Arslan-Egecioglu-Pevzner.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 dynamic programming approach by
        <a href="https://doi.org/10.1093/bioinformatics/17.4.327">
            Abdullah N. Arslan, &Ouml;mer E&gbreve;ecio&gbreve;lu and Pavel A. Pevzner (2001)</a>
        computes the best normalized local alignments of two sequences.
        It identifies the local alignment that is best preserved with respect to the length of the aligned subsequences.
        <br />
        <br />
        The normalized local alignment algorithm tries
        to find the alignment $\mathcal{A}$ of two sequences
        with the highest normalized score
        $$\quad\lambda^* = ns(\mathcal{A}) = \frac{s(\mathcal{A})}{[\![\mathcal{A}]\!] + L}$$
        where $[\![\mathcal{A}]\!]$ is the length
        of the aligned subsequences (excluding gaps) and $L$ is a parameter
        to control the length-dependence of the normalized similarity score.
        <br />
        <br />
        To find a normalized local alignment,
        the fractional programming algorithm by
        <a href="https://doi.org/10.1287/mnsc.13.7.492">Werner Dinkelbach (1967)</a>
        is used in concert with the 'standard' local alignment algorithm by
        <a href="https://doi.org/10.1016/0022-2836(81)90087-5">Smith and Waterman (1981)</a>.
        Fractional programming is a machine learning approach
        to solve problems in which the objective function $ns(\mathcal{A})$ is a fraction as given above.
        <br />
        <br />
		Given a similarity scoring function $s(a_i,b_j)$ for two sequences $a$ and $b$.
		To find $\lambda^*$ and the according best normalized alignment $\mathcal{A}$,
		the <a href="https://doi.org/10.1016/0022-2836(81)90087-5">Smith-Waterman</a>
		algorithm is executed iteratively using an iteration-specific scoring-function $s^r$ (see below).
		In iteration $r>0$, $s^r$ depends on the $\lambda^{r-1}$ estimate of the previous iteration,
		where $\lambda^{0} = 0$.
		Given this, the best local alignment $\mathcal{A}^r$ with score $S^r$ using $s^r$ is identified and
		the new lambda estimate $\lambda^r$ is computed by $\lambda^r= S^r/([\![\mathcal{A}^r]\!]+L)$.
		This is iterated until $\lambda$ converges or a certain number of iterations is exceeded.
        <br />
        <br />
        For a given input, the according tables and scoring functions
        are shown for up to 5 iterations (or convergence) along with a list 
		of according optimal local alignments.
        On table cell selection, optimal traceback options are shown.
    </div>

    <div class="picture">
        <img src="Arslan-Egecioglu-Pevzner-120x90.png" />
    </div>
</div>

<h1>Input:</h1>
<div id="algorithm_input">
	<div class="row">
		<div class="colW100"><label>Sequence $a$:</label></div>
		<div class="colW400"><input class="sequence" data-bind="value: input.sequence1" id="sequence_1" placeholder="EXAMPLE 'ATC'"
									title="Allowed are A-Z and '-'." type="text"></div>
	</div>

	<div class="row">
		<div class="colW100"><label>Sequence $b$:</label></div>
		<div class="colW400"><input class="sequence" data-bind="value: input.sequence2" id="sequence_2" placeholder="EXAMPLE 'AGTC'"
									title="Allowed are A-Z and '-'." type="text"></div>
	</div>

	<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">
				Gap <input class="fx_parameter" data-bind="value: input.gap" id="gap" type="number">
			</span>

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

	<div class="row">
		<div class="colW100"><label>Length-scal. $L$:</label></div>
		<div class="colW600"><input class="fx_parameter" data-bind="value: input.length" id="length" type="number"></div>
	</div>

	<div class="row">
		<div class="colW100">
			<br />
			<br />
			<label>
				Recursion of
				<br />iteration $r=1$:
			</label>
		</div>

