Evolution of Bow-Tie Architectures in Biology
Publication Date
March 23, 2015
Journal
PLOS Computational Biology
Authors
Tamar Friedlander, Avraham E. Mayo, Tsvi Tlusty & Uri Alon
Volume
11
Issue
3
Pages
e1004055
DOI
https://dx.plos.org/10.1371/journal.pcbi.1004055
Publisher URL
http://journals.plos.org/ploscompbiol/article?id=10.1371%2Fjournal.pcbi.1004055
PubMed
http://www.ncbi.nlm.nih.gov/pubmed/25798588
PubMed Central
http://www.ncbi.nlm.nih.gov/pmc/articles/PMC4370773
Europe PMC
http://europepmc.org/abstract/MED/25798588
Web of Science
000352195700006
Scopus
84926370637
Mendeley
http://www.mendeley.com/research/evolution-bowtie-architectures-biology
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Mendeley | Further Information

{"title"=>"Evolution of Bow-Tie Architectures in Biology", "type"=>"journal", "authors"=>[{"first_name"=>"Tamar", "last_name"=>"Friedlander", "scopus_author_id"=>"24464245400"}, {"first_name"=>"Avraham E.", "last_name"=>"Mayo", "scopus_author_id"=>"7004426436"}, {"first_name"=>"Tsvi", "last_name"=>"Tlusty", "scopus_author_id"=>"6601956447"}, {"first_name"=>"Uri", "last_name"=>"Alon", "scopus_author_id"=>"35239724400"}], "year"=>2015, "source"=>"PLoS Computational Biology", "identifiers"=>{"issn"=>"15537358", "arxiv"=>"1404.7715", "scopus"=>"2-s2.0-84926370637", "sgr"=>"84926370637", "pui"=>"603513916", "isbn"=>"1553-7358 (Electronic)\\r1553-734X (Linking)", "pmid"=>"25798588", "doi"=>"10.1371/journal.pcbi.1004055"}, "id"=>"6c2e8ae8-6ecf-3955-b661-308e189c5564", "abstract"=>"Bow-tie or hourglass structure is a common architectural feature found in biological and technological networks. A bow-tie in a multi-layered structure occurs when intermediate layers have much fewer components than the input and output layers. Examples include metabolism where a handful of building blocks mediate between multiple input nutrients and multiple output biomass components, and signaling networks where information from numerous receptor types passes through a small set of signaling pathways to regulate multiple output genes. Little is known, however, about how bow-tie architectures evolve. Here, we address the evolution of bow-tie architectures using simulations of multi-layered systems evolving to fulfill a given input-output goal. We find that bow-ties spontaneously evolve when two conditions are met: (i) the evolutionary goal is rank deficient, where the rank corresponds to the minimal number of input features on which the outputs depend, and (ii) The effects of mutations on interaction intensities between components are described by product rule – namely the mutated element is multiplied by a random number. Product-rule mutations are more biologically realistic than the commonly used sum-rule mutations that add a random number to the mutated element. These conditions robustly lead to bow-tie structures. The minimal width of the intermediate network layers (the waist or knot of the bow-tie) equals the rank of the evolutionary goal. These findings can help explain the presence of bow-ties in diverse biological systems, and can also be relevant for machine learning applications that employ multi-layered networks.", "link"=>"http://www.mendeley.com/research/evolution-bowtie-architectures-biology", "reader_count"=>146, "reader_count_by_academic_status"=>{"Unspecified"=>1, "Professor > Associate Professor"=>10, "Librarian"=>3, "Student > Doctoral Student"=>5, "Researcher"=>41, "Student > Ph. D. Student"=>39, "Student > Postgraduate"=>7, "Other"=>8, "Student > Master"=>9, "Student > Bachelor"=>13, "Lecturer"=>1, "Professor"=>9}, "reader_count_by_user_role"=>{"Unspecified"=>1, "Professor > Associate Professor"=>10, "Librarian"=>3, "Student > Doctoral Student"=>5, "Researcher"=>41, "Student > Ph. D. Student"=>39, "Student > Postgraduate"=>7, "Other"=>8, "Student > Master"=>9, "Student > Bachelor"=>13, "Lecturer"=>1, "Professor"=>9}, "reader_count_by_subject_area"=>{"Unspecified"=>5, "Agricultural and Biological Sciences"=>72, "Arts and Humanities"=>1, "Veterinary Science and Veterinary Medicine"=>1, "Computer Science"=>12, "Engineering"=>7, "Environmental Science"=>1, "Biochemistry, Genetics and Molecular Biology"=>23, "Mathematics"=>4, "Medicine and Dentistry"=>5, "Neuroscience"=>2, "Physics and Astronomy"=>11, "Social Sciences"=>1, "Linguistics"=>1}, "reader_count_by_subdiscipline"=>{"Medicine and Dentistry"=>{"Medicine and Dentistry"=>5}, "Social Sciences"=>{"Social Sciences"=>1}, "Physics and Astronomy"=>{"Physics and Astronomy"=>11}, "Mathematics"=>{"Mathematics"=>4}, "Unspecified"=>{"Unspecified"=>5}, "Environmental Science"=>{"Environmental Science"=>1}, "Arts and Humanities"=>{"Arts and Humanities"=>1}, "Engineering"=>{"Engineering"=>7}, "Neuroscience"=>{"Neuroscience"=>2}, "Agricultural and Biological Sciences"=>{"Agricultural and Biological Sciences"=>72}, "Computer Science"=>{"Computer Science"=>12}, "Linguistics"=>{"Linguistics"=>1}, "Biochemistry, Genetics and Molecular Biology"=>{"Biochemistry, Genetics and Molecular Biology"=>23}, "Veterinary Science and Veterinary Medicine"=>{"Veterinary Science and Veterinary Medicine"=>1}}, "reader_count_by_country"=>{"Republic of Singapore"=>1, "United States"=>7, "Japan"=>3, "United Kingdom"=>1, "Portugal"=>2, "Switzerland"=>1, "Spain"=>3, "Canada"=>2, "Belgium"=>1, "Brazil"=>1, "South Africa"=>1, "Slovenia"=>1, "Germany"=>4}, "group_count"=>3}

CrossRef

Scopus | Further Information

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Figshare

  • {"files"=>["https://ndownloader.figshare.com/files/1983070"], "description"=>"<p>We show simulation results of networks with <i>L</i> = 4 (5 layers of nodes) and 6 nodes in each layer (<i>D</i> = 6). We performed 4 different sets of repeated simulations with goals of different ranks = 1,2,3 or 6. We illustrate the histograms of layer width for each set of runs. Each column in this figure shows simulation results for a different goal, and each row shows a different network layer. The number of active nodes in middle layers varies depending on the goal. The minimal number of nodes in intermediate layers (“waist”) is bounded from below by the rank of the goal. The waist width could be higher than the rank, because not all runs reach the most minimal configuration, but it cannot be lower. For example, it can be as low as 1 if the goal rank equals 1 (left column), but it is always 6 if the goal is full rank, demonstrating that no bow-tie can evolve with a full-rank goal. Simulation parameters: 3000 repeats for rank 1 and 2, 1500 repeats for rank 3 and 700 repeats for rank 6. Only runs that reached a fitness value less than 0.01 from the optimum were considered in the analysis. Product mutations were drawn from a Gaussian distribution with <b>σ</b> = 0.1, element-wise mutation rate <i>p</i> = 0.05 / <i>D</i><sup>2</sup>, tournament selection with <i>s</i> = 4.</p>", "links"=>[], "tags"=>["building blocks", "output biomass components", "simulation", "information", "hourglass structure", "output genes", "rank", "receptor types", "evolution", "input nutrients", "output layers"], "article_id"=>1351456, "categories"=>["Biological Sciences"], "users"=>["Tamar Friedlander", "Avraham E. Mayo", "Tsvi Tlusty", "Uri Alon"], "doi"=>"https://dx.doi.org/10.1371/journal.pcbi.1004055.g002", "stats"=>{"downloads"=>1, "page_views"=>5, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Product_rule_mutations_and_goal_which_is_not_full_rank_can_lead_to_bow_tie_architecture_/1351456", "title"=>"Product-rule mutations and goal which is not full rank can lead to bow-tie architecture.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2015-03-23 05:46:08"}
