Mp

Ma

Sb

a

ARRA

KMD

1

onpflCHttet

c(Zspfi

1d

Computational Biology and Chemistry 35 (2011) 269–281

Contents lists available at ScienceDirect

Computational Biology and Chemistry

jo ur n al homep age: www.elsev ier .com/ locate /compbio lchem

acroscopic simulations of microtubule dynamics predict two steady-staterocesses governing array morphology

árcio Mourãoa, Santiago Schnell a, Sidney L. Shawb,∗

Department of Molecular and Integrative Physiology, Brehm Center for Diabetes Research and Center for Computational Medicine & Bioinformatics, University of Michigan Medicalchool, Ann Arbor, MI 48105, USADepartment of Biology, Indiana University, 1003 East Third St., Bloomington, IN 47405, USA

r t i c l e i n f o

rticle history:eceived 11 March 2011eceived in revised form 10 May 2011ccepted 17 June 2011

eywords:icrotubuleynamic instability

a b s t r a c t

Microtubule polymers typically function through their collective organization into a patterned array. Theformation of the pattern, whether it is a relatively simple astral array or a highly complex mitotic spin-dle, relies on controlled microtubule nucleation and the basal dynamics parameters governing polymergrowth and shortening. We have investigated the interaction between the microtubule nucleation anddynamics parameters, using macroscopic Monte Carlo simulations, to determine how these parameterscontribute to the underlying microtubule array morphology (i.e. polymer density and length distribu-tion). In addition to the well-characterized steady state achieved between free tubulin subunits andmicrotubule polymer, we propose that microtubule nucleation and extinction constitute a second, inter-

dependent steady state process. Our simulation studies show that the magnitude of both nucleation andextinction additively impacts the final steady state free subunit concentration. We systematically variedindividual microtubule dynamics parameters to survey the effects on array morphology and find specificsensitivity to perturbations of catastrophe frequency. Altering the cellular context for the microtubulearray, we find that nucleation template number plays a defining role in shaping the microtubule lengthdistribution and polymer density.

. Introduction

The microtubule cytoskeleton organizes into a broad rangef arrays, critical to cellular function. By controlling microtubuleucleation, assembly rates, and interactions, the cell builds com-lex polymer assemblies required for mitosis, vesicle trafficking,agellar movement, and numerous other tasks (Compton, 2000;onde and Caceres, 2009; Desai and Mitchison, 1997; Walczak andeald, 2008). Both the construction and function of specific micro-

ubule arrays depend upon the macroscopic dynamic properties ofhe polymers. In this manuscript, we explore how the basal prop-rties governing microtubule dynamics and nucleation contributeo the morphology of the resulting microtubule array.

In interphase animal cells, microtubule polymers nucleate prin-ipally from gamma-tubulin templates located at the centrosomeOakley, 2000; Oakley et al., 1990; Xiong and Oakley, 2009;heng et al., 1991). Microtubules polymerize from free tubulin

ubunits forming hollow tubes, in most cases, comprised of 13roto-filaments. Each alpha/beta tubulin dimer binds to the proto-lament in a head to tail association creating the intrinsically

∗ Corresponding author. Tel.: +1 812 856 5001; fax: +1 812 855 6705.E-mail address: SiShaw@Indiana.edu (S.L. Shaw).

476-9271/$ – see front matter © 2011 Published by Elsevier Ltd.oi:10.1016/j.compbiolchem.2011.06.002

© 2011 Published by Elsevier Ltd.

polarized microtubule lattice (Amos, 1995; Downing and Nogales,1998; Li et al., 2002; Mandelkow et al., 1995; Nogales et al., 1999).For a centrosomal system, the array typically forms an astral shapecharacterized by an underlying microtubule length distribution anda density or microtubule number (i.e. array morphology).

The broader mechanisms controlling microtubule nucleationin vivo have not been explicitly determined (Aldaz et al., 2005).Gamma-tubulin complexes and other microtubule templates lowerthe energetic barrier for nucleation in the cell but it is not knownhow the number of templates are specified and activated. Mostinterphase animal microtubules remain anchored near the centro-some and are apparently capped or otherwise inactive at the minusend after nucleation (Rieder et al., 2001). The resulting effect is anupper limit on total microtubule number to roughly the number ofactivated nucleation templates in the cell.

Microtubules show the intriguing property of stochasticallyswitching between states of growth and shortening, a phenomenontermed ‘dynamic instability’ (Burbank and Mitchison, 2006; Desaiand Mitchison, 1997; Mitchison and Kirschner, 1984, 1987). Thebeta-tubulin subunit acts as a guanasine triphosphatase (GTPase)

where end-exposed GTP-bound subunits show a higher bindingaffinity at microtubule ends than GDP-bound subunits (Desai andMitchison, 1997). Free tubulin subunits rapidly exchange GDPfor GTP in solution resulting in a cellular pool of GTP-bound

2 iology

sm1tGhmdbdoia1M1stee

us1wfitmo

pqiMataKpdbaWtie

mancartitlp(

spbsC

70 M. Mourão et al. / Computational B

ubunits that drives polymerization in a concentration dependentanner (Caplow et al., 1989; Cassimeris et al., 1987; Tran et al.,

997; Walker et al., 1988, 1991). Incorporation into the micro-ubule lattice accelerates GTPase activity, resulting in eventualTP hydrolysis. The relative timing of subunit binding and GTP-ydrolysis creates a theorized ‘cap’ of GTP-bound subunits at theicrotubule plus end trailed by GDP-bound subunits in the remain-

er of the microtubule (Desai and Mitchison, 1997). Since subunitinding and GTP hydrolysis are both rates that are subject to ran-om fluctuations, it is postulated that the GTP-tubulin cap will onccasion be lost due to a lull in polymerization and/or an advancen GTP-hydrolysis (Carlier et al., 1984; Chen and Hill, 1985b; Desaind Mitchison, 1997; Dogterom and Leibler, 1993; Gliksman et al.,993; Hill, 1984; Hill and Carlier, 1983; Mandelkow et al., 1995;itchison and Kirschner, 1987; Odde et al., 1995; Walker et al.,

988). This event is theorized to correspond with the observedwitch from growth to shortening (termed catastrophe) owing tohe lower free subunit affinity of the GDP-bound subunits at thexposed end (Brun et al., 2009; Desai and Mitchison, 1997; Oddet al., 1995).

Microtubule depolymerization is relatively insensitive to sub-nit concentration and possibly accelerated by the release of energytored from GTP hydrolysis in the microtubule lattice (Nogales,999; Walker et al., 1988). When free GTP-tubulin dimers associateith the depolymerizing microtubule end, the polymer switches

rom shortening to growth (termed rescue) where in vitro exper-ments indicate some dependence of rescue frequency on freeubulin concentration (Walker et al., 1988). The exact microscopic

echanisms governing both catastrophe and rescue are the subjectf considerable debate (Desai and Mitchison, 1997).

Dynamic instability is observed for microtubules made fromurified tubulin and for microtubules imaged in living cells. Whileualitatively similar, the dynamic properties in live cells differ

n several important ways (Cassimeris et al., 1987; Desai anditchison, 1997; Shelden and Wadsworth, 1993). The assembly

nd transition properties tend to be accelerated in live cells relativeo in vitro preparations denoting the involvement of microtubule-ssociated proteins (MAPs) (Dhamodharan and Wadsworth, 1995;omarova et al., 2009). MAPs have been characterized that alterlus end polymerization velocities and transition frequenciesemonstrating that the underlying GTPase driven dynamic insta-ility can be modulated to change the array morphology (Desaind Mitchison, 1997; Desai et al., 1999; Walczak and Heald, 2008;alczak et al., 1996). In addition to growth and shortening, micro-

ubules in living cells exhibit extended pause phases less typical forn vitro preparations (Rusan et al., 2001; Tran et al., 1997; Walkert al., 1988).

We are interested in discovering how dynamic instability andicrotubule nucleation contribute to the creation of a steady-state

rray with a constant polymer level distributed into a constantumber of microtubules. We consider a microtubule array in aellular system with a finite number of tubulin subunits and nucle-tion sites. Provided that GTP is in excess, the cellular system willeach a steady-state level of polymer through which the micro-ubule density and average microtubule length are related. We arenterested in the ‘cellular’ case because we want to understand howhe eventual competition for tubulin subunits (i.e. global intracel-ular subunit concentration) affects the nucleation and dynamicsarameters as they contribute to the underlying array morphologyGregoretti et al., 2006; Vorobjev and Maly, 2008).

We hypothesize that a cellular microtubule array limited byubunit availability will exhibit two interdependent steady-state

henomena. A first steady-state will arise when the average num-er of subunits incorporated into polymer is balanced by theubunits lost to depolymerization (Gregoretti et al., 2005; Hill andarlier, 1983; Hill and Kirschner, 1982). A second steady state will

and Chemistry 35 (2011) 269–281

be achieved when the number of microtubules depolymerizing tocompletion (termed extinction) is balanced by the average num-ber of new polymers nucleated on a similar time scale. Of criticalinterest is determining how the two steady state phenomena areinterconnected in this complex system. For this work, we develop asystem of linear models relating dynamic instability and nucleationbehaviors to tubulin concentration. We then use Monte Carlo meth-ods to simulate microtubule arrays through time within a specifiedcellular context (Chen and Hill, 1983, 1985a; Chen and Hill, 1985b;Gliksman et al., 1993; Goodson and Gregoretti, 2010; Gregorettiet al., 2005; Martin et al., 1993; Odde and Buettner, 1995). Theresults from these simulations provide an important foundationfor predicting how specific microtubule dynamics or nucleationparameters should generally affect the microtubule array under avariety of assumptions about the cellular context.

