Fusion Tensor Subspace Transformation Framework

Tensor subspace transformation, a commonly used subspace transformation technique, has gained more and more popularity over the past few years because many objects in the real world can be naturally represented as multidimensional arrays, i.e. tensors. For example, a RGB facial image can be represented as a three-dimensional array (or 3rd-order tensor). The first two dimensionalities (or modes) represent the facial spatial information and the third dimensionality (or mode) represents the color space information. Each mode of the tensor may express a different semantic meaning. Thus different transformation strategies should be applied to different modes of the tensor according to their semantic meanings to obtain the best performance. To the best of our knowledge, there are no existing tensor subspace transformation algorithm which implements different transformation strategies on different modes of a tensor accordingly. In this paper, we propose a fusion tensor subspace transformation framework, a novel idea where different transformation strategies are implemented on separate modes of a tensor. Under the framework, we propose the Fusion Tensor Color Space (FTCS) model for face recognition.     

Bitpacking techniques for indexing genomes: II. Enhanced suffix arrays

Background

Suffix arrays and their variants are used widely for representing genomes in search applications. Enhanced suffix arrays (ESAs) provide fast search speed, but require large auxiliary data structures for storing longest common prefix and child interval information. We explore techniques for compressing ESAs to accelerate genomic search and reduce memory requirements.

Results

We evaluate various bitpacking techniques that store integers in fewer than 32 bits each, as well as bytecoding methods that reserve a single byte per integer whenever possible. Our results on the fly, chicken, and human genomes show that bytecoding with an exception guide array is the fastest method for retrieving auxiliary information. Genomic searching can be further accelerated using a data structure called a discriminating character array, which reduces memory accesses to the suffix array and the genome string. Finally, integrating storage of the auxiliary and discriminating character arrays further speeds up genomic search.

Conclusions

The combination of exception guide arrays, a discriminating character array, and integrated data storage provide a 2- to 3-fold increase in speed for genomic searching compared with using bytecoding alone, and is 20 % faster and 40 % more space-efficient than an uncompressed ESA.

     

Prediction of apparent trabecular bone stiffness through fourth-order fabric tensors

The apparent stiffness tensor is an important mechanical parameter for characterizing trabecular bone. Previous studies have modeled this parameter as a function of mechanical properties of the tissue, bone density, and a second-order fabric tensor, which encodes both anisotropy and orientation of trabecular bone. Although these models yield strong correlations between observed and predicted stiffness tensors, there is still space for reducing accuracy errors. In this paper, we propose a model that uses fourth-order instead of second-order fabric tensors. First, the totally symmetric part of the stiffness tensor is assumed proportional to the fourth-order fabric tensor in the logarithmic scale. Second, the asymmetric part of the stiffness tensor is derived from relationships among components of the harmonic tensor decomposition of the stiffness tensor. The mean intercept length (MIL), generalized MIL (GMIL), and fourth-order global structure tensor were computed from images acquired through microcomputed tomography of 264 specimens of the femur. The predicted tensors were compared to the stiffness tensors computed by using the micro-finite element method (\(\upmu \)FE), which was considered as the gold standard, yielding strong correlations (\(R^2\) above 0.962). The GMIL tensor yielded the best results among the tested fabric tensors. The Frobenius error, geodesic error, and the error of the norm were reduced by applying the proposed model by 3.75, 0.07, and 3.16 %, respectively, compared to the model by Zysset and Curnier (Mech Mater 21(4):243–250, 1995) with the second-order MIL tensor. From the results, fourth-order fabric tensors are a good alternative to the more expensive \(\upmu \)FE stiffness predictions.     