  • {"files"=>["https://ndownloader.figshare.com/files/1983089", "https://ndownloader.figshare.com/files/1983090"], "description"=>"<div><p>Bow-tie or hourglass structure is a common architectural feature found in many biological systems. A bow-tie in a multi-layered structure occurs when intermediate layers have much fewer components than the input and output layers. Examples include metabolism where a handful of building blocks mediate between multiple input nutrients and multiple output biomass components, and signaling networks where information from numerous receptor types passes through a small set of signaling pathways to regulate multiple output genes. Little is known, however, about how bow-tie architectures evolve. Here, we address the evolution of bow-tie architectures using simulations of multi-layered systems evolving to fulfill a given input-output goal. We find that bow-ties spontaneously evolve when the information in the evolutionary goal can be compressed. Mathematically speaking, bow-ties evolve when the rank of the input-output matrix describing the evolutionary goal is deficient. The maximal compression possible (the rank of the goal) determines the size of the narrowest part of the network—that is the bow-tie. A further requirement is that a process is active to reduce the number of links in the network, such as product-rule mutations, otherwise a non-bow-tie solution is found in the evolutionary simulations. This offers a mechanism to understand a common architectural principle of biological systems, and a way to quantitate the effective rank of the goals under which they evolved.</p></div>", "links"=>[], "tags"=>["building blocks", "output biomass components", "simulation", "information", "hourglass structure", "output genes", "rank", "receptor types", "evolution", "input nutrients", "output layers"], "article_id"=>1351469, "categories"=>["Biological Sciences"], "users"=>["Tamar Friedlander", "Avraham E. Mayo", "Tsvi Tlusty", "Uri Alon"], "doi"=>["https://dx.doi.org/10.1371/journal.pcbi.1004055.s001", "https://dx.doi.org/10.1371/journal.pcbi.1004055.s002"], "stats"=>{"downloads"=>0, "page_views"=>3, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Evolution_of_Bow_Tie_Architectures_in_Biology_/1351469", "title"=>"Evolution of Bow-Tie Architectures in Biology", "pos_in_sequence"=>0, "defined_type"=>4, "published_date"=>"2015-03-23 05:46:08"}
  • {"files"=>["https://ndownloader.figshare.com/files/1983079"], "description"=>"<p>We show simulation results of a simple nonlinear problem mimicking a 4-pixel retina. <b>(A) Problem definition</b>: The retina has four inputs (one for each pixel, that can be either black or white), four outputs and two internal processing layers. The retina is evolved so that its outputs detect whether there is (i) an object on the left side (at least one pixel in the left column is black), (ii) on the right side (at least one pixel in the right column is black), (iii) left AND right objects, (iv) left OR right objects, correspondingly. Inset: in contrast to previous problems, here each node performs a nonlinear transformation of the sum of weighted inputs: <b>u</b><sup>(<i>l</i>+1)</sup> = <i>f</i> (<b>A</b><sup>(<i>l</i>)</sup><b>u</b><sup>(<i>l</i>)</sup> – <b>T</b><sup>(<i>l</i>+1)</sup>), where <b>A</b><sup>(<i>l</i>)</sup> and <b>T</b><sup>(<i>l</i>)</sup> are the weight matrix and set of thresholds in the <i>l</i>-th layer. <b>(B) Typical example of simulation results</b>. Apparently, two bits of information are sufficient to fully describe the four required outputs in this model. Indeed, the network evolved so that it has only two active nodes in the second layer (red circles).