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Parikh vectors
8UMF_1 5AKV_1 7TQF_1 Letter Amino acid
52 5 9 Y Tyrosine
88 12 29 E Glutamic acid
20 4 2 H Histidine
134 8 17 K Lycine
58 5 6 F Phenylalanine
40 8 8 P Proline
70 12 12 T Threonine
92 12 28 A Alanine
71 3 13 N Asparagine
17 1 7 C Cysteine
74 0 7 Q Glutamine
112 11 10 S Serine
20 2 0 W Tryptophan
82 12 26 V Valine
100 4 13 R Arginine
69 10 28 G Glycine
85 5 22 I Isoleucine
155 7 21 L Leucine
21 1 15 M Methionine
102 5 17 D Aspartic acid 

8UMF_1|Chain A|Cas9|Parasutterella secunda (626947)
>5AKV_1|Chains A, B|TRANSTHYRETIN|Homo sapiens (9606)
>7TQF_1|Chain A|Nitrogenase iron protein 1|Azotobacter vinelandii (354)
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 \(w=f(c)\)
8UMF , Knot 508 1462 0.84 40 329 1213 
GAASMKRTADGSEFESPKKKRKVTVKSTLAVDMGGRYTGIFSYTTDSGFPKAKEARAYVLNMPDNDALTYSMAARTQTRHRIRSQQRFVLARRLTYILIEGKLKRKLSPREKEAISSLLRRRGYSRLESELDLSVLQGVESGFFKCFLPNFDEDENLLTQWTSLTDGYLQNNSDSRRQIQIFLESSKDSKEFLTVVKSQHQDTKEYKNALKVMRDDAESMIEQSMFGHKHRRLYLEAIAQDIPRDSRLKPIIEAFSGVEKFHHFIGNLSNLQLRALRWYFNDPSMKNNVFDKERLKSVLVRAYQFFHYPKDLTQQRAEVLNAYEGATDILETLQTLNPELTIPPYEDQNNRRPPLDQTLWLSPRLLDQRYGDTWEIWVQNLLRSPLSKGIDENLDTILITTDRKARLLERQSGRLIHYTSQKLYHSYVLQRLLDRTVENDAYLLKTLVSSNRGNSNEIHQAQERLTRDLGSQHIKKFLDFVRQYYDEVDKAKRGLWFIVEKPLMERADIHPPMKNDSVILRLVGNILCVSDLVDLSFWTRKVKGQSTVRSLCTAIEKTRKEYGNSFNYLYQRALYLQSKGKKLSAEDKDFIKLQSNVLLVSDVIAEALDIKEEQKKKFANPFSLAQLYNIIETEKSGFISTTLAAVDENAWRNNLQGKARCVQLCADTVRPFDGALRNILDRQAYEIAKLKAEELLSTELKNQTIDLVVLLESNQFAFSASLAEVKKSANTAAIRQKVAKAQKRQQDRWLSKDERIKSASRGLCPYTGKNLGDKGEVDHIIPRSLSMNYMGSILNSEANLIYCSQEGNQLKLNGRKKLSDLADNYLKVVFGTADRGTICKYIEKSVSELTDAKIVQFELLDRSQQDAVRHALFLEDFSEARRRIIRLLGKINTARVNGTQAWFAKSFITKLRELTKEWCANNQITLAFDLYRLDAQTVSQDYRKKFALINKDWAKPDDKKQPIASHAIDAFCVFAAAKDKRNIANVLGVFDEVAEEQNLKTIAQLMPSEVNLISPKRKSILDKNEVGSRALMKEGIFAEHFLPILVRGDDCRIGFDWSESGSVKVKDADKLFGVLDGLLKQSQKRSVNGFETYTVDRIKAFELLHDVFIRPCSQKMLEQAEVLEKLHYITQNISVTSVYDAVNRQFKCREEILKDKDFDIKVDLGNRFGSAKGKITLPAKREWEKLVNRSELKNLIKDKLSDKGSEKTPDGETLIYDIFRSIPVQKLSHKATRRVWSLPKIPSISSGVRIKRKDSNGNDIYQLYMLNDTKCKGFVVNEKGVIDWSSDLVADLYKQPTLTILNGRYLKADQYVRMDQWYEVDCGRDDVIVKMCPGTSGRRYIEITQSKKQFEDWTGYISGSFWNYPVTIKLSSQQIANFVKNSQMPLLGKPRSGQITVITLGNTLKYWYCVESKNSMMNEAYQKAYLVHFNQSGGSASDYKDDDDKKRTADGSEFEPKKKRKV
5AKV , Knot 65 127 0.82 38 101 125 
GPTGTGESKCPLMVKVLDAVRGSPAINVAVHVFRKAADDTWEPFASGKTSESGELHGLTTEEEFVEGIYKVEIDTKSYWKALGISPFHEHAEVVFTANDSGPRRYTIAALLSPYSYSTTAVVTNPKE
7TQF , Knot 124 290 0.80 38 172 272 
MAMRQCAIYGKGGIGKSTTTQNLVAALAEMGKKVMIVGCDPKADSTRLILHSKAQNTIMEMAAEAGTVEDLELEDVLKAGYGGVKCVESGGPEPGVGCAGRGVITAINFLEEEGAYEDDLDFVFYDVLGDVVCGGFAMPIRENKAQEIYIVCSGEMMAMYAANNISKGIVKYANSGSVRLGGLICNSRNTDREDELIIALANKLGTQMIHFVPRDNVVQRAEIRRMTVIEYDPKAKQADEYRALARKVVDNKLLVIPNPITMDELEELLMEFGIMEVEDESIVGKTAEEV

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 \(c\).

\(|P_{f(8UMF_1)}(2) \setminus P_{f(5AKV_1)}(2)|=236\), \(|P_{f(5AKV_1)}(2) \setminus P_{f(8UMF_1)}(2)|=8\). 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 1:1110100010100100100000101000111011100011100000011101001010110110001100011100000001000001111001001110101000101000011001100010001000101011011001110011101000001100100100101000000000101110000000011011000000000000110110001001100011100000101011100110000101110110110010011101001010110101001010001100001001110100110010010000101101001100110010010101011100000000111000111010110000100101110011001100110001001110000010110000101100000010000110011000100010110011000010000100100010001100010011011000000100100111111001110010101110000111011101101001101011000101000100100110000000100100100011010001001010000110100011110011101101000000011011011010011000001110001111000110001010100101010010110111001100010011010100110001000010111110000111010110100010011100011010000000110000010010011010010011001010011100101001101100010110000010010101000100110001011110100101000100010010010110101100000011001111001001000110111010010101001111001100100100010100010111010010100100000001111000110100000111001101101111100000110111110011000010011011100101101000011000011001110011110011111101000011101000101010010011111011100000001011000010010110110011101000011001011001001000101001001100010000011000010101011001101010101110001001100001001100010001000010100110011001110010001000110110110100110100000010010010110000001111000111010001110100010101101001010001010010010010001110101100100010100000010010101010110011010100001101100001111101001010110110010010010000011001000101101000110100000000000010100101000001
Pair \(Z_2\) Length of longest common subsequence
8UMF_1,5AKV_1 244 4
8UMF_1,7TQF_1 189 4
5AKV_1,7TQF_1 155 3

Newick tree

 
[
	8UMF_1:11.78,
	[
		7TQF_1:77.5,5AKV_1:77.5
	]:40.28
]

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 \((0.85)(0.8)(\frac{1589 }{\log_{20} 1589}-\frac{127}{\log_{20}127})=385.\)
Status Protein1 Protein2 d d1/2
Query variables 8UMF_1 5AKV_1 487 263.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]