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| bshanks@0 | 1 Specific aims
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| bshanks@53 | 2 Massivenew datasets obtained with techniques such as in situ hybridization (ISH), immunohistochemistry, in situ transgenic
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| bshanks@53 | 3 reporter, microarray voxelation, and others, allow the expression levels of many genes at many locations to be compared.
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| bshanks@53 | 4 Our goal is to develop automated methods to relate spatial variation in gene expression to anatomy. We want to find marker
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| bshanks@53 | 5 genes for specific anatomical regions, and also to draw new anatomical maps based on gene expression patterns. We have
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| bshanks@53 | 6 three specific aims:
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| bshanks@30 | 7 (1) develop an algorithm to screen spatial gene expression data for combinations of marker genes which selectively target
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| bshanks@30 | 8 anatomical regions
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| bshanks@84 | 9 (2) develop an algorithm to suggest new ways of carving up a structure into anatomically distinct regions, based on
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| bshanks@84 | 10 spatial patterns in gene expression
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| bshanks@33 | 11 (3) create a 2-D “flat map” dataset of the mouse cerebral cortex that contains a flattened version of the Allen Mouse
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| bshanks@35 | 12 Brain Atlas ISH data, as well as the boundaries of cortical anatomical areas. This will involve extending the functionality of
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| bshanks@35 | 13 Caret, an existing open-source scientific imaging program. Use this dataset to validate the methods developed in (1) and (2).
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| bshanks@84 | 14 Although our particular application involves the 3D spatial distribution of gene expression, we anticipate that the methods
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| bshanks@84 | 15 developed in aims (1) and (2) will generalize to any sort of high-dimensional data over points located in a low-dimensional
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| bshanks@84 | 16 space.
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| bshanks@84 | 17 In terms of the application of the methods to cerebral cortex, aim (1) is to go from cortical areas to marker genes,
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| bshanks@84 | 18 and aim (2) is to let the gene profile define the cortical areas. In addition to validating the usefulness of the algorithms,
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| bshanks@84 | 19 the application of these methods to cortex will produce immediate benefits, because there are currently no known genetic
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| bshanks@84 | 20 markers for most cortical areas. The results of the project will support the development of new ways to selectively target
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| bshanks@84 | 21 cortical areas, and it will support the development of a method for identifying the cortical areal boundaries present in small
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| bshanks@84 | 22 tissue samples.
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| bshanks@53 | 23 All algorithms that we develop will be implemented in a GPL open-source software toolkit. The toolkit, as well as the
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| bshanks@30 | 24 machine-readable datasets developed in aim (3), will be published and freely available for others to use.
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| bshanks@30 | 25 Background and significance
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| bshanks@84 | 26 Aim 1: Given a map of regions, find genes that mark the regions
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| bshanks@84 | 27 After defining terms, we will describe a set of principles which determine our strategy to completing this aim.
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| bshanks@84 | 28 Machine learning terminology: supervised learning The task of looking for marker genes for known anatomical
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| bshanks@84 | 29 regions means that one is looking for a set of genes such that, if the expression level of those genes is known, then the
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| bshanks@84 | 30 locations of the regions can be inferred.
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| bshanks@42 | 31 If we define the regions so that they cover the entire anatomical structure to be divided, then instead of saying that we
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| bshanks@42 | 32 are using gene expression to find the locations of the regions, we may say that we are using gene expression to determine to
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| bshanks@42 | 33 which region each voxel within the structure belongs. We call this a classification task, because each voxel is being assigned
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| bshanks@42 | 34 to a class (namely, its region).
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| bshanks@30 | 35 Therefore, an understanding of the relationship between the combination of their expression levels and the locations of
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| bshanks@42 | 36 the regions may be expressed as a function. The input to this function is a voxel, along with the gene expression levels
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| bshanks@42 | 37 within that voxel; the output is the regional identity of the target voxel, that is, the region to which the target voxel belongs.
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| bshanks@42 | 38 We call this function a classifier. In general, the input to a classifier is called an instance, and the output is called a label
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| bshanks@42 | 39 (or a class label).
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| bshanks@30 | 40 The object of aim 1 is not to produce a single classifier, but rather to develop an automated method for determining a
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| bshanks@30 | 41 classifier for any known anatomical structure. Therefore, we seek a procedure by which a gene expression dataset may be
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| bshanks@30 | 42 analyzed in concert with an anatomical atlas in order to produce a classifier. Such a procedure is a type of a machine learning
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| bshanks@33 | 43 procedure. The construction of the classifier is called training (also learning), and the initial gene expression dataset used
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| bshanks@33 | 44 in the construction of the classifier is called training data.
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| bshanks@30 | 45 In the machine learning literature, this sort of procedure may be thought of as a supervised learning task, defined as a
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| bshanks@30 | 46 task in which the goal is to learn a mapping from instances to labels, and the training data consists of a set of instances
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| bshanks@42 | 47 (voxels) for which the labels (regions) are known.
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| bshanks@30 | 48 Each gene expression level is called a feature, and the selection of which genes1 to include is called feature selection.
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| bshanks@33 | 49 Feature selection is one component of the task of learning a classifier. Some methods for learning classifiers start out with
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| bshanks@33 | 50 a separate feature selection phase, whereas other methods combine feature selection with other aspects of training.
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| bshanks@30 | 51 One class of feature selection methods assigns some sort of score to each candidate gene. The top-ranked genes are then
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| bshanks@30 | 52 chosen. Some scoring measures can assign a score to a set of selected genes, not just to a single gene; in this case, a dynamic
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| bshanks@30 | 53 procedure may be used in which features are added and subtracted from the selected set depending on how much they raise
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| bshanks@30 | 54 the score. Such procedures are called “stepwise” or “greedy”.
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| bshanks@30 | 55 Although the classifier itself may only look at the gene expression data within each voxel before classifying that voxel, the
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| bshanks@30 | 56 learning algorithm which constructs the classifier may look over the entire dataset. We can categorize score-based feature
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| bshanks@30 | 57 selection methods depending on how the score of calculated. Often the score calculation consists of assigning a sub-score to
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| bshanks@53 | 58 each voxel, and then aggregating these sub-scores into a final score (the aggregation is often a sum or a sum of squares or
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| bshanks@53 | 59 average). If only information from nearby voxels is used to calculate a voxel’s sub-score, then we say it is a local scoring
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| bshanks@53 | 60 method. If only information from the voxel itself is used to calculate a voxel’s sub-score, then we say it is a pointwise scoring
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| bshanks@53 | 61 method.
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| bshanks@30 | 62 Key questions when choosing a learning method are: What are the instances? What are the features? How are the
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| bshanks@30 | 63 features chosen? Here are four principles that outline our answers to these questions.
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| bshanks@84 | 64 Principle 1: Combinatorial gene expression
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| bshanks@84 | 65 It is too much to hope that every anatomical region of interest will be identified by a single gene. For example, in the
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| bshanks@84 | 66 cortex, there are some areas which are not clearly delineated by any gene included in the Allen Brain Atlas (ABA) dataset.
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| bshanks@84 | 67 However, at least some of these areas can be delineated by looking at combinations of genes (an example of an area for
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| bshanks@84 | 68 which multiple genes are necessary and sufficient is provided in Preliminary Studies, Figure 4). Therefore, each instance
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| bshanks@84 | 69 should contain multiple features (genes).
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| bshanks@84 | 70 Principle 2: Only look at combinations of small numbers of genes
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| bshanks@84 | 71 When the classifier classifies a voxel, it is only allowed to look at the expression of the genes which have been selected
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| bshanks@84 | 72 as features. The more data that are available to a classifier, the better that it can do. For example, perhaps there are weak
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| bshanks@84 | 73 correlations over many genes that add up to a strong signal. So, why not include every gene as a feature? The reason is that
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| bshanks@84 | 74 we wish to employ the classifier in situations in which it is not feasible to gather data about every gene. For example, if we
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| bshanks@84 | 75 want to use the expression of marker genes as a trigger for some regionally-targeted intervention, then our intervention must
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| bshanks@84 | 76 contain a molecular mechanism to check the expression level of each marker gene before it triggers. It is currently infeasible
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| bshanks@84 | 77 to design a molecular trigger that checks the level of more than a handful of genes. Similarly, if the goal is to develop a
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| bshanks@84 | 78 _________________________________________
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| bshanks@63 | 79 1Strictly speaking, the features are gene expression levels, but we’ll call them genes.
