A first-passage-time theory for search and capture of chromosomes by microtubules in mitosis
aa r X i v : . [ q - b i o . S C ] J a n A first-passage-time theory for search and capture ofchromosomes by microtubules in mitosis
Manoj Gopalakrishnan ∗ and Bindu S. Govindan † Department of Physics, Indian Institute of Technology (Madras), Chennai 600036, India (Dated: October 22, 2018)
Abstract
The mitotic spindle is an important intermediate structure in eukaryotic cell division, in whicheach of a pair of duplicated chromosomes is attached through microtubules to centrosomal bodieslocated close to the two poles of the dividing cell. Several mechanisms are at work towards theformation of the spindle, one of which is the ‘capture’ of chromosome pairs, held together bykinetochores, by randomly searching microtubules. Although the entire cell cycle can be up to24 hours long, the mitotic phase typically takes only less than an hour. How does the cell keepthe duration of mitosis within this limit? Previous theoretical studies have suggested that thechromosome search and capture is optimized by tuning the microtubule dynamic parameters tominimize the search time. In this paper, we examine this conjecture. We compute the mean searchtime for a single target by microtubules from a single nucleating site, using a systematic and rigoroustheoretical approach, for arbitrary kinetic parameters. The result is extended to multiple targetsand nucleating sites by physical arguments. Estimates of mitotic time scales are then obtainedfor different cells using experimental data. In yeast and mammalian cells, the observed changesin microtubule kinetics between interphase and mitosis are beneficial in reducing the search time.In
Xenopus extracts, by contrast, the opposite effect is observed, in agreement with the currentunderstanding that large cells use additional mechanisms to regulate the duration of the mitoticphase. ∗ [email protected] † [email protected] ey words : microtubule, chromosome capture, metaphase spindle, Green’s functions I. INTRODUCTION
Microtubules are one class of polymeric filaments in the eukaryotic cell, whose sub-unit is ahetero-dimer of alpha- and beta-tubulin. Microtubules therefore possess structural polarity,and the ends are differentiated as plus and minus ends. A hall-mark of microtubules istheir unique mechanism of assembly and dis-assembly: a polymerizing microtubule canabruptly start shrinking by losing sub-units and vice-versa , a process referred to as dynamicinstability(reviewed in [1]). The stochastic switching process between growth and shrinkageis referred to as catastrophe and the reverse process is called rescue. Between a rescue anda catastrophe, a microtubule grows in length by polymerizing and between a catastropheand rescue, it shrinks. In vivo, a third state called pause is also observed where the lengthremains static. Microtubules usually nucleate from organizing centers called centrosomes,but may also be found free in the cytoplasm.Microtubules play a central role in eukaryotic cell division. An important milestone inthe cell division cycle is the formation of the metaphase spindle, where all the duplicatedchromosome pairs, held together by kinetochores are aligned along the cell ‘equator (the“metaphase plate”) in such a way that each chromosome of a pair is facing one of the polesof the cell, and attached to one or more microtubules emanating from a centrosome locatednear that pole. The spindle starts forming when microtubules nucleating from each cen-trosome randomly searches the surrounding space for chromosomes by alternately growingand shrinking (the random search-and-capture model, and are stabilized upon contact witha kinetochore[2, 3]. Investigations over the last decade or so have revealed that the chro-mosomes do not always remain passive in this process; rather, the kinetochores nucleateand stabilize microtubules in their vicinity, a process facilitated by RanGTP, which thenconnect to the astral microtubules emanating from the chromosomes, assisted by motor pro-teins such as dynein (see [4] for a recent review). In the present paper, we, however, restrictourselves to the situation where chromosomes are passive, and microtubules perform thesearch-and-capture.We now briefly review the previous papers that addressed this problem. A theoreticaland numerical study of the random search-and-capture model was done first by Holy and2eibler[5]. In this paper, the conditions for optimization of search process was investigatedfor a spherical cell of radius R = 50 µ m, with a single stationary target at various distances d < R from the centre. The number of searching polymers was fixed at 250, and was assumedto remain constant with time (effectively infinite nucleation rate). At the cell boundary, thefilaments would stop growing and wait until a catastrophe occurs. By a combination ofintuitive arguments and explicit simulations, it was postulated that (i) the global minimumof the search time occurs when rescue is absent and (ii) the optimized mean time of searchincreases with d , and is less than 10 minutes for d < µ m, but much higher for larger d .In a more recent paper, Wollman et al[6] carried out a more detailed study of the problem,and also investigated the time to capture multiple chromosomes. An optimal catastrophefrequency was first estimated by minimizing a weighted average time of search for multiplechromosomes at variable separations from the nucleating centre. Numerical simulations ofthe problem, using this optimal frequency showed that the search typically took hours tocomplete when the number of targets was large (eg. 46 in humans). However, when abiochemically induced bias in search (a microtubule stabilizing RanGTP gradient aroundchromosomes) was introduced, the search was completed over physiologically reasonabletimes.In the earlier studies, it has generally been assumed, on the basis of probabilistic argu-ments, that the rescue frequency should be optimally zero. However, small, but non-zerorescue frequency is typically observed in mitotic cells, and the existing theoretical resultscannot be used to analyze this case. Also, the earlier studies have generally ignored the finitecell size which limits long searches, and is likely to be crucial at least in small cells. Thenumber of searching microtubules was generally assumed constant, while this is a fluctuatingquantity, controlled by the nucleation rate at the centrosomes.The primary motivation behind the present paper is to present a rigorous theoreticalmethod for calculating the search time for arbitrary rescue frequency, nucleation rate