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author bshanks@bshanks.dyndns.org
date Wed Apr 22 07:35:14 2009 -0700 (16 years ago)
parents 89815d210b5c
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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, and we will present our strategy for solving it. Next we will discuss related work. At the
37 conclusion of each section, we will summarize why our strategy is different from what has been done before. At
38 the end of this section, we will describe the potential impact.
39 Aim 1: Given a map of regions, find genes that mark the regions
40 Machine learning terminology: classifiers The task of looking for marker genes for known anatomical regions
41 means that one is looking for a set of genes such that, if the expression level of those genes is known, then the
42 locations of the regions can be inferred.
43 If we define the regions so that they cover the entire anatomical structure to be subdivided, we may say that
44 we are using gene expression in each voxel to assign that voxel to the proper area. We call this a classification
45 task, because each voxel is being assigned to a class (namely, its region). An understanding of the relationship
46 between the combination of their expression levels and the locations of the regions may be expressed as a
47 function. The input to this function is a voxel, along with the gene expression levels within that voxel; the output is
48 the regional identity of the target voxel, that is, the region to which the target voxel belongs. We call this function
49 a classifier. In general, the input to a classifier is called an instance, and the output is called a label (or a class
50 label).
51 The object of aim 1 is not to produce a single classifier, but rather to develop an automated method for
52 determining a classifier for any known anatomical structure. Therefore, we seek a procedure by which a gene
53 expression dataset may be analyzed in concert with an anatomical atlas in order to produce a classifier. The
54 initial gene expression dataset used in the construction of the classifier is called training data. In the machine
55 learning literature, this sort of procedure may be thought of as a supervised learning task, defined as a task in
56 which the goal is to learn a mapping from instances to labels, and the training data consists of a set of instances
57 (voxels) for which the labels (regions) are known.
58 Each gene expression level is called a feature, and the selection of which genes1 to include is called feature
59 selection. Feature selection is one component of the task of learning a classifier. Some methods for learning
60 classifiers start out with a separate feature selection phase, whereas other methods combine feature selection
61 with other aspects of training.
62 One class of feature selection methods assigns some sort of score to each candidate gene. The top-ranked
63 genes are then chosen. Some scoring measures can assign a score to a set of selected genes, not just to a
64 single gene; in this case, a dynamic procedure may be used in which features are added and subtracted from the
65 selected set depending on how much they raise the score. Such procedures are called “stepwise” or “greedy”.
66 Although the classifier itself may only look at the gene expression data within each voxel before classifying
67 that voxel, the algorithm which constructs the classifier may look over the entire dataset. We can categorize
68 score-based feature selection methods depending on how the score of calculated. Often the score calculation
69 consists of assigning a sub-score to each voxel, and then aggregating these sub-scores into a final score (the
70 aggregation is often a sum or a sum of squares or average). If only information from nearby voxels is used to
71 calculate a voxel’s sub-score, then we say it is a local scoring method. If only information from the voxel itself is
72 used to calculate a voxel’s sub-score, then we say it is a pointwise scoring method.
73 Both gene expression data and anatomical atlases have errors, due to a variety of factors. Individual subjects
74 have idiosyncratic anatomy. Subjects may be improperly registered to the atlas. The method used to measure
75 gene expression may be noisy. The atlas may have errors. It is even possible that some areas in the anatomical
76 1Strictly speaking, the features are gene expression levels, but we’ll call them genes.
77 atlas are “wrong” in that they do not have the same shape as the natural domains of gene expression to which
78 they correspond. These sources of error can affect the displacement and the shape of both the gene expression
79 data and the anatomical target areas. Therefore, it is important to use feature selection methods which are
80 robust to these kinds of errors.
81 Our strategy for Aim 1
82 Key questions when choosing a learning method are: What are the instances? What are the features? How are
83 the features chosen? Here are four principles that outline our answers to these questions.
84 Principle 1: Combinatorial gene expression
85 It is too much to hope that every anatomical region of interest will be identified by a single gene. For example,
86 in the cortex, there are some areas which are not clearly delineated by any gene included in the Allen Brain Atlas
87 (ABA) dataset. However, at least some of these areas can be delineated by looking at combinations of genes
88 (an example of an area for which multiple genes are necessary and sufficient is provided in Preliminary Studies,
89 Figure 4). Therefore, each instance should contain multiple features (genes).
90 Principle 2: Only look at combinations of small numbers of genes
91 When the classifier classifies a voxel, it is only allowed to look at the expression of the genes which have
92 been selected as features. The more data that are available to a classifier, the better that it can do. For example,
93 perhaps there are weak correlations over many genes that add up to a strong signal. So, why not include every
94 gene as a feature? The reason is that we wish to employ the classifier in situations in which it is not feasible to
95 gather data about every gene. For example, if we want to use the expression of marker genes as a trigger for
96 some regionally-targeted intervention, then our intervention must contain a molecular mechanism to check the
97 expression level of each marker gene before it triggers. It is currently infeasible to design a molecular trigger that
98 checks the level of more than a handful of genes. Similarly, if the goal is to develop a procedure to do ISH on
99 tissue samples in order to label their anatomy, then it is infeasible to label more than a few genes. Therefore, we
100 must select only a few genes as features.
101 The requirement to find combinations of only a small number of genes limits us from straightforwardly ap-
102 plying many of the most simple techniques from the field of supervised machine learning. In the parlance of
103 machine learning, our task combines feature selection with supervised learning.
104 Principle 3: Use geometry in feature selection
105 When doing feature selection with score-based methods, the simplest thing to do would be to score the per-
106 formance of each voxel by itself and then combine these scores (pointwise scoring). A more powerful approach
107 is to also use information about the geometric relations between each voxel and its neighbors; this requires non-
108 pointwise, local scoring methods. See Preliminary Studies, figure 3 for evidence of the complementary nature of
109 pointwise and local scoring methods.