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| bshanks@84 | 80 procedure to do ISH on tissue samples in order to label their anatomy, then it is infeasible to label more than a few genes.
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| bshanks@84 | 81 Therefore, we must select only a few genes as features.
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| bshanks@63 | 82 The requirement to find combinations of only a small number of genes limits us from straightforwardly applying many
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| bshanks@63 | 83 of the most simple techniques from the field of supervised machine learning. In the parlance of machine learning, our task
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| bshanks@63 | 84 combines feature selection with supervised learning.
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| bshanks@30 | 85 Principle 3: Use geometry in feature selection
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| bshanks@30 | 86 When doing feature selection with score-based methods, the simplest thing to do would be to score the performance of
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| bshanks@30 | 87 each voxel by itself and then combine these scores (pointwise scoring). A more powerful approach is to also use information
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| bshanks@30 | 88 about the geometric relations between each voxel and its neighbors; this requires non-pointwise, local scoring methods. See
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| bshanks@84 | 89 Preliminary Studies, figure 3 for evidence of the complementary nature of pointwise and local scoring methods.
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| bshanks@30 | 90 Principle 4: Work in 2-D whenever possible
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| bshanks@30 | 91 There are many anatomical structures which are commonly characterized in terms of a two-dimensional manifold. When
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| bshanks@30 | 92 it is known that the structure that one is looking for is two-dimensional, the results may be improved by allowing the analysis
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| bshanks@33 | 93 algorithm to take advantage of this prior knowledge. In addition, it is easier for humans to visualize and work with 2-D
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| bshanks@33 | 94 data.
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| bshanks@30 | 95 Therefore, when possible, the instances should represent pixels, not voxels.
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| bshanks@43 | 96 Related work
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| bshanks@44 | 97 There is a substantial body of work on the analysis of gene expression data, most of this concerns gene expression data
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| bshanks@84 | 98 which are not fundamentally spatial2.
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| bshanks@43 | 99 As noted above, there has been much work on both supervised learning and there are many available algorithms for
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| bshanks@43 | 100 each. However, the algorithms require the scientist to provide a framework for representing the problem domain, and the
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| bshanks@43 | 101 way that this framework is set up has a large impact on performance. Creating a good framework can require creatively
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| bshanks@43 | 102 reconceptualizing the problem domain, and is not merely a mechanical “fine-tuning” of numerical parameters. For example,
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| bshanks@84 | 103 we believe that domain-specific scoring measures (such as gradient similarity, which is discussed in Preliminary Studies) may
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| bshanks@43 | 104 be necessary in order to achieve the best results in this application.
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| bshanks@53 | 105 We are aware of six existing efforts to find marker genes using spatial gene expression data using automated methods.
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| bshanks@53 | 106 [8 ] mentions the possibility of constructing a spatial region for each gene, and then, for each anatomical structure of
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| bshanks@53 | 107 interest, computing what proportion of this structure is covered by the gene’s spatial region.
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| bshanks@53 | 108 GeneAtlas[3] and EMAGE [18] allow the user to construct a search query by demarcating regions and then specifing
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| bshanks@53 | 109 either the strength of expression or the name of another gene or dataset whose expression pattern is to be matched. For the
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| bshanks@53 | 110 similiarity score (match score) between two images (in this case, the query and the gene expression images), GeneAtlas uses
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| bshanks@53 | 111 the sum of a weighted L1-norm distance between vectors whose components represent the number of cells within a pixel3
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| bshanks@53 | 112 whose expression is within four discretization levels. EMAGE uses Jaccard similarity, which is equal to the number of true
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| bshanks@53 | 113 pixels in the intersection of the two images, divided by the number of pixels in their union. Neither GeneAtlas nor EMAGE
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| bshanks@53 | 114 allow one to search for combinations of genes that define a region in concert but not separately.
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| bshanks@53 | 115 [10 ] describes AGEA, ”Anatomic Gene Expression Atlas”. AGEA has three components:
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| bshanks@61 | 116 ∙Gene Finder: The user selects a seed voxel and the system (1) chooses a cluster which includes the seed voxel, (2)
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| bshanks@61 | 117 yields a list of genes which are overexpressed in that cluster. (note: the ABA website also contains pre-prepared lists
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| bshanks@61 | 118 of overexpressed genes for selected structures)
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| bshanks@84 | 119 ∙Correlation: The user selects a seed voxel and the system then shows the user how much correlation there is between
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| bshanks@84 | 120 the gene expression profile of the seed voxel and every other voxel.
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| bshanks@61 | 121 ∙Clusters: will be described later
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| bshanks@43 | 122 Gene Finder is different from our Aim 1 in at least three ways. First, Gene Finder finds only single genes, whereas we
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| bshanks@43 | 123 will also look for combinations of genes. Second, gene finder can only use overexpression as a marker, whereas we will also
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| bshanks@53 | 124 search for underexpression. Third, Gene Finder uses a simple pointwise score4, whereas we will also use geometric scores
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| bshanks@84 | 125 such as gradient similarity (described in Preliminary Studies). Figures 4, 2, and 3 in the Preliminary Studies section contains
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| bshanks@84 | 126 evidence that each of our three choices is the right one.
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| bshanks@53 | 127 [4 ] looks at the mean expression level of genes within anatomical regions, and applies a Student’s t-test with Bonferroni
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| bshanks@51 | 128 correction to determine whether the mean expression level of a gene is significantly higher in the target region. Like AGEA,
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| bshanks@84 | 129 _________________________________________
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| bshanks@84 | 130 2By “fundamentally spatial” we mean that there is information from a large number of spatial locations indexed by spatial coordinates; not
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| bshanks@84 | 131 just data which have only a few different locations or which is indexed by anatomical label.
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| bshanks@84 | 132 3Actually, many of these projects use quadrilaterals instead of square pixels; but we will refer to them as pixels for simplicity.
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| bshanks@84 | 133 4“Expression energy ratio”, which captures overexpression.
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| bshanks@51 | 134 this is a pointwise measure (only the mean expression level per pixel is being analyzed), it is not being used to look for
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| bshanks@51 | 135 underexpression, and does not look for combinations of genes.
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| bshanks@53 | 136 [7 ] describes a technique to find combinations of marker genes to pick out an anatomical region. They use an evolutionary
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| bshanks@46 | 137 algorithm to evolve logical operators which combine boolean (thresholded) images in order to match a target image. Their
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| bshanks@51 | 138 match score is Jaccard similarity.
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| bshanks@84 | 139 In summary, there has been fruitful work on finding marker genes, but only one of the previous projects explores
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| bshanks@51 | 140 combinations of marker genes, and none of these publications compare the results obtained by using different algorithms or
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| bshanks@51 | 141 scoring methods.
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| bshanks@84 | 142 Aim 2: From gene expression data, discover a map of regions
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| bshanks@30 | 143 Machine learning terminology: clustering
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| bshanks@30 | 144 If one is given a dataset consisting merely of instances, with no class labels, then analysis of the dataset is referred to as
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| bshanks@30 | 145 unsupervised learning in the jargon of machine learning. One thing that you can do with such a dataset is to group instances
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| bshanks@46 | 146 together. A set of similar instances is called a cluster, and the activity of finding grouping the data into clusters is called
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| bshanks@46 | 147 clustering or cluster analysis.
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| bshanks@84 | 148 The task of deciding how to carve up a structure into anatomical regions can be put into these terms. The instances
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| bshanks@84 | 149 are once again voxels (or pixels) along with their associated gene expression profiles. We make the assumption that voxels
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| bshanks@84 | 150 from the same anatomical region have similar gene expression profiles, at least compared to the other regions. This means
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| bshanks@84 | 151 that clustering voxels is the same as finding potential regions; we seek a partitioning of the voxels into regions, that is, into
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| bshanks@84 | 152 clusters of voxels with similar gene expression.