		<div class="colW600">
			<span data-bind="text: $root.input.formula"></span>
			<br />
			<br />
			with <br />
			<span data-bind="text: $root.input.subFormula"></span>
			<br />
			<br />
			where $\lambda^0 = 0$
			<br />
			<br />
		</div>
	</div>
</div>

<!-- Dynamic creation of iterations didn't work,
because of several problems with MathJax,
ObservableArrays of arrays and runtime for drawing. -->

<h1>Output:</h1>
<div id="algorithm_output">
	<!-- ko if: $root.output.matrix5().length > 0 -->
		<div class="iteration">
			<h2>Iteration 5</h2>

			<div class="output">
				<div class="main_output">
					<table class="calculation_5">
						<thead>
							<tr>
								<th>$D^5$</th>
								<th></th>
								<!-- ko foreach: $root.input.sequence2 -->
									<th data-bind="drawChar: [$data, $index()+1]"></th>
								<!-- /ko -->
							</tr>
						</thead>

						<tbody>
							<!-- ko foreach: $root.output.matrix5 -->
								<tr>
									<!-- ko if: $index() == 0 -->
										<th></th>
									<!-- /ko -->

									<!-- ko if: $index() > 0 -->
										<th data-bind="drawChar: [$root.input.sequence1()[$index()-1], $index()]"></th>
									<!-- /ko -->

									<!-- ko foreach: $root.output.matrix5[0] -->
										<td class="selectable_entry" data-bind="text: $root.output.matrix5()[$parentContext.$index()][$index()]"></td>
									<!-- /ko -->
								</tr>
							<!-- /ko -->
						</tbody>
					</table>
				</div>

				<div class="parameters">
					Score $s(\mathcal{A}^5)$:
					<span data-bind="text: $root.output.score5"></span> <br />

					Length $[\![\mathcal{A}^5]\!]$:
					<span data-bind="text: $root.output.length5"></span> <br />

					Normalized score $\lambda^5$:
					<span data-bind="text: $root.output.lambda5"></span> <br />
				</div>
			</div>

			<div class="outcome">
				<div class ="ancillary_output">
					<table class="results_header">
						<thead>
							<tr>
								<th>
									Results <br />
									<small>
										The result for which the normalized score was computed is marked blue.
									</small>
								</th>
							</tr>
						</thead>
					</table>

					<div class="results_with_scrollbar">
						<table class="results">
							<tbody>
								<!-- ko foreach: $root.output.alignments5 -->
									<!-- ko if: $index() < 10 -->

										<!-- ko if: $index() == $root.output.alignmentNumber5() -->
											<!-- ko if: $index() == 0 -->
													<tr>
														<td class="entry entry_start selected">
															<code data-bind="text: $root.output.alignments5()[$index()][0]"></code> <br />
															<code data-bind="text: $root.output.alignments5()[$index()][1]"></code> <br />
															<code data-bind="text: $root.output.alignments5()[$index()][2]"></code>
														</td>
													</tr>
											<!-- /ko -->
											<!-- ko if: $index() != 0 -->
													<tr>
														<td class="entry selected">
															<code data-bind="text: $root.output.alignments5()[$index()][0]"></code> <br />
															<code data-bind="text: $root.output.alignments5()[$index()][1]"></code> <br />
															<code data-bind="text: $root.output.alignments5()[$index()][2]"></code>
														</td>
													</tr>
											<!-- /ko -->
										<!-- /ko -->

										<!-- ko if: $index() != $root.output.alignmentNumber5() -->
											<!-- ko if: $index() == 0 -->
													<tr>
														<td class="entry entry_start">
															<code data-bind="text: $root.output.alignments5()[$index()][0]"></code> <br />
															<code data-bind="text: $root.output.alignments5()[$index()][1]"></code> <br />
															<code data-bind="text: $root.output.alignments5()[$index()][2]"></code>
														</td>
													</tr>
											<!-- /ko -->
											<!-- ko if: $index() != 0 -->
													<tr>
														<td class="entry">
															<code data-bind="text: $root.output.alignments5()[$index()][0]"></code> <br />
															<code data-bind="text: $root.output.alignments5()[$index()][1]"></code> <br />
															<code data-bind="text: $root.output.alignments5()[$index()][2]"></code>
														</td>
													</tr>
											<!-- /ko -->
										<!-- /ko -->

