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view grant.html @ 105:6c48f37d0f0c
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author | bshanks@bshanks.dyndns.org |
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date | Wed Apr 22 07:39:32 2009 -0700 (16 years ago) |
parents | d6ecbc494f0b |
children | ffa1390e4f39 |
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1 Specific aims
2 Massive new datasets obtained with techniques such as in situ hybridization (ISH), immunohistochemistry, in
3 situ transgenic reporter, microarray voxelation, and others, allow the expression levels of many genes at many
4 locations to be compared. Our goal is to develop automated methods to relate spatial variation in gene expres-
5 sion to anatomy. We want to find marker genes for specific anatomical regions, and also to draw new anatomical
6 maps based on gene expression patterns. We have three specific aims:
7 (1) develop an algorithm to screen spatial gene expression data for combinations of marker genes which
8 selectively target anatomical regions
9 (2) develop an algorithm to suggest new ways of carving up a structure into anatomically distinct regions,
10 based on spatial patterns in gene expression
11 (3) create a 2-D “flat map” dataset of the mouse cerebral cortex that contains a flattened version of the Allen
12 Mouse Brain Atlas ISH data, as well as the boundaries of cortical anatomical areas. This will involve extending
13 the functionality of Caret, an existing open-source scientific imaging program. Use this dataset to validate the
14 methods developed in (1) and (2).
15 Although our particular application involves the 3D spatial distribution of gene expression, we anticipate that
16 the methods developed in aims (1) and (2) will generalize to any sort of high-dimensional data over points located
17 in a low-dimensional space. In particular, our method could be applied to genome-wide sequencing data derived
18 from sets of tissues and disease states.
19 In terms of the application of the methods to cerebral cortex, aim (1) is to go from cortical areas to marker
20 genes, and aim (2) is to let the gene profile define the cortical areas. In addition to validating the usefulness
21 of the algorithms, the application of these methods to cortex will produce immediate benefits, because there
22 are currently no known genetic markers for most cortical areas. The results of the project will support the
23 development of new ways to selectively target cortical areas, and it will support the development of a method for
24 identifying the cortical areal boundaries present in small tissue samples.
25 All algorithms that we develop will be implemented in a GPL open-source software toolkit. The toolkit, as well
26 as the machine-readable datasets developed in aim (3), will be published and freely available for others to use.
27 The challenge topic
28 This proposal addresses challenge topic 06-HG-101. Massive new datasets obtained with techniques such as
29 in situ hybridization (ISH), immunohistochemistry, in situ transgenic reporter, microarray voxelation, and others,
30 allow the expression levels of many genes at many locations to be compared. Our goal is to develop automated
31 methods to relate spatial variation in gene expression to anatomy. We want to find marker genes for specific
32 anatomical regions, and also to draw new anatomical maps based on gene expression patterns.
33 ______________
34 The Challenge and Potential impact
35 Each of our three aims will be discussed in turn. For each aim, we will develop a conceptual framework for
36 thinking about the task. Next we will discuss related work, and then summarize why our strategy is different from
37 what has been done before. After we have discussed all three aims, we will describe the potential impact.
38 Aim 1: Given a map of regions, find genes that mark the regions
39 Machine learning terminology: classifiers The task of looking for marker genes for known anatomical regions
40 means that one is looking for a set of genes such that, if the expression level of those genes is known, then the
41 locations of the regions can be inferred.
42 If we define the regions so that they cover the entire anatomical structure to be subdivided, we may say that
43 we are using gene expression in each voxel to assign that voxel to the proper area. We call this a classification
44 task, because each voxel is being assigned to a class (namely, its region). An understanding of the relationship
45 between the combination of their expression levels and the locations of the regions may be expressed as a
46 function. The input to this function is a voxel, along with the gene expression levels within that voxel; the output is
47 the regional identity of the target voxel, that is, the region to which the target voxel belongs. We call this function
48 a classifier. In general, the input to a classifier is called an instance, and the output is called a label (or a class
49 label).
50 The object of aim 1 is not to produce a single classifier, but rather to develop an automated method for
51 determining a classifier for any known anatomical structure. Therefore, we seek a procedure by which a gene
52 expression dataset may be analyzed in concert with an anatomical atlas in order to produce a classifier. The
53 initial gene expression dataset used in the construction of the classifier is called training data. In the machine
54 learning literature, this sort of procedure may be thought of as a supervised learning task, defined as a task in
55 which the goal is to learn a mapping from instances to labels, and the training data consists of a set of instances
56 (voxels) for which the labels (regions) are known.
57 Each gene expression level is called a feature, and the selection of which genes1 to include is called feature
58 selection. Feature selection is one component of the task of learning a classifier. Some methods for learning
59 classifiers start out with a separate feature selection phase, whereas other methods combine feature selection
60 with other aspects of training.
61 One class of feature selection methods assigns some sort of score to each candidate gene. The top-ranked
62 genes are then chosen. Some scoring measures can assign a score to a set of selected genes, not just to a
63 single gene; in this case, a dynamic procedure may be used in which features are added and subtracted from the
64 selected set depending on how much they raise the score. Such procedures are called “stepwise” or “greedy”.
65 Although the classifier itself may only look at the gene expression data within each voxel before classifying
66 that voxel, the algorithm which constructs the classifier may look over the entire dataset. We can categorize
67 score-based feature selection methods depending on how the score of calculated. Often the score calculation
68 consists of assigning a sub-score to each voxel, and then aggregating these sub-scores into a final score (the
69 aggregation is often a sum or a sum of squares or average). If only information from nearby voxels is used to
70 calculate a voxel’s sub-score, then we say it is a local scoring method. If only information from the voxel itself is
71 used to calculate a voxel’s sub-score, then we say it is a pointwise scoring method.
72 Both gene expression data and anatomical atlases have errors, due to a variety of factors. Individual subjects
73 have idiosyncratic anatomy. Subjects may be improperly registered to the atlas. The method used to measure
74 gene expression may be noisy. The atlas may have errors. It is even possible that some areas in the anatomical
75 atlas are “wrong” in that they do not have the same shape as the natural domains of gene expression to which
76 1Strictly speaking, the features are gene expression levels, but we’ll call them genes.
