Microtubule dynamics: Microtubule dynamics:

Caps, catastrophes, Caps, catastrophes, and coupled hydrolysisand coupled hydrolysis

Presented byPresented by

XIA,FanXIA,Fan

IntroductionIntroduction MTs are long and extremely rigid, tubular polymers, MTs are long and extremely rigid, tubular polymers,

assembled from tubulin. Each tubulin consists of two closely assembled from tubulin. Each tubulin consists of two closely related polypeptides. They arrange along the microtubule in related polypeptides. They arrange along the microtubule in a head-to-tail pattern, forming a protofilament. Microtubules a head-to-tail pattern, forming a protofilament. Microtubules in living cells usually have 13 protofilaments.in living cells usually have 13 protofilaments.

MTs take part in many important biological process, like MTs take part in many important biological process, like intracellular transportation, cell division and so on.intracellular transportation, cell division and so on.

IntroductionIntroduction Dynamic instabilityDynamic instability• a MT can repeatedly and apparently randomly, switch a MT can repeatedly and apparently randomly, switch

between persistent states of assembly and disassembly in a between persistent states of assembly and disassembly in a constant concentration.constant concentration.

• Hydrolysis of GTP increase the chemical potential of the Hydrolysis of GTP increase the chemical potential of the monomer after assembly, which explains the coexistence of monomer after assembly, which explains the coexistence of the growing and the shrinking states but fails to explain the the growing and the shrinking states but fails to explain the the dynamics of the transitions between these states.the dynamics of the transitions between these states.

• Mitchison and Kirschner: transitions occur as a consequence Mitchison and Kirschner: transitions occur as a consequence of competition between assembly and GTP hydrolysis. A of competition between assembly and GTP hydrolysis. A growing microtubule has a stabilizing cap of GTP tubulin. If growing microtubule has a stabilizing cap of GTP tubulin. If hydrolysis overtakes the addition of new GTP tubulin, the hydrolysis overtakes the addition of new GTP tubulin, the cap is gone and the MT’s end undergoes a change to the cap is gone and the MT’s end undergoes a change to the shrinking state, a so-called catastrophe.shrinking state, a so-called catastrophe.

• GTP hydrolysis may not be the rate-limiting process in the GTP hydrolysis may not be the rate-limiting process in the change to a disassembly-favoring state. Conformational change to a disassembly-favoring state. Conformational changes of tubulin or structural changes of the MT are other changes of tubulin or structural changes of the MT are other candidates.candidates.

IntroductionIntroduction Failure of other cap model: can not explain the following Failure of other cap model: can not explain the following

experiment:experiment:• The observed relations between concentration and frequency The observed relations between concentration and frequency

of catastrophe in a quantitative manner. of catastrophe in a quantitative manner. (catastrophe rate)(catastrophe rate)• In the dilution experiment, the delay time is independent of In the dilution experiment, the delay time is independent of

initial concentration. initial concentration. (delay time)(delay time)• No GTP can be found after 15-20s dead time of the No GTP can be found after 15-20s dead time of the

experiment, in which the microtubule is grown in a manner to experiment, in which the microtubule is grown in a manner to assure maximal GTP contents. assure maximal GTP contents. (GTP content)(GTP content)

Requirement of a successful model:Requirement of a successful model:• Resolve the above contradiction.Resolve the above contradiction.• Explain a range of other observations: Explain a range of other observations: o the distribution of catastrophe times is nearly exponentialthe distribution of catastrophe times is nearly exponentialo a small cap assembled from a nonhydrolyzable GTP analog can a small cap assembled from a nonhydrolyzable GTP analog can

stabilize a microtubulestabilize a microtubuleo cutting a microtubule usually results in a catastrophe and cutting a microtubule usually results in a catastrophe and

others.others.

IntroductionIntroduction Effective theory:Effective theory:• Several rather detailed cap models is not practical Several rather detailed cap models is not practical

since the experimental data available are since the experimental data available are insufficient to determine many free parameters.insufficient to determine many free parameters.

• A theory that is not formulated from in terms of A theory that is not formulated from in terms of fundamental variables and phenomena, but in fundamental variables and phenomena, but in terms of fewer variables on a coarser scale.terms of fewer variables on a coarser scale.

