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

dPk
 aPk1  bPk1 (a b)Pk ckPk  cPj
dt
jk1
• a - Polymerization rate
• b - Induced transition rate
• c - Spontaneous transition rate
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 ) p  rxp r  dyp(y,t)
2
x
v  v g  v AB
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