110 Principle 4: Work in 2-D whenever possible
111 There are many anatomical structures which are commonly characterized in terms of a two-dimensional
112 manifold. When it is known that the structure that one is looking for is two-dimensional, the results may be
113 improved by allowing the analysis algorithm to take advantage of this prior knowledge. In addition, it is easier for
114 humans to visualize and work with 2-D data. Therefore, when possible, the instances should represent pixels,
115 not voxels.
116 Related work
117 There is a substantial body of work on the analysis of gene expression data, most of this concerns gene expres-
118 sion data which are not fundamentally spatial2.
119 As noted above, there has been much work on both supervised learning and there are many available
120 algorithms for each. However, the algorithms require the scientist to provide a framework for representing the
121 problem domain, and the way that this framework is set up has a large impact on performance. Creating a
122 good framework can require creatively reconceptualizing the problem domain, and is not merely a mechanical
123 “fine-tuning” of numerical parameters. For example, we believe that domain-specific scoring measures (such
124 _________________________________________
125 2By “fundamentally spatial” we mean that there is information from a large number of spatial locations indexed by spatial coordinates;
126 not just data which have only a few different locations or which is indexed by anatomical label.
127 as gradient similarity, which is discussed in Preliminary Studies) may be necessary in order to achieve the best
128 results in this application.
129 We now turn to efforts to find marker genes using spatial gene expression data using automated methods.
130 GeneAtlas[5] and EMAGE [26] allow the user to construct a search query by demarcating regions and then
131 specifying either the strength of expression or the name of another gene or dataset whose expression pattern
132 is to be matched. Neither GeneAtlas nor EMAGE allow one to search for combinations of genes that define a
133 region in concert but not separately.
134 [15 ] describes AGEA, ”Anatomic Gene Expression Atlas”. AGEA has three components. Gene Finder: The
135 user selects a seed voxel and the system (1) chooses a cluster which includes the seed voxel, (2) yields a list of
136 genes which are overexpressed in that cluster. Correlation: The user selects a seed voxel and the system then
137 shows the user how much correlation there is between the gene expression profile of the seed voxel and every
138 other voxel. Clusters: will be described later. [6] looks at the mean expression level of genes within anatomical
139 regions, and applies a Student’s t-test with Bonferroni correction to determine whether the mean expression
140 level of a gene is significantly higher in the target region. [15] and [6] differ from our Aim 1 in at least three
141 ways. First, [15] and [6] find only single genes, whereas we will also look for combinations of genes. Second,
142 [15 ] and [6] can only use overexpression as a marker, whereas we will also search for underexpression. Third,
143 [15 ] and [6] use scores based on pointwise expression levels, whereas we will also use geometric scores such
144 as gradient similarity (described in Preliminary Studies). Figures 4, 2, and 3 in the Preliminary Studies section
145 contain evidence that each of our three choices is the right one.
146 [10 ] describes a technique to find combinations of marker genes to pick out an anatomical region. They use
147 an evolutionary algorithm to evolve logical operators which combine boolean (thresholded) images in order to
148 match a target image.
149 In summary, there has been fruitful work on finding marker genes, but only one of the previous projects
150 explores combinations of marker genes, and none of these publications compare the results obtained by using
151 different algorithms or scoring methods.
152 Aim 2: From gene expression data, discover a map of regions
153 Machine learning terminology: clustering
154 If one is given a dataset consisting merely of instances, with no class labels, then analysis of the dataset is
155 referred to as unsupervised learning in the jargon of machine learning. One thing that you can do with such a
156 dataset is to group instances together. A set of similar instances is called a cluster, and the activity of finding
157 grouping the data into clusters is called clustering or cluster analysis.
158 The task of deciding how to carve up a structure into anatomical regions can be put into these terms. The
159 instances are once again voxels (or pixels) along with their associated gene expression profiles. We make
160 the assumption that voxels from the same anatomical region have similar gene expression profiles, at least
161 compared to the other regions. This means that clustering voxels is the same as finding potential regions; we
162 seek a partitioning of the voxels into regions, that is, into clusters of voxels with similar gene expression.
163 It is desirable to determine not just one set of regions, but also how these regions relate to each other. The
164 outcome of clustering may be a hierarchical tree of clusters, rather than a single set of clusters which partition
165 the voxels. This is called hierarchical clustering.
166 Similarity scores A crucial choice when designing a clustering method is how to measure similarity, across
167 either pairs of instances, or clusters, or both. There is much overlap between scoring methods for feature
168 selection (discussed above under Aim 1) and scoring methods for similarity.
169 Spatially contiguous clusters; image segmentation We have shown that aim 2 is a type of clustering
170 task. In fact, it is a special type of clustering task because we have an additional constraint on clusters; voxels
171 grouped together into a cluster must be spatially contiguous. In Preliminary Studies, we show that one can get
172 reasonable results without enforcing this constraint; however, we plan to compare these results against other
173 methods which guarantee contiguous clusters.
174 Image segmentation is the task of partitioning the pixels in a digital image into clusters, usually contiguous
175 clusters. Aim 2 is similar to an image segmentation task. There are two main differences; in our task, there are
176 thousands of color channels (one for each gene), rather than just three3. A more crucial difference is that there
177 are various cues which are appropriate for detecting sharp object boundaries in a visual scene but which are not
178 appropriate for segmenting abstract spatial data such as gene expression. Although many image segmentation
179 algorithms can be expected to work well for segmenting other sorts of spatially arranged data, some of these
180 algorithms are specialized for visual images.