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| bshanks@42 | 153 It is desirable to determine not just one set of regions, but also how these regions relate to each other, if at all; perhaps
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| bshanks@44 | 154 some of the regions are more similar to each other than to the rest, suggesting that, although at a fine spatial scale they
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| bshanks@42 | 155 could be considered separate, on a coarser spatial scale they could be grouped together into one large region. This suggests
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| bshanks@42 | 156 the outcome of clustering may be a hierarchial tree of clusters, rather than a single set of clusters which partition the voxels.
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| bshanks@42 | 157 This is called hierarchial clustering.
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| bshanks@30 | 158 Similarity scores
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| bshanks@30 | 159 A crucial choice when designing a clustering method is how to measure similarity, across either pairs of instances, or
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| bshanks@33 | 160 clusters, or both. There is much overlap between scoring methods for feature selection (discussed above under Aim 1) and
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| bshanks@30 | 161 scoring methods for similarity.
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| bshanks@30 | 162 Spatially contiguous clusters; image segmentation
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| bshanks@33 | 163 We have shown that aim 2 is a type of clustering task. In fact, it is a special type of clustering task because we have
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| bshanks@33 | 164 an additional constraint on clusters; voxels grouped together into a cluster must be spatially contiguous. In Preliminary
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| bshanks@84 | 165 Studies, we show that one can get reasonable results without enforcing this constraint; however, we plan to compare these
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| bshanks@33 | 166 results against other methods which guarantee contiguous clusters.
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| bshanks@30 | 167 Perhaps the biggest source of continguous clustering algorithms is the field of computer vision, which has produced a
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| bshanks@33 | 168 variety of image segmentation algorithms. Image segmentation is the task of partitioning the pixels in a digital image into
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| bshanks@30 | 169 clusters, usually contiguous clusters. Aim 2 is similar to an image segmentation task. There are two main differences; in
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| bshanks@84 | 170 our task, there are thousands of color channels (one for each gene), rather than just three. However, there are imaging
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| bshanks@84 | 171 tasks which use more than three colors, for example multispectral imaging and hyperspectral imaging, which are often used
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| bshanks@33 | 172 to process satellite imagery. A more crucial difference is that there are various cues which are appropriate for detecting
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| bshanks@33 | 173 sharp object boundaries in a visual scene but which are not appropriate for segmenting abstract spatial data such as gene
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| bshanks@33 | 174 expression. Although many image segmentation algorithms can be expected to work well for segmenting other sorts of
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| bshanks@33 | 175 spatially arranged data, some of these algorithms are specialized for visual images.
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| bshanks@51 | 176 Dimensionality reduction In this section, we discuss reducing the length of the per-pixel gene expression feature
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| bshanks@51 | 177 vector. By “dimension”, we mean the dimension of this vector, not the spatial dimension of the underlying data.
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| bshanks@33 | 178 Unlike aim 1, there is no externally-imposed need to select only a handful of informative genes for inclusion in the
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| bshanks@30 | 179 instances. However, some clustering algorithms perform better on small numbers of features. There are techniques which
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| bshanks@30 | 180 “summarize” a larger number of features using a smaller number of features; these techniques go by the name of feature
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| bshanks@30 | 181 extraction or dimensionality reduction. The small set of features that such a technique yields is called the reduced feature
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| bshanks@30 | 182 set. After the reduced feature set is created, the instances may be replaced by reduced instances, which have as their features
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| bshanks@30 | 183 the reduced feature set rather than the original feature set of all gene expression levels. Note that the features in the reduced
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| bshanks@30 | 184 feature set do not necessarily correspond to genes; each feature in the reduced set may be any function of the set of gene
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| bshanks@30 | 185 expression levels.
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| bshanks@51 | 186 Dimensionality reduction before clustering is useful on large datasets. First, because the number of features in the
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| bshanks@84 | 187 reduced dataset is less than in the original dataset, the running time of clustering algorithms may be much less. Second, it
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| bshanks@84 | 188 is thought that some clustering algorithms may give better results on reduced data.
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| bshanks@51 | 189 Another use for dimensionality reduction is to visualize the relationships between regions after clustering. For example,
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| bshanks@84 | 190 one might want to make a 2-D plot upon which each region is represented by a single point, and with the property that
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| bshanks@84 | 191 regions with similar gene expression profiles should be nearby on the plot (that is, the property that distance between
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| bshanks@84 | 192 pairs of points in the plot should be proportional to some measure of dissimilarity in gene expression). It is likely that no
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| bshanks@84 | 193 arrangement of the points on a 2-D plan will exactly satisfy this property; however, dimensionality reduction techniques allow
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| bshanks@84 | 194 one to find arrangements of points that approximately satisfy that property. Note that in this application, dimensionality
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| bshanks@84 | 195 reduction is being applied after clustering; whereas in the previous paragraph, we were talking about using dimensionality
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| bshanks@84 | 196 reduction before clustering.
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| bshanks@30 | 197 Clustering genes rather than voxels
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| bshanks@30 | 198 Although the ultimate goal is to cluster the instances (voxels or pixels), one strategy to achieve this goal is to first cluster
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| bshanks@30 | 199 the features (genes). There are two ways that clusters of genes could be used.
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| bshanks@30 | 200 Gene clusters could be used as part of dimensionality reduction: rather than have one feature for each gene, we could
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| bshanks@30 | 201 have one reduced feature for each gene cluster.
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| bshanks@30 | 202 Gene clusters could also be used to directly yield a clustering on instances. This is because many genes have an expression
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| bshanks@53 | 203 pattern which seems to pick out a single, spatially continguous region. Therefore, it seems likely that an anatomically
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| bshanks@53 | 204 interesting region will have multiple genes which each individually pick it out5. This suggests the following procedure:
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| bshanks@42 | 205 cluster together genes which pick out similar regions, and then to use the more popular common regions as the final clusters.
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| bshanks@84 | 206 In Preliminary Studies, Figure 7, we show that a number of anatomically recognized cortical regions, as well as some
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| bshanks@84 | 207 “superregions” formed by lumping together a few regions, are associated with gene clusters in this fashion.
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| bshanks@51 | 208 The task of clustering both the instances and the features is called co-clustering, and there are a number of co-clustering
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| bshanks@51 | 209 algorithms.
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| bshanks@43 | 210 Related work
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| bshanks@51 | 211 We are aware of five existing efforts to cluster spatial gene expression data.
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| bshanks@53 | 212 [15 ] describes an analysis of the anatomy of the hippocampus using the ABA dataset. In addition to manual analysis,
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| bshanks@43 | 213 two clustering methods were employed, a modified Non-negative Matrix Factorization (NNMF), and a hierarchial recursive
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| bshanks@44 | 214 bifurcation clustering scheme based on correlation as the similarity score. The paper yielded impressive results, proving
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| bshanks@53 | 215 the usefulness of computational genomic anatomy. We have run NNMF on the cortical dataset6 and while the results are
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| bshanks@84 | 216 promising, they also demonstrate that NNMF is not necessarily the best dimensionality reduction method for this application
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| bshanks@84 | 217 (see Preliminary Studies, Figure 6).
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| bshanks@53 | 218 AGEA[10] includes a preset hierarchial clustering of voxels based on a recursive bifurcation algorithm with correlation
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| bshanks@53 | 219 as the similarity metric. EMAGE[18] allows the user to select a dataset from among a large number of alternatives, or by
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| bshanks@53 | 220 running a search query, and then to cluster the genes within that dataset. EMAGE clusters via hierarchial complete linkage
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| bshanks@53 | 221 clustering with un-centred correlation as the similarity score.
