A Stochastic model for dynamics of FtsZ filaments and the formation of Z-ring
EEur. Phys. J. E manuscript No. (will be inserted by the editor)
A stochastic model for dynamics of FtsZ filamentsand the formation of Z -ring Arabind Swain a,1,2,3 , A. V. Anil Kumar b,1,2 ,Sumedha c,d,1,2 School of Physical Sciences, National Institute of Science Education and Research, Jatni -752050, India Homi Bhabha National Institute, Training School Complex, Anushakti Nagar, Mumbai400094, India Present Address:
Department of Physics, Emory College of Arts and Sciences,Atlanta,Georgia 30322 USAReceived: date / Accepted: date
Abstract
Understanding the mechanisms responsible for the formation and growthof FtsZ polymers and their subsequent formation of the Z -ring is important forgaining insight into the cell division in prokaryotic cells. In this work, we presenta minimal stochastic model that qualitatively reproduces in vitro observations ofpolymerization, formation of dynamic contractile ring that is stable for a long timeand depolymerization shown by FtsZ polymer filaments. In this stochastic model,we explore different mechanisms for ring breaking and hydrolysis. In addition tohydrolysis, which is known to regulate the dynamics of other tubulin polymerslike microtubules, we find that the presence of the ring allows for an additionalmechanism for regulating the dynamics of FtsZ polymers. Ring breaking dynamicsin the presence of hydrolysis naturally induce rescue and catastrophe events in thismodel irrespective of the mechanism of hydrolysis. Keywords
Cytoskeletal filaments and networks · Stochastic Modelling
Cell division is one of the most fundamental processes in living cells. In prokaryoticcells, numerous proteins take part in assembling the machinery of cell division,called the divisome. FtsZ(Filamenting temperature-sensitive mutant Z ) is themost important among them. This tubulin homologue polymerizes head to tail,forming dynamic protofilaments. These protofilaments together form the core of astructure called the Z-ring in the division plane of the cell[1,2,3,4,5]. The Z-ringpersists throughout the cell division. Recent studies on septal wall suggest thatFtsZ monomers move around the ring via treadmilling which guides and regulatestheir growth, which in turn controls division [6,7,8,9,10]). This dynamic ring is aphenomenon that is intrinsic to FtsZ and the importance of this contractile ring a e-mail: [email protected] b e-mail: [email protected] c e-mail: [email protected] d corresponding author a r X i v : . [ q - b i o . S C ] J un in cell division is well established by now. Recently, there have been a lot of focusboth in vivo [8,9] and in vitro [11,12] to understand the dynamics of the ring. Butthe mechanism controlling the activity of the ring is still poorly understood.The monomers of FtsZ consist of two independent domains. The N -terminaldomain with its parallel beta sheets connected to alpha helices, provides the bind-ing site for GTP/GDP. The C- terminal region is essential for FtsZ to interactwith other proteins like FtsA and ZipA[13] and also as a membrane tether for theZ-ring formation during cell division[14]. Above a critical concentration of GTP,FtsZ polymerizes cooperatively into filaments which are single stranded and havea head to tail orientation with polymerization at only one end [15,16,17,18,19,20,21,22,23,24,25]. The exact mechanism behind this cooperativity is not well under-stood. Studies have shown that FtsZ can polymerize into multi-stranded bundles,sheets and form chiral vortex on membranes [10,24,26,27,28,29,30] depending onthe assembly buffer, nucleotide and other proteins present.Atomic Force Microscopy(AFM) of filaments adsorbed on Mica surface hasshown the role played by lateral interactions and filament curvature in under-standing the dynamic behavior of FtsZ filaments[31,32]. The study of individualfilaments under AFM showed that the filaments can form rings which are dy-namic in nature. These rings may open up, lose material and close again untilthey open irreversibly and then the filaments depolymerize. It has been shownthat unlike the case of tubulin, where nucleotide exchange is the rate limiting stepfor the reaction kinetics, in the case of FtsZ, it is the hydrolysis which controls thereaction kinetics. The ring formation of FtsZ is studied in the presence of GTP,guanosine-5-[( β, γ )-methyleno-triphosphate (GMPCPP) which reduces the hydrol-ysis as well as depolymerization rates and GTP with glycerol in the buffer whichreduces the depolymerization rates and hence stabilizes the lifetime for irreversibleopening[12]. They found that the rings formed in the presence of GMPCPP hadbroader length distribution, with a higher average length than the rings formedin the presence of GTP. A very recent study by Ramirez-Diaz et al [11] has againfocused on the ring formation in vitro . They made some key observations. Theyfound that the fine tuning of hydrolysis is needed for the formation of stable rings.They also found that the treadmilling results from a directional growth of curvedand polar filaments from the nucleation point at the membrane. The preferentialaddition of GTP subunit to the leading edge establishes a GTP-GDP gradient.FtsZ monomers are very similar to eucaryotic tubulin polymers like micro-tubules(MTs) and actin in their composition and functioning. MTs are semiflex-ible polymers that are a key component of the mitotic spindle[33] in eukaryoticcells and this requires them to be dynamic in nature. They exhibit a phenomena ofdynamic instability, in which they switch from a phase of slow elongation to rapidshortening (catastrophe) and from rapid shortening to growth(rescue). A polymer-izing MT grows until it suffers a catastrophe and starts to depolymerise. Similarly,a depolymerising MT undergoes rescue and starts polymerising again. Attemptsusing stochastic models have been successful in explaining this behaviour[34,35,36,37,38]. Experimental results and theory together have now established thatthe hydrolysis of tubulin monomers is responsible for the dynamic instability ofMTs, though the exact mechanism of hydrolysis is still debated [39]. Similarly,actin filaments[40,41] also exhibit non equilibrium phenomena known as tread-milling. In treadmilling, the new subunits get added at the growing end and oldsubunits leave the polymer from the other end. Mathematical models of MTs and actin have helped our understanding of the phenomena of dynamic instability andtreadmilling. These models bridge scales and hence facilitate our understandingof complex biological phenomena in terms of elementary processes[42]. This helpsin organizing the plethora of information one gets from the biochemistry study ofthese proteins.The basic self-assembly mechanism underlying dynamic instability, assemblymediated by nucleotide phosphate activity, is omnipresent in biological systems.FtsZ falls in the same class of bio-polymers and hence it seems natural that a simi-lar modelling approach capturing the dynamics will help our understanding of theFtsZ. Mechanical models[43,44,45,46], based on torsion and curvature to explainexperimental data have been attempted. But except for a few deterministic ap-proaches to model the FtsZ ring, there have been almost no attemps at modellingFtsZ at the molecular level. Most of these deterministic models aimed to charac-terise theoretically the in vivo and in vitro observations of FtsZ assembly. Thesemodels involve a number of differential equations to be solved simultaneously mak-ing it computationally very costly. The eight equation model proposed by Chen etal. [47,48] described the initial stages of FtsZ polymerization successfully, but failsto handle the whole process of FtsZ assembly. There are other models by Dow etal. [49], Lan et al. [50] and Surovtsev et al. [51] which employ few hundreds of differ-ential equations making the computations very complex. Recently, kinetic modelsbased on average charcteristics of different species and their concentrations havebeen proposed, where the number of differential equations need to be solved hasreduced considerably to 17[52] or 10[53]. These models reduced the computationalcost drastically and were able to predict the time taken to reach the steady state,the concentration of FtsZ in the Z-ring and average dimension of the filaments andbundles, which were in agreement with the experimental observations. However,the dependence of these on factors like rate of hydrolysis could not be obtainedby these models. These deterministic models cannot capture the ring dynamics ,which is stochastic in nature. There have been very few attempts to model the FtsZring formation stochastically[54,55]. These studies concentrated on the clusteringand ring formation of FtsZ polymers, by modeling the attachment and detachmentof these polymers to the cell membrane. For a recent review on modelling FtsZsee [56].MTs and actin form straight filaments while FtsZ form a ring of roughly onemicro-meter diameter in its active state [11]. In this paper we propose a stochasticmodel for FtsZ with treadmilling, which is similar in spirit to known stochasticmodels of MTs and actin[38,39,40,41]. Our aim is to describe the FtsZ dynamicsobserved in the in vitro experiments. The models on MTs and actins take intoaccount three processes mainly: polymerisation, depolymerisation and hydrolysis.Since the in vitro experiments reveal that the ring formation in FtsZ polymers isdynamic in nature, we introduce additional processes to account for ring forma-tion and breaking in the model. We consider two mechanisms for ring opening : a)random breaking of the ring, i.e., the ring opens randomly at any interface and b)non-random breaking of the ring, i.e., rupture can occur only at the interface withatleast one GDP bound FtsZ monomer. We also consider both possible mechanismsof hydrolysis of tubulin monomers: namely the vectorial and stochastic hydrolysis[39]. In vectorial hydrolysis, an unhydrolyzed monomer gets hydrolyzed only if theneighboring monomer is already