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 • 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 TheoryMicroscopic description :thepolymerization 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 TheoryMicroscopic 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 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 cap’s length x happens to decrease to zero, represents a catastrophe.
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.
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 • 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.
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 • 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.