77 they correspond. These sources of error can affect the displacement and the shape of both the gene expression
78 data and the anatomical target areas. Therefore, it is important to use feature selection methods which are
79 robust to these kinds of errors.
80 Our strategy for Aim 1
81 Key questions when choosing a learning method are: What are the instances? What are the features? How are
82 the features chosen? Here are four principles that outline our answers to these questions.
83 Principle 1: Combinatorial gene expression
84 It is too much to hope that every anatomical region of interest will be identified by a single gene. For example,
85 in the cortex, there are some areas which are not clearly delineated by any gene included in the Allen Brain Atlas
86 (ABA) dataset. However, at least some of these areas can be delineated by looking at combinations of genes
87 (an example of an area for which multiple genes are necessary and sufficient is provided in Preliminary Studies,
88 Figure 4). Therefore, each instance should contain multiple features (genes).
89 Principle 2: Only look at combinations of small numbers of genes
90 When the classifier classifies a voxel, it is only allowed to look at the expression of the genes which have
91 been selected as features. The more data that are available to a classifier, the better that it can do. For example,
92 perhaps there are weak correlations over many genes that add up to a strong signal. So, why not include every
93 gene as a feature? The reason is that we wish to employ the classifier in situations in which it is not feasible to
94 gather data about every gene. For example, if we want to use the expression of marker genes as a trigger for
95 some regionally-targeted intervention, then our intervention must contain a molecular mechanism to check the
96 expression level of each marker gene before it triggers. It is currently infeasible to design a molecular trigger that
97 checks the level of more than a handful of genes. Similarly, if the goal is to develop a procedure to do ISH on
98 tissue samples in order to label their anatomy, then it is infeasible to label more than a few genes. Therefore, we
99 must select only a few genes as features.
100 The requirement to find combinations of only a small number of genes limits us from straightforwardly ap-
101 plying many of the most simple techniques from the field of supervised machine learning. In the parlance of
102 machine learning, our task combines feature selection with supervised learning.
103 Principle 3: Use geometry in feature selection
104 When doing feature selection with score-based methods, the simplest thing to do would be to score the per-
105 formance of each voxel by itself and then combine these scores (pointwise scoring). A more powerful approach
106 is to also use information about the geometric relations between each voxel and its neighbors; this requires non-
107 pointwise, local scoring methods. See Preliminary Studies, figure 3 for evidence of the complementary nature of
108 pointwise and local scoring methods.
109 Principle 4: Work in 2-D whenever possible
110 There are many anatomical structures which are commonly characterized in terms of a two-dimensional
111 manifold. When it is known that the structure that one is looking for is two-dimensional, the results may be
112 improved by allowing the analysis algorithm to take advantage of this prior knowledge. In addition, it is easier for
113 humans to visualize and work with 2-D data. Therefore, when possible, the instances should represent pixels,
114 not voxels.
115 Related work
116 There is a substantial body of work on the analysis of gene expression data, most of this concerns gene expres-
117 sion data which are not fundamentally spatial2.
118 As noted above, there has been much work on both supervised learning and there are many available
119 algorithms for each. However, the algorithms require the scientist to provide a framework for representing the
120 problem domain, and the way that this framework is set up has a large impact on performance. Creating a
121 good framework can require creatively reconceptualizing the problem domain, and is not merely a mechanical
122 “fine-tuning” of numerical parameters. For example, we believe that domain-specific scoring measures (such
123 _________________________________________
124 2By “fundamentally spatial” we mean that there is information from a large number of spatial locations indexed by spatial coordinates;
125 not just data which have only a few different locations or which is indexed by anatomical label.
126 as gradient similarity, which is discussed in Preliminary Studies) may be necessary in order to achieve the best
127 results in this application.
128 We now turn to efforts to find marker genes using spatial gene expression data using automated methods.
129 GeneAtlas[5] and EMAGE [26] allow the user to construct a search query by demarcating regions and then
130 specifying either the strength of expression or the name of another gene or dataset whose expression pattern
131 is to be matched. Neither GeneAtlas nor EMAGE allow one to search for combinations of genes that define a
132 region in concert but not separately.
133 [15 ] describes AGEA, ”Anatomic Gene Expression Atlas”. AGEA has three components. Gene Finder: The
134 user selects a seed voxel and the system (1) chooses a cluster which includes the seed voxel, (2) yields a list of
135 genes which are overexpressed in that cluster. Correlation: The user selects a seed voxel and the system then
136 shows the user how much correlation there is between the gene expression profile of the seed voxel and every
137 other voxel. Clusters: will be described later. [6] looks at the mean expression level of genes within anatomical
138 regions, and applies a Student’s t-test with Bonferroni correction to determine whether the mean expression
139 level of a gene is significantly higher in the target region. [15] and [6] differ from our Aim 1 in at least three
140 ways. First, [15] and [6] find only single genes, whereas we will also look for combinations of genes. Second,
141 [15 ] and [6] can only use overexpression as a marker, whereas we will also search for underexpression. Third,
142 [15 ] and [6] use scores based on pointwise expression levels, whereas we will also use geometric scores such
143 as gradient similarity (described in Preliminary Studies). Figures 4, 2, and 3 in the Preliminary Studies section
144 contain evidence that each of our three choices is the right one.
145 [10 ] describes a technique to find combinations of marker genes to pick out an anatomical region. They use
146 an evolutionary algorithm to evolve logical operators which combine boolean (thresholded) images in order to
147 match a target image.
148 In summary, there has been fruitful work on finding marker genes, but only one of the previous projects
149 explores combinations of marker genes, and none of these publications compare the results obtained by using
150 different algorithms or scoring methods.