andcell size/radius. The formalism presented here is based on a set of Greens functions andfirst passage densities for microtubule dynamics, related through a set of convolution equa-tions. As such, this approach permits us to derive an implicit expression for the probabilitydistribution of the search time for a single chromosome/target. Existing theoretical resultsfollow from our more general expressions in the appropriate limits.We believe that this for-malism will be useful in obtaining a great deal of insight into microtubule dynamics in in ivo situations, and may well find applications in other related problems.In the following sections, we discuss the problem and the model, develop the formalismto address the problem, and analyze our results, first in theoretically interesting limits. Theresults are then discussed in the context of available experimental observations. We thenconclude with a summary of our findings and mention a few directions in which this studymay be extended. Two appendices supplement the mathematical part of the paper. II. MATHEMATICAL FORMALISMA. Model details
It is convenient to imagine that during pro-metaphase, prior to division, the shape ofthe cell is close to an ellipsoid. Microtubules nucleate from two centrosomes, which, forsimplicity, may be assumed to be located at the two focal points of the ellipsoid, and theduplicated chromosomes, held together through kinetochores (henceforth called simply ‘tar-gets’) are assumed to be scattered around the equatorial plane. In the rest of this paper, weonly consider capture of the target by microtubules emanating from one of the centrosomes,which is a precursor event to the later ‘bi-oriented’ configuration, where microtubules fromboth centrosomes will bind to a target and engage in ‘tug-of-war’ which ultimately separatesthe individual chromosomes in the pair.It is generally estimated that there are hundreds of nucleating sites in a centrosome.From a vacant site, a microtubules nucleate at rate ν in a random direction and grows bypolymerization as long as it is in the growth phase, while the same microtubule shrinks inlength by depolymerization in the shrinking phase. Catastrophe and rescue frequencies aredenoted by ν c and ν r respectively, and are assumed to be the same everywhere inside thecell, as are the growth and shrinkage velocities, denoted v g and v s . In this paper, as inthe earlier papers which addressed this problem, we will treat all these different dynamicalquantities are independent parameters (see the discussion in the last section, however). Inthe process, the microtubule scans the surrounding space for chromosomes, and is stabilizedwhen the growing end encounters a kinetochore.A microtubule from a certain nucleation site on the centrosome can nucleate in manypossible directions; however, given the finite size of the centrosome, the orientation is likely4o be constrained by the geometry of the centrosome. In an extreme case, one may imaginethat a microtubule will always grow only along the local normal to the surface, but thiscase is pathological when the target is fixed in space, since no microtubule might ever growin the right direction to find it. It is therefore, more realistic to imagine that microtubulesfrom each nucleation site in the centrosome will grow within a certain solid angle ∆Ω, whichdefines a search cone for the corresponding nucleation site. In this case, if the fixed targetfalls inside the cone, and has a cross-sectional area a , it subtends a solid angle a/d at apoint on the centrosome, and therefore a microtubule originating at that particular site hasa probability p = ad − ∆Ω − for nucleating in the right direction, within the search cone ofthe site.For a given search process, ∆Ω is determined by several factors, the most importantbeing (a) orientational constraints on nucleation at a given site in the centrosome and (b)steric hindrance between microtubules in the cytoplasm. In general, one may see thatwhen ∆Ω is large, p is small, and consequently, the search by microtubules from any singlenucleating site becomes inefficient. However, in this case, the search cones of differentmicrotubules overlap (each being large) and more nucleating sites/microtubules will be ableto participate in the search.On the other hand, if ∆Ω is small, only a few microtubules willbe effectively searching, however the search by each is now more efficient; the two effectstherefore compensate each other. For concreteness, we choose ∆Ω = π/ π ). An illustration of the model is shown inFig.1. We take the cross-sectional area of a target to be a = 0 . µ m throughout this paper(corresponding to a radius of ∼ . µ m), therefore, with the previous estimate of ∆Ω, wefind p ≃ . /d , where d is measured in µm .For d = 2 µ m and larger, therefore p ≪ R from the centre, although this is strictly true5nly when ∆Ω is sufficiently small. The cut-off distance R therefore serves as an estimateof the size of the cell. In the last case, we assume that once the microtubule hits the cellboundary, it undergoes catastrophe at a rate ν ′ c , which is generally higher than the value inthe interior [7–9]. In particular, it was reported in [7] that the catastrophe frequency nearthe boundary is 16-fold higher than that in the interior, for certain cells. In cases (ii) and(iii), a new microtubule will nucleate again from the center in a randomly chosen direction,at rate ν .Search cones of many nucleating sites will overlap, and therefore a target will be searchedsimultaneously by many microtubules which will reduce the mean search time. It wasshown in Wollman et. al[6] that the mean time to capture N targets by M microtubules( M nucleating sites in our case) is given by T M,N ≃ T , σ N M (1)where σ N = P Nk =1 k − ∼ log N when N ≫ T M,N < T max1 , σ N M (2)where T max1 , is the search time for the farthest target, which is an upper limit on T , . Inthis paper, we compute T , rigorously, with some simplifying assumptions, but for arbitrarykinetic parameters. We then use Eq.2 to make estimates for multiple targets and parallelsearch, for the sake of comparison with experiments. B. Capture time distribution
