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Parikh vectors
8FAC_1 5CJQ_1 6TME_1 Letter Amino acid
144 20 23 A Alanine
55 5 11 C Cysteine
120 16 23 G Glycine
89 5 16 I Isoleucine
98 18 17 T Threonine
31 4 3 W Tryptophan
47 0 4 M Methionine
77 5 29 F Phenylalanine
158 33 32 S Serine
132 10 18 E Glutamic acid
47 2 3 H Histidine
244 14 38 L Leucine
105 11 22 K Lycine
118 16 22 P Proline
56 8 9 Y Tyrosine
115 6 10 R Arginine
71 6 32 N Asparagine
114 8 26 D Aspartic acid
92 12 9 Q Glutamine
135 16 25 V Valine 

8FAC_1|Chain A|Leucine-rich repeat serine/threonine-protein kinase 1|Homo sapiens (9606)
>5CJQ_1|Chain A[auth L]|CR9114 light chain|Homo sapiens (9606)
>6TME_1|Chains A, B|Pollen-specific leucine-rich repeat extensin-like protein 1|Arabidopsis thaliana (3702)
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 \(w=f(c)\)
8FAC , Knot 690 2048 0.85 40 374 1650 
MDYKDHDGDYKDHDIDYKDDDDKLGLEVLFQGPMAGMSQRPPSMYWCVGPEESAVCPERAMETLNGAGDTGGKPSTRGGDPAARSRRTEGIRAAYRRGDRGGARDLLEEACDQCASQLEKGQLLSIPAAYGDLEMVRYLLSKRLVELPTEPTDDNPAVVAAYFGHTAVVQELLESLPGPCSPQRLLNWMLALACQRGHLGVVKLLVLTHGADPESYAVRKNEFPVIVRLPLYAAIKSGNEDIAIFLLRHGAYFCSYILLDSPDPSKHLLRKYFIEASPLPSSYPGKTALRVKWSHLRLPWVDLDWLIDISCQITELDLSANCLATLPSVIPWGLINLRKLNLSDNHLGELPGVQSSDEIICSRLLEIDISSNKLSHLPPGFLHLSKLQKLTASKNCLEKLFEEENATNWIGLRKLQELDISDNKLTELPALFLHSFKSLNSLNVSRNNLKVFPDPWACPLKCCKASRNALECLPDKMAVFWKNHLKDVDFSENALKEVPLGLFQLDALMFLRLQGNQLAALPPQEKWTCRQLKTLDLSRNQLGKNEDGLKTKRIAFFTTRGRQRSGTEAASVLEFPAFLSESLEVLCLNDNHLDTVPPSVCLLKSLSELYLGNNPGLRELPPELGQLGNLWQLDTEDLTISNVPAEIQKEGPKAMLSYLRAQLRKAEKCKLMKMIIVGPPRQGKSTLLEILQTGRAPQVVHGEATIRTTKWELQRPAGSRAKVESVEFNVWDIGGPASMATVNQCFFTDKALYVVVWNLALGEEAVANLQFWLLNIEAKAPNAVVLVVGTHLDLIEAKFRVERIATLRAYVLALCRSPSGSRATGFPDITFKHLHEISCKSLEGQEGLRQLIFHVTCSMKDVGSTIGCQRLAGRLIPRSYLSLQEAVLAEQQRRSRDDDVQYLTDRQLEQLVEQTPDNDIKDYEDLQSAISFLIETGTLLHFPDTSHGLRNLYFLDPIWLSECLQRIFNIKGSRSVAKNGVIRAEDLRMLLVGTGFTQQTEEQYFQFLAKFEIALPVANDSYLLPHLLPSKPGLDTHGMRHPTANTIQRVFKMSFVPVGFWQRFIARMLISLAEMDLQLFENKKNTKSRNRKVTIYSFTGNQRNRCSTFRVKRNQTIYWQEGLLVTFDGGYLSVESSDVNWKKKKSGGMKIVCQSEVRDFSAMAFITDHVNSLIDQWFPALTATESDGTPLMEQYVPCPVCETAWAQHTDPSEKSEDVQYFDMEDCVLTAIERDFISCPRHPDLPVPLQELVPELFMTDFPARLFLENSKLEHSEDEGSVLGQGGSGTVIYRARYQGQPVAVKRFHIKKFKNFANVPADTMLRHLRATDAMKNFSEFRQEASMLHALQHPCIVALIGISIHPLCFALELAPLSSLNTVLSENARDSSFIPLGHMLTQKIAYQIASGLAYLHKKNIIFCDLKSDNILVWSLDVKEHINIKLSDYGISRQSFHEGALGVEGTPGYQAPEIRPRIVYDEKVDMFSYGMVLYELLSGQRPALGHHQLQIAKKLSKGIRPVLGQPEEVQFRRLQALMMECWDTKPEKRPLALSVVSQMKDPTFATFMYELCCGKQTAFFSSQGQEYTVVFWDGKEESRNYTVVNTEKGLMEVQRMCCPGMKVSCQLQVQRSLWTATEDQKIYIYTLKGMCPLNTPQQALDTPAVVTCFLAVPVIKKNSYLVLAGLADGLVAVFPVVRGTPKDSCSYLCSHTANRSKFSIADEDARQNPYPVKAMEVVNSGSEVWYSNGPGLLVIDCASLEICRRLEPYMAPSMVTSVVCSSEGRGEEVVWCLDDKANSLVMYHSTTYQLCARYFCGVPSPLRDMFPVRPLDTEPPAASHTANPKVPEGDSIADVSIMYSEELGTQILIHQESLTDYCSMSSYSSSPPRQAARSPSSLPSSPASSSSVPFSTDCEDSDMLHTPGAASDRSEHDLTPMDGETFSQHLQAVKILAVRDLIWVPRRGGDVIVIGLEKDSGAQRGRVIAVLKARELTPHGVLVDAAVVAKDTVVCTFENENTEWCLAVWRGWGAREFDIFYQSYEELGRLEACTRKRR
5CJQ , Knot 97 215 0.80 38 139 208 
SALTQPPAVSGTPGQRVTISCSGSDSNIGRRSVNWYQQFPGTAPKLLIYSNDQRPSVVPDRFSGSKSGTSASLAISGLQSEDEAEYYCAAWDDSLKGAVFGGGTQLTVLGQPKAAPSVTLFPPSSEELQANKATLVCLISDFYPGAVTVAWKADSSPVKAGVETTTPSKQSNNKYAASSYLSLTPEQWKSHRSYSCQVTHEGSTVEKTVAPTECS
6TME , Knot 156 372 0.82 40 207 344 
MELTDEEASFLTRRQLLALSENGDLPDDIEYEVDLDLKFANNRLKRAYIALQAWKKAFYSDPFNTAANWVGPDVCSYKGVFCAPALDDPSVLVVAGIDLNHADIFGYLPPELGLLTDVALFHVNSNRFCGVIPKSLSKLTLMYEFDVSNNRFVGPFPTVALSWPSLKFLDIRYNDFEGKLPPEIFDKDLDAIFLNNNRFESTIPETIGKSTASVVTFAHNKFSGCIPKTIGQMKNLNEIVFIGNNLSGCLPNEIGSLNNVTVFDASSNGFVGSLPSTLSGLANVEQMDFSYNKFTGFVTDNICKLPKLSNFTFSYNFFNGEAQSCVPGSSQEKQFDDTSNCLQNRPNQKSAKECLPVVSRPVDCSKDKCAGG