151 Aim 2: From gene expression data, discover a map of regions
152 Machine learning terminology: clustering
153 If one is given a dataset consisting merely of instances, with no class labels, then analysis of the dataset is
154 referred to as unsupervised learning in the jargon of machine learning. One thing that you can do with such a
155 dataset is to group instances together. A set of similar instances is called a cluster, and the activity of finding
156 grouping the data into clusters is called clustering or cluster analysis.
157 The task of deciding how to carve up a structure into anatomical regions can be put into these terms. The
158 instances are once again voxels (or pixels) along with their associated gene expression profiles. We make
159 the assumption that voxels from the same anatomical region have similar gene expression profiles, at least
160 compared to the other regions. This means that clustering voxels is the same as finding potential regions; we
161 seek a partitioning of the voxels into regions, that is, into clusters of voxels with similar gene expression.
162 It is desirable to determine not just one set of regions, but also how these regions relate to each other. The
163 outcome of clustering may be a hierarchical tree of clusters, rather than a single set of clusters which partition
164 the voxels. This is called hierarchical clustering.
165 Similarity scores A crucial choice when designing a clustering method is how to measure similarity, across
166 either pairs of instances, or clusters, or both. There is much overlap between scoring methods for feature
167 selection (discussed above under Aim 1) and scoring methods for similarity.
168 Spatially contiguous clusters; image segmentation We have shown that aim 2 is a type of clustering
169 task. In fact, it is a special type of clustering task because we have an additional constraint on clusters; voxels
170 grouped together into a cluster must be spatially contiguous. In Preliminary Studies, we show that one can get
171 reasonable results without enforcing this constraint; however, we plan to compare these results against other
172 methods which guarantee contiguous clusters.
173 Image segmentation is the task of partitioning the pixels in a digital image into clusters, usually contiguous
174 clusters. Aim 2 is similar to an image segmentation task. There are two main differences; in our task, there are
175 thousands of color channels (one for each gene), rather than just three3. A more crucial difference is that there
176 are various cues which are appropriate for detecting sharp object boundaries in a visual scene but which are not
177 appropriate for segmenting abstract spatial data such as gene expression. Although many image segmentation
178 algorithms can be expected to work well for segmenting other sorts of spatially arranged data, some of these
179 algorithms are specialized for visual images.
180 Dimensionality reduction In this section, we discuss reducing the length of the per-pixel gene expression
181 feature vector. By “dimension”, we mean the dimension of this vector, not the spatial dimension of the underlying
182 data.
183 Unlike aim 1, there is no externally-imposed need to select only a handful of informative genes for inclusion
184 in the instances. However, some clustering algorithms perform better on small numbers of features4. There are
185 techniques which “summarize” a larger number of features using a smaller number of features; these techniques
186 go by the name of feature extraction or dimensionality reduction. The small set of features that such a technique
187 yields is called the reduced feature set. Note that the features in the reduced feature set do not necessarily
188 correspond to genes; each feature in the reduced set may be any function of the set of gene expression levels.
189 Clustering genes rather than voxels Although the ultimate goal is to cluster the instances (voxels or pixels),
190 one strategy to achieve this goal is to first cluster the features (genes). There are two ways that clusters of genes
191 could be used.
192 Gene clusters could be used as part of dimensionality reduction: rather than have one feature for each gene,
193 we could have one reduced feature for each gene cluster.
194 Gene clusters could also be used to directly yield a clustering on instances. This is because many genes have
195 an expression pattern which seems to pick out a single, spatially contiguous region. This suggests the following
196 procedure: cluster together genes which pick out similar regions, and then to use the more popular common
197 regions as the final clusters. In Preliminary Studies, Figure 7, we show that a number of anatomically recognized
198 cortical regions, as well as some “superregions” formed by lumping together a few regions, are associated with
199 gene clusters in this fashion.
200 Related work
201 Some researchers have attempted to parcellate cortex on the basis of non-gene expression data. For example,
202 [18 ], [2 ], [19], and [1] associate spots on the cortex with the radial profile5 of response to some stain ([12] uses
203 MRI), extract features from this profile, and then use similarity between surface pixels to cluster.
204 [23 ] describes an analysis of the anatomy of the hippocampus using the ABA dataset. In addition to manual
205 analysis, two clustering methods were employed, a modified Non-negative Matrix Factorization (NNMF), and
206 a hierarchical recursive bifurcation clustering scheme based on correlation as the similarity score. The paper
207 yielded impressive results, proving the usefulness of computational genomic anatomy. We have run NNMF on
208 the cortical dataset
209 AGEA[15] includes a preset hierarchical clustering of voxels based on a recursive bifurcation algorithm with
210 correlation as the similarity metric. EMAGE[26] allows the user to select a dataset from among a large number
211 of alternatives, or by running a search query, and then to cluster the genes within that dataset. EMAGE clusters
212 via hierarchical complete linkage clustering.
213 [6 ] clusters genes. For each cluster, prototypical spatial expression patterns were created by averaging the
214 genes in the cluster. The prototypes were analyzed manually, without clustering voxels.
215 [10 ] applies their technique for finding combinations of marker genes for the purpose of clustering genes
216 around a “seed gene”.
217 In summary, although these projects obtained clusterings, there has not been much comparison between
218 different algorithms or scoring methods, so it is likely that the best clustering method for this application has not
219 yet been found. The projects using gene expression on cortex did not attempt to make use of the radial profile
220 of gene expression. Also, none of these projects did a separate dimensionality reduction step before clustering
221 _________________________________________
222 3There are imaging tasks which use more than three colors, for example multispectral imaging and hyperspectral imaging, which are
223 often used to process satellite imagery.
224 4First, because the number of features in the reduced dataset is less than in the original dataset, the running time of clustering
225 algorithms may be much less. Second, it is thought that some clustering algorithms may give better results on reduced data.
226 5A radial profile is a profile along a line perpendicular to the cortical surface.
227 pixels, none tried to cluster genes first in order to guide automated clustering of pixels into spatial regions, and
228 none used co-clustering algorithms.