For the rest of this paper, we replace T , simply by T . Let us denote by C ( T ) theprobability density of the capture time T for a single stationary target at a certain distancefrom the centrosome. The mean capture time follows:6 T i = R ∞ dT C ( T ) T R ∞ dT C ( T ) , (3)where, R ∞ dT C ( T ) is the probability that the search will be eventually successful, whichwe will, later, show to be unity.Since the basic process under consideration here is the capture of a certain target by one(or a set) of dynamically unstable filaments, it is natural to base our theory on considerationof first passage probability densities[10, 11]. For this purpose, it is convenient to define aset of three conditional first passage probability densities (CFPD), which will serve as thebasic quantities in terms of which the probability distribution C ( T ) can be expressed. TheseCFPDs are defined below, with the corresponding condition for each given in italics.1. K ( T ) ≡ p Φ( d, T ), where Φ( d, T ) is the CFPD for a freshly nucleated microtubule toreach a distance d for the first time after a time interval T , without ever shrinking backto the origin in between .2. K ( T ) ≡ (1 − p ) Q R ( T ), where Q X ( T ) is the CFPD for shrinking to the origin after alife-time T , without ever reaching a length X in between .3. K ( T ) ≡ (1 − p )Ψ( T ), where Ψ( T ) is the CFPD for return to the origin after a timeinterval T , after encounter with the boundary (and consequent catastrophe) at leastonce (and possibly several times) in between .A successful search event is, in general, preceded by n unsuccessful search events: let usdenote by Ω n ( T ) the probability of n unsuccessful nucleation-search-disappearance eventswithin a time interval T , so that C ( T ) may be written as C ( T ) = ∞ X n =0 Z T Ω n ( T − T ′ ) p Φ( d, T ′ ) νdT ′ (4)Our next task is to write an expression for Ω n ( T ). Let us now assume that among the n unsuccessful nucleation events, there are n events of type 1 (above), where the microtubulenucleated in the right direction, but did not reach the chromosome, n events of type 2(above), where it nucleated in a wrong direction, but shrank back to origin before encoun-tering the boundary, and n = n − n − n events of type 3 (above), where the microtubule7ucleated in a wrong direction, encountered the boundary, underwent catastrophe and thenshrank to the origin.Specifying the total number of events in each class does not completely describe thehistory of the process, as the temporal ordering of the events is still arbitrary. The n eventsof type 1 can be distributed in a total of n in (cid:0) nn (cid:1) different ways, and the n events of type2 can be distributed among the remaining n − n in (cid:0) n − n n (cid:1) different ways. The remaining n − n − n events naturally belong to type 3.Ω n ( T ) may now be expressed as a sum over histories (i.e., a path-integral ) of all theseevents, ordered temporally in all possible ways. This is done as follows: Starting at time T = 0, let the first microtubule nucleation occur at time T ′ , and let this microtubule livefor a time interval T . Then there is a time gap of T ′ until the next nucleation, and themicrotubule nucleated then lasts for a time interval T and so on. The time gap T ′ occurswith a probability exp( − νT ′ ) and the nucleation at the end of it occurs with probability νdT ′ . The probability that a microtubule will last for a time interval T before shrinkingback to the origin may be denoted K ( T ) dT , but K could be K , K or K dependingon whether this event falls into type 1, 2 or 3. For Ω n ( T ), there are a total of n suchnucleation-death events within a time interval T . The resulting mathematical expressioncan be written as a convolution over all these time-intervals, and has the formΩ n ( T ) = n X n =0 n − n X n =0 X per Z T νdT ′ e − νT ′ Z T − T ′ dT K ( T ) ..... Z T − T ′ − ..T n − dT n K ( T n ) e − ν [ T − P nk =1 ( T k + T ′ k )] (5)where per stands for all the possible permutations of events, as far as their temporalorder of occurrence is concerned. The preceding equation has the form of a 2 n -fold con-volution, and it is therefore convenient to use Laplace transforms. We define ˜Ω n ( s ) = R ∞ dT e − sT Ω n ( T ) and similarly for other quantities. A generalized form of the standardconvolution theorem for Laplace transforms may be applied to Eq.5 (see, eg.,[12]), and theresult is˜Ω n ( s ) = 1( s + ν ) (cid:18) νs + ν (cid:19) n n X n =0 n − n X n =0 (cid:18) nn (cid:19)(cid:18) n − n n (cid:19) ˜ K ( s ) n ˜ K ( s ) n ˜ K ( s ) n − n − n (6)8ote that, in the passage from Eq.5 to Eq.6, allowance has been made for the fact thatthe random variable K takes the value K n times, K n times and K n − n − n times.The previous equation is clearly a binomial series, and can be summed immediately. FromEq.4, we find ˜ C ( s ) = νp ˜Φ( d, s ) P ∞ n =0 ˜Ω n ( s ). After substituting the binomial sum from Eq.6,and replacing K , K , K by their original notations, we arrive at the following expression:˜ C ( s ) = νp ˜Φ( d, s ) (cid:20) s + ν (cid:18) − p ˜ Q d ( s ) − (1 − p )[ ˜ Q R ( s ) + ˜Ψ( s )] (cid:19)(cid:21) (7)As a special case, if MT nucleation occurs very fast and therefore not rate-limiting, wemay take the limit ν → ∞ in the above equation, whence the following limiting form isreached: lim ν →∞ ˜ C ( s ) = p ˜Φ( d, s )1 − (cid:20) p ˜ Q d ( s ) + (1 − p ) (cid:18) ˜ Q R ( s ) + ˜Ψ( s ) (cid:19)(cid:21) (8)Eq.7 is the central result of this paper. Using this expression, the mean search time fora single target, and its variance may be expressed as h T i = − ∂ s ˜ C ( s ) | s =0 ˜ C (0) ; h T i = 1˜ C (0) ∂ ˜ C ( s ) ∂s | s =0 , (9)where ˜ C (0) = R dT C ( T ). It is, however clear that ˜Φ( d,
0) + ˜ Q ( d,
0) = 1, since a micro-tubule growing in the right direction will either have to hit the target, or shrink backwithout touching the target. Similarly, for the wrong directions, we have the relation˜ Q ( R,
0) + ˜Ψ(0) = 1 for similar reasons. Substitution of these normalization relations intoEq.7 shows that ˜ C (0) = 1 for all parameters, i.e., the search is always eventually successful.The CFPDs introduced above are now calculated from the Green’s functions for MTkinetics, derived explicitly in the next section. C. Green’s functions
The stochastic state of a MT at a given point in time t is characterized by two variables,its length l and its state of polymerization versus depolymerization, which we denote byan index i ,which takes one of the two values,1 or 0 respectively for growing and shrinkingstates. In this case, therefore, we need to compute four Green’s functions, or propagators,9 ij ( x, t ; x , i, j = 0 ,
