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Title: Microtubule dynamics: Caps, catastrophes, and coupled hydrolysis


1
Microtubule dynamics Caps, catastrophes, and
coupled hydrolysis
  • Presented by
  • XIA,Fan

2
Introduction
  • MTs are long and extremely rigid, tubular
    polymers, assembled from tubulin. Each tubulin
    consists of two closely related polypeptides.
    They arrange along the microtubule in a
    head-to-tail pattern, forming a protofilament.
    Microtubules in living cells usually have 13
    protofilaments.
  • MTs take part in many important biological
    process, like intracellular transportation, cell
    division and so on.

3
Introduction
  • Dynamic instability
  • a MT can repeatedly and apparently randomly,
    switch between persistent states of assembly and
    disassembly in a constant concentration.
  • Hydrolysis of GTP increase the chemical potential
    of the monomer after assembly, which explains the
    coexistence of the growing and the shrinking
    states but fails to explain the the dynamics of
    the transitions between these states.
  • Mitchison and Kirschner transitions occur as a
    consequence of competition between assembly and
    GTP hydrolysis. A growing microtubule has a
    stabilizing cap of GTP tubulin. If hydrolysis
    overtakes the addition of new GTP tubulin, the
    cap is gone and the MTs end undergoes a change
    to the shrinking state, a so-called catastrophe.
  • GTP hydrolysis may not be the rate-limiting
    process in the change to a disassembly-favoring
    state. Conformational changes of tubulin or
    structural changes of the MT are other
    candidates.

4
Introduction
  • Failure of other cap model can not explain the
    following experiment
  • The observed relations between concentration and
    frequency of catastrophe in a quantitative
    manner. (catastrophe rate)
  • In the dilution experiment, the delay time is
    independent of initial concentration. (delay
    time)
  • No GTP can be found after 15-20s dead time of the
    experiment, in which the microtubule is grown in
    a manner to assure maximal GTP contents. (GTP
    content)
  • Requirement of a successful model
  • Resolve the above contradiction.
  • Explain a range of other observations
  • the distribution of catastrophe times is nearly
    exponential
  • a small cap assembled from a nonhydrolyzable GTP
    analog can stabilize a microtubule
  • cutting a microtubule usually results in a
    catastrophe and others.

5
Introduction
  • Effective theory
  • Several rather detailed cap models is not
    practical since the experimental data available
    are insufficient to determine many free
    parameters.
  • A theory that is not formulated from in terms of
    fundamental variables and phenomena, but in terms
    of fewer variables on a coarser scale.
  • Several data sets are available from experiments
    investigating different manifestations of the
    cap. None of the existing models have been able
    to explain more than selected aspects of the
    data. Thus a model should contain only a few free
    parameters if they are to be unambiguously
    determined by the fit.

6
An Effective TheoryMicroscopic description
the polymerization rate constant (can be
calculated) the length contributing to the
polymer by one monomer the average velocity the
end polymer end grow (can be observed)
In a normal polymerization processed, the on rate
kg is usually accompanied by an off rate and the
growth velocity vg is the net effect of the
competition between these two rates. In the case
of microtubules, the off rate is 0.
the hydrolysis rate constant where a tubulin-t
monomer neighbors a tubulin-d monomer. the
average velocity the tubulin-t will hydrolyze
from its borders with tubulin-d. It may depend
on whether the border moves towards the plus of
the minis end. (vectorial hydrolysis) (determined
by fitting) the hydrolysis rate constant inside
a section of polymer that consists of tubulin-t.
(scalar hydrolysis) the rate that the new
boarder forms (per unit length per unit time)
(determined by fitting)
7
An Effective TheoryMicroscopic description
On the average, the cap grows with velocity
vvg-vh. And hydrolysis of its inerior breaks it
into a shorter cap and another section of
tubulin-t at rate rx, where x is the
instantaneous length of the cap. The length of
the resulting shorter cap is any fraction of x
with equal probability.
  • The model does not provide a mechanism for
    rescues, which presumably are due to an entirely
    separate phenomenon. It means that he microtubule
    depolymerizes uninhibited by the patch.
  • This is a random process and the rate constants
    only describe the average outcome. But the
    fluctuations around average

8
An Effective TheoryGetting Rid of the Microscopic
  • Take the limit while keeping r, v, vg,
    and vh at fixed values they are of order zeor in
    .
  • Retain one consequence of microscopic scale
    fluctuations around the average are inevitable,
    but only of order one in
  • the variance of this cap length distribution
    grows in time with a constant rate
  • i.e., the cap length evolves in time as the
    coordinate x of a particle diffusing in one
    dimension with diffusion constant D.
  • Complete description of the model
  • a cap of length x grows steadily with velocity v,
    but also experiences two different stochastic
    process
  • A diffusionlike time evolution with diffusion
    constant D
  • With probability rx per unit time the length x of
    the cap will be reduced to any fraction its
    length with equal probability.
  • The event that a caps length x happens to
    decrease to zero, represents a catastrophe.

