Open AcceResearch articleA general modeling and visualization tool for comparing different members of a group: application to studying tau-mediated regulation of microtubule dynamicsArnab Bhattacharya*1,2, Sasha Levy3, Adria LeBoeuf4, Michelle Gaylord5, Leslie Wilson5, Ambuj K Singh2 and Stuart C Feinstein5
Address: 1Department of Computer Science and Engineering, Indian Institute of Technology (I.I.T.), Kanpur, India, 2Department of Computer Science, University of California at Santa Barbara, Santa Barbara, CA, USA, 3Center for Comparative Functional Genomics, New York University, New York, NY, USA, 4Howard Hughes Medical Institute and Laboratory of Sensory Neuroscience, Rockefeller University, New York, NY, USA and 5Department of Molecular, Cellular and Developmental Biology, University of California at Santa Barbara, Santa Barbara, CA, USA
AbstractBackground: Innumerable biological investigations require comparing collections of molecules, cells ororganisms to one another with respect to one or more of their properties. Almost all of thesecomparisons are performed manually, which can be susceptible to inadvertent bias as well as miss subtleeffects. The development and application of computer-assisted analytical and interpretive tools could helpaddress these issues and thereby dramatically improve these investigations.
Results: We have developed novel computer-assisted analytical and interpretive tools and applied themto recent studies examining the ability of 3-repeat and 4-repeat tau to regulate the dynamic behavior ofmicrotubules in vitro. More specifically, we have developed an automated and objective method to definegrowth, shortening and attenuation events from real time videos of dynamic microtubules, anddemonstrated its validity by comparing it to manually assessed data. Additionally, we have used the samedata to develop a general strategy of building different models of interest, computing appropriatedissimilarity functions to compare them, and embedding them on a two-dimensional plot for visualizationand easy comparison. Application of these methods to assess microtubule growth rates and growth ratedistributions established the validity of the embedding procedure and revealed non-linearity in therelationship between the tau:tubulin molar ratio and growth rate distribution.
Conclusion: This work addresses the need of the biological community for rigorously quantitative andgenerally applicable computational tools for comparative studies. The two-dimensional embedding methodretains the inherent structure of the data, and yet markedly simplifies comparison between models andparameters of different samples. Most notably, even in cases where numerous parameters exist by whichto compare the different samples, our embedding procedure provides a generally applicable computationalstrategy to detect subtle relationships between different molecules or conditions that might otherwiseescape manual analyses.
BackgroundStatement of ProblemInnumerable biological investigations require comparingdifferent members of a collection of entities with respectto one or more properties. The conclusions to be drawnfrom such studies are based on an analysis of the degreeof similarity or dissimilarity among the different mem-bers. For example, one might compare the activity of dif-ferent isoforms or fragments of a protein of interest, orcompare wild type protein(s) with various mutant ver-sions of a protein that causes a disease state. Many addi-tional examples come from comparisons of data setsderived from microarray and proteomics studies, as wellas population genetics. Given the technical advances ofrecombinant DNA technology and the explosion ingenomics over the past few years, it is a certainty that thenumber of these sorts of comparative studies, and thenumber of entities to be compared within each study, willincrease dramatically in the near future. Unfortunately,the vast majority of such comparative studies are currentlyperformed manually, with investigators searching for sim-ilarities and dissimilarities among different test entities"by eye". This is especially difficult when each member ofthe collection is being characterized by multiple criteria.The analytical process is time consuming, likely to misssubtleties and is susceptible to inadvertent bias andhuman errors. Development and application of compu-ter-assisted modeling and visualization can provideextraordinarily valuable data analyses and interpretivetools for assessing relationships among different mem-bers in a study.
Microtubules and Microtubule DynamicsMicrotubules represent one of the three main compo-nents of the eukaryotic cellular cytoskeleton [1]. They arehollow, unbranched cylinders, formed by the non-cova-lent association of αβ tubulin dimer subunits. Microtu-bules serve a wide variety of essential structural andtransport functions, including the segregation of chromo-somes during cell division and the transport of vesicularcargo up and down long axonal processes in neurons.
Microtubules are highly dynamic structures, gaining andlosing tubulin dimer subunits by a stochastic processknown as dynamic instability [2,3]. A large body of data,both pharmacological and somatic cell genetics, has led tothe conclusion that proper regulation of microtubuledynamics is essential in order for microtubules to performtheir many critical cellular functions (for review, see [4]).For example, the effectiveness of the anti-cancer drug taxolderives from its ability to suppress microtubule dynamics,thereby interfering with the ability of cancer cells to pro-liferate [5]. Given the importance of properly regulatedmicrotubule dynamics, it is not surprising that cells haveevolved a host of regulatory proteins that finely tune
microtubule dynamics, including tau, MAP2, MAP4,SCG10 and stathmin.
The Microtubule Associated Protein TauThe microtubule associated protein tau is essential for thenormal development and maintenance of the nervous sys-tem [6-8]. It binds directly to microtubules [9,10], and itsability to regulate microtubule dynamics [11-13] is itselftightly regulated by both alternative RNA splicing [14]and phosphorylation (for review, see [15]). AlternativeRNA splicing leads to the synthesis of two classes of tau,known as 3-repeat tau and 4-repeat tau (See Figure 1 for aschematic). Whereas normal human fetal brain expressesonly 3-repeat tau, adult human brain expresses approxi-mately equal amounts of 3-repeat and 4-repeat tau.Despite this dramatic developmental shift in expressionprofiles, the functional and mechanistic differencesbetween 3-repeat and 4-repeat tau remain poorly under-stood. While it is well-established that 4-repeat tau is amore potent regulator of microtubule dynamics than 3-repeat tau, there have been indications over the years thatthe two classes of tau isoforms may also have inherentqualitative differences as well [12,16-19].