hydrolyzed. This is a highly cooperative mecha-nism and there exists a sharp boundary between the hydrolyzed and unhydrolyzed parts of the FtsZ filaments. In the stochastic hydrolysis, GTP-FtsZ subunit canhydrolyze in a stochastic manner, irrespective of the position of the subunit inthe protofilament. The rate of hydrolysis in this case would be proportional to theamount of unhydrolyzed monomers. We find that the hydrolysis of the monomers isessential for the dynamics of FtsZ polymer. But interestingly the rescue and catas-trophe events are more crucially regulated by the process responsible for openingand closing of the Z-ring. Hence we consider four models in this paper based onthe nature of hydrolysis and the mechanism of ring breaking: vectorial hydrolysiswith random ring breaking, vectorial hydrolysis with non-random ring breaking,stochastic hydrolysis with random ring breaking and stochastic hydrolysis withnon-random ring breaking. The stability of ring improves with randomness andstochastic hydrolysis with random ring breaking gives the most stable rings. Wefind that in the case of random ring breaking, we need to fine tune hydrolysis rateto get a stable ring like structure. On the other hand, the dynamics in the case onnon-random breaking is insensitive to the hydrolysis rate. Also the experimentallyobserved ring length distribution matches with the ones obtained by vectorial hy-drolysis with random ring breaking. Hence we conclude that the non-random ringbreaking mechanism can be ruled out. This agrees with earlier suggestions [12]and also with the suggestions that ring breaks due to tension created on the ringdue to deformation of the membrane [65,66].The plan of the paper is as follows: In Section 2 we describe our model. InSection 3.1 we consider the vectorial hydrolysis with random and non-randombreaking of the ring and find that random ring breaking gives rise to stable dynamicring. In section 3.2 we study the effect of changing the ring breaking rate on thedynamics and conclude that with the random ring breaking one gets a filamentthat remains stable for a long time and then contracts. In section 3.3 we look atthe stochastic hydrolysis with random and non-random ring breaking and compareit with the results of vectorial hydrolysis. We discuss our results in Section 4. FtsZ filament is known to exist both in the form of an open chain and as aclosed ring. In the open form it has a single active end(the arrow head) whichis in contact with a reservoir with GTP-bound monomers. Based on recent ex-perimental evidence [10,11,25], we assume the filament to be directional withpolymerization only possible at its head end. In our model, in the presence of aGTP-bound monomer at its polymerizing end, a GTP-bound monomer gets addedto the polymer at a rate λ . Whenever a GDP-bound monomer is at either endsof an open filament, the polymer can lose a GDP-bound monomer from the endswith a rate µ . Both these processes are independent. This introduces treadmillingin the model. It has been experimentally observed that depolymerization is fasterat the ends as compared to the center[12]. Hence, we have assumed depolymeriza-tion only at the ends. Also, in general the polymerization rate is higher than thedepolymerization rate[10].When polymerized FtsZ in a solution of GTP is exposed to an excess of GDPit depolymerizes quickly[57]. GTP-bound monomers have been found to constitute Fig. 1
Schematics for the different reactions taking place in the system.
80% of filaments under in vitro conditions[17,58]. Though GDP-bound monomershave been found to polymerize in vitro but as compared to polymerization of GTP-bound monomers the equilibrium constant for this process has been found to besignificantly lower[20]. These results suggest that the GDP-bound polymerizationis unlikely to be viable and thus irrelevant in explaining FtsZ dynamics. Hence , inour model we assume that there is no polymerization of GDP-bound monomers.Our minimal stochastic model incorporates the process of polymerization atthe preferred end, depolymerization from both ends, hydrolysis, ring breaking and closing. In Figure 1, we present the schematic of all the processes we include inour model.The cases for both vectorial and stochastic hydrolysis have been consideredindividually. In the case of vectorial hydrolysis, the interface grows by convertingthe GTP-bound monomer associated with a GDP-GTP interface to a GDP-boundmonomer with a rate h . Propensity of a reaction in stochastic processes gives usthe likelihood of a particular reaction happening in a unit time. Reactions withhigher propensities are more likely. Thus the propensity for this reaction is hvec = h × ( Number of GDP − GT P interfaces ) (1)In the case of stochastic hydrolysis, a GTP-bound monomer can get randomlyconverted into a GDP-bound monomer with a rate h . The propensity for thereaction is h rand = h × ( Number of GT P bound monomers ) (2)The hydrolysis reactions can take place both in the open as well as the ring formingfilaments.Following the scheme used in earlier works[51,60] to model the natural curva-ture of FtsZ we assume that the filament has an optimum length( N ) at which ithas the maximum probability of closing and forming a ring. Hence, given a length N , the polymer has a non-zero probability to form a ring. We assume that therate of this process depends on N and is given by a Gaussian distribution thatpeaks at N = N : ν = C σ √ π e − ( N − N σ (3)FtsZ is known to polymerize into single stranded protofilaments in-vitro [16].Since treadmilling occurs at the level of a single protofilament, we have not in-cluded lateral interaction in our present model.Ring opening was assumed to have two possibilities. Firstly, the ring opens atany monomer-monomer interface chosen randomly. If we take δ to be the rate ofrandom breaking, then U = δN is the propensity of this reaction. Secondly, giventhat the GDP associated bonds are much weaker than GTP associated bonds,we also consider the case where the ring breaking is only possible at an interfacecontaining atleast one GDP-bound monomer. So, for a given ring with N GDPbound interfaces, the propensity of breaking is U = δN .The dynamics hence can be described by a set of coupled chemical masterequations for the probability of open and closed polymer. Due to the presence ofring breaking, it is not possible to solve the equations even in the simpler case ofvectorial hydrolysis. Hence we use Gillespie algorithm[61,62] to solve the chemicalmaster equations numerically. Gillespie algorithm offers an elegant way to speedup simulations by doing away with the many rejected trials of the traditionalMonte-Carlo moves. While, traditional Monte Carlo methods check at each stepif each reaction takes place, Gillespie algorithm draws directly the next reactionand time elapsed until that next reaction. The advantage of Gillespie algorithmis that it generates an ensemble of trajectories with the correct statistics. It hasbeen very successful in simulating many chemical and biological reactions [62].In fact, Gillespie alogorithm was employed before in modelling the clustering andring formation of FtsZ polymers[54,55]. P r ( N, t ) represents theprobability of having a closed ring polymer of length N at time t and P o ( N, t )represents the probability of having an open chain polymer of length N at time t .We define p ( t ) as the probability of having a GTP-bound monomer at the activeend in the open ring configuration. Thus the probability of active end having aGDP bound monomer is 1 − p ( t ). Let p ( t ) be the probability of finding a GDPbound monomer at the non-polymerizing end. The probability of observing a GDPmonomer at either end is p ( t ) = p ( t ) + 1 − p ( t ) , As discussed in Section 2, ν = C σ √ π e − ( N − N σ is the rate for an open polymer of length N to close and forma ring. Then for random breaking the system can be described by the followingset of equations: ∂P r ( N, t ) ∂t = νP o ( N, t ) − δNP r ( N, t ) (4) ∂P o ( N, t ) ∂t = − p ( t ) λ ∂P o ( N, t ) ∂N + p ( t ) µ ∂P o ( N, t ) ∂N (5) − νP o ( N, t ) + δNP r ( N, t )In the beginning there are only the processes of polymerization, depolymer-ization from the back end and hydrolysis. Once FtsZ polymer attains sufficientlength, it can form a closed ring. In the ring form the polymer doesn’t have apolymerizing and a non-polymerizing end. In this state there are only two reac-tions going on :hydrolysis and ring opening. Thus polymerization rate does nothave a direct effect on the dynamics of a closed ring. The ring opening being arandom process, it is not possible to tell what will the ends of the polymer bewhen it opens up. Hence, once the ring opens up, it can undergo polymerizationor depolymerization depending on the type of monomers present at the ends. Thisrandomness in the opening of the ring makes it possible for the polymer to undergodynamic catastrophe and rescue events. This makes the ring length and the ratioof GDP/GTP bound monomers in the ring fluctuating parameters, which in turnmakes the calculation of p ( t ) and p ( t ) impossible. Hence, it is not possible tosolve the chemical master equations exactly. One can solve the equations exactlyonly in the trivial case where hydrolysis is much lower than the polymerization rateand hence the probability p ( t ) and p ( t ) can be taken to be 1 and 0 respectively.In this case one gets a gaussian distribution for the ring length distribution. Wehence simulate the system using Gillespie algorithm. We will compare the mecha-nisms of ring opening i.e random breaking and non-random breaking by comparingthe properties of experimentally possible observables like ring length and ring lifetime. We fix depolymerisation rate to be 1 (as we can always fix one of the ratesto be one) and take N = 120 and σ = 15. C is chosen such that C σ √ π is 10. Wewill now study the effect of polymerisation,hydrolysis and ring breaking rate onthe dynamics of the ring.In Figure 2(left) we plot the length of FtsZ polymer as a function of timefrom a typical run for different hydrolysis rates for random breaking of FtsZ rings. , Fig. 2
