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Parikh vectors
9ENS_1 3EFO_1 7ZYX_1 Letter Amino acid
39 17 0 H Histidine
158 75 0 L Leucine
83 50 0 P Proline
110 44 1 T Threonine
49 25 0 Y Tyrosine
82 43 0 R Arginine
79 51 0 G Glycine
18 10 0 W Tryptophan
109 36 0 D Aspartic acid
21 16 5 C Cysteine
126 39 0 E Glutamic acid
150 55 0 S Serine
76 49 0 A Alanine
108 25 0 N Asparagine
138 28 0 K Lycine
66 26 0 M Methionine
66 38 0 F Phenylalanine
129 42 0 V Valine
60 54 0 Q Glutamine
103 42 0 I Isoleucine 

9ENS_1|Chain A|Vitellogenin|Apis cerana (7461)
>3EFO_1|Chain A|Protein transport protein Sec23A|Homo sapiens (9606)
>7ZYX_1|Chains A, B|DNA (5'-D(*TP*(CFL)P*(CFL)P*(CFL)P*(CFL)P*(CFL))-3')|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 \(w=f(c)\)
9ENS , Knot 611 1770 0.86 40 366 1475 
MLLLLTLLLFAGTVAADFQHNWQVGNEYTYLVRSRTLTSLGDLSDVHTGILIKALLTVQAKDSNVLAAKVWNGQYARVQQSMPDGWETEISDQMLELRDLPISGKPFQIRMKHGLIRDLIVDRDVPTWEVNILKSIVGQLQVDTQGENAVKVNSVQVPTDDEPYASFKAMEDSVGGKCEVLYDIAPLSDFVIHRSPELVPMPTLKGDGRHMEVIKIKNFDNCDQRINYHFGMTDNSRLEPGTNKNGKFFSRSSTSRIVISESLKHFTIQSSVTTSKMMVSPRLYDRQNGLVLSRMNLTLAKMEKTSKPLPMVDNPESTGNLVYIYNNPFSDVEERRVSKTAMNSNQIVSDNSLSSSEEKLKQDILNLRTDISSSSSSISSSEENDFWQPKPTLEDAPQNSLLPNFVGYKGKHIGKSGKVDVINAAKELIFQIANELEDASNIPVHATLEKFMILCNLMRTMNRKQISELESNMQISPNELKPNDKSQVIKQNTWTVFRDAITQTGTGPAFLTIKEWIERGTTKSMEAANIMSKLPKTVRTPTDSYIRSFFELLQNPKVSNEQFLNTAATLSFCEMIHNAQVNKRSIHNNYPVHTFGRLTSKHDNSLYDEYIPFLERELRKAHQEKDSPRIQTYIMALGMIGEPKILSVFEPYLEGKQQMTVFQRTLMVGSLGKLTETNPKLARSVLYKIYLNTMESHEVRCTAVFLLMKTNPPLSMLQRMAEFTKLDTNRQVNSAVKSTIQSLMKLKSPEWKDLAKKARSVNHLLTHHEYDYELSRGYIDEKILENQNIITHMILNYVGSEDSVIPRILYLTWYSSNGDIKVPSTKVLAMISSVKSFMELSLRSVKDRETIISAAEKIAEELKIVPEELVPLEGNLMINNKYALKFFPFDKHILDKLPTLISNYIEAVKEGKFMNVNMLDTYESVHSFPTETGLPFVYTFNVIKLTKTSGTVQAQINPDFAFIVNSNLRLTFSKNVQGRVGFVTPFEHRHFISGIDSNLHVYAPLKISLDVNTPKGNMQWKIWPMKGEEKSRLFHYSVVPFVSNHDILNLRPLSMEKGTRPMIPDDNTSLALPKNEGPFRLNVETAKTNEEMWELIDTEKLTDRLPYPWTMDNERYVKVDMYMNLEGEQKDPVIFSTSFDSKVMTRPDTDSENWTPKMMAVEPTDKQANSKTRRQEMMREAGRGIESAKSYVVDVRVHVPGESESETVLTLAWSESNVESKGRLLGFWRVEMPRSNADYEVCIGSQIMVSPETLLSYDEKMDQKPKMDFNVDIRYGKNCGKGERIDMNGKLRQSPRLKELVGATSIIKDCVEDMKRGNKILRTCQKAVVLSMLLDEVDISMEVPSDALIALYSQGLFSLSEIDNLDVSLDVSNPKNAGKKKIDVRAKLNEYLDKADVIVNTPIMDAHFKDVKLSDFGFSTEDILDTADEDLLINNVFYEDETSCMLDKTRAQTFDGKDYPLRLGPCWHAVMTTYPRINPDNHNEKLHIPKDKSVSVLSRENEAGQKEVKVLLGSDKIKFVPGTTSQPEVFVNGEKIVVSRNKAYQKVEENEIIFEIYKMGDRFIGLTSDKFDVSLALDGERVMLKASEDYRYSVRGLCGNFDHDSTNDFVGPKNCLFRKPEHFVASYALISNQCEGDSLNVAKSLQDHDCIRQERTQQRNVISDSESGRLDTEMSTWGYHHNVNKHCTIHRTQVKETDDKICFTMRPVVSCASGCTAVETKSKPYKFHCMEKNEAAMKLKKRIEKGANPDLSQKPVSTTEELTVPFVCKA
3EFO , Knot 297 765 0.86 40 315 709 
MTTYLEFIQQNEERDGVRFSWNVWPSSRLEATRMVVPVAALFTPLKERPDLPPIQYEPVLCSRTTCRAVLNPLCQVDYRAKLWACNFCYQRNQFPPSYAGISELNQPAELLPQFSSIEYVVLRGPQMPLIFLYVVDTCMEDEDLQALKESMQMSLSLLPPTALVGLITFGRMVQVHELGCEGISKSYVFRGTGDLSAKQLQEMLGLSKVPLTQATRGPQVQQPPPSNRFLQPVQKIDMNLTDLLGELQRDPWPVPQGKRPLRSSGVALSIAVGLLECTFPNTGARIMMFIGGPATQGPGMVVGDELKTPIRSWHDIDKDNAKYVKKGTKHFEALANRAATTGHVIDIYACALDQTGLLEMKCCPNLTGGYMVMGDSFNTSLFKQTFQRVFTKDMHGQFKMGFGGTLEIKTSREIKISGAIGPCVSLNSKGPCVSENEIGTGGTCQWKICGLSPTTTLAIYFEVVNQHNAPIPQGGRGAIQFVTQYQHSSGQRRIRVTTIARNWADAQTQIQNIAASFDQEAAAILMARLAIYRAETEEGPDVLRWLDRQLIRLCQKFGEYHKDDPSSFRFSETFSLYPQFMFHLRRSSFLQVFNNSPDESSYYRHHFMRQDLTQSLIMIQPILYAYSFSGPPEPVLLDSSSILADRILLMDTFFQILIYHGETIAQWRKSGYQDMPEYENFRHLLQAPVDDAQEILHSRFPMPRYIDTEHGGSQARFLLSKVNPSQTHNNMYAWGQESGAPILTDDVSLQVFMDHLKKLAVSSAA
7ZYX , Knot 3 6 0.29 4 2 2 
TCCCCC