181 Dimensionality reduction In this section, we discuss reducing the length of the per-pixel gene expression
182 feature vector. By “dimension”, we mean the dimension of this vector, not the spatial dimension of the underlying
183 data.
184 Unlike aim 1, there is no externally-imposed need to select only a handful of informative genes for inclusion
185 in the instances. However, some clustering algorithms perform better on small numbers of features4. There are
186 techniques which “summarize” a larger number of features using a smaller number of features; these techniques
187 go by the name of feature extraction or dimensionality reduction. The small set of features that such a technique
188 yields is called the reduced feature set. Note that the features in the reduced feature set do not necessarily
189 correspond to genes; each feature in the reduced set may be any function of the set of gene expression levels.
190 Clustering genes rather than voxels Although the ultimate goal is to cluster the instances (voxels or pixels),
191 one strategy to achieve this goal is to first cluster the features (genes). There are two ways that clusters of genes
192 could be used.
193 Gene clusters could be used as part of dimensionality reduction: rather than have one feature for each gene,
194 we could have one reduced feature for each gene cluster.
195 Gene clusters could also be used to directly yield a clustering on instances. This is because many genes have
196 an expression pattern which seems to pick out a single, spatially contiguous region. This suggests the following
197 procedure: cluster together genes which pick out similar regions, and then to use the more popular common
198 regions as the final clusters. In Preliminary Studies, Figure 7, we show that a number of anatomically recognized
199 cortical regions, as well as some “superregions” formed by lumping together a few regions, are associated with
200 gene clusters in this fashion.
201 Related work
202 Some researchers have attempted to parcellate cortex on the basis of non-gene expression data. For example,
203 [18 ], [2 ], [19], and [1] associate spots on the cortex with the radial profile5 of response to some stain ([12] uses
204 MRI), extract features from this profile, and then use similarity between surface pixels to cluster.
205 [23 ] describes an analysis of the anatomy of the hippocampus using the ABA dataset. In addition to manual
206 analysis, two clustering methods were employed, a modified Non-negative Matrix Factorization (NNMF), and
207 a hierarchical recursive bifurcation clustering scheme based on correlation as the similarity score. The paper
208 yielded impressive results, proving the usefulness of computational genomic anatomy. We have run NNMF on
209 the cortical dataset
210 AGEA[15] includes a preset hierarchical clustering of voxels based on a recursive bifurcation algorithm with
211 correlation as the similarity metric. EMAGE[26] allows the user to select a dataset from among a large number
212 of alternatives, or by running a search query, and then to cluster the genes within that dataset. EMAGE clusters
213 via hierarchical complete linkage clustering.
214 [6 ] clusters genes. For each cluster, prototypical spatial expression patterns were created by averaging the
215 genes in the cluster. The prototypes were analyzed manually, without clustering voxels.
216 [10 ] applies their technique for finding combinations of marker genes for the purpose of clustering genes
217 around a “seed gene”.
218 In summary, although these projects obtained clusterings, there has not been much comparison between
219 different algorithms or scoring methods, so it is likely that the best clustering method for this application has not
220 yet been found. The projects using gene expression on cortex did not attempt to make use of the radial profile
221 of gene expression. Also, none of these projects did a separate dimensionality reduction step before clustering
222 _________________________________________
223 3There are imaging tasks which use more than three colors, for example multispectral imaging and hyperspectral imaging, which are
224 often used to process satellite imagery.
225 4First, because the number of features in the reduced dataset is less than in the original dataset, the running time of clustering
226 algorithms may be much less. Second, it is thought that some clustering algorithms may give better results on reduced data.
227 5A radial profile is a profile along a line perpendicular to the cortical surface.
228 pixels, none tried to cluster genes first in order to guide automated clustering of pixels into spatial regions, and
229 none used co-clustering algorithms.
230 Aim 3: apply the methods developed to the cerebral cortex
233 Figure 1: Top row: Genes Nfic
234 and A930001M12Rik are the most
235 correlated with area SS (somatosen-
236 sory cortex). Bottom row: Genes
237 C130038G02Rik and Cacna1i are
238 those with the best fit using logistic
239 regression. Within each picture, the
240 vertical axis roughly corresponds to
241 anterior at the top and posterior at the
242 bottom, and the horizontal axis roughly
243 corresponds to medial at the left and
244 lateral at the right. The red outline is
245 the boundary of region SS. Pixels are
246 colored according to correlation, with
247 red meaning high correlation and blue
248 meaning low. Background
249 The cortex is divided into areas and layers. Because of the cortical
250 columnar organization, the parcellation of the cortex into areas can be
251 drawn as a 2-D map on the surface of the cortex. In the third dimension,
252 the boundaries between the areas continue downwards into the cortical
253 depth, perpendicular to the surface. The layer boundaries run parallel
254 to the surface. One can picture an area of the cortex as a slice of a
255 six-layered cake6.
256 It is known that different cortical areas have distinct roles in both
257 normal functioning and in disease processes, yet there are no known
258 marker genes for most cortical areas. When it is necessary to divide a
259 tissue sample into cortical areas, this is a manual process that requires
260 a skilled human to combine multiple visual cues and interpret them in
261 the context of their approximate location upon the cortical surface.
262 Even the questions of how many areas should be recognized in
263 cortex, and what their arrangement is, are still not completely settled.
264 A proposed division of the cortex into areas is called a cortical map.
265 In the rodent, the lack of a single agreed-upon map can be seen by
266 contrasting the recent maps given by Swanson[22] on the one hand,
267 and Paxinos and Franklin[17] on the other. While the maps are cer-
268 tainly very similar in their general arrangement, significant differences
269 remain.