Trajectories for Random(left) and Non-Random Breaking(right) for polymerizationrate 2.0, 4.0 and 8.0 respectively(top to bottom). The depolymerisation rate( µ ) is taken tobe 1 and ring breaking rate ( δ ) is 0 . h ), it continues to grow or forms a dynamic ring. Thering formed stays dynamic for a long time. In the non-random case(right), a ring is formed fornon zero hydrolysis rate and its dynamics is not sensitive to change in hydrolysis rate. We fix the random breaking rate to be δ = 0 . h . After this initialgrowth phase, the polymer makes a transition to a phase where the length getsstabilised for a period. This is the phase where FtsZ ring exists. It is clear fromthe trajectories that FtsZ ring is dynamic as the length fluctuates with time. Thethird region is where the polymer either grows an open chain or depolymerisecompletely.When we consider random breaking, ring can break at any of the interface.When h = 0, hydrolysis does not occur as there are no GDP bound monomers inthe polymer. So when the ring opens up, it cannot depolymerise. Hence, after ashort period of dynamic ring formation and breaking, the chain grows as a linearchain. When the hydrolysis rate is not zero, but small there is a finite probabilityto encounter a GDP bound monomer at one of the ends whenever the ring opensup. In this case, depolymerisation occurs before the chain start polymerising againresulting in a larger lifetime for the dynamic ring. As the hydrolysis rate increases,the probabilty of finding GDP bound monomer increases. In the case where GDPbound monomers are present at the polymerizing end, all of them need to get Fig. 3
Average number of D-T interfaces for random(a) and non-random(b) breaking forpolymerisation rate λ = 4. In the random ring breaking(a), the ring develops many moreinterfaces while active. depolymerized to expose a GTP bound monomer for the polymer to grow again.We find that this results in catastrophe and rescue events in which the polymermay lose as many as 20-30 monomers before recovering again. Additionally, theopening and closing dynamics ensures that we have many interfaces in this case(seeFig. 3(a)). Finally, when the polymer has no more GTP bound monomers and hasonly GDP bound monomers there are no more rescue events and the polymerdepolymerizes completely. We find that as hydrolysis rate increases, initially thering life time increases. However, because of interplay between polymerization,depolymerization, ring closing and opening, ring life time peaks at a finite valueof h and beyond that stability of ring keeps on decreasing.In Figure 2(right), we plot the length of FtsZ polymers in the case of nonran-dom breaking; i.e. the ring opens only at the interface formed by atleast one GDPbound monomer, for different hydrolysis rates. In the zero hydrolysis case once thering gets formed it will never open since GDP bound interfaces are absent in thering. At small values of hydrolysis rate, GTP will get hydrolyzed after a long timeand at some later time it will open. Since GDP bound monomers are present atleast at one of the ends, the chain is exposed for depolymerisation once it opensup. Hence, chances of depolymerization is bigger in this case in contrast to therandom breaking case. This results in the large fluctations seen in the trajectoriescompared to the case of random breaking. The number of interfaces for vectorialhydrolysis remains the same, i.e, 1 throughout the time evolution(recall that forrandom breaking the number of interfaces increase with time). We have plottedthe average number of interfaces as a function of time for different hydrolysis ratesin Fig. 3(b).From Figure 2, it is clear that the ring dynamics and its stability are sensitiveto the hydrolysis rate in the case of random breaking of the ring and insensitiveto hydrolysis rate in the case of non-random breaking. These observations can bemade more quantitative in terms of experimentally measurable quantities like ringlifetime and length distributions. We obtain these quantities by averaging over 10 independent trajectories. In Fig 4(a) and 4(b) we plot distributions of ring size in both random breakingand non-random breaking respectively for different hydrolysis rates. In the case ofrandom breaking the distribution gets narrower with increasing hydrolysis rate. Fig. 4 a) and b): Plot of distribution of ring length for polymerization rate 4.0 for randomand non-random ring breaking respectively. c) and d): Plot of average length for differentpolymerization rates, for random and non-random breaking respectively
In ref. [12] authors studied the ring length distribution of FtsZ polymers in-vitro.They found that rings formed in the presence of slowly hydrolyzing analogue, GM-PCPP, resulted in broader distribution , which was shifted to the right. Our plotsin Fig. 4(a) are consistent with this observation. The ring length distribution getsthinner with increasing hydrolysis rate and the peak shifts to the left, resulting indecreasing average length of the ring. In contrast, in the case of the non-randomring breaking, except for zero hydrolysis rate, the ring length distribution remainsunchanged, suggesting that the ring length is insensitive to the changes in hydrol-ysis rate. In the case of zero hydrolysis rate, the ring never opens up once it formsas there are no GTP-GDP interfaces, and the distribution is just proportional tothe ring closing rate ν .In fig 4(c) and 4(d) we plot the average length of FtsZ rings in the two cases.As the hydrolysis rates increases, the average length of the polymer decreasesgradually in the case of random breaking. In the case of non-random breaking,the average length remains same except for the case of zero hydrolysis rate. Thissuggests that the observation that the dynamics of FtsZ polymer is insensitive tothe hydrolysis rate in the case of non-random breaking, made based on figure 2(b)is indeed true. Near zero hydrolysis rate, the average length of ring, depends onlyon the polymerisation rate as the number of GDP-bound monomers are very lessand depolymerisation does not take place often. So the average length of the ringincreases as the rate of polymerization increases. It is known that the dynamic ring stage of FtsZ polymer is very stable. Thefilament undergoes many catastrophe and rescue event during this time, beforeeventually decaying. Hence, one of the most important observable is the totaltime( τ ) for which the ring exists before it eventually escapes the ring formationand grows as a linear chain or decays completely. For each trajectory the time for Fig. 5
Average lifetime( τ ) for irreversible opening for random(a) and non-random(b) break-ing. Here τ is the time for irreversible opening. For random ring breaking, τ is sensitive to thehydrolysis rate and peaks at a fixed value of h /λ , irreversible ring opening is calculated as τ = T final − T init , where T final is thetime at which the ring opens up finally and afterwhich the polymer grows as alinear chain or decay completely and T init is the time at which the polymer formsa ring for the first time after the initial growth phase. This time is averaged overmany independent trajectories. The time for this irreversible ring opening, τ hasbeen plotted in Figures 5(a) and 5(b) for random and non-random ring breakingrespectively. For random breaking it peaks at a finite value of hydrolysis rate. Thisis because at very low values of hydrolysis the polymer escapes into growth phaseand at large value of hydrolysis, the chances of ring closing again is small. Hencethere is a narrow window of hydrolysis rate where the dynamic ring is most stable.This is consistent with the very recent experiments of Ramirez-Diaz et al [11]where they found that fine tuning of hydrolysis was needed to get stable dynamicrings. The stability of individual filaments which was observed to be very sensitiveto the hydrolysis rate may be a major contributing factor to ring stability observedin experiments. Moreover, we observe that the region of maximum stability occursroughly at a fixed ratio of hydrolysis to polymerization rates. It has been observedthat in the presence of a GTP regeneration system, the dynamic ring can continuefor a long time[57]. This along with the fact that FtsZ remains highly conservedacross species[63], can be used to argue that FtsZ sits on the edge of stability in itsnaturally occurring form. This is clearly demonstrated in the plots for random ringopening in which, across all polymerization rates the region of maximum stabilityhappens at a fixed ratio of hydrolysis and polymerization rates. In contrast tothe case of random ring opening, for non-random breaking the ring is not reallydynamic and hence the lifetime goes down monotonically with the hydrolysis rate. We can also look at the distribution of time elapsed between two openings. Weexpect that there would be many small events where the ring opens with GTP atthe growing end and close back immediately. Figure 6(a) and 6(b) show these dis-tributions for the case of random and non-random ring breaking. The distributionsbecome flatter and flatter as the hydrolysis rate increases for both cases; howeverthe effect is more pronounced in the case of non-random breaking. Polymerizationrate does not affect the lifetime of single openings since the polymerisation doesnot take place when the ring is closed. Such a trend can be observed in the case Fig. 6 a) and b): Plot of distribution of ring lifetime for polymerization rate 4.0 for randomand non random ring breaking respectively. c) and d): Plot of average ring lifetime as a functionof hydrolysis for random and non-random breaking respectively for different polymerisationrate. of random ring opening. But in case of non-random opening, increase in polymer-ization rate leads to an decrease in average ring lifetime for the same hydrolysisrate. Hence, we conclude that in the case of non-random opening polymerizationrate affects the ring opening dynamics(see Fig. 6 (c) and 6(d)).