									<!-- /ko -->
								<!-- /ko -->
							</tbody>
						</table>
					</div>

					<table class="results_footer">
						<tr>
							<th>
								<small>
									<!-- ko if: $root.output.moreTracebacks5 -->
										<b>Hint:</b> Only the first ten results are computed. <br />
									<!-- /ko -->
									<!-- ko if: $root.output.alignments5().length > 1 && $root.output.maxNumberIterations -->
										<b>Hint:</b> Not marked alignments aren't used and only listed for completeness.
									<!-- /ko -->
								</small>
							</th>
						</tr>
					</table>
				</div>
			</div>
		</div>
	<!-- /ko -->

	<!-- ko if: $root.output.matrix4().length > 0 -->
		<div class="iteration">
			<h2>Iteration 4</h2>

			<div class="output">
				<div class="main_output">
					<table class="calculation_4">
						<thead>
							<tr>
								<th>$D^4$</th>
								<th></th>
								<!-- ko foreach: $root.input.sequence2 -->
									<th data-bind="drawChar: [$data, $index()+1]"></th>
								<!-- /ko -->
							</tr>
						</thead>

						<tbody>
							<!-- ko foreach: $root.output.matrix4 -->
								<tr>
									<!-- ko if: $index() == 0 -->
										<th></th>
									<!-- /ko -->

									<!-- ko if: $index() > 0 -->
										<th data-bind="drawChar: [$root.input.sequence1()[$index()-1], $index()]"></th>
									<!-- /ko -->

									<!-- ko foreach: $root.output.matrix4[0] -->
										<td class="selectable_entry" data-bind="text: $root.output.matrix4()[$parentContext.$index()][$index()]"></td>
									<!-- /ko -->
								</tr>
							<!-- /ko -->
						</tbody>
					</table>
				</div>

				<div class="parameters">
					Score $s(\mathcal{A}^4)$:
					<span data-bind="text: $root.output.score4"></span> <br />

					Length $[\![\mathcal{A}^4]\!]$:
					<span data-bind="text: $root.output.length4"></span> <br />

					Normalized score $\lambda^4$:
					<span data-bind="text: $root.output.lambda4"></span> <br />
				</div>
			</div>

			<div class="outcome">
				<div class ="ancillary_output">
					<table class="results_header">
						<thead>
							<tr>
								<th>
									Results <br />
									<small>
										The result for which the normalized score was computed is marked blue.
									</small>
								</th>
							</tr>
						</thead>
					</table>

					<div class="results_with_scrollbar">
						<table class="results">
							<tbody>
								<!-- ko foreach: $root.output.alignments4 -->
									<!-- ko if: $index() < 10 -->

										<!-- ko if: $index() == $root.output.alignmentNumber4() -->
											<!-- ko if: $index() == 0 -->
													<tr>
														<td class="entry entry_start selected">
															<code data-bind="text: $root.output.alignments4()[$index()][0]"></code> <br />
															<code data-bind="text: $root.output.alignments4()[$index()][1]"></code> <br />
															<code data-bind="text: $root.output.alignments4()[$index()][2]"></code>
														</td>
													</tr>
											<!-- /ko -->
											<!-- ko if: $index() != 0 -->
													<tr>
														<td class="entry selected">
															<code data-bind="text: $root.output.alignments4()[$index()][0]"></code> <br />
															<code data-bind="text: $root.output.alignments4()[$index()][1]"></code> <br />
															<code data-bind="text: $root.output.alignments4()[$index()][2]"></code>
														</td>
													</tr>
											<!-- /ko -->
										<!-- /ko -->