2. Methods

2.1. Model

We define a physical ‘cell’ of fixed 2000 �m3 volume and an ini-tial cellular context for the simulations having a 9 �M starting freetubulin concentration, 750 nucleation templates and a maximummicrotubule length of 15 �m. The values were selected to create arelatively compact astral array with limited overall effects comingfrom the cell boundary (Gregoretti et al., 2006; Vorobjev and Maly,2008). A cell boundary was selected that approximates the cellradius to prevent overly long microtubules. We define five variablesto describe the nucleation and dynamic properties of the individualmicrotubule polymers: Eqs. (1)–(5). Each variable is taken as a lin-ear model relative to the global free tubulin subunit concentrationin the cell and is always positive in the calculations. When appliedto the same unit time interval (1 s for the Monte Carlo simulations),each variable value is then taken at each free subunit concentrationas a rate or frequency as described.

We model microtubule nucleation using a constant number ofnucleation templates (Tt) classified as occupied (To) or unoccupied(Tu) by a microtubule. Each unoccupied nucleation site will initiatea new microtubule with a relative frequency per unit time intervalthat is linearly related to free dimer concentration [Df]:

Nc = k1[Df ] − k2 (1)

The global nucleation rate (N, in microtubules s−1) is then theproduct of the number of free templates in the time interval andconcentration dependent expected frequency of nucleation at eachunoccupied template over a unit time interval. Once nucleated,each individual growing microtubule will add tubulin subunits asa growth velocity (Vg) that is linearly proportional to free tubulinconcentration:

Vg = k3[Df ] − k4 (2)

In the above equation, we assume that subunit diffusion androtation are orders of magnitude faster than microtubule endmovement in the cell; our model is homogeneous in space. Growthvelocity models are taken as a first order association rate constantk3 (slope units of t−1), and a zero order dissociation rate constant k4(offset units of �m t−1) with 1634 tubulin subunits per micrometerof growth. We assume that the loss of incorporated subunits frompolymer occurs as a zero order shortening velocity (Vs):

Vs = C1 (3)

Transitions between growth and shortening phases are con-ventionally measured as catastrophe and rescue frequencies. Wereasoned from in vitro measurements (Walker et al., 1988) thatat relatively high free tubulin concentrations, polymerization is

M. Mourão et al. / Computational Biology and Chemistry 35 (2011) 269–281 271

Table 1Parameter values for the Monte-Carlo simulations.

Parameter Value Variations Description

k1 0.1 �M−1 s−1 0.05:0.01:0.14 �M−1 s−1 Nucleation rate slopek2 0.2 s−1 0.15:0.01:0.24 s−1 Nucleation rate ordinate at the origink3 3500 × 10−6 s−1 3000:100:3900 × 10−6 s−1 Polymerization rate slopek4 10 × 10−6 �m s−1 5:1:14 × 10−6 �m s−1 Polymerization rate ordinate at the origink5 0.02 �M−1 s−1 0.01:0.01:0.1 �M−1 s−1 Transition frequency Psg slopek6 0.01 s−1 0.01:0.01:0.1 s−1 Transition frequency Psg ordinate at the origin

−1 −1 1:0.1 � −1 −1

:0.59 s× 10−

fttrssaps

P

Asi

P

rdscwSoao

2

wltttsscgpp

k7 0.06 �M s 0.01:0.0k8 0.55 s−1 0.5:0.01C1 15 × 10−6 �m s−1 10:1:19

avored over depolymerization. And at low free subunit concen-rations, the opposite would hold true. We further assumed thatransitions between phases show no prior dependence other than aelation to global free subunit concentration. We modeled the tran-ition frequencies as having a linear dependence on the free tubulinubunit concentration in the cell. Therefore the relative frequencyt which an individual microtubule will change from a shorteninghase to a growth phase over a unit time interval relative to freeubunit concentration is depicted as:

sg = k5[Df ] − k6 (4)

nd the relative frequency of transition from a growth phase to ahortening phase for an individual microtubule over a unit timenterval relative to the free subunit concentration is depicted as:

gs = −k7[Df ] + k8 (5)

A set of parameter values was selected (Table 1) that produces aelatively compact steady-state microtubule array under the con-itions of our specified cellular context. This reference set of valuespecifies the relationship of each variable to the free subunit con-entration in the model where the final steady-state values fallithin a range that is reasonable for in vivo data (Rusan et al., 2001;

helden and Wadsworth, 1993). A compact array was sought forur reference so that changes to individual dynamics and nucle-tion models would reveal behaviors that are mostly independentf cell border effects.

.2. Monte Carlo simulation

We consider a centrosome based interphase microtubule arrayithin a cell of fixed volume containing a finite number of tubu-

in subunits. As initial conditions, all nucleation templates are seto unoccupied and the free tubulin concentration is set to the totalubulin concentration. Each simulation consists of 20,000 iterationsaken on a 1 s time interval per iteration. The 1 s interval corre-ponds well with many in vivo measurements and we found thathorter iteration times had no significant effect on steady-state out-omes. In each iteration, each microtubule is assessed based on thelobal free tubulin concentration with reference to the 5 specifiedarameter models. For each Monte Carlo iteration, the followingrocedure is followed:

(i) Nucleation process. All nucleation templates are assessed asoccupied or unoccupied. For each unoccupied template, a ran-dom number is generated and compared to the probability oftemplate nucleation at the present free subunit concentrationfor a 1 s time interval. We mark the site as occupied if the ran-dom value is lower than or equal to the value specified by thenucleation probability model. The newly formed microtubule

is set to a growth phase.

(ii) Dynamic instability process. For each existing microtubule, thegrowth phase is assessed and a random number is generated.For each growing microtubule, the number is compared to the

M s Transition frequency Pgs slope−1 Transition frequency Pgs ordinate at the origin

6 �m s−1 Depolymerization rate (constant)

value specified for Pgs at the present free subunit concentra-tion. For a shortening microtubule, the value is compared to thevalue specified for Psg at the present free subunit concentration.The phase of each individual microtubule is either changed orleft standing. For growing microtubules, subunits are addedto the each polymer based on the value of Vg at the presentfree subunit concentration. For shorting microtubules, a num-ber of subunits corresponding to the value of Vs are subtractedfrom the polymer. The total number of subunits incorporatedinto polymer is subtracted from the free subunit pool and thenumber of subunits liberated from polymer is added to the freepool.

(iii) Extinctions and boundary conditions. Following all nucleation,phase change, and growth or shortening events, the length ofeach microtubule is determined. For each growing microtubulethat exceeds the specified boundary limit, the phase is changedfrom growth to shortening. For any shortening microtubulereaching 0 subunits (i.e. extinction), the template is marked asunoccupied.

At the end of each iteration, the free tubulin subunit concentra-tion is calculated from the number of free tubulin dimers in thesimulation using the formula:

[Tubulin]dimers = [Df ]NaV

dimers(6)

Here, Na is Avogadro’s number in mol−1 and V represents volumein liters (L).

At the end of each simulation, we determined a steady statevalue for the following variables: free tubulin concentration, totalmicrotubule number, growing microtubules, shortening micro-tubules, mean microtubule length, growth velocity, shorteningvelocity, rescue frequency, catastrophe frequency, nucleation rateand extinction rate. For each steady state value, we averaged thelast 10% of the iterations (usually the last 2000 data points fromthe total number of iterations in each simulation) to minimize theeffects of noise and oscillations. Averaging more than the last 5%of the simulation values never resulted in more than a 0.1% differ-ence between values in replicate simulation experiments for thereported steady state variables. The final steady state value is takenas the mean from 15 replicate simulations in each case. Error barsindicate that standard deviation of the 15 replicate simulations.

2.3. Analysis of relative change in steady state parameters

The relative change analysis expresses changes in an outputvariable in comparison to its starting value. We use this analysis to

compare the steady state values obtained with different parametervalues to a reference steady state value. We calculate the relativechange from a steady state (SS) to a steady state of reference (SSr)as follows:

2 iology and Chemistry 35 (2011) 269–281

(

R

3

3c

lnm(

Sag((

Sps

Aste

eMGaTsuarbtfmte(

Free Subunit Concentration (µM)

0

1

2

0.5

1.5

Rat

io

Steady-StatePrediction with

N = E = 0

6 6.5 7 7.5 8

Steady-StateSimulation with

Basis Set

[10

µM]

[9 µ

M]

[8 µ

M]

Vg/VsPgs/Psg

Fig. 1. Prediction of steady state free tubulin subunit concentration. Linear mod-els were developed relating the catastrophe (Pgs), rescue (Psg), growth velocity (Vg),and shortening velocity (Vs) to free tubulin subunit concentration over a unit timeinterval. The ratios of catastrophe over rescue (green line) and growth velocity overshortening velocity (blue line) are plotted against free subunit concentration using areference set of parameter values (Table 1). The intersection of the ratio values (reddashed line) predicts a steady state free subunit concentration of 7.65 �M whenmicrotubule nucleation and extinction are set to zero. Using the same referenceset of model parameters, Monte Carlo simulations were performed to estimate thesteady state free tubulin concentration with a fixed number of nucleation templateshaving a concentration dependent probability of microtubule initiation. Inclusion of

72 M. Mourão et al. / Computational B

(i) Run the model with the reference parameter set to obtain thereference steady state values. The result is SSr.

(ii) Run the model with a different parameter value to obtain anew steady state value. The result is SS.

iii) Compute the relative change (dimensionless quantity) withthe following formula:

C = SS − SSr

SSr(7)

. Results

.1. Considerations for predicting the steady-state free tubulinoncentration

We assume that two primary conditions are met for the estab-ishment of a global steady state in our system. First, the averageumber of tubulin dimers incorporated into polymer per unit timeust equal to the average number lost to the free subunit pool (Eq.

