Feng-Doolittle.html

<!--
University of Freiburg WS 2017/2018
Chair for Bioinformatics
Supervisor: Martin Raden
Authors: Alexander Mattheis, Martin Raden
-->

<div id="algorithm_description">
    <div class="description">
        The progressive alignment approach by
        <a href="https://doi.org/10.1007/BF02603120">Da-Fei Feng and Russell F. Doolittle (1987)</a>
        computes a multi-sequence-alignment (MSA) of a set of sequences based on pairwise alignments.
        This approximative approach identifies good MSA solutions in reasonable time.
        <br />
        <br />
	    First, all pairwise alignments with maximal similarity of all input sequence pairs are computed.
        To this end, we use  <a href="https://doi.org/10.1016/0022-2836(82)90398-9">Osamu Gotoh's (1982)</a>
	    approach available in
	    <a href="http://rna.informatik.uni-freiburg.de/Teaching/index.jsp?toolName=Gotoh">our Gotoh implementation</a>.
        <br />
        In order to build a guide tree to generate the MSA, 
	    observed similarity scores $S(a,b)$ of sequence pairs $a,b\in\Sigma^{*}$ are converted into distances using
        $D(a,b) = -ln(S^{eff}_{a,b} \cdot f)$, where we set the scaling factor $f=1$ (originally $f = 100$). <br />
        The 'effective' similarity score $S^{eff}$ is computed by $S^{eff}_{a,b} = (S(a,b) - S^{rand}_{a,b})/(S^{max}_{a,b} - S^{rand}_{a,b})$
        where $S^{max}_{a,b} = (S(a,a)+S(b,b))/2$ provides the best alignment score for both sequences.
	    $S^{rand}$ provides a kind of 'expected score' that can be obtained via sequence randomization.
        Here, we use an alternative approach introduced in
        <a href="https://doi.org/10.1016/s0076-6879(96)66023-6">(Feng&amp;Doolittle, 1996)</a>
        using the approximate formula for a given optimal alignment of $a$ and $b$:

        $$
        \quad
        S^{rand}_{a,b} =
        \frac{1}{L_{a,b}}
        \bigg(
        \sum_{x \in \Sigma} \sum_{y \in \Sigma}
        s(x,y) \cdot N_a(x) \cdot N_b(y)
        \bigg)
        + N_{a,b}^{e} \cdot \beta
        + N_{a,b}^{o} \cdot \alpha
        $$

        where <br />
        <div class="row">
            <div class="colW150">$L_{a,b}$</div>
            <div class="colW400">number of columns in the alignment of $a,b$</div>
	    </div>

        <div class="row">
            <div class="colW150">$s(x,y)$</div>
            <div class="colW400">similarity-score for aligning the characters $x$ and $y$</div>
        </div>

        <div class="row">
            <div class="colW150">$N_a(x), N_b(y)$</div>
            <div class="colW400">number of characters $x$ and $y$ in sequences $a$ and $b$, resp.</div>
        </div>

        <div class="row">
            <div class="colW150">$N_{a,b}^{e}$</div>
            <div class="colW400">number of gaps (enlargements)</div>
        </div>

        <div class="row">
            <div class="colW150">$N_{a,b}^{o}$</div>
            <div class="colW400">number of gap openings (adaptation for Gotoh)</div>
        </div>

        <div class="row">
            <div class="colW150">$\alpha, \beta$</div>
            <div class="colW400">
                parameters from
                <a href="http://rna.informatik.uni-freiburg.de/Teaching/index.jsp?toolName=Gotoh">Gotoh</a> algorithm
            </div>
        </div>

        The resulting distances are then used
        to create a phylogenetic tree that guides the generation of the MSA, i.e. more related (similar) sequences are fused first.
        Here, the <a href="http://rna.informatik.uni-freiburg.de/Teaching/index.jsp?toolName=Agglomerative-Clustering">UPGMA</a> algorithm envisioned
        by <a href="https://archive.org/details/cbarchive_33927_astatisticalmethodforevaluatin1902">Robert R. Sokal and Charles D. Michener (1958)</a> is applied.
        <br />
	    The tree defines the order in which sequences or groups of sequences
        have to be fused to create a multi-sequence-alignment (MSA). 
	    The fusion is done based on the best pairwise alignment between members of the groups to be aligned.
        To simplify the fusion, the 'once a gap, always a gap'-strategy is applied, 
	    i.e. once a gap is introduced in a sequence it is preserved within all subsequent fusions.
        To this end, gaps in subalignments are replaced with 
        a neutral symbol $\#$ before aligning them (i.e. $s(\cdot,\#) = s(\#,\cdot) = 0$).
        <br />
        <br />
        For a given input, the stated computation steps generate
        <ul>
            <li>pairwise alignments</li>
            <li>distance matrix</li>
            <li>phylogenetic tree</li>
            <li>multiple sequence alignment (MSA) with SoP score</li>
        </ul>
        and are represented in reversed order below. <br />

