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Parikh vectors
2PCG_1 7GKC_1 7NYO_1 Letter Amino acid
14 17 4 D Aspartic acid
6 7 3 H Histidine
21 11 2 I Isoleucine
7 10 2 M Methionine
15 24 3 T Threonine
13 11 5 Y Tyrosine
1 14 2 Q Glutamine
22 9 9 E Glutamic acid
29 29 6 L Leucine
23 17 5 A Alanine
8 21 1 N Asparagine
1 12 0 C Cysteine
17 11 1 K Lycine
10 17 3 F Phenylalanine
24 27 5 V Valine
14 11 5 R Arginine
20 26 3 G Glycine
11 13 3 P Proline
8 16 0 S Serine
1 3 1 W Tryptophan 

2PCG_1|Chains A, B|Probable diphthine synthase|Pyrococcus horikoshii (53953)
>7GKC_1|Chains A, B|3C-like proteinase|Severe acute respiratory syndrome coronavirus 2 (2697049)
>7NYO_1|Chains A[auth AAA], B[auth BBB], C[auth CCC], D[auth DDD]|SH3 domain of JNK-interacting Protein 1 (JIP1)|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 \(w=f(c)\)
2PCG , Knot 115 265 0.80 40 162 254 
MVLYFIGLGLYDERDITVKGLEIAKKCDYVFAEFYTSLMAGTTLGRIQKLIGKEIRVLSREDVELNFENIVLPLAKENDVAFLTPGDPLVATTHAELRIRAKRAGVESYVIHAPSIYSAVGITGLHIYKFGKSATVAYPEGNWFPTSYYDVIKENAERGLHTLLFLDIKAEKRMYMTANEAMELLLKVEDMKKGGVFTDDTLVVVLARAGSMNPTIRAGYVKDLIREDFGDPPHILIVPGKLHIVEAEYLVEIAGAPREILRVNV
7GKC , Knot 134 306 0.83 40 196 290 
SGFRKMAFPSGKVEGCMVQVTCGTTTLNGLWLDDVVYCPRHVICTSEDMLNPNYEDLLIRKSNHNFLVQAGNVQLRVIGHSMQNCVLKLKVDTANPKTPKYKFVRIQPGQTFSVLACYNGSPSGVYQCAMRPNFTIKGSFLNGSCGSVGFNIDYDCVSFCYMHHMELPTGVHAGTDLEGNFYGPFVDRQTAQAAGTDTTITVNVLAWLYAAVINGDRWFLNRFTTTLNDFNLVAMKYNYEPLTQDHVDILGPLSAQTGIAVLDMCASLKELLQNGMNGRTILGSALLEDEFTPFDVVRQCSGVTFQ
7NYO , Knot 37 63 0.81 36 55 60 
GHMEQTHRAIFRFVPRHEDELELEVDDPLLVELQAEDYWYEAYNMRTGARGVFPLYYAIEVTK

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 \(c\).

\(|P_{f(2PCG_1)}(2) \setminus P_{f(7GKC_1)}(2)|=63\), \(|P_{f(7GKC_1)}(2) \setminus P_{f(2PCG_1)}(2)|=97\). 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 1:1110111111000001010110110000011101000111100110100111001011000010101001111110000111101101111000101010100111000110110100111101101001100101101010111000001100010011001111010100010101001101110100100111100001111110110101010110100110001101101111110101101001101111100110101
Pair \(Z_2\) Length of longest common subsequence
2PCG_1,7GKC_1 160 3
2PCG_1,7NYO_1 161 3
7GKC_1,7NYO_1 187 3

Newick tree

 
[
	7NYO_1:89.52,
	[
		2PCG_1:80,7GKC_1:80
	]:9.52
]

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 \((0.85)(0.8)(\frac{571 }{\log_{20} 571}-\frac{265}{\log_{20}265})=86.5\)
Status Protein1 Protein2 d d1/2
Query variables 2PCG_1 7GKC_1 108 100
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]