Modeling the effects of drug binding on the dynamic instability of microtubules.

ar
X
iv
:1
01
0.
42
88
v1
[
ph
ys
ics
.bi
o-
ph
]
12
O
ct
20
10
Modeling the Effects of Drug Binding on the
Dynamic Instability of Microtubules
Peter Hinow1, Vahid Rezania2, Manu Lopus3, Mary Ann
Jordan3 and Jack A. Tuszyn´ski4
1Department of Mathematical Sciences, University of Wisconsin – Milwaukee,
P.O. Box 413, Milwaukee, WI 53201, USA
E-mail: hinow@uwm.edu
2Department of Physical Sciences, Grant MacEwan University, Edmonton AB, T5J
4S2, Canada
3Department of Molecular, Cellular, and Developmental Biology and the
Neuroscience Research Institute, University of California, Santa Barbara, CA 93106,
USA
4Cross Cancer Institute and Department of Physics, University of Alberta, Edmonton
AB, T6G 2J1, Canada
Abstract.
We propose a stochastic model that accounts for the growth, catastrophe and
rescue processes of steady state microtubules assembled from MAP-free tubulin. Both
experimentally and theoretically we study the perturbation of microtubule dynamic
instability by S-methyl-D-DM1, a synthetic derivative of the microtubule-targeted
agent maytansine and a potential anticancer agent. We find that to be an effective
suppressor of microtubule dynamics a drug must primarily suppress the loss of GDP
tubulin from the microtubule tip.
Keywords : microtubules, dynamic instability, stochastic modeling
Submitted to: Phys. Biol.
PACS numbers: 87.10.Mn

Drug Binding and Dynamic Instability 2
1. Introduction
Microtubules are hollow and flexible cylindrical polymers of the protein tubulin that
form a major component of the cytoskeleton of eukaryotic cells. They play a central
role in maintenance of structural stability of the cell, intracellular vesicle transport and
chromosome separation during mitosis. The polymerization of tubulin into microtubules
and the subsequent catastrophic depolymerization have been studied extensively both
experimentally and theoretically, see [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14] for just
a few examples. The prevailing model to explain dynamic instability, the lateral cap
model, is that a cap of GTP tubulin at the growing tip is required for stability of the
polymer and that a loss of this GTP cap results in dramatic shortening of microtubules
[15].
In the paper [14] we proposed a partial differential equation model inspired by
dynamics of size-structured populations. The variables of that continuous model are
length distributions of microtubules and the amounts of free tubulin. The major
reactions are the polymerization of free GTP tubulin, the hydrolysis of assembled GTP
tubulin to GDP tubulin, the decay and rescue of microtubules without a GTP cap
and the recycling of free GDP tubulin to GTP tubulin. The model conserves the
total amount of tubulin in all its forms. In addition, we allowed for the nucleation
of fresh microtubules at certain specified (short) lengths. With a small number of
parameters that have clear biochemical interpretations, we were able to reproduce
commonly observed experimental behaviors, such as oscillations in the amount of tubulin
assembled into microtubules. Obviously, the continuous model is not expected to
reproduce the lengths of individual microtubules. To this end, in this paper we present a
stochastic discrete model of microtubule dynamic instability and compare its predictions
to observations of microtubule lengths from in vitro experiments.
Stochastic discrete models of biopolymer dynamics have been investigated in
[4, 5, 16, 17, 18, 12, 13, 19], among others. Here we present a stochastic model
that represents the same set of reactions as our continuous model in [14], except for
the nucleation of fresh microtubules which is an unnecessary complication for the
case studied here. As in [14], we couple the growth velocity of microtubules to the
amount of free GTP tubulin. Our goal is to explain individual length observations
of microtubules growing without microtubule associated proteins (MAPs) in vitro.
Moreover, we investigate the effects of tubulin-binding drugs on microtubule dynamic
instability. These drugs belong roughly to one of two classes, namely those that bind
to assembled microtubules and those that bind to free tubulin [20]. This difference
in behaviors may be due to conformational changes of the α/β-tubulin heterodimer
[21] (the tubulin unit from now on) upon incorporation into the microtubule lattice
that expose or hide the binding site for the drug molecule. Apart from these different
binding modes, the effects on polymerization kinetics can also differ. Some drugs mainly
slow down microtubule formation while others mainly prevent microtubule decay. This
is not a strict dichotomy in that some drugs can have multiple actions, depending on