        The sum-of-pairs (SoP) score of the final MSA is simply the score sum of all pairwise
        alignments within the MSA, i.e. of pairwise row combinations where overlying gap columns are removed
        since we are using an affine gap cost model.
    </div>

    <div class="picture">
        <img src="Feng-Doolittle-120x90.png" />
    </div>
</div>

<h1>Input:</h1>
<div id="algorithm_input">
    <!-- ko foreach: $root.input.sequences() -->
        <div class="row">
            <div class="colW100">
                <label>
                    Sequence <span data-bind="text: $root.input.clusterNames()[$index()]"></span>:
                </label>
            </div>

            <div class="colW225">
                <input class="sequence_multi" data-bind="value: $data" placeholder="EXAMPLE 'ATC'" title="Allowed are A-Z and '-'." type="text">

                <!-- ko if: $index() == $root.input.clusterNames().length-1 -->
                    <div class="group_hint">
                        <b>Hints:</b> <br />
                        The "+" and "-" buttons can be used to add or remove sequences.
                        Duplicates of sequences are excluded from computations.
                        <!-- The distance formula
                        is not defined for two sequences of same type,
                        because the denominator gets zero.
                        Also, this allows a bigger code readability and runtime enhancement:
                        Sequences are stored as keys in hash-tables and
                        two times the same key is not possible in a hash-table!
                        -->
                    </div>
                <!-- /ko -->
            </div>

            <!-- ko if: $index() == $root.input.clusterNames().length-1 -->
                <button class="add_remove" data-bind="click: $root.input.addRow" title="Add Sequence" type="button">+</button>
            <!-- /ko -->

            <!-- ko if: $index() == $root.input.clusterNames().length-1 && $root.input.clusterNames().length > 2 -->
                <button class="add_remove" data-bind="click: $root.input.removeRow" title="Remove Sequence" type="button">-</button>
            <!-- /ko -->
        </div>
    <!-- /ko -->

    <div class="row">
		<div class="colW100"><label>Scoring in $s$:</label></div>

		<div class="colW400">
            <span class="group"> <!-- Microsoft Browsers will fallback on text-fields using following input type -->
                Match <input class="fx_parameter" data-bind="value: input.match" id="match" type="number">
                Mismatch <input class="fx_parameter" data-bind="value: input.mismatch" id="mismatch" type="number">
            </span>
		</div>
    </div>

    <div class="row">
		<div class="colW100"><label>Scoring in $g$:</label></div>

		<div class="colW400">
            <span class="group"> <!-- Microsoft Browsers will fallback on text-fields using following input type -->
                Gap opening $\alpha$ <input class="fx_parameter" data-bind="value: input.baseCosts" id="base_costs" type="number">
                Enlargement $\beta$ <input class="fx_parameter" data-bind="value: input.enlargement" id="enlargement" type="number">
            </span>
            <br />
            <br />
            <span data-bind="text: $root.input.gapFunction"></span>

            <div class="group_hint">
				<b>Hint:</b> <br />
				For similarity maximization, <br /> match scores should be positive and all other scores lower. <br />
			</div>
		</div>
    </div>
</div>

<h1>Output:</h1>
<div id="algorithm_output">
    <div class="final">
        <table class="final_result_header">
            <thead>
                <tr>
                    <th>
                        Final result
                    </th>
                </tr>
            </thead>
        </table>

        <div class="final_result_with_scrollbar">
            <table class="final_result">
                <tbody>
                    <tr>
                        <td class="entry entry_start">
                            <!-- ko foreach: $root.output.progressiveAlignment() -->
                                <code data-bind="text: $root.output.progressiveAlignment()[$index()]"></code> <br />
                            <!-- /ko -->
                        </td>
                    </tr>
                </tbody>
            </table>
        </div>

        <table class="final_result_footer">
            <tr>
                <th>
                    <small>
                        Affine SoP score: <span data-bind="text: $root.output.score()"></span>
                    </small>
                </th>
            </tr>
        </table>
    </div>

    <h2>Intermediate Results</h2>

    <div class="merge_steps">
        <table class="result_header">
            <thead>
                <tr>
                    <th>
                        Fusion steps
                    </th>
                </tr>
            </thead>
        </table>