From our observations on the average time for irreversible opening, it is clear thatmost stable ring is formed by fine tuning the hydrolysis rate in the case of vectorialhydrolysis with random breaking. We find that for random breaking, the averagelife time( τ ) does not change monotonically with the hydrolysis rate( h ) for a givenpolymerisation rate( λ ). In Fig. 7, we have plotted the value of h for which τ ismaximum as a function of polymerisation rate( λ ). We find that h value for which τ becomes maximum increases with increasing λ .3.2 Force generation and the effect of ring breaking rate on the dynamicsWe studied the effect of ring breaking rate on the FtsZ ring for random breakingmodel with vectorial hydrolysis. As the breaking rate increases, the average lengthof the ring for a given hydrolysis and polymerization rate increases(see Fig. 8).Also as the breaking rate increases, the average length for single ring breaking ofthe polymer for a given hydrolysis and polymerization rate decreases. For poly-merization rate 1.0 different breaking rates i.e 0.008 and 0.012 were found to besame. This suggests that for a given polymerization rate beyond a certain value,the breaking rate does not affect the average time for single ring breaking(seeFig. 9). We also looked at the average time of irreversible opening. We find thatas the breaking rate increases, the hydrolysis rate at which the average time forirreversible opening becomes maximum increases( Fig. 10). Fig. 7
The system can exist in 2 states. In one state the polymer keeps growing uncontrollablyand forms a ring for a short time. In another state the polymer forms the ring before depolymer-izing completely. Between these two extremes there are states in which the ring remains stablefor a long time going through multiple events of ring closing, ring opening, polmerizationsand depolymerizations. The plot shows the critical hydrolysis rate ( h ) at which the systemhas stable ring for the longest time before getting completely depolymerized as a function ofpolymerization rate ( λ ). Fig. 8
Average lengths for different ring breaking rates δ . The average ring length are plottedas a function of hydrolysis rates. Data for polymerization rate 1.0 and for polymerization rate4.0 are plotted in (a) and (b) respectively. All the plots are for random breaking and thedifferent breaking rates δ are represented by different colors. h = 0 now.We have seen in the previous section, the ring breaking naturally provides a wayto build rescue and catastrophe in the FtsZ ring. For the sake of completion, weexplore FtsZ dynamics assuming SH in this section. Fig 11 shows trajectory with Fig. 9
The average ring-life time for single ring breaking are plotted as a function of hydrolysisrates. Data for polymerization rate 1.0 and for polymerization rate 4.0 are plotted in (a) and(b) respectively. All the plots are for random breaking and the different breaking rates δ arerepresented by different colors. Fig. 10
The average lifetime ( τ ) of FtsZ for irreversible ring opening are plotted as a functionof hydrolysis rates. Data for polymerization rate 1.0 and for polymerization rate 4.0 are plottedin (a) and (b) respectively. All the plots are for random breaking and the different breakingrates δ are represented by different colors. both random breaking and non-random breaking. In the case of random breaking,there are now two ways to introduce interfaces in the system. As expected, thering is more dynamic in this case and at the critical value of the hydrolysis rate h , the ring stays stable for a very long time with regular opening and closing ofthe ring. In the case of non-random breaking the ring is similar, but the ring isnot as stable as in the case of SH with random breaking.We looked at the ring length distribution in this case. We find that the ringlength distribution is sensitive to change of hydrolysis rate for both , random andnon-random breaking(see Fig. 12). However the non-random breaking case is lesssensitive to the hydrolysis rate(especially for smaller polymerization rate).We also looked at the time for irreversible ring opening(Fig. 13) and interest-ingly we find that unlike the case of vectorial hydrolysis, the average lifetime isinsensitive to the hydrolysis rate for a wide range of hydrolysis rate values. The dynamics of FtsZ ring is being very actively studied in experiments in recenttimes[6,7,8,9]. It is still not possible to observe the microscopic dynamics and onecan at best study the average quantities like the average lifetime and length of thering [12]. In this paper, we have introduced a stochastic model which considersall the essential processes and compares all possible scenarios. This is importantas the nature of hydrolysis and ring breaking is still not well understood, eventhough there is a general consensus that hydrolysis and treadmilling of the FtsZ Fig. 11
Trajectories for random (left) and non-random breaking (right) for polymerizationrate 2.0, 4.0 and 8.0 respectively (top to bottom)
Fig. 12
Distribution of ring length for polymerization rate 4.0 for random(a) and non-random(b) breaking with stochastic hydrolysis.
Fig. 13
Average lifetime of FtsZ rings before irreversible opening for random (a) and non-random (b) breaking with stochastic hydrolysis.6 polymer plays a crucial role in defining its dynamics and functionality. Note thatthe dynamic nature of the ring cannot be captured effectively in a deterministicmodel. Our stochastic model naturally gives rise to catastrophe and rescue of FtsZring, which have been observed in experiments. We find that unlike microtubulesand actin the nature of hydrolysis does not play a crucial role for FtsZ ring.For microtubules and actin if one considers vectorial hydrolysis , then one hasto introduce rescues by hand for the polymer to be dynamic. We find that therescues and catastrophe events are built in our model due to the ring topologyirrespective of the hydrolysis mechanism. This suggests that ring breaking is anatural mechanism for introducing dynamics in the FtsZ filaments.In the case of vectorial hydrolysis with random ring breaking we could quali-tatively reproduced the known behavior of the Z -ring. We found that increasingthe hydrolysis rate in this case narrows the ring length distribution and the av-erage length of the ring decreases with hydrolysis. This is consistent with the in vitro studies of Z-ring by Mateos-Gil et al. [12]. We also find that the timelapsed between ring formation and final decay(time for irreversible ring opening)changes non-trivially with the hydrolysis rate. One needs to fine tune hydrolysisrate to achieve the most stable ring in this case. In contrast, when we considernon-random ring breaking, the behavior of the ring is insensitive to the hydrolysisrate. Initially the polymer has only one interface where the hydrolysis occurs. Butafter ring closure, we find that in the case of random ring breaking, many newinterfaces show up, and hence the ring can potentially break up at multiple places,consistent with the recent experiments [9]. Hence even though we consider a closedring, in the case of random breaking, the ring actually has a patchy structure. Incontrast in the case of non-random ring breaking, the number of interfaces do notchange with time and one typically has only one interface in the Z -ring. All theseobservations suggests that Z -ring breaks randomly. Similar conclusion was alsoreached by Mateos-Gil et al. [12]. Recent works suggest that treadmilling of Z -ring is sufficient to generate the constriction force generated by Z -ring during celldivision. The ring with random breaking is also contractile, as shown in Section3.2.We have tried to keep our model basic so that we could distinguish the effect ofdifferent random processes clearly. The model can be studied by including manyother processes like allowing the attachment /detachment of short polymer andnot just a single monomer during polymerization/ depolymerization process. Wefind that this does not change the picture qualitatively and hence not neededto understand the basic dynamics of the FtsZ polymer. One can also introducelateral interaction to model the situation when the FtsZ concentration is high, forexample to model the dynamics invivo .We have also looked at the ring dynamics with stochastic hydrolysis. In thiscase we find that for both ring breaking mechanisms (random and non-random),we do not need to fine tune the hydrolysis rate to achieve the stable dynamic ring.Recent experiments [11] suggest that fine tuning of hydrolysis rate is required forstable dynamic ring. Behaviour of ring life time as a function of hydrolysis ratecan hence be a distinguishing feature between vectorial and stochastic hydrolysis.Hence, differences in the statistics of the ring can be used to unravel the actualmechanism of hydrolysis in the FtsZ polymer.We find that the randomness makes the ring more dynamic and stable, as seenin the case of stochastic hydrolysis and random breaking. In general it is now established that the stochasticity increases the stability of biological processes [70]and is important to understand the functioning of biological systems. Thoughthe Z-ring has been studied by others using deterministic models, this is the firststochastic model for Z-ring. We find that the model is able to explain a numberof features of FtsZ filament dynamics in vitro , elucidating the crucial role the ringtopology plays in their functioning. The model successfully brings out the role ofhydrolysis and ring breaking mechanisms in forming a contractile dynamic ring.The model is simple and in future can easily be used for more quantitative studiesor involve more complex interactions to study the in vivo dynamics of Z-ring. Acknowledgements
The authors thank Dr. R. Sreenivasan and Dr. S. Roychowdhury for fruitful dis-cussions. This work has been supported by Department of Atomic Energy, Indiathorugh the 12th plan project (12-R&D-NIS-5.02-0100).
