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Parikh vectors
5XGP_1 8QAQ_1 7KED_1 Letter Amino acid
18 0 93 K Lycine
8 0 44 M Methionine
10 0 58 F Phenylalanine
16 0 115 T Threonine
15 0 163 S Serine
12 0 83 R Arginine
10 0 66 Q Glutamine
11 3 115 G Glycine
24 0 143 L Leucine
9 0 58 Y Tyrosine
19 0 98 A Alanine
16 0 21 C Cysteine
18 0 117 E Glutamic acid
5 1 13 W Tryptophan
15 0 124 P Proline
16 5 117 V Valine
9 0 71 N Asparagine
10 0 102 D Aspartic acid
10 0 35 H Histidine
10 3 97 I Isoleucine 

5XGP_1|Chain A|Secreted Xwnt8 inhibitor sizzled|Xenopus laevis (8355)
>8QAQ_1|Chain A[auth Z]|TRP-MVA-ILE-MVA-MVA-SAR-MVA-IML-SAR-VAL-IML-SAR|Omphalotus olearius (72120)
>7KED_1|Chain A|DNA-directed RNA polymerase II subunit RPB1|Saccharomyces cerevisiae (strain ATCC 204508 / S288c) (559292)
Protein code \(c\) LZ-complexity \(\mathrm{LZ}(w)\) Length \(n=|w|\) \(\frac{\mathrm{LZ}(w)}{n /\log_{20} n}\) \(p_w(1)\) \(p_w(2)\) \(p_w(3)\) Sequence \(w=f(c)\)
5XGP , Knot 121 261 0.86 40 186 252 
STKCVTIPTEMAMCNDVGYSEMRLPNLMGHTNMAEVVPKSAEWQNLLQTGCHPYARTFLCSLFAPVCLDTFIQPCRSMCVAVRDSCAPVLACHGHSWPESLDCDRFPAGEDMCLDTLSKEYQYSYKELPKPSCQGCPLIEEFFSHKTVLEAFCDNNFAVKVKLAKKKSASGLYEYETEGPVEFIKQGLLLPYDTRTMIEQWLLINENCAQKLIRTRPTVYVIAGEIHHGKVKVNRIFHWQKKDSQLTLATRRWRHHKCAAA
8QAQ , Knot 7 12 0.48 8 7 8 
WVIVVGVIGVIG
7KED , Knot 556 1733 0.79 40 343 1337 
MVGQQYSSAPLRTVKEVQFGLFSPEEVRAISVAKIRFPETMDETQTRAKIGGLNDPRLGSIDRNLKCQTCQEGMNECPGHFGHIDLAKPVFHVGFIAKIKKVCECVCMHCGKLLLDEHNELMRQALAIKDSKKRFAAIWTLCKTKMVCETDVPSEDDPTQLVSRGGCGNTQPTIRKDGLKLVGSWKKDRATGDADEPELRVLSTEEILNIFKHISVKDFTSLGFNEVFSRPEWMILTCLPVPPPPVRPSISFNESQRGEDDLTFKLADILKANISLETLEHNGAPHHAIEEAESLLQFHVATYMDNDIAGQPQALQKSGRPVKSIRARLKGKEGRIRGNLMGKRVDFSARTVISGDPNLELDQVGVPKSIAKTLTYPEVVTPYNIDRLTQLVRNGPNEHPGAKYVIRDSGDRIDLRYSKRAGDIQLQYGWKVERHIMDNDPVLFNRQPSLHKMSMMAHRVKVIPYSTFRLNLSVTSPYNADFDGDEMNLHVPQSEETRAELSQLCAVPLQIVSPQSNKPCMGIVQDTLCGIRKLTLRDTFIELDQVLNMLYWVPDWDGVIPTPAIIKPKPLWSGKQILSVAIPNGIHLQRFDEGTTLLSPKDNGMLIIDGQIIFGVVEKKTVGSSNGGLIHVVTREKGPQVCAKLFGNIQKVVNFWLLHNGFSTGIGDTIADGPTMREITETIAEAKKKVLDVTKEAQANLLTAKHGMTLRESFEDNVVRFLNEARDKAGRLAEVNLKDLNNVKQMVMAGSKGSFINIAQMSACVGQQSVEGKRIAFGFVDRTLPHFSKDDYSPESKGFVENSYLRGLTPQEFFFHAMGGREGLIDTAVKTAETGYIQRRLVKALEDIMVHYDNTTRNSLGNVIQFIYGEDGMDAAHIEKQSLDTIGGSDAAFEKRYRVDLLNTDHTLDPSLLESGSEILGDLKLQVLLDEEYKQLVKDRKFLREVFVDGEANWPLPVNIRRIIQNAQQTFHIDHTKPSDLTIKDIVLGVKDLQENLLVLRGKNEIIQNAQRDAVTLFCCLLRSRLATRRVLQEYRLTKQAFDWVLSNIEAQFLRSVVHPGEMVGVLAAQSIGEPATQMTLNTFHFAGVASKKVTSGVPRLKEILNVAKNMKTPSLTVYLEPGHAADQEQAKLIRSAIEHTTLKSVTIASEIYYDPDPRSTVIPEDEEIIQLHFSLLDEEAEQSFDQQSPWLLRLELDRAAMNDKDLTMGQVGERIKQTFKNDLFVIWSEDNDEKLIIRCRVVRPKSLDAETEAEEDHMLKKIENTMLENITLRGVENIERVVMMKYDRKVPSPTGEYVKEPEWVLETDGVNLSEVMTVPGIDPTRIYTNSFIDIMEVLGIEAGRAALYKEVYNVIASDGSYVNYRHMALLVDVMTTQGGLTSVTRHGFNRSNTGALMRCSFEETVEILFEAGASAELDDCRGVSENVILGQMAPIGTGAFDVMIDEESLVKYMPEQKITEIEDGQDGGVTPYSNESGLVNADLDVKDELMFSPLVDSGSNDAMAGGFTAYGGADYGEATSPFGAYGEAPTSPGFGVSSPGFSPTSPTYSPTSPAYSPTSPSYSPTSPSYSPTSPSYSPTSPSYSPTSPSYSPTSPSYSPTSPSYSPTSPSYSPTSPSYSPTSPSYSPTSPSYSPTSPSYSPTSPSYSPTSPSYSPTSPAYSPTSPSYSPTSPSYSPTSPSYSPTSPSYSPTSPNYSPTSPSYSPTSPGYSPGSPAYSPKQDEQKHNENENSR