</p>", "links"=>[], "tags"=>["building blocks", "output biomass components", "simulation", "information", "hourglass structure", "output genes", "rank", "receptor types", "evolution", "input nutrients", "output layers"], "article_id"=>1351465, "categories"=>["Biological Sciences"], "users"=>["Tamar Friedlander", "Avraham E. Mayo", "Tsvi Tlusty", "Uri Alon"], "doi"=>"https://dx.doi.org/10.1371/journal.pcbi.1004055.g006", "stats"=>{"downloads"=>0, "page_views"=>5, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Bow_tie_can_evolve_for_a_nonlinear_input_output_relation_too_if_the_input_can_be_more_compactly_represented_with_no_effect_on_the_output_/1351465", "title"=>"Bow-tie can evolve for a nonlinear input-output relation too, if the input can be more compactly represented with no effect on the output.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2015-03-23 05:46:08"}
  • {"files"=>["https://ndownloader.figshare.com/files/1983076"], "description"=>"<p>We show simulation results when the goal consisted of a matrix of deficient rank (1, 2 or 3) to which some level of noise was added (see<a href=\"http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004055#sec008\" target=\"_blank\"> Methods</a>), so mathematically speaking goals had full rank, such that some of the eigenvalues were relatively small. Remarkably, here too a bow-tie architecture evolved, however the width of the waist was not as narrow as if the goal had exact noiseless deficient rank (compare to <a href=\"http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004055#pcbi.1004055.g002\" target=\"_blank\">Fig. 2</a>). For each goal rank we calculated layer activity statistics based on 1500 different runs (each having a different goal, but with same noise statistics). Noise level here was 1% (averaged over all runs analyzed) for all ranks. This result demonstrates that the evolutionary process can expose a deficient goal rank even when noise is added, as is expected to be the case in realistic systems. Other parameters are the same as in <a href=\"http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004055#pcbi.1004055.g002\" target=\"_blank\">Fig. 2</a>.</p>", "links"=>[], "tags"=>["building blocks", "output biomass components", "simulation", "information", "hourglass structure", "output genes", "rank", "receptor types", "evolution", "input nutrients", "output layers"], "article_id"=>1351462, "categories"=>["Biological Sciences"], "users"=>["Tamar Friedlander", "Avraham E. Mayo", "Tsvi Tlusty", "Uri Alon"], "doi"=>"https://dx.doi.org/10.1371/journal.pcbi.1004055.g005", "stats"=>{"downloads"=>0, "page_views"=>7, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Bow_tie_evolves_even_if_the_goal_is_only_approximately_of_deficient_rank_/1351462", "title"=>"Bow-tie evolves even if the goal is only approximately of deficient rank.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2015-03-23 05:46:08"}
  • {"files"=>["https://ndownloader.figshare.com/files/1983072"], "description"=>"<p><b>Top:</b> Median number of nodes at each layer. Different curves represent results for goals of different ranks. Due to symmetry considerations, the waist is most likely to evolve in the middle layer of nodes. Results refer to the same simulations as in the previous figure. Estimation of error in median calculation by bootstrapping resulted in negligible error. <b>Bottom:</b> examples of possible network structures evolved with goals having different ranks 1,2,3 and 6, illustrating how the width of waist depends on the goal rank.</p>", "links"=>[], "tags"=>["building blocks", "output biomass components", "simulation", "information", "hourglass structure", "output genes", "rank", "receptor types", "evolution", "input nutrients", "output layers"], "article_id"=>1351458, "categories"=>["Biological Sciences"], "users"=>["Tamar Friedlander", "Avraham E. Mayo", "Tsvi Tlusty", "Uri Alon"], "doi"=>"https://dx.doi.org/10.1371/journal.pcbi.1004055.g003", "stats"=>{"downloads"=>0, "page_views"=>2, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_The_waist_is_most_likely_to_evolve_in_the_middle_layer_for_equal_number_of_inputs_and_outputs_/1351458", "title"=>"The waist is most likely to evolve in the middle layer (for equal number of inputs and outputs).", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2015-03-23 05:46:08"}