Author Contributions
A.S ran the simulations. S and A.V.A.K developed the model. A.S,S and A.V.A.Kanalysed the results and drafted the manuscript.
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Lecture Notes inMathematics, 2058:3-34. Springer, Berlin, Heidelberg ur. Phys. J. C manuscript No. (will be inserted by the editor)
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Cell division is one of the most fundamental processes inliving cells. In prokaryotic cells numerous proteins takepart in assembling the machinery of cell division, calledthe divisome. FtsZ(Filamenting temperature-sensitivemutant Z ) is the most important among them. Thistubulin homologue polymerizes head to tail, formingdynamic protofilaments. These protofilaments togetherform the core of a structure called the Z-ring in thedivision plane of the cell[1,2,3,4,5]. The Z-ring per-sists throughout the cell division. Recent studies on sep-tal wall suggest that FtsZ monomers move around thering via treadmilling which guides and regulates theirgrowth, which in turn controls division [6,7,8,9,10]).The dynamic ring is a phenomena that is intrinsic toFtsZ and the importance of this contractile ring in celldivision is well established by now. Recently, there havebeen a lot of focus both in vivo [8,9] and in vitro [11,12] to understand the dynamics of the ring. But themechanism controlling the activity of the ring is poorlyunderstood.The monomers of FtsZ consists of two independentdomains. The N -terminal domain with its parallel betasheets connected to alpha helices, provides the bindingsite for GTP/GDP. The C- terminal region is essen-tial for FtsZ to interact with other proteins like FtsAand ZipA[13] and also as a membrane tether for the a e-mail: [email protected] Z-ring to form during cell division[14]. Above a crit-ical concentration of GTP, FtsZ polymerizes coopera-tively into filaments which are single stranded and havea head to tail orientation with polymerization at onlyone end [15,16,17,18,19,20,21,22,23,24,25]. The exactmechanism behind this cooperativity is not well un-derstood. Studies have shown that FtsZ can polymer-ize into multi-stranded bundles, sheets and form chiralvortex on membranes [24,26,27,10,28,29,30] depend-ing on the assembly buffer, nucleotide and other pro-teins present.Atomic Force Microscopy(AFM) of filaments adsorbedon Mica surface have shown the role played by lat-eral interactions and filament curvature in understand-ing the dynamic behavior of FtsZ filaments[31,32]. Thestudy of individual filaments under AFM showed thatthe filaments can form rings which are dynamic andmay open up, lose material and close again until theyopen irreversibly and then the filaments depolymerize.It has been shown that unlike the case of tubulin, wherenucleotide exchange is the rate limiting step for the re-action kinetics, in the case of FtsZ it is the hydrolysiswhich controls the reaction kinetics. The ring forma-tion of FtsZ is studied in presence of GTP, guanosine-5-[( β, γ )-methyleno-triphosphate (GMPCPP) which re-duces the hydrolysis as well as depolymerization ratesand GTP with glycerol in the buffer which reduces thedepolymerization rates and hence stabilize the lifetimesfor irreversible opening[12]. They found that the ringsformed in the presence of GMPCPP had broader lengthdistribution, with higher average length than the ringsformed in the presence of GTP. A very recent study byRamirez-Diaz et al [11] has again focused on the ringformation in vitro . They made some key observations.They found that the fine tuning of hydrolysis is neededfor the formation of stable rings. They also found that a r X i v : . [ q - b i o . S C ] J un the treadmilling results from a directional growth ofcurved and polar filaments from the nucleation pointat the membrane. The preferential addition of GTPsubunit to the leading edge establishes a GTP-GDPgradient.FtsZ monomers are very similar to eucaryotic tubu-lin polymers like microtubules(MTs) and actin in theircomposition and functioning. MTs are semiflexible poly-mers that are a key component of the mitotic spindle[33]in eukaryotic cells and this requires them to be dynamicin nature. They exhibit a phenomena of dynamic insta-bility(DI), in which they switch from a phase of slowelongation to rapid shortening (catastrophe) and fromrapid shortening to growth(rescue). A polymerizing MTgrows until it suffers a catastrophe and starts to de-polymerise. Similarly, a depolymerising MT undergoesrescue and starts polymerising again. Attempts usingstochastic models have been successful in explainingthis behaviour[34, ? ,36,37, ? ]. Experimental results andtheory together have now established that the hydroly-sis of tubulin monomers is responsible for the dynamicinstability of MTs, though the exact mechanism of hy-drolysis is still debated [39]. Similarly, actin filaments[40,41] also exhibit non equilibrium phenomena known astreadmilling. In treadmilling, the new subunits get addedat the growing end and old subunits leave the polymerfrom the other end. Mathematical models of MTs andactin have helped our understanding of the phenomenaof dynamic instability and treadmilling. These modelsbridge scales and hence facilitate our understanding ofcomplex biological phenomena in terms of elementaryprocesses [42]. This helps in organizing the plethora ofinformation one gets from the biochemistry study ofthese proteins.The basic self-assembly mechanism underlying DI,assembly mediated by nucleotide phosphate activity, isomnipresent in biological systems. FtsZ falls in the sameclass of bio-polymers and hence it seems natural that asimilar modelling approach capturing the dynamics willhelp our understanding of the FtsZ. Mechanical models[43,44,45,46], based on torsion and curvature to explainexperimental data have been attempted. But exceptfor a few deterministic approaches to model the FtsZring, there have been almost no attemps at modellingFtsZ at the molecular level. Most of these determin-istic models aimed to characterise theoretically the invivo and in vitro observations of FtsZ assembly. Thesemodels involve a number of differential equations to besolved simultaneously making it computationally verycostly. The eight equation model proposed by Chen etal. [47,48] described the initial stages of FtsZ polymer-ization successfully, but fails to handle the whole pro-cess of FtsZ assembly. There are other models by Dow et al. [49], Lan et al. [50] and Surovtsev et al. [51] whichemploy few hundreds of differential equations makingthe computations very complex. Recently, kinetic mod-els based on average charcteristics of different speciesand their concentrations have been proposed, where thenumber of differential equations need to be solved hasreduced considerably to 17[52] or 10[53]. These mod-els reduced the computational cost drastically and wereable to predict the time taken to reach the steady state,the concentration of FtsZ in the Z-ring and average di-mension of the filaments and bundles, which were inagreement with the experimental observations. How-ever, the dependence of these on factors like rate of hy-drolysis could not be obtained by these models. Thesedeterministic models cannot capture the ring dynamics, which is stochastic in nature. For a recent review onmodelling FtsZ see [54].MTs and actin form straight filaments while FtsZform a ring of roughly one micro-meter diameter in itsactive state [11]. In this paper we propose a stochasticmodel for FtsZ with treadmilling, which is similar inspirit to known stochastic models of MTs and actin[38,39,40,41]. Our aim is to describe the FtsZ dynamics ob-served in the in vitro experiements. The models on MTsand actins take into account three processes mainly:polymerisation, depolymerisation and hydrolysis. Sincethe in vitro experiments reveal that the ring formationin FtsZ polymers is dynamic in nature, we introduceadditional processes to account for ring formation andbreaking in the model. We consider two mechanismsfor ring opening : a) random breaking of the ring, i.e.,the ring opens randomly at any interface and b) non-random breaking of the ring, i.e., rupture can occuronly at the interface with atleast one GDP bound FtsZmonomer. We also consider both possible mechanismsof hydrolysis of tubulin monomers: namely the vectorialand stochastic hydrolysis [39]. In vectorial hydrolysis,an unhydrolyzed monomer gets hydrolyzed only if theneighboring monomer is already hydrolyzed. This is ahighly cooperative mechanism and there exists a sharpboundary between the hydrolyzed and unhydrolyzedparts of the FtsZ filaments. In the stochastic hydrol-ysis, GTP-FtsZ subunit can hydrolyze in a stochasticmanner, irrespective of the position of the subunit inthe protofilament. The rate of hydrolysis in this casewould be proportional to the amount of unhydrolyzedmonomers. We find that the hydrolysis of the monomersis essential for the dynamics of FtsZ polymer. But in-terestingly the rescue and catastrophe events are morecrucially