• Several data sets are available from experiments Several data sets are available from experiments investigating different manifestations of the cap. investigating different manifestations of the cap. None of the existing models have been able to None of the existing models have been able to explain more than selected aspects of the data. explain more than selected aspects of the data. Thus a model should contain only a few free Thus a model should contain only a few free parameters if they are to be unambiguously parameters if they are to be unambiguously determined by the fit.determined by the fit.

An Effective TheoryAn Effective TheoryMicroscopic descriptionMicroscopic 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 TheoryAn Effective TheoryMicroscopic descriptionMicroscopic description

•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

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.

An Effective TheoryAn Effective TheoryGetting Rid of the Getting Rid of the

MicroscopicMicroscopic Take the limit while keeping r, v, vg, and vh at fixed values; Take the limit while keeping r, v, vg, and vh at fixed values; they are of order zeor in .they are of order zeor in .

Retain one consequence of microscopic scale: fluctuations around Retain one consequence of microscopic scale: fluctuations around the average are inevitable, but only of order one in the average are inevitable, but only of order one in

the variance of this cap length distribution grows in time with a the variance of this cap length distribution grows in time with a constant rate constant rate

i.e., the cap length evolves in time as the coordinate x of a particle i.e., the cap length evolves in time as the coordinate x of a particle

diffusing in one dimension with diffusion constant D.diffusing in one dimension with diffusion constant D. Complete description of the model: Complete description of the model: • a cap of length x grows steadily with velocity v, but also experiences a cap of length x grows steadily with velocity v, but also experiences

two different stochastic process: two different stochastic process: o A diffusionlike time evolution with diffusion constant DA diffusionlike time evolution with diffusion constant Do With probability rx per unit time the length x of the cap will be With probability rx per unit time the length x of the cap will be

reduced to any fraction its length with equal probability.reduced to any fraction its length with equal probability.• The event that a cap’s length x happens to decrease to zero, The event that a cap’s length x happens to decrease to zero,

represents a catastrophe.represents a catastrophe.

An Effective TheoryAn Effective TheoryMaster EquationMaster 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 Heuristic Analysis of the modelmodel

Dynamically coupled hydrolysisDynamically coupled hydrolysis

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

Heuristic Analysis of the Heuristic Analysis of the modelmodel

Cap size Cap size (According to different values of three parameters (According to different values of three parameters

v, D and r, three regimes of behavior are v, D and r, three regimes of behavior are expected.)expected.)

• Large-positive-velocityLarge-positive-velocity The cap growths quickly in length and only the The cap growths quickly in length and only the

cutting prevents the cap from becoming too cutting prevents the cap from becoming too large. large. (v, r)(v, r)

• Large-negative-velocityLarge-negative-velocity The cap shrinks on average and only the The cap shrinks on average and only the

fluctuations allow the cap to exist. fluctuations allow the cap to exist. (v, D)(v, D)• Small-velocitySmall-velocity Diffusion and cutting are most important. Diffusion and cutting are most important. (D, r)(D, r)

Heuristic Analysis of the Heuristic Analysis of the modelmodel

Cap SizeCap Size

Heuristic Analysis of the Heuristic Analysis of the modelmodel

Catastrophe rateCatastrophe rate

Heuristic Analysis of the Heuristic Analysis of the modelmodel

Delay time for dilution-induced Delay time for dilution-induced catastrophescatastrophesThe 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 Heuristic Analysis of the modelmodel

Amount of GTP in a microtubuleAmount of GTP in a microtubule• The tubulin-t exists as a cap on each end and a The tubulin-t exists as a cap on each end and a

number of GTP patches. It is convenient to number of GTP patches. It is convenient to treat the two caps as one patch with the treat the two caps as one patch with the caps’s summed length.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 RateCatastrophe Rate Connecting theory and experimentConnecting theory and experiment• Catastrophe rate is the frequency at which microtubules Catastrophe rate is the frequency at which microtubules

change from their growing to their shrinking state.change from their growing to their shrinking state.• Experimentally, it is found as the ratio between the total Experimentally, it is found as the ratio between the total

number of catastrophes observed and the total time spent number of catastrophes observed and the total time spent in the growing state. In the experiment, microtubules are in the growing state. In the experiment, microtubules are grown from seeds and a shrinking microtubule always grown from seeds and a shrinking microtubule always vanishes entirely, whereupon a new microtubule grows vanishes entirely, whereupon a new microtubule grows from the seed.from the seed.