1; by definition, G ij ( x, t ; x , dx gives the probability that agiven MT will have length l between x and x + dx , and will be in state i at time t , providedthat it had a length x and was at state j at an earlier time t = 0.Calculating the above Green’s functions for a physically realistic situation would alsorequire specification of appropriate boundary conditions at the origin (nucleating site) andthis has been done earlier[13]. However, we deem this unnecessary for our purpose, sincewe are only interested in using these Green’s functions to compute the CFPDs introducedabove. Therefore, for the rest of this paper, we will allow the ‘length’ x to be a continuouslyvarying variable between positive and negative values, with no boundary condition imposedon the dynamics at x = 0. The boundary conditions are used in the definition of the CFPDslater.The Dogterom-Leibler[14] rate equations for MT kinetics takes the form ∂ t G j = − v g ∂ x G j + ν r G j − ν c G j ∂ t G j = v s ∂ x G j + ν c G j − ν r G j (10)The equations may be solved together using combined Laplace-Fourier transforms, definedas ˜ G ij ( k, s ; x ) = R ∞−∞ e − ikx dx R ∞ dte − st G ij ( x, t ; x , G ij ( k, s ) = e − ikx [ ν r − δ ij ( ikv s − s )] v s v g [ k − ikA ( s ) + B ( s )] (11)where A ( s ) = [ v s ν c − v g ν r + s ( v s − v g )] /v s v g B ( s ) = [ s ( s + ν r + ν c )] /v s v g . (12)For connection with the CFPD s introduced earlier, it is convenient to define the Green’sfunction in such a way as that they have dimensions of inverse time, and not inverse length.This is done by defining F j = v g G j ; F j = v s G j (13)It will be convenient for later calculations to carry out the inversion k → x explicitly:10 F j ( x, s ; x ) = ν r + sδ j v s [ α s + β s ] (cid:20) e − α s ( x − x ) Θ( x − x ) + e β s ( x − x ) Θ( x − x ) (cid:21) + δ j ( α s + β s ) (cid:20) α s e − α s ( x − x ) Θ( x − x ) − β s e β s ( x − x ) Θ( x − x ) (cid:21) ˜ F j ( x, s ; x ) = ν c + sδ j v g [ α s + β s ] (cid:20) e − α s ( x − x ) Θ( x − x ) + e β s ( x − x ) Θ( x − x ) (cid:21) − δ j ( α s + β s ) (cid:20) α s e − α s ( x − x ) Θ( x − x ) − β s e β s ( x − x ) Θ( x − x ) (cid:21) (14)where α s = A ( s )2 + p B ( s ) + A ( s ) / β s = − A ( s )2 + p B ( s ) + A ( s ) / x ) is the usual step-function: Θ( x ) = 1 for x ≥ D. Calculation of Φ( d, T ) and Q X ( T ) The Green’s functions calculated in the last section may now be used to compute theCFPDs which we used before. For this purpose, it is convenient to define first a set of unconditional first passage densities (denoted FPD) as follows: let C ij ( x, t ; x ,
0) denote theprobability, per unit time, for a MT in state j and with length x at time t = 0, to reach alength x for the first time at time t , and in state i .For l > C ( l, t ; 0 ,
0) is given by the implicit equation F ( l, t ; 0 ,
0) = C ( l, t ; 0 ,
0) +lim ǫ → Z t dt ′ C ( l, t ′ ; 0 , F ( l − ǫ, t ; l, t ′ ) (16)In the above equation (and the following equations), the ǫ -factors take into account thefollowing restriction on its dynamics: starting from a growing state at t = 0, with a length l ,it can return to the same length l at a later time, in a growing state, only from below (whichdecides which of the Θ-functions appearing in Eq.14 is non-zero). Similar restrictions applyto the equations below. 11imilarly, C (0 , T ; d,
0) and C ( d, T ; 0 ,
0) are given by the equations F (0 , T ; d,
0) = C (0 , T ; d,
0) +lim ǫ → Z T dT ′ C (0 , T ′ ; d, F ( ǫ, T ; 0 , T ′ ) (17) F ( d, T ; 0 ,
0) = C ( d, T ; 0 ,
0) +lim ǫ → Z T dT ′ C ( d, T ′ ; 0 , F ( d − ǫ, T ; d, T ′ ) (18)Using these two FPDs, we are now in a position to write down the following relationsbetween the CFPDs introduced earlier: C ( d, t ; 0 ,
0) = Φ( d, t ) + Z t dt ′ Q d (0 , t ′ ) C ( d, t ; 0 , t ′ ) (19) C (0 , T ; 0 ,
0) = Q d ( T ) + Z T dT ′ Φ( d, T ′ ) C (0 , T ; d, T ′ ) (20)Eq.16-20 may now be solved using Laplace transforms. From Eq.16, we find that˜ C ( d, s ; 0) = lim ǫ → ˜ F ( d, s ; 0)1 + ˜ F ( d − ǫ, s ; d ) = e − α s d (21)Similarly, ˜ C ( d, s ; 0) = ν r e − α s d ν r + s + α s v s ; ˜ C (0 , s ; d ) = ν c e − β s d ν c + s + β s v g (22)After solving Eq.19 and Eq.20 together, and using Eq.21,22, we find the explicit expres-sions ˜Φ( d, s ) = D ( s ) e − α s d ν r ν c [1 − e − ( α s + β s ) d ] + D ( s )˜ Q d ( s ) = ν c ν c + s + βv g h − e − β s d ˜Φ( d, s ) i (23)where D ( s ) = ( s + α s v s )( s + β s v g ) + ν r ( s + β s v g ) + ν c ( s + α s v s ) . (24)12 . Calculation of Ψ( T ) : Catastrophes at the cell boundary We assume that when a MT hits the cell boundary by growing, it undergoes catastrophethere at a rate ν ′ c . We now compute Ψ( T ), which is the CFPD of return to origin (i.e.,complete depolymerization) of a MT after a lifetime T , and an encounter with the boundaryat least once.Clearly, along the line of our previous arguments, Ψ( T ) may be given by the expressionΨ( T ) = Z T dT Φ( R, T ) × Z T − T ν ′ c dT e − ν ′ c T χ ( R, T − T − T ) , (25)where χ ( R, T ) gives the FPD of complete depolymerization of a MT, starting at theboundary, at length R in shrinking state, with possibly multiple visits back to the boundaryin between. This quantity may now be expressed implicitly through the equation χ ( R, T ) = Φ ∗ ( R, T ) + Z T dT Q ∗ R ( T ) × Z T − T ν ′ c dT e − ν ′ c T χ ( R, T − T − T ) (26)where Q ∗ R ( T ) is a ‘mirror’ image, or dynamic inverse of the quantity Q R ( T ) introducedearlier, and represents the CPFD of a return to boundary over a time interval T , withoutever reaching the origin (i.e., shrinking to zero) in between. Similarly, Φ ∗ ( R, T ) is the‘inverse’ of Φ(
R, T ), and gives the CFPD of complete depolymerization of a MT starting atthe boundary, without ever returning to the boundary in between.Eq.25 and Eq.26 may now be solved together using Laplace transforms, and we find˜Ψ( s ) = ν ′ c ˜Φ( R, s ) ˜Φ ∗ ( R, s ) s + ν ′ c (cid:18) − ˜ Q ∗ R ( s ) (cid:19) (27)The inverse quantities ˜Φ ∗ ( R, s ) and ˜ Q ∗ R ( s ) may be obtained from ˜Φ( R, s ) and Q R ( s )respectively by the transformations v s ↔ v g and ν r ↔ ν c . From Eq.15, this has the effectof replacing α s by β s and vice-versa , while D ( s ) defined in Eq.24 remains invariant under13hese transformations. Therefore, using Eq.23, we arrive at the following expressions forthese ‘inverse’ quantities:˜ Q ∗ R ( s ) = ν r ν r + s + α s v s (cid:20) − e − α s R ˜Φ ∗ ( R, s ) (cid:21) (28)˜Φ ∗ ( R, s ) = D ( s ) e − β s R ν r ν c [1 − e − ( α s + β s ) R ] + D ( s ) (29)Eq.29 completes the list of quantities that we need to compute the mean search time. Wenow start from Eq.9, and after a few elementary calculations and rearrangement of terms,it turns out that the complete expression for the mean search time may be written out asfollows(see Appendix A for details): h T i = N s [ pt d + (1 − p ) t R + t ν ] = T d + 1 − pp T R + T ν (30)where N s = [ p ˜Φ d (0)] − gives the mean number of unsuccessful search events before eachsuccessful one, and pt d + (1 − p ) t R + t ν gives the weighted mean lifetime per event. t d = − ˜Φ ′ d (0) − ˜ Q ′ ( d,