regulated by the process responsible for open-ing and closing of the Z-ring. Hence we consider fourmodels in this paper based on the nature of hydrolysisand the mechanism of ring breaking: vectorial hydroly- sis with random ring breaking, vectorial hydrolysis withnon-random ring breaking, stochastic hydrolysis withrandom ring breaking and stochastic hydrolysis withnon-random ring breaking. The stability of ring im-proves with randomness and stochastic hydrolysis withrandom ring breaking gives the most stable rings. Wefind that in the case of random ring breaking, we needto fine tune hydrolysis rate to get a stable ring likestructure. On the other hand, the dynamics in the caseon non-random breaking is insensitive to the hydrolysisrate. Also the experimentally observed ring length dis-tribution matches with the ones obtained by vectorialhydrolysis with random ring breaking. Hence we con-clude that the non-random ring breaking mechanismcan be ruled out. This agrees with earlier suggestions[12] and also with the suggestions that ring breaks dueto tension created on the ring due to deformation of themembrane [63,64].The plan of the paper is as follows: In Section 2 wedescribe our model. In Section 3.1 we consider the vec-torial hydrolysis with random and non-random break-ing of the ring and find that random ring breaking givesrise to stable dynamic ring. In section 3.2 we study theeffect of changing the ring breaking rate on the dynam-ics and conclude that with random ring breaking onegets a filament that remains stable for a long time andthen contracts. In section 3.3 we look at the stochastichydrolysis with random and non-random ring breakingand compare it with the results of vectorial hydrolysis.We discuss our results in Section 4. FtsZ filament is known to exist both in the form ofan open chain and as a closed ring. In the open formit has a single active end(the arrow head) which is incontact with a reservoir with GTP-bound monomers.Based on recent experimental evidence [10,11,25], weassume the filament to be directional with polymeriza-tion only possible at its head end. In our model, in thepresence of a GTP-bound monomer at its polymerizingend, a GTP-bound monomer gets added to the poly-mer at a rate λ . Whenever a GDP-bound monomer isat either ends of an open filament, the polymer can losea GDP-bound monomer from the ends with a rate µ .Both these processes are independent. This introducestreadmilling in the model. It has been experimentallyobserved that depolymerization is faster at the ends ascompared to the center[12]. Hence, we have assumeddepolymerization only at the ends. Also, in general the Fig. 1
Schematics for the different reactions taking place inthe system. polymerisation rate is higher than the depolymerizationrate[10].When polymerized FtsZ in a solution of GTP is ex-posed to an excess of GDP it depolymerizes quickly[55].GTP-bound monomers have been found to constitute80% of filaments under in vitro conditions[17,56]. ThoughGDP-bound monomers have been found to polymerizein vitro but as compared to polymerization of GTP-bound monomers the equilibrium constant for this pro-cess has been found to be significantly lower[20]. Theseresults suggest that the GDP-bound polymerization isunlikely to be viable and thus irrelevant in explainingFtsZ dynamics. Hence , in our model we assume thatthere is no polymerization of GDP-bound monomers.Our minimal stochastic model incorporates the pro-cess of polymerization at the preferred end, depolymer-ization from both ends, hydrolysis, ring breaking and closing. In Figure 1, we present the schematic of all theprocesses we include in our model.The cases for both vectorial and stochastic hydroly-sis have been considered individually. In the case of vec-torial hydrolysis in the presence of a GDP-GTP inter-face, the interface grows by converting the GTP-boundmonomer associated with it to a GDP-bound monomerwith a rate h . Propensity of a reaction in stochasticreactions gives us the likelihood of a particular reac-tion happening in a unit time. Reactions with higherpropensities are more likely. Thus the propensity forthis reaction is hvec = h × ( N umber of GDP − GT P interf aces )(1)In the case of stochastic hydrolysis, a GTP-bound monomercan get randomly converted into a GDP-bound monomerwith a rate h . The propensity for the reaction is h rand = h × ( N umber of GT P bound monomers ) (2)The hydrolysis reactions can take place both in the openas well as ring forming filaments.Following the scheme used in earlier works[51,58]to model the natural curvature of FtsZ we assume thatthe filament has an optimum length( N ) at which ithas the maximum probability of closing and forming aring. Hence, given a length N , the polymer has a non-zero probability to form a ring. We assume the rate ofthis process depends on N and is given by a Gaussiandistribution that peaks at N = N : ν = C σ √ π e − ( N − N σ (3)Ring opening was assumed to have two possibilities.One in which the ring opening is completely random.If we take δ to be the rate of random breaking, then U = δN is the propensity of this reaction. Given thatthe GDP associated bonds are much weaker than GTPassociated bonds, we also consider the case where thering breaking is only possible at an interface contain-ing atleast one GDP-bound monomer. So, for a givenring with N GDP bound interfaces, the propensity ofbreaking is U = δN .The dynamics hence can be described by a set ofcoupled chemical master equations for the probabilityof open and closed polymer. Due to the presence of ringbreaking, it is not possible to solve the equations evenin the simpler case of vectorial hydrolysis. Hence we useGillespie algorithm[59,60] to solve the chemical masterequations numerically. Gillespie algorithm offers an el-egant way to speed up simulations by doing away withthe many rejected trials of the traditional Monte-Carlomoves. While, traditional Monte-Carlo methods checkat each step if each reaction takes place, Gillespie algo-rithm draws directly the next reaction and time elapsed until that next reaction. The advantage of Gillespie al-gorithm is that it generates an ensemble of trajectorieswith the correct statistics. It has been very successful insimulating many chemical and biological reactions [60]. P r ( N, t ) represent the proba-bility of having a closed ring polymer of length N attime t and P o ( N, t ) represent the probability of hav-ing an open chain polymer of length N at time t . Wedefine p ( t ) as the probability of having a GTP-boundmonomer at the active end in the open ring configura-tion. Thus the probability of active end having a GDPbound monomer is 1 − p ( t ). Let p ( t ) be the proba-bility of finding a GDP bound monomer at the non-polymerizing end. The probability of observing a GDPmonomer at either end is p ( t ) = p ( t ) + 1 − p ( t ) ,Also as discuseed in Section 2, ν = C σ √ π e − ( N − N σ isthe rate for an open polymer of length N to close upinto a ring. Then for random breaking the system canbe described by the following set of equations: ∂P r ( N, t ) ∂t = νP o ( N, t ) − δN P r ( N, t ) (4) ∂P o ( N, t ) ∂t = − p ( t ) λ ∂P o ( N, t ) ∂N + p ( t ) µ ∂P o ( N, t ) ∂N − νP o ( N, t )+ δN P r ( N, t )(5)In the beginning there are only the processes ofpolymerization, depolymerization from the back endand hydrolysis. Once the polymer takes a ring form,it doesn’t have a polymerizing and a non-polymerizingend. In this state for sometime there are only two reac-tions going on :hydrolysis and ring opening. Thus poly-merization rate does not have a direct effect on thedynamics of a closed ring. The ring opening being arandom process, it is not possible to tell what will theends of the polymer be when it opens up. Once the ringopens up, it can again undergo polymerization and de-polymerization. Depending on the tip of the open poly-mer, the polymer undergoes dynamic catastrophe andrescue events. This makes the ring length and the ratioof GDP/GTP bound monomers in the ring a fluctuat-ing parameter, which makes calculating p ( t ) and p ( t )impossible. Hence, it is not possible to solve the chem-ical master equations exactly. One can solve the equa-tions exactly only in the trivial case where hydrolysis ismuch lower than the polymerization rate and hence theprobability p ( t ) and p ( t ) can be taken to be 1 and 0 respectively. In this case one gets a gaussian distribu-tion for the ring length distribution. We hence simulatethe system using Gillespie algorithm. We will comparethe mechanisms of ring opening i.e random breakingand non-random breaking by comparing the propertiesof experimentally possible observables like ring lengthand ring life time. We fix depolymerisation rate to be1 (as we can always fix one of the rates to be one) andtake N = 120 and σ = 15. C is chosen such that C σ √ π is 10. We will now study the effect of polymerisa-tion,hydrolysis and ring breaking rate on the dynamicsof the ring.In Figure 2(left) we plot length of FtsZ polymeras a function