8)):

dDi

dt= dDf

dt(8)

econd, we assume that the number of microtubules remainspproximately constant in the cell at steady state. We express thelobal rate of free tubulin dimer incorporation into polymer (Eq.9)) as a product of the number of individual growing microtubulesMg) and the growth velocity (Vg) in a unit time interval:

dDi

dt= VgMg (9)

imilarly, free subunits enter the cytoplasmic pool (Eq. (10)) as aroduct of the number of shortening microtubules (Ms) and thehortening velocity (Vs):

dDf

dt= VsMs (10)

t steady state, we reason that the total number of growing andhortening microtubules must remain constant and proportionalo one another (Eq. (11)), if the total change in free subunits is toqual zero:

Vg

Vs= Ms

Mg(11)

Microtubules stochastically switch between growth and short-ning phases in a process termed dynamic instability (Desai anditchison, 1997; Dogterom and Leibler, 1993; Flyvbjerg et al., 1996;regoretti et al., 2005; Hill, 1984; Hill and Chen, 1984; Mitchisonnd Kirschner, 1984; Odde and Buettner, 1995; Verde et al., 1992).he probability that a microtubule will switch from a growth tohortening phase (Pgs) or a shortening to growth phase (Psg) in anit time interval is dependent upon various intrinsic (e.g. GTPasectivity) and extrinsic (e.g. MAPs) properties, but is ultimatelyesponsive to free subunit concentration. The change in the num-er of growing microtubules per time interval will be the numberransiting from shortening to growth, minus the number transitingrom growth to shortening, and including any newly nucleating (N)

icrotubules (Eq. (12)). The change in number of shortening micro-ubules follows the same logic except that microtubules going toxtinction (E) will be subtracted from the shortening populationEq. (13)):

dMg

dt= MsPsg − MgPgs + N (12)

dMs

dt= MgPgs − MsPsg − E (13)

nucleation and extinction shifts the predicted steady state free subunit concentra-tion to a lower value (black dashed line) with a clear dependence on the startingsubunit concentration (black lines).

To preserve a constant ratio of growing to shortening micro-tubules at steady state, the number of polymers switching phaseshould equal to zero for both Eqs. (12) and (13). We observe that ifnucleation and extinction approach zero (N ≈ 0, E ≈ 0), the ratio ofcatastrophe to rescue (Pgs/Psg) should be equal to the ratio of short-ening to growing microtubules (Ms/Mg), and from Eq. (11), equalto the ratio of velocities (Vg/Vs). Therefore, if we plot the ratio ofvelocities and the ratio of transition probabilities over free subunitconcentration for our reference set of model parameters (Table 1),the steady state free subunit concentration is predicted by the inter-secting point (Fig. 1) where the ratios of the microtubule dynamicsparameters are equal (Vg/Vs = Pgs/Psg). Hence, when nucleationand extinction rates approach zero, the macroscopic microtubuledynamics parameters establish a baseline steady state free subunitconcentration (MDss), yielding a minimal polymer mass (Fig. 1) andan expected ratio of shortening to growing microtubules. While wedo not take this state to be generally physiologically relevant, theMDss concept is useful in illustrating the general effect of nucleationand extinction on the final steady state array.

3.2. A steady-state system with microtubule nucleation andextinction

Owing to the stochastic nature of the microtubule dynamics, allmicrotubules will have a probability of depolymerizing to comple-tion. We assume that nucleation must eventually equal extinctionto maintain a constant number of microtubules. From Eqs. (12) and(13), we observe that nucleation adds to the number of growingmicrotubules while extinction removes shortening microtubulesfrom the system. Therefore, even when nucleation and extinctioncome to equivalent (i.e. offsetting) rates, we observe that they addi-tively bias the ratio of shortening to growing microtubules (Ms/Mg)in favor of growing polymers relative to MDss. To offset the impact

of nucleation and extinction, we propose that the system must finda new steady state free subunit concentration where the transitionfrequencies can create a constant ratio of growing to shorteningpolymers.

iology

ms

Rste

FetaTsfTss(seta

gottbc

acobat(fcgrre2dpde

adete

M. Mourão et al. / Computational B

Assuming that the ratio of growing and shortening polymersust remain constant at steady state, we can combine Mg and Ms

olutions with Eqs. (12) and (13), this yields:

dDi

dt= Vg

[2MsPsg + N + E

2Pgs

](14)

dDf

dt= Vs

[2MgPgs − N − E

2Psg

](15)

earranging the equations and accounting for subunit exchange atteady-state, we can examine the relationship between the micro-ubule dynamics parameters and the properties of nucleation andxtinction:

Vg

Vs= Pgs(2MgPgs − N − E)

Psg(2MsPsg + N + E)(16)

rom Eq. (16), we predict that with increasing nucleation andxtinction rates, the ratio of velocities (Vg/Vs) should decrease andhe ratio of transition frequencies (Pgs/Psg) should increase to create

sustainable ratio of shortening to growing microtubule polymers.he increase in catastrophe, coupled with the decrease in rescue,hould shift polymers into the shortening state to offset the biasor growing microtubules resulting from nucleation and extinction.his result suggests that when nucleation and extinction are con-idered, the steady state will be established at a lower relative freeubunit concentration and higher polymer mass relative to MDss

Fig. 1). Assuming nucleation and extinction are equal at steadytate, we can simplify Eqs. (12) and (13) (using only N here) toxpress the ratio of shortening to growing polymers (Ms/Mg) in rela-ion to the transition frequencies when nucleation and extinctionre included (Eq. (17)):

Ms

Mg= Pgs

Psg− N

MgPsg(17)

We predict that the system will seek a final steady state wherelobal nucleation and extinction rates are equal and where the ratiof growing to shortening microtubules remains constant to offsethe magnitude of N and E. Eqs. (16) and (17) provide clear predic-ions for the effect of increasing N and E on the steady state systemut provide little information about how nucleation and extinctionome to an equal value.

Microtubule nucleation is controlled in cells primarily by nucle-tion templates that initiate polymers at a steady state free subunitoncentration where significant spontaneous nucleation does notccur. We propose that the steady state nucleation rate (N) wille a product of the concentration dependent probability of nucle-tion (Nc) per unit time and the number of unoccupied nucleationemplates (Tu) within the total template (Tt) population. From Eq.13), we propose that the extinction rate (E) will be equal to the dif-erence between the number of growing microtubules undergoingatastrophe and the number of shortening microtubules rescuing torowth. Nucleation has typically been modeled as an instantaneousecovery of extinct polymers, acting as a spatially bounded form ofescue (Dogterom et al., 1995; Gliksman et al., 1993; Gregorettit al., 2006; Mitchison and Kirschner, 1987; Vorobjev and Maly,008). However, we speculate that when nucleation is made depen-ent upon the free subunit concentration, regimes will arise with aool of unoccupied templates that lead to a more natural polymerensity owing to the establishment of a steady state nucleation andxtinction rate.

Our algebraic relationship of microtubule dynamics and nucle-tion parameters provides a general set of predictions for

etermining a steady state for the microtubule dynamics. How-ver, we could not derive an analytical expression to determinehe steady state extinction rate (Dogterom and Leibler, 1993; Verdet al., 1992; Yarahmadian et al., 2010). We find that the nucleation

and Chemistry 35 (2011) 269–281 273

rate, occupancy rate of nucleation templates and the extinction rateare not well-determined measurements from cellular systems formodeling (Bre et al., 1990; Buxton et al., 2010; Ebbinghaus andSanten, 2011; Goodson and Gregoretti, 2010; Nedelec et al., 2003;Piehl et al., 2004; Rubin, 1988; Walczak and Heald, 2008). We there-fore feel compelled to explore several possible cellular regimes fornucleation that could exist for determining the steady state. Wemove to simulation studies to investigate how the steady statecondition arises from different sets of nucleation and dynamicsparameters.

3.3. Macroscopic Monte Carlo simulations for evaluation ofsteady state values

We performed Monte Carlo simulations of a compact astralmicrotubule system using our reference set of parameter values(Table 1) and the starting cellular context values. The time-evolution of the simulated microtubule array shows a rapidestablishment of a steady state free subunit concentration wherethe subunits incorporated into polymer are balanced by thesubunits being released (Fig. 2A). A steady state free subunit con-centration of 7.05 �M is obtained for the reference set of parametervalues (Fig. 1). This value is less than the 7.65 �M value (MDss)predicted from the parameter models when N and E are takenas zero (Fig. 1) and is in agreement with our general predictionsfrom Eq. (11). The average run length times for growth and short-ening events (Fig. 2B) come to a steady state later in time thanthe dimer flux (i.e. >100 s vs 25 s), revealing an initial settling ofthe microtubule length distribution after the initial steady stateis established. Nucleation and extinction rates are initially vari-able but achieve approximately equal steady state values by 100 s(Fig. 2C), slightly later than the predicted steady state for subunitassociation and dissociation.

To visualize the time-evolution of the array morphology, weplotted the mean microtubule length versus microtubule numberfor each iteration of the simulation (Fig. 1D – initial simula-tion). The plot reveals an initial burst of nucleation occupyingnearly all of the 750 nucleation templates. As tubulin subunitsare incorporated into polymer, the average microtubule lengthrapidly increases and the number of microtubules settles to a valuecloser to 700. We observe that at this proportion of total tubulinsubunits to nucleation sites, 93% of the available nucleation tem-plates are occupied and 25 microtubules (3.5% of population) aregained and lost per time interval for the reference set. The micro-tubules reach a stable length distribution (Fig. 2E) characterizedby an exponential decrease in the number of longer microtubules(Fig. 2F).

After establishing a global steady state, we instantaneouslyincreased the catastrophe frequency (variable k7) by changing theparameter model (Fig. 2D – Perturbation 1). We observed that thearray morphology changes rapidly, first losing microtubules andthen reaching a steady state average length. Reverting the catas-trophe model back to the initial reference set value results in thearray returning to the initial steady state morphology (Fig. 2D – Per-turbation 2). The array morphology follows essentially the samepath as the initial simulation, rapidly increasing the microtubulenumber and then more slowly settling to a final mean length withfewer microtubules. These data show that the final steady statearray morphology is determined intrinsically for the variable setand is not affected by the starting array morphology. These datafurther support the general prediction that there are two steadystate processes working interdependently to achieve the steady

state morphology. The dimer flux appears to reach a numericalequilibrium first followed by the eventual balancing of nucleationand extinction to create a stable polymer number of constant meanlength.