0) gives the mean time of a search in the right direction, the first termcorresponding to the single successful search event and the second giving the mean of allunsuccessful events. t R = − ˜ Q ′ R (0) − ˜ ψ ′ (0) is the mean lifetime of an unsuccessful search eventin the wrong direction, the first term corresponding to events not reaching the boundarywhile the second corresponds to events which hit the boundary at least once. Note that t R is the mean lifetime of microtubules. Finally, t ν = ν − is the mean time between thedisappearance of one microtubule and nucleation of a new one at a site, and is the only termthat depends on the nucleation rate ν .Various quantities such as the mean, standard deviation and higher moments (if neces-sary) of the search time, as well as other quantities such as the mean lifetime of microtubulesmay now be calculated using the set of equations presented in this section, and parameterssuch as catastrophe and rescue frequencies as well as growth and shortening velocities takenfrom experiments. We found it convenient to use Mathematica (Version 7, Wolfram Re-search) to carry out the explicit computations. The results will be discussed in the followingsection. 14 II. RESULTS
In this section, we will analyze some experimental observations using the results from ourmodel.Experimental measurements of the microtubule kinetic parameters show that distinctchanges occur as the cell progresses from interphase to mitosis[15], see Table I. BuddingYeast cells show a reduction in both catastrophe and rescue frequencies, but the changesare relatively small. In mammalian cells, which are typically larger, there is a marked fallin rescue frequency between interphase and mitosis, and a two-fold increase in catastrophefrequency. In
Xenopus oocytes (frog egg cells, which are large and almost 1 mm in radius),the effects are somewhat different: the rescue frequency, while small, is almost doubled, butmore remarkably, there is a sharp, seven-fold rise in the catastrophe frequency.Given that mitosis occupies only a small fraction of the total time duration of a cellcycle, we first seek to determine whether the changes in microtubule kinetics are beneficialto reduce the mean time of search. For all cases discussed below, we choose ν ′ c = 10 ν c to beroughly consistent with experiments[7]. However, reducing or increasing ν ′ c by an order ofmagnitude does not significantly affect the results. A. Yeast and mammalian cells show features that are consistent with the randomsearch and capture model, but
Xenopus oocytes do not
Fig.2 shows a comparison for the mean time of search, between interphase and mitosisvalues in yeast, for a range of target distance d . Since yeast undergoes closed mitosis, therelevant boundary cut-off length scale is the nuclear radius, which we take to be R = 2 µ m.The mitosis parameters clearly reduces the time scale relative to the interphase parameters,though understandably, given the small cut-off radius of search, the effect is small. Whenthe centrosomal microtubule nucleation frequency is chosen to be ν = 0 . − per site (seenext paragraph) h T i in yeast is found to vary from 100-400 minutes, for d between 1 µ m and2 µ m. Taking T max1 , ∼
400 min, and using N = 32 in budding yeast, we see from Eq.2 thatthe mean time to complete search is 24 min, with 50 searching microtubules.We now turn to the case of mammalian cells. Experiments by Piehl et. al[16] measureda nucleation rate of ∼ − , per centrosome in kidney epithelial (LLCPK) cells.The total number of nucleation sites is unknown, but from the measured surface area of15entrosomes in metaphase ( ∼ µ m ), and the base area of a single microtubule ( πr ≃ × − µ m , with 2 r = 25 nm ), a total of almost 10 nucleation sites are possible, which isclearly too large; as a conservative estimate, we may assume a total of 1000 nucleation sites.Then, the nucleation rate per site may be roughly estimated as ν ≃ . − .Let us now perform our analysis on a mammalian cell of radius R = 20 µ m, and comparethe search time between interphase and mitosis. As seen in Fig.3, where the logarithm of thetime is shown against the distance d , the mitosis values significantly reduce the mean searchtime, almost by 4 orders of magnitude! For d = 6 µ m, the time computed from mitosis valuesis T max1 , ≃ M = 100 kinetochoremicrotubules will be actively searching for one target, and since N >
10 typically (46 inhumans), using Eq.2, we arrive at an estimate of 35-40 minutes for the total mean searchtime, which is reasonable. However, this is only the average time, and may not represent atypical value. We have observed that the standard deviation of T is typically of the sameorder as h T i ; therefore, in an individual experiment, the search could take twice as long.In Xenopus extracts, the situation is very different (see Fig.4). In this case, surprisingly,it is the interphase parameter values that give the lower mean search time, and mitosistime is typically 1-2 orders of magnitude larger! Even the interphase mean search time isquite large, ranging from 2000-10,000 minutes for d between 5 and 10 µ m, and it wouldneed almost 1000 actively searching microtubules to bring the total search time down toacceptable values. The anomalously large value of the catastrophe frequency in mitosis ispuzzling, since targets that are far are more effectively searched when ν c is small. . How dowe understand this discrepancy? One reason, as is now becoming increasingly clear, couldbe that the random search and capture mechanism is simply inefficient in such large cells.Indeed, it is now well-established that large cells like oocytes require additional mechanismsof search like actin contractile ring[17] and guided search through RanGTP gradient aroundchromosomes[6, 18]. In the latter case, microtubules are preferentially stabilized close tochromosomes hy a gradient of the protein Ran, and the increased catastrophe frequencymight simply serve to enhance the turnover rate, and hence the dynamicity of microtubules.16 . Mathematical analysis show that ν r = 0 is a minimum condition when R ≫ d In this section, we take a closer look at the theoretical expression for the mean searchtime derived earlier, and try to understand some general features, from the point of view ofoptimization of the search process, i.e., minimization of the search time. Without any lossof generality, we may put ν = ∞ here, as the only effect of a finite ν is to add a timescaleof ν − to the mean time (see Eq.30).It is instructive to look at the mean search time separately for parameter regimes demar-cated by the conditions A (0) > A (0) <