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| bshanks@53 | 222 [4 ] clustered genes, starting out by selecting 135 genes out of 20,000 which had high variance over voxels and which were
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| bshanks@53 | 223 highly correlated with many other genes. They computed the matrix of (rank) correlations between pairs of these genes, and
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| bshanks@53 | 224 ordered the rows of this matrix as follows: “the first row of the matrix was chosen to show the strongest contrast between
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| bshanks@53 | 225 the highest and lowest correlation coefficient for that row. The remaining rows were then arranged in order of decreasing
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| bshanks@53 | 226 similarity using a least squares metric”. The resulting matrix showed four clusters. For each cluster, prototypical spatial
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| bshanks@53 | 227 expression patterns were created by averaging the genes in the cluster. The prototypes were analyzed manually, without
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| bshanks@53 | 228 clustering voxels
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| bshanks@53 | 229 In an interesting twist, [7] applies their technique for finding combinations of marker genes for the purpose of clustering
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| bshanks@46 | 230 genes around a “seed gene”. The way they do this is by using the pattern of expression of the seed gene as the target image,
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| bshanks@46 | 231 and then searching for other genes which can be combined to reproduce this pattern. Those other genes which are found
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| bshanks@53 | 232 are considered to be related to the seed. The same team also describes a method[17] for finding “association rules” such as,
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| bshanks@46 | 233 “if this voxel is expressed in by any gene, then that voxel is probably also expressed in by the same gene”. This could be
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| bshanks@46 | 234 useful as part of a procedure for clustering voxels.
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| bshanks@46 | 235 In summary, although these projects obtained clusterings, there has not been much comparison between different algo-
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| bshanks@51 | 236 rithms or scoring methods, so it is likely that the best clustering method for this application has not yet been found. Also,
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| bshanks@53 | 237 none of these projects did a separate dimensionality reduction step before clustering pixels, none tried to cluster genes first
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| bshanks@53 | 238 in order to guide automated clustering of pixels into spatial regions, and none used co-clustering algorithms.
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| bshanks@63 | 239 _________________________________________
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| bshanks@63 | 240 5This would seem to contradict our finding in aim 1 that some cortical areas are combinatorially coded by multiple genes. However, it is
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| bshanks@63 | 241 possible that the currently accepted cortical maps divide the cortex into regions which are unnatural from the point of view of gene expression;
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| bshanks@63 | 242 perhaps there is some other way to map the cortex for which each region can be identified by single genes. Another possibility is that, although
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| bshanks@63 | 243 the cluster prototype fits an anatomical region, the individual genes are each somewhat different from the prototype.
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| bshanks@63 | 244 6We ran “vanilla” NNMF, whereas the paper under discussion used a modified method. Their main modification consisted of adding a soft
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| bshanks@63 | 245 spatial contiguity constraint. However, on our dataset, NNMF naturally produced spatially contiguous clusters, so no additional constraint was
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| bshanks@63 | 246 needed. The paper under discussion also mentions that they tried a hierarchial variant of NNMF, which we have not yet tried.
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| bshanks@30 | 247 Aim 3
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| bshanks@30 | 248 Background
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| bshanks@84 | 249 The cortex is divided into areas and layers. Because of the cortical columnar organization, the parcellation of the cortex
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| bshanks@84 | 250 into areas can be drawn as a 2-D map on the surface of the cortex. In the third dimension, the boundaries between the
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| bshanks@84 | 251 areas continue downwards into the cortical depth, perpendicular to the surface. The layer boundaries run parallel to the
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| bshanks@84 | 252 surface. One can picture an area of the cortex as a slice of a six-layered cake7.
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| bshanks@30 | 253 Although it is known that different cortical areas have distinct roles in both normal functioning and in disease processes,
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| bshanks@84 | 254 there are no known marker genes for most cortical areas. When it is necessary to divide a tissue sample into cortical areas,
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| bshanks@30 | 255 this is a manual process that requires a skilled human to combine multiple visual cues and interpret them in the context of
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| bshanks@30 | 256 their approximate location upon the cortical surface.
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| bshanks@33 | 257 Even the questions of how many areas should be recognized in cortex, and what their arrangement is, are still not
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| bshanks@53 | 258 completely settled. A proposed division of the cortex into areas is called a cortical map. In the rodent, the lack of a single
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| bshanks@53 | 259 agreed-upon map can be seen by contrasting the recent maps given by Swanson[14] on the one hand, and Paxinos and
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| bshanks@53 | 260 Franklin[11] on the other. While the maps are certainly very similar in their general arrangement, significant differences
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| bshanks@30 | 261 remain in the details.
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| bshanks@36 | 262 The Allen Mouse Brain Atlas dataset
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| bshanks@84 | 263 The Allen Mouse Brain Atlas (ABA) data were produced by doing in-situ hybridization on slices of male, 56-day-old
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| bshanks@36 | 264 C57BL/6J mouse brains. Pictures were taken of the processed slice, and these pictures were semi-automatically analyzed
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| bshanks@36 | 265 in order to create a digital measurement of gene expression levels at each location in each slice. Per slice, cellular spatial
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| bshanks@36 | 266 resolution is achieved. Using this method, a single physical slice can only be used to measure one single gene; many different
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| bshanks@36 | 267 mouse brains were needed in order to measure the expression of many genes.
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| bshanks@36 | 268 Next, an automated nonlinear alignment procedure located the 2D data from the various slices in a single 3D coordinate
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| bshanks@36 | 269 system. In the final 3D coordinate system, voxels are cubes with 200 microns on a side. There are 67x41x58 = 159,326
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| bshanks@53 | 270 voxels in the 3D coordinate system, of which 51,533 are in the brain[10].
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| bshanks@53 | 271 Mus musculus, the common house mouse, is thought to contain about 22,000 protein-coding genes[20]. The ABA contains
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| bshanks@36 | 272 data on about 20,000 genes in sagittal sections, out of which over 4,000 genes are also measured in coronal sections. Our
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| bshanks@84 | 273 dataset is derived from only the coronal subset of the ABA, because the sagittal data do not cover the entire cortex, and
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| bshanks@53 | 274 also has greater registration error[10]. Genes were selected by the Allen Institute for coronal sectioning based on, “classes
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| bshanks@53 | 275 of known neuroscientific interest... or through post hoc identification of a marked non-ubiquitous expression pattern”[10].
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| bshanks@53 | 276 The ABA is not the only large public spatial gene expression dataset. Other such resources include GENSAT[6],
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| bshanks@84 | 277 GenePaint[19], its sister project GeneAtlas[3], BGEM[9], EMAGE[18], EurExpress8, EADHB9, MAMEP10, Xenbase11,
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| bshanks@84 | 278 ZFIN[13], Aniseed12, VisiGene13, GEISHA[2], Fruitfly.org[16], COMPARE14 GXD[12], GEO[1]15. With the exception of
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| bshanks@53 | 279 the ABA, GenePaint, and EMAGE, most of these resources have not (yet) extracted the expression intensity from the ISH
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| bshanks@53 | 280 images and registered the results into a single 3-D space, and to our knowledge only ABA and EMAGE make this form of
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| bshanks@84 | 281 data available for public download from the website16. Many of these resources focus on developmental gene expression.
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| bshanks@46 | 282 Significance
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| bshanks@43 | 283 The method developed in aim (1) will be applied to each cortical area to find a set of marker genes such that the
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| bshanks@42 | 284 combinatorial expression pattern of those genes uniquely picks out the target area. Finding marker genes will be useful for
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| bshanks@30 | 285 drug discovery as well as for experimentation because marker genes can be used to design interventions which selectively
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| bshanks@30 | 286 target individual cortical areas.
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| bshanks@30 | 287 The application of the marker gene finding algorithm to the cortex will also support the development of new neuroanatom-
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| bshanks@33 | 288 ical methods. In addition to finding markers for each individual cortical areas, we will find a small panel of genes that can
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| bshanks@33 | 289 find many of the areal boundaries at once. This panel of marker genes will allow the development of an ISH protocol that
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| bshanks@30 | 290 will allow experimenters to more easily identify which anatomical areas are present in small samples of cortex.
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| bshanks@53 | 291 The method developed in aim (2) will provide a genoarchitectonic viewpoint that will contribute to the creation of
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| bshanks@33 | 292 a better map. The development of present-day cortical maps was driven by the application of histological stains. It is
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| bshanks@33 | 293 conceivable that if a different set of stains had been available which identified a different set of features, then the today’s
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| bshanks@33 | 294 cortical maps would have come out differently. Since the number of classes of stains is small compared to the number of
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| bshanks@84 | 295 _________________________________________
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| bshanks@84 | 296 7Outside of isocortex, the number of layers varies.
