1.1 --- a/grant.txt Tue Apr 21 14:28:12 2009 -0700 1.2 +++ b/grant.txt Tue Apr 21 14:50:10 2009 -0700 1.3 @@ -222,6 +222,21 @@ 1.4 1.5 == Significance == 1.6 1.7 +\begin{wrapfigure}{L}{0.35\textwidth}\centering 1.8 +%%\includegraphics[scale=.27]{singlegene_SS_corr_top_1_2365_jet.eps}\includegraphics[scale=.27]{singlegene_SS_corr_top_2_242_jet.eps}\includegraphics[scale=.27]{singlegene_SS_corr_top_3_654_jet.eps} 1.9 +%%\\ 1.10 +%%\includegraphics[scale=.27]{singlegene_SS_lr_top_1_654_jet.eps}\includegraphics[scale=.27]{singlegene_SS_lr_top_2_685_jet.eps}\includegraphics[scale=.27]{singlegene_SS_lr_top_3_724_jet.eps} 1.11 +%%\caption{Top row: Genes Nfic, A930001M12Rik, C130038G02Rik are the most correlated with area SS (somatosensory cortex). Bottom row: Genes C130038G02Rik, Cacna1i, Car10 are those with the best fit using logistic regression. Within each picture, the vertical axis roughly corresponds to anterior at the top and posterior at the bottom, and the horizontal axis roughly corresponds to medial at the left and lateral at the right. The red outline is the boundary of region SS. Pixels are colored according to correlation, with red meaning high correlation and blue meaning low.} 1.12 + 1.13 +\includegraphics[scale=.27]{singlegene_SS_corr_top_1_2365_jet.eps}\includegraphics[scale=.27]{singlegene_SS_corr_top_2_242_jet.eps} 1.14 +\\ 1.15 +\includegraphics[scale=.27]{singlegene_SS_lr_top_1_654_jet.eps}\includegraphics[scale=.27]{singlegene_SS_lr_top_2_685_jet.eps} 1.16 + 1.17 +\caption{Top row: Genes $Nfic$ and $A930001M12Rik$ are the most correlated with area SS (somatosensory cortex). Bottom row: Genes $C130038G02Rik$ and $Cacna1i$ are those with the best fit using logistic regression. Within each picture, the vertical axis roughly corresponds to anterior at the top and posterior at the bottom, and the horizontal axis roughly corresponds to medial at the left and lateral at the right. The red outline is the boundary of region SS. Pixels are colored according to correlation, with red meaning high correlation and blue meaning low.} 1.18 +\label{SScorrLr}\end{wrapfigure} 1.19 + 1.20 + 1.21 + 1.22 The method developed in aim (1) will be applied to each cortical area to find a set of marker genes such that the combinatorial expression pattern of those genes uniquely picks out the target area. Finding marker genes will be useful for drug discovery as well as for experimentation because marker genes can be used to design interventions which selectively target individual cortical areas. 1.23 1.24 The application of the marker gene finding algorithm to the cortex will also support the development of new neuroanatomical methods. In addition to finding markers for each individual cortical areas, we will find a small panel of genes that can find many of the areal boundaries at once. This panel of marker genes will allow the development of an ISH protocol that will allow experimenters to more easily identify which anatomical areas are present in small samples of cortex. 1.25 @@ -247,6 +262,12 @@ 1.26 new anatomical subregions in various structures, which will widely 1.27 impact all areas of biology. 1.28 1.29 +\begin{wrapfigure}{L}{0.2\textwidth}\centering 1.30 +\includegraphics[scale=.27]{holeExample_2682_SS_jet.eps} 1.31 +\caption{Gene $Pitx2$ is selectively underexpressed in area SS.} 1.32 +\label{hole}\end{wrapfigure} 1.33 + 1.34 + 1.35 Although our particular application involves the 3D spatial 1.36 distribution of gene expression, we anticipate that the methods 1.37 developed in aims (1) and (2) will not be limited to gene expression 1.38 @@ -257,19 +278,6 @@ 1.39 1.40 1.41 == The approach: Preliminary Studies == 1.42 -\begin{wrapfigure}{L}{0.35\textwidth}\centering 1.43 -%%\includegraphics[scale=.27]{singlegene_SS_corr_top_1_2365_jet.eps}\includegraphics[scale=.27]{singlegene_SS_corr_top_2_242_jet.eps}\includegraphics[scale=.27]{singlegene_SS_corr_top_3_654_jet.eps} 1.44 -%%\\ 1.45 -%%\includegraphics[scale=.27]{singlegene_SS_lr_top_1_654_jet.eps}\includegraphics[scale=.27]{singlegene_SS_lr_top_2_685_jet.eps}\includegraphics[scale=.27]{singlegene_SS_lr_top_3_724_jet.eps} 1.46 -%%\caption{Top row: Genes Nfic, A930001M12Rik, C130038G02Rik are the most correlated with area SS (somatosensory cortex). Bottom row: Genes C130038G02Rik, Cacna1i, Car10 are those with the best fit using logistic regression. Within each picture, the vertical axis roughly corresponds to anterior at the top and posterior at the bottom, and the horizontal axis roughly corresponds to medial at the left and lateral at the right. The red outline is the boundary of region SS. Pixels are colored according to correlation, with red meaning high correlation and blue meaning low.