Tensor regularized total variation for denoising of third harmonic generation images of brain tumors

Third harmonic generation (THG) microscopy shows great potential for instant pathology of brain tissue during surgery. However, the rich morphologies contained and the noise associated makes image restoration, necessary for quantification of the THG images, challenging. Anisotropic diffusion filtering (ADF) has been recently applied to restore THG images of normal brain, but ADF is hard‐to‐code, time‐consuming and only reconstructs salient edges. This work overcomes these drawbacks by expressing ADF as a tensor regularized total variation model, which uses the Huber penalty and the L₁ norm for tensor regularization and fidelity measurement, respectively. The diffusion tensor is constructed from the structure tensor of ADF yet the tensor decomposition is performed only in the non‐flat areas. The resulting model is solved by an efficient and easy‐to‐code primal‐dual algorithm. Tests on THG brain tumor images show that the proposed model has comparable denoising performance as ADF while it much better restores weak edges and it is up to 60% more time efficient.      

Study of Plasma Flow Modes in Imploding Nested Arrays

Results from experimental studies of implosion of nested wire and fiber arrays at currents of up to 4 МА at the Angara-5-1 facility are presented. Depending on the ratio between the radii of the inner and outer arrays, different modes of the plasma flow in the space between the inner and outer arrays were implemented: the sub-Alfvénic (V _r < V _А ) and super-Alfvénic (V _r > V _А ) modes and a mode with the formation of the transition shock wave (SW) region between the cascades. By varying the material of the outer array (tungsten wires or kapron fibers), it is shown that the plasma flow mode between the inner and outer arrays depends on the ratio between the plasma production rates ?_in /?_out in the inner and outer arrays. The obtained experimental results are compared with the results of one-dimensional MHD simulation of the plasma flow between the arrays. Stable implosion of the inner array plasma was observed in experiments with combined nested arrays consisting of a fiber outer array and a tungsten inner array. The growth rates of magnetic Rayleigh?Taylor (MRT) instability in the inner array plasma at different numbers of fibers in the outer array and different ratios between the radii of the inner and outer arrays are compared. Suppression of MRT instability during the implosion of the inner array plasma results in the formation of a stable compact Z-pinch and generation of a soft X-ray pulse. A possible scenario of interaction between the plasmas of the inner and outer arrays is offered. The stability of the inner array plasma in the stage of final compression depends on the character of interaction of plasma jets from the outer array with the magnetic field of the inner array.     

Recursive N-Way Partial Least Squares for Brain-Computer Interface

In the article tensor-input/tensor-output blockwise Recursive N-way Partial Least Squares (RNPLS) regression is considered. It combines the multi-way tensors decomposition with a consecutive calculation scheme and allows blockwise treatment of tensor data arrays with huge dimensions, as well as the adaptive modeling of time-dependent processes with tensor variables. In the article the numerical study of the algorithm is undertaken. The RNPLS algorithm demonstrates fast and stable convergence of regression coefficients. Applied to Brain Computer Interface system calibration, the algorithm provides an efficient adjustment of the decoding model. Combining the online adaptation with easy interpretation of results, the method can be effectively applied in a variety of multi-modal neural activity flow modeling tasks.     

Refining gene signatures: a Bayesian approach

Background  

In high density arrays, the identification of relevant genes for disease classification is complicated by not only the curse of dimensionality but also the highly correlated nature of the array data. In this paper, we are interested in the question of how many and which genes should be selected for a disease class prediction. Our work consists of a Bayesian supervised statistical learning approach to refine gene signatures with a regularization which penalizes for the correlation between the variables selected.     

Study of Implosion of Combined Nested Arrays

New experimental data on the implosion of plasma of nested kapron?tungsten arrays are obtained at the Angara-5-1 facility. The mode of plasma implosion is implemented in which a shock wave region forms in the space between the inner and outer arrays where a transition from the super-Alfvénic (V _r > V _A ) to sub-Alfvénic (V _r < V _A ) plasma flow takes place. Specific features of the formation and decay of the shock region are studied using laser shadow imaging and X-ray frame photography. The plasma density in the transition region is estimated. By comparing the experimental data with the results of simulations of quasi-steady implosion of a nested array with allowance for extended plasma production, the physical conditions are determined at which the implosion mode with the formation the shock region takes place. Stable compression of the plasma of the inner array was observed during the implosion of combined nested arrays with a fiber outer array and tungsten inner array. Suppression of magnetic Rayleigh-Taylor instability during the compression of the inner array plasma results in the formation of a compact radiating Z-pinch and generation of a soft X-ray pulse with a peak power of 4 TW and duration of about 5 ns.     

