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Parikh vectors
9FUC_1 9KLN_1 8INE_1 Letter Amino acid
71 184 10 A Alanine
27 124 16 R Arginine
20 28 1 H Histidine
37 53 4 I Isoleucine
18 21 3 M Methionine
28 90 8 S Serine
15 16 4 C Cysteine
19 54 5 Q Glutamine
48 120 2 E Glutamic acid
25 30 3 N Asparagine
36 100 1 D Aspartic acid
50 109 7 G Glycine
61 142 9 L Leucine
46 113 16 K Lycine
30 38 1 F Phenylalanine
33 63 4 P Proline
37 61 6 T Threonine
52 75 12 V Valine
5 41 0 W Tryptophan
11 34 5 Y Tyrosine 

9FUC_1|Chain A|Carbon monoxide dehydrogenase|Carboxydothermus hydrogenoformans (129958)
>9KLN_1|Chain A|C2c1 CRISPR-Cas endonuclease RuvC-like domain-containing protein|Candidatus Hydrogenedentes bacterium ADurb.Bin170 (1852848)
>8INE_14|Chain N[auth F]|60S ribosomal protein L34|Homo sapiens (9606)
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 \(w=f(c)\)
9FUC , Knot 260 669 0.84 40 276 609 
PRFRDLEHTSKPSKADRVWEPKNRKRTIDPAALEMLEKAEKDGVKTAFDRFVEMQPQCQFGYKGLCCRFCLQGPCRLPNDDPSKKGICGASAWTIAARSVGTLILTGAAAHNEHARHIAHALKELAEGKAPDYKITDPDKLRRIAQRLGLDTQGKDDMTLAKEVAELALEDFARLPGFGENLWIKTTLNKERLEKYDECNIMPSGIFGDISDLLAQAHIGNDDDPVNITFSALRVALTDYAGMHIATDFSDVLFGTPKPIVTEANLGVLDANKVNIAVHGHNPLLSEKVVDAAKELEEEAKAAGAEGINIVGMCCTGNEVLMRRGVHLATSFASSELAIVTGAMDAVVVDVQCIMPGLKQVTECYHTRLITTSNIAKMPGTYHVPFHIENALESAKEIVRLGIEAFKQRVGKPVHIPEVKHKVVAGFSFEALMEIFAHVNQENPIRVLNDAILSGQLKGVVLFAGCNNLKRPQDESHITILKEMLKNDVFVVTTGCSAQAFAKHGFLRPEALELAGEGLKSFIKMLEEKAGLQGQLPPAFFMGSCVDNTRASDILVAMAKDLGVDTPKVPFVASAPEAMSGKAVSIGTWFVTLGVPVHVGTMPPLEGSELFYSITTQIASDVYGGYFMFEVDPVVAARKILNALEYRTWKLGVHKQTAEKFETALCQNY
9KLN , Knot 514 1496 0.83 40 335 1206 
PADDLSTQRAYTLRLQGTDPEDQSWRDALWMTHEAVNAGGRAFGDWLLTLRGGIAHELADTPVKGKKDITDELRKKRRILLALSWLSVESRRGAPDKFIVAGGEEPAGSRNEKVLQALKEILKRRGLSAEESESWMSDCRASLSAAIRDDAVWVNRSAAFDDAQVRIGASLTREDIWDMLDPFFGSREAYLTPAKKKKEDEDSSEGTGEEKAKDLVQKAGQWLSSRFGTGKGANFDAMAEVYSKISEWAGTAQEGVSGKEGIKNLADALAAFSPVSQNLEGVLKLISGPGYKSATRNLLGELDSLPVVSRDHLSALHEKAAEDTVKCKESTGTKGRRPYADAILNDVEKRCGFTYLTDSDNRSVSILDTSEFPSDYKWGTARHSEFAVILDHAARRISVAHSWIKLAEAERDRCEEDAAKVYDLPDKVKEWLDTFCSNRSDISGAQGEGYRIRRKAIEGWKEVVASWGRSSCITAEDRVAAARALQDDPEIDKFGAIQLFEILAQDEALCVWHKDGDVAKSPDAQMLIDYVLASDAESKKRRFKVPAYRHPDALLHPIFCDFGNSRWDITYDIHGARGKKKAKRGSKKEEAMPRGVAMKLWTGSDVLSVSLRWQSKKLAADLALDQEAEEVTDTAAVSRADRLGRAAAGIDRGAGVTIAGLFEEAHWNGRLQAPRQQLEAIAAVRDNQKLSSEERERRIAFMKDRIRWLVTFSAKLRPQGPWHSYAPTQGLQSDPKYWPHSEINKKRKGQAKLILSRLPGLRILSVDLGHRFAAACAVWETMSSEAIQEACRLANHQLPAPADLYLHLKRTVQKNLIDGEKTVEESTVYRRIGADRLPDGTAHPAPWARLDRQFLIKLQGEEKVREASNEEVWQVHLMESALGLSFPLIDRLVYAGWGGTEKQAARLEALREKGWKPTGTPADQDEEGGGYKPSLAVDELMFSAVRTLRLALKYHGDRARIAFALTADYKPMPGDTRYYFSEAKDRSSGADAAEREAKHKDYLLDMLLLWHDLAFSRKWRDEEAKELWNLHIAALPGYQAPAAPIQEEAGQGRKKAREEARAKMTPAAEALLADGTLREKLHGLWKERWEKDDAQWKKHLRWMKDGILPRGGRAATPSIRYVGGLSLTRLATLTEFRRKVQVGFYTRLFPSGEKREIKEAFGQTALDALERLREQRVKQLASRIAEAALGAGRVSRTALKQDPKRPEARVDAACHAVIIENLEHYRPEETRTRRENRGLMNWASSKVKKYLSEACQLHGLFLREVPAGYTSRQDSRTGAPGMRCQDVTVKTFLNSPFWQKQCVQAQKNKSTARDRFLCALKEAVAQGGMEEEKKMGPIRVPVPGGEVFVSADAASPAAKGLQADLNAAANIGLRALLDPDWPGKWWYVPCDRKTAYPAKEKVEGSAAVDVKQALPFVLPEEKENKGKTKGGKKGKGEVMNLWRDVSAEPLMTGQWLDYTAYRKEVENRVIQVLTAQLKARNPLRFGNLGDEEEIPY
8INE , Knot 57 117 0.77 38 88 111 
MVQRLTYRRRLSYNTASNKTRLSRTPGNRIVYLYTKKVGKAPKSACGVCPGRLRGVRAVRPKVLMRLSKTKKHVSRAYGGSMCAKCVRDRIKRAFLIEEQKIVVKVLKAQAQSQKAK

