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Parikh vectors
8QFV_1 7BAW_1 1ADB_1 Letter Amino acid
48 4 28 A Alanine
21 1 8 Q Glutamine
15 0 26 S Serine
13 0 39 V Valine
18 0 12 R Arginine
15 0 8 N Asparagine
0 0 14 C Cysteine
0 0 7 H Histidine
5 8 24 I Isoleucine
6 0 20 P Proline
14 6 38 G Glycine
7 0 9 M Methionine
15 0 24 T Threonine
1 1 2 W Tryptophan
16 0 17 D Aspartic acid
27 4 21 E Glutamic acid
34 0 25 L Leucine
16 6 30 K Lycine
5 0 18 F Phenylalanine
1 0 4 Y Tyrosine 

8QFV_1|Chain A[auth 0]|Protein sll0617|Synechocystis sp. PCC 6803 (1148)
>7BAW_1|Chains A, B, C, D, E, F, G|CC-Type2-(GgIaId)4|synthetic construct (32630)
>1ADB_1|Chains A, B|ALCOHOL DEHYDROGENASE|Equus caballus (9796)
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 \(w=f(c)\)
8QFV , Knot 114 277 0.77 36 144 248 
MGLFDRLGRVVRANLNDLVSKAEDPEKVLEQAVIDMQEDLVQLRQAVARTIAEEKRTEQRLNQDTQEAKKWEDRAKLALTNGEENLAREALARKKSLTDTAAAYQTQLAQQRTMSENLRRNLAALEAKISEAKTKKNMLQARAKAAKANAELQQTLGGLGTSSATSAFERMENKVLDMEATSQAAGELAGFGIENQFAQLEASSGVEDELAALKASMAGGALPGTSAATPQLEAAPVDSSVPANNASQDDAVIDQELDDLRRRLNNLAALEVLFQGP
7BAW , Knot 12 32 0.43 16 14 17 
XGEIAQGIKEIAKGIKEIAWGIKEIAKGIKGX
1ADB , Knot 158 374 0.83 40 210 358 
STAGKVIKCKAAVLWEEKKPFSIEEVEVAPPKAHEVRIKMVATGICRSDDHVVSGTLVTPLPVIAGHEAAGIVESIGEGVTTVRPGDKVIPLFTPQCGKCRVCKHPEGNFCLKNDLSMPRGTMQDGTSRFTCRGKPIHHFLGTSTFSQYTVVDEISVAKIDAASPLEKVCLIGCGFSTGYGSAVKVAKVTQGSTCAVFGLGGVGLSVIMGCKAAGAARIIGVDINKDKFAKAKEVGATECVNPQDYKKPIQEVLTEMSNGGVDFSFEVIGRLDTMVTALSCCQEAYGVSVIVGVPPDSQNLSMNPMLLLSGRTWKGAIFGGFKSKDSVPKLVADFMAKKFALDPLITHVLPFEKINEGFDLLRSGESIRTILTF

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 \(c\).

\(|P_{f(8QFV_1)}(2) \setminus P_{f(7BAW_1)}(2)|=138\), \(|P_{f(7BAW_1)}(2) \setminus P_{f(8QFV_1)}(2)|=8\). 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 1:1111001101101010011001001001100111010001101001110011000000001000000100100010111001000110011100001000111000011000010001000111101010010000011010101101010100011111000100110010001101010001110111111000110101001100011110101111111100110101011110001110010000111000100100010011110111011
Pair \(Z_2\) Length of longest common subsequence
8QFV_1,7BAW_1 146 2
8QFV_1,1ADB_1 160 4
7BAW_1,1ADB_1 206 2

Newick tree

 
[
	1ADB_1:97.79,
	[
		8QFV_1:73,7BAW_1:73
	]:24.79
]

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 \((0.85)(0.8)(\frac{309 }{\log_{20} 309}-\frac{32}{\log_{20}32})=90.9\)
Status Protein1 Protein2 d d1/2
Query variables 8QFV_1 7BAW_1 109 58.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]