Transcript Document

Microtubule dynamics:
Caps, catastrophes,
and coupled hydrolysis
Presented by
XIA,Fan
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.
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 MT’s 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.
Introduction

•
•
•

•
•
o
o
o
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.
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.
An Effective Theory
Microscopic 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)
An Effective Theory
Microscopic description
On the average, the cap grows with velocity
v=vg-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
An Effective Theory
Getting Rid of the Microscopic



•
o
o
•
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 cap’s length x happens to decrease to zero, represents a catastrophe.
An Effective Theory
Master 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.
Heuristic Analysis of the model

Dynamically coupled hydrolysis
The total rate of hydrolysis at each microtubule
end is dynamically coupled to its growth rate.
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)

Heuristic Analysis of the model

Cap Size
Heuristic Analysis of the model

Catastrophe rate
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.
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 caps’s 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.
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:
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
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.)
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.
Dilution Experiment

Initial condition
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.

In the case of strong dilution (v’ = - vh)
The distribution in time of catastrophes.
The average lifetime upon dilution.
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.
Combined Fit
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.
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 cap’s 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.
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.