  • {"files"=>["https://ndownloader.figshare.com/files/1983081"], "description"=>"<p>examples for networks having bow-tie (hourglass) architecture.</p>", "links"=>[], "tags"=>["building blocks", "output biomass components", "simulation", "information", "hourglass structure", "output genes", "rank", "receptor types", "evolution", "input nutrients", "output layers"], "article_id"=>1351467, "categories"=>["Biological Sciences"], "users"=>["Tamar Friedlander", "Avraham E. Mayo", "Tsvi Tlusty", "Uri Alon"], "doi"=>"https://dx.doi.org/10.1371/journal.pcbi.1004055.t001", "stats"=>{"downloads"=>0, "page_views"=>5, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_examples_for_networks_having_bow_tie_hourglass_architecture_/1351467", "title"=>"examples for networks having bow-tie (hourglass) architecture.", "pos_in_sequence"=>0, "defined_type"=>3, "published_date"=>"2015-03-23 05:46:08"}
  • {"files"=>["https://ndownloader.figshare.com/files/1983074"], "description"=>"<p><b>A</b> We tested the existence and width of bow-ties under a broad range of parameter values. We illustrate here the mean and standard deviation of the bow-tie width for various values of mutation rate, mutation size, population size and selection intensity. Bow-ties were obtained in all cases. The width of the bow-tie showed little sensitivity to the parameter values. Each point is based on 50 independent repeats of the simulation. Parameter values tested: population size = [50, 100, 250, 500]; mutation size = [0.01, 0.05, 0.1, 0.2, 0.5, 1]; mutation rate = [1, 0.25, 0.1, 0.05]/<i>LD</i><sup>2</sup>, tournament size <i>s</i> = [2, 4, 6, 8].</p>", "links"=>[], "tags"=>["building blocks", "output biomass components", "simulation", "information", "hourglass structure", "output genes", "rank", "receptor types", "evolution", "input nutrients", "output layers"], "article_id"=>1351460, "categories"=>["Biological Sciences"], "users"=>["Tamar Friedlander", "Avraham E. Mayo", "Tsvi Tlusty", "Uri Alon"], "doi"=>"https://dx.doi.org/10.1371/journal.pcbi.1004055.g004", "stats"=>{"downloads"=>0, "page_views"=>7, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_bow_tie_architecture_is_obtained_under_a_broad_range_of_evolutionary_parameters_/1351460", "title"=>"bow-tie architecture is obtained under a broad range of evolutionary parameters.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2015-03-23 05:46:08"}
  • {"files"=>["https://ndownloader.figshare.com/files/1983069"], "description"=>"<p><b>(A)</b> Bow-tie in a multi-layered network means that the network is capable of processing many different inputs, by converting them into a small set of universal building blocks and then re-using these building blocks to construct a wide range of outputs. <b>(B) Multi-layered networks are represented by interaction intensities between components:</b> Our model represents a multi-layered information transmission network, by the values of interaction intensities between nodes in consecutive layers. In this schematic figure we illustrate networks with 3 layers of nodes, connected by <i>L</i> = 2 layers of interactions. It is convenient to recapitulate these interactions by <i>L</i> = 2 matrices, where the <i>A</i><sub><i>ij</i></sub> term in the <i>l</i>-th matrix represents the interaction between the <i>j</i>-th component in node layer <i>l</i> to the <i>i</i>-th component in node