Dynamics of Microtubules

Microtubules can be modeled[1] as long chains composed of $ \oplus$ and $ \ominus$ monomers (guanosin triphosphate (GTP+) and guanosin diphosphate (GDP-) tubulin complexes).

Their dynamics is a combination of three types of moves:

  1. Growth: a $ \oplus$ monomer attaches to the active end of the microtubule:
    $ \left\vert\,\ldots \oplus\,\right>\qquad\Longrightarrow\qquad\left\vert\,\ldots \oplus\oplus\,\right>$     with rate $ \lambda^+$, if the extreme monomer was a $ \oplus$ monomer;
    $ \left\vert\,\ldots \ominus\,\right>\qquad\Longrightarrow\qquad\left\vert\,\ldots \ominus\oplus\,\right>$     with rate $ \lambda^-$, if the extreme monomer was a $ \ominus$ monomer;

  2. Conversion: a $ \oplus$ monomer hydrolyses:
    $ \left\vert\,\ldots \oplus\dots\,\right>\qquad\Longrightarrow\qquad\left\vert\,\ldots \ominus\dots\,\right>$      with rate 1, independently of all other $ \oplus$ monomers;

  3. Shrinking: the extreme $ \ominus$ monomer detaches from the active end of the microtubule:
    $ \left\vert\,\ldots \ominus\,\right>\qquad\Longrightarrow\qquad\left\vert\,\ldots \,\right>$     with rate $ \mu$.

Dynamics of microtubules has many interesting properties [2,3] and even exhibits the so-called reentrant transtion [4].

The following applet simulates the long time evolution of microtubules as a function of these parameters: here $ \oplus$ monomers are shown in black and $ \ominus$ monomers are grey; the initial condition is a long chain of $ \ominus$ monomers. Your browser is not Java enabled.

[1] Antal, T., Krapivsky, P.L., Redner, S., Mailman, M., and Chakraborty, B. (2007). Dynamics of an Idealized Model of Microtubule Growth and Catastrophe. Phys. Rev. E76, 041907.

[2] Hryniv, O. and Menshikov, M. (2010). Long-time behaviour in a model of microtubule growth. Advances in Applied Probability 42(1): 268-291. DOI:10.1239/aap/1269611153

[3] Hryniv, O. (2012). Regular phase in a model of microtubule growth. Markov Processes and Related Fields 18(2): 177-201.

[4] Hryniv, O., Letcher, A. and Sheard, D. Reentrant phase transition in a model of microtubule growth. [in preparation]

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