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 \(c\).

\(|P_{f(9FUC_1)}(2) \setminus P_{f(9KLN_1)}(2)|=32\), \(|P_{f(9KLN_1)}(2) \setminus P_{f(9FUC_1)}(2)|=91\). 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 1:101001000001001001101000000101111011001000110011001101010001100110001010110011000100011011011011100110111011110000100110110011010110001001001001100111000100010110011011100110111110011100010000100000001110111101001110101100001101010110111000111011001001111010111001011110100101110100111000110110010001011110110111100010011100110110011000111101110111101001111100100000001100001101110001110100110010011011101100011011011010001111101011101110100001101100111010101111111000100100000101100110001111001001011100111010110111011001101100011101011111111001000010011111100111001011111011011010110110111011111011011110100110010001100101101110101111100110110000101110000100100110000
Pair \(Z_2\) Length of longest common subsequence
9FUC_1,9KLN_1 123 4
9FUC_1,8INE_1 228 4
9KLN_1,8INE_1 265 4

Newick tree

 
[
	8INE_1:13.22,
	[
		9FUC_1:61.5,9KLN_1:61.5
	]:76.72
]

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 \((0.85)(0.8)(\frac{2165 }{\log_{20} 2165}-\frac{669}{\log_{20}669})=364.\)
Status Protein1 Protein2 d d1/2
Query variables 9FUC_1 9KLN_1 459 330.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]