} 1.47 - 1.48 -\includegraphics[scale=.27]{singlegene_SS_corr_top_1_2365_jet.eps}\includegraphics[scale=.27]{singlegene_SS_corr_top_2_242_jet.eps} 1.49 -\\ 1.50 -\includegraphics[scale=.27]{singlegene_SS_lr_top_1_654_jet.eps}\includegraphics[scale=.27]{singlegene_SS_lr_top_2_685_jet.eps} 1.51 - 1.52 -\caption{Top row: Genes $Nfic$ and $A930001M12Rik$ are the most correlated with area SS (somatosensory cortex). Bottom row: Genes $C130038G02Rik$ and $Cacna1i$ are those with the best fit using logistic regression. Within each picture, the vertical axis roughly corresponds to anterior at the top and posterior at the bottom, and the horizontal axis roughly corresponds to medial at the left and lateral at the right. The red outline is the boundary of region SS. Pixels are colored according to correlation, with red meaning high correlation and blue meaning low.} 1.53 -\label{SScorrLr}\end{wrapfigure} 1.54 - 1.55 1.56 1.57 1.58 @@ -290,10 +298,6 @@ 1.59 1.60 At this point, the data are in the form of a number of 2-D matrices, all in registration, with the matrix entries representing a grid of points (pixels) over the cortical surface: 1.61 1.62 -\begin{wrapfigure}{L}{0.2\textwidth}\centering 1.63 -\includegraphics[scale=.27]{holeExample_2682_SS_jet.eps} 1.64 -\caption{Gene $Pitx2$ is selectively underexpressed in area SS.} 1.65 -\label{hole}\end{wrapfigure} 1.66 1.67 1.68 * A 2-D matrix whose entries represent the regional label associated with each surface pixel 1.69 @@ -301,38 +305,6 @@ 1.70 1.71 1.72 1.73 - 1.74 -We created a normalized version of the gene expression data by subtracting each gene's mean expression level (over all surface pixels) and dividing the expression level of each gene by its standard deviation. 1.75 - 1.76 -The features and the target area are both functions on the surface pixels. They can be referred to as scalar fields over the space of surface pixels; alternately, they can be thought of as images which can be displayed on the flatmapped surface. 1.77 - 1.78 -To move beyond a single average expression level for each surface pixel, we plan to create a separate matrix for each cortical layer to represent the average expression level within that layer. Cortical layers are found at different depths in different parts of the cortex. In preparation for extracting the layer-specific datasets, we have extended Caret with routines that allow the depth of the ROI for volume-to-surface projection to vary. 1.79 - 1.80 -In the Research Plan, we describe how we will automatically locate the layer depths. For validation, we have manually demarcated the depth of the outer boundary of cortical layer 5 throughout the cortex. 1.81 - 1.82 - 1.83 - 1.84 - 1.85 - 1.86 - 1.87 - 1.88 -=== Feature selection and scoring methods === 1.89 - 1.90 - 1.91 - 1.92 -\vspace{0.3cm}**Underexpression of a gene can serve as a marker** 1.93 -Underexpression of a gene can sometimes serve as a marker. See, for example, Figure \ref{hole}. 1.94 - 1.95 - 1.96 - 1.97 -\vspace{0.3cm}**Correlation** 1.98 -Recall that the instances are surface pixels, and consider the problem of attempting to classify each instance as either a member of a particular anatomical area, or not. The target area can be represented as a boolean mask over the surface pixels. 1.99 - 1.100 -One class of feature selection scoring methods contains methods which calculate some sort of "match" between each gene image and the target image. Those genes which match the best are good candidates for features. 