270 The Allen Mouse Brain Atlas dataset
271 The Allen Mouse Brain Atlas (ABA) data were produced by doing in-
272 situ hybridization on slices of male, 56-day-old C57BL/6J mouse brains.
273 Pictures were taken of the processed slice, and these pictures were
274 semi-automatically analyzed to create a digital measurement of gene
275 expression levels at each location in each slice. Per slice, cellular spa-
276 tial resolution is achieved. Using this method, a single physical slice
277 can only be used to measure one single gene; many different mouse brains were needed in order to measure
278 the expression of many genes.
279 An automated nonlinear alignment procedure located the 2D data from the various slices in a single 3D
280 coordinate system. In the final 3D coordinate system, voxels are cubes with 200 microns on a side. There are
281 67x41x58 = 159,326 voxels in the 3D coordinate system, of which 51,533 are in the brain[15].
282 Mus musculus is thought to contain about 22,000 protein-coding genes[28]. The ABA contains data on about
283 20,000 genes in sagittal sections, out of which over 4,000 genes are also measured in coronal sections. Our
284 dataset is derived from only the coronal subset of the ABA7.
285 The ABA is not the only large public spatial gene expression dataset. However, with the exception of the ABA,
286 GenePaint, and EMAGE, most of the other resources have not (yet) extracted the expression intensity from the
287 ISH images and registered the results into a single 3-D space.
288 Related work
289 [15 ] describes the application of AGEA to the cortex. The paper describes interesting results on the structure
290 of correlations between voxel gene expression profiles within a handful of cortical areas. However, this sort
291 _________________________________________
292 6Outside of isocortex, the number of layers varies.
293 7The sagittal data do not cover the entire cortex, and also have greater registration error[15]. Genes were selected by the Allen
294 Institute for coronal sectioning based on, “classes of known neuroscientific interest... or through post hoc identification of a marked
295 non-ubiquitous expression pattern”[15].
296 of analysis is not related to either of our aims, as it neither finds marker genes, nor does it suggest a cortical
297 map based on gene expression data. Neither of the other components of AGEA can be applied to cortical
298 areas; AGEA’s Gene Finder cannot be used to find marker genes for the cortical areas; and AGEA’s hierarchical
299 clustering does not produce clusters corresponding to the cortical areas8.
300 In summary, for all three aims, (a) only one of the previous projects explores combinations of marker genes,
301 (b) there has been almost no comparison of different algorithms or scoring methods, and (c) there has been no
302 work on computationally finding marker genes for cortical areas, or on finding a hierarchical clustering that will
303 yield a map of cortical areas de novo from gene expression data.
304 Our project is guided by a concrete application with a well-specified criterion of success (how well we can
305 find marker genes for / reproduce the layout of cortical areas), which will provide a solid basis for comparing
306 different methods.
307 Significance
309 Figure 2: Gene Pitx2
310 is selectively underex-
311 pressed in area SS. The method developed in aim (1) will be applied to each cortical area to find a set of
312 marker genes such that the combinatorial expression pattern of those genes uniquely
313 picks out the target area. Finding marker genes will be useful for drug discovery as
314 well as for experimentation because marker genes can be used to design interventions
315 which selectively target individual cortical areas.
316 The application of the marker gene finding algorithm to the cortex will also support
317 the development of new neuroanatomical methods. In addition to finding markers for
318 each individual cortical areas, we will find a small panel of genes that can find many of
319 the areal boundaries at once. This panel of marker genes will allow the development of
320 an ISH protocol that will allow experimenters to more easily identify which anatomical
321 areas are present in small samples of cortex.
322 The method developed in aim (2) will provide a genoarchitectonic viewpoint that will contribute to the creation
323 of a better map. The development of present-day cortical maps was driven by the application of histological
324 stains. If a different set of stains had been available which identified a different set of features, then today’s
325 cortical maps may have come out differently. It is likely that there are many repeated, salient spatial patterns
326 in the gene expression which have not yet been captured by any stain. Therefore, cortical anatomy needs to
327 incorporate what we can learn from looking at the patterns of gene expression.
328 While we do not here propose to analyze human gene expression data, it is conceivable that the methods
329 we propose to develop could be used to suggest modifications to the human cortical map as well. In fact, the
330 methods we will develop will be applicable to other datasets beyond the brain.
331 _______________________________
332 The approach: Preliminary Studies
333 Format conversion between SEV, MATLAB, NIFTI
334 We have created software to (politely) download all of the SEV files9 from the Allen Institute website. We have
335 also created software to convert between the SEV, MATLAB, and NIFTI file formats, as well as some of Caret’s
336 file formats.
337 Flatmap of cortex
338 We downloaded the ABA data and applied a mask to select only those voxels which belong to cerebral cortex.
339 We divided the cortex into hemispheres. Using Caret[7], we created a mesh representation of the surface of the
340 selected voxels. For each gene, and for each node of the mesh, we calculated an average of the gene expression
341 of the voxels “underneath” that mesh node. We then flattened the cortex, creating a two-dimensional mesh. We
342 sampled the nodes of the irregular, flat mesh in order to create a regular grid of pixel values. We converted this
343 grid into a MATLAB matrix. We manually traced the boundaries of each of 49 cortical areas from the ABA coronal
344 reference atlas slides. We then converted these manual traces into Caret-format regional boundary data on the
345 8In both cases, the cause is that pairwise correlations between the gene expression of voxels in different areas but the same layer
346 are often stronger than pairwise correlations between the gene expression of voxels in different layers but the same area. Therefore, a
347 pairwise voxel correlation clustering algorithm will tend to create clusters representing cortical layers, not areas.
348 9SEV is a sparse format for spatial data. It is the format in which the ABA data is made available.