Drug Binding and Dynamic Instability 3
their concentration. For example, vinblastine inhibits microtubule formation at high
drug concentrations, and inhibits microtubule decay at low concentrations [20]. In any
case, within a living cell exposed to anti-mitotic agents, mitosis cannot be completed
and the cell dies. This property of tubulin-binding drugs leads to many successful anti-
cancer chemotherapeutics such as paclitaxel, vincristine, vinblastine to name but a few.
Maytansine and its derivatives are known to suppress microtubule dynamics in vitro and
in cells [22]. We focus on the potential anticancer agent S-methyl-D-DM1, a synthetic
derivative of the microtubule-binding agent maytansine. Antibody-DM1 conjugates are
currently under clinical trials with promising results [23].
2. The stochastic model
We consider a linear model of the microtubule and disregard the fact that it actually
consists of 13-17 protofilaments arranged in a helical lattice. Tubulin can be added to
the microtubule in form of small oligomers of varying sizes [24].
Let m ≥ 1 be the number of simultaneously growing and shrinking microtubules.
The state of each microtubule is represented as a word vk = (. . . , vk2 , vk1 , vk0), k = 1, . . . , m
on the binary alphabet {0, 1} where the letter 1 stands for a position occupied by a GTP
tubulin unit and 0 stands for a position occupied by a GDP tubulin unit. The length
of the microtubule vk is denoted by |vk|. The number of GTP tubulin units within
vk is denoted by I(vk). The “tip” of the microtubule is the letter vk0 and this is the
only position where growth or shrinkage can occur. Consecutive strings of 0s and 1s
are called GDP zones and GTP zones, respectively. The number of boundaries between
such zones is denoted by B(vk). The numbers of free GTP tubulin and GDP tubulin
are denoted by NT and ND, respectively. The following reactions occur.
(i) Growth by attachment of GTP tubulin(s)
vk 7→ [vk 1 . . . 1︸ ︷︷ ︸
l
], NT 7→ NT − l,
at rate
λNT (vk0 + p(1− vk0 )). (1)
The number of added GTP tubulin units l can be set to a fixed value (say, 1) or
drawn from a Poisson distribution with parameter L. The dimensionless parameter
p ≥ 0 is the propensity of a rescue event when a GDP tubulin unit at the tip of
the microtubule is exposed. In the simplest case, p = 1, attachment of a new GTP
tubulin is independent of the tip status.
(ii) Loss of a GDP tubulin
vk 7→ (. . . , vk2 , vk1), ND 7→ ND + 1,
at rate µGDP (1− vk0).
(iii) Loss of a GTP tubulin
vk 7→ (. . . , vk2 , vk1), NT 7→ NT + 1,

Drug Binding and Dynamic Instability 4
at rate µGTPvk0 .
(iv) Hydrolytic conversion of a bound GTP tubulin vk∗ 7→ v˜k∗ at rate δsc
∑m
k=1 I(vk).
The index k∗ is chosen uniformly in the set {1, . . . , m} and a random position of
vk
∗ that is occupied by 1 is changed to 0 (this hydrolysis mechanism is called scalar
hydrolysis in [8]).
(v) Hydrolytic conversion of a bound GTP tubulin vk∗ 7→ v˜k∗ at rate δvec
∑m
k=1 B(vk).
Again, the index k∗ is chosen uniformly in the set {1, . . . , m} and the word v˜k∗
is created by selecting randomly a position of vk∗ where a 1 neighbors a 0 and
changing that 1 to 0 (this hydrolysis mechanism is called vectorial hydrolysis in [8],
see Figure 1, left panel).
(vi) Recycling of free GDP tubulin to GTP tubulin
ND 7→ ND − 1, NT 7→ NT + 1,
at rate κND. It is assumed that a sufficient amount of chemical energy in the form
of free GTP is always present.
This scheme can be simplified by setting some parameter values to zero. For
example, one may disregard the possibility of a bound GTP tubulin to be lost again
(µGTP = 0, cf. [8]), although other authors argue that this may take place in up to
90% of all binding events [25]. The hydrolysis reaction iv picks any bound GTP tubulin
and changes it to a GDP tubulin, thereby creating islands of GTP tubulin within the
length of the microtubule. That this is possible and important for the rescue process was
recently shown by Dimitrov et al. [26]. Both hydrolysis mechanisms in concert provide
an indirect coupling of the hydrolysis reaction to the addition of new GTP tubulin units
[8].
Tubulin-binding drugs can bind to tubulin in one of two states, whether it is free
or bound within a microtubule. Here, we consider drugs that suppress microtubule
dynamic instability by specifically binding to microtubules. For every microtubule
encoded by a word v, we introduce a second word w = (. . . , w2, w1, w0) over the alphabet
{0, 1} (the drug state), of equal length as v. Here wi = 1, if the tubulin unit at position
vi is occupied by a drug molecule and wi = 0 otherwise. There are binding events of
drug molecules to unoccupied sites and release of drug molecules from the microtubule.
Let E(w) be the number of available sites for drug binding and let F (w) be the number
of drug occupied sites. The latter is always the sum of the entries 1 in w while the
former may be only a subset of entries 0 in w. We have the association and dissociation
events
7 Binding of drug to tubulin units within the microtubule
wk
∗ 7→ w˜k∗, D 7→ D − 1,
at rate ρD∑mk=1 E(wk). The new word w˜
k∗ is obtained by selecting randomly one
letter 0 among the sites available for binding and changing it to 1. This set may be
the set of all entries 0 or the entries 0 that are within a certain distance from the
tips or the unoccupied tips alone.