1.101 - 1.102 -One of the simplest methods in this class is to use correlation as the match score. We calculated the correlation between each gene and each cortical area. The top row of Figure \ref{SScorrLr} shows the three genes most correlated with area SS. 1.103 - 1.104 - 1.105 \begin{wrapfigure}{L}{0.35\textwidth}\centering 1.106 %%\includegraphics[scale=.27]{singlegene_AUD_lr_top_1_3386_jet.eps}\includegraphics[scale=.27]{singlegene_AUD_lr_top_2_1258_jet.eps}\includegraphics[scale=.27]{singlegene_AUD_lr_top_3_420_jet.eps} 1.107 %% 1.108 @@ -344,14 +316,36 @@ 1.109 \caption{The top row shows the two genes which (individually) best predict area AUD, according to logistic regression. The bottom row shows the two genes which (individually) best match area AUD, according to gradient similarity. From left to right and top to bottom, the genes are $Ssr1$, $Efcbp1$, $Ptk7$, and $Aph1a$.} 1.110 \label{AUDgeometry}\end{wrapfigure} 1.111 1.112 -\vspace{0.3cm}**Conditional entropy** 1.113 -An information-theoretic scoring method is to find features such that, if the features (gene expression levels) are known, uncertainty about the target (the regional identity) is reduced. Entropy measures uncertainty, so what we want is to find features such that the conditional distribution of the target has minimal entropy. The distribution to which we are referring is the probability distribution over the population of surface pixels. 1.114 - 1.115 -The simplest way to use information theory is on discrete data, so we discretized our gene expression data by creating, for each gene, five thresholded boolean masks of the gene data. For each gene, we created a boolean mask of its expression levels using each of these thresholds: the mean of that gene, the mean minus one standard deviation, the mean minus two standard deviations, the mean plus one standard deviation, the mean plus two standard deviations. 1.116 - 1.117 -Now, for each region, we created and ran a forward stepwise procedure which attempted to find pairs of gene expression boolean masks such that the conditional entropy of the target area's boolean mask, conditioned upon the pair of gene expression boolean masks, is minimized. 1.118 - 1.119 -This finds pairs of genes which are most informative (at least at these discretization thresholds) relative to the question, "Is this surface pixel a member of the target area?". Its advantage over linear methods such as logistic regression is that it takes account of arbitrarily nonlinear relationships; for example, if the XOR of two variables predicts the target, conditional entropy would notice, whereas linear methods would not. 1.120 +We created a normalized version of the gene expression data by subtracting each gene's mean expression level (over all surface pixels) and dividing the expression level of each gene by its standard deviation. 1.121 + 1.122 +The features and the target area are both functions on the surface pixels. They can be referred to as scalar fields over the space of surface pixels; alternately, they can be thought of as images which can be displayed on the flatmapped surface. 1.123 + 1.124 +To move beyond a single average expression level for each surface pixel, we plan to create a separate matrix for each cortical layer to represent the average expression level within that layer. Cortical layers are found at different depths in different parts of the cortex. In preparation for extracting the layer-specific datasets, we have extended Caret with routines that allow the depth of the ROI for volume-to-surface projection to vary. 1.125 + 1.126 +In the Research Plan, we describe how we will automatically locate the layer depths. For validation, we have manually demarcated the depth of the outer boundary of cortical layer 5 throughout the cortex. 1.127 + 1.128 + 1.129 + 1.130 + 1.131 + 1.132 + 1.133 + 1.134 +=== Feature selection and scoring methods === 1.135 + 1.136 + 1.137 + 1.138 +\vspace{0.3cm}**Underexpression of a gene can serve as a marker** 1.139 +Underexpression of a gene can sometimes serve as a marker. See, for example, Figure \ref{hole}. 1.140 + 1.141 + 1.142 + 1.143 +\vspace{0.3cm}**Correlation** 1.144 +Recall that the instances are surface pixels, and consider the problem of attempting to classify each instance as either a member of a particular anatomical area, or not. The target area can be represented as a boolean mask over the surface pixels. 