Let \(P_w(n)\) be the set of distinct subwords (intervals) in a word \(w\). Let \(p_w(n)\) be the cardinality of \(P_w(n)\). Let \(f(c)\) be the sequence in FASTA with 4-symbol Protein Data Bank code \(c\).

\(|P_{f(5XGP_1)}(2) \setminus P_{f(8QAQ_1)}(2)|=183\), \(|P_{f(8QAQ_1)}(2) \setminus P_{f(5XGP_1)}(2)|=4\). Let \( Z_k(x,y)=|P_x(k)\setminus P_y(k)|+|P_y(k)\setminus P_x(k)| \) be a LZ76 style (set of subwords) Jaccard distance numerator for \(P(k)\).Hydrophobic-polar version of Sequence 1:000010110011100011000101101110001101110010100110010010100110011111010011010001011100001111100100110010000111100101001000000000011010001011100110000110110000111010110000101100000011101100111110000011001111000010011000101011110100101010011010000001011000100000111
Pair \(Z_2\) Length of longest common subsequence
5XGP_1,8QAQ_1 187 2
5XGP_1,7KED_1 191 5
8QAQ_1,7KED_1 336 3

Newick tree

 
[
	7KED_1:14.26,
	[
		5XGP_1:93.5,8QAQ_1:93.5
	]:54.76
]

Let d be the Otu--Sayood distance d.
Let d1 be the Otu--Sayood distance d1. (This makes the 4TYN sequence AAAAAA a close match...)
A roughly speaking expected distance is \((0.85)(0.8)(\frac{273 }{\log_{20} 273}-\frac{12}{\log_{20}12})=89.3\)
Status Protein1 Protein2 d d1/2
Query variables 5XGP_1 8QAQ_1 117 60.5
Was not able to put for d
Was not able to put for d1

In notation analogous to [Theorem 16, Kjos-Hanssen, Niraula and Yoon (2022)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]