0. Physically, these conditions correspond toa bounded growth regime for the filaments, with finite mean length and unbounded growthregime with mean length linearly increasing with time[14]. After rather lengthy algebraiccalculations, an explicit expression for h T i can be obtained, and the terms therein could beclassified into two: (i) those that remain finite, or diverge as R → ∞ and (ii) those thatinvolve only terms of the form e − R | A (0) | which disappear for large R . In the following, weignore terms that fall into (ii), since they are not crucial for our analysis here. Bounded growth regime (BG); A (0) > d,
0) = D e − A (0) d / [ D + ν r ν c (1 − e − A (0) d )], so that the mean number of trials N s diverges exponentially with d . The breakup of the total time in the limit R | A (0) | → ∞ ,is as below: T d = β ′ + v − g A (0) [ e A (0) d − − β ′ dT R ≃ (cid:18) − ν r v g ν c v s e − A (0) d (cid:19) e A (0) d ν c v s ( v s + v g )( v s v g A (0)) (31)where β ′ = ( ν r + ν c ) /v s v g A (0). In this regime, T d and T R diverge exponentially with d , but only linearly with R , with the R -dependence entirely disappearing for R ≫ A (0) − (terms now shown in the equation). This is physically reasonable, since A (0) − is indeed themean length of the filaments in this regime. We also observe that T R diverges as A (0) − and T d diverges as A (0) − as A (0) →
0. The intuitive explanation is that, at this ‘critical point’in parameter space, the tip of a microtubule performs a pure one-dimensional random walk,whose mean time of return to the origin is infinite, as is well-known.The R | A (0) | ≫ A (0) < nbounded growth regime(UBG): A (0) < d,
0) = D / ( D + ν r ν c [1 − e −| A (0) | d ], which is finite as d → ∞ . Thisis natural, because under conditions of unbounded growth, the microtubules are able toreach out to larger distances with fewer attempts compared to the bounded growth regime.Similar to the previous case, we may now calculate the breakup of the terms in the limit R | A (0) | ≫ ν ′ c → ∞ . The results are given below: T d = | A (0) | ν r v g F ( d ) − ν c v s ν r v g β ′ d + ν c ( v s − v g ) v g v s | A (0) | [1 − e −| A (0) | d ] T R ≃ ν r v g H ( d ) (cid:26) v s ν r | A (0) | F ( R ) + F ∗ ( R ) − ν c v s ν r v g β ′ R + ν c ( v s − v g ) v g v s | A (0) | + v g + v s v g v s | A (0) | (cid:20) e | A (0) | R (cid:18) ν r v s | A (0) | (cid:19) − (cid:21)(cid:27) (32)where H ( d ) = ν c v s − ν r v g e −| A (0 | d F ( R ) = − D ′ D + D ′ D + ν r ν c + α ′ RF ∗ ( R ) = F ( R ) + ( β ′ − α ′ ) Rα ′ = ν r + ν c v g v s | A (0) | ; β ′ = v s ν c − v g ν r ( v s v g ) | A (0) | (33)In contrast to the previous case, T R diverges exponentially with R , while T d divergesonly linearly with d . As | A (0) | →
0, the mean time again diverges as | A (0) | − as before.The exponential divergence with R arises solely from the boundary-interaction term ˜Ψ ′ (0);the return-to-origin term ˜ Q ′ (0) has a finite limit as R → ∞ .This result is in quantitativeagreement with the corresponding results in [13] for the mean time of return to origin of amicrotubule in unbounded (as well as bounded) growth phase.The analogy between the dynamics of the tip of a microtubule and a one-dimensionalbiased random walk as developed in [13] is helpful in understanding these results. It hasbeen shown that the bias (drift) of this random walk is proportional to v g ν r − v s ν c , i.e.,the bias is negative when (in our notation) A (0) > A (0) < A (0) = 0. In the limit R → ∞ , T R and T d diverge as A (0) → p ≪
1, in which case, the T R termdominates over T d in Eq.30. If R ≫ d , the exponential divergence of T R with R in the UBGregime makes it less favourable compared to the bounded growth regime. For the lattercase, at least when R ≫ A (0) − , it is proved in Appendix B that T R is a monotonicallyincreasing function of ν r , i.e., it is minimized for ν r = 0. Therefore, we conclude that if thecell boundary is sufficiently far in comparison with d , for a single target, search is optimal ifthe microtubules are in a bounded growth phase, at zero rescue frequency. This conclusionis in agreement with the previous authors[5, 6].Interestingly, these conclusions hold even if R is only about thrice as large as d , as shown in Fig.5, or even when d = R (Fig.6).In a more realistic situation where a number of chromosomes are distributed randomly inthe cytoplasm at varying distances, p itself effectively becomes a dynamic variable, startingat a large value and progressively decreasing with time as targets are captured one by one. Inthis situation, it is likely that search is optimized at a small, but non-zero rescue frequency.Preliminary results shown in Fig.6 suggest that when the target is far from the centrosome,non-zero rescue does not increase the mean search time significantly, and in addition, couldproduce a more robust minimum. C. Non-zero rescue is likely to be a compromise between ν r = 0 and ν → ∞ If the search time is minimized at zero rescue frequency, as shown by the previous argu-ments, why is not the observed rescue frequency in mitosis even smaller ? We believe thatthis could possibly reflect a compromise between minimizing rescue and maximizing nucle-ation. Both rescue frequency and nucleation rate depend directly on the concentration offree tubulin in cytoplasm. Experimental observations by Walker et. al.[19] have shown thatrescue frequency is an almost linearly increasing function of free GTP-tubulin concentration,and nucleation rate is an even more strongly increasing function of concentration. Therefore,the observed rescue frequency in mitosis could probably be understood as the result of amore general optimization exercise also involving nucleation and catastrophe frequencies, aswell as growth velocity, all of which depend on free GTP-tubulin concentration.19 . Microtubule turnover time is much smaller in mitosis
In Fig.7A, we show the mean lifetime (defined in Eq.30) of microtubules searching in thewrong directions, as a function of cell size R , in both interphase and mitosis. The lifetimein mitosis is several orders of magnitude smaller than interphase, and varies little with cellsize (a direct consequence of being in the BG regime discussed earlier). However, mitoticmicrotubules are also more dynamic: the standard deviation of the lifetime as a fraction ofthe mean, is larger in mitosis compared to interphase (Fig.7B). Experimental observationsin mammalian cells have shown that microtubule turnover in mitosis is 18-fold higher thanin interphase[20]. IV. CONCLUSIONS
