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PlusTipTracker: Quantitative image analysis software for the measurement of microtubule dynamics
Abstract
Here we introduce plusTipTracker, a Matlab-based open source software package that combines automated tracking, data analysis, and visualization tools for movies of fluorescently-labeled microtubule (MT) plus end binding proteins (+TIPs). Although +TIPs mark only phases of MT growth, the plusTipTracker software allows inference of additional MT dynamics, including phases of pause and shrinkage, by linking collinear, sequential growth tracks. The algorithm underlying the reconstruction of full MT trajectories relies on the spatially and temporally global tracking framework described in Jaqaman et al. (2008). Post-processing of track populations yields a wealth of quantitative phenotypic information about MT network architecture that can be explored using several visualization modalities and bioinformatics tools included in plusTipTracker. Graphical user interfaces enable novice Matlab users to track thousands of MTs in minutes. In this paper, we describe the algorithms used by plusTipTracker and show how the package can be used to study regional differences in the relative proportion of MT subpopulations within a single cell. The strategy of grouping +TIP growth tracks for the analysis of MT dynamics has been introduced before (Matov et al., 2010). The numerical methods and analytical functionality incorporated in plusTipTracker substantially advance this previous work in terms of flexibility and robustness. To illustrate the enhanced performance of the new software we thus compare computer-assembled +TIP-marked trajectories to manually-traced MT trajectories from the same movie used in Matov et al. (2010).
... To assay the dynamics of these microtubules, we imaged cells stably expressing EB3-mEGFP at different phases of mitosis staged by chromosome morphology (Fig. 3B). We tracked microtubule tips using the plusTipTracker software to quantify microtubule dynamics (Applegate et al., 2011). In control cells, this analysis revealed dynamics consistent with data published earlier (Ferenz and Wadsworth, 2007), showing an increase in plus-tip velocities as cells progress from G2 to prophase and then to prometaphase (Fig. 3B). ...
... For measuring Eevee-spCDK biosensor FRET, the same Knime workflow was used with some modifications, where the YFP and CFP mean fluorescence intensities were measured, background subtracted, and divided by each other at every time frame for each nucleus. For quantifying microtubule dynamics using EB3 comets, we used the uTrack software (Jaqaman et al., 2008) and its PlusTipTracker package (Applegate et al., 2011) in MatLab (MathWorks Inc., Version 9.11 (R2021b)). The 2D microtubule plus-ends object tracking was used for comet detection with a maximum of 4 frames gap-closing. ...
Polo-like kinase 1 (plk1) is a conserved regulator of cell division. During prophase, plk1 phosphorylates direct substrates and is involved in activation of the cyclin-dependent kinase 1 (cdk1). How-ever, the exact functions of plk1 in prophase remain incompletely understood. By testing several cell lines and small-molecule inhibitors, we confirm that plk1 inhibition causes a delay in mitotic entry. We show that cells are delayed in a prophase-like state displaying progressively condensing chromosomes, increased microtubule dynamics, reorganization of the actin cortex, while the nu-clear envelope remains intact. We show that during this prolonged prophase cdk1 activity increases gradually over several hours with individual cells stochastically reaching the entry threshold, ex-plaining the highly variable timing of mitotic entry. We then use phosphoproteomics to characterize this prolonged prophase state revealing for the first time phosphosites specific to prophase including several regulators of chromatin organization and the cytoskeleton. Together, we show that plk1 functions as a catalyst of prophase to prometaphase transition, and by using plk1 inhibition as a tool, we identify early changes in the phosphoproteome as the cell prepares for division.
... To analyze EB1-GFP comet velocity, which was used as a proxy for the microtubule growth rate, we employed analysis software from the Danuser lab (Applegate et al., 2011). In short, we collected multiple 100 by 100-pixel movies of LLC-Pk1 cells treated with DMSO or transfected with DARPin from three separate biological replicates and loaded them into the Danuser code software, using constant parameters for thresholding and water shedding. ...