349 mesh surface. We projected the regions onto the 2-d mesh, and then onto the grid, and then we converted the
350 region data into MATLAB format.
351 At this point, the data are in the form of a number of 2-D matrices, all in registration, with the matrix entries
352 representing a grid of points (pixels) over the cortical surface. There is one 2-D matrix whose entries represent
353 the regional label associated with each surface pixel. And for each gene, there is a 2-D matrix whose entries
354 represent the average expression level underneath each surface pixel. We created a normalized version of the
355 gene expression data by subtracting each gene’s mean expression level (over all surface pixels) and dividing the
356 expression level of each gene by its standard deviation. The features and the target area are both functions on
357 the surface pixels. They can be referred to as scalar fields over the space of surface pixels; alternately, they can
358 be thought of as images which can be displayed on the flatmapped surface.
359 To move beyond a single average expression level for each surface pixel, we plan to create a separate matrix
360 for each cortical layer to represent the average expression level within that layer. Cortical layers are found at
361 different depths in different parts of the cortex. In preparation for extracting the layer-specific datasets, we have
362 extended Caret with routines that allow the depth of the ROI for volume-to-surface projection to vary. In the
363 Research Plan, we describe how we will automatically locate the layer depths. For validation, we have manually
364 demarcated the depth of the outer boundary of cortical layer 5 throughout the cortex.
365 Feature selection and scoring methods
368 Figure 3: The top row shows the two
369 genes which (individually) best predict
370 area AUD, according to logistic regres-
371 sion. The bottom row shows the two
372 genes which (individually) best match
373 area AUD, according to gradient sim-
374 ilarity. From left to right and top to
375 bottom, the genes are Ssr1, Efcbp1,
376 Ptk7, and Aph1a. Underexpression of a gene can serve as a marker Underexpression
377 of a gene can sometimes serve as a marker. See, for example, Figure
378 2.
379 Correlation Recall that the instances are surface pixels, and con-
380 sider the problem of attempting to classify each instance as either a
381 member of a particular anatomical area, or not. The target area can be
382 represented as a boolean mask over the surface pixels.
383 We calculated the correlation between each gene and each cortical
384 area. The top row of Figure 1 shows the three genes most correlated
385 with area SS.
386 Conditional entropy
387 For each region, we created and ran a forward stepwise procedure
388 which attempted to find pairs of gene expression boolean masks such
389 that the conditional entropy of the target area’s boolean mask, condi-
390 tioned upon the pair of gene expression boolean masks, is minimized.
391 This finds pairs of genes which are most informative (at least at
392 these discretization thresholds) relative to the question, “Is this surface
393 pixel a member of the target area?”. Its advantage over linear methods
394 such as logistic regression is that it takes account of arbitrarily nonlin-
395 ear relationships; for example, if the XOR of two variables predicts the
396 target, conditional entropy would notice, whereas linear methods would
397 not.
398 Gradient similarity We noticed that the previous two scoring methods, which are pointwise, often found
399 genes whose pattern of expression did not look similar in shape to the target region. For this reason we designed
400 a non-pointwise scoring method to detect when a gene had a pattern of expression which looked like it had a
401 boundary whose shape is similar to the shape of the target region. We call this scoring method “gradient
402 similarity”. The formula is:
403 ∑
404 pixel<img src="cmsy8-32.png" alt="&#x2208;" />pixels cos(abs(&#x2220;&#x2207;1 -&#x2220;&#x2207;2)) &#x22C5;|&#x2207;1| + |&#x2207;2|
405 2 &#x22C5; pixel_value1 + pixel_value2
406 2
407 where &#x2207;1 and &#x2207;2 are the gradient vectors of the two images at the current pixel; &#x2220;&#x2207;i is the angle of the
408 gradient of image i at the current pixel; |&#x2207;i| is the magnitude of the gradient of image i at the current pixel; and
409 pixel valuei is the value of the current pixel in image i.
410 The intuition is that we want to see if the borders of the pattern in the two images are similar; if the borders
411 are similar, then both images will have corresponding pixels with large gradients (because this is a border) which
412 are oriented in a similar direction (because the borders are similar).
413 Gradient similarity provides information complementary to correlation
416 Figure 4: Upper left: wwc1. Upper
417 right: mtif2. Lower left: wwc1 + mtif2
418 (each pixel&#8217;s value on the lower left is
419 the sum of the corresponding pixels in
420 the upper row). To show that gradient similarity can provide useful information that
421 cannot be detected via pointwise analyses, consider Fig. 3. The
422 pointwise method in the top row identifies genes which express more
423 strongly in AUD than outside of it; its weakness is that this includes
424 many areas which don&#8217;t have a salient border matching the areal bor-
425 der. The geometric method identifies genes whose salient expression
426 border seems to partially line up with the border of AUD; its weakness
427 is that this includes genes which don&#8217;t express over the entire area.
428 Areas which can be identified by single genes Using gradient
429 similarity, we have already found single genes which roughly identify
430 some areas and groupings of areas. For each of these areas, an ex-
431 ample of a gene which roughly identifies it is shown in Figure 5. We
432 have not yet cross-verified these genes in other atlases.
433 In addition, there are a number of areas which are almost identified
434 by single genes: COAa+NLOT (anterior part of cortical amygdalar area,
435 nucleus of the lateral olfactory tract), ENT (entorhinal), ACAv (ventral
436 anterior cingulate), VIS (visual), AUD (auditory).
437 These results validate our expectation that the ABA dataset can be
438 exploited to find marker genes for many cortical areas, while also validating the relevancy of our new scoring
439 method, gradient similarity.