In this paper, we studied the capture of a target by dynamically unstable microtubulesusing a novel and mathematically rigorous first passage time-based formalism. Compared toearlier studies, the principal new features in our model are (a) estimation of the mean timeof capture at non-zero rescue frequency (b) introduction of the cell size as a parameter inthe theory and (c) explicit comparison with experimental observations in different mitoticcells. Although the model was formulated for the purpose of understanding chromosomecapture in mitosis, the formalism itself is very general. In particular, the technique couldbe directly applied to the study of cortical capture of microtubules(see, eg.[23]) and othersimilar problems.Several in vivo experiments have shown distinct and significant changes in microtubulesdynamics in different cells, as the cell proceeds from interphase to mitosis. We sought todetermine whether these changes are beneficial to the search and capture of chromosomes.Our analysis shows that in yeast and mammalian cells, the mean search time for a singletarget is reduced in mitosis compared to interphase. In
Xenopus oocytes, by contrast, theexperimental observations could not be reconciled with the observed changes in microtubuledynamics between interphase and mitosis, suggesting that the basic strategy of search maybe strongly modified by additional mechanisms.Although this was not our main interest, we also tried to determine theoretically theconditions for minimization of the mean search time. We showed rigorously that when the20arget is well inside the boundary, the time is minimized at zero rescue and an optimalcatastrophe frequency, in agreement with previous authors. However,when the target isclose to the boundary, although ν r = 0 is still the absolute minimum, it was observed thata small, but non-zero ν r produced a more robust minimum (with respect to change in ν c ),and therefore could be preferred by cells.The present study was only concerned with a single target, while in all realistic situa-tions, multiple chromosome pairs have to be captured. Unfortunately, to extend the presentanalysis to multiple targets would require more detailed knowledge of C ( T ), but given thecomplexity of the mathematical form for ˜ C ( s ), this is not too easy(see, however, [6], wherethis analysis was done for exponentially decaying C ( T ) at ν r = 0). We are presently workingon extracting information about C ( T ) from our formalism, and extending our analysis formultiple targets. In particular, it is not immediately clear whether the optimization criteriafor multiple targets will be the same as for a single target, especially when multiple targetsare at variable separations from the centrosome. Further, as discussed earlier, the variousdynamic parameters for microtubule dynamics are not generally independent (eg. nucleationand rescue could be related, and detailed GTP cap theories suggest that growth velocity isrelated to catastrophe frequency[24]). A more general optimization scheme has to take thesepossibilities into account, and could produce a non-zero optimal rescue frequency. We leavethese ideas to a future study.Other possible extensions of this study involve including (i) chromosome diffusion (ii)side-capture of microtubules by chromosomes through intermediaries like kinesin-13 motorproteins, followed by one-dimensional diffusive or directed motion to the tip[25] and (iii)microtubule nucleation close to the chromosomes. The latter is a possible alternative to theend-capture mechanism studied in this paper and it would be interesting to look at its effecton the mitotic time-scales and its optimization. ACKNOWLEDGMENTS
BG acknowledges financial support under a Young Scientist-Fast Track Research Fellow-ship, from the Department of Science and Technology, Government of India. MG wouldlike to thank Mohan Balasubramanian (NUS, Singapore) for a brief but illuminating con-versation on mitosis in yeast. MG’s work was supported in part by a grant from the Centre21or Industrial Consultancy and Sponsored Research, IIT Madras. The authors are thankfulfor the hospitality under the Visitors Program at the Max-Planck Institute for Physics ofComplex Systems, Dresden, Germany, where this work was initiated. [1] Desai A and Mitchison T J (1997) Microtubule polymerization dynamics
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A guide to First-Passage Processes (Cambridge University Press).[12] M. Gopalakrishnan, P. Borowski, F. J¨ulicher and M. Zapotocky (2007) Response and fluctu-ations of a two-state signaling module with feedback
Phys. Rev. E. Stat. Nonlin. Soft Matter hys. Phys. Rev. E. Stat. Nonlin. Soft Matter Phys. Phys. Rev. Lett. (9) 1347.[15] Rusan N M, Fagerstrom C J, Yvon, A C and Wadsworth P (2001) Cell cycle-dependentchanges in microtubule dynamics in living cells expressing green fluroscent protein- α tubulin Mol. Biol. Cell Proc.Natl. Acad. Sci. USA (6) 1584.[17] Lenart P, Bacher C P, Daigle N, Hand A R, Eils R, Terasaki M and Ellenberg J (2005) Acontractile nuclear actin network drives chromosome congression in oocytes
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Cell Science
Phys. Rev. E.Stat. Nonlin. Soft Matter Phys. Curr. Opin.Cell. Biol. . APPENDIX I General expression for the mean search time :After a series of calculations, the following general expression is reached for the meansearch time from Eq.7. h T i = T d + 1 − pp T R + 1 p T ν (34)with T d = − [ ˜Φ( d, − h ˜Φ ′ ( d,
0) + ˜ Q ′ ( d, i T R = − [ ˜Φ( d, − h ˜ Q ′ ( R,
0) + ˜Ψ ′ (0) i T ν = 1 ν [ ˜Φ( d, − (35)where ˜Φ ′ ( d,
0) = ∂ s ˜Φ( d, s ) | s =0 , ˜ Q ′ ( d,
0) = ∂ s ˜ Q ( d, s ) | s =0 , ˜ Q ′ ( R,
0) = ∂ s ˜ Q ( R, s ) | s =0 and˜Ψ ′ (0) = ∂ s ˜Ψ( s ) | s =0 . Here, T d , T R and T ν represent, respectively, the mean time spent insearching in the right direction, wrong directions and between successive nucleations.Wenote that the last term disappears in the (theoretical) ν → ∞ limit, where the nucleationhappens infinitely fast. Also, for small p , T R and T ν dominate over T d , since, in this limit,it is the unsuccessful search events that take up most of the time spent on search.The exact analytical forms for these functions are as given below:˜Φ ′ ( X,
0) = ˜Φ( X, (cid:20) D ′ D − D ′ + ν r ν c ( α ′ + β ′ ) Xe − γ X D + ν r ν c (1 − e − γ X ) − α ′ X (cid:21) ˜ Q ′ ( X,
0) = − ˜ Q ( X, " β ′ v g ν c + β v g + e − β X [ ˜Φ ′ ( X, − β ′ X ˜Φ( X, − ˜Φ( X, e − β X ] ˜Ψ ′ (0) = ˜Ψ(0) " ˜Φ ′ ( R, R,
0) + ˜Φ ∗′ ( R, ∗ ( R, − [1 − ν ′ c ˜ Q ∗′ ( R, ν ′ c [1 − ˜ Q ∗ ( R, (36)and ˜Φ ∗′ ( R,
0) = ˜Φ ′ ( R, v g → v s , ν r → ν c )˜ Q ∗′ ( R,