Drug Binding and Dynamic Instability 5
8 Release of drug from the occupied sites of the microtubule
wk
∗ 7→ w˜k∗, D 7→ D + 1,
at rate σ
∑m
k=1 F (wk). The new word w˜
k∗ is obtained by changing one randomly
selected letter 1 in any of the drug words to 0.
The reactions i–vi have the same outcomes as far as changes in numerical quantities
are concerned, however the rates of reactions i, ii and iii have a more complicated
dependence upon the status of the microtubule tip. Since the tip can now have four
different states, the attachment process i (to microtubule k) occurs at rate
λNT
(
(vk0 + p(1− vk0 ))(1− wk0) + r(vk0 + p(1− vk0))wk0
)
, (2)
where the dimensionless non-negative constant r modulates the attachment propensity
compared to the drug-free tip, see Equation (1). Small values of r would mean that
attachment of new GTP tubulin units is hindered by drug molecules bound to the
tip. On the other hand, values r > 1 would increase microtubule polymerization. The
shrinking reactions ii and iii occur at rates
µGDP (1− vk0 )((1− wk0) + qwk0), and µGTPvk0 ((1− wk0) + qwk0), (3)
where a small value of q ≥ 0 implies a high level of protection afforded by a drug
molecule bound to the tip. If a drug bound tubulin can fall off a microtubule (i.e. if
q > 0), then the drug-tubulin compound is assumed to dissociate immediately.
3. Materials and Methods
Tubulin (15µM), phosphocellulose purified, MAP-free, was assembled on the ends of
sea urchin (Strongylocentrotus purpuratus) axoneme fragments at 30◦C in 87mmol/L
Pipes, 36mmol/L Mes, 1.4mmol/L MgCl2, 1mmol/L EGTA, pH 6.8 (PMME
buffer) containing 2mmol/L GTP for 30 min to achieve steady state. We used
a 100nmol/L concentration of S-methyl-D-DM1 (N2′ -deacetyl-N2′ -(3-thiomethyl-1-
oxopropyl)-D-maytansine [27], see Figure 1, right panel), which had no considerable
effect on microtubule polymer mass, to analyze their individual effects on dynamic
instability. Time-lapse images of microtubule plus ends were obtained at 30◦C by
video-enhanced differential interference contrast microscopy at a spatial resolution
of 0.3µm using an Olympus IX71 inverted microscope with a 100 × oil immersion
objective (NA = 1.4). The end of an axoneme that possesses more, faster growing, and
longer microtubules than the other end was designated as the plus end as described
previously [28, 29]. Microtubule dynamics were recorded for 40 min at 30◦C, capturing
∼ 10min long videos for each area under observation. The microtubules were tracked
using RTMII software, and the life-history data were obtained using IgorPro software
(MediaCybernetics, Bethesda, MD) [30].
We have programmed the reactions i–8 using the Gillespie Algorithm [31] (available
from the corresponding author upon request). This algorithm simulates the chemical
reactions as collisions of particles in real time and its parameters are the actual