229 Aim 3: apply the methods developed to the cerebral cortex
232 Figure 1: Top row: Genes Nfic
233 and A930001M12Rik are the most
234 correlated with area SS (somatosen-
235 sory cortex). Bottom row: Genes
236 C130038G02Rik and Cacna1i are
237 those with the best fit using logistic
238 regression. Within each picture, the
239 vertical axis roughly corresponds to
240 anterior at the top and posterior at the
241 bottom, and the horizontal axis roughly
242 corresponds to medial at the left and
243 lateral at the right. The red outline is
244 the boundary of region SS. Pixels are
245 colored according to correlation, with
246 red meaning high correlation and blue
247 meaning low. Background
248 The cortex is divided into areas and layers. Because of the cortical
249 columnar organization, the parcellation of the cortex into areas can be
250 drawn as a 2-D map on the surface of the cortex. In the third dimension,
251 the boundaries between the areas continue downwards into the cortical
252 depth, perpendicular to the surface. The layer boundaries run parallel
253 to the surface. One can picture an area of the cortex as a slice of a
254 six-layered cake6.
255 It is known that different cortical areas have distinct roles in both
256 normal functioning and in disease processes, yet there are no known
257 marker genes for most cortical areas. When it is necessary to divide a
258 tissue sample into cortical areas, this is a manual process that requires
259 a skilled human to combine multiple visual cues and interpret them in
260 the context of their approximate location upon the cortical surface.
261 Even the questions of how many areas should be recognized in
262 cortex, and what their arrangement is, are still not completely settled.
263 A proposed division of the cortex into areas is called a cortical map.
264 In the rodent, the lack of a single agreed-upon map can be seen by
265 contrasting the recent maps given by Swanson[22] on the one hand,
266 and Paxinos and Franklin[17] on the other. While the maps are cer-
267 tainly very similar in their general arrangement, significant differences
268 remain.
269 The Allen Mouse Brain Atlas dataset
270 The Allen Mouse Brain Atlas (ABA) data were produced by doing in-
271 situ hybridization on slices of male, 56-day-old C57BL/6J mouse brains.
272 Pictures were taken of the processed slice, and these pictures were
273 semi-automatically analyzed to create a digital measurement of gene
274 expression levels at each location in each slice. Per slice, cellular spa-
275 tial resolution is achieved. Using this method, a single physical slice
276 can only be used to measure one single gene; many different mouse brains were needed in order to measure
277 the expression of many genes.
278 An automated nonlinear alignment procedure located the 2D data from the various slices in a single 3D
279 coordinate system. In the final 3D coordinate system, voxels are cubes with 200 microns on a side. There are
280 67x41x58 = 159,326 voxels in the 3D coordinate system, of which 51,533 are in the brain[15].
281 Mus musculus is thought to contain about 22,000 protein-coding genes[28]. The ABA contains data on about
282 20,000 genes in sagittal sections, out of which over 4,000 genes are also measured in coronal sections. Our
283 dataset is derived from only the coronal subset of the ABA7.
284 The ABA is not the only large public spatial gene expression dataset. However, with the exception of the ABA,
285 GenePaint, and EMAGE, most of the other resources have not (yet) extracted the expression intensity from the
286 ISH images and registered the results into a single 3-D space.
287 Related work
288 [15 ] describes the application of AGEA to the cortex. The paper describes interesting results on the structure
289 of correlations between voxel gene expression profiles within a handful of cortical areas. However, this sort
290 _________________________________________
291 6Outside of isocortex, the number of layers varies.
292 7The sagittal data do not cover the entire cortex, and also have greater registration error[15]. Genes were selected by the Allen
293 Institute for coronal sectioning based on, “classes of known neuroscientific interest... or through post hoc identification of a marked
294 non-ubiquitous expression pattern”[15].
295 of analysis is not related to either of our aims, as it neither finds marker genes, nor does it suggest a cortical
296 map based on gene expression data. Neither of the other components of AGEA can be applied to cortical
297 areas; AGEA’s Gene Finder cannot be used to find marker genes for the cortical areas; and AGEA’s hierarchical
298 clustering does not produce clusters corresponding to the cortical areas8.
299 In summary, for all three aims, (a) only one of the previous projects explores combinations of marker genes,
300 (b) there has been almost no comparison of different algorithms or scoring methods, and (c) there has been no
301 work on computationally finding marker genes for cortical areas, or on finding a hierarchical clustering that will
302 yield a map of cortical areas de novo from gene expression data.
303 Our project is guided by a concrete application with a well-specified criterion of success (how well we can
304 find marker genes for / reproduce the layout of cortical areas), which will provide a solid basis for comparing
305 different methods.
306 Significance
308 Figure 2: Gene Pitx2
309 is selectively underex-
310 pressed in area SS. The method developed in aim (1) will be applied to each cortical area to find a set of
311 marker genes such that the combinatorial expression pattern of those genes uniquely
312 picks out the target area. Finding marker genes will be useful for drug discovery as
313 well as for experimentation because marker genes can be used to design interventions
314 which selectively target individual cortical areas.
315 The application of the marker gene finding algorithm to the cortex will also support
316 the development of new neuroanatomical methods. In addition to finding markers for
317 each individual cortical areas, we will find a small panel of genes that can find many of
318 the areal boundaries at once. This panel of marker genes will allow the development of
319 an ISH protocol that will allow experimenters to more easily identify which anatomical
320 areas are present in small samples of cortex.
321 The method developed in aim (2) will provide a genoarchitectonic viewpoint that will contribute to the creation
322 of a better map. The development of present-day cortical maps was driven by the application of histological
323 stains. If a different set of stains had been available which identified a different set of features, then today’s
324 cortical maps may have come out differently. It is likely that there are many repeated, salient spatial patterns
325 in the gene expression which have not yet been captured by any stain. Therefore, cortical anatomy needs to
326 incorporate what we can learn from looking at the patterns of gene expression.
327 While we do not here propose to analyze human gene expression data, it is conceivable that the methods
328 we propose to develop could be used to suggest modifications to the human cortical map as well. In fact, the
329 methods we will develop will be applicable to other datasets beyond the brain.