Abnormal tau action has long been correlated with neuro-degeneration. Indeed, the classic neurofibrillary tanglepathology of Alzheimers and many related dementias arecomposed primarily of aberrant tau (for example, see[20]). In 1998, a direct cause and effect relationshipbetween errors in tau action and/or regulation and neuro-degeneration was established by the genetic linkagebetween mutations in the tau gene and FTDP-17, a fronto-temporal dementia with many similarities to Alzheimersdisease [21-23]. Two classes of tau mutations have beendescribed. The first collection of mutations are structuralin nature, caused by various amino acid substitutions intau. The second class of mutations are especially subtleand provocative – they are caused by errors in alternativetau RNA splicing that alter the expression ratio of other-wise normal 4-repeat and 3-repeat tau molecules. Specifi-cally, rather than a ~50/50 ratio in adult human brain, themutant ratio is closer to ~75/25. In both the structural and
Isoforms of tauFigure 1Isoforms of tau. Schematic of two classes of tau isoforms – 4-repeat tau and 3-repeat tau. In 3-repeat tau, the region between repeats 1 and 2 and the second repeat structures are missing by virtue of alternative RNA splicing.
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regulatory mutations, the result is early onset of neuronalcell death and dementia.
Unfortunately, the molecular mechanisms underlyingtau-mediated neuronal cell death remain unclear. Onewidely held model suggests that errors in tau action leadto the aggregation of tau into neurofibrillary tangles,which are in turn cytotoxic [24]. An alternative model sug-gests that tau-mediated neuronal cell death results fromthe inability of tau to properly maintain microtubuledynamics within a narrow range of activities required forcell viability [4,13,19,25]. Additional models have alsobeen proposed (see http://www.alzforum.org/res/adh/cur/default.asp).
Computational PerspectivesTo quantitatively investigate the regulation of microtu-bule dynamics under varying conditions (for example,with different tau isoforms or tau:tubulin molar ratios),cell biologists employ video microscopy to visualize andrecord images of dynamic microtubules in real time. Foreach condition being assessed, many different individualmicrotubules must be imaged, tracked and analyzed [19].From the resulting microtubule "life history plots" (Figure2), the dynamic behaviors of similarly treated microtu-bules can be determined, such as average growth or short-ening rates. Subsequently, the behavior of microtubulesunder different conditions can be compared.
Computer-assisted methods are especially attractive fortime series investigations of this sort. In the specific caseof analyzing the regulation of microtubule dynamics,inadvertent bias and non-reproducibility in data interpre-
tation among different labs and different investigatorswhen defining the beginning and end points of individualgrowth, shortening or attenuation events could becomesignificant. Despite the fact that these events are explicitlydefined, investigators must make many judgment calls. Incontrast, computer-assisted methods offer a faster andmore objective assessment of the data. More importantly,these methods can also provide analytical tools as muchas determine the fit of the data to various statistical mod-els, thereby testing various conceptual representations ofthe underlying molecular mechanisms of action of thesystem under study. Modeling can also generate testablemechanistic predictions for subsequent investigations. Ina general sense, sophisticated computational tools havethe potential to make major contributions to many areasof biological research.
ResultsThe main goal of this work is to develop general compu-tational tools to quantitatively assess the differencesamong samples of interest and to visualize those differ-ences in a manner that facilitates their comparison. Thedata being analyzed is derived from an earlier work inwhich video microscopy was used to visualize and assessthe abilities of 3-repeat and 4-repeat tau to regulate vari-ous parameters of microtubule dynamics in in vitro reac-tions (Levy et al., 2005) [19]. Samples contained purifiedtubulin dimers and purified recombinant human tau. Thetwo primary variables were (i) the presence of 3-repeattau, 4-repeat tau or no tau, and (ii) the molar ratio of tauto tubulin. In vivo, the molar ratio of tau:tubulin variesfrom cell to cell. Further, the ratio can vary even amongdifferent regions within single cells, such as cell body ver-sus axon versus growth cone. The different ratios exam-ined are likely to span the range of biologicallymeaningful values [26].
Automated Life History Plot AnalysisWe first developed an automated method to identify thedifferent events – growth, shortening, and attenuation("pause") – on microtubule life-history plots, using a setof pre-defined rules (see the Methods section for details).We then compared the ability of this automated methodto determine average microtubule growth rates withstandard manual analysis, using data sets assessing theability of tau to regulate microtubule dynamics from [19].In this earlier work, microtubule dynamics were assayedunder nine different experimental conditions. As seen inTable 1, the deviation between automatic and manuallydetermined rates ranges from 3.33% to 12.68%, with anaverage deviation of 6.59%. Statistically, the differencesbetween the manually determined values and the auto-matically measured values are not significant (as shownby the p-values), except for one condition. The p-value iscomputed by performing a t-test of the manually deter-
Example life history plotFigure 2Example life history plot. An example of a typical "life his-tory" plot of a microtubule, i.e., microtubule length as a func-tion of time plot. The microtubule shown here is from the 3-repeat tau sample at a tau:tubulin molar ratio of 1:38.
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mined growth rates against the automatically computedones for each condition (details are in the Methods sec-tion). More importantly, the relative order of the condi-tions do not change, and the degrees of separation are wellmaintained. Table 2 shows the comparison for the sameset of conditions using a different tubulin preparationfrom [19]. Again, our automated method accurately reca-pitulates manual analysis with increased objectivity. Addi-tionally, it markedly reduces the time required to conductthese investigations.