274 M. Mourão et al. / Computational Biology and Chemistry 35 (2011) 269–281

0 50 100 1500

2

4

6

8

10

IncorporatedFree

0 50 100 1500

10

20

30

40

50

GrowthShortening

0 50 100 1500

10

20

30

40

NucleationsExtinctions

Time (s) Time (s)Time (s)

Tubu

lin C

onc.

(µM

)

Mic

rotu

bule

s (s

-1)

Pha

se R

un T

ime

(s)

A CB

1 2 3 4 5 6 7 8 9 10 11 12 13 140

50

100

150

200

250

300

Microtubule Length (µm)

Num

ber o

f Mic

rotu

bule

s

EF

0 5 10 15−2

0

2

4

6

ln M

T nu

mbe

r

Microtubule Length (µm)

D

550

600

650

700

750

0 0.5 1 1.5 2 2.5

Mean Microtubule Length (µm)

Num

ber o

f Mic

rotu

bule

s

500

Initial SimulationPerturbation 1Perturbation 2

Fig. 2. Monte Carlo simulation results for the reference parameter set. The mean concentration values for free (red) and incorporated (blue) tubulin subunits from simulationsof the reference set parameters are plotted over the first 150 s of simulation (A). Each point in A–C represents the mean and standard deviation from 15 simulations withevery 10th iteration plotted. The mean contiguous run times (B) for growth (blue) and shortening (green) events are plotted over time and come to a steady state after asteady state tubulin concentration is established. The mean number of microtubules that are nucleated (red) and lost to extinction (blue) per iteration are plotted againsttime (C) and come to a steady state after the subunit steady state. The time-evolution of array morphology is graphed as a scatterplot of microtubule number over the meanlength of the microtubule array (D). The initial simulation (blue ×) starts with zero microtubules and rapidly accumulates microtubules (blue arrow indicates progression ofarray morphology) until the array settles to a global steady state morphology. The steady state distribution of microtubule lengths (E) forms an exponential distribution (F).Increasing the catastrophe frequency (D, Perturbation 1) shifts the array morphology (green arrow) to fewer microtubules of shorter mean length. Reverting the catastrophep y (red

(scsasdsnsic

3m

dstsaVi1

robability to the reference set parameter value (D, Perturbation 2) returns the arra

We evaluated the analytical relationships proposed in Eqs.16) and (17) using steady state parameter estimates from ourimulations of the reference set parameter models. Numeri-ally evaluating both sides of Eq. (16) for fifteen individualimulations, we find an average value of 0.528 with an aver-ge difference of 0.028 ± 6.35e−4. Numerically evaluating bothides of Eq. (17), we find an average of 0.514 and an averageifference of 8.35e−4 ± 0.0027. In both cases we find that theimulations support the previously made conjecture that whenucleation and extinction are explicitly added to the macro-copic model for these compact arrays, subunits are shiftednto polymer, lowering the apparent steady state free subunitoncentration.

.4. Microtubule dynamics parameters and steady state arrayorphology

We investigated the effect of the individual microtubuleynamics parameters on microtubule arrays using Monte Carloimulations. Using the original cellular context variables (i.e. 9 �Mubulin, 750 nucleation sites, 15 �m length), we incremented thelope value of each reference set parameter model individually over

range of 10 values, bracketing the original reference set (Table 1 –ariations). All other parameters were taken from the reference set

n each simulation yielding 5 sets (one for each parameter model) of0 simulations. We first determined how changes in each parame-

arrow) back to the original mean length and microtubule number.

ter model influenced the steady state value of the other microtubuledynamics parameters in this system where the parameter modelsare explicitly coupled to the same finite subunit pool (Fig. 3). Forcomparisons, the difference was calculated for each final steadystate parameter value (i.e. Vg, Vs, Nc, Psg, and Pgs) relative to thevalue obtained with the reference set parameters (Fig. 3).

Incrementing the growth velocity from lower to higher values(Fig. 3A, Vg – thick line) produced a positively correlated changein catastrophe probability with little change to the other param-eters at steady state. Increasing the shortening velocity produceda positively correlated change in the nucleation probability with anegatively correlated effect on catastrophes (Fig. 3B). We observethat altering the relative frequency of nucleation over this range(Fig. 3C) has little predicted effect on any of the other steady stateparameter values.

Increasing the probability of rescue in these simulations drivesa proportionately small increase in catastrophe with a negativelycorrelated effect on both growth velocity and template nucleationprobability (Fig. 3D). Altering the probability model for catastropheproduced a dramatic positively correlated change in the probabilityof individual template nucleation (Fig. 3E). This effect is explainedby the observation that increased catastrophe leads to a higher

extinction rate and therefore creates more unoccupied templateswith more opportunity for nucleation. The steady state values forboth growth velocity and rescue probability are predicted to showpositively correlated changes with catastrophe (Fig. 3E).

M. Mourão et al. / Computational Biology

A

B

C

0

1

2

3D

1 2 3 4 5 6 7 8 9

0

1

2

3E

Growth Velocity Parameter (k3)

VgVs NcPgsPsg

(SS

-SS

r)/S

Sr

(SS

-SS

r)/S

Sr

(SS

-SS

r)/S

Sr

(SS

-SS

r)/S

Sr

(SS

-SS

r)/S

Sr

30 3837363534333231

10 1817161514131211

1 2 3 4 5 6 7 8 9

11 12 135 6 7 8 9 10Rescue Parameter (k5)

Shortening Velocity Parameter (C1)

Catastrophe Parameter (k7)

Nucleation Parameter (k1)

−0.20−0.15−0.10−0.05

0.000.050.100.15

0.0

−0.3

−0.2

−0.1

0.1

0.2

−0.20

−0.15

−0.10

−0.05

0.00

0.05

(x10-4)

(x10-6 )

(x10-2)

(x10-2 )

(x10-2 )

Fig. 3. Effects of incrementing individual parameter values. We incremented theslope for each parameter model (Eqs. (1)–(5)) over a range of 10 values bracketingthe reference set value in each case (Table 1). The steady state values for all dynam-ics variables were then obtained through Monte Carlo simulations and compared tothe steady state value obtained using the reference set of parameter values. Incre-menting the growth velocity model parameter k3 (Eq. (2)) alters the final steadystate growth velocity (A, thick blue line) with a positively correlated effect on catas-trophe (A, pink line). Incrementing the shortening velocity model variable C1 (Eq.(3); thick black line in B) results in a correlated effect on steady state nucleationfrequency (B, red line) with an opposing effect on catastrophe (B, pink line). Notethat shortening velocity is modeled as a constant rate, relative to free subunit con-centration, resulting in no relative change in steady state value for A, C, D, and E. Thefrequency of nucleation at unoccupied templates (C, thick red line) has relativelylittle effect on other steady state parameters. Catastrophe (D, thick green line) andrescue (E, thick pink line) show relatively strong effects on the parameter values(note the change in ordinate scale).

and Chemistry 35 (2011) 269–281 275

We next examined the predicted effect of the single parameterson microtubule array morphology. The steady state microtubulenumber is plotted against the average microtubule length for eachparameter (Fig. 4). Each point on the plot represents an average of15 simulations where the numbers 1–10 represent the increasingvalues for each parameter model. Using the steady state polymermass from the reference set parameter values, a trend line wascreated to indicate the partitioning of the total polymer length intoarrays of different average microtubule length (Fig. 4).

Changing the growth and shortening velocities relative to freesubunit concentration has opposing effects on average micro-tubule length (Fig. 4). Increasing growth velocity increases averagemicrotubule length with a relatively small reduction in averagemicrotubule number. The predicted length increase resulting fromfaster subunit addition shifts free subunits into polymer lead-ing to a lower steady state free subunit concentration. In turn,catastrophes increase and rescues decrease (Fig. 3A), lesseningthe overall effect on microtubule length and leading to increasedextinctions. Increasing the shortening velocity reduces averagemicrotubule length and also reduces microtubule number. Asshortening velocity is increased, the average microtubule lengthshortens and more free tubulin is liberated per unit time into thesystem. However, the increased amount of free tubulin is not suf-ficient under these conditions to offset the increased probabilityof extinction, owing to the number of shorter microtubules andthe effect of the faster rate on taking short microtubules to extinc-tion.

We varied the concentration dependent probability that anunoccupied nucleation template would initiate a new microtubule.As expected, we find that as the probability of template nucle-ation is increased, the total number of microtubules increases(Fig. 4). We observe that increasing nucleation probability doesnot overtly shift the final steady state free subunit concentrationat this template number. The polymer mass remains roughly con-stant as nucleation rate increases where the increased numberof microtubules is compensated for by a gradual decrease in theaverage microtubule length. We observe that nucleation rate hasa substantial effect on the final array morphology but only mod-estly affects the steady state value of other dynamics parameters(Fig. 3C).

Altering the transition probabilities shifts the steady state par-titioning of dimers and dramatically alters the morphology ofthe microtubule array, predominantly through changes to averagepolymer length (Fig. 4). As the probability of rescue is increased rel-ative to free subunit concentration, both the average microtubulelength and the average number of microtubules increase almostproportionally. As rescue probability gets higher, the decrease infree tubulin subunits increases the catastrophe rate and decreasesthe growth velocity (Fig. 3C). We also observe that the occupancy ofnucleation templates becomes limiting for nucleation and micro-tubule number.