0) = ˜ Q ′ ( R, v g → v s , ν r → ν c ) (37)24re the mirror-image quantities defined in text. The time-integrated probabilities˜Φ( d, , ˜ Q ( X, X = d, R ) and ˜Ψ(0) are given by˜Φ( X,
0) = D e − α X D + ν r ν c [1 − e − γ X ]˜ Q ( X,
0) = ν c ν c + β v g h − ˜Φ( X, e − β X i ˜Ψ(0) = ˜Φ( R,
0) ˜Φ ∗ ( R, − ˜ Q ∗ ( R, . (38)The cofficients appearing in the above expressions are defined as follows: α ≡ α (0); α ′ = ∂ s α ( s ) | s =0 β ≡ β (0); β ′ = ∂ s β ( s ) | s =0 D ≡ D (0); D ′ = ∂ s D ( s ) | s =0 (39)and θ = ∂ s A ( s ) | s =0 = ( v s − v g ) /v s v g , θ ′ = ∂ s B ( s ) | s =0 = ( ν r + ν c ) /v s v g . VI. APPENDIX II
Some simple special cases in BG regime If p ≪ T R and T ν dominate over T d , and, from Eq.30 and Eq.31 we have h T i ≃ e A (0) d p (cid:18) ν r (1 − e − A (0) d ) v s A (0) (cid:19) (cid:20) v s + v g v s v g A (0) + 1 ν (cid:21) ( p ≪ , R → ∞ ) (40)It can be shown that this expression is a monotonically increasing function of ν r , where0 ≤ ν r < ( v g /v s ) ν c in the BG regime. In order to see this, let us define x = ν c v s and y = ν r v g , and x − y > x and y , the mean search timemay be expressed in the form h T i x,y = [ xe δ ( x − y ) − y ]( x − y ) (cid:20) a ( x − y ) + b (cid:21) ≥ a ( x − y ) + b (41)25here a , b and δ are positive constants, and the inequality follows because e δ ( x − y ) ≥ y in the applicablerange [0: x ]. We then conclude that h T i itself is an increasing function of y , and hence ν r .Therefore, ν r = 0 is a necessary condition for a minimum. In this case, Eq.40 reduces to h T i = (cid:20) ν + 1 ν c (1 + v g /v s ) (cid:21) e νcdvg p − dv s ( p ≪ , R → ∞ , ν r = 0) (42)which is minimized ν c = ν min c , where ν min c = 2 v g d (cid:20) q v g dν (1+ v g /v s ) (cid:21) ( p ≪ , R → ∞ , ν r = 0) (43)Finally, in the limit ν → ∞ , ν min c = v g /d , and the optimized search time is h T i min = Γ d − dv s ; ( p ≪ , R → ∞ , ν r = 0 , ν → ∞ ) (44)where Γ = e ∆Ω / [ a ( v − g + v − s )] from Eq.1.The expression in Eq.42 above may be approximately reproduced by physical argumentsas follows[5]. The probability that a microtubule will nucleate in the right direction, andwill not undergo catastrophe until it reaches the target is given by p s = pe − ν c d/v g , and itwill take at least N ∼ p − s unsuccessful attempts before this is accomplished. Each of theseunsuccessful search events lasts a time τ ∼ ν − c , and therefore, the total search time is T s ∼ N ν − c = e ν c d/v g pν c . (45)Note that Eq.42 reduces to Eq.45 in the limit p → ν → ∞ and v g ≪ v s .26 udding Yeast (I) mammalian (II) Xenopus extracts(III) ν r (0.42)0.12 min − (10.5)2.7 min − (0.66)1.2 min − ν c (0.48)0.24 min − ((1.56)3.48 min − (1.08)7.2 min − v g µ m min − µ m min − µ m min − v s µ m min − µ m min − µ m min − R µ m 20 µ m 500 µ mTABLE I. Experimental values of microtubule kinetics in the mitotic phase. The values in paran-theses are interphase values, prior to the cell entering mitosis, for Yeast[21], mammalian[15] and Xenopus extracts[22]. The cell radii given are only rough estimates. For theoretical and numericalanalysis, we used v g = 2 . v s = 3 . µ m min − for I, and v g = 12 . v s = 14 . µ m min − for II and III. The experimental data for different cell sizes are summarized in [15]. ∆ Ω d FIG. 1. A schematic illustration of the geometry of our model is shown here. Microtubules nucleatefrom nucleating sites on the centrosome, and search for a stationary target at a distance d . ∆Ω isthe solid angle of the ‘search cone’ for a certain nucleating site depicted in the picture. The searchis curtailed by the cell boundary. The search cones of neighbouring nucleating sites may overlap(not shown here), which accelerates the process by ‘parallel’ search. .5 1.0 1.5 2.0 d H Μ m L < T > H min L MitosisInterphase
FIG. 2. The mean time of capture (in minutes) of a single chromosome in budding yeast, bymicrotubules from a single nucleating site, for various target distance d is shown here. The thickblack like corresponds to interphase values of ν r and ν c . We assume the nuclear radius to be R = 2 µm . The other parameters are v d = 3 µ m min − , v g = 2 µ m min − , ν = 0 . − and ν ′ c = 10 ν c . H Μ m L < T > MitosisInterphase
FIG. 3. Similar to the previous figure, but for mammalian cells. The cell radius is taken as R = 20 µ m. The other parameters are v d = 14 µ m min − , v g = 12 µ m min − , ν = 0 . − and ν ′ c = 10 ν c . H Μ m L < T > MitosisInterphase
FIG. 4. Similar to the previous figures, but for
Xenopus oocyte cells. The cell radius is taken as R = 500 µ m. The other parameters are v d = 15 µ m min − , v g = 12 µ m min − , ν = 0 . − and ν ′ c = 10 ν c . Note that, unlike the previous figures, mitosis values appear to increase the search timecompared to interphase, which suggests that the random search and capture mechanism might beinefficient in large cells. Ν c H (cid:144) min L Ν r H (cid:144) m i n L A. Ν c H (cid:144) min L < T > H hours L Ν r = Ν r = Ν r = B.FIG. 5. A. Contour plot for the mean search time (expressed in hours) of a single chromosome at d = 6 µ m, by a single nucleating site, in a mammalian cell with radius R = 20 µ m. B. Cross-sectionsof the same plot at three values of ν r . The parameters are v d = 14 µ m min − , v g = 12 µ m min − , ν = 0 . − and ν ′ c = 10 ν c . .5 1.0 1.5 2.0 Ν c H (cid:144) min L < T > H hours L Ν r = Ν r = Ν r = FIG. 6. The mean time of search for a target close to the boundary, at d = 20 µ m, in a cell ofradius R = 20 µ m for three different ν r . The other parameters are chosen as v s = 12 µ m min − , v g = 10 µ m min − , ν = 1min − and ν ′ c = 10 ν c . Note that slightly larger values of ν r produce amore robust minimum as a function of ν c , at the cost of a small increase in time. H Μ m L t R MitosisInterphase A.
15 20 25 30 R H Μ m L D R (cid:144) t R MitosisInterphase
B.FIG. 7. A. The mean lifetime t R , defined as the mean lifetime of microtubules nucleating in direc-tions away from the target (Eq.30), is plotted as a function of the cell radius R for interphase andmitotic parameter values. The lifetime in interphase is larger by several orders of magnitude.Theparameter values are chosen as in Fig.4. B. The relative fluctuation in the lifetime, defined as theratio of standard deviation ∆ R to the mean t R , as a function of R , for the same set of param-eter values. Fluctuations in the mitotic phase are larger than in interphase, signaling increaseddynamicity of the microtubules., for the same set of param-eter values. Fluctuations in the mitotic phase are larger than in interphase, signaling increaseddynamicity of the microtubules.