{"id":2265,"date":"2014-12-09T18:09:32","date_gmt":"2014-12-09T18:09:32","guid":{"rendered":"http:\/\/www.blopig.com\/blog\/?p=2265"},"modified":"2014-12-09T18:09:56","modified_gmt":"2014-12-09T18:09:56","slug":"looking-for-a-null-model-of-ppi-ego-networks","status":"publish","type":"post","link":"https:\/\/www.blopig.com\/blog\/2014\/12\/looking-for-a-null-model-of-ppi-ego-networks\/","title":{"rendered":"Looking for a null model of PPI ego-networks"},"content":{"rendered":"<div class=\"page\" title=\"Page 11\">\n<div class=\"layoutArea\">\n<div class=\"column\">\n<p>Protein-protein interaction (PPI) networks describe how proteins are connected to one another in terms of physical interactions. They can be used to aid our understanding of the individual roles of proteins (Sarajli \u0301c et al., 2013), the co-functioning properties of sets of proteins (West et al., 2013) and even the operation of the complete system (Janowski et al., 2014).<\/p>\n<div class=\"page\" title=\"Page 11\">\n<div class=\"layoutArea\">\n<div class=\"column\">\n<p>Different approaches have been proposed to analyse, describe and predict these PPI networks, such as network summary statistics, clustering methods, random graph models and machine learning methods. However, despite the large biological, computational and statistical interest in PPI net- works, current models insufficiently describe PPI networks (Winterbach et al., 2013; Ali et al., 2014; Rito et al., 2010). It is commonly accepted that proteins perform functions usually in conjunction with other proteins, forming a functional module (Lewis et al., 2010). Hence local structure is found to be important in protein-protein interaction networks (Planas-Iglesias et al., 2013).<\/p>\n<p>Here we address the modelling problem locally by modelling the ego-networks of PPI networks by means of random graph models.<\/p>\n<p><strong>Random graph models<\/strong><\/p>\n<p>Loosely speaking, a random graph model is a set of rules that define an edge generation process among a set of nodes. Usually this\u00a0set of rules relate to particular characteristics that are embedded in the network generation process. Here are three examples of such characteristics:<\/p>\n<ul>\n<li><span style=\"color: #000000\">\u00a0<\/span>Independence<span style=\"color: #0000ff\">\u00a0<\/span>\u00a0(each edge has a probability <img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/s0.wp.com\/latex.php?latex=p&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"p\" class=\"latex\" \/> of being present).<\/li>\n<li>Preferential attachment (nodes form edges with highly interacting nodes).<\/li>\n<li>Space-based interactions (an edge is present between two nodes if the distance between them small).<\/li>\n<\/ul>\n<p>A classical model representing an independence structure is the\u00a0ER(nv,p) model. This is a random graph on nv nodes, and where edges are present independently at random with probability p.<\/p>\n<p><a href=\"https:\/\/i0.wp.com\/www.blopig.com\/blog\/wp-content\/uploads\/2014\/12\/ER3.jpg?ssl=1\"><img data-recalc-dims=\"1\" decoding=\"async\" loading=\"lazy\" class=\"alignnone  wp-image-2266 aligncenter\" src=\"https:\/\/i0.wp.com\/www.blopig.com\/blog\/wp-content\/uploads\/2014\/12\/ER3.jpg?resize=363%2C363&#038;ssl=1\" alt=\"ER3\" width=\"363\" height=\"363\" srcset=\"https:\/\/i0.wp.com\/www.blopig.com\/blog\/wp-content\/uploads\/2014\/12\/ER3.jpg?resize=1024%2C1024&amp;ssl=1 1024w, https:\/\/i0.wp.com\/www.blopig.com\/blog\/wp-content\/uploads\/2014\/12\/ER3.jpg?resize=150%2C150&amp;ssl=1 150w, https:\/\/i0.wp.com\/www.blopig.com\/blog\/wp-content\/uploads\/2014\/12\/ER3.jpg?resize=300%2C300&amp;ssl=1 300w, https:\/\/i0.wp.com\/www.blopig.com\/blog\/wp-content\/uploads\/2014\/12\/ER3.jpg?resize=624%2C624&amp;ssl=1 624w, https:\/\/i0.wp.com\/www.blopig.com\/blog\/wp-content\/uploads\/2014\/12\/ER3.jpg?w=1050&amp;ssl=1 1050w\" sizes=\"auto, (max-width: 363px) 100vw, 363px\" \/><\/a><\/p>\n<p>Now, the preferential attachment characteristic can be illustrated by the\u00a0Chung-Lu model. That is, given an expected sequence of weights <img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/s0.wp.com\/latex.php?latex=%5C%7Bd_1%2Cd_2%2C...