Catastrophe frequency has a relatively large effect on the com-pact array morphology (Fig. 4). As the probability of catastrophedecreases (increasing numbers in Fig. 4), the average microtubulelength increases and the probability that new microtubules willpersist increases. When combined, this effect drives subunits intopolymer and proportionally increases average length and the den-sity of microtubules analogous to increasing rescue frequency.Increasing the catastrophe frequency has the opposite effect. Theaverage microtubule length and the total number of microtubulesdecrease. Though increasing the catastrophe frequency results inan increased free subunit concentration, at some point, most of

the newly nucleated microtubules never elongate and are rapidlydepolymerized to extinction (Fig. 4, Pgs values 1–3). This propertyof dominantly affecting average microtubule length down to thepoint of truncating most newly made microtubules suggests that

276 M. Mourão et al. / Computational Biology and Chemistry 35 (2011) 269–281

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5550

600

650

700

750

1 2 3 4 57 8 910

12345

8910

1

2

3

45

78910

1

3 4 5 6 7 8 9 10

1

2

3

4

5

78

910

Mean Microtubule Length (µm)

Num

ber o

f Mic

rotu

bule

s

Catastrophe

Nucleation

RescueVs

Vg

Vg (k3)Vs (C1) Nuc (k1)Pgs (k7)Psg (k5)

Fig. 4. The effect of individual microtubule dynamics parameters on the steady state microtubule array morphology. Parameter values for each of the dynamics models wereindividually incremented over 10 values above and below the reference set value (Table 1). The steady state microtubule number was plotted against mean microtubulelength (averaged values from 15 simulations) for each parameter increment (numbered 1–10 for each parameter) and connected with a line related to the parameter (seei state

d

cm

3m

finntvicotfpiio

tbsiutsoltttts

TV

nset key). A trend line (gray dashed line) was constructed using the total steady

ivided into arrays of increasing microtubule number.

atastrophe frequency will play a dominant role in controlling arrayorphology.

.5. The role of cellular context in shaping microtubule arrayorphology

We investigated the predicted effect of cellular context on thenal steady state array morphology. Beginning with our origi-al cellular context values, we varied the total subunit number,ucleation template number, and the maximum allowed micro-ubule length. In each case, we selected one value above and onealue below the original cellular context value (Table 2) creat-ng a 3 × 3 × 3 matrix of cellular context values for simulation. Toompare the effects of cellular context on the final array morphol-gy, we plot the steady state number of microtubules achieved inhe system against the average microtubule length for the 27 dif-erent cellular context simulations (Fig. 5). A central trend line islotted indicating the expected partitioning of total polymer mass

nto arrays of increasing average microtubule length and decreas-ng microtubule number for the reference set of parameters andriginal cellular context values (position 14 in Fig. 5).

Our simulations indicate that the initial molar concentration ofubulin subunits in the system should have a significant impact onoth the steady state free subunit concentration and the steadytate polymer mass. The steady state free subunit concentrations predicted to scale positively with the total number of sub-nits in the system (Fig. 1). Therefore, the overall dynamicity ofhe system (i.e. subunit turnover) should increase with increasingubunit concentration. The effects of initial subunit concentrationn array morphology are highlighted in Fig. 5 with broad trendines. When all other variables are held the same, increasing theotal subunit number has the general effect of shifting the micro-

ubules to a longer average length with only a modest increase inhe average number of microtubules. Decreasing initial concen-ration has the exact opposite effect (Fig. 5). Thus, total tubulinubunit number is predicted to have a large effect on average

able 2alues for the cellular context variables.

Initial free tubulin 8 �M 9 �M 10 �MMTs max number 500 750 1000MTs max length 10 �m 15 �m 20 �m

polymer length for the reference set simulation (point of intersection in the plot)

microtubule length but less effect on the steady state number ofmicrotubules.

We varied the number of nucleation templates in the systemover a range from 500 to 1000. The probability of any unoccupiedtemplate nucleating a new microtubule was determined using thereference set parameter model for all simulations. Thereby, the rateof steady state nucleation was determined by both the number ofunoccupied sites and the concentration dependent probability ofnucleation. We observe that as the total number of nucleation tem-plates increases, the number of microtubules also increases (Fig. 5).However, we see that the percentage of occupied nucleation sitesat steady state actually decreases mildly with increasing templatenumber. This trend in occupancy rate is modified by the total num-ber of subunits in the system suggesting that a significant factor indetermining microtubule number will be the proportion of nucle-ation sites to total subunit number in the system. We observe thatas the number of total nucleation sites increases, the average micro-tubule length decreases. This average length decrease follows ageneral trend where the total polymer mass (i.e. total length ofpolymer) is divided by a total number of microtubules to producea roughly hyperbolic relationship of average microtubule length(trend line in Fig. 5). Therefore, the number of nucleation templatesis expected to modify the array morphology substantively throughshifting polymer density while having a relatively modest effect onthe steady state free subunit concentration (Fig. 5).

We reasoned that microtubules in most somatic cells do notreach lengths that are substantially longer than the cell radius. Wetherefore created an artificial cell boundary that triggers a catas-trophe when reached by a microtubule end. The boundary shouldhave significant effects on the time evolution of the pre-steadystate system when many early microtubules would reach extraor-dinary lengths before the catastrophe probability reaches a steadystate value. We instituted maximum lengths of 10, 15, or 20 �m.We observed that for our reference set of parameters, producinga relatively compact astral array, restricting the maximum micro-tubule length had little predicted effect on the steady state arraymorphology (Fig. 5).

In sum, we find that simulating the same reference set ofmicrotubule dynamics parameter models within different cellularcontexts leads to significantly different array morphologies. Criti-cally, we predict a strong interaction between total subunit number

and the total number of microtubule nucleation sites present in thesystem.

M. Mourão et al. / Computational Biology and Chemistry 35 (2011) 269–281 277

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5300

400

500

600

700

800

900

1000

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

Mean Microtubule Length (µm)

10

1

12

13

14

15

16

1

19

0

21

22

233

24

25

262

2

[8 µM] [9 µM][10 µM]

500 Nuc

750 Nuc

1000 Nuc

Num

ber o

f Mic

rotu

bule

s

Fig. 5. The effect of cellular context on the steady state microtubule array morphology. The starting tubulin concentration, the number of nucleation templates, and themaximum polymer length were independently incremented for simulations using the reference set of parameter values. The steady state microtubule number was plottedagainst mean microtubule length (averaged values from 15 simulations) for each cellular context change. Results for 8 �M (1–9), 9 �M (10–18), and 10 �M (19–27) initialconcentrations are plotted with numeric labels and grouped using three broad trend lines. Circled regions group simulations with incremented nucleation templates: 500(points 1,4,7,10,13,16,19,22,25), 750 (points 2,5,8,11,14,17,20,23,26) and 1000 (point 3,6,9,12,15,18,21,24,27). Each cluster of 3 numeric labels shows the effect of changingt dashe(

3a

ttpmptpc

tphrawTaibtisdat

ntaaaWtMmf

he maximum microtubule length (10 �m, 15 �m, and 20 �m). The trend line (graypoint 14 in the plot) divided into arrays of increasing microtubule number.

.6. Nucleation probability and template number shape the finalrray morphology

We further investigated how nucleation probability andemplate number combine to influence the final steady state micro-ubule array. Using the reference set of microtubule dynamicsarameters, we varied template number from 100 to 1300 in incre-ents of 200 templates. We additionally altered the nucleation

robability model by changing the y-intercept value (variable k2)o create 3 independent regimes of progressively higher nucleationrobability over the expected range of final steady state free subunitoncentrations.

We observed increasing matched rates of nucleation and extinc-ion with increasing template number (Fig. 6A). The nucleationrobability model had a significant effect on the final rates, withigher probabilities leading to higher nucleation and extinctionates. The Nc model having the highest initial nucleation prob-bility leads to a trend upwards with increasing template valuehile the medium probability model looks nearly linear (Fig. 6A).

he Nc model having the lowest initial probability trends towardsymptotic with increasing template number. We see an increasen the total number of microtubules with increased template num-er and nucleation probability (Fig. 6B). The percent occupancy ofhe templates was strongly influenced by the nucleation probabil-ty model, where higher frequency resulted in a higher number ofteady state microtubules. The percent occupancy of the templatesecreased with increasing template number for all nucleation prob-bility models indicating a progressively larger pool of unoccupiedemplates.

We observed the combined effects of template number anducleation probability on microtubule array morphology by plot-ing the average microtubule number at steady state against theverage length of the microtubule population (Fig. 6C). We created

trend line by dividing the total length of microtubule polymert MDss by increasing microtubule number to get average length.e observe that with 100 templates, the microtubules have a rela-

ively long average length and sit close to the trend line estimating

Dss polymer levels. With increasing template number, the averageicrotubule length decreases proportionately but also moves away

rom the MDss trend line. This result is in general agreement with

d line) indicates total steady state polymer length for the reference set simulation

our prediction that increased nucleation and extinction will resultin increased total polymer, relative to MDss. The additional micro-tubules created with higher nucleation probability models havea progressively smaller effect on the average array length, owingto the near hyperbolic relationship of average length to polymernumber.

Plotting the final steady state values for the velocity and transi-tion probability ratios, we see the shift of values to a lower steadystate free subunit concentration predicted from Eq. (16) (Fig. 6D).The ratio of shortening to growing microtubules is further plottedon the same graph and clearly follows the trend line establishedby the velocity ratios. These data support the general predictionthat as the template and nucleation probabilities increase, result-ing in an increased rate of nucleation and extinction, the systemwill seek a lower free subunit concentration where a stable ratio ofgrowing to shortening microtubules can be found. Contrary to ouroriginal predictions, we observe that at the level of 100 templates,the simulations predict a slightly higher free subunit concentra-tion with less total polymer (Fig. 6D). This result can be explainedin the simulations by our imposition of a length limit at 15 �m.The boundary-induced increase in catastrophe frequency resultsin a bias for shortening microtubules that cannot be balanced byincreasing the nucleation rate. Therefore, we predict that the sys-tem seeks a higher steady state free subunit concentration wheremore growing microtubules can be created.