It is also important to note that inherent biological varia-bility exists in the microtubule growth rate data. Thislikely results from biochemical variations between differ-ent tubulin preparations, such as different tubulin iso-form expression ratios and/or varying degrees of post-translational modifications, such as phosphorylation,
acetylation or tyrosination. For example, assuming eachgrowth rate determination is within a variation of ± 0.1μm/min, the rank order of the conditions for each of thetwo data sets is quite similar, although the 4R-1:55 and3R-1:38 conditions are reversed (see Table 3). This inher-ent biological variability could limit the utility of somesophisticated and highly sensitive statistical models tomake testable predictions regarding mechanisms underly-ing the regulation of microtubule dynamics. At the mini-mum, multiple data sets might be necessary and allpredictions would need to be considered tentative untiltested directly by other biological means.
Modeling and Embedding StrategyNext, we sought to develop mathematical and statisticalmodels to capture different dynamic aspects of microtu-bule behavior and to embed them in a two-dimensionalspace for visualization and easy comparison of differentconditions. We used the Sammon projection method [27]for embedding and visualization. In short, the embeddingprocess displays each experimental condition with an (x,y) position; the relative distance between the (x, y) posi-tions of any pair of experimental conditions correspondsto their relative degree of relatedness (details are in theMethods section). The conditions of interest can be com-pared based on numerous parameters and the computa-tional method is applicable to all kinds of numericalparameters.
The outline of our method is as follows. First, the experi-mental measurements are analyzed based on an appropri-ate mathematical model. Then, an appropriatedissimilarity function is applied to calculate the relativedistances between the models of each pair of conditions.Finally, the conditions are embedded on a two-dimen-sional space such that the inherent structure of the data isapproximately preserved. This is achieved by assigningpoints (x and y coordinates) to the models such that theEuclidean distance between any pair of points in thisspace is as close to the original dissimilarity measurebetween their models as possible. Unlike principal com-ponent analysis (PCA) [28], this method works with anydistance matrix. The quality of the embedding is meas-ured by distortion. For ideal embeddings, where all dissim-ilarity values are maintained exactly as Euclideandistances in the embedded space, the distortion is 1. Thedetails of the models, the dissimilarity functions, theembedding algorithm, and the distortion computationsare presented in the Methods section.
Two-Dimensional Embedding AnalysisMicrotubule Growth RateAs a proof-of-principle exercise, we used the automaticallymeasured values from Table 1 and applied our embed-ding strategy to compare the abilities of each tau isoform
Table 1: Growth rates of tau conditions (sample 1)
The manual data is reproduced from Table III of (Levy et al., 2005) [19]. The third column shows growth rates (in μm/min) automatically computed from the manually tracked tips of the microtubules using an objective set of rules with no human interference. The relative ranks of the conditions remain the same.
Table 2: Growth rates of tau conditions (sample 2)
The manual data is reproduced from Table II of (Levy et al., 2005) [19]. The third column shows growth rates (in μm/min) automatically computed from the manually tracked tips of the microtubules using an objective set of rules with no human interference. The relative ranks of the conditions remain the same.
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to regulate the growth rate of microtubules. Since thesegrowth rate data are one-dimensional, the distortion is 1,and the embedding procedure should yield a straight line.Figure 3 shows the two-dimensional embedding of theconditions. The requirement for the distances are fulfilledand the points are on a straight line. Additionally, consist-ent with [19], we observe that very low ratios of 3-repeatand 4-repeat tau:tubulin have opposite effects on thedynamic behavior of microtubules. More specifically,while 1:55 and 1:45 3-repeat tau and 1:55 4-repeat tau areall relatively close to the control (no-tau) point, the two 3-repeat tau conditions decrease the microtubule growthrate while the 1:55 4-repeat tau condition increases it ascompared to the no-tau condition. Additionally, any
increase in the tau:tubulin ratio beyond these low levelscauses a relatively large increase in growth rate, since thedistance between the no-tau point and all other tau pointsis relatively large. Thus, there are two clusters of growthrates rather than a simple linear relationship, consistentwith a threshold effect. Further, as the tau:tubulin ratioincreases (for both tau isoform classes), the differencewith the no-tau point increases. Finally, for any giventau:tubulin ratio, 4-repeat tau is always more distant fromthe no-tau point than 3-repeat tau is; this demonstratesthat 4-repeat tau is a more potent regulator of microtubuledynamics than 3-repeat tau. Thus, these data establish thevalidity of our automated life history analytical methodand the two-dimensional embedding method.
Figure 4 shows the plot for another set of samples corre-sponding to the values in Table 2. This second sample cor-responds to tubulin preparation 1 mentioned in Table IIof [19]. The low ratios of 3-repeat and 4-repeat tau:tubulinbehave similar to the control (no-tau) point. The higherratios cluster separately.
Microtubule Growth Rate Distribution HistogramNext, we used two-dimensional embedding to comparethe effects of 3-repeat tau and 4-repeat tau upon the distri-bution of growth rates within the growing population ofmicrotubules. As demonstrated in [19], a histogram anal-ysis of control populations of growing microtubulesyields two pools – a more abundant and slower growingpool and a less abundant and faster growing pool. Basedon fitting mixture of two Gaussians to the histograms, theauthors concluded that both tau isoforms cause anincrease in the abundance of the faster growing pool anda decrease in the abundance of the slower growing pool,with 4-repeat causing the population change at lowertau:tubulin ratios than 3-repeat tau. Other than these con-clusions, there was no other comparison possible betweenthe histograms.
Table 3: Rank order of different conditions
Rank Condition (growth rate)
Table III of (Levy et al., 2005) [19] (Table 1 of this paper) Table II of (Levy et al., 2005) [19] (Table 2 of this paper)
The ranks of the conditions remain almost the same across the two samples (considering a variation of 0.1 μm/min) except that the 4R-1:55 and the 3R-1:38 conditions are reversed.
