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Parikh vectors
8UHE_1 5MYA_1 5THN_1 Letter Amino acid
5 14 8 Y Tyrosine
12 13 7 R Arginine
7 22 10 N Asparagine
8 17 11 Q Glutamine
0 14 12 H Histidine
4 9 12 F Phenylalanine
6 25 17 P Proline
4 1 1 C Cysteine
13 25 9 I Isoleucine
7 16 24 K Lycine
4 2 2 M Methionine
12 24 18 S Serine
9 28 17 V Valine
16 16 13 A Alanine
7 18 22 G Glycine
5 20 12 T Threonine
8 13 19 D Aspartic acid
16 22 13 E Glutamic acid
13 29 26 L Leucine
2 5 7 W Tryptophan 

8UHE_1|Chains A, E, G, M, O, Q|ApcD5|Synechococcus sp. PCC 7335 (91464)
>5MYA_1|Chains A, B|Angiopoietin-1 receptor|Homo sapiens (9606)
>5THN_1|Chain A|Carbonic anhydrase 2|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 \(w=f(c)\)
8UHE , Knot 77 158 0.82 38 129 156 
MSLVTELILSADSEARYPAPKELRIFQDFVKTGEQRVRIAKALAANEERIVQNGSQKFWERCPNTPSNSGVDRKTASCQRDQGWYVRLIAYSILAGSERPLEDIGTVGIKEMYNNLEIPIRNIAECMRCLKEEAMAVLSDEDAQEVAAYFDLIIQSLS
5MYA , Knot 146 333 0.85 40 197 314 
MKFLVNVALVFMVVYISYIYADPVLPKPLNAPNVIDTGHNFAVINISSEPYFGDGPIKSKKLLYKPVNHYEAWQHIQVTNEIVTLNYLEPRTEYELCVQLVRRGEGGEGHPGPVRRFTTASIGLPPPRGLNLLPKSQTTLNLTWQPIFPSSEDDFYVEVERRSVQKSDQQNIKVPGNLTSVLLNNLHPREQYVVRARVNTKAQGEWSEDLTAWTLSDILPPQPENIKISNITHSSAVISWTILDGYSISSITIRYKVQGKNEDQHVDVKIKNATITQYQLKGLEPETAYQVDIFAENNIGSSNPAFSHELVTLPESQAPADLGIEGRHHHHHH
5THN , Knot 112 260 0.79 40 176 249 
MSHHWGYGKHNGPEHWHKDFPIAKGERQSPVDIDTHTAKYDPSLKPLSVSYDQATSLRILNNGHAFNVEFDDSQDKAVLKGGPLDGTYRLIQFHFHWGSLDGQGSEHTVDKKKYAAELHLVHWNTKYGDFGKAVQQPDGLAVLGIFLKVGSAKPGLQKVVDVLDSIKTKGKSADFTNFDPRGLLPESLDYWTYPGSLTTPPLLECVTWIVLKEPISVSSEQVLKFRKLNFNGEGEPEELMVDNWRPAQPLKNRQIKASFK

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 \(c\).

\(|P_{f(8UHE_1)}(2) \setminus P_{f(5MYA_1)}(2)|=58\), \(|P_{f(5MYA_1)}(2) \setminus P_{f(8UHE_1)}(2)|=126\). 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 1:10110011101000100111001011001100100010110111100001100100011000100100011000010000001101011100111100011001101110010001011100110010010001111100001001110101110010
Pair \(Z_2\) Length of longest common subsequence
8UHE_1,5MYA_1 184 3
8UHE_1,5THN_1 169 3
5MYA_1,5THN_1 155 4

Newick tree

 
[
	8UHE_1:91.65,
	[
		5THN_1:77.5,5MYA_1:77.5
	]:14.15
]

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 \((0.85)(0.8)(\frac{491 }{\log_{20} 491}-\frac{158}{\log_{20}158})=97.8\)
Status Protein1 Protein2 d d1/2
Query variables 8UHE_1 5MYA_1 125 92
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]