We plotted the probability of extinction for all combinations oftemplate number and Nc model by dividing the steady state extinc-tion rate by the number of shortening microtubules (Fig. 6E). Weobserve that all values, except those for 100 templates, fall on asingle trajectory when plotted over the steady state free subunitconcentration. Rearranging Eq. (13) to plot expected extinctionprobability (i.e. E/Ms = Mg/[MsPgs] − Psg), taken over the same freesubunit concentration, we observe a clear correlation for templatenumbers greater than 100. The ratio of shortening and grow-ing microtubules was assumed for this plot to be equal to theratio of growth to shortening velocity. This result suggests thatas nucleation probability and template number combine to raise

the nucleation rate and lower the free subunit concentration, theextinction probability will fall on a trajectory defined principallyby the microtubule dynamics parameters.

278 M. Mourão et al. / Computational Biology

0 200 400 600 800 1000 1200 14000

10

20

30

40

50

60

70

80E

xtin

ctio

n or

Nuc

leat

ion

Rat

e (s

-1)

Templates

0 200 600 1000 14000

200

400

600

800

1000

1200

1400

Templates

Mic

rotu

bule

s

0 5 10 150

200

400

600

800

1000

1200

1400

Average Microtubule Length (µm)

Mic

rotu

bule

s

LowMedHigh

6.5 7 7.5 80.30.40.50.60.70.80.91.01.11.21.3

Rat

ios

Free Subunit Conc. (µM)

Vg/VsPgs/PsgMs/Mg

6.5 7 7. 5 8−0.05

0

0.05

0.1

0.15

0.2

Free Subunit Conc. (µM)

Ext

inct

ion

(s-1

)

LowMedHigh

LowMedHigh

100 Templates

100 Templates

100 Templates

AB

C

D

E

Fig. 6. Combined effect of nucleation template number and nucleation probability.The total number of nucleation templates was varied from 100 to 1300 in incre-ments of 200 for Monte Carlo simulations using the reference set of microtubuledynamics parameters. The nucleation probability model for unoccupied templateswas varied to yield three probability ranges denoted as low, med, and high (k2 = 0.70,0.65, and 0, respectively). Steady state extinction and nucleation rates were equal inall cases and increased with increasing template number and nucleation probability(A). Total microtubule number also increased with increasing template number andnucleation probability (B) while the percent template occupancy declined (dashedline in B represent 100% occupancy). The steady state microtubule number is plot-ted against average microtubule length depicting array morphology for all templatenumbers and nucleation probability models (C). Template number is lowest to theright of the plot (i.e. 100 templates), and moves progressively to the left for eachprobability model. The array morphology follows the general trend expected forthe partitioning of the steady state polymer mass at MDss (solid line) except thatmore polymer is found after 100 templates for all models (i.e. plot to the right ofthe trend line). The final steady state values for the microtubule dynamics parame-ters are plotted as ratios for each template number and probability model (D). Theratios are lowest to the right of the plot (i.e. 100 templates) and move sequentiallyto the left indicating a progressively lower steady state free subunit concentration.The ratios of shortening to growing microtubules follow the predicted path of thevelocity ratios. Dashed lines are plots of the reference set parameter ratios. Theextinction probabilities (E/Ms) for all template numbers and nucleation models (pro-gressive from right to left) follow the trajectory predicted from rearranging Eq. (13)as described in Section 3.6.

and Chemistry 35 (2011) 269–281

4. Conclusions

4.1. Microtubule arrays exhibit two interdependent steady statephenomena

Our macroscopic simulation study examines the question ofhow microtubules come to a final steady state array morphologyin a cell with template-based nucleation. We focus on developingexpectations for the system in the absence of specific or localizedMAP-based activities (Cheerambathur et al., 2007; Cytrynbaumet al., 2004; Loughlin et al., 2010; Nedelec et al., 1997). We makethe simplifying assumptions that the dynamics parameters andnucleation probability have a linear relationship to the global freesubunit concentration in a fixed cell volume (Gliksman et al., 1993;Gregoretti et al., 2006; Mitchison and Kirschner, 1987; Vorobjevand Maly, 2008). The principal concept arising from this study isthat both nucleation and extinction bias the array in favor of grow-ing polymers. Provided that the cell does not create a mechanismto prevent extinctions, we predict that the microtubule system willcompensate by finding a lower steady state free subunit concentra-tion to offset this bias in growing versus shortening polymers. Oursimulations indicate that nucleation and extinction alter the steadystate free subunit concentration established by the microtubuledynamics parameters as predicted from our model.

We consider template-based nucleation to be a steady state pro-cess, rather than an equilibrium system, owing to the energeticrequirements for creating the templates and in maintaining the freeGTP-tubulin pool. We therefore propose that there are minimallytwo macroscopic and interdependent steady state phenomenadriving the system to a global steady state. The microtubule dynam-ics parameters partition the subunits into free and incorporatedpools. Nucleation and the availability of nucleation templates setthe polymer density and balance the loss of polymers brought onby extinction. The two steady state phenomena are interconnectedbecause the magnitude of nucleation and extinction will affect thefinal partitioning of subunits at steady state.

The time-evolution of the simulations suggests that the twosteady state processes can be temporally distinct. We proposethat for a system undergoing a sudden perturbation, throughsignaling or introduction of a new modifying activity, the parti-tioning of tubulin subunits into free and incorporated pools occursrelatively rapidly. However, balancing nucleation and extinctionshould require more time owing to the slower processes of adjust-ing the length distribution and ratio of growing to shorteningpolymers. The transition from interphase to mitotic arrays in ani-mal systems, for example, undergoes a dramatic rearrangementunderpinned by significant changes to the microtubule dynamics(Desai and Mitchison, 1997; Walczak and Heald, 2008). Our resultspoint to the likelihood that the subunit partitioning is establishedbefore the eventual length distribution and microtubule numberare defined.

4.2. Array morphology is shaped by both dynamics andnucleation parameters

Our macroscopic simulation data provide a set of predictionsfor how changes to the basal microtubule dynamics and nucleationparameters should affect steady state array morphology. We findthat the roughly hyperbolic scaling of the steady state microtubulemass between microtubule number and average length emergesas an important conceptual framework from this study. We seethat the dynamics parameters, when applied to the total subunits

in the cell volume, lead to an initial prediction for polymer mass.The interplay between the available nucleation templates and thedynamics parameters distributes that polymer mass into a basearray morphology that is constrained by this basic relationship. In

iology

btnTinatp

tsimicmaaqamta

motssbVrfiratttsw(nsc

4

mnatiptiras

naImi

M. Mourão et al. / Computational B

road terms, we observe that the microtubule dynamics parame-ers tend to alter the average length of the microtubule array whileucleation properties more directly alter the microtubule number.hese effects arise because changes to the dynamics parametersnherently shift the steady state free subunit concentration whileucleation primarily changes the capacity for microtubule numbernd only secondary changes free subunit concentration. We assumehat in a cellular system, combinations of parameters are altered toroduce refined effects on the final array.

We find that the morphology of the microtubule array is par-icularly sensitive to catastrophe frequency. Our simulation resultshow that transition frequency from growth to shortening primar-ly changes the length distribution of the array with less effect on

icrotubule number up to a point. The range of values we exam-ned suggests that relatively high catastrophe frequency can have aompounding effect on the array if many new microtubules depoly-erize to extinction within the first few seconds. In this case, the

rray is reduced to short microtubules, rapidly nucleating and dyingt high free subunit concentration. Lowering the catastrophe fre-uency leads to longer excursion distances and arrays of longerverage length. We point out that a significant proportion of theicrotubule-associated proteins thus far identified directly affect

he catastrophe frequency (Desai and Mitchison, 1997; Walczaknd Heald, 2008).

We included a cell boundary in our simulations to preventicrotubules from getting longer than the cell radius. For the range

f starting tubulin concentrations that we used in these simula-ions, the length-induced catastrophe had little effect on the finalteady state array morphology for the majority of the parameterets examined. The effects of driving the microtubules into a lengthoundary have been explored theoretically (Gregoretti et al., 2006;orobjev and Maly, 2008) and provide a potentially importantationale for arrays showing persistently long microtubules. We dond that when the nucleation template number is held very low,elative to total subunit number, there is a pronounced effect onrray morphology (Margolin et al., 2006). In our case, we observedhat with very few templates, the microtubules grow long enoughhat the number of induced catastrophes creates an imbalance inhe ratio of shortening to growing microtubules. In the simulationtudies, the system migrated to a higher free subunit concentrationhere the concentration dependent catastrophe probability is less

Fig. 6D). These data indicate that cell boundary can exert a pro-ounced effect on the system and suggest the possibility of steadytate free subunit concentrations higher than MDss under specialircumstances.

.3. Extinction rate

We have focused on the critical issue of how the rate oficrotubule extinction arises from the microtubule dynamics and

ucleation properties. We expect from the biological system thatvailable nucleation templates, and a nucleation probability greaterhan zero for each template, leads to an addition of new polymersn a growth phase. The nucleation probability per template, we pro-ose, must be above zero to avoid collapse of the system. We findhat the total number of templates and the probability of nucleationn a unit time interval, combine to create at least two predictedegimes for the microtubule system; one that is template limitednd another where a pool of unoccupied templates exist at steadytate.

We found in our simulation studies that a system with very fewucleation templates relative to polymer mass at MDss creates rel-

tively long microtubules, having an extremely low extinction rate.n this template-limited regime, the likelihood of replacing a lost

icrotubule before losing a second microtubule was high, produc-ng a near 100% occupancy rate for templates. The final steady state

and Chemistry 35 (2011) 269–281 279

free subunit concentration was nearly identical to MDss except thatthe imposed cell boundary led to a slight shift upward to accom-modate the high rate of imposed catastrophe. We found that theaverage microtubule length was approximately equal to the poly-mer mass specified at MDss divided by the total template number.In this regime, we saw that the extinction rate for the reference setof dynamics parameters was relatively low because the total poly-mer mass was divided into only a few nucleation templates. Finally,we observe that when templates are extremely limiting, the nucle-ation rate is explicitly limited by the extinction rate (Gregorettiet al., 2006; Vorobjev and Maly, 2008).