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| bshanks@84 | 297 8http://www.eurexpress.org/ee/; EurExpress data are also entered into EMAGE
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| bshanks@84 | 298 9http://www.ncl.ac.uk/ihg/EADHB/database/EADHB_database.html
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| bshanks@84 | 299 10http://mamep.molgen.mpg.de/index.php
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| bshanks@84 | 300 11http://xenbase.org/
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| bshanks@84 | 301 12http://aniseed-ibdm.univ-mrs.fr/
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| bshanks@84 | 302 13http://genome.ucsc.edu/cgi-bin/hgVisiGene ; includes data from some the other listed data sources
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| bshanks@84 | 303 14http://compare.ibdml.univ-mrs.fr/
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| bshanks@84 | 304 15GXD and GEO contain spatial data but also non-spatial data. All GXD spatial data are also in EMAGE.
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| bshanks@84 | 305 16without prior offline registration
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| bshanks@33 | 306 genes, it is likely that there are many repeated, salient spatial patterns in the gene expression which have not yet been
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| bshanks@63 | 307 captured by any stain. Therefore, current ideas about cortical anatomy need to incorporate what we can learn from looking
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| bshanks@63 | 308 at the patterns of gene expression.
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| bshanks@63 | 309 While we do not here propose to analyze human gene expression data, it is conceivable that the methods we propose to
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| bshanks@63 | 310 develop could be used to suggest modifications to the human cortical map as well.
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| bshanks@63 | 311 Related work
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| bshanks@63 | 312 [10 ] describes the application of AGEA to the cortex. The paper describes interesting results on the structure of correlations
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| bshanks@63 | 313 between voxel gene expression profiles within a handful of cortical areas. However, this sort of analysis is not related to either
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| bshanks@46 | 314 of our aims, as it neither finds marker genes, nor does it suggest a cortical map based on gene expression data. Neither of
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| bshanks@46 | 315 the other components of AGEA can be applied to cortical areas; AGEA’s Gene Finder cannot be used to find marker genes
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| bshanks@84 | 316 for the cortical areas; and AGEA’s hierarchial clustering does not produce clusters corresponding to the cortical areas17.
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| bshanks@46 | 317 In summary, for all three aims, (a) only one of the previous projects explores combinations of marker genes, (b) there has
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| bshanks@43 | 318 been almost no comparison of different algorithms or scoring methods, and (c) there has been no work on computationally
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| bshanks@43 | 319 finding marker genes for cortical areas, or on finding a hierarchial clustering that will yield a map of cortical areas de novo
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| bshanks@43 | 320 from gene expression data.
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| bshanks@53 | 321 Our project is guided by a concrete application with a well-specified criterion of success (how well we can find marker
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| bshanks@53 | 322 genes for / reproduce the layout of cortical areas), which will provide a solid basis for comparing different methods.
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| bshanks@53 | 323 _________________________________________
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| bshanks@84 | 324 17In both cases, the root cause is that pairwise correlations between the gene expression of voxels in different areas but the same layer are
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| bshanks@44 | 325 often stronger than pairwise correlations between the gene expression of voxels in different layers but the same area. Therefore, a pairwise voxel
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| bshanks@46 | 326 correlation clustering algorithm will tend to create clusters representing cortical layers, not areas. This is why the hierarchial clustering does not
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| bshanks@84 | 327 find cortical areas (there are clusters which presumably correspond to the intersection of a layer and an area, but since one area will have many
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| bshanks@84 | 328 layer-area intersection clusters, further work is needed to make sense of these). The reason that Gene Finder cannot the find marker genes for
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| bshanks@84 | 329 cortical areas is that in Gene Finder, although the user chooses a seed voxel, Gene Finder chooses the ROI for which genes will be found, and it
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| bshanks@84 | 330 creates that ROI by (pairwise voxel correlation) clustering around the seed.
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| bshanks@84 | 331 Preliminary Studies
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| bshanks@75 | 332
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| bshanks@75 | 333
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| bshanks@75 | 334 Figure 1: Top row: Genes Nfic and
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| bshanks@69 | 335 A930001M12Rik are the most correlated with
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| bshanks@69 | 336 area SS (somatosensory cortex). Bottom row:
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| bshanks@78 | 337 Genes C130038G02Rik and Cacna1i are those
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| bshanks@69 | 338 with the best fit using logistic regression. Within
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| bshanks@78 | 339 each picture, the vertical axis roughly corresponds
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| bshanks@78 | 340 to anterior at the top and posterior at the bot-
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| bshanks@78 | 341 tom, and the horizontal axis roughly corresponds
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| bshanks@78 | 342 to medial at the left and lateral at the right. The
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| bshanks@78 | 343 red outline is the boundary of region SS. Pixels are
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| bshanks@78 | 344 colored according to correlation, with red meaning
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| bshanks@78 | 345 high correlation and blue meaning low. Format conversion between SEV, MATLAB, NIFTI
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| bshanks@84 | 346 We have created software to (politely) download all of the SEV files18
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| bshanks@84 | 347 from the Allen Institute website. We have also created software to con-
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| bshanks@84 | 348 vert between the SEV, MATLAB, and NIFTI file formats, as well as
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| bshanks@84 | 349 some of Caret’s file formats.
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| bshanks@75 | 350 Flatmap of cortex
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| bshanks@75 | 351 We downloaded the ABA data and applied a mask to select only those
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| bshanks@75 | 352 voxels which belong to cerebral cortex. We divided the cortex into hemi-
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| bshanks@75 | 353 spheres.
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| bshanks@75 | 354 Using Caret[5], we created a mesh representation of the surface of the
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| bshanks@75 | 355 selected voxels. For each gene, for each node of the mesh, we calculated
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| bshanks@75 | 356 an average of the gene expression of the voxels “underneath” that mesh
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| bshanks@75 | 357 node. We then flattened the cortex, creating a two-dimensional mesh.
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| bshanks@75 | 358 We sampled the nodes of the irregular, flat mesh in order to create
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| bshanks@75 | 359 a regular grid of pixel values. We converted this grid into a MATLAB
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| bshanks@75 | 360 matrix.
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| bshanks@75 | 361 We manually traced the boundaries of each of 49 cortical areas from
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| bshanks@75 | 362 the ABA coronal reference atlas slides. We then converted these manual
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| bshanks@75 | 363 traces into Caret-format regional boundary data on the mesh surface.
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| bshanks@75 | 364 We projected the regions onto the 2-d mesh, and then onto the grid, and
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| bshanks@75 | 365 then we converted the region data into MATLAB format.
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| bshanks@84 | 366 At this point, the data are in the form of a number of 2-D matrices,
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| bshanks@75 | 367 all in registration, with the matrix entries representing a grid of points
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| bshanks@75 | 368 (pixels) over the cortical surface:
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| bshanks@75 | 369 ∙ A 2-D matrix whose entries represent the regional label associated with
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| bshanks@75 | 370 each surface pixel
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| bshanks@75 | 371 ∙ For each gene, a 2-D matrix whose entries represent the average expres-
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| bshanks@75 | 372 sion level underneath each surface pixel
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| bshanks@75 | 373
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| bshanks@78 | 374 Figure 2: Gene Pitx2
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| bshanks@75 | 375 is selectively underex-
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| bshanks@77 | 376 pressed in area SS. We created a normalized version of the gene expression data by subtracting each gene’s mean
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| bshanks@84 | 377 expression level (over all surface pixels) and dividing the expression level of each gene by its
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| bshanks@84 | 378 standard deviation.
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| bshanks@75 | 379 The features and the target area are both functions on the surface pixels. They can be referred
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| bshanks@75 | 380 to as scalar fields over the space of surface pixels; alternately, they can be thought of as images
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| bshanks@75 | 381 which can be displayed on the flatmapped surface.
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| bshanks@75 | 382 To move beyond a single average expression level for each surface pixel, we plan to create a
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| bshanks@75 | 383 separate matrix for each cortical layer to represent the average expression level within that layer.
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| bshanks@75 | 384 Cortical layers are found at different depths in different parts of the cortex. In preparation for
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| bshanks@75 | 385 extracting the layer-specific datasets, we have extended Caret with routines that allow the depth
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| bshanks@75 | 386 of the ROI for volume-to-surface projection to vary.
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| bshanks@75 | 387 In the Research Plan, we describe how we will automatically locate the layer depths. For
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| bshanks@75 | 388 validation, we have manually demarcated the depth of the outer boundary of cortical layer 5
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| bshanks@84 | 389 throughout the cortex.