Compression dynamics of quasi-spherical wire arrays with different linear mass profiles

Results of experimental studies of the implosion of quasi-spherical wire (or metalized fiber) arrays are presented. The goal of the experiments was to achieve synchronous three-dimensional compression of the plasma produced in different regions of a quasi-spherical array into its geometrical center. To search for optimal synchronization conditions, quasi-spherical arrays with different initial profiles of the linear mass were used. The following dependences of the linear mass on the poloidal angle were used: m _l (θ) ∝ sin^–1θ and m _l (θ) ∝ sin^–2θ. The compression dynamics of such arrays was compared with that of quasi-spherical arrays without linear mass profiling, m _l (θ) = const. To verify the experimental data, the spatiotemporal dynamics of plasma compression in quasi-spherical arrays was studied using various diagnostics. The experiments on three-dimensional implosion of quasi-spherical arrays made it possible to study how the frozen-in magnetic field of the discharge current penetrates into the array. By measuring the magnetic field in the plasma of a quasi-spherical array, information is obtained on the processes of plasma production and formation of plasma flows from the wire/fiber regions with and without an additionally deposited mass. It is found that penetration of the magnetic flux depends on the initial linear mass profile m _l (θ) of the quasi-spherical array. From space-resolved spectral measurements and frame imaging of plasma X-ray emission, information is obtained on the dimensions and shape of the X-ray source formed during the implosion of a quasi-spherical array. The intensity of this source is estimated and compared with that of the Z-pinch formed during the implosion of a cylindrical array.     

Serum Ceruloplasmin Levels Correlate Negatively with Liver Fibrosis in Males with Chronic Hepatitis B: A New Noninvasive Model for Predicting Liver Fibrosis in HBV-Related Liver Disease

Aims

This study aimed to investigate associations between ceruloplasmin (CP) levels, inflammation grade and fibrosis stages in patients with chronic hepatitis B (CHB) and to establish a noninvasive model to predict cirrhosis.

Methods

Liver biopsy samples and sera were collected from 198 CHB patients randomized into a training group (n=109) and a validation group (n=89). CP levels were determined using nephelometric immunoassays. Relationships between CP and liver inflammation and fibrosis were analyzed by Spearman rank correlation. Receiver operator characteristic (ROC) curves were used to evaluate the diagnostic value of CP for determining liver fibrosis in CHB. The liver pathology-predictive model was built using multivariate logistic regression analysis to identify relevant indicators.

Results

CP levels were lower in males than in females, lower in patients with inflammation stage G4 compared to other stages and lower in cirrhotic compared to non-cirrhotic patients. Using area under the curve (AUC) values, CP levels distinguished different stages of inflammation and fibrosis. Multivariate analysis showed that CP levels were all significantly associated with cirrhosis in males. A model was developed combining routine laboratory markers APPCI (alpha-fetoprotein [AFP], prothrombin time, and platelets [PLT] with CP) to predict fibrosis in CHB patients. The APPCI had a significantly greater AUC than FIB-4 (aspartate aminotransferase [AST]/ alanine aminotransferase [ALT]/PLT/age), APRI (AST/PLT ratio index), GPI (globin/PLT), and APGA (AST/PLT/gammaglutamyl transpeptidase [GGT]) models (all P-values<0.001).

Conclusions

CP levels correlate negatively and indirectly with inflammation and fibrosis stages in male CHB patients. The APPCI model uses routine laboratory variables with CP to accurately predict liver fibrosis in CHB.     