1.145 + 1.146 +One class of feature selection scoring methods contains methods which calculate some sort of "match" between each gene image and the target image. Those genes which match the best are good candidates for features. 1.147 + 1.148 +One of the simplest methods in this class is to use correlation as the match score. We calculated the correlation between each gene and each cortical area. The top row of Figure \ref{SScorrLr} shows the three genes most correlated with area SS. 1.149 + 1.150 1.151 1.152 \begin{wrapfigure}{L}{0.35\textwidth}\centering 1.153 @@ -361,23 +355,15 @@ 1.154 \caption{Upper left: $wwc1$. Upper right: $mtif2$. Lower left: wwc1 + mtif2 (each pixel's value on the lower left is the sum of the corresponding pixels in the upper row).} 1.155 \label{MOcombo}\end{wrapfigure} 1.156 1.157 - 1.158 -\vspace{0.3cm}**Gradient similarity** 1.159 -We noticed that the previous two scoring methods, which are pointwise, often found genes whose pattern of expression did not look similar in shape to the target region. For this reason we designed a non-pointwise local scoring method to detect when a gene had a pattern of expression which looked like it had a boundary whose shape is similar to the shape of the target region. We call this scoring method "gradient similarity". 1.160 - 1.161 -One might say that gradient similarity attempts to measure how much the border of the area of gene expression and the border of the target region overlap. However, since gene expression falls off continuously rather than jumping from its maximum value to zero, the spatial pattern of a gene's expression often does not have a discrete border. Therefore, instead of looking for a discrete border, we look for large gradients. Gradient similarity is a symmetric function over two images (i.e. two scalar fields). It is is high to the extent that matching pixels which have large values and large gradients also have gradients which are oriented in a similar direction. The formula is: 1.162 - 1.163 -\begin{align*} 1.164 -\sum_{pixel \in pixels} cos(abs(\angle \nabla_1 - \angle \nabla_2)) \cdot \frac{\vert \nabla_1 \vert + \vert \nabla_2 \vert}{2} \cdot \frac{pixel\_value_1 + pixel\_value_2}{2} 1.165 -\end{align*} 1.166 - 1.167 -where $\nabla_1$ and $\nabla_2$ are the gradient vectors of the two images at the current pixel; $\angle \nabla_i$ is the angle of the gradient of image $i$ at the current pixel; $\vert \nabla_i \vert$ is the magnitude of the gradient of image $i$ at the current pixel; and $pixel\_value_i$ is the value of the current pixel in image $i$. 1.168 - 1.169 -The intuition is that we want to see if the borders of the pattern in the two images are similar; if the borders are similar, then both images will have corresponding pixels with large gradients (because this is a border) which are oriented in a similar direction (because the borders are similar). 1.170 - 1.171 -Most of the genes in Figure \ref{singleSoFar} were identified via gradient similarity. 1.172 - 1.173 -\vspace{0.3cm}**Gradient similarity provides information complementary to correlation** 1.174 +\vspace{0.3cm}**Conditional entropy** 1.175 +An information-theoretic scoring method is to find features such that, if the features (gene expression levels) are known, uncertainty about the target (the regional identity) is reduced. Entropy measures uncertainty, so what we want is to find features such that the conditional distribution of the target has minimal entropy. The distribution to which we are referring is the probability distribution over the population of surface pixels. 1.176 + 1.177 +The simplest way to use information theory is on discrete data, so we discretized our gene expression data by creating, for each gene, five thresholded boolean masks of the gene data. For each gene, we created a boolean mask of its expression levels using each of these thresholds: the mean of that gene, the mean minus one standard deviation, the mean minus two standard deviations, the mean plus one standard deviation, the mean plus two standard deviations. 