440 Combinations of multiple genes are useful and necessary for some areas
441 In Figure 4, we give an example of a cortical area which is not marked by any single gene, but which
442 can be identified combinatorially. According to logistic regression, gene wwc1 is the best fit single gene for
443 predicting whether or not a pixel on the cortical surface belongs to the motor area (area MO). The upper-left
444 picture in Figure 4 shows wwc1&#8217;s spatial expression pattern over the cortex. The lower-right boundary of MO is
445 represented reasonably well by this gene, but the gene overshoots the upper-left boundary. This flattened 2-D
446 representation does not show it, but the area corresponding to the overshoot is the medial surface of the cortex.
447 MO is only found on the dorsal surface. Gene mtif2 is shown in the upper-right. Mtif2 captures MO&#8217;s upper-left
448 boundary, but not its lower-right boundary. Mtif2 does not express very much on the medial surface. By adding
449 together the values at each pixel in these two figures, we get the lower-left image. This combination captures
450 area MO much better than any single gene.
451 This shows that our proposal to develop a method to find combinations of marker genes is both possible and
452 necessary.
453 Multivariate supervised learning
454 Forward stepwise logistic regression Logistic regression is a popular method for predictive modeling of cate-
455 gorical data. As a pilot run, for five cortical areas (SS, AUD, RSP, VIS, and MO), we performed forward stepwise
456 logistic regression to find single genes, pairs of genes, and triplets of genes which predict areal identify. This is
457 an example of feature selection integrated with prediction using a stepwise wrapper. Some of the single genes
458 found were shown in various figures throughout this document, and Figure 4 shows a combination of genes
459 which was found.
460 SVM on all genes at once
461 In order to see how well one can do when looking at all genes at once, we ran a support vector machine to
462 classify cortical surface pixels based on their gene expression profiles. We achieved classification accuracy of
463 about 81%10. This shows that the genes included in the ABA dataset are sufficient to define much of cortical
464 anatomy. However, as noted above, a classifier that looks at all the genes at once isn&#8217;t as practically useful as a
465 classifier that uses only a few genes.
466 Data-driven redrawing of the cortical map
471 Figure 5: From left to right and top
472 to bottom, single genes which roughly
473 identify areas SS (somatosensory pri-
474 mary + supplemental), SSs (supple-
475 mental somatosensory), PIR (piriform),
476 FRP (frontal pole), RSP (retrosple-
477 nial), COApm (Cortical amygdalar, pos-
478 terior part, medial zone). Grouping
479 some areas together, we have also
480 found genes to identify the groups
481 ACA+PL+ILA+DP+ORB+MO (anterior
482 cingulate, prelimbic, infralimbic, dor-
483 sal peduncular, orbital, motor), poste-
484 rior and lateral visual (VISpm, VISpl,
485 VISI, VISp; posteromedial, posterolat-
486 eral, lateral, and primary visual; the
487 posterior and lateral visual area is dis-
488 tinguished from its neighbors, but not
489 from the entire rest of the cortex). The
490 genes are Pitx2, Aldh1a2, Ppfibp1,
491 Slco1a5, Tshz2, Trhr, Col12a1, Ets1. We have applied the following dimensionality reduction algorithms
492 to reduce the dimensionality of the gene expression profile associ-
493 ated with each pixel: Principal Components Analysis (PCA), Simple
494 PCA, Multi-Dimensional Scaling, Isomap, Landmark Isomap, Laplacian
495 eigenmaps, Local Tangent Space Alignment, Stochastic Proximity Em-
496 bedding, Fast Maximum Variance Unfolding, Non-negative Matrix Fac-
497 torization (NNMF). Space constraints prevent us from showing many of
498 the results, but as a sample, PCA, NNMF, and landmark Isomap are
499 shown in the first, second, and third rows of Figure 6.
500 After applying the dimensionality reduction, we ran clustering algo-
501 rithms on the reduced data. To date we have tried k-means and spec-
502 tral clustering. The results of k-means after PCA, NNMF, and landmark
503 Isomap are shown in the last row of Figure 6. To compare, the leftmost
504 picture on the bottom row of Figure 6 shows some of the major subdivi-
505 sions of cortex. These results clearly show that different dimensionality
506 reduction techniques capture different aspects of the data and lead to
507 different clusterings, indicating the utility of our proposal to produce a
508 detailed comparison of these techniques as applied to the domain of
509 genomic anatomy.
510 Many areas are captured by clusters of genes We also clustered
511 the genes using gradient similarity to see if the spatial regions defined
512 by any clusters matched known anatomical regions. Figure 7 shows, for
513 ten sample gene clusters, each cluster&#8217;s average expression pattern,
514 compared to a known anatomical boundary. This suggests that it is
515 worth attempting to cluster genes, and then to use the results to cluster
516 pixels.
517 The approach: what we plan to do
518 Flatmap cortex and segment cortical layers
519 There are multiple ways to flatten 3-D data into 2-D. We will compare
520 mappings from manifolds to planes which attempt to preserve size
521 (such as the one used by Caret[7]) with mappings which preserve an-
522 gle (conformal maps). Our method will include a statistical test that
523 warns the user if the assumption of 2-D structure seems to be wrong.
524 We have not yet made use of radial profiles. While the radial pro-
525 files may be used &#8220;raw&#8221;, for laminar structures like the cortex another
526 strategy is to group together voxels in the same cortical layer; each sur-
527 face pixel would then be associated with one expression level per gene
528 per layer. We will develop a segmentation algorithm to automatically
529 identify the layer boundaries.
530 Develop algorithms that find genetic markers for anatomical re-
531 gions
532 Scoring measures and feature selection We will develop scoring
533 methods for evaluating how good individual genes are at marking ar-
534 eas. We will compare pointwise, geometric, and information-theoretic
535 _________________________________________
536 105-fold cross-validation.