Identification of the novel localization of tenascinX in the monkey choroid plexus and comparison with the mouse

Tenascin-X (Tn-X) belongs to the tenascin family of glycoproteins and has been reported to be significantly associated with schizophrenia in a single nucleotide polymorphism analysis in humans. This finding indicates an important role of Tn-X in the central nervous system (CNS). However, details of Tn-X localization are not clear in the primate CNS. Using immunohistochemical techniques, we found novel localizations of Tn-X in the interstitial connective tissue and around blood vessels in the choroid plexus (CP) in macaque monkeys. To verify the reliability of Tn-X localization, we compared the Tn-X localization with the tenascin-C (Tn-C) localization in corresponding regions using neighbouring sections. Localization of Tn-C was not observed in CP. This result indicated consistently restricted localization of Tn-X in CP. Comparative investigations using mouse tissues showed equivalent results. Our observations provide possible insight into specific roles of Tn-X in CP for mammalian CNS function.Key words: tenascin-X, choroid plexus, monkey, mouse, Ehlers-Danlos syndrome, schizophrenia.The tenascins (Tn) are a family of four glyco-protein members – tenascin-C (Tn-C), tenascin-R (Tn-R), tenascin-W (Tn-W) and tenascin-X (Tn-X) – found diversely in the extra-cellular matrix of vertebrate organs (Hsia and Schwarzbauer, 2005; Tucker and Chiquet-Ehrismann, 2009). Important functions of Tn have been investigated in developmental cell adhesion modulation and pathological conditions such as wound healing and tumourigenesis (Adams and Watt, 1993; Hsia and Schwarzbauer, 2005; Tucker and Chiquet-Ehrismann, 2009). Tn-C and Tn-R are prominent in the nervous system and play a role in the development of neurite outgrowth and postnatal synaptic plasticity (Yamaguchi, 2000; Chiquet-Ehrismann and Tucker, 2004; Dityatev and Schachner, 2006). Tn-W is found abundantly in the developing bone and stroma of certain tumours (Chiquet-Ehrismann and Tucker, 2004; Tucker and Chiquet-Ehrismann, 2009). Tn-X is the first tenascin member shown to be clearly associated with the human connective tissue disorder Ehlers–Danlos syndrome (EDS; Burch et al., 1997). Patients with a Tn-X deficiency suffer from skin hyperextensibility, joint hypermobility and poor wound healing ability (Bristow et al., 2005). These symptoms are caused by the occurrence of abnormal irregular collagen fibres. Tn-X plays a role in collagen fibrillogenesis by directly binding to collagen (Mao et al. 2002; Minamitani et al. 2004). Mice with a Tn-X deficiency also showed skin symptoms comparable with those of EDS (Mao et al., 2002).Interestingly, in an analysis of human single nucleotide polymorphisms, Tn-X was reported to be significantly associated with schizophrenia (Wei and Hemmings, 2004; Tochigi et al., 2007). However, thus far, there have been no neuroanatomical reports on the involvement of Tn-X in schizophrenia. In the mammalian central nervous system (CNS), Tn-X mRNA expression has only been shown in the rat meninges of the olfactory bulb (Deckner et al., 2000). Recently, we found novel Tn-X localizations in the adult mouse leptomeninges trabecula in the cerebral cortex and in the connective tissue in the lateral ventricle choroid plexus (CP; Imura and Sato, 2008). Our finding of Tn-X localization in CP, which produces cerebrospinal fluid (CSF), might be a key factor in the investigation of the association between CSF metabolism and enlarged ventricles in schizophrenia. Enlarged ventricles are typical structural abnormalities associated with schizophrenia (Staal et al., 1999). Furthermore, CP secretes biologically active molecules into the CSF for brain development, activity and protection (Strazielle and Ghersi-Egea, 2000; Brown et al., 2004; Thouvenot et al., 2006; Johanson et al., 2008). In these molecules, for instance, there is a brain-derived neurotrophic factor (BDNF), the gene expression level and polymorphism of which have been analysed in relation to the pathogenesis of schizophrenia (Buckley et al., 2007). One study reported that BDNF is able to stimulate Tn-X expression in vitro (Takeda et al., 2005).The validity and limitations of animal models (rodents and monkeys) for use in the study of schizophrenia have been discussed (Tordjman et al., 2007). The authors concluded that monkeys appear to be an interesting social interaction model, more so than rodents, because of their complex well-organized social structure. In addition to differences in social structure, the dopaminergic system of rats and monkeys is quite different (García-Cabezas et al., 2009), and dysfunction of the dopaminergic system is related to schizophrenia (Wang et al. 2008).The CSF outflow system has been studied in some animal models (Kapoor et al., 2008). An anatomical difference in arachnoid granulations has been shown between rodents and monkeys (Krisch, 1988). Arachnoid granulations in monkeys are structurally similar to those in humans (Cooper, 1958; Krisch, 1988). In contrast, arachnoid granulations in rodents are similar to those of cats and dogs (Krisch, 1988). It is possible that Tn-X localization in CP is different between rodents and monkeys.Therefore, details concerning Tn-X localization in monkey CP need to be clarified. In the present study, we compared the immunohistochemistry of Tn-X in monkey CP with that in mouse CP. Subsequently, to verify the reliability of Tn-X localization, we compared it with Tn-C localization in corresponding regions using neighbouring sections.     