1.178 + 1.179 +Now, for each region, we created and ran a forward stepwise procedure which attempted to find pairs of gene expression boolean masks such that the conditional entropy of the target area's boolean mask, conditioned upon the pair of gene expression boolean masks, is minimized. 1.180 + 1.181 +This finds pairs of genes which are most informative (at least at these discretization thresholds) relative to the question, "Is this surface pixel a member of the target area?". Its advantage over linear methods such as logistic regression is that it takes account of arbitrarily nonlinear relationships; for example, if the XOR of two variables predicts the target, conditional entropy would notice, whereas linear methods would not. 1.182 + 1.183 1.184 1.185 \begin{wrapfigure}{L}{0.35\textwidth}\centering 1.186 @@ -389,6 +375,25 @@ 1.187 \caption{From left to right and top to bottom, single genes which roughly identify areas SS (somatosensory primary \begin{latex}+\end{latex} supplemental), SSs (supplemental somatosensory), PIR (piriform), FRP (frontal pole), RSP (retrosplenial), COApm (Cortical amygdalar, posterior part, medial zone). Grouping some areas together, we have also found genes to identify the groups ACA+PL+ILA+DP+ORB+MO (anterior cingulate, prelimbic, infralimbic, dorsal peduncular, orbital, motor), posterior and lateral visual (VISpm, VISpl, VISI, VISp; posteromedial, posterolateral, lateral, and primary visual; the posterior and lateral visual area is distinguished from its neighbors, but not from the entire rest of the cortex). The genes are $Pitx2$, $Aldh1a2$, $Ppfibp1$, $Slco1a5$, $Tshz2$, $Trhr$, $Col12a1$, $Ets1$.} 1.188 \label{singleSoFar}\end{wrapfigure} 1.189 1.190 +\vspace{0.3cm}**Gradient similarity** 1.191 +We noticed that the previous two scoring methods, which are pointwise, often found genes whose pattern of expression did not look similar in shape to the target region. For this reason we designed a non-pointwise local scoring method to detect when a gene had a pattern of expression which looked like it had a boundary whose shape is similar to the shape of the target region. We call this scoring method "gradient similarity". 1.192 + 1.193 +One might say that gradient similarity attempts to measure how much the border of the area of gene expression and the border of the target region overlap. However, since gene expression falls off continuously rather than jumping from its maximum value to zero, the spatial pattern of a gene's expression often does not have a discrete border. Therefore, instead of looking for a discrete border, we look for large gradients. Gradient similarity is a symmetric function over two images (i.e. two scalar fields). It is is high to the extent that matching pixels which have large values and large gradients also have gradients which are oriented in a similar direction. The formula is: 1.194 + 1.195 +\begin{align*} 1.196 +\sum_{pixel \in pixels} cos(abs(\angle \nabla_1 - \angle \nabla_2)) \cdot \frac{\vert \nabla_1 \vert + \vert \nabla_2 \vert}{2} \cdot \frac{pixel\_value_1 + pixel\_value_2}{2} 1.197 +\end{align*} 1.198 + 1.199 +where $\nabla_1$ and $\nabla_2$ are the gradient vectors of the two images at the current pixel; $\angle \nabla_i$ is the angle of the gradient of image $i$ at the current pixel; $\vert \nabla_i \vert$ is the magnitude of the gradient of image $i$ at the current pixel; and $pixel\_value_i$ is the value of the current pixel in image $i$. 1.200 + 1.201 +The intuition is that we want to see if the borders of the pattern in the two images are similar; if the borders are similar, then both images will have corresponding pixels with large gradients (because this is a border) which are oriented in a similar direction (because the borders are similar). 1.202 + 1.203 +Most of the genes in Figure \ref{singleSoFar} were identified via gradient similarity. 