										<!-- ko if: $index() != $root.output.alignmentNumber4() -->
											<!-- ko if: $index() == 0 -->
													<tr>
														<td class="entry entry_start">
															<code data-bind="text: $root.output.alignments4()[$index()][0]"></code> <br />
															<code data-bind="text: $root.output.alignments4()[$index()][1]"></code> <br />
															<code data-bind="text: $root.output.alignments4()[$index()][2]"></code>
														</td>
													</tr>
											<!-- /ko -->
											<!-- ko if: $index() != 0 -->
													<tr>
														<td class="entry">
															<code data-bind="text: $root.output.alignments4()[$index()][0]"></code> <br />
															<code data-bind="text: $root.output.alignments4()[$index()][1]"></code> <br />
															<code data-bind="text: $root.output.alignments4()[$index()][2]"></code>
														</td>
													</tr>
											<!-- /ko -->
										<!-- /ko -->

									<!-- /ko -->
								<!-- /ko -->
							</tbody>
						</table>
					</div>

					<table class="results_footer">
						<tr>
							<th>
								<small>
									<!-- ko if: $root.output.moreTracebacks4 -->
										<b>Hint:</b> Only the first ten results are computed. <br />
									<!-- /ko -->
									<!-- ko if: $root.output.alignments4().length > 1 && $root.output.alignments5().length > 0 -->
										<b>Hint:</b> Not marked alignments aren't used and only listed for completeness.
									<!-- /ko -->
								</small>
							</th>
						</tr>
					</table>
				</div>
			</div>
		</div>
	<!-- /ko -->

	<!-- ko if: $root.output.matrix3().length > 0 -->
		<div class="iteration">
			<h2>Iteration 3</h2>

			<div class="output">
				<div class="main_output">
					<table class="calculation_3">
						<thead>
							<tr>
								<th>$D^3$</th>
								<th></th>
								<!-- ko foreach: $root.input.sequence2 -->
									<th data-bind="drawChar: [$data, $index()+1]"></th>
								<!-- /ko -->
							</tr>
						</thead>

						<tbody>
							<!-- ko foreach: $root.output.matrix3 -->
								<tr>
									<!-- ko if: $index() == 0 -->
										<th></th>
									<!-- /ko -->

									<!-- ko if: $index() > 0 -->
										<th data-bind="drawChar: [$root.input.sequence1()[$index()-1], $index()]"></th>
									<!-- /ko -->

									<!-- ko foreach: $root.output.matrix3[0] -->
										<td class="selectable_entry" data-bind="text: $root.output.matrix3()[$parentContext.$index()][$index()]"></td>
									<!-- /ko -->
								</tr>
							<!-- /ko -->
						</tbody>
					</table>
				</div>

				<div class="parameters">
					Score $s(\mathcal{A}^3)$:
					<span data-bind="text: $root.output.score3"></span> <br />

					Length $[\![\mathcal{A}^3]\!]$:
					<span data-bind="text: $root.output.length3"></span> <br />

					Normalized score $\lambda^3$:
					<span data-bind="text: $root.output.lambda3"></span> <br />
				</div>
			</div>

			<div class="outcome">
				<div class ="ancillary_output">
					<table class="results_header">
						<thead>
							<tr>
								<th>
									Results <br />
									<small>
										The result for which the normalized score was computed is marked blue.
									</small>
								</th>
							</tr>
						</thead>
					</table>

					<div class="results_with_scrollbar">
						<table class="results">
							<tbody>
								<!-- ko foreach: $root.output.alignments3 -->
									<!-- ko if: $index() < 10 -->

										<!-- ko if: $index() == $root.output.alignmentNumber3() -->
											<!-- ko if: $index() == 0 -->
													<tr>
														<td class="entry entry_start selected">
															<code data-bind="text: $root.output.alignments3()[$index()][0]"></code> <br />
															<code data-bind="text: $root.output.alignments3()[$index()][1]"></code> <br />
															<code data-bind="text: $root.output.alignments3()[$index()][2]"></code>
														</td>
													</tr>
											<!-- /ko -->
											<!-- ko if: $index() != 0 -->
													<tr>
														<td class="entry selected">
															<code data-bind="text: $root.output.alignments3()[$index()][0]"></code> <br />
															<code data-bind="text: $root.output.alignments3()[$index()][1]"></code> <br />
															<code data-bind="text: $root.output.alignments3()[$index()][2]"></code>
														</td>
													</tr>
											<!-- /ko -->
										<!-- /ko -->