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| bshanks@77 | 390 Feature selection and scoring methods
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| bshanks@75 | 391 Underexpression of a gene can serve as a marker Underexpression of a gene can sometimes serve as a marker. See,
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| bshanks@75 | 392 for example, Figure 2.
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| bshanks@75 | 393 Correlation Recall that the instances are surface pixels, and consider the problem of attempting to classify each instance
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| bshanks@75 | 394 as either a member of a particular anatomical area, or not. The target area can be represented as a boolean mask over the
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| bshanks@75 | 395 surface pixels.
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| bshanks@84 | 396 _____________________________
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| bshanks@84 | 397 18SEV is a sparse format for spatial data. It is the format in which the ABA data is made available.
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| bshanks@84 | 398 One class of feature selection scoring methods contains methods which calculate some sort of “match” between each gene
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| bshanks@84 | 399 image and the target image. Those genes which match the best are good candidates for features.
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| bshanks@75 | 400 One of the simplest methods in this class is to use correlation as the match score. We calculated the correlation between
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| bshanks@75 | 401 each gene and each cortical area. The top row of Figure 1 shows the three genes most correlated with area SS.
|
| bshanks@69 | 402
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| bshanks@69 | 403
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| bshanks@69 | 404 Figure 3: The top row shows the two genes which
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| bshanks@69 | 405 (individually) best predict area AUD, according
|
| bshanks@69 | 406 to logistic regression. The bottom row shows the
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| bshanks@69 | 407 two genes which (individually) best match area
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| bshanks@69 | 408 AUD, according to gradient similarity. From left
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| bshanks@69 | 409 to right and top to bottom, the genes are Ssr1,
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| bshanks@69 | 410 Efcbp1, Ptk7, and Aph1a. Conditional entropy An information-theoretic scoring method is
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| bshanks@69 | 411 to find features such that, if the features (gene expression levels) are
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| bshanks@69 | 412 known, uncertainty about the target (the regional identity) is reduced.
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| bshanks@69 | 413 Entropy measures uncertainty, so what we want is to find features such
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| bshanks@69 | 414 that the conditional distribution of the target has minimal entropy. The
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| bshanks@69 | 415 distribution to which we are referring is the probability distribution over
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| bshanks@69 | 416 the population of surface pixels.
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| bshanks@69 | 417 The simplest way to use information theory is on discrete data, so
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| bshanks@69 | 418 we discretized our gene expression data by creating, for each gene, five
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| bshanks@69 | 419 thresholded boolean masks of the gene data. For each gene, we created a
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| bshanks@69 | 420 boolean mask of its expression levels using each of these thresholds: the
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| bshanks@69 | 421 mean of that gene, the mean minus one standard deviation, the mean
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| bshanks@69 | 422 minus two standard deviations, the mean plus one standard deviation,
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| bshanks@69 | 423 the mean plus two standard deviations.
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| bshanks@69 | 424 Now, for each region, we created and ran a forward stepwise pro-
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| bshanks@69 | 425 cedure which attempted to find pairs of gene expression boolean masks
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| bshanks@69 | 426 such that the conditional entropy of the target area’s boolean mask, con-
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| bshanks@69 | 427 ditioned upon the pair of gene expression boolean masks, is minimized.
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| bshanks@69 | 428 This finds pairs of genes which are most informative (at least at these
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| bshanks@69 | 429 discretization thresholds) relative to the question, “Is this surface pixel
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| bshanks@69 | 430 a member of the target area?”. Its advantage over linear methods such
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| bshanks@69 | 431 as logistic regression is that it takes account of arbitrarily nonlinear re-
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| bshanks@69 | 432 lationships; for example, if the XOR of two variables predicts the target,
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| bshanks@69 | 433 conditional entropy would notice, whereas linear methods would not.
|
| bshanks@69 | 434
|
| bshanks@69 | 435
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| bshanks@69 | 436 Figure 4: Upper left: wwc1. Upper right: mtif2.
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| bshanks@69 | 437 Lower left: wwc1 + mtif2 (each pixel’s value on
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| bshanks@69 | 438 the lower left is the sum of the corresponding pix-
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| bshanks@69 | 439 els in the upper row). Gradient similarity We noticed that the previous two scoring
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| bshanks@69 | 440 methods, which are pointwise, often found genes whose pattern of ex-
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| bshanks@69 | 441 pression did not look similar in shape to the target region. For this
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| bshanks@69 | 442 reason we designed a non-pointwise local scoring method to detect when
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| bshanks@69 | 443 a gene had a pattern of expression which looked like it had a boundary
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| bshanks@69 | 444 whose shape is similar to the shape of the target region. We call this
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| bshanks@69 | 445 scoring method “gradient similarity”.
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| bshanks@69 | 446 One might say that gradient similarity attempts to measure how
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| bshanks@69 | 447 much the border of the area of gene expression and the border of the
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| bshanks@69 | 448 target region overlap. However, since gene expression falls off continu-
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| bshanks@69 | 449 ously rather than jumping from its maximum value to zero, the spatial
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| bshanks@69 | 450 pattern of a gene’s expression often does not have a discrete border.
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| bshanks@69 | 451 Therefore, instead of looking for a discrete border, we look for large
|
| bshanks@69 | 452 gradients. Gradient similarity is a symmetric function over two images
|
| bshanks@69 | 453 (i.e. two scalar fields). It is is high to the extent that matching pixels
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| bshanks@69 | 454 which have large values and large gradients also have gradients which
|
| bshanks@69 | 455 are oriented in a similar direction. The formula is:
|
| bshanks@69 | 456 ∑
|
| bshanks@69 | 457 pixel<img src="cmsy7-32.png" alt="∈" />pixels cos(abs(∠∇1 -∠∇2)) ⋅|∇1| + |∇2|
|
| bshanks@41 | 458 2 ⋅ pixel_value1 + pixel_value2
|
| bshanks@41 | 459 2
|
| bshanks@69 | 460 where ∇1 and ∇2 are the gradient vectors of the two images at the
|
| bshanks@69 | 461 current pixel; ∠∇i is the angle of the gradient of image i at the current pixel; |∇i| is the magnitude of the gradient of image
|
| bshanks@69 | 462 i at the current pixel; and pixel_valuei is the value of the current pixel in image i.
|
| bshanks@40 | 463 The intuition is that we want to see if the borders of the pattern in the two images are similar; if the borders are similar,
|
| bshanks@40 | 464 then both images will have corresponding pixels with large gradients (because this is a border) which are oriented in a
|
| bshanks@40 | 465 similar direction (because the borders are similar).
|
| bshanks@69 | 466 Most of the genes in Figure 5 were identified via gradient similarity.
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| bshanks@43 | 467 Gradient similarity provides information complementary to correlation
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| bshanks@41 | 468 To show that gradient similarity can provide useful information that cannot be detected via pointwise analyses, consider
|
| bshanks@84 | 469 Fig. 3. The top row of Fig. 3 displays the 3 genes which most match area AUD, according to a pointwise method19. The
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| bshanks@84 | 470 bottomrow displays the 3 genes which most match AUD according to a method which considers local geometry20 The
|
| bshanks@46 | 471 pointwise method in the top row identifies genes which express more strongly in AUD than outside of it; its weakness is
|
| bshanks@46 | 472 that this includes many areas which don’t have a salient border matching the areal border. The geometric method identifies
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| bshanks@46 | 473 genes whose salient expression border seems to partially line up with the border of AUD; its weakness is that this includes
|
| bshanks@46 | 474 genes which don’t express over the entire area. Genes which have high rankings using both pointwise and border criteria,
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| bshanks@46 | 475 such as Aph1a in the example, may be particularly good markers. None of these genes are, individually, a perfect marker
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| bshanks@46 | 476 for AUD; we deliberately chose a “difficult” area in order to better contrast pointwise with geometric methods.
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| bshanks@84 | 477 Areas which can be identified by single genes Using gradient similarity, we have already found single genes which
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| bshanks@84 | 478 roughly identify some areas and groupings of areas. For each of these areas, an example of a gene which roughly identifies
|
| bshanks@84 | 479 it is shown in Figure 5. We have not yet cross-verified these genes in other atlases.