When template number was increased relative to total polymermass at MDss, we observed that the extinction rate and nucleationrate both increased. We see in this study that with the increasedavailability of nucleation templates the nucleation rate eventuallybecomes self-limiting, resulting in a pool of unoccupied templates(Tu). In this regime, the nucleation probability (Nc) drives subunitsinto polymer resulting in a lower free subunit concentration aspredicted by Eq. (16). Lowering the free subunit concentration, inturn, has two predicted self-limiting effects on the steady state sys-tem. First, the increased catastrophe, lower rescue probability, andslower growth velocity resulting from lower subunit concentration(Fig. 1) should increase the extinction rate (i.e. MgPgs > MsPsg). Sec-ond, nucleation probability (Nc) per template is directly coupledto free subunit concentration and should fall with the decrease infree subunit availability. Therefore, when templates are not limit-ing, the nucleation probability starting at MDss drives the system toa new free subunit concentration, ultimately resulting in increasedextinction and an attenuated nucleation probability. We proposefrom our analysis that these are the relevant properties of thesystem that equalize extinction rate with nucleation rate in thecell.

We find that the final extinction probability (i.e. extinctionrate/shortening microtubules) is related to the transition probabil-ities and to the proportion of growing to shortening microtubulesthrough steady state free subunit concentration (Fig. 6E). Predict-ing a specific extinction rate, given the total subunit and nucleationtemplate values, requires that we determine either the steady statenumber of shortening microtubules or the steady state extinc-tion probability. Further work using higher order mathematicalapproaches may reveal this underlying association (Ebbinghausand Santen, 2011; Flyvbjerg et al., 1996; Maly, 2002; VanBuren etal., 2005; Yarahmadian et al., 2010).

4.4. Application to live-cell studies

A typical mammalian tissue culture cell organizes the inter-phase microtubules into an astral array with the dynamic plus endstoward the cell margins and minus ends associated with the cen-trosome. Through regulation of the microtubules, the cell can alterthat basal astral array morphology to create spindles, axonal bun-dles, and other specialized array types (Budde et al., 2006; Condeand Caceres, 2009; Cortes et al., 2006; Dogterom et al., 1995; Faivre-Moskalenko and Dogterom, 2002; Karsenti et al., 2006; Komarovaet al., 2009; Loughlin et al., 2010; Malikov et al., 2005; Nedelec et al.,1997, 2003; Papaseit et al., 1999; Rusan et al., 2001; Seetapun andOdde, 2010; Surrey et al., 2001; Walczak and Heald, 2008). In thiswork, we have focused on the interaction between nucleation andthe principal dynamics parameters to determine how the underly-ing array morphology is created. The observations provide a broadset of general predictions for how the individual nucleation anddynamics parameters should act on the array before the local or

global action of specific MAPs.

When measured in live cells, the 4 principle microtubuledynamic instability parameters rarely yield values that produce anet equality of subunits moving into and out of polymer (Vorobjev

2 iology

asss(fMw

olhoibttittlfm

rnclmmtcucmnbtam

A

fNawut

R

A

AB

B

B

B

B

80 M. Mourão et al. / Computational B

nd Maly, 2008). Taking the average time spent in growth andhortening, and the related rates of subunit gain and loss, the mea-urements typically suggest persistent gain or loss of polymer,ometimes referred to as a positive or negative drift coefficientMaly, 2002). The rationale presented for the perceived variationrom steady state is often couched in the widely held view that

APs control most of the relevant dynamic properties of the systemith potentially restricted spatial domains in the cell.

Recent theoretical work has focused on determining how muchf the dynamic instability process in cells can be attributed to cellu-ar context before consideration of MAP activities. Several studiesave made the case that persistent microtubule growth propertiesbserved in some cell types, and related aspects of the dynam-cs measurements, can be explained by the relationship of the celloundary to the average length of microtubule polymers in the sys-em (Gregoretti et al., 2006; Vorobjev and Maly, 2008). Assuminghat growing microtubules switch to shortening at the cell cortex,t was shown that a system making lots of otherwise long micro-ubules builds up free subunits from the induced catastrophes athe cell boundary. In theory, the increased free subunit pool wouldower the catastrophe frequency and the system would appear,rom the dynamic instability measures alone, to be gaining poly-

er.Our analysis suggests that nucleation phenomena confer a

elated property on to the microtubule array. We show that whenucleation sites are in excess, relative to total cellular tubulin con-entration, there is a non-linear effect on the average microtubuleength and number density. Our model further suggests that the

agnitude of nucleation and extinction will naturally shift theeasured catastrophe and rescue frequencies away from a state

hat would produce a perceived steady-state polymer mass whenalculated from the measured dynamic instability parameter val-es alone (i.e. a negative drift coefficient). Taken together withell border effects, a general case can be made from theory thateasurements of dynamic instability from live cells should appear

on-steady state owing to cellular context, irrespective of MAP-ased modulation of the dynamics. In grossly simplifying terms,here is added catastrophe at the cell border and added rescuet the centrosome that are typically not accounted for in live-celleasurements.

cknowledgements

The authors wish to thank Dr. Shantia Yarahmadian for help-ul advice related to this manuscript. SLS was funded by theational Science Foundation (MCB-0920555 to SLS) and The Indi-na Metabolomics and Cytomics Initiative (METACyt). MM and SSere funded by a grant from the James S. McDonnell Foundationnder the 21st Century Science Initiative, Studying Complex Sys-ems Program.

eferences

ldaz, H., Rice, L.M., Stearns, T., Agard, D.A., 2005. Insights into microtubule nucle-ation from the crystal structure of human gamma-tubulin. Nature 435, 523–527.

mos, L.A., 1995. The microtubule lattice—20 years on. Trends Cell Biol. 5, 48–51.re, M.H., Pepperkok, R., Hill, A.M., Levilliers, N., Ansorge, W., Stelzer, E.H., Karsenti,

E., 1990. Regulation of microtubule dynamics and nucleation during polarizationin MDCK II cells. J. Cell Biol. 111, 3013–3021.

run, L., Rupp, B., Ward, J.J., Nedelec, F., 2009. A theory of microtubule catastrophesand their regulation. Proc. Natl. Acad. Sci. U.S.A. 106, 21173–21178.

udde, P.P., Desai, A., Heald, R., 2006. Analysis of microtubule polymerization in vitroand during the cell cycle in Xenopus egg extracts. Methods 38, 29–34.

urbank, K.S., Mitchison, T.J., 2006. Microtubule dynamic instability. Curr. Biol. 16,R516–R517.

uxton, G.A., Siedlak, S.L., Perry, G., Smith, M.A., 2010. Mathematical modeling ofmicrotubule dynamics: insights into physiology and disease. Prog. Neurobiol.92, 478–483.

and Chemistry 35 (2011) 269–281

Caplow, M., Ruhlen, R., Shanks, J., Walker, R.A., Salmon, E.D., 1989. Stabilization ofmicrotubules by tubulin-GDP-Pi subunits. Biochemistry 28, 8136–8141.

Carlier, M.F., Hill, T.L., Chen, Y., 1984. Interference of GTP hydrolysis in the mech-anism of microtubule assembly: an experimental study. Proc. Natl. Acad. Sci.U.S.A. 81, 771–775.

Cassimeris, L.U., Walker, R.A., Pryer, N.K., Salmon, E.D., 1987. Dynamic instability ofmicrotubules. Bioessays 7, 149–154.

Cheerambathur, D.K., Civelekoglu-Scholey, G., Brust-Mascher, I., Sommi, P.,Mogilner, A., Scholey, J.M., 2007. Quantitative analysis of an anaphase B switch:predicted role for a microtubule catastrophe gradient. J. Cell Biol. 177, 995–1004.

Chen, Y., Hill, T.L., 1983. Use of Monte Carlo calculations in the study of microtubulesubunit kinetics. Proc. Natl. Acad. Sci. U.S.A. 80, 7520–7523.

Chen, Y., Hill, T.L., 1985a. Theoretical treatment of microtubules disappearing insolution. Proc. Natl. Acad. Sci. U.S.A. 82, 4127–4131.

Chen, Y.D., Hill, T.L., 1985b. Monte Carlo study of the GTP cap in a five-start helixmodel of a microtubule. Proc. Natl. Acad. Sci. U.S.A. 82, 1131–1135.

Compton, D.A., 2000. Spindle assembly in animal cells. Annu. Rev. Biochem. 69,95–114.

Conde, C., Caceres, A., 2009. Microtubule assembly, organization and dynamics inaxons and dendrites. Nat. Rev. Neurosci. 10, 319–332.

Cortes, S., Glade, N., Chartier, I., Tabony, J., 2006. Microtubule self-organisation byreaction-diffusion processes in miniature cell-sized containers and phospho-lipid vesicles. Biophys. Chem. 120, 168–177.

Cytrynbaum, E.N., Rodionov, V., Mogilner, A., 2004. Computational model of dynein-dependent self-organization of microtubule asters. J. Cell Sci. 117, 1381–1397.

Desai, A., Mitchison, T.J., 1997. Microtubule polymerization dynamics. Annu. Rev.Cell Dev. Biol. 13, 83–117.

Desai, A., Verma, S., Mitchison, T.J., Walczak, C.E., 1999. Kin I kinesins aremicrotubule-destabilizing enzymes. Cell 96, 69–78.

Dhamodharan, R., Wadsworth, P., 1995. Modulation of microtubule dynamic insta-bility in vivo by brain microtubule associated proteins. J. Cell Sci. 108 (Pt 4),1679–1689.

Dogterom, M., Leibler, S., 1993. Physical aspects of the growth and regulation ofmicrotubule structures. Phys. Rev. Lett. 70, 1347–1350.

Dogterom, M., Maggs, A.C., Leibler, S., 1995. Diffusion and formation of microtubuleasters: physical processes versus biochemical regulation. Proc. Natl. Acad. Sci.U.S.A. 92, 6683–6688.

Downing, K.H., Nogales, E., 1998. Tubulin and microtubule structure. Curr. Opin. CellBiol. 10, 16–22.

