along the MT (distinct from the energy landscape experienced by an unzippering PF shown in Fig. ). This model assumes that th... |
). To move to the left (towards negative [MATH] ) it must partially unbind from the MT, to move to the right (positive [MATH] ) it must do work against the applied force. The potential gradients experienced by the Dam1 ring determine the load force [MATH] (while on the MT, [MATH] ) and the resultant force [MATH] (while... |
[MATH] ). The force on the ring, adopting a sign convention where a positive force acts in the direction of positive [MATH] , is therefore given by |
[EQUATION] where [MATH] is the unbinding region. If the ring is in the region [MATH] then it is lost, and if lost we assume it never returns, hence we have [MATH] for [MATH] |
Symmetry from electron microscopy 10 and copy number 32 experiments suggest 16 complexes are required to form the Dam1 ring, however, the total bond energy may not be additive and this should therefore be regarded as an extreme upper bound on the total binding energy. |
The detachment of the ring can be cast as a classical Kramers escape problem 35 . To solve Eq. with Eq. we followed the method in 36 37 . In this way we obtain the lifetime of the metastable state directly from the Laplace transformed version of Eq. , with initial condition |
[MATH] , where [MATH] is the Dirac delta function, although the precise form of this initial condition is unimportant. The mean time the ring remains on the MT is the runtime [MATH] |
[EQUATION] where [MATH] is a small distance. Runtime: Protofilament model The protofilament model involves a ring diffusing on a MT according to Eq. leading to a depolymerization velocity Eq. , as before. However, in order to detach from the MT end the ring has to wait until all protofilaments have broken (depolymerize... |
. We no longer require the Dam1 ring to overcome a Dam1-MT binding energy. Electron microscopy reveals that short, separated PFs splay outwards at the depolymerizing MT end |
18 19 and it is quite plausible that these block the escape of the ring; the elastic energy required to straighten a curled PF 38 follows from measurements of their rigidity |
39 40 and is of the order of tens of [MATH] per subunit, i.e. very large. The “frayed” PFs near the end of the MT are curved and laterally separate. The unzipping (depolymerization) of the MT lattice (see Fig. A) is most accurately described as a process which transfers length from the polymerized MT into separated PFs... |
41 . When not constrained by lateral bonds, PFs relax into a curved state. We model unzipping as a Poisson process with rate [MATH] . Each unzipping event extends every PF curl by some microscopic, or subunit, length |
[MATH] , leading to a depolymerization velocity [MATH] . This microscopic length might be the tubulin dimer repeat distance [MATH] , if the MT splits between a particular pair of PFs, or otherwise smaller than this. When the ring is within a small length [MATH] unzipping is inhibited. To more carefully analyze this pro... |
[EQUATION] The distribution of the ring position in this model follows Eq. . Detachment occurs when all PF curl lengths reach zero . Since [MATH] is a function of the applied force [MATH] , according to Eq. [MATH] increases under load. From Eq. , we have that the characteristic distance of the ring from the tip is [MAT... |
We assume that tubulin subunits on the frayed PFs break independently according to a Poisson process with rate [MATH] . The depolymerization of PFs then follows from the loss of all PF material beyond the break, as in previous computational models 42 . A PF curl reaches zero length if the axial bond nearest to the unzi... |
. Since this occurs with a rate [MATH] the waiting time [MATH] for PF curl [MATH] to break off completely is an exponential random variable. The wait time for all [MATH] PF curls breaking is the order statistic [MATH] . The distribution function for this time is [MATH] and the mean wait time (see section 4.6 from 43 or... |
[EQUATION] where [MATH] is the harmonic number, roughly [MATH] for [MATH] as can be seen by converting the sum to an integral, [MATH] |
denotes the ensemble average. The Dam1 ring will therefore no longer be secured to the MT end and will detach after a time [MATH] provided that no unzipping events having taken place during the time [MATH] . If the MT has unzipped then the PFs extend (from their base), effectively “restarting” the waiting process. |
Fundamentally we are interested in the mean runtime [MATH] , this being the time taken for the curled PFs to all depolymerize completely even while the MT is simultaneously undergoing stochastic unzipping events. [MATH] can be found by counting the number of unzipping events [MATH] that occur before the PFs all success... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.