9
An Effective TheoryMaster Equation
The ensemble density of microtubules with caps of
length x at time t.
Microtubules with caps of length longer than x.
The total number of microtubules with caps at
time t. The equation to the left shows how the
number of capped microtubules evolves in time.
To ensure the diffusive loss will be infinite.
The catastrophe rate, the rate per capped
microtubule at which caps are lost.
10
Heuristic Analysis of the model
  • Dynamically coupled hydrolysis

The total rate of hydrolysis at each microtubule
end is dynamically coupled to its growth rate.
11
Heuristic Analysis of the model
  • Cap size
  • (According to different values of three
    parameters v, D and r, three regimes of behavior
    are expected.)
  • Large-positive-velocity
  • The cap growths quickly in length and only
    the cutting prevents the cap from becoming too
    large. (v, r)
  • Large-negative-velocity
  • The cap shrinks on average and only the
    fluctuations allow the cap to exist. (v, D)
  • Small-velocity
  • Diffusion and cutting are most important. (D,
    r)

12
Heuristic Analysis of the model
  • Cap Size

13
Heuristic Analysis of the model
  • Catastrophe rate

14
Heuristic Analysis of the model
  • Delay time for dilution-induced catastrophes

The length is short enough that the negative
growth velocity causes it to disappear before the
next cutting event.
The delay time for a dilution induced catastrophe.
15
Heuristic Analysis of the model
  • Amount of GTP in a microtubule
  • The tubulin-t exists as a cap on each end and a
    number of GTP patches. It is convenient to treat
    the two caps as one patch with the capss summed
    length.

The total number of patches at time t.
The total length of tubulin-t left at time t.
The loss term describes the rate at which patches
disappear by shrinking to zero length. It depends
on the patch length distribution. It is rather
complicated.
16
Catastrophe Rate
  • Connecting theory and experiment
  • Catastrophe rate is the frequency at which
    microtubules change from their growing to their
    shrinking state.
  • Experimentally, it is found as the ratio between
    the total number of catastrophes observed and the
    total time spent in the growing state. In the
    experiment, microtubules are grown from seeds and
    a shrinking microtubule always vanishes entirely,
    whereupon a new microtubule grows from the seed.
  • Initial condition each cap is initially created
    with 0 length.
  • Boundary condition
  • Catastrophe rate

17
Catastrophe Rate
  • Characteristic features of theoretical result

When v is big
f seems to be constant for higher tubulin
concentrations, while f increases rapidly if vg
is decreased to small values.
Dots with error bars represent experimental
result. The full curve represents the theoretical
expression.The dashed curve represents the
theoretical approximate expression from the above
equation. All three theoretical expressions were
fitted to the experimental results, using
18
Catastrophe Rate
  • Comparing the theory to experimental results for
    the catastrophe rate
  • Satisfactory agreement between theory and
    experimental result for the catastrophe rate for
    plus ends by treating vh() and r as fitting
    parameters.
  • Though the values for vg and vh are different for
    plus and minus ends, when vg is rather big, the
    catastrophe rate is the same for both ends. This
    prediction is consistent with experimental
    results. (These results are not that precise,
    however, and the validity of this prediction is
    another experimental acid test of the model. To
    the extent the model survives the test, such an
    experiment is a very direct way to measure the
    parameter r.)

19
Dilution Experiment
  • Motivation
  • Extended cap model (uncoupled vectorial
    hydrolysis) long delay times upon dilution were
    expected for high growth rate.
  • Experiment catastrophe rate is essentially
    independent of the growth rate.

20
Dilution Experiment
  • Initial condition
  • In the case of strong dilution (v - vh)

Before dilution, the microtubule is grown at high
tubulin concentration. Then we can neglect the
diffusive term in our master equation. Then the
steady-state solution to the master equation is
found.
The distribution in time of catastrophes.
The average lifetime upon dilution.
21
Dilution Experiment
  • Experiment

Left, plus end right, minus end top, delay as a
function of initial growth velocity. Curves are
theoretical mean and standard deviation of the
delay from the theoretical calculation. Bottom,
histogram from the experimental data. The curves
are fits of the theoretical calculation.
22
Combined Fit
23
Combined Fit
  • The difference is not radical but nevertheless
    significant. This reemphasized the desirability
    of
  • having both types of experimental data taken
    under the same condition.
  • Since the combined fit overdetermines the three
    parameters. We use the excess of information
  • available to fit also the value of . The
    result is close enough to the true one to give
    good
  • support for the model.

24
Experiments visualization the GTP cap
  • Experiment a minimal size for a cap that will
    stabilize a microtubule is estimated roughly to
    contain 40 tubulin dimers.
  • There is no way to define a minimal cap size that
    will stabilize a growing microtubule because of
    the fluctuations in the cap size and hence no
    absolute stability.
  • We expect that a microtubule must grow faster
    than the cap hydrolyzes from its trailing edge
    for the cap to exist. parameterizes the
    relative importance of the various processes
    contributing to the caps dynamics at large
    values catastrophes are rare and the microtubule
    is stable.. Chose as the separator of
    stabilized microtubules from unstable ones.
  • Use the parameter values obtained from fit, we
    find that the minimal cap contians 26 tubulin-t
    dimers.

25
Conlusion
  • Self-consistency It was assumed that . Use
    the parameter values from the fit,
  • Different Microscopic interpretation of the model
  • Rescue another model is needed. But much less
    data has been collected on rescues than on
    catastrophes.
  • Issue for future experiment
  • More experiment would overdetermined the
    parameters and provide a rigorous test of the
    model.
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