Stochastic Microtubule Dynamics Revisited

Richard Yamada

Yoichiro Mori

Maya Mincheva

(Alex Mogilner and Baochi Nguyen)

What are Microtubules?• Protein structures with a diameter of

approximately 24 nm, and with a length up to several millimeters in some cells

• Microtubules consist of polymers of tubulin, 13 protofilaments of which which are formed into a hollow cylinder

• Microtubules have plus and minus ends• Highly dynamic - capable of polymerizing and

depolymerizing within a time scale of seconds to minutes

What Do Microtubules Look Like?

Why Are Microtubules Important?

• Microtubules are involved in many fundamental biological functions/processes, among them:

1) segregating the chromosomes and to orient the plane of cleavage during cell division

2) organize cytoplasm by positioning the organelles

3) serve as the principal structural element of flagella and cilia

Dynamic Instability

Dynamic Instability

Additional Assumptions

Kinetic Equations

• a - Polymerization rate

• b - Induced transition rate

• c - Spontaneous transition rate

€

dPkdt= aPk−1 + bPk+1 − (a+ b)Pk − ckPk + cPj

j= k+1

∞

∑

Integro-Differential Equations(Continuum Limit)

• The ensemble density of microtubules with caps of lengths x at time t is governed by a integro-differential equation:

€

∂t p = −v∂x p+ D(∂x )2 p− rxp+ r dyp(y, t)

x

∞

∫

v = vg − vAB

Numerical Methods

2 ways to simulate Kinetic Equations:• Trapezoidal rule for integration of ODEs

(Deterministic)• Gillespie Method (Stochastic)

all events (hydrolysis,induced,spontaneous) are possible but are weighted by rate constants along with a random number

( e.g. -log(random)/(rate constant))

Experimental Method - Dilution

Dilution Simulations - Same Cap Lengths

Dilution Simulations - Different Cap Lengths

Dilution Simulations - Initial Growth versus Delay Time

Microtubule Structure

Results of Simulations - 2D Dynamic Instability

Summary• No use of continuum limit equations -

instead our approach started from kinetic equations, using 2 numerical methods to investigate dynamic instability

• Incorporation of dilution washout effects

• Results are consistent with previous published results

Future Directions

• Stochastic methods may provide additional statistical measures to validate theory

• Compare results to more recent/different experimental data

• More realistic 2D cap simulations• Incorporation of our model into a general

framework of dynamic instability