• Initial condition: each cap is initially created with 0 length.Initial condition: each cap is initially created with 0 length.

• Boundary condition:Boundary condition:

• Catastrophe rate:Catastrophe rate:

Catastrophe RateCatastrophe Rate

Characteristic features of theoretical Characteristic features of theoretical resultresult

f seems to be constant for higher tubulin concentrations, while f increases rapidly if vg is decreased to small values.

When v is big

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 RateCatastrophe Rate Comparing the theory to experimental results for the Comparing the theory to experimental results for the

catastrophe ratecatastrophe rate• Satisfactory agreement between theory and Satisfactory agreement between theory and

experimental result for the catastrophe rate for plus experimental result for the catastrophe rate for plus ends by treating vends by treating vhh

(+) (+) and r as fitting parameters.and r as fitting parameters.• Though the values for vThough the values for vgg and v and vhh are different for plus are different for plus

and minus ends, when vand minus ends, when vgg is rather big, the catastrophe is rather big, the catastrophe rate is the same for both ends. This prediction is rate is the same for both ends. This prediction is consistent with experimental results. consistent with experimental results. (These results (These results are not that precise, however, and the validity of this are not that precise, however, and the validity of this prediction is another experimental acid test of the prediction is another experimental acid test of the model. To the extent the model survives the test, such model. To the extent the model survives the test, such an experiment is a very direct way to measure the an experiment is a very direct way to measure the parameter r.)parameter r.)

Dilution ExperimentDilution Experiment

MotivationMotivation• Extended cap model (uncoupled Extended cap model (uncoupled

vectorial hydrolysis) : long delay vectorial hydrolysis) : long delay times upon dilution were expected times upon dilution were expected for high growth rate.for high growth rate.

• Experiment: catastrophe rate is Experiment: catastrophe rate is essentially independent of the essentially independent of the growth rate.growth rate.

Dilution ExperimentDilution Experiment

Initial conditionInitial condition

In the case of strong dilution (v’ = - In the case of strong dilution (v’ = - vvhh))

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.

Dilution ExperimentDilution Experiment

ExperimentExperiment

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 FitCombined Fit

Combined FitCombined 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 goodsupport for the model.

Experiments visualization the Experiments visualization the GTP cap GTP cap

Experiment: a minimal size for a cap that will stabilize Experiment: a minimal size for a cap that will stabilize a microtubule is estimated roughly to contain 40 a microtubule is estimated roughly to contain 40 tubulin dimers.tubulin dimers.

There is no way to define a minimal cap size that will There is no way to define a minimal cap size that will stabilize a growing microtubule because of the stabilize a growing microtubule because of the fluctuations in the cap size and hence no absolute fluctuations in the cap size and hence no absolute stability. stability.

We expect that a microtubule must grow faster than We expect that a microtubule must grow faster than the cap hydrolyzes from its trailing edge for the cap to the cap hydrolyzes from its trailing edge for the cap to exist. parameterizes the relative importance of the exist. parameterizes the relative importance of the various processes contributing to the cap’s dynamics; various processes contributing to the cap’s dynamics; at large values catastrophes are rare and the at large values catastrophes are rare and the microtubule is stable.. Chose as the separator of microtubule is stable.. Chose as the separator of stabilized microtubules from unstable ones. stabilized microtubules from unstable ones.

Use the parameter values obtained from fit, we find Use the parameter values obtained from fit, we find that the minimal cap contians 26 tubulin-t dimers.that the minimal cap contians 26 tubulin-t dimers.

ConlusionConlusion

Self-consistency: It was assumed that . Use Self-consistency: It was assumed that . Use the parameter values from the fit, the parameter values from the fit,

Different Microscopic interpretation of the Different Microscopic interpretation of the modelmodel

Rescue: another model is needed. But much Rescue: another model is needed. But much less data has been collected on rescues than less data has been collected on rescues than on catastrophes.on catastrophes.

Issue for future experimentIssue for future experiment• More experiment would overdetermined the More experiment would overdetermined the

parameters and provide a rigorous test of the parameters and provide a rigorous test of the model. model.