EB1 is a key cellular protein that delivers regulatory molecules throughout the cell via the tip-tracking of growing microtubule plus-ends. Thus, it is important to understand the mechanism for how EB1 efficiently tracks growing microtubule plus-ends. It is widely accepted that EB1 binds with higher affinity to GTP-tubulin subunits at the growing microtubule tip, relative to GDP-tubulin along the microtubule length. However, it is unclear whether this difference in affinity alone is sufficient to explain the tip-tracking of EB1 at growing microtubule tips. Previously, we found that EB1 binds to exposed microtubule protofilament-edge sites at a ~70 fold faster rate than to closed-lattice sites, due to diffusional steric hindrance to binding. Thus, we asked whether rapid protofilament-edge binding could contribute to efficient EB1 tip tracking. A computational simulation with differential EB1 on-rates based on closed-lattice or protofilament-edge binding, and with EB1 off-rates that were dependent on the tubulin hydrolysis state, robustly recapitulated experimental EB1 tip tracking. To test this model, we used cell-free biophysical assays, as well as live-cell imaging, in combination with a Designed Ankyrin Repeat Protein (DARPin) that binds exclusively to protofilament-edge sites, and whose binding site partially overlaps with the EB1 binding site. We found that DARPin blocked EB1 protofilament-edge binding, which led to a decrease in EB1 tip tracking on dynamic microtubules. We conclude that rapid EB1 binding to microtubule protofilament-edge sites contributes to robust EB1 tip tracking at the growing microtubule plus-end.
... To achieve subpixel resolution, a 2-dimensional Gaussian function was fitted to every peak. Then, the particles were tracked with uTrack 2.3 [100][101][102] . The derivative of the position vector was calculated to determine the instantaneous velocity of the particles, which was subsequently filtered with a symmetric median filter of ±15 s (which amounts to 31 subsequent frames) to reduce positioning noise. ...
Due to the abundance of microplastics in the environment, research about its possible adverse effects is increasing exponentially. Most studies investigating the effect of microplastics on cells still rely on commercially available polystyrene microspheres. However, the choice of these model microplastic particles can affect the outcome of the studies, as even nominally identical model microplastics may interact differently with cells due to different surface properties such as the surface charge. Here, we show that nominally identical polystyrene microspheres from eight different manufacturers significantly differ in their ζ-potential, which is the electrical potential of a particle in a medium at its slipping plane. The ζ-potential of the polystyrene particles is additionally altered after environmental exposure. We developed a microfluidic microscopy platform to demonstrate that the ζ-potential determines particle-cell adhesion strength. Furthermore, we find that due to this effect, the ζ-potential also strongly determines the internalization of the microplastic particles into cells. Therefore, the ζ-potential can act as a proxy of microplastic-cell interactions and may govern adverse effects reported in various organisms exposed to microplastics.
... To segment and track subcellular structures such as actin filaments or microtubules, many tools are available, such as plug-ins in Fiji (Ershov et al., 2022;Kapoor et al., 2019;Tinevez et al., 2017), ilastik (Berg et al., 2019), Matlab (Applegate et al., 2011) or Imaris (https://imaris.oxinst. com/packages). ...
Understanding complex living systems, which are fundamentally constrained by physical phenomena, requires combining experimental data with theoretical physical and mathematical models. To develop such models, collaborations between experimental cell biologists and theoreticians are increasingly important but these two groups often face challenges achieving mutual understanding. To help navigate these challenges, this Perspective discusses different modelling approaches, including bottom-up hypothesis-driven and top-down data-driven models, and highlights their strengths and applications. Using cell mechanics as an example, we explore the integration of specific physical models with experimental data from the molecular, cellular and tissue level up to multiscale input. We also emphasize the importance of constraining model complexity and outline strategies for crosstalk between experimental design and model development. Furthermore, we highlight how physical models can provide conceptual insights and produce unifying and generalizable frameworks for biological phenomena. Overall, this Perspective aims to promote fruitful collaborations that advance our understanding of complex biological systems.