330 _______________________________
331 The approach: Preliminary Studies
332 Format conversion between SEV, MATLAB, NIFTI
333 We have created software to (politely) download all of the SEV files9 from the Allen Institute website. We have
334 also created software to convert between the SEV, MATLAB, and NIFTI file formats, as well as some of Caret’s
335 file formats.
336 Flatmap of cortex
337 We downloaded the ABA data and applied a mask to select only those voxels which belong to cerebral cortex.
338 We divided the cortex into hemispheres. Using Caret[7], we created a mesh representation of the surface of the
339 selected voxels. For each gene, and for each node of the mesh, we calculated an average of the gene expression
340 of the voxels “underneath” that mesh node. We then flattened the cortex, creating a two-dimensional mesh. We
341 sampled the nodes of the irregular, flat mesh in order to create a regular grid of pixel values. We converted this
342 grid into a MATLAB matrix. We manually traced the boundaries of each of 49 cortical areas from the ABA coronal
343 reference atlas slides. We then converted these manual traces into Caret-format regional boundary data on the
344 8In both cases, the cause is that pairwise correlations between the gene expression of voxels in different areas but the same layer
345 are often stronger than pairwise correlations between the gene expression of voxels in different layers but the same area. Therefore, a
346 pairwise voxel correlation clustering algorithm will tend to create clusters representing cortical layers, not areas.
347 9SEV is a sparse format for spatial data. It is the format in which the ABA data is made available.
348 mesh surface. We projected the regions onto the 2-d mesh, and then onto the grid, and then we converted the
349 region data into MATLAB format.
350 At this point, the data are in the form of a number of 2-D matrices, all in registration, with the matrix entries
351 representing a grid of points (pixels) over the cortical surface. There is one 2-D matrix whose entries represent
352 the regional label associated with each surface pixel. And for each gene, there is a 2-D matrix whose entries
353 represent the average expression level underneath each surface pixel. We created a normalized version of the
354 gene expression data by subtracting each gene’s mean expression level (over all surface pixels) and dividing the
355 expression level of each gene by its standard deviation. The features and the target area are both functions on
356 the surface pixels. They can be referred to as scalar fields over the space of surface pixels; alternately, they can
357 be thought of as images which can be displayed on the flatmapped surface.
358 To move beyond a single average expression level for each surface pixel, we plan to create a separate matrix
359 for each cortical layer to represent the average expression level within that layer. Cortical layers are found at
360 different depths in different parts of the cortex. In preparation for extracting the layer-specific datasets, we have
361 extended Caret with routines that allow the depth of the ROI for volume-to-surface projection to vary. In the
362 Research Plan, we describe how we will automatically locate the layer depths. For validation, we have manually
363 demarcated the depth of the outer boundary of cortical layer 5 throughout the cortex.
364 Feature selection and scoring methods
367 Figure 3: The top row shows the two
368 genes which (individually) best predict
369 area AUD, according to logistic regres-
370 sion. The bottom row shows the two
371 genes which (individually) best match
372 area AUD, according to gradient sim-
373 ilarity. From left to right and top to
374 bottom, the genes are Ssr1, Efcbp1,
375 Ptk7, and Aph1a. Underexpression of a gene can serve as a marker Underexpression
376 of a gene can sometimes serve as a marker. See, for example, Figure
377 2.
378 Correlation Recall that the instances are surface pixels, and con-
379 sider the problem of attempting to classify each instance as either a
380 member of a particular anatomical area, or not. The target area can be
381 represented as a boolean mask over the surface pixels.
382 We calculated the correlation between each gene and each cortical
383 area. The top row of Figure 1 shows the three genes most correlated
384 with area SS.
385 Conditional entropy
386 For each region, we created and ran a forward stepwise procedure
387 which attempted to find pairs of gene expression boolean masks such
388 that the conditional entropy of the target area’s boolean mask, condi-
389 tioned upon the pair of gene expression boolean masks, is minimized.
390 This finds pairs of genes which are most informative (at least at
391 these discretization thresholds) relative to the question, “Is this surface
392 pixel a member of the target area?”. Its advantage over linear methods
393 such as logistic regression is that it takes account of arbitrarily nonlin-
394 ear relationships; for example, if the XOR of two variables predicts the
395 target, conditional entropy would notice, whereas linear methods would
396 not.
397 Gradient similarity We noticed that the previous two scoring methods, which are pointwise, often found
398 genes whose pattern of expression did not look similar in shape to the target region. For this reason we designed
399 a non-pointwise scoring method to detect when a gene had a pattern of expression which looked like it had a
400 boundary whose shape is similar to the shape of the target region. We call this scoring method “gradient
401 similarity”. The formula is:
402 ∑
403 pixel<img src="cmsy8-32.png" alt="∈" />pixels cos(abs(∠∇1 -∠∇2)) ⋅|∇1| + |∇2|
404 2 ⋅ pixel_value1 + pixel_value2
405 2
406 where ∇1 and ∇2 are the gradient vectors of the two images at the current pixel; ∠∇i is the angle of the
407 gradient of image i at the current pixel; |∇i| is the magnitude of the gradient of image i at the current pixel; and
408 pixel valuei is the value of the current pixel in image i.
409 The intuition is that we want to see if the borders of the pattern in the two images are similar; if the borders
410 are similar, then both images will have corresponding pixels with large gradients (because this is a border) which
411 are oriented in a similar direction (because the borders are similar).
412 Gradient similarity provides information complementary to correlation
415 Figure 4: Upper left: wwc1. Upper
416 right: mtif2. Lower left: wwc1 + mtif2
417 (each pixel’s value on the lower left is
418 the sum of the corresponding pixels in
419 the upper row). To show that gradient similarity can provide useful information that
420 cannot be detected via pointwise analyses, consider Fig. 3. The
421 pointwise method in the top row identifies genes which express more
422 strongly in AUD than outside of it; its weakness is that this includes
423 many areas which don’t have a salient border matching the areal bor-
424 der. The geometric method identifies genes whose salient expression
425 border seems to partially line up with the border of AUD; its weakness
426 is that this includes genes which don’t express over the entire area.