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| bshanks@84 | 480 In addition, there are a number of areas which are almost identified by single genes: COAa+NLOT (anterior part of
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| bshanks@84 | 481 cortical amygdalar area, nucleus of the lateral olfactory tract), ENT (entorhinal), ACAv (ventral anterior cingulate), VIS
|
| bshanks@84 | 482 (visual), AUD (auditory).
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| bshanks@84 | 483 These results validate our expectation that the ABA dataset can be exploited to find marker genes for many cortical
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| bshanks@84 | 484 areas, while also validating the relevancy of our new scoring method, gradient similarity.
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| bshanks@84 | 485 Combinations of multiple genes are useful and necessary for some areas
|
| bshanks@84 | 486 In Figure 4, we give an example of a cortical area which is not marked by any single gene, but which can be identified
|
| bshanks@84 | 487 combinatorially. Acccording to logistic regression, gene wwc1 is the best fit single gene for predicting whether or not a
|
| bshanks@84 | 488 pixel on the cortical surface belongs to the motor area (area MO). The upper-left picture in Figure 4 shows wwc1’s spatial
|
| bshanks@84 | 489 expression pattern over the cortex. The lower-right boundary of MO is represented reasonably well by this gene, but the
|
| bshanks@84 | 490 gene overshoots the upper-left boundary. This flattened 2-D representation does not show it, but the area corresponding
|
| bshanks@84 | 491 to the overshoot is the medial surface of the cortex. MO is only found on the dorsal surface. Gene mtif2 is shown in the
|
| bshanks@84 | 492 upper-right. Mtif2 captures MO’s upper-left boundary, but not its lower-right boundary. Mtif2 does not express very much
|
| bshanks@84 | 493 on the medial surface. By adding together the values at each pixel in these two figures, we get the lower-left image. This
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| bshanks@84 | 494 combination captures area MO much better than any single gene.
|
| bshanks@84 | 495 This shows that our proposal to develop a method to find combinations of marker genes is both possible and necessary.
|
| bshanks@84 | 496 Feature selection integrated with prediction As noted earlier, in general, any predictive method can be used for
|
| bshanks@84 | 497 feature selection by running it inside a stepwise wrapper. Also, some predictive methods integrate soft constraints on number
|
| bshanks@84 | 498 of features used. Examples of both of these will be seen in the section “Multivariate Predictive methods”.
|
| bshanks@84 | 499 Multivariate Predictive methods
|
| bshanks@84 | 500 Forward stepwise logistic regression Logistic regression is a popular method for predictive modeling of categorial data.
|
| bshanks@84 | 501 As a pilot run, for five cortical areas (SS, AUD, RSP, VIS, and MO), we performed forward stepwise logistic regression to
|
| bshanks@84 | 502 find single genes, pairs of genes, and triplets of genes which predict areal identify. This is an example of feature selection
|
| bshanks@84 | 503 integrated with prediction using a stepwise wrapper. Some of the single genes found were shown in various figures throughout
|
| bshanks@84 | 504 this document, and Figure 4 shows a combination of genes which was found.
|
| bshanks@84 | 505 We felt that, for single genes, gradient similarity did a better job than logistic regression at capturing our subjective
|
| bshanks@84 | 506 impression of a “good gene”.
|
| bshanks@84 | 507 _________________
|
| bshanks@84 | 508 19For each gene, a logistic regression in which the response variable was whether or not a surface pixel was within area AUD, and the predictor
|
| bshanks@84 | 509 variable was the value of the expression of the gene underneath that pixel. The resulting scores were used to rank the genes in terms of how well
|
| bshanks@84 | 510 they predict area AUD.
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| bshanks@84 | 511 20For each gene the gradient similarity between (a) a map of the expression of each gene on the cortical surface and (b) the shape of area AUD,
|
| bshanks@84 | 512 was calculated, and this was used to rank the genes.
|
| bshanks@84 | 513
|
| bshanks@69 | 514
|
| bshanks@69 | 515
|
| bshanks@69 | 516
|
| bshanks@69 | 517
|
| bshanks@69 | 518 Figure 5: From left to right and top to bot-
|
| bshanks@69 | 519 tom, single genes which roughly identify ar-
|
| bshanks@77 | 520 eas SS (somatosensory primary + supplemen-
|
| bshanks@77 | 521 tal), SSs (supplemental somatosensory), PIR (pir-
|
| bshanks@77 | 522 iform), FRP (frontal pole), RSP (retrosplenial),
|
| bshanks@69 | 523 COApm (Cortical amygdalar, posterior part, me-
|
| bshanks@69 | 524 dial zone). Grouping some areas together, we
|
| bshanks@69 | 525 have also found genes to identify the groups
|
| bshanks@69 | 526 ACA+PL+ILA+DP+ORB+MO (anterior cingu-
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| bshanks@69 | 527 late, prelimbic, infralimbic, dorsal peduncular, or-
|
| bshanks@69 | 528 bital, motor), posterior and lateral visual (VISpm,
|
| bshanks@69 | 529 VISpl, VISI, VISp; posteromedial, posterolateral,
|
| bshanks@69 | 530 lateral, and primary visual; the posterior and lat-
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| bshanks@69 | 531 eral visual area is distinguished from its neigh-
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| bshanks@69 | 532 bors, but not from the entire rest of the cortex).
|
| bshanks@69 | 533 The genes are Pitx2, Aldh1a2, Ppfibp1, Slco1a5,
|
| bshanks@84 | 534 Tshz2, Trhr, Col12a1, Ets1.
|
| bshanks@84 | 535 SVM on all genes at once
|
| bshanks@84 | 536 In order to see how well one can do when looking at all genes at once, we ran a support vector machine to classify cortical
|
| bshanks@84 | 537 surface pixels based on their gene expression profiles. We achieved classification accuracy of about 81%21. This shows that
|
| bshanks@84 | 538 the genes included in the ABA dataset are sufficient to define much of cortical anatomy. However, as noted above, a classifier
|
| bshanks@84 | 539 that looks at all the genes at once isn’t as practically useful as a classifier that uses only a few genes.
|
| bshanks@84 | 540 Data-driven redrawing of the cortical map
|
| bshanks@84 | 541 We have applied the following dimensionality reduction algorithms to reduce the dimensionality of the gene expression
|
| bshanks@84 | 542 profile associated with each voxel: Principal Components Analysis (PCA), Simple PCA (SPCA), Multi-Dimensional Scaling
|
| bshanks@84 | 543 (MDS), Isomap, Landmark Isomap, Laplacian eigenmaps, Local Tangent Space Alignment (LTSA), Hessian locally linear
|
| bshanks@84 | 544 embedding, Diffusion maps, Stochastic Neighbor Embedding (SNE), Stochastic Proximity Embedding (SPE), Fast Maximum
|
| bshanks@84 | 545 Variance Unfolding (FastMVU), Non-negative Matrix Factorization (NNMF). Space constraints prevent us from showing
|
| bshanks@63 | 546 _________________________________________
|
| bshanks@84 | 547 215-fold cross-validation.
|
| bshanks@84 | 548 many of the results, but as a sample, PCA, NNMF, and landmark Isomap are shown in the first, second, and third rows of
|
| bshanks@84 | 549 Figure 6.
|
| bshanks@60 | 550
|
| bshanks@69 | 551
|
| bshanks@69 | 552
|
| bshanks@69 | 553
|
| bshanks@69 | 554 Figure 6: First row: the first 6 reduced dimensions, using PCA. Second
|
| bshanks@69 | 555 row: the first 6 reduced dimensions, using NNMF. Third row: the first
|
| bshanks@69 | 556 six reduced dimensions, using landmark Isomap. Bottom row: examples
|
| bshanks@69 | 557 of kmeans clustering applied to reduced datasets to find 7 clusters. Left:
|
| bshanks@69 | 558 19 of the major subdivisions of the cortex. Second from left: PCA. Third
|
| bshanks@69 | 559 from left: NNMF. Right: Landmark Isomap. Additional details: In the
|
| bshanks@69 | 560 third and fourth rows, 7 dimensions were found, but only 6 displayed. In
|
| bshanks@69 | 561 the last row: for PCA, 50 dimensions were used; for NNMF, 6 dimensions
|
| bshanks@84 | 562 were used; for landmark Isomap, 7 dimensions were used.