Differentially private distributed logistic regression using private and public data

Background

Privacy protecting is an important issue in medical informatics and differential privacy is a state-of-the-art framework for data privacy research. Differential privacy offers provable privacy against attackers who have auxiliary information, and can be applied to data mining models (for example, logistic regression). However, differentially private methods sometimes introduce too much noise and make outputs less useful. Given available public data in medical research (e.g. from patients who sign open-consent agreements), we can design algorithms that use both public and private data sets to decrease the amount of noise that is introduced.

Methodology

In this paper, we modify the update step in Newton-Raphson method to propose a differentially private distributed logistic regression model based on both public and private data.

Experiments and results

We try our algorithm on three different data sets, and show its advantage over: (1) a logistic regression model based solely on public data, and (2) a differentially private distributed logistic regression model based on private data under various scenarios.

Conclusion

Logistic regression models built with our new algorithm based on both private and public datasets demonstrate better utility than models that trained on private or public datasets alone without sacrificing the rigorous privacy guarantee.

     

On the identifiability of metabolic network models

A major problem for the identification of metabolic network models is parameter identifiability, that is, the possibility to unambiguously infer the parameter values from the data. Identifiability problems may be due to the structure of the model, in particular implicit dependencies between the parameters, or to limitations in the quantity and quality of the available data. We address the detection and resolution of identifiability problems for a class of pseudo-linear models of metabolism, so-called linlog models. Linlog models have the advantage that parameter estimation reduces to linear or orthogonal regression, which facilitates the analysis of identifiability. We develop precise definitions of structural and practical identifiability, and clarify the fundamental relations between these concepts. In addition, we use singular value decomposition to detect identifiability problems and reduce the model to an identifiable approximation by a principal component analysis approach. The criterion is adapted to real data, which are frequently scarce, incomplete, and noisy. The test of the criterion on a model with simulated data shows that it is capable of correctly identifying the principal components of the data vector. The application to a state-of-the-art dataset on central carbon metabolism in Escherichia coli yields the surprising result that only $4$ out of $31$ reactions, and $37$ out of $100$ parameters, are identifiable. This underlines the practical importance of identifiability analysis and model reduction in the modeling of large-scale metabolic networks. Although our approach has been developed in the context of linlog models, it carries over to other pseudo-linear models, such as generalized mass-action (power-law) models. Moreover, it provides useful hints for the identifiability analysis of more general classes of nonlinear models of metabolism.     

SNPchiMp v.3: integrating and standardizing single nucleotide polymorphism data for livestock species

Background

In recent years, the use of genomic information in livestock species for genetic improvement, association studies and many other fields has become routine. In order to accommodate different market requirements in terms of genotyping cost, manufacturers of single nucleotide polymorphism (SNP) arrays, private companies and international consortia have developed a large number of arrays with different content and different SNP density. The number of currently available SNP arrays differs among species: ranging from one for goats to more than ten for cattle, and the number of arrays available is increasing rapidly. However, there is limited or no effort to standardize and integrate array- specific (e.g. SNP IDs, allele coding) and species-specific (i.e. past and current assemblies) SNP information.