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 \(c\).

\(|P_{f(9ENS_1)}(2) \setminus P_{f(3EFO_1)}(2)|=70\), \(|P_{f(3EFO_1)}(2) \setminus P_{f(9ENS_1)}(2)|=19\). 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 1:111110111111011101000101100000110000100110100100111101110101000011110110100101000110110001000110100111010110101001110011100011010101100111010100010011010010110000101010110001110001100111100111000101111101010100101101001000000100011100000101100001011000000011100010010100010000111010100000111100101011010000011111001000101101000110010000100011000011000010000001000110100010000001000000011010101001100011101110010011001010110110011101100100100111010100111100110010000100100010101001010000011000010110011000101111101001100100001011011001100100100001001101100101000011001101010011001010000100001100110100000001000011110001001000000101000111111110101101101010100010110001111011010000101100110010100100001000111111000111011001101001000001001100010011010010100110010010011000000001001010001100001100111001100001110110101000010101100011111001001101010010000011011001100101110011110101110000110111100011001101100010110010110101100000100110001111100101101000010101010101111100010101000101011110110000110110001010111010101001010101011110100000110001111100001101011010010011110000011110001110101001000001101100001000110110100000101010101010000111100010001100100000010101111010000100000000110011011001000110101011100000011011100001000101111101011000100010110011101001100000100010101010100100010100101010100010100111100110001001001001100000111101110010101011001111100011101001001010101001001100010101010001001011100111010100101001110000110010001110011000000011000010010100011011101011100010101000000101100001011000001100010111100010111100001011101001110000100010000111010011001111000010101110100111010000000101101010000000111100011001001110011100000100101100100000100000000011000001010001001100001000001000010000001010101110010100110000010010010000111010001001101010001100000101111001
Pair \(Z_2\) Length of longest common subsequence
9ENS_1,3EFO_1 89 4
9ENS_1,7ZYX_1 366 2
3EFO_1,7ZYX_1 313 2

Newick tree

 
[
	7ZYX_1:19.92,
	[
		9ENS_1:44.5,3EFO_1:44.5
	]:15.42
]

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 \((0.85)(0.8)(\frac{2535 }{\log_{20} 2535}-\frac{765}{\log_{20}765})=424.\)
Status Protein1 Protein2 d d1/2
Query variables 9ENS_1 3EFO_1 544 389.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]