%2Cd_%7Bn_v%7D%5C%7D&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"&#92;{d_1,d_2,...,d_{n_v}&#92;}\" class=\"latex\" \/>. The probability of obtaining an edge between nodes i and j\u00a0is given by \u00a0<img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/s0.wp.com\/latex.php?latex=P%28%28i%2Cj%29%5Cin+E%29%3Dd_id_j+%2F+%5Csum_j+d_j.&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"P((i,j)&#92;in E)=d_id_j \/ &#92;sum_j d_j.\" class=\"latex\" \/><\/p>\n<div class=\"page\" title=\"Page 8\">\n<div class=\"layoutArea\">\n<div class=\"column\">\n<p><a href=\"https:\/\/i0.wp.com\/www.blopig.com\/blog\/wp-content\/uploads\/2014\/12\/Screen-Shot-2014-12-09-at-16.22.47.png?ssl=1\"><img data-recalc-dims=\"1\" decoding=\"async\" loading=\"lazy\" class=\"alignnone size-large wp-image-2270\" src=\"https:\/\/i0.wp.com\/www.blopig.com\/blog\/wp-content\/uploads\/2014\/12\/Screen-Shot-2014-12-09-at-16.22.47.png?resize=625%2C268&#038;ssl=1\" alt=\"Screen Shot 2014-12-09 at 16.22.47\" width=\"625\" height=\"268\" srcset=\"https:\/\/i0.wp.com\/www.blopig.com\/blog\/wp-content\/uploads\/2014\/12\/Screen-Shot-2014-12-09-at-16.22.47.png?resize=1024%2C440&amp;ssl=1 1024w, https:\/\/i0.wp.com\/www.blopig.com\/blog\/wp-content\/uploads\/2014\/12\/Screen-Shot-2014-12-09-at-16.22.47.png?resize=300%2C128&amp;ssl=1 300w, https:\/\/i0.wp.com\/www.blopig.com\/blog\/wp-content\/uploads\/2014\/12\/Screen-Shot-2014-12-09-at-16.22.47.png?resize=624%2C268&amp;ssl=1 624w, https:\/\/i0.wp.com\/www.blopig.com\/blog\/wp-content\/uploads\/2014\/12\/Screen-Shot-2014-12-09-at-16.22.47.png?w=1922&amp;ssl=1 1922w, https:\/\/i0.wp.com\/www.blopig.com\/blog\/wp-content\/uploads\/2014\/12\/Screen-Shot-2014-12-09-at-16.22.47.png?w=1250&amp;ssl=1 1250w\" sizes=\"auto, (max-width: 625px) 100vw, 625px\" \/><\/a><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>Finally, a model representing a spaced based network generation process could be the Geometric model. Here, nodes are placed uniformly at random in a d-dimensional square <img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/s0.wp.com\/latex.php?latex=%5B0%2C1%5D%5Ed&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"[0,1]^d\" class=\"latex\" \/>. Now, given a radius or threshold distance (r), edges are drawn among nodes <img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/s0.wp.com\/latex.php?latex=v_i%2C%5C%2Cv_j&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"v_i,&#92;,v_j\" class=\"latex\" \/> <img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/s0.wp.com\/latex.php?latex=i%5Cneq+j&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"i&#92;neq j\" class=\"latex\" \/> \u00a0if <img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/s0.wp.com\/latex.php?latex=d%28v_i%2Cv_j%29%5Cleq+r&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"d(v_i,v_j)&#92;leq r\" class=\"latex\" \/>.<\/p>\n<p style=\"text-align: center\"><a href=\"https:\/\/i0.wp.com\/www.blopig.com\/blog\/wp-content\/uploads\/2014\/12\/Screen-Shot-2014-12-09-at-16.11.01.png?ssl=1\"><img data-recalc-dims=\"1\" decoding=\"async\" loading=\"lazy\" class=\"alignnone  wp-image-2268\" src=\"https:\/\/i0.wp.com\/www.blopig.com\/blog\/wp-content\/uploads\/2014\/12\/Screen-Shot-2014-12-09-at-16.11.01.png?resize=367%2C373&#038;ssl=1\" alt=\"Screen Shot 2014-12-09 at 16.11.01\" width=\"367\" height=\"373\" srcset=\"https:\/\/i0.wp.com\/www.blopig.com\/blog\/wp-content\/uploads\/2014\/12\/Screen-Shot-2014-12-09-at-16.11.01.png?w=580&amp;ssl=1 580w, https:\/\/i0.wp.com\/www.blopig.com\/blog\/wp-content\/uploads\/2014\/12\/Screen-Shot-2014-12-09-at-16.11.01.png?resize=294%2C300&amp;ssl=1 294w\" sizes=\"auto, (max-width: 367px) 100vw, 367px\" \/><\/a><\/p>\n<p>From the latter figures it can be seen that\u00a0different models often lead to different network structures. Thus, although standard random graph models do not reproduce a sufficiently similar network structure to the one of PPI networks, they could still be good approximations for different local regions in a\u00a0PPI network.<\/p>\n<hr \/>\n<p>&nbsp;<\/p>\n<p><strong>Finding a null model for PPI ego-networks<\/strong><\/p>\n<p>Our approach consist in finding local regions of the PPI networks that could be represented well by the random graph models. To do so, we propose to extract all 2-step ego-networks and classifying them according to some simple characteristic, network density for example.