        <div class="result_with_scrollbar">
            <table class="result_categories">
                <thead>
                    <tr>
                        <th>
                            <small><b>Group 1</b></small>
                        </th>

                        <th>
                            <small><b>Group 2</b></small>
                        </th>

                        <th>
                            <small><b>Guide Alignment</b></small>
                        </th>

                        <th>
                             <small><b>Result</b></small>
                        </th>
                    </tr>
                </thead>

                <tbody>
                    <!-- ko foreach: $root.output.guideAlignments() -->
                        <tr>
                            <td class="entry entry_start">
                                <code data-bind="text: $root.output.firstGroupsNames()[$index()]"></code>: <br />
                                <!-- ko foreach: $root.output.firstGroups()[$index()] -->
                                    <code data-bind="text: $data"></code> <br />
                                <!-- /ko -->
                            </td>

                            <td class="entry entry_start">
                                <code data-bind="text: $root.output.secondGroupsNames()[$index()]"></code>: <br />
                                <!-- ko foreach: $root.output.secondGroups()[$index()] -->
                                    <code data-bind="text: $data"></code> <br />
                                <!-- /ko -->
                            </td>

                            <td class="entry entry_start">
                                <code data-bind="text: $root.output.guideAlignmentsNames()[$index()]"></code>: <br />
                                <code data-bind="text: $root.output.guideAlignments()[$index()][0]"></code> <br />
                                <code data-bind="text: $root.output.guideAlignments()[$index()][1]"></code> <br />
                                <code data-bind="text: $root.output.guideAlignments()[$index()][2]"></code>
                            </td>

                            <td class="entry entry_start">
                                <code data-bind="text: $root.output.joinedGroupNames()[$index()]"></code>: <br />
                                <!-- ko foreach: $root.output.joinedGroups()[$index()] -->
                                    <code data-bind="text: $data"></code> <br />
                                <!-- /ko -->
                            </td>
                        </tr>
                    <!-- /ko -->
                </tbody>
            </table>
        </div>
    </div>

    <h2>Phylogenetic Tree</h2>

    <div class="newick_tree">
        <table class="result_header">
            <thead>
                <tr>
                    <th>
                        Newick Tree
                    </th>
                </tr>
            </thead>
        </table>

        <div class="result_with_scrollbar">
            <table class="result">
                <tbody>
                    <tr>
                        <td class="entry entry_start">
                            <code data-bind="text: $root.output.newickString()"></code>
                        </td>
                    </tr>
                </tbody>
            </table>
        </div>
    </div>

    <div class="tree_container"> <!-- allows to delete and reinsert the div -->
        <div id="phylogenetic_tree"></div>
    </div>

    <!-- ko if: $root.output.newickString().length !== 1 && $root.output.newickString().indexOf(SYMBOLS.MINUS) === -1 -->
        <div class="group_hint">
            <b>Visualization done with</b> <br />
            Smits SA, Ouverney CC, 2010. jsPhyloSVG: <br />
            A Javascript Library for Visualizing Interactive and Vector-Based Phylogenetic Trees on the Web. <br />
            <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0012267">
                PLoS ONE 5(8): e12267. doi:10.1371/journal.pone.0012267
            </a>
        </div>
    <!-- /ko -->

    <div class="group_hint">
        <b>Hint: </b> Visualization is only shown, if there are no negative distances.
    </div>

    <h2>Distance Table</h2>

    <!-- ko if: $root.output.progressiveAlignment()[0].length > 0 -->
        <div class="distance_table">
            <table class="distances">
                <thead>
                    <tr>
                        <th></th>
                        <!-- ko foreach: $root.output.remainingClusters()[0] -->
                            <th data-bind="drawChar: [$data, undefined]"></th>
                        <!-- /ko -->
                    </tr>
                </thead>

                <tbody>
                    <!-- ko foreach: $root.output.distanceMatrices()[0][0]--> <!-- to get i-indexes = $parentContext.$index() -->
                        <tr>
                            <th data-bind="drawChar: [$root.output.remainingClusters()[0][$index()], undefined]"></th>

                            <!-- ko foreach: $root.output.distanceMatrices()[0][0] --> <!-- to get j-indexes = $index() -->
                                <!-- ko if: $index() >= $parentContext.$index() -->
                                    <td class="non_selectable_entry"
                                        data-bind="text: $root.output.distanceMatrices()[0][$parentContext.$index()][$index()]">
                                    </td>
                                <!-- /ko -->

                                <!-- ko if: $index() < $parentContext.$index() -->
                                    <td class="non_selectable_entry gray"
                                        data-bind="text: $root.output.distanceMatrices()[0][$parentContext.$index()][$index()]">
                                    </td>
                                <!-- /ko -->
                            <!-- /ko -->
                        </tr>
                    <!-- /ko -->