Ebbinghaus, M., Santen, L., 2011. Theoretical modeling of aging effects in micro-tubule dynamics. Biophys. J. 100, 832–838.

Faivre-Moskalenko, C., Dogterom, M., 2002. Dynamics of microtubule asters inmicrofabricated chambers: the role of catastrophes. Proc. Natl. Acad. Sci. U.S.A.99, 16788–16793.

Flyvbjerg, H., Holy, T.E., Leibler, S., 1996. Microtubule dynamics: caps, catastrophes,and coupled hydrolysis. Phys. Rev. E: Stat. Phys. Plasmas Fluids Relat. Interdiscip.Top. 54, 5538–5560.

Gliksman, N.R., Skibbens, R.V., Salmon, E.D., 1993. How the transition frequencies ofmicrotubule dynamic instability (nucleation, catastrophe, and rescue) regulatemicrotubule dynamics in interphase and mitosis: analysis using a Monte Carlocomputer simulation. Mol. Biol. Cell 4, 1035–1050.

Goodson, H.V., Gregoretti, I.V., 2010. Using computational modeling to under-stand microtubule dynamics: a primer for cell biologists. Method Cell Biol. 95,175–188.

Gregoretti, I.V., Chaturvedi, R., Sozinova, O., Alber, M., Goodson, H.V., 2005. Modelingmicrotubule dynamic instability. Biophys. J. 88, 642A.

Gregoretti, I.V., Margolin, G., Alber, M.S., Goodson, H.V., 2006. Insights into cytoskele-tal behavior from computational modeling of dynamic microtubules in a cell-likeenvironment. J. Cell Sci. 119, 4781–4788.

Hill, T.L., 1984. Introductory analysis of the GTP-cap phase-change kinetics at theend of a microtubule. Proc. Natl. Acad. Sci. U.S.A. 81, 6728–6732.

Hill, T.L., Carlier, M.F., 1983. Steady-state theory of the interference of GTP hydrol-ysis in the mechanism of microtubule assembly. Proc. Natl. Acad. Sci. U.S.A. 80,7234–7238.

Hill, T.L., Chen, Y., 1984. Phase changes at the end of a microtubule with a GTP cap.Proc. Natl. Acad. Sci. U.S.A. 81, 5772–5776.

Hill, T.L., Kirschner, M.W., 1982. Bioenergetics and kinetics of microtubule and actinfilament assembly–disassembly. Int. Rev. Cytol. 78, 1–125.

Karsenti, E., Nedelec, F., Surrey, T., 2006. Modelling microtubule patterns. Nat. CellBiol. 8, 1204–1211.

Komarova, Y., De Groot, C.O., Grigoriev, I., Gouveia, S.M., Munteanu, E.L., Schober,J.M., Honnappa, S., Buey, R.M., Hoogenraad, C.C., Dogterom, M., et al., 2009. Mam-malian end binding proteins control persistent microtubule growth. J. Cell Biol.184, 691–706.

Li, H., DeRosier, D.J., Nicholson, W.V., Nogales, E., Downing, K.H., 2002. Microtubulestructure at 8 A resolution. Structure 10, 1317–1328.

Loughlin, R., Heald, R., Nedelec, F., 2010. A computational model predicts Xenopusmeiotic spindle organization. J. Cell Biol. 191, 1239–1249.

Malikov, V., Cytrynbaum, E.N., Kashina, A., Mogilner, A., Rodionov, V., 2005. Centeringof a radial microtubule array by translocation along microtubules spontaneously

nucleated in the cytoplasm. Nat. Cell Biol. 7, 1213–1218.

Maly, I.V., 2002. Diffusion approximation of the stochastic process of microtubuleassembly. Bull. Math. Biol. 64, 213–238.

Mandelkow, E., Song, Y.H., Mandelkow, E.M., 1995. The microtubulelattice—dynamic instability of concepts. Trends Cell Biol. 5, 262–266.

iology

M

M

M

M

N

N

N

N

OO

O

O

P

P

R

R

R

M. Mourão et al. / Computational B

argolin, G., Gregoretti, I.V., Goodson, H.V., Alber, M.S., 2006. Analysis of a meso-scopic stochastic model of microtubule dynamic instability. Phys. Rev. E 74.

artin, S.R., Schilstra, M.J., Bayley, P.M., 1993. Dynamic instability of microtubules:Monte Carlo simulation and application to different types of microtubule lattice.Biophys. J. 65, 578–596.

itchison, T., Kirschner, M., 1984. Dynamic instability of microtubule growth.Nature 312, 237–242.

itchison, T.J., Kirschner, M.W., 1987. Some thoughts on the partitioning of tubulinbetween monomer and polymer under conditions of dynamic instability. CellBiophys. 11, 35–55.

edelec, F., Surrey, T., Karsenti, E., 2003. Self-organisation and forces in the micro-tubule cytoskeleton. Curr. Opin. Cell Biol. 15, 118–124.

edelec, F.J., Surrey, T., Maggs, A.C., Leibler, S., 1997. Self-organization of micro-tubules and motors. Nature 389, 305–308.

ogales, E., 1999. A structural view of microtubule dynamics. Cell. Mol. Life Sci. 56,133–142.

ogales, E., Whittaker, M., Milligan, R.A., Downing, K.H., 1999. High-resolutionmodel of the microtubule. Cell 96, 79–88.

akley, B.R., 2000. gamma-Tubulin. Curr. Top. Dev. Biol. 49, 27–54.akley, B.R., Oakley, C.E., Yoon, Y., Jung, M.K., 1990. Gamma-tubulin is a component

of the spindle pole body that is essential for microtubule function in Aspergillusnidulans. Cell 61, 1289–1301.

dde, D.J., Buettner, H.M., 1995. Time series characterization of simulated micro-tubule dynamics in the nerve growth cone. Ann. Biomed. Eng. 23, 268–286.

dde, D.J., Cassimeris, L., Buettner, H.M., 1995. Kinetics of microtubule catastropheassessed by probabilistic analysis. Biophys. J. 69, 796–802.

apaseit, C., Vuillard, L., Tabony, J., 1999. Reaction-diffusion microtubule concentra-tion patterns occur during biological morphogenesis. Biophys. Chem. 79, 33–39.

iehl, M., Tulu, U.S., Wadsworth, P., Cassimeris, L., 2004. Centrosome maturation:measurement of microtubule nucleation throughout the cell cycle by using GFP-tagged EB1. Proc. Natl. Acad. Sci. U.S.A. 101, 1584–1588.

ieder, C.L., Faruki, S., Khodjakov, A., 2001. The centrosome in vertebrates: morethan a microtubule-organizing center. Trends Cell Biol. 11, 413–419.

ubin, R.J., 1988. Mean lifetime of microtubules attached to nucleating sites. Proc.Natl. Acad. Sci. U.S.A. 85, 446–448.

usan, N.M., Fagerstrom, C.J., Yvon, A.M., Wadsworth, P., 2001. Cell cycle-dependentchanges in microtubule dynamics in living cells expressing green fluorescentprotein-alpha tubulin. Mol. Biol. Cell 12, 971–980.

and Chemistry 35 (2011) 269–281 281

Seetapun, D., Odde, D.J., 2010. Cell-length-dependent microtubule accumulationduring polarization. Curr. Biol. 20, 979–988.

Shelden, E., Wadsworth, P., 1993. Observation and quantification of individualmicrotubule behavior in vivo: microtubule dynamics are cell-type specific. J.Cell Biol. 120, 935–945.

Surrey, T., Nedelec, F., Leibler, S., Karsenti, E., 2001. Physical properties determiningself-organization of motors and microtubules. Science 292, 1167–1171.

Tran, P.T., Walker, R.A., Salmon, E.D., 1997. A metastable intermediate state of micro-tubule dynamic instability that differs significantly between plus and minusends. J. Cell Biol. 138, 105–117.

VanBuren, V., Cassimeris, L., Odde, D.J., 2005. Mechanochemical model of micro-tubule structure and self-assembly kinetics. Biophys. J. 89, 2911–2926.

Verde, F., Dogterom, M., Stelzer, E., Karsenti, E., Leibler, S., 1992. Control of micro-tubule dynamics and length by cyclin A- and cyclin B-dependent kinases inXenopus egg extracts. J. Cell Biol. 118, 1097–1108.

Vorobjev, I.A., Maly, I.V., 2008. On the relation of the microtubule length and dynam-ics: boundary effect and properties of an extended radial array. Cell Tissue Biol.2, 272–281.

Walczak, C.E, Heald, R., 2008. Mechanisms of mitotic spindle assembly and function.Int. Rev. Cytol. 265, 111–158.

Walczak, C.E., Mitchison, T.J., Desai, A., 1996. XKCM1: a Xenopus kinesin-relatedprotein that regulates microtubule dynamics during mitotic spindle assembly.Cell 84, 37–47.

Walker, R.A., O’Brien, E.T., Pryer, N.K., Soboeiro, M.F., Voter, W.A., Erickson, H.P.,Salmon, E.D., 1988. Dynamic instability of individual microtubules analyzed byvideo light microscopy: rate constants and transition frequencies. J. Cell Biol.107, 1437–1448.

Walker, R.A., Pryer, N.K., Salmon, E.D., 1991. Dilution of individual microtubulesobserved in real time in vitro: evidence that cap size is small and independentof elongation rate. J. Cell Biol. 114, 73–81.

Xiong, Y., Oakley, B.R., 2009. In vivo analysis of the functions of gamma-tubulin-complex proteins. J. Cell Sci. 122, 4218–4227.

Yarahmadian, S., Barker, B., Zumbrun, K., Shaw, S.L., 2010. Existence and stability of

steady states of a reaction convection diffusion equation modeling microtubuleformation. J. Math. Biol..

Zheng, Y., Jung, M.K., Oakley, B.R., 1991. Gamma-tubulin is present in Drosophilamelanogaster and Homo sapiens and is associated with the centrosome. Cell 65,817–823.