... Variations in these genes' expression levels have correlated to the effects of specific MTAs on cells and whether or not a drug is able to kill the cancer cells [29][30][31][32][33][34][35][36][37] . We will use a comprehensive approach to measure MT dynamic signatures, a panel of 12 parameters that comprehensively describe MT dynamics 1,38 , and correlate changes in the signatures that result from treatment with chemotherapeutics. Traditionally, MT dynamics have been evaluated by means of manual tracking (recording x,y coordinates in data files for each frame over the time-lapse sequence) or kymograph analysis, where a band of pixels around the desired MT end is stitched onto a single panel thus aggregating the evolution of pixel intensities over the time-lapse sequence. ...
(See also the 8 PDF files appended below) The treatment of prostate cancer (PCa) has been impeded by the lack of clinically relevant disease models. There are only three PCa cell lines as available models of this heterogeneous disease out of the over 1,000 publicly available cancer cell lines; patient-derived
xenografts (PDX) have proven rarely possible to establish thus far. My previous work in PCa cell lines has demonstrated that we can associate the alterations in a particular parameter of microtubule (MT) dynamics to taxane resistance in ERG overexpressing cells. MT regulating genes such as chTOG, CLIP170, MAP4, MCAK, TPX2 and Op18 to name a few are implicated in the process of conferring drug resistance and susceptibility. Because these genes are also involved in the regulation of MT dynamics, my central hypothesis is that computational analysis of MT dynamics (MT signature) can serve as a real-time readout of cell susceptibility to drug action, which will allow to discover mechanisms of resistance. Therefore, to test my hypotheses in patient-derived cells, I propose to perform two specific aims in patient-derived PCa organoids: Specific Aim 1: Define and validate cancer-specific MT dynamics signatures of MT-targeting agents (MTA) resistance. I will image and measure endogenous and drug-induced signatures (consisting of 12 descriptive parameters) of MT dynamics alterations in patient-
derived cancer organoids using ClusterTrack. For each organoid type, I will measure populations of cells before and after treatment with MTAs, and will perform statistical analysis. I will use, as a
starting point, samples with known patient treatment outcome for specific MTAs (Docetaxel, Cabazitaxel, Eribulin, etc.), which will allow me to validate the MT signatures and classify them into
sensitive and resistant groups using clustering. Specific Aim 2: Define cancer-specific molecular mechanisms of MTA resistance to elucidate targeting strategies. I will extract RNA and will perform differential expression analysis of MT regulators in patient-derived cancer organoids as well as pathway analysis using iPAGE. I will correlate the RNA-seq MT signatures to the validated MT dynamics signatures obtained in Aim 1 using pattern matching. Correlated signatures will contain multiple up- or down-regulated MT regulator genes. I will test the proteins in these lists as candidates for personalized targets in knockdown and overexpression experiments. This approach could identify novel candidates for effective targeted therapies in metastatic disease within the MT-interacting transcriptome. I will further test these novel organoids into PDX models of PCa to confirm sensitization and therapeutic susceptibility in vivo. Further validation, of both prognostic and predictive markers, will be done by comparing ex vivo tumor growth kinetics and treatment predictions to outcomes of patients with similar genetic profiles, after drug treatment, enrolled in clinical trials. See: github.com/amatov/SegmentationBiomarkerCTC,
github.com/amatov/AntibodyTextureMorphology,
github.com/amatov/DataSetTracker
Microtubules regulate cell polarity and migration via local activation of focal adhesion turnover, but the mechanism of this process is insufficiently understood. Molecular complexes containing KANK family proteins connect microtubules with talin, the major component of focal adhesions. Here, local optogenetic activation of KANK1-mediated microtubule/talin linkage promoted microtubule targeting to an individual focal adhesion and subsequent withdrawal, resulting in focal adhesion centripetal sliding and rapid disassembly. This sliding is preceded by a local increase of traction force due to accumulation of myosin-II and actin in the proximity of the focal adhesion. Knockdown of the Rho activator GEF-H1 prevented development of traction force and abolished sliding and disassembly of focal adhesions upon KANK1 activation. Other players participating in microtubule-driven, KANK-dependent focal adhesion disassembly include kinases ROCK, PAK, and FAK, as well as microtubules/focal adhesion-associated proteins kinesin-1, APC, and αTAT. Based on these data, we develop a mathematical model for a microtubule-driven focal adhesion disruption involving local GEF-H1/RhoA/ROCK-dependent activation of contractility, which is consistent with experimental data.