Growth rate (sample 1)Figure 3Growth rate (sample 1). Embedding of the growth rates of tau conditions for sample 1 (corresponding to Table 1). The distortion is 1, indicating no error in embedding. The automatically computed growth rates maintain the relation-ship of the conditions as described in the Results section and in [19]. Distortion = 1.00.
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We subjected the growth rate distribution data to our two-dimensional embedding analysis method (Figure 5). Eachdistribution histogram had 19 bins (similar to the analy-sis in [19]), and the dissimilarities among the histogramswere computed by the match distance [29]. Conceptually,the match distance takes into account both the height of a
histogram bin and the spatial position of the bin in thehistogram; two histograms that differ in far-off bins aremore distant than histograms that differ in adjacent bins.The details of the procedure are presented in the Methodssection. As was true for the growth rates (Figure 3), the his-togram distribution data reveals that there are only minordifferences among the control (no-tau) sample and lowlevels of tau (3-repeat tau at both 1:55 and 1:45 tau:tubu-lin ratio and 4-repeat tau at a 1:55 tau:tubulin ratio).Moreover, in a manner parallel to the growth rates, 3-repeat and 4-repeat tau regulate microtubule dynamics indifferent directions, as indicated by the fact that the 4-repeat tau (1:55 ratio) is closer to the no-tau point than itis to either of the 3-repeat tau (1:55 or 1:45) samples.
Additionally, it is also clear that 4-repeat tau is morepotent than 3-repeat tau at any given tau:tubulin ratio(i.e., the distance between the 4-repeat tau point and theno-tau point is greater than the distance between the 3-repeat tau point and the no-tau point for all molar ratios).Finally, similar to the growth rate analysis in Figure 3,there are two clusters of behaviors rather than a contin-uum. One cluster contains the no-tau point and the lowertau:tubulin ratio samples and the other cluster containsthe higher tau:tubulin ratio samples. Such non-linearitycoupled with different functional effects could have sig-nificant mechanistic effects in the alternative RNA splicingclass of tau FTDP-17 mutations in which relatively subtleincreases in the 4-repeat tau concentration have dramaticconsequences. By assessing the histogram landscape ofthe conditions, the two-dimensional embedding proce-dure complements the previous analyses using Gaussianmixture models [19]. The two-dimensional embeddingplot is more sensitive in picking out the differencesbetween a pair of conditions or among multiple condi-tions; on the other hand, it shows distances that lack phys-ical meaning.
Figure 6 shows the corresponding embedding plot of thegrowth rate distribution histograms for another set ofsamples. This second sample corresponds to tubulin prep-aration 1 mentioned in Table II of [19]. Similar to the casepresented in Figure 5, the low ratios of 3-repeat and 4-repeat tau:tubulin cluster together with the control (no-tau) point. The higher ratios of tau:tubulin induce shiftsin the growth rates.
Additional file 1 shows the effect of the number of bins onthe embedding plots. Histograms were generated by vary-ing the number of bins from 4 to 29 in variations of 5. Theplots show only minor differences. In all of them, thelower tau:tubulin ratios (4-repeat tau at 1:55, and 3-repeattau at 1:55 and 1:45) and the control (no-tau) point arefar away from the higher tau:tubulin ratios.
Growth rate (sample 2)Figure 4Growth rate (sample 2). Embedding of the growth rates of tau conditions for sample 2 (corresponding to Table 2). The distortion is 1, indicating no error in embedding. The automatically computed growth rates maintain the relation-ship of the conditions as described in the Results section and in [19]. Distortion = 1.00.
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Growth rate distribution histogram (sample 1)Figure 5Growth rate distribution histogram (sample 1). Embedding of the microtubule growth rate distributions at varying tau:tubulin molar ratios for sample 1. The growth behavior of microtubules for low molar ratios of tau:tubulin for both 4-repeat and 3-repeat taus are similar to those in no-tau conditions. In higher molar ratios, however, the behavior is quite different. Distortion = 1.84.
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Microtubule Dynamics and Haar WaveletsFinally, we compared the two-dimensional embeddingsof the Haar wavelet features [30] to 3-repeat tau, 4-repeattau and the control (no-tau) samples. Wavelets [31] arepowerful statistical tools that are used for a wide range ofapplications, including signal description and data com-pression. One of the main advantages of wavelets is thatthey offer a simultaneous localization in both time andfrequency domains. Further, they can provide a multi-res-olution view of the original time-series by changing thewidth of the "window" over which the coefficients arecomputed. Haar wavelets [30] are the simplest and thefastest to calculate among all the different types of waveletfunctions. The specific window sizes and the details ofhow the dissimilarities among the conditions are com-puted are described in the Methods section. Additionalfile 2 shows the plots for the two different samples. Thedisparity in the two plots likely arises from the inherentvariability in the biological data. The first plot (corre-sponding to the data set presented in Table 1) suggeststwo distinct clusters, one corresponding to the 3-repeattau conditions and the other to the 4-repeat tau condi-tions, consistent with the notion that 3-repeat and 4-repeat tau might interact with microtubules in qualita-tively distinct manners. The lack of similar behavior forthe second data set (see Table 2) makes the conclusionsfrom the plots tentative, requiring independent corrobo-ration.