										<!-- ko if: $index() != $root.output.alignmentNumber3() -->
											<!-- ko if: $index() == 0 -->
													<tr>
														<td class="entry entry_start">
															<code data-bind="text: $root.output.alignments3()[$index()][0]"></code> <br />
															<code data-bind="text: $root.output.alignments3()[$index()][1]"></code> <br />
															<code data-bind="text: $root.output.alignments3()[$index()][2]"></code>
														</td>
													</tr>
											<!-- /ko -->
											<!-- ko if: $index() != 0 -->
													<tr>
														<td class="entry">
															<code data-bind="text: $root.output.alignments3()[$index()][0]"></code> <br />
															<code data-bind="text: $root.output.alignments3()[$index()][1]"></code> <br />
															<code data-bind="text: $root.output.alignments3()[$index()][2]"></code>
														</td>
													</tr>
											<!-- /ko -->
										<!-- /ko -->

									<!-- /ko -->
								<!-- /ko -->
							</tbody>
						</table>
					</div>

					<table class="results_footer">
						<tr>
							<th>
								<small>
									<!-- ko if: $root.output.moreTracebacks3 -->
										<b>Hint:</b> Only the first ten results are computed. <br />
									<!-- /ko -->
									<!-- ko if: $root.output.alignments3().length > 1 && $root.output.alignments4().length > 0 -->
										<b>Hint:</b> Not marked alignments aren't used and only listed for completeness.
									<!-- /ko -->
								</small>
							</th>
						</tr>
					</table>
				</div>
			</div>
		</div>
	<!-- /ko -->

	<!-- ko if: $root.output.matrix2().length > 0 -->
		<div class="iteration">
			<h2>Iteration 2</h2>

			<div class="output">
				<div class="main_output">
					<table class="calculation_2">
						<thead>
							<tr>
								<th>$D^2$</th>
								<th></th>
								<!-- ko foreach: $root.input.sequence2 -->
									<th data-bind="drawChar: [$data, $index()+1]"></th>
								<!-- /ko -->
							</tr>
						</thead>

						<tbody>
							<!-- ko foreach: $root.output.matrix2 -->
								<tr>
									<!-- ko if: $index() == 0 -->
										<th></th>
									<!-- /ko -->

									<!-- ko if: $index() > 0 -->
										<th data-bind="drawChar: [$root.input.sequence1()[$index()-1], $index()]"></th>
									<!-- /ko -->

									<!-- ko foreach: $root.output.matrix2[0] -->
										<td class="selectable_entry" data-bind="text: $root.output.matrix2()[$parentContext.$index()][$index()]"></td>
									<!-- /ko -->
								</tr>
							<!-- /ko -->
						</tbody>
					</table>
				</div>

				<div class="parameters">
					Score $s(\mathcal{A}^2)$:
					<span data-bind="text: $root.output.score2"></span> <br />

					Length $[\![\mathcal{A}^2]\!]$:
					<span data-bind="text: $root.output.length2"></span> <br />

					Normalized score $\lambda^2$:
					<span data-bind="text: $root.output.lambda2"></span> <br />
				</div>
			</div>

			<div class="outcome">
				<div class ="ancillary_output">
					<table class="results_header">
						<thead>
							<tr>
								<th>
									Results <br />
									<small>
										The result for which the normalized score was computed is marked blue.
									</small>
								</th>
							</tr>
						</thead>
					</table>

					<div class="results_with_scrollbar">
						<table class="results">
							<tbody>
								<!-- ko foreach: $root.output.alignments2 -->
									<!-- ko if: $index() < 10 -->