427 Areas which can be identified by single genes Using gradient
428 similarity, we have already found single genes which roughly identify
429 some areas and groupings of areas. For each of these areas, an ex-
430 ample of a gene which roughly identifies it is shown in Figure 5. We
431 have not yet cross-verified these genes in other atlases.
432 In addition, there are a number of areas which are almost identified
433 by single genes: COAa+NLOT (anterior part of cortical amygdalar area,
434 nucleus of the lateral olfactory tract), ENT (entorhinal), ACAv (ventral
435 anterior cingulate), VIS (visual), AUD (auditory).
436 These results validate our expectation that the ABA dataset can be
437 exploited to find marker genes for many cortical areas, while also validating the relevancy of our new scoring
438 method, gradient similarity.
439 Combinations of multiple genes are useful and necessary for some areas
440 In Figure 4, we give an example of a cortical area which is not marked by any single gene, but which
441 can be identified combinatorially. According to logistic regression, gene wwc1 is the best fit single gene for
442 predicting whether or not a pixel on the cortical surface belongs to the motor area (area MO). The upper-left
443 picture in Figure 4 shows wwc1’s spatial expression pattern over the cortex. The lower-right boundary of MO is
444 represented reasonably well by this gene, but the gene overshoots the upper-left boundary. This flattened 2-D
445 representation does not show it, but the area corresponding to the overshoot is the medial surface of the cortex.
446 MO is only found on the dorsal surface. Gene mtif2 is shown in the upper-right. Mtif2 captures MO’s upper-left
447 boundary, but not its lower-right boundary. Mtif2 does not express very much on the medial surface. By adding
448 together the values at each pixel in these two figures, we get the lower-left image. This combination captures
449 area MO much better than any single gene.
450 This shows that our proposal to develop a method to find combinations of marker genes is both possible and
451 necessary.
452 Multivariate supervised learning
453 Forward stepwise logistic regression Logistic regression is a popular method for predictive modeling of cate-
454 gorical data. As a pilot run, for five cortical areas (SS, AUD, RSP, VIS, and MO), we performed forward stepwise
455 logistic regression to find single genes, pairs of genes, and triplets of genes which predict areal identify. This is
456 an example of feature selection integrated with prediction using a stepwise wrapper. Some of the single genes
457 found were shown in various figures throughout this document, and Figure 4 shows a combination of genes
458 which was found.
459 SVM on all genes at once
460 In order to see how well one can do when looking at all genes at once, we ran a support vector machine to
461 classify cortical surface pixels based on their gene expression profiles. We achieved classification accuracy of
462 about 81%10. This shows that the genes included in the ABA dataset are sufficient to define much of cortical
463 anatomy. However, as noted above, a classifier that looks at all the genes at once isn’t as practically useful as a
464 classifier that uses only a few genes.
465 Data-driven redrawing of the cortical map
470 Figure 5: From left to right and top
471 to bottom, single genes which roughly
472 identify areas SS (somatosensory pri-
473 mary + supplemental), SSs (supple-
474 mental somatosensory), PIR (piriform),
475 FRP (frontal pole), RSP (retrosple-
476 nial), COApm (Cortical amygdalar, pos-
477 terior part, medial zone). Grouping
478 some areas together, we have also
479 found genes to identify the groups
480 ACA+PL+ILA+DP+ORB+MO (anterior
481 cingulate, prelimbic, infralimbic, dor-
482 sal peduncular, orbital, motor), poste-
483 rior and lateral visual (VISpm, VISpl,
484 VISI, VISp; posteromedial, posterolat-
485 eral, lateral, and primary visual; the
486 posterior and lateral visual area is dis-
487 tinguished from its neighbors, but not
488 from the entire rest of the cortex). The
489 genes are Pitx2, Aldh1a2, Ppfibp1,
490 Slco1a5, Tshz2, Trhr, Col12a1, Ets1. We have applied the following dimensionality reduction algorithms
491 to reduce the dimensionality of the gene expression profile associ-
492 ated with each pixel: Principal Components Analysis (PCA), Simple
493 PCA, Multi-Dimensional Scaling, Isomap, Landmark Isomap, Laplacian
494 eigenmaps, Local Tangent Space Alignment, Stochastic Proximity Em-
495 bedding, Fast Maximum Variance Unfolding, Non-negative Matrix Fac-
496 torization (NNMF). Space constraints prevent us from showing many of
497 the results, but as a sample, PCA, NNMF, and landmark Isomap are
498 shown in the first, second, and third rows of Figure 6.
499 After applying the dimensionality reduction, we ran clustering algo-
500 rithms on the reduced data. To date we have tried k-means and spec-
501 tral clustering. The results of k-means after PCA, NNMF, and landmark
502 Isomap are shown in the last row of Figure 6. To compare, the leftmost
503 picture on the bottom row of Figure 6 shows some of the major subdivi-
504 sions of cortex. These results clearly show that different dimensionality
505 reduction techniques capture different aspects of the data and lead to
506 different clusterings, indicating the utility of our proposal to produce a
507 detailed comparison of these techniques as applied to the domain of
508 genomic anatomy.
509 Many areas are captured by clusters of genes We also clustered
510 the genes using gradient similarity to see if the spatial regions defined
511 by any clusters matched known anatomical regions. Figure 7 shows, for
512 ten sample gene clusters, each cluster’s average expression pattern,
513 compared to a known anatomical boundary. This suggests that it is
514 worth attempting to cluster genes, and then to use the results to cluster
515 pixels.
516 The approach: what we plan to do
517 Flatmap cortex and segment cortical layers
518 There are multiple ways to flatten 3-D data into 2-D. We will compare
519 mappings from manifolds to planes which attempt to preserve size
520 (such as the one used by Caret[7]) with mappings which preserve an-
521 gle (conformal maps). Our method will include a statistical test that
522 warns the user if the assumption of 2-D structure seems to be wrong.