We also used two-dimensional embedding to comparethe effects of 3-repeat and 4-repeat tau with respect to the
Markov Chain models. A Markov Chain (MC) [32] cap-tures the underlying dynamics of the physical phenomenaor entity by a generative model that emits a sequence ofsymbols. The primary advantage of Markov Chains overother models of time-series data is their ability to charac-terize an entire family of sequences. MCs are fairly easy tobuild, require a small set of sequences and allow very fastsearching and comparison. There was no obvious cluster-ing of points with respect to either the tau:tubulin ratio or3-repeat tau versus 4-repeat tau (plot not shown). Weused other time-series models as well, like the Lomb-Scar-gle periodograms [33,34] that can assess periodic behav-iors (akin to Fourier analysis [35]) even in the presence ofmissing data and unequal sampling frequencies. Unfortu-nately, the embedding plot did not reveal any clear pat-terns, with the exception that the control (no-tau) pointwas on a distant corner of the plot and the tau sampleswith lower molar ratios of tau:tubulin are closer to the no-tau point than the samples with higher ratios (data notshown). Another class of models – the auto-regressivemoving average (ARMA) models [36] – has often beenused in analyzing time-series data. These models assumethat the data is stationary, i.e., both the mean and the var-iance is fixed. Since the microtubules are clearly growing,we did not consider these models.
DiscussionThis work addresses the need of the biological researchcommunity for rigorously quantitative and generallyapplicable computational tools to compare the complexbehaviors of individual members of groups of molecules,cells or even organisms. Presently, the vast majority ofsuch comparisons are performed manually, or "by eye".As such, they are time-consuming, susceptible to inadvert-ent bias and errors and can be insensitive to subtleties.Using the regulatory effects of the microtubule associatedprotein tau upon the dynamic behavior of microtubulesas a system of study, we have developed a novel modelingand visualization strategy allowing investigators (i) toassess the relative degree of similarity/dissimilarity amongindividual tau isoforms with respect to numerous param-eters of interest under varying experimental conditions,and (ii) to visualize all the conditions with respect to eachother. More importantly, the same computational strategyshould be generally applicable to a great many otherapplications.
The validity of the two-dimensional embedding strategypresented in this paper is established by comparing theplot presented in Figure 3 with the growth rate data inTable 1. The relative positions of all points in Figure 3 arein complete agreement with the quantitative growth ratedata determined both automatically and manually. Addi-tionally, the semi-quantitative analysis of the histogramsshown in Figure 4 of [19] are confirmed and extended bythe more rigorous quantitative analysis leading to the
Growth rate distribution histogram (sample 2)Figure 6Growth rate distribution histogram (sample 2). Embedding of the microtubule growth rate distributions at varying tau:tubulin molar ratios for sample 2. The growth behavior of microtubules for low molar ratios of tau:tubulin for both 4-repeat and 3-repeat taus are similar to those in no-tau conditions. In higher molar ratios, however, the behavior is quite different. Distortion = 1.69.
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two-dimensional embedding plot shown in Figure 5. Inthis case, 19 different bins of microtubule growth rateswere integrated into the analysis for each of the nineexperimental conditions tested. The resulting two-dimen-sional plot in Figure 5 presents the investigator with novelperspectives on the data set, including the existence of twoclusters of histogram distributions based on growth rate aswell as the distinct behavior of low ratios of 3-repeattau:tubulin relative to all other tested reactions.
Finally, although the molecular mechanisms underlyingbehaviors suggested by various statistical models may notbe clear, these models could suggest mechanisms thatcould not be drawn using the standard manual analyticalmethods generally utilized in biological investigations.Indeed, one of the most important and generally applica-ble features of our computational strategy is the ability todetect subtle relationships between different molecules orconditions that might escape manual investigation.
ConclusionIn this manuscript, we present (i) an automated methodfor quantitatively characterizing microtubule dynamics asa function of time, and (ii) a novel and generally applica-ble computational tool for two-dimensional visualizationand modeling of entities of interest for comparative stud-ies. Comparison of our automated tracking method withmanually acquired data demonstrates its accuracy. Thistool greatly increases the rate at which microtubule track-ing data can be acquired as well as improve upon its objec-tivity and accuracy. Our embedding strategy accuratelyrecapitulates and extends previous biological observa-tions that were collected and analyzed manually. Impor-tantly, our methods facilitate the integration ofsophisticated statistical modeling with biological investi-gations, which should promote novel and deeper mecha-nistic insights into biological phenomena as well as thedevelopment of testable hypotheses for subsequent inves-tigation. In the future, we anticipate applying these meth-ods to compare wild-type tau action versus various taumutants causing neurodegeneration and dementia, seek-ing to identify novel mechanistic effects. Additionally, weenvision using new models and embedding strategies.
MethodsModelingThe different models described in this section capture dif-ferent characteristics of microtubule dynamicity. Compar-ison of conditions across these models highlightsdifferent features of tau action.
Microtubule EventsThree kinds of events are used to characterize microtubuledynamics: growth, shortening and attenuation ("pause").Each kind of event can be simple or complex. An event is
simple when it is characterized between two consecutivetracked time-points. Simple events are coalesced togetherto form bigger complex events. Complex events, therefore,can be defined over a contiguous set of more than twotime-points. Identification of simple events are easy, butidentifying the start and end points of complex eventsrequire sub-sequence analysis.
The simple events are classified in the following manner.The different parameters for defining these events used inthis particular study are indicated next. An event is a simplegrowth if and only if: (i) the rate of increase of microtubulelength is at least 0.5 μm/min, and (ii) the increase inlength is at least 0.05 μm. The corresponding parametersfor a simple shortening event are: (i) the rate of decrease oflength is at least 0.5 μm/min, and (ii) the decrease inlength is at least 0.5 μm. A simple attenuation event musthave (i) a rate of change of length outside the range forsimple growth and simple shortening events, i.e., between-0.5 μm/min and +0.5 μm/min, and (ii) a total time dura-tion of at least 4 s. Any event that does not fall in any ofthe above categories are excluded from the analysis. Due toerrors in human tracking and image resolution issues,such events are likely to be part of the input noise, and arehence, discarded. Simple events are used for MarkovChain analysis.