										<!-- ko if: $index() == $root.output.alignmentNumber2() -->
											<!-- ko if: $index() == 0 -->
													<tr>
														<td class="entry entry_start selected">
															<code data-bind="text: $root.output.alignments2()[$index()][0]"></code> <br />
															<code data-bind="text: $root.output.alignments2()[$index()][1]"></code> <br />
															<code data-bind="text: $root.output.alignments2()[$index()][2]"></code>
														</td>
													</tr>
											<!-- /ko -->
											<!-- ko if: $index() != 0 -->
													<tr>
														<td class="entry selected">
															<code data-bind="text: $root.output.alignments2()[$index()][0]"></code> <br />
															<code data-bind="text: $root.output.alignments2()[$index()][1]"></code> <br />
															<code data-bind="text: $root.output.alignments2()[$index()][2]"></code>
														</td>
													</tr>
											<!-- /ko -->
										<!-- /ko -->

										<!-- ko if: $index() != $root.output.alignmentNumber2() -->
											<!-- ko if: $index() == 0 -->
													<tr>
														<td class="entry entry_start">
															<code data-bind="text: $root.output.alignments2()[$index()][0]"></code> <br />
															<code data-bind="text: $root.output.alignments2()[$index()][1]"></code> <br />
															<code data-bind="text: $root.output.alignments2()[$index()][2]"></code>
														</td>
													</tr>
											<!-- /ko -->
											<!-- ko if: $index() != 0 -->
													<tr>
														<td class="entry">
															<code data-bind="text: $root.output.alignments2()[$index()][0]"></code> <br />
															<code data-bind="text: $root.output.alignments2()[$index()][1]"></code> <br />
															<code data-bind="text: $root.output.alignments2()[$index()][2]"></code>
														</td>
													</tr>
											<!-- /ko -->
										<!-- /ko -->

									<!-- /ko -->
								<!-- /ko -->
							</tbody>
						</table>
					</div>

					<table class="results_footer">
						<tr>
							<th>
								<small>
									<!-- ko if: $root.output.moreTracebacks2 -->
										<b>Hint:</b> Only the first ten results are computed. <br />
									<!-- /ko -->
									<!-- ko if: $root.output.alignments2().length > 1 && $root.output.alignments3().length > 0 -->
										<b>Hint:</b> Not marked alignments aren't used and only listed for completeness.
									<!-- /ko -->
								</small>
							</th>
						</tr>
					</table>
				</div>
			</div>
		</div>
	<!-- /ko -->

	<!-- ko if: $root.output.matrix1().length > 0 -->
		<div class="iteration">
			<h2>Iteration 1</h2>

			<div class="output">
				<div class="main_output">
					<table class="calculation_1">
						<thead>
							<tr>
								<th>$D^1$</th>
								<th></th>
								<!-- ko foreach: $root.input.sequence2 -->
									<th data-bind="drawChar: [$data, $index()+1]"></th>
								<!-- /ko -->
							</tr>
						</thead>

						<tbody>
							<!-- ko foreach: $root.output.matrix1 -->
								<tr>
									<!-- ko if: $index() == 0 -->
										<th></th>
									<!-- /ko -->

									<!-- ko if: $index() > 0 -->
										<th data-bind="drawChar: [$root.input.sequence1()[$index()-1], $index()]"></th>
									<!-- /ko -->

									<!-- ko foreach: $root.output.matrix1[0] -->
										<td class="selectable_entry" data-bind="text: $root.output.matrix1()[$parentContext.$index()][$index()]"></td>
									<!-- /ko -->
								</tr>
							<!-- /ko -->
						</tbody>
					</table>
				</div>

				<div class="parameters">
					Score $s(\mathcal{A}^1)$:
					<span data-bind="text: $root.output.score1"></span> <br />

					Length $[\![\mathcal{A}^1]\!]$:
					<span data-bind="text: $root.output.length1"></span> <br />

					Normalized score $\lambda^1$:
					<span data-bind="text: $root.output.lambda1"></span> <br />
				</div>
			</div>

			<div class="outcome">
				<div class ="ancillary_output">
					<table class="results_header">
						<thead>
							<tr>
								<th>
									Results <br />
									<small>
										The result for which the normalized score was computed is marked blue.
									</small>
								</th>
							</tr>
						</thead>
					</table>