|
| bshanks@71 | 563
|
| bshanks@71 | 564 Figure 7: Prototypes corresponding to sample gene clusters, clustered by
|
| bshanks@71 | 565 gradient similarity. Region boundaries for the region that most matches
|
| bshanks@71 | 566 each prototype are overlayed. After applying the dimensionality reduc-
|
| bshanks@71 | 567 tion, we ran clustering algorithms on the re-
|
| bshanks@71 | 568 duced data. To date we have tried k-means and
|
| bshanks@71 | 569 spectral clustering. The results of k-means after
|
| bshanks@71 | 570 PCA, NNMF, and landmark Isomap are shown
|
| bshanks@71 | 571 in the last row of Figure 6. To compare, the
|
| bshanks@71 | 572 leftmost picture on the bottom row of Figure
|
| bshanks@71 | 573 6 shows some of the major subdivisions of cor-
|
| bshanks@71 | 574 tex. These results clearly show that different di-
|
| bshanks@71 | 575 mensionality reduction techniques capture dif-
|
| bshanks@71 | 576 ferent aspects of the data and lead to differ-
|
| bshanks@71 | 577 ent clusterings, indicating the utility of our pro-
|
| bshanks@71 | 578 posal to produce a detailed comparion of these
|
| bshanks@71 | 579 techniques as applied to the domain of genomic
|
| bshanks@71 | 580 anatomy.
|
| bshanks@71 | 581 Many areas are captured by clusters of genes We also clustered the genes using gradient similarity to see if the
|
| bshanks@72 | 582 spatial regions defined by any clusters matched known anatomical regions. Figure 7 shows, for ten sample gene clusters, each
|
| bshanks@72 | 583 cluster’s average expression pattern, compared to a known anatomical boundary. This suggests that it is worth attempting
|
| bshanks@72 | 584 to cluster genes, and then to use the results to cluster voxels.
|
| bshanks@84 | 585 Research Design and Methods
|
| bshanks@42 | 586 Further work on flatmapping
|
| bshanks@42 | 587 In anatomy, the manifold of interest is usually either defined by a combination of two relevant anatomical axes (todo),
|
| bshanks@42 | 588 or by the surface of the structure (as is the case with the cortex). In the former case, the manifold of interest is a plane, but
|
| bshanks@42 | 589 in the latter case it is curved. If the manifold is curved, there are various methods for mapping the manifold into a plane.
|
| bshanks@42 | 590 In the case of the cerebral cortex, it remains to be seen which method of mapping the manifold into a plane is optimal
|
| bshanks@53 | 591 for this application. We will compare mappings which attempt to preserve size (such as the one used by Caret[5]) with
|
| bshanks@42 | 592 mappings which preserve angle (conformal maps).
|
| bshanks@42 | 593 Although there is much 2-D organization in anatomy, there are also structures whose shape is fundamentally 3-dimensional.
|
| bshanks@42 | 594 If possible, we would like the method we develop to include a statistical test that warns the user if the assumption of 2-D
|
| bshanks@42 | 595 structure seems to be wrong.
|
| bshanks@30 | 596 todo amongst other things:
|
| bshanks@63 | 597 layerfinding
|
| bshanks@30 | 598 Develop algorithms that find genetic markers for anatomical regions
|
| bshanks@30 | 599 1.Develop scoring measures for evaluating how good individual genes are at marking areas: we will compare pointwise,
|
| bshanks@30 | 600 geometric, and information-theoretic measures.
|
| bshanks@30 | 601 2.Develop a procedure to find single marker genes for anatomical regions: for each cortical area, by using or combining
|
| bshanks@30 | 602 the scoring measures developed, we will rank the genes by their ability to delineate each area.
|
| bshanks@30 | 603 3.Extend the procedure to handle difficult areas by using combinatorial coding: for areas that cannot be identified by any
|
| bshanks@30 | 604 single gene, identify them with a handful of genes. We will consider both (a) algorithms that incrementally/greedily
|
| bshanks@30 | 605 combine single gene markers into sets, such as forward stepwise regression and decision trees, and also (b) supervised
|
| bshanks@33 | 606 learning techniques which use soft constraints to minimize the number of features, such as sparse support vector
|
| bshanks@30 | 607 machines.
|
| bshanks@33 | 608 4.Extend the procedure to handle difficult areas by combining or redrawing the boundaries: An area may be difficult
|
| bshanks@33 | 609 to identify because the boundaries are misdrawn, or because it does not “really” exist as a single area, at least on the
|
| bshanks@30 | 610 genetic level. We will develop extensions to our procedure which (a) detect when a difficult area could be fit if its
|
| bshanks@30 | 611 boundary were redrawn slightly, and (b) detect when a difficult area could be combined with adjacent areas to create
|
| bshanks@30 | 612 a larger area which can be fit.
|
| bshanks@51 | 613 # Linear discriminant analysis
|
| bshanks@64 | 614 Decision trees todo
|
| bshanks@64 | 615 For each cortical area, we used the C4.5 algorithm to find a pruned decision tree and ruleset for that area. We achieved
|
| bshanks@64 | 616 estimated classification accuracy of more than 99.6% on each cortical area (as evaluated on the training data without
|
| bshanks@64 | 617 cross-validation; so actual accuracy is expected to be lower). However, the resulting decision trees each made use of many
|
| bshanks@64 | 618 genes.
|
| bshanks@30 | 619 Apply these algorithms to the cortex
|
| bshanks@30 | 620 1.Create open source format conversion tools: we will create tools to bulk download the ABA dataset and to convert
|
| bshanks@30 | 621 between SEV, NIFTI and MATLAB formats.
|
| bshanks@30 | 622 2.Flatmap the ABA cortex data: map the ABA data onto a plane and draw the cortical area boundaries onto it.
|
| bshanks@30 | 623 3.Find layer boundaries: cluster similar voxels together in order to automatically find the cortical layer boundaries.
|
| bshanks@30 | 624 4.Run the procedures that we developed on the cortex: we will present, for each area, a short list of markers to identify
|
| bshanks@30 | 625 that area; and we will also present lists of “panels” of genes that can be used to delineate many areas at once.
|
| bshanks@30 | 626 Develop algorithms to suggest a division of a structure into anatomical parts
|
| bshanks@60 | 627 # mixture models, etc
|
| bshanks@30 | 628 1.Explore dimensionality reduction algorithms applied to pixels: including TODO
|
| bshanks@30 | 629 2.Explore dimensionality reduction algorithms applied to genes: including TODO
|
| bshanks@30 | 630 3.Explore clustering algorithms applied to pixels: including TODO
|
| bshanks@30 | 631 4.Explore clustering algorithms applied to genes: including gene shaving, TODO
|
| bshanks@30 | 632 5.Develop an algorithm to use dimensionality reduction and/or hierarchial clustering to create anatomical maps
|
| bshanks@30 | 633 6.Run this algorithm on the cortex: present a hierarchial, genoarchitectonic map of the cortex
|
| bshanks@51 | 634 # Linear discriminant analysis
|
| bshanks@51 | 635 # jbt, coclustering
|
| bshanks@51 | 636 # self-organizing map
|
| bshanks@53 | 637 # confirm with EMAGE, GeneAtlas, GENSAT, etc, to fight overfitting
|
| bshanks@53 | 638 # compare using clustering scores
|
| bshanks@64 | 639 # multivariate gradient similarity
|
| bshanks@66 | 640 # deep belief nets
|
| bshanks@66 | 641 # note: slice artifact
|
| bshanks@33 | 642 Bibliography & References Cited
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| bshanks@33 | 731
|
| bshanks@33 | 732 _______________________________________________________________________________________________________
|
| bshanks@30 | 733 stuff i dunno where to put yet (there is more scattered through grant-oldtext):
|
| bshanks@16 | 734 Principle 4: Work in 2-D whenever possible
|
| bshanks@33 | 735 —
|
| bshanks@33 | 736 note:
|
| bshanks@36 | 737 two hemis
|
| bshanks@33 | 738
|
| bshanks@33 | 739
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