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Parikh vectors
8ZFM_1 1PGG_1 1QZL_1 Letter Amino acid
40 24 0 K Lycine
21 15 0 M Methionine
30 46 0 P Proline
45 31 2 T Threonine
21 34 0 R Arginine
34 35 0 E Glutamic acid
29 37 0 F Phenylalanine
23 27 0 Y Tyrosine
31 19 0 N Asparagine
26 27 0 Q Glutamine
24 44 2 G Glycine
47 27 0 I Isoleucine
43 27 0 S Serine
45 26 1 A Alanine
30 25 0 D Aspartic acid
7 13 2 C Cysteine
11 18 0 H Histidine
67 62 0 L Leucine
13 9 0 W Tryptophan
33 30 0 V Valine 

8ZFM_1|Chains A, B, C, D, E|Betaine receptor acr-23,Soluble cytochrome b562|Caenorhabditis elegans (6239)
>1PGG_1|Chains A, B|PROSTAGLANDIN H2 SYNTHASE-1|Ovis aries (9940)
>1QZL_1|Chain A[auth X]|5'-D(*GP*CP*AP*TP*GP*CP*T)-3'|
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 \(w=f(c)\)
8ZFM , Knot 242 620 0.83 40 283 581 
MHRIYTFLIFISQLALGLSNNPDIPIQYELANNIMENYQKGLIPKVRKGSPINVTLSLQLYQIIQVNEPQQYLLLNAWAVERWVDQMLGWDPSEFDNETEIMARHDDIWLPDTTLYNSLEMDDSASKKLTHVKLTTLGKNQGAMVELLYPTIYKISCLLNLKYFPFDTQTCRMTFGSWSFDNSLIDYFPRTFTNGPIGLANFLENDAWSVLGTKVNREEKKYTCCPVNYTLLHYDVVIQRKPLYYVLNLIAPTAVITFISIIGFFTSSSVHDLRQEKITLGITTLLSMSIMIFMVSDKMPSTSTCVPLIALFYTLMITIISVGTLAASSVIFVQKLGSIGNPPASKTMKWTHRIAPFVLIQMPLVMKQAYAKRAKEEKHRKRMSRKMAEAGAMADLEDNWETLNDNLKVIEKADNAAQVKDALTKMRAAALDAQKATPPKLEDKSPDSPEMKDFRHGFDILVGQIDDALKLANEGKVKEAQAAAEQLKTTRNAYIQKYLSQTSETFAAPMDTSFTESLHIPELNRVASSNSIQSVLKPTEIQLTPYCTRNIVELEWDWVAAVLERVFLIFFTICFLFSAIGINLYGWYIWYTENHFLFLEGTKLVPRGSSSGWSHPQFEK
1PGG , Knot 239 576 0.88 40 280 555 
ADPGAPAPVNPCCYYPCQHQGICVRFGLDRYQCDCTRTGYSGPNCTIPEIWTWLRTTLRPSPSFIHFLLTHGRWLWDFVNATFIRDTLMRLVLTVRSNLIPSPPTYNIAHDYISWESFSNVSYYTRILPSVPRDCPTPMGTKGKKQLPDAEFLSRRFLLRRKFIPDPQGTNLMFAFFAQHFTHQFFKTSGKMGPGFTKALGHGVDLGHIYGDNLERQYQLRLFKDGKLKYQMLNGEVYPPSVEEAPVLMHYPRGIPPQSQMAVGQEVFGLLPGLMLYATIWLREHNRVCDLLKAEHPTWGDEQLFQTARLILIGETIKIVIEEYVQQLSGYFLQLKFDPELLFGAQFQYRNRIAMEFNQLYHWHPLMPDSFRVGPQDYSYEQFLFNTSMLVDYGVEALVDAFSRQPAGRIGGGRNIDHHILHVAVDVIKESRVLRLQPFNEYRKRFGMKPYTSFQELTGEKEMAAELEELYGDIDALEFYPGLLLEKCHPNSIFGESMIEMGAPFSLKGLLGNPICSPEYWKASTFGGEVGFNLVKTATLKKLVCLNTKTCPYVSFHVPDPRQEDRPGVERPPTEL
1QZL , Knot 5 7 0.46 8 5 5 
GCATGCT

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 \(c\).

\(|P_{f(8ZFM_1)}(2) \setminus P_{f(1PGG_1)}(2)|=72\), \(|P_{f(1PGG_1)}(2) \setminus P_{f(8ZFM_1)}(2)|=69\). 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 1:10010011111001111100010111000110011000001111010010110101010100110100100011101111001100111101001000001110000111100010001010001000100101001100011110110101001001101001110000001011010100011001100100111111011000110111001000000000011000110001110001100110111101110110111110000100100001011100110101111110001100000111111100111011011011100111100110110111000101000111111101111100101001000000001000110111110100010010001011001001101001100101111010010110100001001010010011011110100110110010100101110010000010100010000001111100010001011010011000010011010010101000001101010111111001111110101110111101011011000001111010011101000110010100
Pair \(Z_2\) Length of longest common subsequence
8ZFM_1,1PGG_1 141 3
8ZFM_1,1QZL_1 284 2
1PGG_1,1QZL_1 279 2

Newick tree

 
[
	1QZL_1:15.35,
	[
		8ZFM_1:70.5,1PGG_1:70.5
	]:86.85
]

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 \((0.85)(0.8)(\frac{1196 }{\log_{20} 1196}-\frac{576}{\log_{20}576})=159.\)
Status Protein1 Protein2 d d1/2
Query variables 8ZFM_1 1PGG_1 206 199.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]