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 \(c\).

\(|P_{f(8FAC_1)}(2) \setminus P_{f(5CJQ_1)}(2)|=237\), \(|P_{f(5CJQ_1)}(2) \setminus P_{f(8FAC_1)}(2)|=2\). 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 1:10000001000000100000000111011101111110001101010111000110100110010111001101000110111000000110110001001110011001000010010010110111101010110011000110110010000111111011001110011001111001001101111110001011110111100110100011000011111011101110010001111110011010001110010100011000110101110001100110101001011110101110100010010101001101101111111010010100001101111000001100011010100001001111110100100101000010011000010011110010010100001001111110010010010100001011101110110000100011001100111110001001010001100111111010111110101001111110001000010010100001100001100001111000100001001101101111100010110100001001110101100100101100111001110110110110100001010011101000110111001010100100001101111111001000110110010110110101010000101001110010100101011011111011010001100011011110111100111010111101010110111111100101101010100110101011110001010010111010100100100001010011001110100010011001100011101110001010011110000000000100100001001100010001000001001101110010110110000110010110111100010011010100011001110100101111101100000000101110101111110000111011100111000110010100100110101111111001110111011010101100000000000010100101000000001010000010100111101011010100001010000011101100001001011111000100110011111010000101110001101100011100001000000100101000110110001100100101111100111011100111011100001000000101110110101100100010111100101001001101110011001010011001001000101101100101111111010110111011110010011000100001111101100011001101110100001110010000111101010001010100011000010011111010110011010101100001011001111001101001111000101100100110111101001010010111100100010001111011001001011011001001000111000100001111010000000011000011101001001110100010100011010000010100101101100100110011110011111110000011111110111111111010100000010000100001011000100010110110110010011000111111100101010001010111011001100001010011101000100111000000010100101110110011110110001111000101011010011010110000110011100001000001000000110011001001100110000111000000001100111100000001011010010001011011110011111001101111110000110010111110100101011110111110001100100000010111101111001011000000110101000000
Pair \(Z_2\) Length of longest common subsequence
8FAC_1,5CJQ_1 239 5
8FAC_1,6TME_1 177 6
5CJQ_1,6TME_1 146 3

Newick tree

 
[
	8FAC_1:11.86,
	[
		6TME_1:73,5CJQ_1:73
	]:40.86
]

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 \((0.85)(0.8)(\frac{2263 }{\log_{20} 2263}-\frac{215}{\log_{20}215})=515.\)
Status Protein1 Protein2 d d1/2
Query variables 8FAC_1 5CJQ_1 658 362
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]