1.204 + 1.205 +\vspace{0.3cm}**Gradient similarity provides information complementary to correlation** 1.206 + 1.207 + 1.208 + 1.209 To show that gradient similarity can provide useful information that cannot be detected via pointwise analyses, consider Fig. \ref{AUDgeometry}. The top row of Fig. \ref{AUDgeometry} displays the 3 genes which most match area AUD, according to a pointwise method\footnote{For each gene, a logistic regression in which the response variable was whether or not a surface pixel was within area AUD, and the predictor variable was the value of the expression of the gene underneath that pixel. The resulting scores were used to rank the genes in terms of how well they predict area AUD.}. The bottom row displays the 3 genes which most match AUD according to a method which considers local geometry\footnote{For each gene the gradient similarity between (a) a map of the expression of each gene on the cortical surface and (b) the shape of area AUD, was calculated, and this was used to rank the genes.} The pointwise method in the top row identifies genes which express more strongly in AUD than outside of it; its weakness is that this includes many areas which don't have a salient border matching the areal border. The geometric method identifies genes whose salient expression border seems to partially line up with the border of AUD; its weakness is that this includes genes which don't express over the entire area. Genes which have high rankings using both pointwise and border criteria, such as $Aph1a$ in the example, may be particularly good markers. None of these genes are, individually, a perfect marker for AUD; we deliberately chose a "difficult" area in order to better contrast pointwise with geometric methods. 1.210 1.211 1.212 @@ -459,6 +464,7 @@ 1.213 After applying the dimensionality reduction, we ran clustering algorithms on the reduced data. To date we have tried k-means and spectral clustering. The results of k-means after PCA, NNMF, and landmark Isomap are shown in the last row of Figure \ref{dimReduc}. To compare, the leftmost picture on the bottom row of Figure \ref{dimReduc} shows some of the major subdivisions of cortex. These results clearly show that different dimensionality reduction techniques capture different aspects of the data and lead to different clusterings, indicating the utility of our proposal to produce a detailed comparion of these techniques as applied to the domain of genomic anatomy. 1.214 1.215 1.216 + 1.217 \begin{wrapfigure}{L}{0.5\textwidth}\centering 1.218 \includegraphics[scale=.2]{cosine_similarity1_rearrange_colorize.eps} 1.219 \caption{Prototypes corresponding to sample gene clusters, clustered by gradient similarity. Region boundaries for the region that most matches each prototype are overlayed.} 1.220 @@ -500,6 +506,8 @@ 1.221 1.222 We will develop scoring methods for evaluating how good individual genes are at marking areas. We will compare pointwise, geometric, and information-theoretic measures. We already developed one entirely new scoring method (gradient similarity), but we may develop more. Scoring measures that we will explore will include the L1 norm, correlation, expression energy ratio, conditional entropy, gradient similarity, Jaccard similarity, Dice similarity, Hough transform, and statistical tests such as Student's t-test, and the Mann-Whitney U test (a non-parametric test). In addition, any predictive procedure induces a scoring measure on genes by taking the prediction error when using that gene to predict the target. 1.223 1.224 + 1.225 + 1.226 Using some combination of these measures, we will develop a procedure to find single marker genes for anatomical regions: for each cortical area, we will rank the genes by their ability to delineate each area. 1.227 1.228 Some cortical areas have no single marker genes but can be identified by combinatorial coding. This requires multivariate scoring measures and feature selection procedures. Many of the measures, such as expression energy, gradient similarity, Jaccard, Dice, Hough, Student's t, and Mann-Whitney U are univariate. We will extend these scoring measures for use in multivariate feature selection, that is, for scoring how well combinations of genes, rather than individual genes, can distinguish a target area. There are existing multivariate forms of some of the univariate scoring measures, for example, Hotelling's T-square is a multivariate analog of Student's t.