523 We have not yet made use of radial profiles. While the radial pro-
524 files may be used “raw”, for laminar structures like the cortex another
525 strategy is to group together voxels in the same cortical layer; each sur-
526 face pixel would then be associated with one expression level per gene
527 per layer. We will develop a segmentation algorithm to automatically
528 identify the layer boundaries.
529 Develop algorithms that find genetic markers for anatomical re-
530 gions
531 Scoring measures and feature selection We will develop scoring
532 methods for evaluating how good individual genes are at marking ar-
533 eas. We will compare pointwise, geometric, and information-theoretic
534 _________________________________________
535 105-fold cross-validation.
536 measures. We already developed one entirely new scoring method (gradient similarity), but we may develop
537 more. Scoring measures that we will explore will include the L1 norm, correlation, expression energy ratio, con-
538 ditional entropy, gradient similarity, Jaccard similarity, Dice similarity, Hough transform, and statistical tests such
539 as Student’s t-test, and the Mann-Whitney U test (a non-parametric test). In addition, any classifier induces a
540 scoring measure on genes by taking the prediction error when using that gene to predict the target.
545 Figure 6: First row: the first 6 reduced dimensions, using PCA. Sec-
546 ond row: the first 6 reduced dimensions, using NNMF. Third row:
547 the first six reduced dimensions, using landmark Isomap. Bottom
548 row: examples of kmeans clustering applied to reduced datasets
549 to find 7 clusters. Left: 19 of the major subdivisions of the cortex.
550 Second from left: PCA. Third from left: NNMF. Right: Landmark
551 Isomap. Additional details: In the third and fourth rows, 7 dimen-
552 sions were found, but only 6 displayed. In the last row: for PCA,
553 50 dimensions were used; for NNMF, 6 dimensions were used; for
554 landmark Isomap, 7 dimensions were used. Using some combination of these mea-
555 sures, we will develop a procedure to
556 find single marker genes for anatomical
557 regions: for each cortical area, we will
558 rank the genes by their ability to delineate
559 each area. We will quantitatively compare
560 the list of single genes generated by our
561 method to the lists generated by previous
562 methods which are mentioned in Aim 1 Re-
563 lated Work.
564 Some cortical areas have no single
565 marker genes but can be identified by com-
566 binatorial coding. This requires multivari-
567 ate scoring measures and feature selec-
568 tion procedures. Many of the measures,
569 such as expression energy, gradient sim-
570 ilarity, Jaccard, Dice, Hough, Student’s t,
571 and Mann-Whitney U are univariate. We
572 will extend these scoring measures for use
573 in multivariate feature selection, that is, for
574 scoring how well combinations of genes,
575 rather than individual genes, can distin-
576 guish a target area. There are existing
577 multivariate forms of some of the univariate
578 scoring measures, for example, Hotelling’s
579 T-square is a multivariate analog of Stu-
580 dent’s t.
581 We will develop a feature selection pro-
582 cedure for choosing the best small set of
583 marker genes for a given anatomical area. In addition to using the scoring measures that we develop, we will
584 also explore (a) feature selection using a stepwise wrapper over “vanilla” classifiers such as logistic regression,
585 (b) supervised learning methods such as decision trees which incrementally/greedily combine single gene mark-
586 ers into sets, and (c) supervised learning methods which use soft constraints to minimize number of features
587 used, such as sparse support vector machines (SVMs).
588 Since errors of displacement and of shape may cause genes and target areas to match less than they should,
589 we will consider the robustness of feature selection methods in the presence of error. Some of these methods,
590 such as the Hough transform, are designed to be resistant in the presence of error, but many are not. We will
591 consider extensions to scoring measures that may improve their robustness; for example, a wrapper that runs a
592 scoring method on small displacements and distortions of the data adds robustness to registration error at the
593 expense of computation time.
594 An area may be difficult to identify because the boundaries are misdrawn in the atlas, or because the shape
595 of the natural domain of gene expression corresponding to the area is different from the shape of the area as
596 recognized by anatomists. We will extend our procedure to handle difficult areas by combining areas or redrawing
597 their boundaries. We will develop extensions to our procedure which (a) detect when a difficult area could be
598 fit if its boundary were redrawn slightly11, and (b) detect when a difficult area could be combined with adjacent
599 _________________________________________
600 11Not just any redrawing is acceptable, only those which appear to be justified as a natural spatial domain of gene expression by
601 multiple sources of evidence. Interestingly, the need to detect “natural spatial domains of gene expression” in a data-driven fashion
602 means that the methods of Aim 2 might be useful in achieving Aim 1, as well – particularly discriminative dimensionality reduction.
603 areas to create a larger area which can be fit.
604 A future publication on the method that we develop in Aim 1 will review the scoring measures and quantita-
605 tively compare their performance in order to provide a foundation for future research of methods of marker gene
606 finding. We will measure the robustness of the scoring measures as well as their absolute performance on our
607 dataset.
608 Classifiers We will explore and compare different classifiers. As noted above, this activity is not separate
609 from the previous one, because some supervised learning algorithms include feature selection, and any clas-
610 sifier can be combined with a stepwise wrapper for use as a feature selection method. We will explore logistic
611 regression (including spatial models[16]), decision trees12, sparse SVMs, generative mixture models (including
612 naive bayes), kernel density estimation, instance-based learning methods (such as k-nearest neighbor), genetic
613 algorithms, and artificial neural networks.
614 Develop algorithms to suggest a division of a structure into anatomical parts
616 Figure 7: Prototypes corresponding to sample gene
617 clusters, clustered by gradient similarity. Region bound-
618 aries for the region that most matches each prototype
619 are overlaid. Dimensionality reduction on gene expression pro-
620 files We have already described the application of
621 ten dimensionality reduction algorithms for the pur-
622 pose of replacing the gene expression profiles, which
623 are vectors of about 4000 gene expression levels,
624 with a smaller number of features. We plan to fur-
625 ther explore and interpret these results, as well as to
626 apply other unsupervised learning algorithms, includ-
627 ing independent components analysis, self-organizing
628 maps, and generative models such as deep Boltz-
629 mann machines. We will explore ways to quantitatively
630 compare the relevance of the different dimensionality
631 reduction methods for identifying cortical areal bound-
632 aries.