The complex events have their corresponding parametercut-offs as well. However, the more important considera-tion in the analysis of complex events is the identificationof where it starts and where it ends. A survey of such meth-ods from the time-series literature can be found in [37].These methods have been successfully used to segmenttime-series streams into different partitions in variousapplication domains, most notably for stock market anal-ysis. An interesting way to combine different segmenta-tion outputs has been proposed in [38]. However, none ofthese methods have employed priority rules to analyzeadjacency relationships.
We now describe our bottom-up approach of identifyingcomplex events by merging together simple events. First,all consecutive simple events of the same type are mergedtogether to form a longer complex event of the same type.Next, each complex event is subjected to the rule set forclassifying into growth, shortening, and attenuation. Anevent that does not pass any of the three rule sets is classi-fied temporarily as an error. Also, the cause of its failure isnoted. More specifically, any event where there is anincrease in length but which cannot be classified as agrowth is assigned into two kinds of failure: (i) rate, whereit did not pass the growth rate threshold, and (ii) length,where it did not pass the growth length threshold. Thefailed shortening events are classified similarly. Note thatthere are no attenuation failure classes.
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The priority rules are applied next. A growth rate failureevent is most likely to be part of an attenuation. Thus, itsneighbors are examined and if possible, it is combinedwith adjacent attenuation events to form a bigger attenu-ation event. If this fails, then attempts to incorporate withneighboring growth events, if any, are made. If, however,the growth failure event is due to the length cutoff and notthe rate, then this event is most likely to be part of agrowth event. The error in length may be due to humantracking and image resolution issues. Hence, attempts tocombine this with neighboring growth events are first car-ried out. The rules for absorbing the shortening failureevents are similar.
The complex event cut-offs are: (i) Growth: rate ≥ 0.5 μm/min, length ≥ 0.06 μm; (ii) Shortening: rate ≤ -0.5 μm/min,length ≤ -0.6 μm; and (iii) Attenuation: rate between -0.5μm/min and +0.5 μm/min, time ≥ 30 s. The growth ratesand the growth distributions are calculated using the com-plex events.
Figure 2 shows a comparison of the manually markedcomplex events and the automatically measured ones. Thesolid line indicates the simple events. As evident from thefigure, these tracked lengths are noisy. The complex eventsget rid of the noise by smoothing over a range of simpleevents. However, while the automatic method marksthree events – two growth events separated by a shorten-ing event – a human may simply mark the entire time-his-tory as a single growth event. Clearly, this human bias willdiffer from one experimenter to another, and may evenvary from time to time. Note that this explains whygrowth rates obtained from the automatic measurementsvary (become slightly higher) from those obtainedthrough the manual method.
The parameters for the different events have been chosenempirically by biologists based on experimenting withdifferent kinds of microtubule samples. The event defini-tions have been used consistently and have become the defacto "industry standard," as evident from [4].
Growth RateThe growth rate for a particular experimental conditionwas calculated as the average of the growth rates of all thecomplex growth events of the microtubules for that con-dition.
In order to understand whether the differences betweenthe automatically computed growth rate values using theabove event analysis technique and the manually meas-ured ones are statistically significant, we calculated the p-values in the following way. Two groups were formed, onewith the automatically identified growth events, and theother with the manually marked growth events. We then
performed a t-test [39] to determine whether the means ofthe growth rates of the two groups are different. The p-val-ues thus obtained are reported in Tables 1 and 2.
Growth Rate Distribution Histogram
For each condition, a growth rate distribution histogramwas computed in the following manner. The rates for thecomplex growth events were divided into 18 bins of width
0.4 μm/min each (consistent with analysis by [19]), start-
ing from 0.5 μm/min up to 7.7 μm/min. Once again, theseparameters conform to the standards set in the microtu-bule event analysis literature [4]. All the higher growthrate events were collected in another bin. Thus, the histo-grams had 19 bins in total. The bin heights were normal-ized such that they add up to 1, yielding a growth ratedistribution. In order to generate histograms with a fixednumber of bins, say b, the width of each bin was specified
as .
Haar WaveletsWavelets are mathematical functions that describe time-series data in terms of various frequency components withresolutions matched to their scales [31]. The orthonormalbasis vectors, called the mother wavelets, that describe thevarious wavelet components are given by:
ψs,l(x) = 2-s/2ψ(2-s x - l)
where s denote the scaling factor and l the localization intime. The Haar wavelet basis functions [30] are the sim-plest:
Haar wavelets are also the fastest to calculate with respectto other wavelet bases. They work by progressively retain-ing the most important parameters of a signal. The firstcoefficient is the "sum" (actually, scaled average) of theentire signal and the next gives the "detail" (difference ofthe two halves) of the signal. The later coefficients givemore and more details about each half of the signal theymodel.
In general, more wavelet parameters mean more detaileddescription; however, they also mean more data and morenoise. Further, the error in embedding is directly propor-tional to the number of parameters of the original data.Thus, 16 coefficients of each microtubule time-series wereretained. These 16 coefficients were then averaged over allthe microtubules from a particular experimental condi-tion to yield the coefficients for that condition.
7 7 0 5. . /−b m minm
y ( )
/
/x
x
x=+ ≤ <− ≤ <
⎧⎨⎪
⎩⎪
1 0 1 2
1 1 2 1
0 otherwise
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Markov ChainA Markov Chain (MC) [32] is a discrete time stochasticprocess that models the observations of a dynamic system(such as the growth or the shortening of a microtubule) asthe states of the system. The number of states is finite andthere is a state corresponding to each observation symbol.In a first-order MC, the probability of occurrence of thefuture state (or observation) depends only on the currentstate; past states are inconsequential. This property iscalled the Markov property. (In a kth order Markov Chain,the future state depends on the current state and k - 1 paststates.)