We describe u-track3D, a software package that extends the versatile u-track framework established in 2D to address the specific challenges of 3D particle tracking. First, we present the performance of the new package in quantifying a variety of intracellular dynamics imaged by multiple 3D microcopy platforms and on the standard 3D test dataset of the particle tracking challenge. These analyses indicate that u-track3D presents a tracking solution that is competitive to both conventional and deep-learning-based approaches. We then present the concept of dynamic region of interest (dynROI), which allows an experimenter to interact with dynamic 3D processes in 2D views amenable to visual inspection. Third, we present an estimator of trackability that automatically defines a score for every trajectory, thereby overcoming the challenges of trajectory validation by visual inspection. With these combined strategies, u-track3D provides a complete framework for unbiased studies of molecular processes in complex volumetric sequences.
Bu çalışmada basketbol branşında aktif olarak alt yapılarda müsabakalara katılan sporcuların branşa yönelik tutum ve beslenme alışkanlıklarının incelenmesi amaçlanmaktadır. Araştırma evreni Türkiye basketbol alt yapı liglerinde basketbol oynayan sporcular olurken örneklem grubu ise rasgele seçilmiş gönüllülük esasına göre katılmak isteyen 378 kişiden oluşmaktadır. Elde edilen bulguların İstatiksel analizde SPSS 22 programı kullanılmıştır. Örneklem büyüklüğü tespiti için ise G*Power (1.3.9.4) Araştırmada iki farklı ölçek kullanılmıştır. Öncü ve ark. (2015) tarafından geliştirilen 21 soru ve Algı, İlgi alt boyutlarından oluşan Basketbol Tutum Ölçeği (BTÖ) kullanılmıştır. Demir ve Cicioğlu (2019) birlikte geliştirmiş oldukları Sağlıklı Beslenmeye İlişkin Tutum Ölçeği (SBYİTÖ) uygulanmıştır. Ölçekte ise; 4 alt boyuttan beslenme hakkında bilgi (BHB), beslenmeye yönelik davranış (BYD), kötü beslenme (KB) olumlu beslenme (OB) olmak üzere toplam 21 sorudan oluşmaktadır Araştırmada tanımlayıcı istatistiklerin yanında bağımsız örneklem t test ve Man-Wtitney U, ikiden fazla ortalamaların karşılaştırılması ise tek yönlü varyans analizi testi ve Kruskal- Wallis testleri kullanılarak gerçekleşmiştir. Çalışma sonunda cinsiyet değişkeni skorlarına göre BYO, KB, Algı; okul türüne göre BHB, İlgi; Aile gelir durumuna göre BHB, BYD, İlgi, Algı alt boyutları bakımından anlamlı farklılaşma olduğu bulgusuna ulaşılmıştır.
Several microtubule binding proteins, including CLIP-170 (cytoplasmic linker protein-170), CLIP-115, and EB1 (end-binding protein 1), have been shown to associate specifically with the ends of growing microtubules in non-neuronal cells, thereby regulating microtubule dynamics and the binding of microtubules to protein complexes, organelles, and membranes. When fused to GFP (green fluorescent protein), these proteins, which collectively are called +TIPs (plus end tracking proteins), also serve as powerful markers for visualizing microtubule growth events. Here we demonstrate that endogenous +TIPs are present at distal ends of microtubules in fixed neurons. Using EB3-GFP as a marker of microtubule growth in live cells, we subsequently analyze microtubule dynamics in neurons. Our results indicate that microtubules grow slower in neurons than in glia and COS-1 cells. The average speed and length of EB3-GFP movements are comparable in cell bodies, dendrites, axons, and growth cones. In the proximal region of differentiated dendrites ∼65% of EB3-GFP movements are directed toward the distal end, whereas 35% are directed toward the cell body. In more distal dendritic regions and in axons most EB3-GFP dots move toward the growth cone. This difference in directionality of EB3-GFP movements in dendrites and axons reflects the highly specific microtubule organization in neurons. Together, these results suggest that local microtubule polymerization contributes to the formation of the microtubule network in all neuronal compartments. We propose that similar mechanisms underlie the specific association of CLIPs and EB1-related proteins with the ends of growing microtubules in non-neuronal and neuronal cells.