633 Dimensionality reduction on pixels Instead of applying dimensionality reduction to the gene expression
634 profiles, the same techniques can be applied instead to the pixels. It is possible that the features generated in
635 this way by some dimensionality reduction techniques will directly correspond to interesting spatial regions.
636 Clustering and segmentation on pixels We will explore clustering and segmentation algorithms in order to
637 segment the pixels into regions. We will explore k-means, spectral clustering, gene shaving[9], recursive division
638 clustering, multivariate generalizations of edge detectors, multivariate generalizations of watershed transforma-
639 tions, region growing, active contours, graph partitioning methods, and recursive agglomerative clustering with
640 various linkage functions. These methods can be combined with dimensionality reduction.
641 Clustering on genes We have already shown that the procedure of clustering genes according to gradient
642 similarity, and then creating an averaged prototype of each cluster’s expression pattern, yields some spatial
643 patterns which match cortical areas. We will further explore the clustering of genes.
644 In addition to using the cluster expression prototypes directly to identify spatial regions, this might be useful
645 as a component of dimensionality reduction. For example, one could imagine clustering similar genes and then
646 replacing their expression levels with a single average expression level, thereby removing some redundancy from
647 the gene expression profiles. One could then perform clustering on pixels (possibly after a second dimensionality
648 reduction step) in order to identify spatial regions. It remains to be seen whether removal of redundancy would
649 help or hurt the ultimate goal of identifying interesting spatial regions.
650 Co-clustering There are some algorithms which simultaneously incorporate clustering on instances and on
651 features (in our case, genes and pixels), for example, IRM[11]. These are called co-clustering or biclustering
652 _________________________________________
653 12Actually, we have already begun to explore decision trees. For each cortical area, we have used the C4.5 algorithm to find a decision
654 tree for that area. We achieved good classification accuracy on our training set, but the number of genes that appeared in each tree was
655 too large. We plan to implement a pruning procedure to generate trees that use fewer genes.
656 algorithms.
657 Radial profiles We wil explore the use of the radial profile of gene expression under each pixel.
658 Compare different methods In order to tell which method is best for genomic anatomy, for each experimental
659 method we will compare the cortical map found by unsupervised learning to a cortical map derived from the Allen
660 Reference Atlas. We will explore various quantitative metrics that purport to measure how similar two clusterings
661 are, such as Jaccard, Rand index, Fowlkes-Mallows, variation of information, Larsen, Van Dongen, and others.
662 Discriminative dimensionality reduction In addition to using a purely data-driven approach to identify
663 spatial regions, it might be useful to see how well the known regions can be reconstructed from a small number
664 of features, even if those features are chosen by using knowledge of the regions. For example, linear discriminant
665 analysis could be used as a dimensionality reduction technique in order to identify a few features which are the
666 best linear summary of gene expression profiles for the purpose of discriminating between regions. This reduced
667 feature set could then be used to cluster pixels into regions. Perhaps the resulting clusters will be similar to the
668 reference atlas, yet more faithful to natural spatial domains of gene expression than the reference atlas is.
669 Apply the new methods to the cortex
670 Using the methods developed in Aim 1, we will present, for each cortical area, a short list of markers to identify
671 that area; and we will also present lists of “panels” of genes that can be used to delineate many areas at once.
672 Because in most cases the ABA coronal dataset only contains one ISH per gene, it is possible for an unrelated
673 combination of genes to seem to identify an area when in fact it is only coincidence. There are two ways we will
674 validate our marker genes to guard against this. First, we will confirm that putative combinations of marker genes
675 express the same pattern in both hemispheres. Second, we will manually validate our final results on other gene
676 expression datasets such as EMAGE, GeneAtlas, and GENSAT[8].
677 Using the methods developed in Aim 2, we will present one or more hierarchical cortical maps. We will identify
678 and explain how the statistical structure in the gene expression data led to any unexpected or interesting features
679 of these maps, and we will provide biological hypotheses to interpret any new cortical areas, or groupings of
680 areas, which are discovered.
681 ____________________________________________________________________________
682 Timeline and milestones
683 Finding marker genes
684 September-November 2009: Develop an automated mechanism for segmenting the cortical voxels into layers
685 November 2009 (milestone): Have completed construction of a flatmapped, cortical dataset with information
686 for each layer
687 October 2009-April 2010: Develop scoring and supervised learning methods.
688 January 2010 (milestone): Submit a publication on single marker genes for cortical areas
689 February-July 2010: Continue to develop scoring methods and supervised learning frameworks. Extend tech-
690 niques for robustness. Compare the performance of techniques. Validate marker genes. Prepare software
691 toolbox for Aim 1.
692 June 2010 (milestone): Submit a paper describing a method fulfilling Aim 1. Release toolbox.
693 July 2010 (milestone): Submit a paper describing combinations of marker genes for each cortical area, and a
694 small number of marker genes that can, in combination, define most of the areas at once
695 Revealing new ways to parcellate a structure into regions
696 June 2010-March 2011: Explore dimensionality reduction algorithms. Explore clustering algorithms. Adapt
697 clustering algorithms to use radial profile information. Compare the performance of techniques.
698 March 2011 (milestone): Submit a paper describing a method fulfilling Aim 2. Release toolbox.
699 February-May 2011: Using the methods developed for Aim 2, explore the genomic anatomy of the cortex,
700 interpret the results. Prepare software toolbox for Aim 2.
701 May 2011 (milestone): Submit a paper on the genomic anatomy of the cortex, using the methods developed in
702 Aim 2
703 May-August 2011: Revisit Aim 1 to see if what was learned during Aim 2 can improve the methods for Aim 1.
704 Possibly submit another paper.
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