Formally, an MC λ is defined as:
λ = {n, π, τ}
where n is the number of states, π is the start state probabilityvector of length n, and τ is the n × n transition matrix; π(u)denotes the probability of being in state u in the first time-step; and, τ (u, v) denotes the probability of reaching statev from state u in a single time-step.
In the work of [19], microtubules were in a non-equilib-rium phase, exhibiting very little shortening and manymicrotubules never shortened at all. Therefore, the micro-tubule events were discretized into two symbols: G forgrowth, and N for non-growth (shortening or attenua-tion). The Markov Chains were built with these two states– growth and non-growth. Since shortening events werevery rare, modeling it as a separate state would havelacked statistical validity.
The transition probabilities for the MCs were estimated inthe following manner. Every microtubule time-series wasdenoted as a string of symbols, with each symbol repre-senting a simple event. Then, pairs of consecutive symbols(states) were read and appropriate entries in the transitionmatrix were incremented. When all the microtubules inan experimental condition were processed, the transitionmatrix was normalized such that the sum of transitionprobabilities from each state form a probability distribu-tion (adds up to 1). The start state probabilities were esti-mated in a similar manner by reading the first symbol forevery microtubule time-series; if it is growth, the entry forG is incremented, otherwise that for N is incremented.Finally, normalization was performed such that the prob-abilities add up to 1. Since most of the microtubulesstarted with growth, these vectors were very close to [1, 0].
Lomb-Scargle PeriodogramsThe periodicity analysis of the microtubule data was per-formed by extracting Lomb-Scargle coefficients [33,34]from each time-series. Lomb-Scargle periodograms cap-ture the different frequency components in a time-seriesand can handle missing values and unequal sampling
intervals. Four low frequency components (correspond-ing to periodicities of 4, 8, 16, and 32 s) were retained foreach microtubule. The Lomb-Scargle coefficients for thecondition were computed as the average of the corre-sponding coefficients of the individual microtubules.
Dissimilarity FunctionsIn order to compare a pair of models, an appropriate sim-ilarity or dissimilarity function is necessary. The dissimi-larity or distance measure is used to compute the distancematrix among the conditions; this distance matrix is thenembedded in a two-dimensional vector space as describedlater in the Visualization section.
Growth RateThe dissimilarity between a pair of conditions with respectto the growth rates was measured by their difference. Thedifference can be also viewed as a Minkowski form of dis-tance or Lk norm. The Lk norm between two vectors p andq of length k each is defined as:
Growth Rate Distribution HistogramGrowth rate histograms can be viewed as vectors and Lknorms can be employed to capture the dissimilaritybetween a pair of histograms. These measures, however,do not capture the relationship among the different histo-gram bins. For example, suppose there are three bins ineach histogram corresponding to low rate of growth,medium rate of growth and high rate of growth. If A = [1,0, 0], B = [0, 1, 0] and C = [0, 0, 1], then Lk norms treatthese histograms as equidistant from each other, eventhough A should be more different from C than B. To cap-ture such spatial properties of the bins, match distance[29,40] was employed.
To calculate the match distance between a pair of histo-grams p and q, a distance matrix among the bins of thehistogram are specified – the distance between two bins iand j is cij = |i - j|. The match distance is defined as the min-imum work required to be done in order to transform thehistogram p into the histogram q by moving values or"flows" from the bins of p to those in q and vice versa.Having a flow fij from bin i of p to bin j of q or vice versais considered as cij·fij amount of work. Finding the matchdistance then reduces to finding the flows fij such that thetotal work done is minimum. The minimum work done isthe match distance:
L p q p qk i ik
i
k
( , )
/
= −⎡
⎣⎢⎢
⎤
⎦⎥⎥∀
∑1
MD p q c ff
ij ij
i j
( , ) min,
=⎧⎨⎪
⎩⎪
⎫⎬⎪
⎭⎪∑
Page 10 of 13(page number not for citation purposes)
For the example histograms A, B, and C mentioned above,the match distances are MD(A, B) = 1, MD(B, C) = 1, andMD(A, C) = 2. Clearly, this captures the relatively largerdissimilarity of A from C as compared to that from B.
For one-dimensional histograms where the sum of the binvalues add up to the same number (here, 1), match dis-tance can be calculated more easily as the L1 distancebetween the cumulative bin values of the two histograms:
MD(p, q) = L1(P, Q)
where and are the cumulative
histogram bin values.
Haar WaveletsSince the relative importance of the wavelet parametersdiffer, a simple distance function such as L1 would beinappropriate. Coefficients that summarize the entiretime-series, such as the "sum" value and the overall"detail" value is more important, and therefore, shouldget higher weights than the coefficients describing parts ofthe time-series.
Thus, in order to determine the dissimilarity between twoconditions with respect to their Haar wavelet coefficients,we used the weighted L1 norm or the weighted Manhattandistance. The levels of the wavelet tree were weighted suchthat the overall sum and the overall detail coefficient werethe most significant values, the next level detail coeffi-cients getting an exponentially lower weight and so on.The weight vector, of length 16, was [8, 8, 4, 4, 2, 2, 2 2, 1,1, 1, 1, 1, 1, 1, 1]. For two vectors p and q, and a weightvector w, all of length k, the weighted L1 distance betweenp and q is measured as:
The L2 norm or the Euclidean distance was applied tomeasure the distances between a pair of conditions forboth the Markov Chain parameters and the Lomb-Scarglecoefficients.
VisualizationThe distances among the experimental conditions, calcu-lated by using the above methods, were visualized by plot-ting the conditions onto a two-dimensional vector space.This allows for easy comparison of the conditions andimmediate comprehension of the structure of the data.The aim of the embedding method is to assign coordi-nates such that the Euclidean distance between any pair ofconditions in the embedded space is as close as possibleto the dissimilarity calculated between their models. Themethod can embed a given set of points into any dimen-
sional space; here, we have chosen two for easy visualiza-tion purposes.