of time from a typical run for differenthydrolysis rates for random breaking of FtsZ rings. Wefix the random breaking rate to be δ = 0 . h . After theinitial growth phase, the polymer makes a transition toa phase where the length gets stabilised for a period.This is the phase where FtsZ ring exists. It is clearfrom the trajectories that FtsZ ring is dynamic as thelength fluctuates with time. The third region is wherethe polymer either grows an open chain or depolymerisecompletely.When we consider random breaking, ring can breakat any of the interface. Since h = 0, there is no hydrol-ysis taking place, there are no GDP bound monomersin the polymer. So when the ring opens up, it cannotdepolymerise. Hence, after a short period of dynamicring formation and breaking, the chain grows as a lin-ear chain. When the hydrolysis rate is not zero, butsmall there is finite probability to encounter a GDPbound monomer at either of the ends whenever the ringopens up. In this case, depolymerisation occurs beforethe chain start polymerising again resulting in a largerlifetime for the dynamic ring. As the hydrolysis rate in-creases, the probabilty of finding GDP bound monomerincreases. In the case where GDP bound monomers arepresent at the polymerizing end, all of them need toget depolymerized to expose a GTP bound monomerfor the polymer to grow again. We find that this resultsin rescue and catastrophe events in which the polymermay lose as many as 20-30 monomers before recoveringagain. Additionally, the opening and closing dynam-ics ensures that we have a lot many interfaces in thiscase(see Fig. 3(a)). Finally, when the polymer has nomore GTP bound monomers and only has GDP boundmonomers there are no more rescue events and the poly-mer depolymerizes completely. Thus at very low hydrol- , Fig. 2
Trajectories for Random(left) and Non-RandomBreaking(right) for polymerization rate 2.0, 4.0 and 8.0 re-spectively(top to bottom)
Fig. 3
Average number of D-T interfaces for random(a) andnon-random(b) breaking for polymerisation rate λ = 4. ysis rates the polymer forms a ring for a short time andthen escapes. We find that as hydrolysis rate increases,initially the ring life time increases. Because of interplaybetween polymerization, depolymerization, ring closingand opening, ring life time peaks at a finite value of h and beyond that stability of ring keeps on decreasing.In Figure 2(right), we plot the length of FtsZ poly-mers in the case of nonrandom breaking; i.e. the ringopens only at the interface formed by atleast one GDPbound monomer, for different hydrolysis rates. In thezero hydrolysis case once the ring gets formed it willnever open as there is no GDP bound interface present.At small values of hydrolysis rate, GTP will get hy-drolyzed after a long time and at some subsequent timeit will open. Since GDP bound monomers are presentat least at one of the ends, the chain is exposed for de-polymerisation once it opens up. Hence, chances of de-polymerization is bigger in this case in contrast to therandom breaking case. This results in the large flucta-tions seen in the trajectories compared to the case ofrandom breaking. The number of interfaces for vectorialhydrolysis remains the same, i.e, 1 throughout the timeevolution(recall that for random breaking the numberof interfaces increase with time). We have plotted theaverage number of interfaces as a function of time fordifferent hydrolysis rates in Fig. 3(b).From Figure 2, it is clear that the ring dynamics andstability is sensitive to the hydrolysis rate in the caseof random breaking of the ring and insensitive to hy-drolysis rate in the case of non-random breaking. Theseobservations can be made more quantitative in terms ofexperimentally measurable quantities like ring lifetimeand length distributions. We obtain these quantities byaveraging over 10 independent trajectories. In Fig 4(a) and 4(b) we plot distribution of ring sizein both random breaking and non-random breaking re-spectively for different hydrolysis rates. In the case ofrandom breaking the distribution gets narrower withincreasing hydrolysis rate. In ref. [12] authors studiedthe ring length distribution of FtsZ polymers in-vitro.They found that rings formed in the presence of slowlyhydrolyzing analogue, GMPCPP, resulted in broaderdistribution , which was shifted to the right. Our plotsin Fig. 4(a) are consistent with this observation. Thering length distribution gets thinner with increasing hy-drolysis rate and the peak shifts to the left, resulting indecreasing average length of the ring. In contrast, in thecase of the non-random ring breaking, except for zerohydrolysis rate, the ring length distribution remains un-changed, suggesting that the ring length is insensitiveto the changes in hydrolysis rate. In the case of zero hy-drolysis rate, the ring never opens up once it forms asthere are no GTP-GDP interfaces, and the distributionis just proportional to the ring closing rate ν .In fig 4(c) and 4(d) we plot the average length inthe two cases. As the hydrolysis rates increased, theaverage length of the polymer was found to decreasegradually in the case of random breaking. In the case ofnon-random breaking, the average length remains sameexcept for the case of zero hydrolysis rate. This suggeststhat the observation that the dynamics of FtsZ poly-mer is insensitive to the hydrolysis rate in the case ofnon-random breaking, made based on figure 2(b) is in-deed true. Near zero hydrolysis rate, the average lengthof ring, depends only on the polymerisation rate as thenumber of GDP-bound monomers are very less and de-polymerisation does not take place often. So the averagelength of the ring increases as the rate of polymerizationincreases. It is known that the dynamic ring stage of FtsZ polymeris very stable. The filament undergoes many catastro-phe and rescue event during this time, before eventuallydecaying. Hence, one of the most important observableis the total time for which the ring exists before it even-tually escapes the ring formation and grows as a linearchain or decays completely. Average time for this irre-versible ring opening has been plotted in Figures 5(a)and 5(b) for random and non-random ring breakingrespectively. For random breaking it peaks at a finitevalue of hydrolysis rate. This is because at very low val-ues of hydrolysis the polymer escapes into growth phaseand at large value of hydrolysis, the chances of ring
Fig. 4 a) and b): Plot of distribution of ring length for poly-merization rate 4.0 for random and non-random ring break-ing respectively. c) and d): Plot of average length for differentpolymerization rates, for random and non-random breakingrespectively closing again is small. Hence there is a narrow windowof hydrolysis rate where the dynamic ring is most sta-ble. This is consistent with the very recent experimentsof Ramirez-Diaz et al [11] where they found that finetuning of hydrolysis was needed to get stable dynamicrings. The stability of individual filaments which wasobserved to be very sensitive to the hydrolysis rate maybe a major contributing factor to ring stability observedin experiments. Moreover, we observe that the region ofmaximum stability occurs roughly at a fixed ratio of hy-drolysis to polymerization rates. It has been observedthat in the presence of a GTP regeneration system,the dynamic ring can continue for a long time[55]. Thisalong with the fact that FtsZ remains highly conservedacross species[61], can be used to argue that FtsZ sits
Fig. 5
Average lifetime( τ ) for irreversible opening for ran-dom(a) and non-random(b) breaking. Here τ is the time forirreversible opening. on the edge of stability in its naturally occurring form.This is clearly demonstrated in the plots for randomring opening in which, across all polymerization ratesthe region of maximum stability happens at a fixed ra-tio of hydrolysis and polymerization rates. In contrastto the case of random ring opening, for non-randombreaking the ring is not really dynamic and hence thelifetime goes down monotonically with the hydrolysisrate. We can also look at the distribution of time elapsedbetween two openings. We expect that there would bemany small events where the ring opens with GTP atthe growing end and close back immediately. Figure6(a) and 6(b) shows these distributions for the case ofrandom and non-random ring breaking. The distribu-tions become flatter and flatter as the hydrolysis rateincreases for both cases; however the effect is more pro-nounced in the case of non-random breaking. Polymer-ization rate does not affect the lifetime of single open-ings since the polymerisation does not take place whenthe ring is closed. Such a trend can be observed in thecase of random ring opening. But in case of non-randomopening, increase in polymerization rate leads to an de-crease in average ring lifetime for the same hydrolysisrate. Hence, we conclude that in the case of non-randomopening polymerization rate affects the ring openingdynamics(see Fig. 6 (c) and 6(d)).
