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Parikh vectors
2QTS_1 6NLW_1 1ZSB_1 Letter Amino acid
10 7 1 M Methionine
3 10 7 W Tryptophan
27 20 13 A Alanine
18 16 12 Q Glutamine
8 7 12 H Histidine
25 16 24 K Lycine
26 11 12 F Phenylalanine
18 14 18 S Serine
23 15 17 V Valine
19 12 7 R Arginine
35 14 12 E Glutamic acid
24 17 9 I Isoleucine
25 11 12 T Threonine
25 6 8 Y Tyrosine
18 16 10 N Asparagine
25 15 19 D Aspartic acid
17 0 1 C Cysteine
28 14 22 G Glycine
44 18 26 L Leucine
20 7 17 P Proline 

2QTS_1|Chains A, B, C, D, E, F|Acid-sensing ion channel|Gallus gallus (9031)
>6NLW_1|Chains A, B, C, D|Beta-lactamase|Shewanella oneidensis (70863)
>1ZSB_1|Chain A|CARBONIC ANHYDRASE II|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 \(w=f(c)\)
2QTS , Knot 185 438 0.85 40 254 414 
STLHGISHIFSYERLSLKRVVWALCFMGSLALLALVCTNRIQYYFLYPHVTKLDEVAATRLTFPAVTFCNLNEFRFSRVTKNDLYHAGELLALLNNRYEIPDTQTADEKQLEILQDKANFRNFKPKPFNMLEFYDRAGHDIREMLLSCFFRGEQCSPEDFKVVFTRYGKCYTFNAGQDGKPRLITMKGGTGNGLEIMLDIQQDEYLPVWGETDETSFEAGIKVQIHSQDEPPLIDQLGFGVAPGFQTFVSCQEQRLIYLPPPWGDCKATTGDSEFYDTYSITACRIDCETRYLVENCNCRMVHMPGDAPYCTPEQYKECADPALDFLVEKDNEYCVCEMPCNVTRYGKELSMVKIPSKASAKYLAKKYNKSEQYIGENILVLDIFFEALNYETIEQKKAYEVAGLLGDIGGQMGLFIGASILTVLELFDYAYEVIKHR
6NLW , Knot 115 246 0.85 38 172 241 
SNANEWQEKPSWNTHFSEHKAQGVIVLWNENKQQGFTNNLKRANQAFLPASTFKIPNSLIALDLGVVKDEHQVFKWDGQTRDIAAWNRDHDLITAMKYSVVPVYQEFARQIGQARMSKMLHAFDYGNEDISGNLDSFWLDGGIRISATEQVAFLRKLYHNKLHVSERSQRIVKQAMLTEANSDYIIRAKTGYSTRIEPQIGWWVGWVELDDNVWFFAMNMDMPTADGLGLRQAITKEVLKQEKIIP
1ZSB , Knot 111 259 0.79 40 175 248 
SHHWGYGKHNGPEHWHKDFPIAKGERQSPVDIDTHTAKYDPSLKPLSVSYDQATSLRILNNGHAFNVEFDDSQDKAVLKGGPLDGTYRLIQFHFHWGSLDGQGSEHTVDKKKYAAQLHLVHWNTKYGDFGKAVQQPDGLAVLGIFLKVGSAKPGLQKVVDVLDSIKTKGKSADFTNFDPRGLLPESLDYWTYPGSLTTPPLLECVTWIVLKEPISVSSEQVLKFRKLNFNGEGEPEELMVDNWRPAQPLKNRQIKASFK

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 \(c\).

\(|P_{f(2QTS_1)}(2) \setminus P_{f(6NLW_1)}(2)|=143\), \(|P_{f(6NLW_1)}(2) \setminus P_{f(2QTS_1)}(2)|=61\). 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 1:001011001100001010011111011101111111000010001101010010011100101111010010010100100001001101111100000110000100001011000101001010110110100011001001110011010000100101110001000010110010101101011010110111010000011111000000101110101000001111001111111110011000000110111111000100100010000010100100000011000000110111011000100000010111011100000001001100100010010110110010100110000000001100111101110110000100001001111110111011111110110110110010011000
Pair \(Z_2\) Length of longest common subsequence
2QTS_1,6NLW_1 204 3
2QTS_1,1ZSB_1 177 4
6NLW_1,1ZSB_1 183 3

Newick tree

 
[
	6NLW_1:99.53,
	[
		2QTS_1:88.5,1ZSB_1:88.5
	]:11.03
]

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 \((0.85)(0.8)(\frac{684 }{\log_{20} 684}-\frac{246}{\log_{20}246})=122.\)
Status Protein1 Protein2 d d1/2
Query variables 2QTS_1 6NLW_1 155 121
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]