537 measures. We already developed one entirely new scoring method (gradient similarity), but we may develop
538 more. Scoring measures that we will explore will include the L1 norm, correlation, expression energy ratio, con-
539 ditional entropy, gradient similarity, Jaccard similarity, Dice similarity, Hough transform, and statistical tests such
540 as Student&#8217;s t-test, and the Mann-Whitney U test (a non-parametric test). In addition, any classifier induces a
541 scoring measure on genes by taking the prediction error when using that gene to predict the target.
546 Figure 6: First row: the first 6 reduced dimensions, using PCA. Sec-
547 ond row: the first 6 reduced dimensions, using NNMF. Third row:
548 the first six reduced dimensions, using landmark Isomap. Bottom
549 row: examples of kmeans clustering applied to reduced datasets
550 to find 7 clusters. Left: 19 of the major subdivisions of the cortex.
551 Second from left: PCA. Third from left: NNMF. Right: Landmark
552 Isomap. Additional details: In the third and fourth rows, 7 dimen-
553 sions were found, but only 6 displayed. In the last row: for PCA,
554 50 dimensions were used; for NNMF, 6 dimensions were used; for
555 landmark Isomap, 7 dimensions were used. Using some combination of these mea-
556 sures, we will develop a procedure to
557 find single marker genes for anatomical
558 regions: for each cortical area, we will
559 rank the genes by their ability to delineate
560 each area. We will quantitatively compare
561 the list of single genes generated by our
562 method to the lists generated by previous
563 methods which are mentioned in Aim 1 Re-
564 lated Work.
565 Some cortical areas have no single
566 marker genes but can be identified by com-
567 binatorial coding. This requires multivari-
568 ate scoring measures and feature selec-
569 tion procedures. Many of the measures,
570 such as expression energy, gradient sim-
571 ilarity, Jaccard, Dice, Hough, Student&#8217;s t,
572 and Mann-Whitney U are univariate. We
573 will extend these scoring measures for use
574 in multivariate feature selection, that is, for
575 scoring how well combinations of genes,
576 rather than individual genes, can distin-
577 guish a target area. There are existing
578 multivariate forms of some of the univariate
579 scoring measures, for example, Hotelling&#8217;s
580 T-square is a multivariate analog of Stu-
581 dent&#8217;s t.
582 We will develop a feature selection pro-
583 cedure for choosing the best small set of
584 marker genes for a given anatomical area. In addition to using the scoring measures that we develop, we will
585 also explore (a) feature selection using a stepwise wrapper over &#8220;vanilla&#8221; classifiers such as logistic regression,
586 (b) supervised learning methods such as decision trees which incrementally/greedily combine single gene mark-
587 ers into sets, and (c) supervised learning methods which use soft constraints to minimize number of features
588 used, such as sparse support vector machines (SVMs).
589 Since errors of displacement and of shape may cause genes and target areas to match less than they should,
590 we will consider the robustness of feature selection methods in the presence of error. Some of these methods,
591 such as the Hough transform, are designed to be resistant in the presence of error, but many are not. We will
592 consider extensions to scoring measures that may improve their robustness; for example, a wrapper that runs a
593 scoring method on small displacements and distortions of the data adds robustness to registration error at the
594 expense of computation time.
595 An area may be difficult to identify because the boundaries are misdrawn in the atlas, or because the shape
596 of the natural domain of gene expression corresponding to the area is different from the shape of the area as
597 recognized by anatomists. We will extend our procedure to handle difficult areas by combining areas or redrawing
598 their boundaries. We will develop extensions to our procedure which (a) detect when a difficult area could be
599 fit if its boundary were redrawn slightly11, and (b) detect when a difficult area could be combined with adjacent
600 _________________________________________
601 11Not just any redrawing is acceptable, only those which appear to be justified as a natural spatial domain of gene expression by
602 multiple sources of evidence. Interestingly, the need to detect &#8220;natural spatial domains of gene expression&#8221; in a data-driven fashion
603 means that the methods of Aim 2 might be useful in achieving Aim 1, as well &#8211; particularly discriminative dimensionality reduction.
604 areas to create a larger area which can be fit.
605 A future publication on the method that we develop in Aim 1 will review the scoring measures and quantita-
606 tively compare their performance in order to provide a foundation for future research of methods of marker gene
607 finding. We will measure the robustness of the scoring measures as well as their absolute performance on our
608 dataset.
609 Classifiers We will explore and compare different classifiers. As noted above, this activity is not separate
610 from the previous one, because some supervised learning algorithms include feature selection, and any clas-
611 sifier can be combined with a stepwise wrapper for use as a feature selection method. We will explore logistic
612 regression (including spatial models[16]), decision trees12, sparse SVMs, generative mixture models (including
613 naive bayes), kernel density estimation, instance-based learning methods (such as k-nearest neighbor), genetic
614 algorithms, and artificial neural networks.
615 Develop algorithms to suggest a division of a structure into anatomical parts
617 Figure 7: Prototypes corresponding to sample gene
618 clusters, clustered by gradient similarity. Region bound-
619 aries for the region that most matches each prototype
620 are overlaid. Dimensionality reduction on gene expression pro-
621 files We have already described the application of
622 ten dimensionality reduction algorithms for the pur-
623 pose of replacing the gene expression profiles, which
624 are vectors of about 4000 gene expression levels,
625 with a smaller number of features. We plan to fur-
626 ther explore and interpret these results, as well as to
627 apply other unsupervised learning algorithms, includ-
628 ing independent components analysis, self-organizing
629 maps, and generative models such as deep Boltz-
630 mann machines. We will explore ways to quantitatively
631 compare the relevance of the different dimensionality
632 reduction methods for identifying cortical areal bound-
633 aries.