Fig. 6 a) and b): Plot of distribution of ring lifetime for poly-merization rate 4.0 for random and non random ring break-ing respectively. c) and d): Plot of average ring lifetime as afunction of hydrolysis for random and non-random breakingrespectively for different polymerisation rate.
From our observations on the average time for irre-versible opening, it is clear that most stable ring isformed by fine tuning the hydrolysis rate in the case ofvectorial hydrolysis with random breaking. We call thisvalue of hydrolysis rate h critical , and find that it changeswith the polymerization rate(see Fig 7). h critical is thevalue where the life time distribution has peak in theFig 4. Fig. 7
Phase diagram for random ring opening h = 0 now.We have seen in the previous section, the ring break-ing naturally provides a way to build rescue and catas-trophe in the FtsZ ring. For the sake of completion,we explore FtsZ dynamics assuming SH in this section.Fig 11 shows trajectory with both random breaking andnon-random breaking. In the case of random breaking,there are now two ways to introduce interfaces in the Fig. 8
Average lengths for different ring breaking rates fordifferent values of hydrolysis rates for polymerization rates1.0(a) and 4.0 (b) for random breaking
Fig. 9
Average ring-life time for single break for differentring breaking rates for different values of hydrolysis rates forpolymerization rates 1.0 (a) and 4.0 (b) for random breaking system. As expected, the ring is more dynamic in thiscase and at the critical value of the hydrolysis rate h ,the ring stays stable for a very long time with regularopening and closing of the ring. In the case of non-random breaking the ring is similar, but the ring is notas stable as in the case of SH with random breaking. Fig. 10
Average lifetime for irreversible opening for poly-merisation rates 1.0 (a) and 4.0 (b).
We looked at the ring length distribution in thiscase. We find that the ring length distribution is sensi-tive to change of hydrolysis rate for both , random andnon-random breaking(see Fig. 12). Though the non-random breaking case is less sensitive to the hydrolysisrate(especially for smaller polymerization rate).We also looked at the time for irreversible ring open-ing(Fig. 13) and interestingly we find that unlike thecase of vectorial hydrolysis, the average lifetime is in-sensitive to hydrolysis rate for a wide range of hydrol-ysis rate values.
The dynamics of FtsZ ring is being very actively stud-ied in experiments in recent times[6,7,8,9]. It is still notpossible to observe the microscopic dynamics and onecan at best study the average quantities like the aver-age lifetime and length of the ring [12]. In this paper,we have introduced a stochastic model which consid-ers all the essential processes and compares all possiblescenarios. This is important as the nature of hydrolysisand ring breaking is still not understood, even thoughthere is a general consensus that hydrolysis and tread-milling of the FtsZ polymer plays a crucial role of defin-ing its dynamics and functionality. Our analysis givesclear differences in the trend of measurable quantities.Note that the dynamic nature of the ring cannot be cap-tured effectively in a deterministic model. Our stochas-tic model naturally gives rise to catastrophe and rescue Fig. 11
Trajectories for random (left) and non-randombreaking (right) for polymerization rate 2.0, 4.0 and 8.0 re-spectively (top to bottom)
Fig. 12
Distribution of ring length in ring form for polymer-ization rate 4.0 for random(a) and non-random (b) breaking.
Fig. 13
Average lifetime for irreversible opening for random(a) and non-random (b) breaking of FtsZ ring, which have been observed in experiments.We find that unlike microtubules and actin the natureof hydrolysis does not play a crucial role for FtsZ ring.For microtubules and actin if one considers vectorial hy-drolysis , then one has to introduce rescues by hand forthe polymer to be dynamic. We find that the rescuesand catastrophe events are built in our model due tothe ring topology irrespective of the hydrolysis mecha-nism, suggesting ring breaking as a natural mechanismfor introducing dynamics in the ring. This may be the reason behind the fragmented nature of Z-ring observedin in vivo experiments.In the case of vectorial hydrolysis with random ringbreaking we could qualitatively reproduce the knownbehavior of the Z -ring. We found that increasing the hy-drolysis rate in this case narrows the ring length distri-bution and the average length of the ring decreases withhydrolysis. This is consistent with the invitro studiesof Z-ring in [12]. We find that the time lapsed betweenring formation and final decay(time for irreversible ringopening) changes non-trivially with the hydrolysis rate.One needs to fine tune hydrolysis rate to achieve themost stable ring in this case. In contrast, when we con-sider non-random ring breaking, the behavior of thering is insensitive to the hydrolysis rate. Initially thepolymer has only one interface where the hydrolysisoccurs. But after ring closure, we find that in the caseof random ring breaking, many new interfaces showup, and hence the ring can potentially break up atmultiple places, consistent with the recent experiments[9]. Hence even though we consider a closed ring, inthe case of random breaking, the ring actually has apatchy structure. In contrast in the case of non-randomring breaking, the number of interfaces do not changewith time and one typically has only one interface inthe Z -ring. All these observations suggests that Z -ringbreaks randomly. Similar conclusion was also reachedby [12]. Recent works suggest that treadmilling of Z -ring is sufficient to generate the constriction force gen-erated by Z -ring during cell division. The ring withrandom breaking is also contractile, as shown in Sec-tion 3.2.We have tried to keep our model basic so that wecould distinguish the effect of different random pro-cesses clearly. The model can be studied by includingmany other processes like allowing the attachment/detachmentof short polymer and not just a single monomer dur-ing polymerization/ depolymerization process. We findthat this does not change the picture qualitatively andhence not needed to understand the basic dynamics ofFtsZ polymer. For the sake of completion we also lookedat the ring dynamics with stochastic hydrolysis. In thiscase we find that for both ring breaking mechanisms(random and non-random), we do not need to fine tunethe hydrolysis rate to achieve the stable dynamic ring.Recent experiments [11] suggest that fine tuning of hy-drolysis rate is required for stable dynamic ring. Be-haviour of ring life time as a function of hydrolysis ratecan hence be a distinguishing feature between vecto-rial and stochastic hydrolysis. Hence, differences in thestatistics of the ring can be used to unravel the actualmechanism of hydrolysis in the FtsZ polymer. We find that the randomness makes the ring moredynamic and stable, as seen in the case of stochastichydrolysis and random breaking. In general it is nowestablished that the stochasticity increases the stabil-ity of biological processes [68] and is important to un-derstand the functioning of biological systems. Thoughthe Z-ring has been studied by others using determinis-tic models, this is the first stochastic model for Z-ring.We find that the model is able to explain a number offeatures of FtsZ filament dynamics in vitro , elucidatingthe crucial role the ring topology plays in their func-tioning. The model successfully brings out the role ofhydrolysis and ring breaking mechanisms in forming acontractile dynamic ring. The model is simple and infuture can easily be used for more quantitative studiesor involve more complex interactions to study the invivo dynamics of Z-ring. Acknowledgements
The authors thank Dr. R. Sreenivasan and Dr. S. Roy-chowdhury for fruitful discussions. This work has beensupported by Department of Atomic Energy, India tho-rugh the 12th plan project (12-R&D-NIS-5.02-0100).
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