Formally, suppose there are two models, α and β, and thedissimilarity between them is d(α, β) according to somedissimilarity function d. If the embeddings of these twomodels in the two-dimensional vector space (x, y space) isgiven by e(α) = (xα, yα) and e(β) = (xβ, yβ), then the aim ofthe embedding function is to choose the coordinates e(α)and e(β) such that the relative difference between L2(e(α),e(β)) and d(α, β) is minimum. When there are n suchmodels, the embedding function should be chosen suchthat the relative cumulative difference for all the n(n - 1)/2 pairs is minimized.
Principal component analysis (PCA) [28] can also be usedto project data onto a two-dimensional space. PCAchooses the axes along which the original data shows thehighest variance. It does not take into account the dis-tances among the points. More importantly, PCA cannotwork with any general distance matrix and is used mostlyas a dimensionality reduction technique.
We used the Sammon projection method [27] as theembedding procedure. This method has been successfullyused to embed proteins on a two-dimensional space forclustering purposes [41]. The method starts with a ran-dom point (random x, y coordinates) for each model. Ineach iteration, the points are updated according to a steep-est descent algorithm such that the following error functionE(x, y), which measures the relative differences betweenthe original distances and the embedded distances, isdecreased.
where
In each iteration, a correction step is added to everydimension of every point. The direction of the correctionis towards the gradient of the error. The coordinates of thepoint α in iteration i + 1 are updated as follows:
P pi ij
i= =∑ 1Q qi ij
i= =∑ 1
L p q w p qi i i
i
k
1
1
( , ) = −=∑
E x yc
d L e e
d( , )
( , ) ( ( ), ( ))
( , ), ,
=−( )⎡
⎣
⎢⎢
⎤
⎦
⎥⎥
∀ ≠∑1 2
2a b a ba b
a b a b
c d=∀∑ ( , )
,
a ba b
x x fE i x
E i x
y y fE i y
i i
i i
a a
a a
( ) ( )
( ) ( )
( ) /( ) /
( ) /
+
+
= − × ∂ ∂
∂ ∂
= − × ∂ ∂
1
1
2 2
∂∂ ∂2 2E i y( ) /
Page 11 of 13(page number not for citation purposes)
where E(i) is the error after iteration i and f is a factor tocontrol the step sizes. We used f = 0.2. The method stopsafter a certain number of steps or when there is no signif-icant improvement in the error. We stopped the iterationseither when the change in error went below 0.01% or upto a maximum of 1000 steps. The number of steps was setas an additional check in order to come out of any localerror problems, e.g., oscillating error values. In practice,after 250–300 iterations, the error stopped changing, andthe algorithms stopped. In addition, in order to counterthe problem of bad initialization, the algorithms were run5 times for each embedding and the one with the lowesterror was picked. The final coordinates or the directions ofthe axes do not have any significance; only the Euclideandistances among the embedded points matter.
For any dimensionality reduction or embedding tech-nique, an important measure of quality is distortion. Dis-tortion measures the largest amount of discrepancy froman original distance value to the corresponding embeddeddistance. It is measured as
where original dist refers to an original dissimilarity meas-ure between two models and embedded dist refers to theEuclidean distance between the corresponding embeddedpoints. For ideal embeddings, where all the original dis-tances have been maintained exactly, the distortion is 1.For others, the distortion is greater than 1. In general,lower the distortion value, better the embedding. Theindividual distortions are measured by the ratio of embed-ded dist to original dist.
For models with only a single parameter, any dissimilaritybetween a pair of them is equal to the difference betweentheir single parameters. Such distance matrices can bealways embedded into two dimensions with distortionequal to 1. The models have to be simply embedded aspoints on a straight line with the order and the distances
maintained, e.g., as (parameter, 0) or (parameter/ ,
parameter/ ) points. In our implementation, we havenot forced this explicitly; the method itself converges to astraight line plot. For models with two parameters, if thedissimilarity function is Euclidean, then, it is again possi-ble to devise an embedding with distortion 1. The originalparameter values will form the coordinates in the embed-ded space. For higher number of parameters or with otherdissimilarity functions, in general, it is not possible todesign embeddings with distortion 1. The distortions of
each of the graphs are mentioned in the captions. Addi-tional file 3 reports the individual distortions for each ofthe distances for all the embeddings.
Authors' contributionsAB built the models, computed the pairwise distances,and embedded them on two dimensions for comparisons.AL, SL and MG collected the data and manually trackedthe microtubules. AKS advised on the choice of the mod-els and the distance functions. SCF and LW provided thebiological interpretations. AB and SCF wrote the manu-script with inputs from the other authors. All the authorsread and approved the final manuscript.
Additional material
AcknowledgementsWe acknowledge grants from the NSF, USA (NSF ITR-0331697 to SCF, AKS, and LW), the NIH, USA (NIH NS-35010 to SCF and NIH NS-13560 to LW), and IIT, Kanpur, India (INI/IITK/CSE/20080069 to AB) for support-ing this work. We also thank the anonymous referees whose comments helped improve the paper immensely.
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Additional file 1This file shows the effect of number of bins on the growth rate distribution histograms.Click here for file[http://www.biomedcentral.com/content/supplementary/1471-2105-9-339-S1.pdf]
Additional file 2This file shows the two-dimensional embedding plots of the Haar wavelet coefficients of microtubules with varying tau:tubulin molar ratios for both the samples.Click here for file[http://www.biomedcentral.com/content/supplementary/1471-2105-9-339-S2.pdf]
Additional file 3This file shows the individual distortions of all the pairwise distances between the conditions for all the different models described.Click here for file[http://www.biomedcentral.com/content/supplementary/1471-2105-9-339-S3.pdf]
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