634 Dimensionality reduction on pixels Instead of applying dimensionality reduction to the gene expression
635 profiles, the same techniques can be applied instead to the pixels. It is possible that the features generated in
636 this way by some dimensionality reduction techniques will directly correspond to interesting spatial regions.
637 Clustering and segmentation on pixels We will explore clustering and segmentation algorithms in order to
638 segment the pixels into regions. We will explore k-means, spectral clustering, gene shaving[9], recursive division
639 clustering, multivariate generalizations of edge detectors, multivariate generalizations of watershed transforma-
640 tions, region growing, active contours, graph partitioning methods, and recursive agglomerative clustering with
641 various linkage functions. These methods can be combined with dimensionality reduction.
642 Clustering on genes We have already shown that the procedure of clustering genes according to gradient
643 similarity, and then creating an averaged prototype of each cluster&#8217;s expression pattern, yields some spatial
644 patterns which match cortical areas. We will further explore the clustering of genes.
645 In addition to using the cluster expression prototypes directly to identify spatial regions, this might be useful
646 as a component of dimensionality reduction. For example, one could imagine clustering similar genes and then
647 replacing their expression levels with a single average expression level, thereby removing some redundancy from
648 the gene expression profiles. One could then perform clustering on pixels (possibly after a second dimensionality
649 reduction step) in order to identify spatial regions. It remains to be seen whether removal of redundancy would
650 help or hurt the ultimate goal of identifying interesting spatial regions.
651 Co-clustering There are some algorithms which simultaneously incorporate clustering on instances and on
652 features (in our case, genes and pixels), for example, IRM[11]. These are called co-clustering or biclustering
653 _________________________________________
654 12Actually, we have already begun to explore decision trees. For each cortical area, we have used the C4.5 algorithm to find a decision
655 tree for that area. We achieved good classification accuracy on our training set, but the number of genes that appeared in each tree was
656 too large. We plan to implement a pruning procedure to generate trees that use fewer genes.
657 algorithms.
658 Radial profiles We wil explore the use of the radial profile of gene expression under each pixel.
659 Compare different methods In order to tell which method is best for genomic anatomy, for each experimental
660 method we will compare the cortical map found by unsupervised learning to a cortical map derived from the Allen
661 Reference Atlas. We will explore various quantitative metrics that purport to measure how similar two clusterings
662 are, such as Jaccard, Rand index, Fowlkes-Mallows, variation of information, Larsen, Van Dongen, and others.
663 Discriminative dimensionality reduction In addition to using a purely data-driven approach to identify
664 spatial regions, it might be useful to see how well the known regions can be reconstructed from a small number
665 of features, even if those features are chosen by using knowledge of the regions. For example, linear discriminant
666 analysis could be used as a dimensionality reduction technique in order to identify a few features which are the
667 best linear summary of gene expression profiles for the purpose of discriminating between regions. This reduced
668 feature set could then be used to cluster pixels into regions. Perhaps the resulting clusters will be similar to the
669 reference atlas, yet more faithful to natural spatial domains of gene expression than the reference atlas is.
670 Apply the new methods to the cortex
671 Using the methods developed in Aim 1, we will present, for each cortical area, a short list of markers to identify
672 that area; and we will also present lists of &#8220;panels&#8221; of genes that can be used to delineate many areas at once.
673 Because in most cases the ABA coronal dataset only contains one ISH per gene, it is possible for an unrelated
674 combination of genes to seem to identify an area when in fact it is only coincidence. There are two ways we will
675 validate our marker genes to guard against this. First, we will confirm that putative combinations of marker genes
676 express the same pattern in both hemispheres. Second, we will manually validate our final results on other gene
677 expression datasets such as EMAGE, GeneAtlas, and GENSAT[8].
678 Using the methods developed in Aim 2, we will present one or more hierarchical cortical maps. We will identify
679 and explain how the statistical structure in the gene expression data led to any unexpected or interesting features
680 of these maps, and we will provide biological hypotheses to interpret any new cortical areas, or groupings of
681 areas, which are discovered.
682 ____________________________________________________________________________
683 Timeline and milestones
684 Finding marker genes
685 September-November 2009: Develop an automated mechanism for segmenting the cortical voxels into layers
686 November 2009 (milestone): Have completed construction of a flatmapped, cortical dataset with information
687 for each layer
688 October 2009-April 2010: Develop scoring and supervised learning methods.
689 January 2010 (milestone): Submit a publication on single marker genes for cortical areas
690 February-July 2010: Continue to develop scoring methods and supervised learning frameworks. Extend tech-
691 niques for robustness. Compare the performance of techniques. Validate marker genes. Prepare software
692 toolbox for Aim 1.
693 June 2010 (milestone): Submit a paper describing a method fulfilling Aim 1. Release toolbox.
694 July 2010 (milestone): Submit a paper describing combinations of marker genes for each cortical area, and a
695 small number of marker genes that can, in combination, define most of the areas at once
696 Revealing new ways to parcellate a structure into regions
697 June 2010-March 2011: Explore dimensionality reduction algorithms. Explore clustering algorithms. Adapt
698 clustering algorithms to use radial profile information. Compare the performance of techniques.
699 March 2011 (milestone): Submit a paper describing a method fulfilling Aim 2. Release toolbox.
700 February-May 2011: Using the methods developed for Aim 2, explore the genomic anatomy of the cortex,
701 interpret the results. Prepare software toolbox for Aim 2.
702 May 2011 (milestone): Submit a paper on the genomic anatomy of the cortex, using the methods developed in
703 Aim 2
704 May-August 2011: Revisit Aim 1 to see if what was learned during Aim 2 can improve the methods for Aim 1.
705 Possibly submit another paper.
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