Results

Here we present SNPchiMp v.3, a solution to these issues for the six major livestock species (cow, pig, horse, sheep, goat and chicken). Original data was collected directly from SNP array producers and specific international genome consortia, and stored in a MySQL database. The database was then linked to an open-access web tool and to public databases. SNPchiMp v.3 ensures fast access to the database (retrieving within/across SNP array data) and the possibility of annotating SNP array data in a user-friendly fashion.

Conclusions

This platform allows easy integration and standardization, and it is aimed at both industry and research. It also enables users to easily link the information available from the array producer with data in public databases, without the need of additional bioinformatics tools or pipelines. In recognition of the open-access use of Ensembl resources, SNPchiMp v.3 was officially credited as an Ensembl E!mpowered tool. Availability at http://bioinformatics.tecnoparco.org/SNPchimp.     

Distributed Shared Arrays: Portable Shared-Memory Programming Interface for Multiple Computer Systems

This paper describes the design and implementation of a parallel programming environment called Distributed Shared Array (DSA), which provides a shared global array abstract across different machines connected by a network. In DSA, users can define and use global arrays that can be accessed uniformly from any machines in the network. Explicit management of array area allocation, replication, and migration is achieved by explicit calls for array manipulation: defining array regions, reading and writing array regions, synchronization, and control of replication and migration. The DSA is integrated with Grid (Globus) services. This paper also describes the use of our model for gene cluster analysis, multiple alignment and molecular dynamics simulation. In these applications, global arrays are used for storing the distance matrix, alignment matrix and atom coordinates, respectively. Large array areas, which cannot be stored in the memory of individual machines, are made available by the DSA. Scalable performance of DSA was obtained compared to that of conventional parallel programs written in MPI.     