By co-injecting fluorescent tubulin and vinculin into fish fibroblasts we have revealed a “cross talk” between microtubules and early sites of substrate contact. This mutuality was first indicated by the targeting of vinculin-rich foci by microtubules during their growth towards the cell periphery. In addition to passing directly over contact sites, the ends of single microtubules could be observed to target several contacts in succession or the same contact repetitively, with intermittent withdrawals. Targeting sometimes involved side-stepping, or the major re-routing of a microtubule, indicative of a guided, rather than a random process. The paths that microtubules followed into contacts were unrelated to the orientation of stress fiber assemblies and targeting occurred also in mouse fibroblasts that lacked a system of intermediate filaments. Further experiments with microtubule inhibitors showed that adhesion foci can: (a) capture microtubules and stabilize them against disassembly by nocodazole; and (b), act as preferred sites of microtubule polymerization, during either early recovery from nocodazole, or brief treatment with taxol. From these and other findings we speculate that microtubules are guided into substrate contact sites and through the motor-dependent delivery of signaling molecules serve to modulate their development. It is further proposed this modulation provides the route whereby microtubules exert their influence on cell shape and polarity.
Microtubule dynamics were investigated in CHO and NRK cells by novel experimental approaches designed to evaluate the microtubule behavior in the cell interior. These approaches were: (1) laser photobleaching of a path through the centrosome; (2) direct observation of microtubules in centrosome-containing cytoplasts; (3) GFP-CLIP-170 expression as a marker for microtubule plus end growth; and (iv) sequential subtraction analysis. The combination of these approaches allowed us to obtain data where the density of microtubules had previously prevented conventional methods to be applicable.
In the steady state, nascent microtubules grew persistently from the centrosome towards the cell margin. Frequently, they arrived at the cell margin without undergoing any transition to the shortening phase. In contrast to the growth of microtubules, shortening of the plus ends from the periphery was non-persistent; that is, rescue was frequent and the extent of shortening showed a distribution of lengths reflecting a stochastic process. The combination of persistent growth and a cell boundary led to a difference in apparent microtubule behavior in the cell interior compared with that near the cell margin. Whereas microtubules in the cell interior showed asymmetric transition frequencies, their behavior near the cell margin showed frequent fluctuations between phases of shortening and growth. Complete microtubule turnover was accomplished by the relatively rare episodes of shortening back to the centrosome. Release from the centrosome with subsequent minus end shortening also occurred but was a minor mechanism for microtubule turnover compared with the plus end pathway.
We propose a life cycle for a microtubule which consists of rapid growth from the centrosome to the cell margin followed by an indefinite period of fluctuations of phases of shortening and growth. We suggest that persistent growth and asymmetric transition frequencies serve the biological function of providing a mechanism by which microtubules may rapidly accommodate to the changing shape and advancing edge of motile cells.