					<div class="results_with_scrollbar">
						<table class="results">
							<tbody>
								<!-- ko foreach: $root.output.alignments1 -->
									<!-- ko if: $index() < 10 -->

										<!-- ko if: $index() == $root.output.alignmentNumber1() -->
											<!-- ko if: $index() == 0 -->
													<tr>
														<td class="entry entry_start selected">
															<code data-bind="text: $root.output.alignments1()[$index()][0]"></code> <br />
															<code data-bind="text: $root.output.alignments1()[$index()][1]"></code> <br />
															<code data-bind="text: $root.output.alignments1()[$index()][2]"></code>
														</td>
													</tr>
											<!-- /ko -->
											<!-- ko if: $index() != 0 -->
													<tr>
														<td class="entry selected">
															<code data-bind="text: $root.output.alignments1()[$index()][0]"></code> <br />
															<code data-bind="text: $root.output.alignments1()[$index()][1]"></code> <br />
															<code data-bind="text: $root.output.alignments1()[$index()][2]"></code>
														</td>
													</tr>
											<!-- /ko -->
										<!-- /ko -->

										<!-- ko if: $index() != $root.output.alignmentNumber1() -->
											<!-- ko if: $index() == 0 -->
													<tr>
														<td class="entry entry_start">
															<code data-bind="text: $root.output.alignments1()[$index()][0]"></code> <br />
															<code data-bind="text: $root.output.alignments1()[$index()][1]"></code> <br />
															<code data-bind="text: $root.output.alignments1()[$index()][2]"></code>
														</td>
													</tr>
											<!-- /ko -->
											<!-- ko if: $index() != 0 -->
													<tr>
														<td class="entry">
															<code data-bind="text: $root.output.alignments1()[$index()][0]"></code> <br />
															<code data-bind="text: $root.output.alignments1()[$index()][1]"></code> <br />
															<code data-bind="text: $root.output.alignments1()[$index()][2]"></code>
														</td>
													</tr>
											<!-- /ko -->
										<!-- /ko -->

									<!-- /ko -->
								<!-- /ko -->
							</tbody>
						</table>
					</div>

					<table class="results_footer">
						<tr>
							<th>
								<small>
									<!-- ko if: $root.output.moreTracebacks1 -->
										<b>Hint:</b> Only the first ten results are computed. <br />
									<!-- /ko -->
									<!-- ko if: $root.output.alignments1().length > 1 && $root.output.alignments2().length > 0 -->
										<b>Hint:</b> Not marked alignments aren't used and only listed for completeness.
									<!-- /ko -->
								</small>
							</th>
						</tr>
					</table>
				</div>
			</div>
		</div>
	<!-- /ko -->

	<div class="iteration_footer">
		<b>Hints:</b> <br />
		<!-- ko if: $root.output.maxNumberIterations -->
			Only first five iterations are computed. <br />
		<!-- /ko -->
		Scores $s(\mathcal{A})$ and normalized scores $\lambda$ are rounded to four digits.
	</div>

    <div class="download_area">
		<!-- ko if: $root.output.matrix5().length > 0 -->
        	<a class="table_download_5" href="#">Download Iteration 5 Table</a> <br />
		<!-- /ko -->

		<!-- ko if: $root.output.matrix4().length > 0 -->
        	<a class="table_download_4" href="#">Download Iteration 4 Table</a> <br />
		<!-- /ko -->

		<!-- ko if: $root.output.matrix3().length > 0 -->
        	<a class="table_download_3" href="#">Download Iteration 3 Table</a> <br />
		<!-- /ko -->

		<!-- ko if: $root.output.matrix2().length > 0 -->
        	<a class="table_download_2" href="#">Download Iteration 2 Table</a> <br />
		<!-- /ko -->

		<!-- ko if: $root.output.matrix1().length > 0 -->
        	<a class="table_download_1" href="#">Download Iteration 1 Table</a>
		<!-- /ko -->
    </div>
</div>