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Parikh vectors
5FUE_1 1AES_1 9EDN_1 Letter Amino acid
16 24 10 K Lycine
26 17 12 A Alanine
21 10 12 R Arginine
20 19 20 N Asparagine
22 20 13 E Glutamic acid
29 26 26 G Glycine
24 10 21 I Isoleucine
49 25 22 L Leucine
19 18 19 F Phenylalanine
8 6 5 W Tryptophan
12 9 16 Q Glutamine
36 15 23 S Serine
28 14 17 V Valine
11 1 3 C Cysteine
16 6 7 H Histidine
8 6 5 M Methionine
26 15 29 P Proline
32 25 17 D Aspartic acid
19 14 24 T Threonine
24 14 9 Y Tyrosine 

5FUE_1|Chains A, B, C, D|HISTONE DEACETYLASE 8|SCHISTOSOMA MANSONI (6183)
>1AES_1|Chain A|CYTOCHROME C PEROXIDASE|Saccharomyces cerevisiae (4932)
>9EDN_1|Chains A, B|VP1|Norovirus GII (122929)
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 \(w=f(c)\)
5FUE , Knot 189 446 0.86 40 246 427 
MSVGIVYGDQYRQLCCSSPKFGDRYALVMDLINAYKLIPELSRVPPLQWDSPSRMYEAVTAFHSTEYVDALKKLQMLHCEEKELTADDELLMDSFSLNYDCPGFPSVFDYSLAAVQGSLAAASALICRHCEVVINWGGGWHHAKRSEASGFCYLNDIVLAIHRLVSSTPPETSPNRQTRVLYVDLDLHHGDGVEEAFWYSPRVVTFSVHHASPGFFPGTGTWNMVDNDKLPIFLNGAGRGRFSAFNLPLEEGINDLDWSNAIGPILDSLNIVIQPSYVVVQCGADCLATDPHRIFRLTNFYPNLNLDSDCDSECSLSGYLYAIKKILSWKVPTLILGGGGYNFPDTARLWTRVTALTIEEVKGKKMTISPEIPEHSYFSRYGPDFELDIDYFPHESHNKTLDSIQKHHRRILEQLRNYADLNKLIYDYDQVYQLYNLTGMGSLVPR
1AES , Knot 133 294 0.85 40 194 287 
MKTLVHVASVEKGRSYEDFQKVYNAIALKLREDDEYDNYIGYGPVLVRLAWHISGTWDKHDNTGGSYGGTYRFKKEFNDPSNAGLQNGFKFLEPIHKEFPWISSGDLFSLGGVTAVQEMQGPKIPWRCGRVDTPEDTTPDNGRLPDADKDAGYVRTFFQRLNMNDREVVALMGAHALGKTHLKNSGYEGPGGAANNVFTNEFYLNLLNEDWKLEKNDANNEQWDSKSGYMMLPTDYSLIQDPKYLSIVKEYANDQDKFFKDFSKAFEKLLEDGITFPKDAPSPFIFKTLEEQGL
9EDN , Knot 131 310 0.80 40 194 298 
SKPFTIPILTIGEMTNSRFPVPIDMLHTSPTENIVVQPQNGRCTIDGELLGTTQLITSNICSFRGSISGHEGNDDQHLWHFNITNPNGTPFDPTEDVPAPLGTPDFRGQLYGVISQRDREGTPSENNQKANRSHQGVISTISPKFVPKLGSVMIGTWTTDDIKNQPSKFTPIGLNDDDNFKQWELPNYSGALTLNMGLAPSVFPTYPGEQILFFRSYLPLKGGYGSPFIDCLVPQEWVAHFYQESAPAQTDVALIRYVNPETGRVLFEAKLHRQGFITVAKSGDSPINVPANGYFRFDSWVNQFYSLAPM

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 \(c\).

\(|P_{f(5FUE_1)}(2) \setminus P_{f(1AES_1)}(2)|=107\), \(|P_{f(1AES_1)}(2) \setminus P_{f(5FUE_1)}(2)|=55\). 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 1:10111101000001000010110001111011010011101001111010010010011011000001011001011000000101000111001010000111101100011110101111011100000111011111001000010110010011111001100011000100000110101010010110011100101101010010111111010101100001111101110101011011100110010100111111001011101001110011001100100110100101010100000000010101011001101011011111110011001011001011010010100101010110000100011010101001100000001001000000110010001010011000001001001011101110
Pair \(Z_2\) Length of longest common subsequence
5FUE_1,1AES_1 162 4
5FUE_1,9EDN_1 176 4
1AES_1,9EDN_1 168 4

Newick tree

 
[
	9EDN_1:87.63,
	[
		5FUE_1:81,1AES_1:81
	]:6.63
]

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 \((0.85)(0.8)(\frac{740 }{\log_{20} 740}-\frac{294}{\log_{20}294})=122.\)
Status Protein1 Protein2 d d1/2
Query variables 5FUE_1 1AES_1 158 129
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]