                    <tr>
                        <th class="hint" colspan=100%>  <!-- HINT: move colspan into "hint"-class when browsers are ready! -->
                            <small>
                                <b>Minimum:</b> <span data-bind="text: $root.output.minimums()[0]"></span>
                            </small>
                        </th>
                    </tr>
                </tbody>
            </table>
        </div>

        <a href="#" data-bind="click: $root.output.toggleVisibility, text: $root.output.toggleLinkText"></a>
    <!-- /ko -->

    <div class="matrices" data-bind="visible: $root.output.showMatrices">
        <!-- ko foreach: $root.output.distanceMatrices() -->
            <!-- ko if: $index() > 0 -->
                <h3 class="header">Iteration <span data-bind="text: $index()"></span></h3>

                <div class="distance_table">
                    <table class="distances">
                        <thead>
                            <tr>
                                <th></th>
                                <!-- ko foreach: $root.output.remainingClusters()[$index()] -->
                                    <th data-bind="drawChar: [$data, undefined]"></th>
                                <!-- /ko -->
                            </tr>
                        </thead>

                        <tbody>
                            <!-- ko foreach: $root.output.distanceMatrices()[$index()] --> <!-- to get i-indexes = $parentContext.$index() -->
                                <tr>
                                    <th data-bind="drawChar: [$root.output.remainingClusters()[$parentContext.$index()][$index()], undefined]"></th>
                                    <!-- ko foreach: $root.output.distanceMatrices()[$parentContext.$index()][$index()] --> <!-- to get j-indexes = $index() -->
                                        <!-- ko if: $index() >= $parentContext.$index() -->
                                            <td class="non_selectable_entry"
                                                data-bind="text: $root.output.distanceMatrices()[$parentContext.$parentContext.$index()][$parentContext.$index()][$index()]">
                                            </td>
                                        <!-- /ko -->

                                        <!-- ko if: $index() < $parentContext.$index() -->
                                            <td class="non_selectable_entry gray"
                                                data-bind="text: $root.output.distanceMatrices()[$parentContext.$parentContext.$index()][$parentContext.$index()][$index()]">
                                            </td>
                                        <!-- /ko -->
                                    <!-- /ko -->
                                </tr>
                            <!-- /ko -->

                            <!-- ko if: $index() < $root.output.distanceMatrices().length-1 --> <!-- The last minimum is not computed -->
                                <tr>
                                    <th class="hint" colspan=100%>  <!-- HINT: move colspan into "hint"-class when browsers are ready! -->
                                        <small>
                                            <b>Minimum:</b> <span data-bind="text: $root.output.minimums()[$index()]"></span>
                                        </small>
                                    </th>
                                </tr>
                            <!-- /ko -->
                        </tbody>
                    </table>
                </div>
            <!-- /ko -->
        <!-- /ko -->
    </div>


    <h2>Pairwise Data</h2>

    <div class="pairwise_data">
        <div class="result_with_scrollbar">
            <table class="result_categories top_border">
                <thead>
                    <tr>
                        <th>
                            <small><b>Sequence Pair</b></small>
                        </th>

                        <th>
                            <small><b>Length $L$</b></small>
                        </th>

                        <th>
                            <small><b>Score $S$</b></small>
                        </th>

                        <th>
                            <small><b>Gaps $N^e$</b></small>
                        </th>

                        <th>
                            <small><b>Gap openings $N^o$</b></small>
                        </th>
                    </tr>
                </thead>

                <tbody>
                    <!-- ko foreach: $root.output.sequencePairNames() -->
                        <tr>
                            <td class="entry entry_start">
                                <code>(</code><code data-bind="text: $data[0]"></code><code>,</code>
                                <code data-bind="text: $data[1]"></code><code>)</code>
                                <br />
                            </td>

                            <td class="entry entry_start">
                                <code data-bind="text: $root.output.alignmentLengths()[$index()]"></code> <br />
                            </td>

                            <td class="entry entry_start">
                                <code data-bind="text: $root.output.similarities()[$index()]"></code> <br />
                            </td>

                            <td class="entry entry_start">
                                <code data-bind="text: $root.output.gapNumbers()[$index()]"></code> <br />
                            </td>

                            <td class="entry entry_start">
                                <code data-bind="text: $root.output.gapStarts()[$index()]"></code> <br />
                            </td>
                        </tr>
                    <!-- /ko -->
                </tbody>
            </table>
        </div>
    </div>
</div>