A novel mechanism important for the alignment of microtubules

Using a live-cell imaging approach to study individual micro-tubules, we have compared microtubule behavior between net-like and aligned cortical arrays. In contrast to previous studies, a steep angled collision between the growing end of a microtubule and a preexisting microtubule was found to favor crossover. Frequencies of microtubule crossovers, bundling and catastrophes are similar regardless of whether the cell exhibited a net-like or aligned microtubule array. In the predominantly aligned array of petiole cells, severing occurs at the sites of microtubule crossovers and serves to remove unaligned microtubules and to increase microtubule density. Severing was observed to be rare in net-like arrays. Microtubule severing is carried out by the katanin enzyme. In this addendum, we present new insights into the possible mechanism of crossing over and preliminary data looking at organization of the array in a katanin mutant.Key words: cortical array, collision, catastrophe, severing, katanin, petiole, pavement, cotyledons, YFPIn plants, proper cell expansion requires the transformation of the cortical microtubule array from a net-like configuration to an aligned and predominantly transverse configuration. The mechanisms underlying this transition have been, until recently, poorly understood. Previous observations in tobacco BY2 cells suggested that the outcome of a collision between a growing microtubule plus end and an existing microtubule determined the configuration of the array.1 In microtubule arrays of BY2 cells, steep angled collisions favor rapid depolymerization, termed a catastrophe, whereas shallow angled collisions promote bundling. These different collision outcomes are depicted in figures 1a and 1b. By applying these experimental observations to a computer model of BY2 microtubules, a disordered array could be seen to gradually transform into an aligned array.The cotyledons of Arabidopsis seedlings are well suited for determining how interactions between individual microtubules affect the configuration of the whole array.² Pavement cells exhibit net-like arrays whereas petiole cells exhibit aligned arrays. By using a yellow fluorescent protein-based microtubule reporter that exclusively labels arrays of the outermost cell layer of the cotyledon, the behavior of microtubules were examined in pavement and petiole cells and compared with the data from the BY2 study.¹In both Arabidopsis and BY2 cells, shallow angle collisions promote bundling and we expected catastrophe frequency from steep-angled collisions to be similar in BY2 cells and Arabidopsis. However, although steep angled collisions at the growing ends of microtubules in BY2 cells exhibit a catastrophe frequency of 60%, this is much reduced in Arabidopsis. The net-like arrays of pavement cells and the aligned arrays of petiole cells exhibit catastrophe frequencies of 24% and 9%, respectively.² The data suggests that in Arabidopsis the transition from net-like to aligned microtubule arrangement is accompanied by a decrease in the catastrophe frequency, with most steep-angled collisions resulting in a crossover event (Fig. 1C). A crossover event occurs when the growing leading end of a microtubule overcomes an obstacle in the form of an existing microtubule. The mechanism that allows a growing microtubule end to form a crossover is not fully understood, however, the end of the growing face of the microtubule is likely to be in contact with the obstruction because crossovers are accompanied by a reduction in the rate of microtubule growth.² These data suggest that, for Arabidopsis, growing ends are somehow stabilized and resist the catastrophes observed in BY2 cells. In Xenopus, XMAP215 increases microtubule growth rates and reduces the catastrophe frequency in egg extracts during interphase³ as well as increasing crossovers similar to those seen in Arabidopsis (http://www.mpi-cbg.de/downloads/movie/interRT.mov). Consistent with this observation, depletion of XMAP215 in Xenopus extracts results in an increase in the microtubule catastrophe frequency. A temperature sensitive mutant allele of mor1, the Arabidopsis homolog of XMAP215 results in shorter cortical microtubules and loss of parallel alignment.⁴ MOR1 is therefore a strong candidate for a stabilizing factor during microtubule crossover.Open in a separate windowFigure 1Schematic representations of collision outcomes (A) catastrophe (B) bundling (C) crossover.Both the net-like arrays of pavement cells and the aligned arrays of petiole cells exhibit high levels of microtubule crossover, suggesting that some other mechanism contributes to array alignment. A detailed analysis of microtubule dynamics in the aligned arrays of the petiole cells revealed a single behavior that was markedly increased compared to the pavement cells—severing at the sites of microtubule crossover.² A severing event from an aligning array is shown in Figure 2. Severing results in removal of the posterior portion of the unaligned microtubule whilst the anterior portion can grow in an arc and bundle with the rest of the array. The posterior portion either depolymerizes completely or forms a new microtubule that grows in a new direction in an attempt to align with the rest of the array. This means that severing at crossovers has two roles, removing those microtubules that are not aligned with the rest of the array and increasing microtubule density. This results in a densely packed, aligned array as seen in petiole cells.Open in a separate windowFigure 2A microtubule crossover before (left panel) and immediately after (right panel) severing. Site of crossover is shown by an arrow. Bar = 1 µm.In higher eukaryotes, severing is carried out by the enzyme katanin.5 In Arabidopsis, several katanin mutants have been described and exhibit phenotypes consistent with a defect in cell expansion.^6–9 In one such mutant, called botero1, roots fail to make the transition from a net-like array in the division zone to the aligned array in the elongation zone. Historically, the relationship between katanin function and array alignment has been hard to interpret because katanin function has only been associated with severing at the sites of microtubule nucleation. According to our data, alignment is dependent upon severing at crossovers, therefore, in a katanin mutant, petiole cells are predicted to contain only net-like arrays. Figure 3 compares microtubule arrays in pavement and petiole cells between wild-type Arabidopsis and the botero1–7 mutant. The cortical arrays of pavement cells are very similar between WT and mutant suggesting katanin function is not important for the generation of a net-like array. However, in petiole cells of botero1–7 the microtubules fail to adopt the aligned configuration observed in WT. This failure of array alignment in the petiole cells has been observed previously for another katanin mutant, fra2.⁸Open in a separate windowFigure 3Comparison of microtubule arrays between WT (ecotype Ws) and botero1–7 plants. Note the alignment defect in petioles of botero1–7. Bar = 10 µm.To determine the relative effects of severing and other behaviors of microtubules upon array organization, accurate modeling of micro- tubules will be essential. Work is currently underway to mimic, in detail, microtubule dynamics using advanced computer modeling techniques (Marshall R, Department of Physics and Astronomy, University of Manchester, personal communication). This modeling will include user changeable parameters such as severing frequency, catastrophe frequency and the outcome of a catastrophe. Comparing the models with biological systems will increase our understanding of the importance of the various behaviours in array organisation.