Migrating cells display a characteristic polarization of the actin cytoskeleton. Actin filaments polymerise in the protruding front of the cell whereas actin filament bundles contract in the cell body, which results in retraction of the cell’s rear. The dynamic organization of the actin cytoskeleton provides the force for cell motility and is regulated by small GTPases of the Rho family, in particular Rac1, RhoA and Cdc42. Although the microtubule cytoskeleton is also polarized in a migrating cell, and microtubules are essential for the directed migration of many cell types, their role in cell motility is not well understood at a molecular level. Here, we discuss the potential molecular mechanisms for interplay of microtubules, actin and Rho GTPase signalling in cell polarization and motility. Recent evidence suggests that microtubules locally modulate the activity of Rho GTPases and, conversely, Rho GTPases might be responsible for the initial polarization of the microtubule cytoskeleton. Thus, microtubules might be part of a positive feedback mechanism that maintains the stable polarization of a directionally migrating cell.
Microtubules are known to be required for locomotion of mammalian cells, and recent experiments demonstrate that suppression of microtubule dynamic turnover reduces the rate of cell motility and induces wandering of growth cones [Liao et al., 1995: J Cell Sci. 108:3473–3483; Tanaka et al., 1995: J Cell Biol. 128:139–155]. To determine how microtubule dynamic instability behavior contributes to directed cell locomotion, the behavior of individual microtubules has been directly observed and quantified at leading and lateral edges of hepatocyte growth factor-treated motile cells. Microtubules extended into newly formed protrusions at the leading edge; these “pioneer” microtubules [Waterman-Storer and Salmon, 1997: J Cell Biol. 139:417–434] showed persistent growth when compared with microtubules in non-leading, lateral edges. The percentage of total observation time spent in the growth phase was 68.2% at the leading edge compared with 32.0% in non-leading edges, and net microtubule elongation was observed in lamellipodia at the leading edge. The frequency of catastrophe transitions was threefold greater and the average number of transitions/microtubule/min was twofold greater in non-leading edges, as compared with the leading edge. These observations demonstrate that pioneer microtubules that enter newly formed lamellipodia at the leading edge of motile cells are characterized by persistent growth excursions, and directly demonstrate that the frequency of catastrophe transitions can be regionally regulated in polarized motile cells. The data indicate that region specific differences in the organization and dynamics of actin filaments may regulate microtubule dynamic instability behavior in vivo. Cell Motil. Cytoskeleton 42:48–59, 1999.
This paper aims at describing the state of the art on linear assignment problems (LAPs). Besides sum LAPs it discusses also problems with other objective functions like the bottleneck LAP, the lexicographic LAP, and the more general algebraic LAP. We consider different aspects of assignment problems, starting with the assignment polytope and the relationship between assignment and matching problems, and focusing then on deterministic and randomized algorithms, parallel approaches, and the asymptotic behaviour. Further, we describe different applications of assignment problems, ranging from the well know personnel assignment or assignment of jobs to parallel machines, to less known applications, e.g. tracking of moving objects in the space. Finally, planar and axial three-dimensional assignment problems are considered, and polyhedral results, as well as algorithms for these problems or their special cases are discussed. The paper will appear in the Handbook of Combinatorial Optimization to be published
EB1 targets to polymerizing microtubule ends, where it is favorably positioned to regulate microtubule polymerization and confer molecular recognition of the microtubule end. In this study, we focus on two aspects of the EB1–microtubule interaction: regulation of microtubule dynamics by EB1 and the mechanism of EB1 association with microtubules. Immunodepletion of EB1 from cytostatic factor-arrested M-phaseXenopus egg extracts dramatically reduced microtubule length; this was complemented by readdition of EB1. By time-lapse microscopy, EB1 increased the frequency of microtubule rescues and decreased catastrophes, resulting in increased polymerization and decreased depolymerization and pausing. Imaging of EB1 fluorescence revealed a novel structure: filamentous extensions on microtubule plus ends that appeared during microtubule pauses; loss of these extensions correlated with the abrupt onset of polymerization. Fluorescent EB1 localized to comets at the polymerizing plus ends of microtubules in cytostatic factor extracts and uniformly along the lengths of microtubules in interphase extracts. The temporal decay of EB1 fluorescence from polymerizing microtubule plus ends predicted a dissociation half-life of seconds. Fluorescence recovery after photobleaching also revealed dissociation and rebinding of EB1 to the microtubule wall with a similar half-life. EB1 targeting to microtubules is thus described by a combination of higher affinity binding to polymerizing ends and lower affinity binding along the wall, with continuous dissociation. The latter is likely to be attenuated in interphase. The highly conserved effect of EB1 on microtubule dynamics suggests it belongs to a core set of regulatory factors conserved in higher organisms, and the complex pattern of EB1 targeting to microtubules could be exploited by the cell for coordinating microtubule behaviors.
