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.

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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)
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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

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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.

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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.
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Heuristic Analysis of the model

• Dynamically coupled hydrolysis

The total rate of hydrolysis at each microtubule
end is dynamically coupled to its growth rate.
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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)

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Heuristic Analysis of the model

• Cap Size

13
Heuristic Analysis of the model

• Catastrophe rate

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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.
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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.
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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

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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
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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.)

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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.

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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.
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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.
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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.

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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.

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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.