<\/p>\n<p>Now, once the ego-networks of the PPI network have been extracted and binned\u00a0according to their network density (ego-density). We\u00a0assess the fit of the model to the PPI networks by\u00a0comparing each bin of PPI ego-networks to the ego-networks extracted from a random graph model. This comparison is made by comparing the difference in the resulting number of subgraph counts, triangles for example,\u00a0in each of the ego-networks within each\u00a0bin.<\/p>\n<p>The following figure illustrates the underlying idea of this procedure:<\/p>\n<p>&nbsp;<\/p>\n<p><a href=\"https:\/\/i0.wp.com\/www.blopig.com\/blog\/wp-content\/uploads\/2014\/12\/Screen-Shot-2014-12-09-at-16.44.40.png?ssl=1\"><img data-recalc-dims=\"1\" decoding=\"async\" loading=\"lazy\" class=\"alignnone size-large wp-image-2273\" src=\"https:\/\/i0.wp.com\/www.blopig.com\/blog\/wp-content\/uploads\/2014\/12\/Screen-Shot-2014-12-09-at-16.44.40.png?resize=625%2C411&#038;ssl=1\" alt=\"Screen Shot 2014-12-09 at 16.44.40\" width=\"625\" height=\"411\" srcset=\"https:\/\/i0.wp.com\/www.blopig.com\/blog\/wp-content\/uploads\/2014\/12\/Screen-Shot-2014-12-09-at-16.44.40.png?resize=1024%2C674&amp;ssl=1 1024w, https:\/\/i0.wp.com\/www.blopig.com\/blog\/wp-content\/uploads\/2014\/12\/Screen-Shot-2014-12-09-at-16.44.40.png?resize=300%2C197&amp;ssl=1 300w, https:\/\/i0.wp.com\/www.blopig.com\/blog\/wp-content\/uploads\/2014\/12\/Screen-Shot-2014-12-09-at-16.44.40.png?resize=624%2C410&amp;ssl=1 624w, https:\/\/i0.wp.com\/www.blopig.com\/blog\/wp-content\/uploads\/2014\/12\/Screen-Shot-2014-12-09-at-16.44.40.png?w=1604&amp;ssl=1 1604w, https:\/\/i0.wp.com\/www.blopig.com\/blog\/wp-content\/uploads\/2014\/12\/Screen-Shot-2014-12-09-at-16.44.40.png?w=1250&amp;ssl=1 1250w\" sizes=\"auto, (max-width: 625px) 100vw, 625px\" \/><\/a><\/p>\n<p>Following this approach we aim to find bins\u00a0for which, possibly different models,\u00a0reproduce similar subgraph counts as the ones obtained in the PPI ego-networks. However we expect to fin bins for which none of the standard models fit.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>Protein-protein interaction (PPI) networks describe how proteins are connected to one another in terms of physical interactions. They can be used to aid our understanding of the individual roles of proteins (Sarajli \u0301c et al., 2013), the co-functioning properties of sets of proteins (West et al., 2013) and even the operation of the complete system [&hellip;]<\/p>\n","protected":false},"author":26,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"nf_dc_page":"","wikipediapreview_detectlinks":true,"_monsterinsights_skip_tracking":false,"_monsterinsights_sitenote_active":false,"_monsterinsights_sitenote_note":"","_monsterinsights_sitenote_category":0,"ngg_post_thumbnail":0,"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[10],"tags":[],"ppma_author":[514],"class_list":["post-2265","post","type-post","status-publish","format-standard","hentry","category-groupmeetings"],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"authors":[{"term_id":514,"user_id":26,"is_guest":0,"slug":"luis","display_name":"Luis Ospina Forero","avatar_url":"https:\/\/secure.gravatar.com\/avatar\/310cef32cd5dac5a383fe35d2e6fa0ed40cb03d0712d2b5a5ef81092db812b3e?s=96&d=mm&r=g","0":null,"1":"","2":"","3":"","4":"","5":"","6":"","7":"","8":""}],"_links":{"self":[{"href":"https:\/\/www.blopig.com\/blog\/wp-json\/wp\/v2\/posts\/2265","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.blopig.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.blopig.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.blopig.com\/blog\/wp-json\/wp\/v2\/users\/26"}],"replies":[{"embeddable":true,"href":"https:\/\/www.blopig.com\/blog\/wp-json\/wp\/v2\/comments?post=2265"}],"version-history":[{"count":9,"href":"https:\/\/www.blopig.com\/blog\/wp-json\/wp\/v2\/posts\/2265\/revisions"}],"predecessor-version":[{"id":2280,"href":"https:\/\/www.blopig.com\/blog\/wp-json\/wp\/v2\/posts\/2265\/revisions\/2280"}],"wp:attachment":[{"href":"https:\/\/www.blopig.com\/blog\/wp-json\/wp\/v2\/media?parent=2265"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.blopig.com\/blog\/wp-json\/wp\/v2\/categories?post=2265"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.blopig.com\/blog\/wp-json\/wp\/v2\/tags?post=2265"},{"taxonomy":"author","embeddable":true,"href":"https:\/\/www.blopig.com\/blog\/wp-json\/wp\/v2\/ppma_author?post=2265"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}