layer <i>l</i> + 1. Node layer 1 is the input signal, and node layer <i>L</i> + 1 is the output. In general, every node could be connected to every node in the next layer—as in the rightmost scheme. A bow-tie is a situation in which in one or more of the middle layers some nodes are disconnected from the rest of the network. This forms a narrow layer, termed “waist”—as exemplified in the left and middle schemes. A bow-tie architecture is captured by interaction matrices in which some rows/ columns are zero. The number of non-zero rows/ columns corresponds to the width of the waist layer. <b>(C) An example of bow-tie networks (simulation results):</b> An example of simulation results with <i>L</i> = 4 interaction layers (5 node layers), demonstrating a bow-tie of width 1 at the middle layer. The network structure is shown on the left (only active nodes shown) and the interaction intensities are shown on the right using a color code (white—no interaction, black—strong interaction).</p>", "links"=>[], "tags"=>["building blocks", "output biomass components", "simulation", "information", "hourglass structure", "output genes", "rank", "receptor types", "evolution", "input nutrients", "output layers"], "article_id"=>1351455, "categories"=>["Biological Sciences"], "users"=>["Tamar Friedlander", "Avraham E. Mayo", "Tsvi Tlusty", "Uri Alon"], "doi"=>"https://dx.doi.org/10.1371/journal.pcbi.1004055.g001", "stats"=>{"downloads"=>0, "page_views"=>5, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Model_description_/1351455", "title"=>"Model description.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2015-03-23 05:46:08"}

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  • {"unique-ip"=>"16", "full-text"=>"16", "pdf"=>"0", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"5", "supp-data"=>"0", "cited-by"=>"0", "year"=>"2019", "month"=>"10"}
  • {"unique-ip"=>"19", "full-text"=>"24", "pdf"=>"8", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"3", "supp-data"=>"0", "cited-by"=>"0", "year"=>"2019", "month"=>"12"}
  • {"unique-ip"=>"22", "full-text"=>"20", "pdf"=>"3", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"0", "supp-data"=>"0", "cited-by"=>"2", "year"=>"2020", "month"=>"2"}
  • {"unique-ip"=>"25", "full-text"=>"28", "pdf"=>"10", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"0", "supp-data"=>"0", "cited-by"=>"0", "year"=>"2020", "month"=>"3"}
  • {"unique-ip"=>"17", "full-text"=>"20", "pdf"=>"3", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"0", "supp-data"=>"2", "cited-by"=>"0", "year"=>"2020", "month"=>"4"}
  • {"unique-ip"=>"17", "full-text"=>"21", "pdf"=>"3", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"0", "supp-data"=>"0", "cited-by"=>"0", "year"=>"2020", "month"=>"5"}
  • {"unique-ip"=>"15", "full-text"=>"13", "pdf"=>"4", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"0", "supp-data"=>"0", "cited-by"=>"0", "year"=>"2020", "month"=>"6"}
  • {"unique-ip"=>"12", "full-text"=>"14", "pdf"=>"1", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"0", "supp-data"=>"0", "cited-by"=>"0", "year"=>"2020", "month"=>"7"}
  • {"unique-ip"=>"15", "full-text"=>"14", "pdf"=>"2", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"0", "supp-data"=>"0", "cited-by"=>"0", "year"=>"2020", "month"=>"8"}
  • {"unique-ip"=>"11", "full-text"=>"9", "pdf"=>"7", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"0", "supp-data"=>"0", "cited-by"=>"0", "year"=>"2020", "month"=>"9"}
  • {"unique-ip"=>"25", "full-text"=>"23", "pdf"=>"5", "scanned-summary"=>"0", "scanned-page-browse"=>"0", "figure"=>"0", "supp-data"=>"0", "cited-by"=>"0", "year"=>"2020", "month"=>"10"}

Relative Metric

{"start_date"=>"2015-01-01T00:00:00Z", "end_date"=>"2015-12-31T00:00:00Z", "subject_areas"=>[]}
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