Heat-stable brain microtubule associated proteins (MAPs) and purified microtubule associated protein 2 (MAP-2) were microinjected into cultured BSC-1 cells which had been previously injected with rhodamine-labeled tubulin. The dynamic instability behavior of individual microtubules was then examined using low-light-level fluorescence microscopy and quantitative microtubule tracking methods. Both MAP preparations suppressed microtubule dynamics in vivo, by reducing the average rate and extent of both growing and shortening events. The average duration of growing events was not affected. When measured as events/unit time, heat-stable MAPs and MAP-2 did not significantly alter the frequency of rescue; the frequency of catastrophe was decreased approximately two-fold by heat-stable MAPs and MAP-2. When transition frequencies were calculated as events/unit distance, both MAP preparations increased the frequency of rescue, without altering the frequency of catastrophe. The percentage of total time spent in the phases of growth, shrink and pause was determined. Both MAP-2 and heat-stable MAPs decreased the percentage of time spent shortening, increased the percentage of time spent paused, and had no effect on percentage of time spent growing. Heat-stable MAPs increased the average pause duration, decreased the average number of events per minute per microtubule and increased the probability that a paused microtubule would switch to growing rather than shortening. The results demonstrate that addition of MAPs to living cells reduces the dynamic behavior of individual microtubules primarily by suppressing the magnitude of dynamic events and increasing the time spent in pause, where no change in the microtubule length can be detected. The results further suggest that the expression of MAPs directly contributes to cell type-specific microtubule dynamic behavior.
We develop a shortest augmenting path algorithm for the linear assignment problem. It contains new initialization routines and a special implementation of Dijkstra's shortest path method. For both dense and sparse problems computational experiments show this algorithm to be uniformly faster than the best algorithms from the literature. A Pascal implementation is presented.Wir entwickeln einen Algorithmus mit krzesten alternierenden Wegen fr das lineare Zuordnungsproblem. Er enthlt neue Routinen fr die Anfangswerte und eine spezielle Implementierung der Krzesten-Wege-Methode von Dijkstra. Sowohl fr dichte als auch fr dnne Probleme zeigen Testlufe, da unser Algorithmus gleichmig schneller als die besten Algorithmen aus der Literatur ist. Eine Implementierung in Pascal wird angegeben.
There are two formalisms for mathematically describing the time behavior of a spatially homogeneous chemical system: The deterministic approach regards the time evolution as a continuous, wholly predictable process which is governed by a set of coupled, ordinary differential equations (the "reaction-rate equations"); the stochastic approach regards the time evolution as a kind of random-walk process which is governed by a single differential-difference equation (the "master equation"). Fairly simple kinetic theory arguments show that the stochastic formulation of chemical kinetics has a firmer physical basis than the deterministic formulation, but unfortunately the stochastic master equation is often mathematically intractable. There is, however, a way to make exact numerical calculations within the framework of the stochastic formulation without having to deal with the master equation directly. It is a relatively simple digital computer algorithm which uses a rigorously derived Monte Carlo procedure to numerically simulate the time evolution of the given chemical system. Like the master equation, this "stochastic simulation algorithm" correctly accounts for the inherent fluctuations and correlations that are necessarily ignored in the deterministic formulation. In addition, unlike most procedures for numerically solving the deterministic reaction-rate equations, this algorithm never approximates infinitesimal time increments dt by finite time steps Δt. The feasibility and utility of the simulation algorithm are demonstrated by applying it to several well-known model chemical systems, including the Lotka model, the Brusselator, and the Oregonator.